Eur. Phys. J. C (2022) 82:869 https://doi.org/10.1140/epjc/s10052-022-10536-1 Review FLAG Review 2021 Flavour Lattice Averaging Group (FLAG) Y. Aoki1, T. Blum2,3, G. Colangelo4, S. Collins5, M. Della Morte6, P. Dimopoulos7,8, S. Dürr9,10, X. Feng11,12,13,14, H. Fukaya15, M. Golterman16, Steven Gottlieb17, R. Gupta18, S. Hashimoto19,20, U. M. Heller21, G. Herdoiza22, P. Hernandez23, R. Horsley24, A. Jüttner25,26,27,a, T. Kaneko19,20, E. Lunghi17, S. Meinel28, C. Monahan29,30, A. Nicholson31, T. Onogi15, C. Pena22, P. Petreczky32, A. Portelli24, A. Ramos23, S. R. Sharpe33, J. N. Simone34, S. Simula35, S. Sint36, R. Sommer37,38, N. Tantalo39, R. Van de Water34, U. Wenger4,27, H. Wittig40 1 RIKEN Center for Computational Science, Kobe 650-0047, Japan 2 Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA 3 RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA 4 Albert Einstein Center for Fundamental Physics, Institut für Theoretische Physik, Universität Bern, Sidlerstr. 5, 3012 Bern, Switzerland 5 Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany 6 CP3-Origins and IMADA, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark 7 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, 43124 Parma, Italy 8 INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/a (Campus), 43124 Parma, Italy 9 University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany 10 Jülich Supercomputing Center, Forschungszentrum Jülich, 52425 Jülich, Germany 11 School of Physics, Peking University, Beijing 100871, China 12 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 13 Center for High Energy Physics, Peking University, Beijing 100871, China 14 State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 15 Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan 16 Department of Physics and Astronomy, San Francisco State University, San Francisco, CA 94132, USA 17 Department of Physics, Indiana University, Bloomington, IN 47405, USA 18 Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, NM 87545, USA 19 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan 20 School of High Energy Accelerator Science, The Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan 21 American Physical Society (APS), One Research Road, Ridge, NY 11961, USA 22 Instituto de Física Teórica UAM/CSIC and Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain 23 IFIC (CSIC-UVEG), Parc Científic de la Universitat de València, 46980 Paterna, Spain 24 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, UK 25 School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK 26 STAG Research Center, University of Southampton, Highfield, Southampton SO17 1BJ, UK 27 CERN, Theoretical Physics Department, Geneva, Switzerland 28 Department of Physics, University of Arizona, Tucson, AZ 85721, USA 29 Department of Physics, The College of William and Mary, Williamsburg, VA 23187, USA 30 Theory Center, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA 31 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27516-3255, USA 32 Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA 33 Physics Department, University of Washington, Seattle, WA 98195-1560, USA 34 Fermi National Accelerator Laboratory, Batavia, IL 60510, USA 35 INFN, Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy 36 School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland 37 John von Neumann Institute for Computing (NIC), DESY, Platanenallee 6, 15738 Zeuthen, Germany 38 Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany 39 INFN, Sezione di Tor Vergata, c/o Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy 40 PRISMA Cluster of Excellence, Institut für Kernphysik and Helmholtz Institute Mainz, University of Mainz, 55099 Mainz, Germany Received: 3 February 2022 / Accepted: 31 May 2022 © Crown 2022 Abstract We review lattice results related to pion, kaon, D-meson, B-meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of a e-mail: a.juttner@soton.ac.uk (corresponding author) 0123456789().: V,-vol 123 869 Page 2 of 296 Eur. Phys. J. C (2022) 82:869 the light-quark masses, the form factor f+(0) arising in the semileptonic K → π transition at zero momentum transfer, as well as the decay constant ratio fK / fπ and its consequences for the CKM matrix elements Vus and Vud . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU (2)L × SU (2)R and SU (3)L × SU (3)R Chiral Perturbation Theory. We review the determination of the BK parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for mc and mb as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant αs . We consider nucleon matrix elements, and review the determinations of the axial, scalar and tensor bilinears, both isovector and flavor diagonal. Finally, in this review we have added a new section reviewing determinations of scale-setting quantities. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 FLAG composition, guidelines and rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Citation policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quality criteria, averaging and error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Systematic errors and colour code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Systematic effects and rating criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Heavy-quark actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Conventions for the figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Averages and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Averaging procedure and error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Averaging – generic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Nested averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Masses of the light quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The physical point and isospin symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Ambiguities in the separation of isospin-breaking contributions . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Inclusion of electromagnetic effects in lattice-QCD simulations . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Lattice determination of ms and mud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Lattice determinations of ms/mud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Lattice determination of mu and md . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7 Estimates for R and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Charm-quark mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 N f = 2 + 1 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 N f = 2 + 1 + 1 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Lattice determinations of the ratio mc/ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bottom-quark mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 N f = 2 + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 N f = 2 + 1 + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Leptonic and semileptonic kaon and pion decay and |Vud | and |Vus | . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Experimental information concerning |Vud |, |Vus |, f+(0) and fK±/ fπ± . . . . . . . . . . . . . . . . . . . . . . 4.2 Lattice results for f+(0) and fK±/ fπ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Direct determination of f+(0) and fK±/ fπ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Results for f+(0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Results for fK±/ fπ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Extraction of |Vud | and |Vus | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Tests of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Analysis within the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Direct determination of fK± and fπ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Eur. Phys. J. C (2022) 82:869 Page 3 of 296 869 5 Low-energy constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Chiral perturbation theory and lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 ππ scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 πK and K K scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Extraction of SU(2) low-energy constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 New results for individual LO SU(2) LECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 New results for individual NLO SU(2) LECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 New results for an SU(2) linear combination linked to ππ scattering . . . . . . . . . . . . . . . . . . . . 5.2.4 LO and NLO SU(2) estimates and averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Extraction of SU(3) low-energy constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 New results for individual LO SU(3) LECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 New results for individual NLO SU(3) LECs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Results for SU(3) linear combinations linked to πK , K K scattering . . . . . . . . . . . . . . . . . . . . 5.3.4 Implication on Zweig rule violations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 LO and NLO SU (3) estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Kaon mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Indirect CP violation and K in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Lattice-QCD studies of the K → (ππ)I decay amplitudes, ξ and ′/ . . . . . . . . . . . . . . . . . . . . . . . 6.3 Lattice computation of BK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Kaon BSM B-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Charm hadron decay constants and form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Leptonic decay constants fD and fDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Form factors for D → πν and D → Kν semileptonic decays . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Form factors for c and c semileptonic decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Form factors for charm semileptonic decays with heavy spectator quarks . . . . . . . . . . . . . . . . . . . . . . 7.5 Determinations of |Vcd | and |Vcs | and test of second-row CKM unitarity . . . . . . . . . . . . . . . . . . . . . . 8 Bottom hadron decays and mixings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Leptonic decay constants fB and fBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Neutral B-meson mixing matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Semileptonic form factors for B decays to light flavours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Form factors for B → πν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Form factors for Bs → Kν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Form factors for rare and radiative B-semileptonic decays to light flavours . . . . . . . . . . . . . . . . . 8.4 Semileptonic form factors for B(s) → D ∗(s)ν and B(s) → D(s)ν . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 B(s) → D(s) decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Lepton-flavour-universality ratios R(D) and R(Ds) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Fragmentation fraction ratio fs/ fd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 B → D∗(s) (s) decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Semileptonic form factors for Bc → (ηc, J/ψ)ν decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Semileptonic form factors for (∗)b → (p,c )ν̄ decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Semileptonic form factors for  → (∗)b  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Determination of |Vub| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Determination of |Vcb| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Determination of |Vub/Vcb| from b decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Determination of |Vub/Vcb| from Bs decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12 Summary: |Vub| and |Vcb| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The strong coupling αs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Scheme and scale dependence of αs and QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Overview of the review of αs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Additions with respect to the FLAG 19 report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 General issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Discussion of criteria for computations entering the averages . . . . . . . . . . . . . . . . . . . . . . . . 123 869 Page 4 of 296 Eur. Phys. J. C (2022) 82:869 9.2.2 Physical scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Studies of truncation errors of perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 αs from step-scaling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The decoupling method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 αs from the potential at short distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 αs from the light-quark vacuum polarization in momentum/position space . . . . . . . . . . . . . . . . . . . . . 9.6.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Definitions in position space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Vacuum polarization in position space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 αs from observables at the lattice spacing scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 αs from heavy-quark current two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 αs from QCD vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.2 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 αs from the eigenvalue spectrum of the Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.2 Discussion of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.1 The present situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.2 Our range for (5)α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MS 9.11.3 Ranges for [r ](N0 f ) and MS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Nucleon matrix elements (NME) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Isovector and flavour diagonal charges of the nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Technical aspects of the calculations of nucleon matrix elements . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Controlling excited-state contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Renormalization and Symanzik improvement of local currents . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Extrapolations in a, Mπ and Mπ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Quality criteria for nucleon matrix elements and averaging procedure . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Isovector charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Results for gu−dA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Results for gu−dS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Results for gu−dT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Flavour diagonal charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Results for gu,d,sA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Results for gu,d,sS from direct and hybrid calculations of the matrix elements . . . . . . . . . . . . . . . . 10.4.3 Results for gu,d,sS using the Feynman–Hellmann theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Summary of results for gu,d,sS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Results for gu,d,sT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Scale setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scale setting as part of hadronic renormalization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Eur. Phys. J. C (2022) 82:869 Page 5 of 296 869 11.2.1 Theory scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Isospin breaking, electromagnetism, and definition of hadronic schemes . . . . . . . . . . . . . . . . . . . . . . 11.3.1 The approximate nature of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Hadronic renormalization of QCD + QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Hadronic definition of QCD and of QED corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Physical scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 The mass of the baryon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Pion and kaon leptonic decay rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Other physics scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Theory scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Potential scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Gradient flow scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Other theory scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 List of computations and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Gradient flow scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Potential scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.3 Ratios of scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Observations and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowlegement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A List of acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Parameterizations of semileptonic form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Notes to Sect. 3 on quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Notes to Sect. 4 on |Vud | and |Vus | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Notes to Sect. 5 on low-energy constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Notes to Sect. 6 on kaon mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4.1 K → ππ decay amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4.2 Kaon B-parameter BK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4.3 Kaon BSM B-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Notes to Sect. 7 on D-meson decay constants and form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5.1 Form factors for semileptonic decays of charmed hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 Notes to Sect. 8 on B-meson decay constants, mixing parameters and form factors . . . . . . . . . . . . . . . . . C.6.1 B(s)-meson decay constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6.2 B(s)-meson mixing matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6.3 Form factors entering determinations of |Vub| (B → πν, Bs → Kν, b → pν̄) . . . . . . . . . . . . . C.6.4 Form factors for rare decays of beauty hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6.5 Form factors entering determinations of |V | (B → D(∗) (∗)cb (s) (s) ν, b → c ν̄) and R(D(s)) . . . . . . . . C.7 Notes to Sect. 9 on the strong coupling αs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7.1 Renormalization scale and perturbative behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7.2 Continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.8 Notes to Sect. 10 on nucleon matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.9 Notes to Sect. 11 on scale setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Flavour physics provides an important opportunity for exploring the limits of the Standard Model of particle physics and for constraining possible extensions that go beyond it. As the LHC explores a new energy frontier and as experiments continue 123 869 Page 6 of 296 Eur. Phys. J. C (2022) 82:869 to extend the precision frontier, the importance of flavour physics will grow, both in terms of searches for signatures of new physics through precision measurements and in terms of attempts to construct the theoretical framework behind direct discoveries of new particles. Crucial to such searches for new physics is the ability to quantify strong-interaction effects. Large-scale numerical simulations of lattice QCD allow for the computation of these effects from first principles. The scope of the Flavour Lattice Averaging Group (FLAG) is to review the current status of lattice results for a variety of physical quantities that are important for flavour physics. Set up in November 2007, it comprises experts in Lattice Field Theory, Chiral Perturbation Theory and Standard Model phenomenology. Our aim is to provide an answer to the frequently posed question “What is currently the best lattice value for a particular quantity?” in a way that is readily accessible to those who are not expert in lattice methods. This is generally not an easy question to answer; different collaborations use different lattice actions (discretizations of QCD) with a variety of lattice spacings and volumes, and with a range of masses for the u- and d-quarks. Not only are the systematic errors different, but also the methodology used to estimate these uncertainties varies between collaborations. In the present work, we summarize the main features of each of the calculations and provide a framework for judging and combining the different results. Sometimes it is a single result that provides the “best” value; more often it is a combination of results from different collaborations. Indeed, the consistency of values obtained using different formulations adds significantly to our confidence in the results. The first four editions of the FLAG review were made public in 2010 [1], 2013 [2], 2016 [3], and 2019 [4] (and will be referred to as FLAG 10, FLAG 13, FLAG 16, and FLAG 19, respectively). The fourth edition reviewed results related to both light (u-, d- and s-), and heavy (c- and b-) flavours. The quantities related to pion and kaon physics were light-quark masses, the form factor f+(0) arising in semileptonic K → π transitions (evaluated at zero momentum transfer), the decay constants fK and fπ , the BK parameter from neutral kaon mixing, and the kaon mixing matrix elements of new operators that arise in theories of physics beyond the Standard Model. Their implications for the CKM matrix elements Vus and Vud were also discussed. Furthermore, results were reported for some of the low-energy constants of SU (2)L × SU (2)R and SU (3)L × SU (3)R Chiral Perturbation Theory. The quantities related to D- and B-meson physics that were reviewed were the masses of the charm and bottom quarks together with the decay constants, form factors, and mixing parameters of B- and D-mesons. These are the heavy-light quantities most relevant to the determination of CKM matrix elements and the global CKM unitarity-triangle fit. The current status of lattice results on the QCD coupling αs was reviewed. Last but not least, we reviewed calculations of nucleon matrix elements of flavor nonsinglet and singlet bilinear operators, including the nucleon axial charge gA and the nucleon sigma term. These results are relevant for constraining Vud , for searches for new physics in neutron decays and other processes, and for dark matter searches. In the present paper we provide updated results for all the above-mentioned quantities, but also extend the scope of the review by adding a section on scale setting, Sect. 11. The motivation for adding this section is that uncertainties in the value of the lattice spacing a are a major source of error for the calculation of a wide range of quantities. Thus we felt that a systematic compilation of results, comparing the different approaches to setting the scale, and summarizing the present status, would be a useful resource for the lattice community. An additional update is the inclusion, in Sect. 6.2, of a brief description of the status of lattice calculations of K → ππ decay amplitudes. Although some aspects of these calculations are not yet at the stage to be included in our averages, they are approaching this stage, and we felt that, given their phenomenological relevance, a brief review was appropriate. For the most precisely determined quantities, isospin breaking – both from the up-down quark mass difference and from QED – must be included. A short review of methods used to include QED in lattice-QCD simulations is given in Sect. 3.1.3. An important issue here is that, in the context of a QED+QCD theory, the separation into QED and QCD contributions to a given physical quantity is ambiguous – there are several ways of defining such a separation. This issue is discussed from different viewpoints in the section on quark masses – see Sect. 3.1.1 – and that on scale setting – see Sect. 11. We stress, however, that the physical observable in QCD + QED is defined unambiguously. Any ambiguity only arises because we are trying to separate a well-defined, physical quantity into two unphysical parts that provide useful information for phenomenology. Our main results are collected in Tables 1, 2, 3, 4 and 5. As is clear from the tables, for most quantities there are results from ensembles with different values for N f . In most cases, there is reasonable agreement among results with N f = 2, 2+1, and 2 + 1 + 1. As precision increases, we may some day be able to distinguish among the different values of N f , in which case, presumably 2+ 1+ 1 would be the most realistic. (If isospin violation is critical, then 1+ 1+ 1 or 1+ 1+ 1+ 1 might be desired.) At present, for some quantities the errors in the N f = 2 + 1 results are smaller than those with N f = 2 + 1 + 1 (e.g., for mc), while for others the relative size of the errors is reversed. Our suggestion to those using the averages is to take whichever of the N f = 2+1 or N f = 2+1+1 results has the smaller error. We do not recommend using the N f = 2 results, except for studies of the N f -dependence of condensates and αs , as these have an uncontrolled systematic error coming from quenching the strange quark. 123 Eur. Phys. J. C (2022) 82:869 Page 7 of 296 869 Table 1 Summary of the main results of this review concerning quark masses, light-meson decay constants, and hadronic kaon-decay and kaon- mixing parameters. These are grouped in terms of N f , the number of dynamical quark flavours in lattice simulations. Quark masses are given in the MS scheme at running scale μ = 2 GeV or as indicated. BSM bag parameters B2,3,4,5 are given in the MS scheme at scale μ = 3 GeV. Further specifications of the quantities are given in the quoted sections. Results for N f = 2 quark masses are unchanged since FLAG 16 [3], and are not included here. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors Quantity Sections N f = 2 + 1 + 1 Refs. N f = 2 + 1 Refs. N f = 2 Refs. mud [MeV] 3.1.4 3.410(43) [6,7] 3.381(40) [8–12] ms [MeV] 3.1.4 93.40(57) [6,7,13,14] 92.2(1.0) [8–11,15] ms/mud 3.1.5 27.23(10) [7,16,17] 27.42(12) [8–10,15,18] mu [MeV] 3.1.6 2.14(8) [6,19] 2.27(9) [20] md [MeV] 3.1.6 4.70(5) [6,19] 4.67(9) [20] mu/md 3.1.6 0.465(24) [19,21] 0.485(19) [20] mc(3 GeV) [GeV] 3.2.2 0.988(11) [6,7,14,22,23] 0.992(5) [11,24–26] mc/ms 3.2.3 11.768(34) [6,7,14] 11.82(16) [24,27] mb(mb) [GeV] 3.3 4.203(11) [6,28–31] 4.171(20) [11] f+(0) 4.3 0.9698(17) [32,33] 0.9677(27) [34,35] 0.9560(57)(62) [36] fK±/ fπ± 4.3 1.1932(21) [16,37–39] 1.1917(37) [8,40–44] 1.205(18) [45] fπ± [MeV] 4.6 130.2(8) [8,40,41] fK± [MeV] 4.6 155.7(3) [17,37,38] 155.7(7) [8,40,41] 157.5(2.4) [45] Re(A2) [GeV] 6.2 1.50(4)(14)× 10−8 [46] Im(A2) [GeV] 6.2 −8.34(1.03)× 10−13 [46] B̂K 6.3 0.717(18)(16) [47] 0.7625(97) [8,48–50] 0.727(22)(12) [51] B2 6.4 0.46(1)(3) [47] 0.502(14) [50,52] 0.47(2)(1) [51] B3 6.4 0.79(2)(5) [47] 0.766(32) [50,52] 0.78(4)(2) [51] B4 6.4 0.78(2)(4) [47] 0.926(19) [50,52] 0.76(2)(2) [51] B5 6.4 0.49(3)(3) [47] 0.720(38) [50,52] 0.58(2)(2) [51] Our plan is to continue providing FLAG updates, in the form of a peer reviewed paper, roughly on a triennial basis. This effort is supplemented by our more frequently updated website http://flag.unibe.ch [5], where figures as well as pdf-files for the individual sections can be downloaded. The papers reviewed in the present edition have appeared before the closing date 30 April 2021.1 This review is organized as follows. In the remainder of Sect. 1 we summarize the composition and rules of FLAG and discuss general issues that arise in modern lattice calculations. In Sect. 2, we explain our general methodology for evaluating the robustness of lattice results. We also describe the procedures followed for combining results from different collaborations in a single average or estimate (see Sect. 2.2 for our definition of these terms). The rest of the paper consists of sections, each dedicated to a set of closely connected physical quantities, or, for the final section, to the determination of the lattice scale. Each of these sections is accompanied by an Appendix with explicatory notes.2 In previous editions, we have provided, in an appendix, a glossary summarizing some standard lattice terminology and describing the most commonly used lattice techniques and methodologies. Since no significant updates in this information have occurred since our previous edition, we have decided, in the interests of reducing the length of this review, to omit this glossary, and refer the reader to FLAG 19 for this information [4]. This appendix also contained, in previous versions, a tabulation of the actions used in the papers that were reviewed. Since this information is available in the discussions in the 1 Working groups were given the option of including papers submitted to arxiv.org before the closing date but published after this date. This flexibility allows this review to be up-to-date at the time of submission. A single paper of this type was included. 2 In some cases, in order to keep the length of this review within reasonable bounds, we have dropped these notes for older data, since they can be found in previous FLAG reviews [1–4]. 123 869 Page 8 of 296 Eur. Phys. J. C (2022) 82:869 Table 2 Summary of the main results of this review concerning heavy-light mesons and the strong coupling constant. These are grouped in terms of N f , the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors Quantity Section N f = 2 + 1 + 1 Refs. N f = 2 + 1 Refs. N f = 2 Refs. fD [MeV] 7.1 212.0(7) [16,38] 209.0(2.4) [53–55] 208(7) [56] fDs [MeV] 7.1 249.9(5) [16,38] 248.0(1.6) [24,54,55,57] 246(4) [56,58] fDs f 7.1 1.1783(16) [16,38] 1.174(7) [53–55] 1.20(2) [56]D f+Dπ (0) 7.2 0.612(35) [59] 0.666(29) [60] f+DK (0) 7.2 0.7385(44) [59,61] 0.747(19) [62] fB [MeV] 8.1 190.0(1.3) [16,30,63,64] 192.0(4.3) [54,65–68] 188(7) [56,69] fBs [MeV] 8.1 230.3(1.3) [16,30,63,64] 228.4(3.7) [54,65–68] 225.3(6.6) [56,58,69] fBs f 8.1 1.209(5) [16,30,63,64] 1.201(16) [54,65–68,70] 1.206(23) [56,69]B √ fBd B̂bd [MeV] 8.2 210.6(5.5) [71] 225(9) [67,72,73] 216(10) [56]√ fBs B̂Bs [MeV] 8.2 256.1(5.7) [71] 274(8) [67,72,73] 262(10) [56] B̂Bd 8.2 1.222(61) [71] 1.30(10) [67,72,73] 1.30(6) [56] B̂Bs 8.2 1.232(53) [71] 1.35(6) [67,72,73] 1.32(5) [56] ξ 8.2 1.216(16) [71] 1.206(17) [67,73] 1.225(31) [56] BBs /BBd 8.2 1.008(25) [71] 1.032(38) [67,73] 1.007(21) [56] Quantity Section N f = 2 + 1 and N f = 2 + 1 + 1 Refs. (5) α (MZ ) 9.11 0.1184(8) [11,14,74–79]MS (5)  [MeV] 9.11 214(10) [11,14,74–79] MS (4)  [MeV] 9.11 297(12) [11,14,74–79] MS (3)  [MeV] 9.11 339(12) [11,14,74–79] MS separate sections, and is time-consuming to collect from the sections, we have dropped these tables. We have, however, kept a short appendix, Appendix B.1, describing the parameterizations of semileptonic form factors that are used in Sect. 8. Moreover, in Appendix A, we have added a summary and explanations of acronyms introduced in the manuscript. Collaborations referred to by an acronym can be identified through the corresponding bibliographic reference. 1.1 FLAG composition, guidelines and rules FLAG strives to be representative of the lattice community, both in terms of the geographical location of its members and the lattice collaborations to which they belong. We aspire to provide the nuclear- and particle-physics communities with a single source of reliable information on lattice results. In order to work reliably and efficiently, we have adopted a formal structure and a set of rules by which all FLAG members abide. The collaboration presently consists of an Advisory Board (AB), an Editorial Board (EB), and nine Working Groups (WG). The rôle of the Advisory Board is to provide oversight of the content, procedures, schedule and membership of FLAG, to help resolve disputes, to serve as a source of advice to the EB and to FLAG as a whole, and to provide a critical assessment of drafts. They also give their approval of the final version of the preprint before it is rendered public. The Editorial Board coordinates the activities of FLAG, sets priorities and intermediate deadlines, organizes votes on FLAG procedures, writes the introductory sections, and takes care of the editorial work needed to amalgamate the sections written by the individual working groups into a uniform and coherent review. The working groups concentrate on writing the review of the physical quantities for which they are responsible, which is subsequently circulated to the whole collaboration for critical evaluation. 123 Eur. Phys. J. C (2022) 82:869 Page 9 of 296 869 Table 3 Summary of the main results of this review concerning LECs, grouped in terms of N f , the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors Quantity Section N f = 2 + 1 + 1 Refs. N f = 2 + 1 Refs. N f = 2 Refs. 1/3 [MeV] 5.2.4 286(23) [80,81] 272(5) [12,82–86] 266(10) [80,87–89] Fπ/F 5.2.4 1.077(3) [90] 1.062(7) [41,82–84,91] 1.073(15) [87–89,92] ̄3 5.2.4 3.53(26) [90] 3.07(64) [41,82–84,91] 3.41(82) [87,88,92] ̄4 5.2.4 4.73(10) [90] 4.02(45) [41,82–84,91] 4.40(28) [87,88,92,93] ̄6 5.2.4 15.1(1.2) [88,92] a20 Mπ 5.2.4 −0.0441(4) [94] −0.04385(47) [95] 1/3 0 [MeV] 5.3.5 245(8) [15] / 0 5.3.5 1.48(16) [15] F0 [MeV] 5.3.5 80.3(6.0) [41] F/F0 5.3.5 1.104(41) [15] B/B0 5.3.5 1.21(7) [15] L4 5.3.5 +0.09(34)× 10−3 [37] −0.02(56)× 10−3 [41] L5 5.3.5 +1.19(25)× 10−3 [37] +0.95(41)× 10−3 [41] L6 5.3.5 +0.16(20)× 10−3 [37] +0.01(34)× 10−3 [41] L8 5.3.5 +0.55(15)× 10−3 [37] +0.43(28)× 10−3 [41] a1/20 μπK 5.3.5 0.127(2) [96] a3/20 μπK 5.3.5 −0.0463(17) [96] a10 MK 5.3.5 −0.388(20) [97] Table 4 Summary of the main results of this review concerning nuclear matrix elements, grouped in terms of N f , the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors Quantity Section N f = 2 + 1 + 1 Refs. N f = 2 + 1 Refs. N f = 2 Refs. gu−dA 10.3.1 1.246(28) [98–100] 1.248(23) [101,102] gu−dS 10.3.2 1.02(10) [98] 1.13(14) [102] gu−dT 10.3.3 0.989(34) [98] 0.965(61) [102] guA 10.4.1 0.777(25)(30) [103] 0.847(18)(32) [101] gdA 10.4.1 −0.438(18)(30) [103] −0.407(16)(18) [101] gsA 10.4.1 −0.053(8) [103] −0.035(6)(7) [101] σπN [MeV] 10.4.4 64.9(1.5)(13.2) [22] 39.7(3.6) [104–106] 37(8)(6) [107] σs [MeV] 10.4.4 41.0(8.8) [108] 52.9(7.0) [104–106,108,109] guT 10.4.5 0.784(28)(10) [110] gdT 10.4.5 −0.204(11)(10) [110] gsT 10.4.5 −0.0027(16) [110] The current list of FLAG members and their Working Group assignments is: • Advisory Board (AB): G. Colangelo, M. Golterman, P. Hernandez, T. Onogi, and R. Van de Water • Editorial Board (EB): S. Gottlieb, A. Jüttner, S. Hashimoto, S.R. Sharpe, and U. Wenger 123 869 Page 10 of 296 Eur. Phys. J. C (2022) 82:869 Table 5 Summary of the main results of this review concerning setting of the lattice scale, grouped in terms of N f , the number of dynamical quark flavours in lattice simulations. The quantities listed are specified in the quoted sections. For each result we list the references that enter the FLAG average or estimate, and we stress again the importance of quoting these original works when referring to FLAG results. From the entries in this column one can also read off the number of results that enter our averages for each quantity. We emphasize that these numbers only give a very rough indication of how thoroughly the quantity in question has been explored on the lattice and recommend consulting the detailed tables and figures in the relevant section for more significant information and for explanations on the source of the quoted errors Quantity Section N f = 1 + 1 Refs. N f = 2 + 1 + 1 Refs. N f = 2 + 1 Refs. N f > 2 + 1 Refs. + 1 + 1 √ t0 [fm] 11.5.2 0.14186(88) [37,111,112] 0.14464(87) [8,113,114] w0 [fm] 11.5.2 0.17236(70) [115] 0.17128(107) [37,111,112] 0.17355(92) [8,114,116] 0.17177(67) [37,111,112,115] r0 [fm] 11.5.2 0.474(14) [7] 0.4701(36) [24,116–119] r1 [fm] 11.5.2 0.3112(30) [37] 0.3127(30) [41,117–120] • Working Groups (coordinator listed first): – Quark masses T. Blum, A. Portelli, and A. Ramos – Vus, Vud T. Kaneko, J. N. Simone, S. Simula, and N. Tantalo – LEC S. Dürr, H. Fukaya, and U.M. Heller – BK P. Dimopoulos, X. Feng, and G. Herdoiza – fB s , fD s , BB Y. Aoki, M. Della Morte, and C. Monahan( ) ( ) – b and c semileptonic and radiative decays E. Lunghi, S. Meinel, and C. Pena – αs S. Sint, R. Horsley, and P. Petreczky – NME R. Gupta, S. Collins, A. Nicholson, and H. Wittig – Scale setting R. Sommer, N. Tantalo, and U. Wenger The most important FLAG guidelines and rules are the following: • the composition of the AB reflects the main geographical areas in which lattice collaborations are active, with members from America, Asia/Oceania, and Europe; • the mandate of regular members is not limited in time, but we expect that a certain turnover will occur naturally; • whenever a replacement becomes necessary this has to keep, and possibly improve, the balance in FLAG, so that different collaborations, from different geographical areas are represented; • in all working groups the members must belong to different lattice collaborations; • a paper is in general not reviewed (nor colour-coded, as described in the next section) by any of its authors; • lattice collaborations will be consulted on the colour coding of their calculation; • there are also internal rules regulating our work, such as voting procedures. As for FLAG 19, for this review we sought the advice of external reviewers once a complete draft of the review was available. For each review section, we have asked one lattice expert (who could be a FLAG alumnus/alumna) and one nonlattice phenomenologist for a critical assessment. The one exception is the scale-setting section, where only a lattice expert has been asked to provide input. This is similar to the procedure followed by the Particle Data Group in the creation of the Review of Particle Physics. The reviewers provide comments and feedback on scientific and stylistic matters. They are not anonymous, and enter into a discussion with the authors of the WG. Our aim with this additional step is to make sure that a wider array of viewpoints enter into the discussions, so as to make this review more useful for its intended audience. 1.2 Citation policy We draw attention to this particularly important point. As stated above, our aim is to make lattice-QCD results easily accessible to those without lattice expertise, and we are well aware that it is likely that some readers will only consult the present paper and not the original lattice literature. It is very important that this paper not be the only one cited when our results are quoted. We strongly suggest that readers also cite the original sources. In order to facilitate this, in Tables 1, 2, 3, 4, and 5, besides summarizing the main results of the present review, we also cite the original references from which they have been obtained. 123 Eur. Phys. J. C (2022) 82:869 Page 11 of 296 869 In addition, for each figure we make a bibtex file available on our webpage [5] which contains the bibtex entries of all the calculations contributing to the FLAG average or estimate. The bibliography at the end of this paper should also make it easy to cite additional papers. Indeed, we hope that the bibliography will be one of the most widely used elements of the whole paper. 1.3 General issues Several general issues concerning the present review are thoroughly discussed in Sec. 1.1 of our initial 2010 paper [1], and we encourage the reader to consult the relevant pages. In the remainder of the present subsection, we focus on a few important points. Though the discussion has been duly updated, it is similar to that of Sec. 1.2 in the previous three reviews [2–4]. The present review aims to achieve two distinct goals: first, to provide a description of the relevant work done on the lattice; and, second, to draw conclusions on the basis of that work, summarizing the results obtained for the various quantities of physical interest. The core of the information about the work done on the lattice is presented in the form of tables, which not only list the various results, but also describe the quality of the data that underlie them. We consider it important that this part of the review represents a generally accepted description of the work done. For this reason, we explicitly specify the quality requirements used and provide sufficient details in appendices so that the reader can verify the information given in the tables.3 On the other hand, the conclusions drawn on the basis of the available lattice results are the responsibility of FLAG alone. Preferring to err on the side of caution, in several cases we draw conclusions that are more conservative than those resulting from a plain weighted average of the available lattice results. This cautious approach is usually adopted when the average is dominated by a single lattice result, or when only one lattice result is available for a given quantity. In such cases, one does not have the same degree of confidence in results and errors as when there is agreement among several different calculations using different approaches. The reader should keep in mind that the degree of confidence cannot be quantified, and it is not reflected in the quoted errors. Each discretization has its merits, but also its shortcomings. For most topics covered in this review we have an increasingly broad database, and for most quantities lattice calculations based on totally different discretizations are now available. This is illustrated by the dense population of the tables and figures in most parts of this review. Those calculations that do satisfy our quality criteria indeed lead, in almost all cases, to consistent results, confirming universality within the accuracy reached. The consistency between independent lattice results, obtained with different discretizations, methods, and simulation parameters, is an important test of lattice QCD, and observing such consistency also provides further evidence that systematic errors are fully under control. In the sections dealing with heavy quarks and with αs , the situation is not the same. Since the b-quark mass can barely be resolved with current lattice spacings, most lattice methods for treating b quarks use effective field theory at some level. This introduces additional complications not present in the light-quark sector. An overview of the issues specific to heavy-quark quantities is given in the introduction of Sect. 8. For B- and D-meson leptonic decay constants, there already exists a good number of different independent calculations that use different heavy-quark methods, but there are only a few independent calculations of semileptonic B, b, and D form factors and of B− B̄ mixing parameters. For αs , most lattice methods involve a range of scales that need to be resolved and controlling the systematic error over a large range of scales is more demanding. The issues specific to determinations of the strong coupling are summarized in Sect. 9. Number of sea quarks in lattice simulations Lattice-QCD simulations currently involve two, three or four flavours of dynamical quarks. Most simulations set the masses of the two lightest quarks to be equal, while the strange and charm quarks, if present, are heavier (and tuned to lie close to their respective physical values). Our notation for these simulations indicates which quarks are nondegenerate, e.g., N f = 2 + 1 if mu = md < ms and N f = 2 + 1 + 1 if mu = md < ms < mc. Calculations with N f = 2, i.e., two degenerate dynamical flavours, often include strange valence quarks interacting with gluons, so that bound states with the quantum numbers of the kaons can be studied, albeit neglecting strange sea-quark fluctuations. The quenched approximation (N f = 0), in which all sea-quark contributions are omitted, has uncontrolled systematic errors and is no longer used in modern lattice simulations with relevance to phenomenology. Accordingly, we will review results obtained with N f = 2, N f = 2+1, and N f = 2+1+1, but omit earlier results with N f = 0. The only exception concerns the QCD coupling constant αs . Since this observable does not require valence light quarks, it is theoretically well defined also in the N f = 0 theory, which is simply pure gluodynamics. 3 We also use terms like “quality criteria”, “rating”, “colour coding”, etc., when referring to the classification of results, as described in Sect. 2. 123 869 Page 12 of 296 Eur. Phys. J. C (2022) 82:869 The N f -dependence of αs , or more precisely of the related quantity r0MS, is a theoretical issue of considerable interest; here r0 is a quantity with the dimension of length that sets the physical scale, as discussed in Sect. 11. We stress, however, that only results with N f ≥ 3 are used to determine the physical value of αs at a high scale. Lattice actions, simulation parameters, and scale setting The remarkable progress in the precision of lattice calculations is due to improved algorithms, better computing resources, and, last but not least, conceptual developments. Examples of the latter are improved actions that reduce lattice artifacts and actions that preserve chiral symmetry to very good approximation. A concise characterization of the various discretizations that underlie the results reported in the present review is given in Appendix A.1 of FLAG 19. Physical quantities are computed in lattice simulations in units of the lattice spacing so that they are dimensionless. For example, the pion decay constant that is obtained from a simulation is fπa, where a is the spacing between two neighboring lattice sites. (All simulations with results quoted in this review use hypercubic lattices, i.e., with the same spacing in all four Euclidean directions.) To convert these results to physical units requires knowledge of the lattice spacing a at the fixed values of the bare QCD parameters (quark masses and gauge coupling) used in the simulation. This is achieved by requiring agreement between the lattice calculation and experimental measurement of a known quantity, which thus “sets the scale” of a given simulation. Given the central importance of this procedure, we include in this edition of FLAG a dedicated section, Sect. 11, discussing the issues and results. Renormalization and scheme dependence Several of the results covered by this review, such as quark masses, the gauge coupling, and B-parameters, are for quantities defined in a given renormalization scheme and at a specific renormalization scale. The schemes employed (e.g., regularization- independent MOM schemes) are often chosen because of their specific merits when combined with the lattice regularization. For a brief discussion of their properties, see Appendix A.3 of FLAG 19. The conversion of the results obtained in these so-called intermediate schemes to more familiar regularization schemes, such as the MS-scheme, is done with the aid of perturbation theory. It must be stressed that the renormalization scales accessible in simulations are limited, because of the presence of an ultraviolet (UV) cutoff of ∼ π/a. To safely match to MS, a scheme defined in perturbation theory, Renormalization Group (RG) running to higher scales is performed, either perturbatively or nonperturbatively (the latter using finite-size scaling techniques). Extrapolations Because of limited computing resources, lattice simulations are often performed at unphysically heavy pion masses, although results at the physical point have become increasingly common. Further, numerical simulations must be done at nonzero lattice spacing, and in a finite (four-dimensional) volume. In order to obtain physical results, lattice data are obtained at a sequence of pion masses and a sequence of lattice spacings, and then extrapolated to the physical pion mass and to the continuum limit. In principle, an extrapolation to infinite volume is also required. However, for most quantities discussed in this review, finite- volume effects are exponentially small in the linear extent of the lattice in units of the pion mass, and, in practice, one often verifies volume independence by comparing results obtained on a few different physical volumes, holding other parameters fixed. To control the associated systematic uncertainties, these extrapolations are guided by effective theories. For light-quark actions, the lattice-spacing dependence is described by Symanzik’s effective theory [121,122]; for heavy quarks, this can be extended and/or supplemented by other effective theories such as Heavy-Quark Effective Theory (HQET). The pion-mass dependence can be parameterized with Chiral Perturbation Theory (χPT), which takes into account the Nambu–Goldstone nature of the lowest excitations that occur in the presence of light quarks. Similarly, one can use Heavy-Light Meson Chiral Perturbation Theory (HMχPT) to extrapolate quantities involving mesons composed of one heavy (b or c) and one light quark. One can combine Symanzik’s effective theory with χPT to simultaneously extrapolate to the physical pion mass and the continuum; in this case, the form of the effective theory depends on the discretization. See Appendix A.4 of FLAG 19 for a brief description of the different variants in use and some useful references. Finally, χPT can also be used to estimate the size of finite-volume effects measured in units of the inverse pion mass, thus providing information on the systematic error due to finite-volume effects in addition to that obtained by comparing simulations at different volumes. Excited-state contamination In all the hadronic matrix elements discussed in this review, the hadron in question is the lightest state with the chosen quantum numbers. This implies that it dominates the required correlation functions as their extent in Euclidean time is increased. Excited-state contributions are suppressed by e−Eτ , where E is the gap between the ground and excited states, and τ the relevant separation in Euclidean time. The size of E depends on the hadron in question, and in general is a multiple of the pion mass. In practice, as discussed at length in Sect. 10, the contamination of signals due to excited-state 123 Eur. Phys. J. C (2022) 82:869 Page 13 of 296 869 contributions is a much more challenging problem for baryons than for the other particles discussed here. This is in part due to the fact that the signal-to-noise ratio drops exponentially for baryons, which reduces the values of τ that can be used. Critical slowing down The lattice spacings reached in recent simulations go down to 0.05 fm or even smaller. In this regime, long autocorrelation times slow down the sampling of the configurations [123–132]. Many groups check for autocorrelations in a number of observables, including the topological charge, for which a rapid growth of the autocorrelation time is observed with decreasing lattice spacing. This is often referred to as topological freezing. A solution to the problem consists in using open boundary conditions in time [133], instead of the more common antiperiodic ones. More recently, two other approaches have been proposed, one based on a multiscale thermalization algorithm [134,135] and another based on defining QCD on a nonorientable manifold [136]. The problem is also touched upon in Sect. 9.2.1, where it is stressed that attention must be paid to this issue. While large scale simulations with open boundary conditions are already far advanced [137], few results reviewed here have been obtained with any of the above methods. It is usually assumed that the continuum limit can be reached by extrapolation from the existing simulations, and that potential systematic errors due to the long autocorrelation times have been adequately controlled. Partially or completely frozen topology would produce a mixture of different θ vacua, and the difference from the desired θ = 0 result may be estimated in some cases using chiral perturbation theory, which gives predictions for the θ -dependence of the physical quantity of interest [138,139]. These ideas have been systematically and successfully tested in various models in [140,141], and a numerical test on MILC ensembles indicates that the topology dependence for some of the physical quantities reviewed here is small, consistent with theoretical expectations [142]. Simulation algorithms and numerical errors Most of the modern lattice-QCD simulations use exact algorithms such as those of Refs. [143,144], which do not produce any systematic errors when exact arithmetic is available. In reality, one uses numerical calculations at double (or in some cases even single) precision, and some errors are unavoidable. More importantly, the inversion of the Dirac operator is carried out iteratively and it is truncated once some accuracy is reached, which is another source of potential systematic error. In most cases, these errors have been confirmed to be much less than the statistical errors. In the following we assume that this source of error is negligible. Some of the most recent simulations use an inexact algorithm in order to speed up the computation, though it may produce systematic effects. Currently available tests indicate that errors from the use of inexact algorithms are under control [145]. 2 Quality criteria, averaging and error estimation The essential characteristics of our approach to the problem of rating and averaging lattice quantities have been outlined in our first publication [1]. Our aim is to help the reader assess the reliability of a particular lattice result without necessarily studying the original article in depth. This is a delicate issue, since the ratings may make things appear simpler than they are. Nevertheless, it safeguards against the possibility of using lattice results, and drawing physics conclusions from them, without a critical assessment of the quality of the various calculations. We believe that, despite the risks, it is important to provide some compact information about the quality of a calculation. We stress, however, the importance of the accompanying detailed discussion of the results presented in the various sections of the present review. 2.1 Systematic errors and colour code The major sources of systematic error are common to most lattice calculations. These include, as discussed in detail below, the chiral, continuum, and infinite-volume extrapolations. To each such source of error for which systematic improvement is possible we assign one of three coloured symbols: green star, unfilled green circle (which replaced in Ref. [2] the amber disk used in the original FLAG review [1]) or red square. These correspond to the following ratings:  the parameter values and ranges used to generate the data sets allow for a satisfactory control of the systematic uncertainties; ◦ the parameter values and ranges used to generate the data sets allow for a reasonable attempt at estimating systematic uncertainties, which however could be improved;  the parameter values and ranges used to generate the data sets are unlikely to allow for a reasonable control of systematic uncertainties. 123 869 Page 14 of 296 Eur. Phys. J. C (2022) 82:869 The appearance of a red tag, even in a single source of systematic error of a given lattice result, disqualifies it from inclusion in the global average. Note that in the first two editions [1,2], FLAG used the three symbols in order to rate the reliability of the systematic errors attributed to a given result by the paper’s authors. Starting with FLAG 16 [3] the meaning of the symbols has changed slightly – they now rate the quality of a particular simulation, based on the values and range of the chosen parameters, and its aptness to obtain well-controlled systematic uncertainties. They do not rate the quality of the analysis performed by the authors of the publication. The latter question is deferred to the relevant sections of the present review, which contain detailed discussions of the results contributing (or not) to each FLAG average or estimate. For most quantities the colour-coding system refers to the following sources of systematic errors: (i) chiral extrapolation; (ii) continuum extrapolation; (iii) finite volume. As we will see below, renormalization is another source of systematic uncertainties in several quantities. This we also classify using the three coloured symbols listed above, but now with a different rationale: they express how reliably these quantities are renormalized, from a field-theoretic point of view (namely, nonperturbatively, or with 2-loop or 1-loop perturbation theory). Given the sophisticated status that the field has attained, several aspects, besides those rated by the coloured symbols, need to be evaluated before one can conclude whether a particular analysis leads to results that should be included in an average or estimate. Some of these aspects are not so easily expressible in terms of an adjustable parameter such as the lattice spacing, the pion mass or the volume. As a result of such considerations, it sometimes occurs, albeit rarely, that a given result does not contribute to the FLAG average or estimate, despite not carrying any red tags. This happens, for instance, whenever aspects of the analysis appear to be incomplete (e.g., an incomplete error budget), so that the presence of inadequately controlled systematic effects cannot be excluded. This mostly refers to results with a statistical error only, or results in which the quoted error budget obviously fails to account for an important contribution. Of course, any colour coding has to be treated with caution; we emphasize that the criteria are subjective and evolving. Sometimes, a single source of systematic error dominates the systematic uncertainty and it is more important to reduce this uncertainty than to aim for green stars for other sources of error. In spite of these caveats, we hope that our attempt to introduce quality measures for lattice simulations will prove to be a useful guide. In addition, we would like to stress that the agreement of lattice results obtained using different actions and procedures provides further validation. 2.1.1 Systematic effects and rating criteria The precise criteria used in determining the colour coding are unavoidably time-dependent; as lattice calculations become more accurate, the standards against which they are measured become tighter. For this reason FLAG reassesses criteria with each edition and as a result some of the quality criteria (the one on chiral extrapolation for instance) have been tightened up over time [1–4]. In the following, we present the rating criteria used in the current report. While these criteria apply to most quantities without modification there are cases where they need to be amended or additional criteria need to be defined. For instance, when discussing results obtained in the -regime of chiral perturbation theory in Sect. 5 the finite volume criterion listed below for the p-regime is no longer appropriate.4 Similarly, the discussion of the strong coupling constant in Sect. 9 requires tailored criteria for renormalization, perturbative behaviour, and continuum extrapolation. Finally, in the section on nuclear matrix elements, Sect. 10, the chiral extrapolation criterion is made slightly stronger, and a new criterion is adopted for excited-state contributions. In such cases, the modified criteria are discussed in the respective sections. Apart from only a few exceptions the following colour code applies in the tables: • Chiral extrapolation:  Mπ,min < 200 MeV, with three or more pion masses used in the extrapolation or two values of Mπ with one lying within 10 MeV of 135 MeV (the physical neutral pion mass) and the other one below 200 MeV ◦ 200 MeV ≤ Mπ,min ≤ 400 MeV, with three or more pion masses used in the extrapolation or two values of Mπ with Mπ,min < 200 MeV or a single value of Mπ , lying within 10 MeV of 135 MeV (the physical neutral pion mass)  otherwise 4 We refer to Sect. 5.1 for an explanation of the various regimes of chiral perturbation theory. 123 Eur. Phys. J. C (2022) 82:869 Page 15 of 296 869 This criterion is unchanged from FLAG 19. In Sect. 10 the upper end of the range for Mπ,min in the green circle criterion is lowered to 300 MeV, as in FLAG 19. • Continuum extrapolation:  at least three lattice spacings and at least two points below 0.1 fm and a range of lattice spacings satisfying [a 2 ◦ max /amin] ≥ 2 at least two lattice spacings and at least one point below 0.1 fm and a range of lattice spacings satisfying [a 2max/amin] ≥ 1.4  otherwise It is assumed that the lattice action is O(a)-improved (i.e., the discretization errors vanish quadratically with the lattice spacing); otherwise this will be explicitly mentioned. For unimproved actions an additional lattice spacing is required. This condition is unchanged from FLAG 19. • Finite-volume effects: The finite-volume colour code used for a result is chosen to be the worse of the QCD and the QED codes, as described below. If only QCD is used the QED colour code is ignored. – For QCD:  [Mπ,min/Mπ,fid]2 exp{4 − Mπ,min[L(Mπ,min)]max} < 1, or at least three volumes◦ [Mπ,min/Mπ,fid]2 exp{3 − Mπ,min[L(Mπ,min)]max} < 1, or at least two volumes  otherwise where we have introduced [L(Mπ,min)]max, which is the maximum box size used in the simulations performed at the smallest pion mass Mπ,min, as well as a fiducial pion mass Mπ,fid, which we set to 200 MeV (the cutoff value for a green star in the chiral extrapolation). It is assumed here that calculations are in the p-regime of chiral perturbation theory, and that all volumes used exceed 2 fm. The rationale for this condition is as follows. Finite volume effects contain the universal factor exp{−L Mπ }, and if this were the only contribution a criterion based on the values of Mπ,minL would be appropriate. However, as pion masses decrease, one must also account for the weakening of the pion couplings. In particular, 1-loop chiral perturbation theory [146] reveals a behaviour proportional to M2π exp{−L Mπ }. Our condition includes this weakening of the coupling, and ensures, for example, that simulations with Mπ,min = 135 MeV and L Mπ,min = 3.2 are rated equivalently to those with Mπ,min = 200 MeV and L Mπ,min = 4. – For QED (where applicable):  1/([M n ◦ π,min L(Mπ,min)]max) min < 0.02, or at least four volumes 1/([M min L(M nmin)]max) minπ, π, < 0.04, or at least three volumes  otherwise Because of the infrared-singular structure of QED, electromagnetic finite-volume effects decay only like a power of the inverse spatial extent. In several cases like mass splittings [147,148] or leptonic decays [149], the leading corrections are known to be universal, i.e., independent of the structure of the involved hadrons. In such cases, the leading universal effects can be directly subtracted exactly from the lattice data. We denote n 1min the smallest power of L at which such a subtraction cannot be done. In the widely used finite-volume formulation QEDL , one always has nmin ≤ 3 due to the nonlocality of the theory [150]. The QED criteria are used here only in Sect. 3. Both QCD and QED criteria are unchanged from FLAG 19. • Isospin breaking effects (where applicable):  all leading isospin breaking effects are included in the lattice calculation ◦ isospin breaking effects are included using the electro-quenched approximation  otherwise This criterion is used for quantities which are breaking isospin symmetry or which can be determined at the sub-percent accuracy where isospin breaking effects, if not included, are expected to be the dominant source of uncertainty. In the current edition, this criterion is only used for the up- and down-quark masses, and related quantities (, Q2 and R2). The criteria for isospin breaking effects are unchanged from FLAG 19. 123 869 Page 16 of 296 Eur. Phys. J. C (2022) 82:869 • Renormalization (where applicable):  nonperturbative ◦ 1-loop perturbation theory or higher with a reasonable estimate of truncation errors  otherwise In Ref. [1], we assigned a red square to all results which were renormalized at 1-loop in perturbation theory. In FLAG 13 [2], we decided that this was too restrictive, since the error arising from renormalization constants, calculated in perturbation theory at 1-loop, is often estimated conservatively and reliably. These criteria have remained unchanged since then. • Renormalization Group (RG) running (where applicable): For scale-dependent quantities, such as quark masses or BK , it is essential that contact with continuum perturbation theory can be established. Various different methods are used for this purpose (cf. Appendix A.3 in FLAG 19 [4]): Regularization-independent Momentum Subtraction (RI/MOM), the Schrödinger functional, and direct comparison with (resummed) perturbation theory. Irrespective of the particular method used, the uncertainty associated with the choice of intermediate renormalization scales in the construction of physical observables must be brought under control. This is best achieved by performing comparisons between nonperturbative and perturbative running over a reasonably broad range of scales. These comparisons were initially only made in the Schrödinger functional approach, but are now also being performed in RI/MOM schemes. We mark the data for which information about nonperturbative running checks is available and give some details, but do not attempt to translate this into a colour code. The pion mass plays an important role in the criteria relevant for chiral extrapolation and finite volume. For some of the regularizations used, however, it is not a trivial matter to identify this mass. In the case of twisted-mass fermions, discretization effects give rise to a mass difference between charged and neutral pions even when the up- and down-quark masses are equal: the charged pion is found to be the heavier of the two for twisted-mass Wilson fermions (cf. Ref. [151]). In early works, typically referring to N f = 2 simulations (e.g., Refs. [151] and [87]), chiral extrapolations are based on chiral perturbation theory formulae which do not take these regularization effects into account. After the importance of accounting for isospin breaking when doing chiral fits was shown in Ref. [152], later works, typically referring to N f = 2 + 1 + 1 simulations, have taken these effects into account [7]. We use Mπ± for Mπ,min in the chiral-extrapolation rating criterion. On the other hand, we identify Mπ,min with the root mean square (RMS) of Mπ+ , Mπ− and Mπ0 in the finite-volume rating criterion. In the case of staggered fermions, discretization effects give rise to several light states with the quantum numbers of the pion.5 The mass splitting among these “taste” partners represents a discretization effect of O(a2), which can be significant at large lattice spacings but shrinks as the spacing is reduced. In the discussion of the results obtained with staggered quarks given in the following sections, we assume that these artifacts are under control. We conservatively identify Mπ,min with the root mean square (RMS) average of the masses of all the taste partners, both for chiral-extrapolation and finite-volume criteria. In some of the simulations, the fermion formulations employed for the valence quarks are different from those used for the sea quarks. Even when the fermion formulations are the same, there are cases where the sea and valence quark masses differ. In such cases, we use the smaller of the valence-valence and valence-sea Mπmin values in the finite-volume criteria, since either of these channels may give the leading contribution depending on the quantity of interest at the one-loop level of chiral perturbation theory. For the chiral-extrapolation criteria, on the other hand, we use the unitary point, where the sea and valence quark masses are the same, to define Mπmin . The strong coupling αs is computed in lattice QCD with methods differing substantially from those used in the calculations of the other quantities discussed in this review. Therefore, we have established separate criteria for αs results, which will be discussed in Sect. 9.2.1. In the section on nuclear matrix elements, Sect. 10, an additional criterion is used. This concerns the level of control over contamination from excited states, which is a more challenging issue for nucleons than for mesons. In response to an improved understanding of the impact of this contamination, the excited-state contamination criterion has been made more stringent compared to that in FLAG 19. 5 We refer the interested reader to a number of reviews on the subject [153–157]. 123 Eur. Phys. J. C (2022) 82:869 Page 17 of 296 869 2.1.2 Heavy-quark actions For the b quark, the discretization of the heavy-quark action follows a very different approach from that used for light flavours. There are several different methods for treating heavy quarks on the lattice, each with its own issues and considerations. Most of these methods use Effective Field Theory (EFT) at some point in the computation, either via direct simulation of the EFT, or by using EFT as a tool to estimate the size of cutoff errors, or by using EFT to extrapolate from the simulated lattice quark masses up to the physical b-quark mass. Because of the use of an EFT, truncation errors must be considered together with discretization errors. The charm quark lies at an intermediate point between the heavy and light quarks. In our earlier reviews, the calculations involving charm quarks often treated it using one of the approaches adopted for the b quark. Since FLAG 16 [3], however, most calculations simulate the charm quark using light-quark actions. This has become possible thanks to the increasing availability of dynamical gauge field ensembles with fine lattice spacings. But clearly, when charm quarks are treated relativistically, discretization errors are more severe than those of the corresponding light-quark quantities. In order to address these complications, the heavy-quark section adds an additional, bipartite, treatment category to the rating system. The purpose of this criterion is to provide a guideline for the level of action and operator improvement needed in each approach to make reliable calculations possible, in principle. A description of the different approaches to treating heavy quarks on the lattice can be found in Appendix A.1.3 of FLAG 19 [4]. For truncation errors we use HQET power counting throughout, since this review is focused on heavy-quark quantities involving B and D mesons rather than bottomonium or charmonium quantities. Here we describe the criteria for how each approach must be implemented in order to receive an acceptable rating () for both the heavy-quark actions and the weak operators. Heavy-quark implementations without the level of improvement described below are rated not acceptable (  ). The matching is evaluated together with renormalization, using the renormalization criteria described in Sect. 2.1.1. We emphasize that the heavy-quark implementations rated as acceptable and described below have been validated in a variety of ways, such as via phenomenological agreement with experimental measurements, consistency between independent lattice calculations, and numerical studies of truncation errors. These tests are summarized in Sect. 8. Relativistic heavy-quark actions  at least tree-level O(a) improved action and weak operators This is similar to the requirements for light-quark actions. All current implementations of relativistic heavy-quark actions satisfy this criterion. NRQCD  tree-level matched through O(1/mh) and improved through O(a2) The current implementations of NRQCD satisfy this criterion, and also include tree-level corrections of O(1/m2h) in the action. HQET  tree-level matched through O(1/mh) with discretization errors starting at O(a2) The current implementation of HQET by the ALPHA collaboration satisfies this criterion, since both action and weak operators are matched nonperturbatively through O(1/mh). Calculations that exclusively use a static-limit action do not satisfy this criterion, since the static-limit action, by definition, does not include 1/mh terms. We therefore include static computations in our final estimates only if truncation errors (in 1/mh) are discussed and included in the systematic uncertainties. Light-quark actions for heavy quarks  discretization errors starting at O(a2) or higher This applies to calculations that use the twisted-mass Wilson action, a nonperturbatively improved Wilson action, domain wall fermions or the HISQ action for charm-quark quantities. It also applies to calculations that use these light quark actions in the charm region and above together with either the static limit or with an HQET-inspired extrapolation to obtain results at the physical b-quark mass. In these cases, the continuum-extrapolation criteria described earlier must be applied to the entire range of heavy-quark masses used in the calculation. 2.1.3 Conventions for the figures For a coherent assessment of the present situation, the quality of the data plays a key role, but the colour coding cannot be carried over to the figures. On the other hand, simply showing all data on equal footing might give the misleading impression 123 869 Page 18 of 296 Eur. Phys. J. C (2022) 82:869 that the overall consistency of the information available on the lattice is questionable. Therefore, in the figures we indicate the quality of the data in a rudimentary way, using the following symbols:  corresponds to results included in the average or estimate (i.e., results that contribute to the black square below);  corresponds to results that are not included in the average but pass all quality criteria;  corresponds to all other results;  corresponds to FLAG averages or estimates; they are also highlighted by a gray vertical band. The reason for not including a given result in the average is not always the same: the result may fail one of the quality criteria; the paper may be unpublished; it may be superseded by newer results; or it may not offer a complete error budget. Symbols other than squares are used to distinguish results with specific properties and are always explained in the caption.6 Often, nonlattice data are also shown in the figures for comparison. For these we use the following symbols: • corresponds to nonlattice results;  corresponds to Particle Data Group (PDG) results. 2.2 Averages and estimates FLAG results of a given quantity are denoted either as averages or as estimates. Here we clarify this distinction. To start with, both averages and estimates are based on results without any red tags in their colour coding. For many observables there are enough independent lattice calculations of good quality, with all sources of error (not merely those related to the colour-coded criteria), as analyzed in the original papers, appearing to be under control. In such cases, it makes sense to average these results and propose such an average as the best current lattice number. The averaging procedure applied to this data and the way the error is obtained is explained in detail in Sect. 2.3. In those cases where only a sole result passes our rating criteria (colour coding), we refer to it as our FLAG average, provided it also displays adequate control of all other sources of systematic uncertainty. On the other hand, there are some cases in which this procedure leads to a result that, in our opinion, does not cover all uncertainties. Systematic errors are by their nature often subjective and difficult to estimate, and may thus end up being underestimated in one or more results that receive green symbols for all explicitly tabulated criteria. Adopting a conservative policy, in these cases we opt for an estimate (or a range), which we consider as a fair assessment of the knowledge acquired on the lattice at present. This estimate is not obtained with a prescribed mathematical procedure, but reflects what we consider the best possible analysis of the available information. The hope is that this will encourage more detailed investigations by the lattice community. There are two other important criteria that also play a role in this respect, but that cannot be colour coded, because a systematic improvement is not possible. These are: (i) the publication status, and (ii) the number of sea-quark flavours N f . As far as the former criterion is concerned, we adopt the following policy: we average only results that have been published in peer-reviewed journals, i.e., they have been endorsed by referee(s). The only exception to this rule consists in straightforward updates of previously published results, typically presented in conference proceedings. Such updates, which supersede the corresponding results in the published papers, are included in the averages. Note that updates of earlier results rely, at least partially, on the same gauge-field-configuration ensembles. For this reason, we do not average updates with earlier results. Nevertheless, all results are listed in the tables,7 and their publication status is identified by the following symbols: • Publication status: A published or plain update of published results P preprint C conference contribution 6 For example, for quark-mass results we distinguish between perturbative and nonperturbative renormalization, for low-energy constants we distinguish between the p- and -regimes, and for heavy-flavour results we distinguish between those from leptonic and semi-leptonic decays. 7 Whenever figures turn out to be overcrowded, older, superseded results are omitted. However, all the most recent results from each collaboration are displayed. 123 Eur. Phys. J. C (2022) 82:869 Page 19 of 296 869 In the present edition, the publication status on the 30th of April 2021 is relevant. If the paper appeared in print after that date, this is accounted for in the bibliography, but does not affect the averages.8 As noted above, in this review we present results from simulations with N f = 2, N f = 2+ 1 and N f = 2+ 1+ 1 (except for r0MS where we also give the N f = 0 result). We are not aware of an a priori way to quantitatively estimate the difference between results produced in simulations with a different number of dynamical quarks. We therefore average results at fixed N f separately; averages of calculations with different N f are not provided. To date, no significant differences between results with different values of N f have been observed in the quantities listed in Tables 1, 2, 3, 4, and 5. In particular, differences between results from simulations with N f = 2 and N f = 2+1 would reflect Zweig-rule violations related to strange-quark loops. Although not of direct phenomenological relevance, the size of such violations is an interesting theoretical issue per se, and one that can be quantitatively addressed only with lattice calculations. It remains to be seen whether the status presented here will change in the future, since this will require dedicated N f = 2 and N f = 2 + 1 calculations, which are not a priority of present lattice work. The question of differences between results with N f = 2+ 1 and N f = 2 + 1+ 1 is more subtle. The dominant effect of including the charm sea quark is to shift the lattice scale, an effect that is accounted for by fixing this scale nonperturbatively using physical quantities. For most of the quantities discussed in this review, it is expected that residual effects are small in the continuum limit, suppressed by αs(mc) and powers of 2/m2c . Here  is a hadronic scale that can only be roughly estimated and depends on the process under consideration. Note that the 2/m2c effects have been addressed in Refs. [158–162], and found to be small for the quantities considered. Assuming that such effects are generically small, it might be reasonable to average the results from N f = 2 + 1 and N f = 2 + 1 + 1 simulations, although we do not do so here. 2.3 Averaging procedure and error analysis In the present report, we repeatedly average results obtained by different collaborations, and estimate the error on the resulting averages. Here we provide details on how averages are obtained. 2.3.1 Averaging – generic case We follow the procedure of the previous two editions [2,3], which we describe here in full detail. One of the problems arising when forming averages is that not all of the data sets are independent. In particular, the same gauge-field configurations, produced with a given fermion discretization, are often used by different research teams with different valence-quark lattice actions, obtaining results that are not really independent. Our averaging procedure takes such correlations into account. Consider a given measurable quantity Q, measured by M distinct, not necessarily uncorrelated, numerical experiments (simulations). The result of each of these measurement is expressed as Qi = x ± (1) (2)i σi ± σi ± · · · ± (E)σi , (1) where xi is the value obtained by the i th experiment (i = 1 (α), . . . , M) and σi (for α = 1, . . . , E) are the various errors. Typically (1)σi stands for the statistical error and (α) σi (α ≥ 2) are the different systematic errors from various sources. For each individual result, we estimate the total error σi by adding statistical and systematic errors in quadrature: Qi = √xi ± σi , √√∑E [ ] σi ≡ √ 2 (α) σi . (2) α=1 With the weight factor of each total error estimated in standard fashion, σ−2 ω = ∑ ii − , (3)M 2 i=1 σi 8 As noted above in footnote 1, one exception to this deadline was made, Ref. [61]. 123 869 Page 20 of 296 Eur. Phys. J. C (2022) 82:869 the central value of the average over all simulations is given by ∑M xav = xi ωi . (4) i=1 The above central value corresponds to a χ2min weighted average, evaluated by adding statistical and systematic errors in quad√rature. If the fit is not of good quality (χ2min/dof > 1), the statistical and systematic error bars are stretched by a factor S = χ2/dof. Next, we examine error budgets for individual calculations and look for potentially correlated uncertainties. Specific problems encountered in connection with correlations between different data sets are described in the text that accompanies the averaging. If there is reason to believe that a source of error is correlated between two calculations, a 100% correla- tion is assumed. The correlation matrix Ci j for the set of correlated lattice results is estimated by a prescription due to Schmelling [163]. This consists in defining √∑′[ ]= 2; (α)σi j σi , (5) α ∑ with ′α running only over those errors of xi that are correlated with the corresponding errors of the measurement x j . This expresses the part of the uncertainty in xi that is correlated with the uncertainty in x j . If no such correlations are known to exist, then we take σi; j = 0. The diagonal and off-diagonal elements of the correlation matrix are then taken to be Cii = σ 2i (i = 1, . . . , M), Ci j = σi; j σ j;i (i = j). (6) Finally, the error of the average is estimated by ∑M ∑M σ 2av = ωi ω j Ci j , (7) i=1 j=1 and the FLAG average is Qav = xav ± σav. (8) 2.3.2 Nested averaging We have encountered one case where the correlations between results are more involved, and a nested averaging scheme is required. This concerns the B-meson bag parameters discussed in Sect. 8.2. In the following, we describe the details of the nested averaging scheme. This is an updated version of the section added in the web update of the FLAG 16 report. The issue arises for a quantity Q that is given by a ratio, Q = Y/Z . In most simulations, both Y and Z are calculated, and the error in Q can be obtained in each simulation in the standard way. However, in other simulations only Y is calculated, with Z taken from a global average of some type. The issue to be addressed is that this average value Z has errors that are correlated with those in Q. In the example that arises in Sect. 8.2, Q = BB , Y = B f 2B B and Z = f 2B . In one of the simulations that contribute to the average, Z is replaced by Z , the PDG average for f 2B [164] (obtained with an averaging procedure similar to that used by FLAG). This simulation is labeled with i = 1, so that = Y1Q1 . (9) Z The other simulations have results labeled Q j , with j ≥ 2. In this set up, the issue is that Z is correlated with the Q j , j ≥ 2.9 9 There is also a small correlation between Y1 and Z , but we follow the original Ref. [73] and do not take this into account. Thus, the error in Q1 is obtained by simple error propagation from those in Y1 and Z . Ignoring this correlation is conservative, because, as in the calculation of BK , the correlations between B f 2B B and f 2 B tend to lead to a cancellation of errors. By ignoring this effect we are making a small overestimate of the error in Q1. 123 Eur. Phys. J. C (2022) 82:869 Page 21 of 296 869 We begin by decomposing the error in Q1 in the same schematic form as above, (1) (2) (E) σ Q Y11 = x1 ± ± σY1 ± · · · ± σY1 ± Y1σZ2 . (10)Z Z Z Z Here the last term represents the error propagating from that in Z , while the others arise from errors in Y1. For the remaining Q j ( j ≥ 2) the decomposition is as in Eq. (1). The total error of Q1 then reads ( (1 ) ( ) ( )) 2 (2) 2 (E) 2 ( ) σ 2 2 Y σ σ 1 Y1 1 = + + · · · + σY1 + Y12 σ 2, (11)Z Z Z ZZ while that for the Q j ( j ≥ 2) is ( )2 ( )2 ( )2 2 = (1) (2) (E)σ j σ j + σ j + · · · + σ j . (12) Correlations between Q j and Qk ( j, k ≥ 2) are taken care of by Schmelling’s prescription, as explained above. What is new here is how the correlations between Q1 and Q j ( j ≥ 2) are taken into account. To proceed, we recall from Eq. (7) that σZ is given by ∑M ′ σ 2 = ω[Z ]i ′ω[Z ] j ′C[Z ]i ′ j ′ . (13)Z i ′, j ′=1 Here the indices i ′ and j ′ run over the M ′ simulations that contribute to Z , which, in general, are different from those contributing to the results for Q. The weightsω[Z ] and correlation matrix C[Z ] are given an explicit argument Z to emphasize that they refer to the calculation of this quantity and not to that of Q. C[Z ] is calculated using the Schmelling prescription [Eqs. (5)–(7)] in terms of the errors, [Z ](α)σ i ′ , taking into account the correlations between the different calculations of Z . We now generalize Schmelling’s prescription for σi; j , Eq. (5), to that for σ1;k (k ≥ 2), i.e., the part of the error in Q1 that is correlated with Qk . We take √ √√ √ ∑′1 [ ]2 Y 2 ∑M ′(α) σ 11;k = √ 2 σY + ω[Z ]i ′4 ω[Z ] j ′C[Z ]i ′ j ′↔k . (14)1 Z (α)↔k Z i ′, j ′ The first term under the square root sums those sources of error in Y1 that are correlated with Qk . Here we are using a more explicit notation from that in Eq. (5), with (α) ↔ k indicating that the sum is restricted to the values of α for which the error (α) σY is correlated with Qk . The second term accounts for the correlations within Z with Qk , and is the nested part of the1 present scheme. The new matrix C[Z ]i ′ j ′↔k is a restriction of the full correlation matrix C[Z ], and is defined as follows. Its diagonal elements are given by C[Z ]i ′i ′↔k = (σ [Z ]i ′↔k)2 (i ′ = 1, . . . , M ′), (15) ∑′ (σ [Z ] ′↔ )2 = [Z ](α)i k (σ i ′ )2, (16) (α)↔k ∑ where the summation ′(α)↔k over (α) is restricted to those σ [Z ](α)i ′ that are correlated with Qk . The off-diagonal elements are C[Z ]i ′ j ′↔k = σ√[Z ]i ′; j ′↔k σ [Z ] ′ j ′;i ′↔k (i = j ′), (17) √ ∑′ σ [Z ] √′; ′↔ = √ (σ [Z ](α)′ )2i j k i , (18) (α)↔ j ′k ∑ where the summation ′(α)↔ j ′k over (α) is restricted to [Z ](α)σ i ′ that are correlated with both Z j ′ and Qk . 123 869 Page 22 of 296 Eur. Phys. J. C (2022) 82:869 The last quantity that we need to define is σk;1. √√ ′ √√ ∑ [ ]= 2(α)σk;1 σk , (19) (α)↔1 ∑ where the summation ′(α)↔1 is restricted to those (α) σk that are correlated with one of the terms in Eq. (11). In summary, we construct the correlation matrix Ci j using Eq. (6), as in the generic case, except the expressions for σ1;k and σk;1 are now given by Eqs. (14) and (19), respectively. All other σi; j are given by the original Schmelling pre- scription, Eq. (5). In this way we extend the philosophy of Schmelling’s approach while accounting for the more involved correlations. 3 Quark masses Authors: T. Blum, A. Portelli, A. Ramos Quark masses are fundamental parameters of the Standard Model. An accurate determination of these parameters is important for both phenomenological and theoretical applications. The bottom- and charm-quark masses, for instance, are important sources of parametric uncertainties in several Higgs decay modes. The up-, down- and strange-quark masses govern the amount of explicit chiral symmetry breaking in QCD. From a theoretical point of view, the values of quark masses provide information about the flavour structure of physics beyond the Standard Model. The Review of Particle Physics of the Particle Data Group contains a review of quark masses [165], which covers light as well as heavy flavours. Here we also consider light- and heavy-quark masses, but focus on lattice results and discuss them in more detail. We do not discuss the top quark, however, because it decays weakly before it can hadronize, and the nonperturbative QCD dynamics described by present day lattice simulations is not relevant. The lattice determination of light- (up, down, strange), charm- and bottom-quark masses is considered below in Sects. 3.1, 3.2, and 3.3, respectively. Quark masses cannot be measured directly in experiment because quarks cannot be isolated, as they are confined inside hadrons. From a theoretical point of view, in QCD with N f flavours, a precise definition of quark masses requires one to choose a particular renormalization scheme. This renormalization procedure introduces a renormalization scale μ, and quark masses depend on this renormalization scale according to the Renormalization Group (RG) equations. In mass-independent renormalization schemes the RG equations read dm̄i (μ) μ = m̄i (μ)τ(ḡ), (20) dμ where the function τ(ḡ) is the anomalous dimension, which depends only on the value of the strong coupling αs = ḡ2/(4π). Note that in QCD τ(ḡ) is the same for all quark flavours. The anomalous dimension is scheme dependent, but its perturbative expansion → ( )ḡ 0 τ(ḡ) ∼ − ḡ2 d + d ḡ20 1 + · · · (21) has a leading coefficient d0 = 8/(4π)2, which is scheme independent.10 Equation (20), being a first order differential equation, can be solved exactly by using Eq. (21) as the boundary condition. The formal solution of the RG equation reads { ∫ }ḡ(μ) [ ] = τ(x) d0Mi m̄i (μ)[2b0 ḡ2(μ)]−d0/(2b0) exp − dx − , (22) 0 β(x) b0x where b0 = (11 − 2N f /3)/(4π)2 is the universal leading perturbative coefficient in the expansion of the β-function, β(ḡ) ≡ dḡ2/d logμ2, which governs the running of the strong coupling constant near the scale μ. The renormalization group invariant (RGI) quark masses Mi are formally integration constants of the RG Eq. (20). They are scale independent, and due to the universality of the coefficient d0, they are also scheme independent. Moreover, they are nonperturbatively defined 10 We follow the conventions of Gasser and Leutwyler [166]. 123 Eur. Phys. J. C (2022) 82:869 Page 23 of 296 869 by Eq. (22). They only depend on the number of flavours N f , making them a natural candidate to quote quark masses and compare determinations from different lattice collaborations. Nevertheless, it is customary in the phenomenology community to use the MS scheme at a scale μ = 2 GeV to compare different results for light-quark masses, and use a scale equal to its own mass for the charm and bottom quarks. In this review, we will quote the final averages of both quantities. Results for quark masses are always quoted in the four-flavour theory. N f = 2+ 1 results have to be converted to the four- flavour theory. Fortunately, the charm quark is heavy (QCD/m )2c < 1, and this conversion can be performed in perturbation theory with negligible (∼ 0.2%) perturbative uncertainties. Nonperturbative corrections in this matching are more difficult to estimate. Since these effects are suppressed by a factor of 1/Nc, and a factor of the strong coupling at the scale of the charm mass, naive power counting arguments would suggest that the effects are ∼ 1%. In practice, numerical nonperturbative studies [158,160,167] have found this power counting argument to be an overestimate by one order of magnitude in the determination of simple hadronic quantities or the -parameter. Moreover, lattice determinations do not show any significant deviation between the N f = 2 + 1 and N f = 2 + 1 + 1 simulations. For example, the difference in the final averages for the mass of the strange quark ms between N f = 2 + 1 and N f = 2 + 1+ 1 determinations is about 1.3%, or about one standard deviation. We quote all final averages at 2 GeV in the MS scheme and also the RGI values (in the four-flavour theory). We use the exact RG Eq. (22). Note that to use this equation we need the value of the strong coupling in the MS scheme at a scale μ = 2 GeV. All our results are obtained from the RG equation in the MS scheme and the 5-loop beta function together with the value of the -parameter in the four-flavour theory (4) = 294(12)MeV obtained in this review (see Sect. 9). In the MS uncertainties of the RGI masses we separate the contributions from the determination of the quark masses and the propagation of the uncertainty of (4) . These are identified with the subscripts m and , respectively. MS Conceptually, all lattice determinations of quark masses contain three basic ingredients: 1. Tuning the lattice bare-quark masses to match the experimental values of some quantities. Pseudo-scalar meson masses provide the most common choice, since they have a strong dependence on the values of quark masses. In pure QCD with N f quark flavours these values are not known, since the electromagnetic interactions affect the experimental values of meson masses. Therefore, pure QCD determinations use model/lattice information to determine the location of the physical point. This is discussed at length in Sect. 3.1.1. 2. Renormalization of the bare-quark masses. Bare-quark masses determined with the above-mentioned criteria have to be renormalized. Many of the latest determinations use some nonperturbatively defined scheme. One can also use perturbation theory to connect directly the values of the bare-quark masses to the values in the MS scheme at 2 GeV. Experience shows that 1-loop calculations are unreliable for the renormalization of quark masses: usually at least two loops are required to have trustworthy results. 3. If quark masses have been nonperturbatively renormalized, for example, to some MOM/SF scheme, the values in this scheme must be converted to the phenomenologically useful values in the MS scheme (or to the scheme/scale independent RGI masses). Either option requires the use of perturbation theory. The larger the energy scale of this matching with perturbation theory, the better, and many recent computations in MOM schemes do a nonperturbative running up to 3– 4 GeV. Computations in the SF scheme allow us to perform this running nonperturbatively over large energy scales and match with perturbation theory directly at the electro-weak scale ∼ 100 GeV. Note that many lattice determinations of quark masses make use of perturbation theory at a scale of a few GeV. We mention that lattice-QCD calculations of the b-quark mass have an additional complication which is not present in the case of the charm and light quarks. At the lattice spacings currently used in numerical simulations the direct treatment of the b quark with the fermionic actions commonly used for light quarks is very challenging. Only two determinations of the b-quark mass use this approach, reaching the physical b-quark mass region at two lattice spacings with aM ∼ 1. There are a few widely used approaches to treat the b quark on the lattice, which have been already discussed in the FLAG 13 review (see Sec. 8 of Ref. [2]). Those relevant for the determination of the b-quark mass will be briefly described in Sect. 3.3. 3.1 Masses of the light quarks Light-quark masses are particularly difficult to determine because they are very small (for the up and down quarks) or small (for the strange quark) compared to typical hadronic scales. Thus, their impact on typical hadronic observables is minute, and it is difficult to isolate their contribution accurately. 123 869 Page 24 of 296 Eur. Phys. J. C (2022) 82:869 Fortunately, the spontaneous breaking of SU (3)L × SU (3)R chiral symmetry provides observables which are particularly sensitive to the light-quark masses: the masses of the resulting Nambu–Goldstone bosons (NGB), i.e., pions, kaons, and eta. Indeed, the Gell–Mann–Oakes–Renner relation [168] predicts that the squared mass of a NGB is directly proportional to the sum of the masses of the quark and antiquark which compose it, up to higher-order mass corrections. Moreover, because these NGBs are light, and are composed of only two valence particles, their masses have a particularly clean statistical signal in lattice-QCD calculations. In addition, the experimental uncertainties on these meson masses are negligible. Thus, in lattice calculations, light-quark masses are typically obtained by renormalizing the input quark mass and tuning them to reproduce NGB masses, as described above. 3.1.1 The physical point and isospin symmetry As mentioned in Sect. 2.1, the present review relies on the hypothesis that, at low energies, the LagrangianLQCD+LQED describes nature to a high degree of precision. However, most of the results presented below are obtained in pure QCD calculations, which do not include QED. Quite generally, when comparing QCD calculations with experiment, radiative corrections need to be applied. In pure QCD simulations, where the parameters are fixed in terms of the masses of some of the hadrons, the electromagnetic contributions to these masses must be discussed. How the matching is done is generally ambiguous because it relies on the unphysical separation of QCD and QED contributions. In this section, and in the following, we discuss this issue in detail. A related discussion, in the context of scale setting, is given in Sect. 11.3. Of course, once QED is included in lattice calculations, the subtraction of electromagnetic contributions is no longer necessary. Let us start from the unambiguous case of QCD+QED. As explained in the introduction of this section, the physical quark masses are the parameters of the Lagrangian such that a given set of experimentally measured, dimensionful hadronic quantities are reproduced by the theory. Many choices are possible for these quantities, but in practice many lattice groups use pseudoscalar meson masses, as they are easily and precisely obtained both by experiment, and through lattice simulations. For example, in the four-flavour case, one can solve the system Mπ+(mu,m m exp. d , s,mc, α) = Mπ+ , (23) M + m exp.K ( u,md ,ms,mc, α) = MK+ , (24) M exp.K 0(mu,md ,ms,mc, α) = MK 0 , (25) M exp.D0(mu,md ,ms,mc, α) = M , (26)D0 where we assumed that • all the equations are in the continuum and infinite-volume limits; • the overall scale has been set to its physical value, generally through some lattice-scale setting procedure involving a fifth dimensionful input (see the discussion in Sect. 11.3); • the quark masses mq are assumed to be renormalized from the bare, lattice ones in some given continuum renormalization scheme; • α = e24 is the fine-structure constant expressed as function of the positron charge e, generally set to the Thomson limitπ α = 0.007297352 . . . [165]; • the mass Mh(mu,md ,ms,mc, α) of the meson h is a function of the quark masses and α. The functional dependence is generally obtained by choosing an appropriate parameterization and performing a global fit to the lattice data; • the superscript exp. indicates that the mass is an experimental input, lattice groups use in general the values in the Particle Data Group review [165]. However, ambiguities arise with simulations of QCD only. In that case, there is no experimentally measurable quantity that emerges from the strong interaction only. The missing QED contribution is tightly related to isospin-symmetry breaking effects. Isospin symmetry is explicitly broken by the differences between the up- and down-quark masses δm = mu − md , and electric charges δQ = Qu − Qd . These effects are, respectively, of order O(δm/QCD) and O(α), and are expected to be O(1%) of a typical isospin-symmetric hadronic quantity. Strong and electromagnetic isospin-breaking effects are of the same order and therefore cannot, in principle, be evaluated separately without introducing strong ambiguities. Because these effects are small, they can be treated as a perturbation, 123 Eur. Phys. J. C (2022) 82:869 Page 25 of 296 869 X (mu,md ,ms,mc, α) = X̄(mud ,ms,mc)+ δm AX (mud ,ms,mc)+ αBX (mud ,ms,mc) , (27) for a given hadronic quantity X , where m 1ud = 2 (mu +md) is the average light-quark mass. There are several things to notice here. Firstly, the neglected higher-orderO(δm2, αδm, α2) corrections are expected to beO(10−4) relatively to X , which at the moment is way beyond the relative statistical accuracy that can be delivered by a lattice calculation. Secondly, this is not strictly speaking an expansion around the isospin-symmetric point, the electromagnetic interaction has also symmetric contributions. From this last expression the previous statements about ambiguities become clearer. Indeed, the only unambiguous prediction one can perform is to solve Eqs. (23)–(26) and use the resulting parameters to obtain a prediction for X , which is represented by the left-hand side of Eq. (27). This prediction will be the sum of the QCD isospin-symmetric part X̄ , the strong isospin- breaking effects X SU (2) = δm AX , and the electromagnetic effects Xγ = αBX . Obtaining any of these terms individually requires extra, unphysical conditions to perform the separation. To be consistent with previous editions of FLAG, we also define X̂ = X̄ + X SU (2) to be the α → 0 limit of X . With pure QCD simulations, one typically solves Eqs. (23)–(26) by equating the QCD isospin-symmetric part of a hadron mass M̄h , result of the simulations, with its experimental value M exp. h . This will result in an O(δm, α) mis-tuning of the theory parameters which will propagate as an error on predicted quantities. Because of this, in general, one cannot predict hadronic quantities with a relative accuracy higher than O(1%) from pure QCD simulations, independently on how the target X is sensitive to isospin-breaking effects. If one performs a complete lattice prediction of the physical value of X , it can be of phenomenological interest to define in some way X̄ , X SU (2), and Xγ . If we keep mud , ms and mc at their physical values in physical units, for a given renormalization scheme and scale, then these three quantities can be extracted by setting successively and simultaneously α and δm to 0. This is where the ambiguity lies: in general the δm = 0 point will depend on the renormalization scheme used for the quark masses. In the next section, we give more details on that particular aspect and discuss the order of scheme ambiguities. 3.1.2 Ambiguities in the separation of isospin-breaking contributions In this section, we discuss the ambiguities that arise in the individual determination of the QED contribution Xγ and the strong- isospin correction X SU (2) defined in the previous section. Throughout this section, we assume that the isospin-symmetric quark masses mud , ms and mc are always kept fixed in physical units to the values they take at the QCD+QED physical point in some given renormalization scheme. Let us assume that both up and down masses have been renormalized in an identical mass-independent scheme which depends on some energy scaleμ. We also assume that the renormalization procedure respects chiral symmetry so that quark masses renormalize multiplicatively. The renormalization constants of the quark masses are identical for α = 0 and therefore the renormalized mass of a quark has the general form m 2 (0) (1) 2 (2)q(μ) = Zm(μ)[1 + αQtot.δZ (μ)+ αQtot.QqδZ (μ)+ αQqδZ (μ)]mq,0 , (28) up to O(α2) corrections, where m is the bare-quark mass, Q and Q2q,0 tot. tot. are the sum of all quark charges and squared charges, respectively, and Qq is the quark charge, all in units of in units of the positron charge e. Throughout this section, a subscript ud generally denotes the average between up and down quantities and δ the difference between the up and the down quantities. The source of the ambiguities described in the previous section is the mixing of the isospin-symmetric mass mud and the difference δm through renormalization. Using Eq. (28) one can make this mixing explicit at leading order in α: ( ) ( ) mud(μ) = Z (μ)[1 + αQ2 (0)δ (μ)+ αM (1)(μ)+ αM (2) ] mud,0(μ) (29) δm m(μ) tot. Z δm0 with the mixing matrices ( 1 ) ( 2 1 )2 M (1) = (1) QQ ud(μ) δ (μ) tot 4δQ Q δQZ . and M (2) = (2)(μ) δZ (μ) ud 4δQ Q δQ2 Q2 , (30)ud ud where Q = 1ud 2 (Qu + Qd) and δQ = Qu − Qd are the average and difference of the up and down charges, and similarly Q2 1 2 2ud = 2 (Qu + Qd) and δQ2 = Q2u − Q2d for the squared charges. Now let us assume that for the purpose of determining the 123 869 Page 26 of 296 Eur. Phys. J. C (2022) 82:869 different components in Eq. (27), one starts by tuning the bare masses to obtain equal up and down masses, for some small coupling α0 at some scale μ0, i.e., δm(μ0) = 0. At this specific point, one can extract the pure QCD, and the QED corrections to a given quantity X by studying the slope of α in Eq. (27). From these quantities the strong-isospin contribution can then readily be extracted using a nonzero value of δm(μ0). However, if now the procedure is repeated at another coupling α and scale μ with the same bare masses, it appears from Eq. (29) that δm(μ) = 0. More explicitly, Zm(μ) δm(μ) = mud(μ0) [αZ (μ)− α0Z (μ0)] , (31) Zm(μ0) with Z (μ) = Qtot.δQ (1)δZ (μ)+ Q2 (2)δ δZ (μ) , (32) up to higher-order corrections in α and α0. In other words, the definitions of X̄ , X SU (2), and Xγ depend on the renormalization scale at which the separation was made. This dependence, of course, has to cancel in the physical sum X . One can notice that at no point did we mention the renormalization of α itself, which, in principle, introduces similar ambiguities. However, the corrections coming from the running of α are O(α2) relatively to X , which, as justified above, can be safely neglected. Finally, important information is provided by Eq. (31): the scale ambiguities are O(αmud). For physical quark masses, one generally has mud δm. So by using this approximation in the first-order expansion Eq. (27), it is actually possible to define unambiguously the components of X up to second-order isospin-breaking corrections. Therefore, in the rest of this review, we will not keep track of the ambiguities in determining pure QCD or QED quantities. However, in the context of lattice simulations, it is crucial to notice that mud δm is only accurate at the physical point. In simulations at larger-than-physical pion masses, scheme ambiguities in the separation of QCD and QED contributions are generally large. Once more, the argument made here assumes that the isospin-symmetric quark masses mud , ms , and mc are kept fixed to their physical value in a given scheme while varying α. Outside of this assumption there is an additional isospin-symmetric O(αmq) ambiguity between X̄ and Xγ . Such separation in lattice QCD+QED simulation results appeared for the first time in RBC 07 [169] and Blum 10 [170], where the scheme was implicitly defined around the χPT expansion. In that setup, the δm(μ0) = 0 point is defined in pure QCD, i.e., α0 = 0 in the previous discussion. The QCD part of the kaon-mass splitting from the first FLAG review [1] is used as an input in RM123 11 [171], which focuses on QCD isospin corrections only. It therefore inherits from the convention that was chosen there, which is also to set δm(μ0) = 0 at zero QED coupling. The same convention was used in the follow-up works RM123 13 [172] and RM123 17 [19]. The BMW collaboration was the first to introduce a purely hadronic scheme in its electro-quenched study of the baryon octet mass splittings [173]. In this work, the quark mass difference δm(μ) is swapped with the mass splitting M2 between the connected ūu and d̄d pseudoscalar masses. Although unphysical, this quantity is proportional [174] to δm(μ) up to O(αmud) chiral corrections. In this scheme, the quark masses are assumed to be equal at M2 = 0, and the O(αmud) corrections to this statement are analogous to the scale ambiguities mentioned previously. The same scheme was used for the determination of light-quark masses in BMW 16 [20] and in the recent BMW prediction of the leading hadronic contribution to the muon magnetic moment [115]. The BMW collaboration used a different hadronic scheme for its determination of the nucleon-mass splitting in BMW 14 [147] using full QCD+QED simulations. In this work, the δm = 0 point was fixed by imposing the baryon splitting M + − M − to cancel. This scheme is quite different from the other ones presented here, in the sense that its intrinsic ambiguity is not O(αmud). What motivates this choice here is that M + − M − = 0 in the limit where these baryons are point particles, so the scheme ambiguity is suppressed by the compositeness of the baryons. This may sound like a more difficult ambiguity to quantify, but this scheme has the advantage of being defined purely by measurable quantities. Moreover, it has been demonstrated numerically in BMW 14 [147] that, within the uncertainties of this study, the M + − M − = 0 scheme is equivalent to the M2 = 0 one, explicitly M + − M − = −0.18(12)(6)MeV at M2 = 0. The calculation QCDSF/UKQCD 15 [175] uses a “Dashen scheme,” where quark masses are tuned such that flavour-diagonal mesons have equal masses in QCD and QCD+QED. Although not explicitly mentioned by the authors of the paper, this scheme is simply a reformulation of the M2 = 0 scheme mentioned previously. Finally, MILC 18 [21] also used the M2 = 0 scheme and noticed its connection to the “Dashen scheme” from QCDSF/UKQCD 15. Before the previous edition of this review, the contributions X̄ , X SU (2), and Xγ were given for pion and kaon masses based on phenomenological information. Considerable progress has been achieved by the lattice community to include isospin-breaking effects in calculations, and it is now possible to determine these quantities precisely directly from a lattice calculation. However, these quantities generally appear as intermediate products of a lattice analysis, and are rarely directly 123 Eur. Phys. J. C (2022) 82:869 Page 27 of 296 869 communicated in publications. These quantities, although unphysical, have a phenomenological interest, and we encourage the authors of future calculations to quote them explicitly. 3.1.3 Inclusion of electromagnetic effects in lattice-QCD simulations El∫ectrom∑agnetism on a lattice can be formulated using a naive discretization of the Maxwell action S[Aμ] =1 4 d 4x μ,ν[∂μAν(x) − ∂ν A 2μ(x)] . Even in its noncompact form, the action remains gauge invariant. This is not the case for non-Abelian theories for which one uses the traditional compact Wilson gauge action (or an improved version of it). Compact actions for QED feature spurious photon-photon interactions which vanish only in the continuum limit. This is one of the main reason why the noncompact action is the most popular so far. It was used in all the calculations presented in this review. Gauge-fixing is necessary for noncompact actions because of the usual infinite measure of equivalent gauge orbits which contribute to the path integral. It was shown [176,177] that gauge-fixing is not necessary with compact actions, including in the construction of interpolating operators for charged states. Although discretization is straightforward, simulating QED in a finite volume is more challenging. Indeed, the long range nature of the interaction suggests that important finite-size effects have to be expected. In the case of periodic boundary conditions, the situation is even more critical: a naive implementation of the theory features an isolated zero-mode singularity in the photon propagator. It was first proposed in [178] to fix the global zero-mode of the photon field Aμ(x) in order to remove it from the dynamics. This modified theory is generally named QEDTL. Although this procedure regularizes the theory and has the right classical infinite-volume limit, it is nonlocal because of the zero-mode fixing. As first discussed in [147], the nonlocality in time of QEDTL prevents the existence of a transfer matrix, and therefore a quantum-mechanical interpretation of the theory. Another prescription named QEDL, proposed in [179], is to remove the zero-mode of Aμ(x) independently for each time slice. This theory, although still nonlocal in space, is local in time and has a well-defined transfer matrix. Whether these nonlocalities constitute an issue to extract infinite-volume physics from lattice-QCD+QEDL simulations is, at the time of this review, still an open question. However, it is known through analytical calculations of electromagnetic finite- size effects at O(α) in hadron masses [147,148,150,172,179–181], meson leptonic decays [181], and the hadronic vacuum polarization [182] that QEDL does not suffer from a problematic (e.g., UV divergent) coupling of short- and long-distance physics due to its nonlocality. Another strategy, first proposed in [183] and used by the QCDSF collaboration, is to bound the zero-mode fluctuations to a finite range. Although more minimal, it is still a nonlocal modification of the theory and so far finite-size effects for this scheme have not been investigated. More recently, two proposals for local formulations of finite- volume QED emerged. The first one described in [184] proposes to use massive photons to regulate zero-mode singularities, at the price of (softly) breaking gauge invariance. The second one presented in [177], based on earlier works [185,186], avoids the zero-mode issue by using anti-periodic boundary conditions for Aμ(x). In this approach, gauge invariance requires the fermion field to undergo a charge conjugation transformation over a period, breaking electric charge conservation. These local approaches have the potential to constitute cleaner approaches to finite-volume QED. All the calculations presented in this review used QEDL or QEDTL, with the exception of QCDSF. Once a finite-volume theory for QED is specified, there are various ways to compute QED effects themselves on a given hadronic quantity. The most direct approach, first used in [178], is to include QED directly in the lattice simulations and assemble correlation functions from charged quark propagators. Another approach proposed in [172], is to exploit the perturbative nature of QED, and compute the leading-order corrections directly in pure QCD as matrix elements of the electromagnetic current. Both approaches have their advantages and disadvantages and as shown in [19], are not mutually exclusive. A critical comparative study can be found in [187]. Finally, most of the calculations presented here made the choice of computing electromagnetic corrections in the electro- quenched approximation. In this limit, one assumes that only valence quarks are charged, which is equivalent to neglecting QED corrections to the fermionic determinant. This approximation reduces dramatically the cost of lattice-QCD+QED calculations since it allows the reuse of previously generated QCD configurations. If QED is introduced perturbatively through current insertions, the electro-quenched approximation avoids computing disconnected contributions coming from the electromagnetic current in the vacuum, which are generally challenging to determine precisely. The electromagnetic contributions from sea quarks to hadron-mass splittings are known to be flavour-SU (3) and large-Nc suppressed, thus electro-quenched simulations are expected to have an O(10%) accuracy for the leading electromagnetic effects. This suppression is in principle rather weak and results obtained from electro-quenched simulations might feature uncontrolled systematic errors. For this reason, the use of the electro-quenched approximation constitutes the difference between and ◦ in the FLAG criterion for the inclusion of isospin-breaking effects. 123 869 Page 28 of 296 Eur. Phys. J. C (2022) 82:869 3.1.4 Lattice determination of ms and mud We now turn to a review of the lattice calculations of the light-quark masses and begin with ms , the isospin-averaged up- and down-quark mass mud , and their ratio. Most groups quote only mud , not the individual up- and down-quark masses. We then discuss the ratio mu/md and the individual determinations of mu and md . Quark masses have been calculated on the lattice since the mid-nineties. However, early calculations were performed in the quenched approximation, leading to unquantifiable systematics. Thus, in the following, we only review modern, unquenched calculations, which include the effects of light sea quarks. Tables 6 and 7 list the results of N f = 2+1 and N f = 2+1+1 lattice calculations of ms and mud . These results are given in the MS scheme at 2 GeV, which is standard nowadays, though some groups are starting to quote results at higher scales (e.g., Ref. [188]). The tables also show the colour coding of the calculations leading to these results. As indicated earlier in this review, we treat calculations with different numbers, N f , of dynamical quarks separately. N f = 2 + 1 lattice calculations We turn now to N f = 2 + 1 calculations. These and the corresponding results for mud and ms are summarized in Table 6. Given the very high precision of a number of the results, with total errors on the order of 1%, it is important to consider the effects neglected in these calculations. Isospin-breaking and electromagnetic effects are small on mud and ms , and have been approximately accounted for in the calculations that will be retained for our averages. We have already commented that the effect of the omission of the charm quark in the sea is expected to be small, below our current precision, and we do not add any additional uncertainty due to these effects in the final averages. The only new computation since the previous FLAG edition is the determination of light-quark masses by the ALPHA col- laboration [18]. This work uses nonperturbativelyO(a) improved Wilson fermions (a subset of the CLS ensembles [137]). The renormalization is performed nonperturbatively in the SF scheme from 200 MeV up to the electroweak scale∼ 100 GeV [203]. This nonperturbative running over such large energy scales avoids any use of perturbation theory at low energy scales, but adds a cost in terms of uncertainty: the running alone propagates to ≈ 1% of the error in quark masses. This turns out to be one of the dominant pieces of uncertainty for the case of ms . On the other hand, for the case of mud , the uncertainty is dominated by the chiral extrapolations. The ensembles used include four values of the lattice spacing below 0.09 fm, which qualifies for a in the continuum extrapolation, and pion masses down to 200 MeV. This value lies just at the boundary of the rating, but since the chiral extrapolation is a substantial source of systematic uncertainty, we opted to rate the work with a ◦. In any case, this work enters in the average and their results show a reasonable agreement with the FLAG average. We now comment in some detail on previous works that also contribute to the averages. RBC/UKQCD 14 [8] significantly improves on their RBC/UKQCD 12B [188] work by adding three new domain wall fermion simulations to three used previously. Two of the new simulations are performed at essentially physical pion masses (Mπ 139 MeV) on lattices of about 5.4 fm in size and with lattice spacings of 0.114 fm and 0.084 fm. It is complemented by a third simulation with Mπ 371 MeV, a 0.063 fm and a rather small L 2.0 fm. Altogether, this gives them six simulations with six unitary (msea = mval) Mπ ’s in the range of 139 to 371 MeV, and effectively three lattice spacings from 0.063 to 0.114 fm. They perform a combined global continuum and chiral fit to all of their results for the π and K masses and decay constants, the baryon mass and two Wilson-flow parameters. Quark masses in these fits are renormalized and run nonperturbatively in the RI-SMOM scheme. This is done by computing the relevant renormalization constant for a reference ensemble, and determining those for other simulations relative to it by adding appropriate parameters in the global fit. This calculation passes all of our selection criteria. N f = 2 + 1 MILC results for light-quark masses go back to 2004 [197,198]. They use rooted staggered fermions. By 2009 their simulations covered an impressive range of parameter space, with lattice spacings going down to 0.045 fm, and valence-pion masses down to approximately 180 MeV [15]. The most recent MILC N f = 2+ 1 results, i.e., MILC 10A [12] and MILC 09A [15], feature large statistics and 2-loop renormalization. Since these data sets subsume those of their previous calculations, these latest results are the only ones that need to be kept in any world average. The BMW 10A, 10B [9,10] calculation still satisfies our stricter selection criteria. They reach the physical up- and down- quark mass by interpolation instead of by extrapolation. Moreover, their calculation was performed at five lattice spacings ranging from 0.054 to 0.116 fm, with full nonperturbative renormalization and running and in volumes of up to (6 fm)3, guaranteeing that the continuum limit, renormalization, and infinite-volume extrapolation are controlled. It does neglect, however, isospin-breaking effects, which are small on the scale of their error bars. Finally, we come to another calculation which satisfies our selection criteria, HPQCD 10 [11]. It updates the staggered- fermions calculation of HPQCD 09A [27]. In these papers, the renormalized mass of the strange quark is obtained by 123 Eur. Phys. J. C (2022) 82:869 Page 29 of 296 869 Table 6 N f = 2 + 1 lattice results for the masses mud and ms (MeV) Collaboration Refs. mud ms ALPHA 19 [18] A ◦    e 3.54(12)(9) 95.7(2.5)(2.4) Maezawa 16 [189] A     d – 92.0(1.7) RBC/UKQCD 14B [8] A     d 3.31(4)(4) 90.3(0.9)(1.0) RBC/UKQCD 12 [188] A  ◦   d 3.37(9)(7)(1)(2) 92.3(1.9)(0.9)(0.4)(0.8) PACS-CS 12 [190] A     b 3.12(24)(8) 83.60(0.58)(2.23) Laiho 11 [49] C ◦   ◦ − 3.31(7)(20)(17) 94.2(1.4)(3.2)(4.7) BMW 10A, 10B+ [9,10] A     c 3.469(47)(48) 95.5(1.1)(1.5) PACS-CS 10 [191] A     b 2.78(27) 86.7(2.3) MILC 10A [12] C ◦   ◦ − 3.19(4)(5)(16) – HPQCD 10∗∗ [11] A ◦   − − 3.39(6) 92.2(1.3) RBC/UKQCD 10A [117] A ◦ ◦   a 3.59(13)(14)(8) 96.2(1.6)(0.2)(2.1) Blum 10† [170] A ◦  ◦  − 3.44(12)(22) 97.6(2.9)(5.5) PACS-CS 09 [192] A     b 2.97(28)(3) 92.75(58)(95) HPQCD 09A⊕ [27] A ◦   − − 3.40(7) 92.4(1.5) MILC 09A [15] C ◦   ◦ − 3.25 (1)(7)(16)(0) 89.0(0.2)(1.6)(4.5)(0.1) MILC 09 [157] A ◦   ◦ − 3.2(0)(1)(2)(0) 88(0)(3)(4)(0) PACS-CS 08 [193] A     − 2.527(47) 72.72(78) RBC/UKQCD 08 [194] A ◦    − 3.72(16)(33)(18) 107.3(4.4)(9.7)(4.9) CP-PACS/JLQCD 07 [195] A     − 3.55(19)(+56) 90.1(4.3)(+16.7−20 −4.3 ) HPQCD 05 [196] A ◦ ◦ ◦ ◦ − 3.2(0)(2)(2)(0)‡ 87(0)(4)(4)(0)‡ MILC 04, HPQCD/ [197,198] A ◦ ◦ ◦  − 2.8(0)(1)(3)(0) 76(0)(3)(7)(0) MILC/UKQCD 04 The results are given in the MS scheme at 3 instead of 2 GeV. We run them down to 2 GeV using numerically integrated 4-loop running [199,200] with N f = 3 and with the values of αs(MZ ), mb , and mc taken from Ref. [201]. The running factor is 1.106. At three loops it is only 0.2% smaller, indicating that perturbative running uncertainties are small. We neglect them here The calculation includes electromagnetic and mu = md effects through reweighting +The fermion action used is tree-level improved ∗∗ms is obtained by combining mc and HPQCD 09A’s mc/ms = 11.85(16) [27]. Finally, mud is determined from ms with the MILC 09 result for ms/mud . Since mc/ms is renormalization group invariant in QCD, the renormalization and running of the quark masses enter indirectly through that of mc (see below) †The calculation includes quenched electromagnetic effects ⊕What is calculated is mc/ms = 11.85(16). ms is then obtained by combining this result with the determination mc(mc) = 1.268(9) GeV from Ref. [202]. Finally, mud is determined from ms with the MILC 09 result for ms/mud ‡The bare numbers are those of MILC 04. The masses are simply rescaled, using the ratio of the 2-loop to 1-loop renormalization factors a The masses are renormalized nonperturbatively at a scale of 2 GeV in a couple of N f = 3 RI-SMOM schemes. A careful study of perturbative matching uncertainties has been performed by comparing results in the two schemes in the region of 2 GeV to 3 GeV [117] b The masses are renormalized and run nonperturbatively up to a scale of 40 GeV in the N f = 3 SF scheme. In this scheme, nonperturbative and NLO running for the quark masses are shown to agree well from 40 GeV all the way down to 3 GeV [191] c The masses are renormalized and run nonperturbatively up to a scale of 4 GeV in the N f = 3 RI-MOM scheme. In this scheme, nonperturbative and N3LO running for the quark masses are shown to agree from 6 GeV down to 3 GeV to better than 1% [10] d All required running is performed nonperturbatively e Running is performed nonperturbatively from 200 MeV to the electroweak scale ∼ 100 GeV combining the result of a precise calculation of the renormalized charm-quark mass, mc, with the result of a calculation of the quark-mass ratio, mc/ms . As described in Ref. [202] and in Sect. 3.2, HPQCD determines mc by fitting Euclidean- time moments of the c̄c pseudoscalar density two-point functions, obtained numerically in lattice QCD, to fourth-order, 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume Renormalization Running 869 Page 30 of 296 Eur. Phys. J. C (2022) 82:869 Table 7 N f = 2 + 1 + 1 lattice results for the masses mud and ms (MeV) Collaboration Refs. mud ms ETM 21A [204] P     − 3.636(66)(+60−57) 98.7(2.4)(+4.0−3.2) HPQCD 18† [13] A     − 94.49(96) FNAL/MILC/TUMQCD 18 [6] A     − 3.404(14)(21) 92.52(40)(56) HPQCD 14A ⊕ [14] A    − − 93.7(8) ETM 14⊕ [7] A ◦    − 3.70(13)(11) 99.6(3.6)(2.3) †Bare-quark masses are renormalized nonperturbatively in the RI-SMOM scheme at scales μ ∼ 2−5 GeV for different lattice spacings and translated to the MS scheme. Perturbative running is then used to run all results to a reference scale μ = 3 GeV ⊕As explained in the text, ms is obtained by combining the results mc(5 GeV; N f = 4) = 0.8905(56) GeV and (mc/ms)(N f = 4) = 11.652(65), determined on the same data set. A subsequent scale and scheme conversion, performed by the authors, leads to the value 93.6(8). In the table, we have converted this to ms(2 GeV; N f = 4), which makes a very small change continuum perturbative expressions. These moments are normalized and chosen so as to require no renormalization with staggered fermions. Since mc/ms requires no renormalization either, HPQCD’s approach displaces the problem of lattice renormalization in the computation of ms to one of computing continuum perturbative expressions for the moments. To calculate mud HPQCD 10 [11] use the MILC 09 determination of the quark-mass ratio ms/mud [157]. HPQCD 09A [27] obtains mc/ms = 11.85(16) [27] fully nonperturbatively, with a precision slightly larger than 1%. HPQCD 10’s determination of the charm-quark mass, m 11c(mc) = 1.268(6), is even more precise, achieving an accuracy better than 0.5%. This discussion leaves us with five results for our final average for ms : ALPHA 19 [18], MILC 09A [15], BMW 10A, 10B [9,10], HPQCD 10 [11] and RBC/UKQCD 14 [8]. Assuming that the result from HPQCD 10 is 100% correlated with that of MILC 09A, as it is based on a subset of the MILC 09A configurations, we find ms = 92.2(1.1)MeV with a χ2/dof = 1.65. For the light-quark mass mud , the results satisfying our criteria are ALPHA 19, RBC/UKQCD 14B, BMW 10A, 10B, HPQCD 10, and MILC 10A. For the error, we include the same 100% correlation between statistical errors for the latter two as for the strange case, resulting in the following (at scale 2 GeV in the MS scheme, and χ2/dof = 1.4), = + m = 3.381(40) MeV Refs. [8–12,18],N f 2 1 : udms = (33)92.2(1.0) MeV Refs. [8–11,15,18], and the RGI values MRGI= + : ud = 4.695(56)m(54) MeV Refs. [8–12,18],N f 2 1 (34) MRGIs = 128.1(1.4)m(1.5) MeV Refs. [8–11,15,18]. N f = 2 + 1 + 1 lattice calculations Since the previous review a new computation of ms,mud has appeared, ETM 21A [204]. Using twisted-mass fermions with an added clover-term to suppress O(a2) effects between the neutral and charged pions, this work represents a significant improvement over ETM 14 [7]. Renormalization is performed nonperturbatively in the RI-MOM scheme. Their ensembles comprise three lattice spacings (0.095, 0.082, and 0.069 fm), two volumes for the finest lattice spacings with pion masses reaching down to the physical point in the two finest lattices allowing a controlled chiral extrapolation. Their volumes are 11 To obtain this number, we have used the conversion from μ = 3 GeV to mc given in Ref. [202]. 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume Renormalization Running Eur. Phys. J. C (2022) 82:869 Page 31 of 296 869 large, with mπ L between four and five. These characteristics of their ensembles pass the most stringent FLAG criteria in all categories. This work extracts quark masses from two different quantities, one based on the meson spectrum and the other based on the baryon spectrum. Results obtained with these two methods agree within errors. The latter agrees well with the FLAG average while the former is high in comparison (there is good agreement with their previous results, ETM 14 [7]). Since ETM 21A was not published by the FLAG deadline, it is not included in the averages. There are three other works that enter in light-quark mass averages: FNAL/MILC/TUMQCD 18 [6] (which contributes both to the average of mud and ms), and the mud determinations in HPQCD 18 [13] and HPQCD 14A [14]. While the results of HPQCD 14A and HPQCD 18 agree well (using different methods), there are several tensions in the determination of ms . The most significant discrepancy is between ETM 21A and the FLAG average. But also two recent and very precise determinations (HPQCD 18 and FNAL/MILC/TUMQCD 18) show a tension. Overall there is a rough agreement between the different determinations with χ2/dof = 1.7 (that we apply to our average according to the standard FLAG averaging procedure). In the case of mud on the other hand only two works contribute to the average: ETM 14 and FNAL/MILC/TUMQCD 18. They disagree, with the FNAL/MILC/TUMQCD 18 value basically matching the N f = 2 + 1 result. The large χ2/dof ≈ 1.7 increases significantly the error of the average. These large values of the χ2 are difficult to understand in terms of a statistical fluctuation. On the other hand the N f = 2+ 1 and N f = 2+ 1+ 1 averages show a good agreement, which increases our confidence in the averages quoted below. The N f = 2+1+1 results are summarized in Table 7. Note that the results of Ref. [14] are reported as ms(2 GeV; N f = 3) and those of Ref. [7] as mud(s)(2 GeV; N f = 4). We convert the former to N f = 4 and obtain ms(2 GeV; N f = 4) = 93.7(8)MeV. The average of FNAL/MILC/TUMQCD 18, HPQCD 18, ETM 14 and HPQCD 14A is 93.43(70)MeV with χ2/dof = 1.7. For the light-quark average we use ETM 14 and FNAL/MILC/TUMQCD 18 with an average 3.410(43)MeV and a χ2/dof = 1.7. We note these χ2 values are large. For the case of the light-quark masses this is mostly due to ETM 14 masses lying significantly above the rest, but in the case of ms there is also some tension between the recent and very precise results of HPQCD 18 and FNAL/MILC/TUMQCD 18. Also note that the 2+1-flavour values are consistent with the four-flavour ones, so in all cases we have decided to simply quote averages according to FLAG rules, including stretching factors for the errors based on χ2 values of our fits: mud = 3.410(43) MeV Refs. [6,7], N f = 2 + 1 + 1 : (35)ms = 93.40(57) MeV Refs. [6,7,13,14], and the RGI values MRGI= + + : ud = 4.736(60)m(55) MeV Refs. [6,7],N f 2 1 1 (36) MRGIs = 129.7(0.8)m(1.5) MeV Refs. [6,7,13,14]. In Figs. 1 and 2 the lattice results listed in Tables 6 and 7 and the FLAG averages obtained at each value of N f are presented and compared with various phenomenological results. 3.1.5 Lattice determinations of ms/mud The lattice results for ms/mud are summarized in Table 8. In the ratio ms/mud , one of the sources of systematic error – the uncertainties in the renormalization factors – drops out. Also other systematic effects (like the effect of the scale setting) are reduced in these ratios. This might explain that despite the discrepancies that are present in the individual quark mass determinations, the ratios show an overall very good agreement. N f = 2 + 1 lattice calculations ALPHA 19 [18], discussed already, is the only new result for this section. The other works contributing to this average are RBC/UKQCD 14B, which replaces RBC/UKQCD 12 (see Sect. 3.1.4), and the results of MILC 09A and BMW 10A, 10B. The results show very good agreement with a χ2/dof = 0.14. The final uncertainty (≈ 0.5%) is smaller than the ones of the quark masses themselves. At this level of precision, the uncertainties in the electromagnetic and strong isospin-breaking corrections might not be completely negligible. Nevertheless, we decided not to add any uncertainty associated with this 123 869 Page 32 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 1 MS mass of the strange quark (at 2 GeV scale) in MeV. The upper two panels show the lattice results listed in Tables 6 and 7, while the bottom panel collects sum rule results [205–209]. Diamonds and squares represent results based on perturbative and nonperturbative renormalization, respectively. The black squares and the grey bands represent our averages (33) and (35). The significance of the colours is explained in Sect. 2 Fig. 2 Mean mass of the two lightest quarks, mud = 12 (mu + md ). The bottom panel shows results based on sum rules [205,208,210] (for more details see Fig. 1) effect. The main reason is that most recent determinations try to estimate this uncertainty themselves and found an effect smaller than naive power counting estimates (see N f = 2 + 1 + 1 section), N f = 2 + 1 : ms/mud = 27.42 (12) Refs. [8–10,15,18]. (37) N f = 2 + 1 + 1 lattice calculations For N f = 2 + 1 + 1 there are three results, MILC 17 [16], ETM 14 [7] and FNAL/MILC 14A [17], all of which satisfy our selection criteria. 123 Eur. Phys. J. C (2022) 82:869 Page 33 of 296 869 Table 8 Lattice results for the ratio ms/mud Collaboration Refs. N f ms/mud ETM 21A [204] 2+1+1 P    27.17(32)+56−38 MILC 17 ‡ [16] 2+1+1 A    27.178(47)+86−57 FNAL/MILC 14A [17] 2+1+1 A    27.35(5)+10 ETM 14 [7] 2+1+1 A ◦ −7 ◦ 26.66(32)(2) ALPHA 19 [18] 2+1 A ◦   27.0(1.0)(0.4) RBC/UKQCD 14B [8] 2+1 A    27.34(21) RBC/UKQCD 12 [188] 2+1 A  ◦  27.36(39)(31)(22) PACS-CS 12 [190] 2+1 A    26.8(2.0) Laiho 11 [49] 2+1 C ◦   28.4(0.5)(1.3) BMW 10A, 10B+ [9,10] 2+1 A    27.53(20)(8) RBC/UKQCD 10A [117] 2+1 A ◦ ◦  26.8(0.8)(1.1) Blum 10† [170] 2+1 A ◦  ◦ 28.31(0.29)(1.77) PACS-CS 09 [192] 2+1 A    31.2(2.7) MILC 09A [15] 2+1 C ◦   27.41(5)(22)(0)(4) MILC 09 [157] 2+1 A ◦   27.2(1)(3)(0)(0) PACS-CS 08 [193] 2+1 A    28.8(4) RBC/UKQCD 08 [194] 2+1 A ◦   28.8(0.4)(1.6) MILC 04, HPQCD/MILC/UKQCD 04 [197,198] 2+1 A ◦ ◦ ◦ 27.4(1)(4)(0)(1) ‡The calculation includes electromagnetic effects The errors are statistical, chiral and finite volume The calculation includes electromagnetic and mu = md effects through reweighting+The fermion action used is tree-level improved †The calculation includes quenched electromagnetic effects All these works have been discussed in the previous FLAG edition [4], except the new result ETM 21A, that we have already examined (and anyway does not appear in the average because it was unpublished at the deadline). The fit has χ2/dof ≈ 2.5, and the result shows reasonable agreement with the N f = 2 + 1 result. N f = 2 + 1 + 1 : ms/mud = 27.23 (10) Refs. [7,16,17], (38) which corresponds to an overall uncertainty equal to 0.4%. It is worth noting that [16] estimates the EM effects in this quantity to be ∼ 0.18% (or 0.049 which is less than the quoted error above). All the lattice results listed in Table 8 as well as the FLAG averages for each value of N f are reported in Fig. 3 and compared with χPT and sum rules. 3.1.6 Lattice determination of mu and md In addition to reviewing computations of individual mu and md quark masses, we will also determine FLAG averages for the parameter  related to the violations of Dashen’s theorem = (M 2 K −M2π )γ 2 , (39)Mπ 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume 869 Page 34 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 3 Results for the ratio ms/mud . The upper part indicates the lattice results listed in Table 8 together with the FLAG averages for each value of N f . The lower part shows results obtained from χPT and sum rules [208,211–215] where M2 = M2+ − M2π 0 and M2 2 2π π K = MK+ − M 0 are the pion and kaon squared mass splittings, respectively. TheK superscript γ , here and in the following, denotes corrections that arise from electromagnetic effects only. This parameter is often a crucial intermediate quantity in the extraction of the individual light-quark masses. Indeed, it can be shown, using the G-parity symmetry of the pion triplet, that M2π does not receive O(δm) isospin-breaking corrections. In other words (M2 )γ M2 = (M2 )γ and  = Kπ π 2 − 1, (40)Mπ at leading-order in the isospin-breaking expansion. The difference (M2 )SU (2)π was estimated in previous editions of FLAG through the m parameter. However, consistent with our leading-order truncation of the isospin-breaking expansion, it is simpler to ignore this term. Once known,  allows one to consistently subtract the electromagnetic part of the kaon-mass splitting to obtain the QCD splitting (M2 SU (2)K ) . In contrast with the pion, the kaon QCD splitting is sensitive to δm, and, in particular, proportional to it at leading order in χPT. Therefore, the knowledge of  allows for the determination of δm from a chiral fit to lattice-QCD data. Originally introduced in another form in [216],  vanishes in the SU (3) chiral limit, a result known as Dashen’s theorem. However, in the 1990s numerous phenomenological papers pointed out that  might be an O(1) number, indicating a significant failure of SU (3) χPT in the description of electromagnetic effects on light-meson masses. However, the phenomenological determinations of  feature some level of controversy, leading to the rather imprecise estimate  = 0.7(5) given in the first edition of FLAG. Starting with the FLAG 19 edition of the review, we quote more precise averages for , directly obtained from lattice-QCD+QED simulations. We refer the reader to earlier editions of FLAG and to the review [217] for discussions of the phenomenological determinations of . The quality criteria regarding finite-volume effects for calculations including QED are presented in Sect. 2.1.1. Due to the long-distance nature of the electromagnetic interaction, these effects are dominated by a power law in the lattice spatial size. The coefficients of this expansion depend on the chosen finite-volume formulation of QED. For QEDL, these effects on the squared mass M2 of a charged meson are given by [147,148,150] { [ ]} 2 = 2 c1 + 2c1 1FV M αM 2 +O 3 , (41)M L (M L) (M L) with c1 −2.83730. It has been shown in [147] that the two first orders in this expansion are exactly known for hadrons, and are equal to the pointlike case. However, the O[1/(M L)3] term and higher orders depend on the structure of the hadron. The universal corrections for QEDTL can also be found in [147]. In all this part, for all computations using such universal formulae, the QED finite-volume quality criterion has been applied with nmin = 3, otherwise nmin = 1 was used. Since FLAG 19, six new results have been reported for nondegenerate light-quark masses. In the N f = 2 + 1 + 1 sector, MILC 18 [21] computed  using N f = 2 + 1 asqtad electro-quenched QCD+QEDTL simulations and extracted the ratio 123 Eur. Phys. J. C (2022) 82:869 Page 35 of 296 869 mu/md from a new set of N f = 2 + 1 + 1 HISQ QCD simulations. Although  comes from N f = 2 + 1 simulations, (M2 SU (2) 2 γK ) , which is about three times larger than (MK ) , has been determined in the N f = 2 + 1 + 1 theory. We therefore chose to classify this result as a four-flavour one. This result is explicitly described by the authors as an update of MILC 17 [16]. In MILC 17 [16], mu/md is determined as a side-product of a global analysis of heavy-meson decay constants, using a preliminary version of  from MILC 18 [21]. In FNAL/MILC/TUMQCD 18 [6] the ratio mu/md from MILC 17 [16] is used to determine the individual masses mu and md from a new calculation of mud . The work RM123 17 [19] is the continuation of the N f = 2 work named RM123 13 [172] in the previous edition of FLAG. This group now uses N f = 2 + 1 + 1 ensembles from ETM 10 [218], however, still with a rather large minimum pion mass of 270 MeV, leading to the ◦ rating for chiral extrapolations. In the N f = 2 + 1 sector, BMW 16 [20] reuses the data set produced from their determination of the light-baryon octet-mass splittings [173] using electro-quenched QCD+QEDTL smeared clover fermion simulations. Finally, MILC 16 [219], which is a preliminary result for the value of  published in MILC 18 [21], also provides a N f = 2 + 1 computation of the ratio mu/md . MILC 09A [15] uses the mass difference between K 0 and K+, from which they subtract electromagnetic effects using Dashen’s theorem with corrections, as discussed in the introduction of this section. The up and down sea quarks remain degenerate in their calculation, fixed to the value of mud obtained from Mπ0 . To determine mu/md , BMW 10A, 10B [9,10] follow a slightly different strategy. They obtain this ratio from their result for ms/mud combined with a phenomenological determination of the isospin-breaking quark-mass ratio Q = 22.3(8), from η → 3π decays [220] (the decay η → 3π is very sensitive to QCD isospin breaking, but fairly insensitive to QED isospin breaking). Instead of subtracting electromag- netic effects using phenomenology, RBC 07 [169] and Blum 10 [170] actually include a quenched electromagnetic field in their calculation. This means that their results include corrections to Dashen’s theorem, albeit only in the presence of quenched electromagnetism. Since the up and down quarks in the sea are treated as degenerate, very small isospin cor- rections are neglected, as in MILC’s calculation. PACS-CS 12 [190] takes the inclusion of isospin-breaking effects one step further. Using reweighting techniques, it also includes electromagnetic and mu − md effects in the sea. However, they do not correct for the large finite-volume effects coming from electromagnetism in their Mπ L ∼ 2 simulations, but provide rough estimates for their size, based on Ref. [179]. QCDSF/UKQCD 15 [221] uses QCD+QED dynamical simu- lations performed at the SU (3)-flavour-symmetric point, but at a single lattice spacing, so they do not enter our average. The smallest partially quenched (msea = mval) pion mass is greater than 200 MeV, so our chiral-extrapolation criteria require a ◦ rating. Concerning finite-volume effects, this work uses three spatial extents L of 1.6 fm, 2.2 fm, and 3.3 fm. QCDSF/UKQCD 15 claims that the volume dependence is not visible on the two largest volumes, leading them to assume that finite-size effects are under control. As a consequence of that, the final result for quark masses does not feature a finite-volume extrapolation or an estimation of the finite-volume uncertainty. However, in their work on the QED corrections to the hadron spectrum [221] based on the same ensembles, a volume study shows some level of compatibility with the QEDL finite-volume effects derived in [148]. We see two issues here. Firstly, the analytical result quoted from [148] predicts large, O(10%) finite-size effects from QED on the meson masses at the values of Mπ L considered in QCDSF/UKQCD 15, which is inconsistent with the statement made in the paper. Secondly, it is not known that the zero-mode regularization scheme used here has the same volume scaling as QEDL. We therefore chose to assign the  rating for finite volume to QCDSF/UKQCD 15. Finally, for N f = 2+ 1+ 1, ETM 14 [7] uses simulations in pure QCD, but determines mu −md from the slope ∂M2K /∂mud and the physical value for the QCD kaon-mass splitting taken from the phenomenological estimate in FLAG 13. Lattice results for mu , md and mu/md are summarized in Table 9. The colour coding is specified in detail in Sect. 2.1. Considering the important progress in the last years on including isospin-breaking effects in lattice simulations, we are now in a position where averages for mu and md can be made without the need of phenomenological inputs. Therefore, lattice calculations of the individual quark masses using phenomenological inputs for isospin-breaking effects will be coded  . We start by recalling the N f = 2 FLAG average for the light-quark masses, entirely coming from RM123 13 [172], mu = 2.40(23)MeV Ref. [172], N f = 2 : md = 4.80(23)MeV Ref. [172], (42) mu/md = 0.50(4) Ref. [172], with errors of roughly 10%, 5% and 8%, respectively. In these results, the errors are obtained by combining the lattice statistical and systematic errors in quadrature. For N f = 2+1, the only result, which qualifies for entering the FLAG average for quark masses, is BMW 16 [20], 123 869 Page 36 of 296 Eur. Phys. J. C (2022) 82:869 123 Table 9 Lattice results for mu , md (MeV) and for the ratio mu/md . The values refer to the MS scheme at scale 2 GeV. The top part of the table lists the results obtained with N f = 2 + 1 + 1, while the lower part presents calculations with N f = 2 + 1 Collaboration Refs. mu md mu/md MILC 18 [21] A    ◦  − 0.4529(48)(+150−67 ) FNAL/MILC/TUMQCD 18∗ [6] A    ◦  − 2.118(17)(32)(12)(03) 4.690(30)(36)(26)(06) MILC 17† [16] A    ◦  − 0.4556(55)(+114−67 )(13) RM123 17 [19] A ◦   ◦  b 2.50(15)(8)(2) 4.88(18)(8)(2) 0.513(18)(24)(6) ETM 14 [7] A      b 2.36(24) 5.03(26) 0.470(56) BMW 16 [20] A    ◦  − 2.27(6)(5)(4) 4.67(6)(5)(4) 0.485(11)(8)(14) MILC 16 [219] C ◦   ◦  − 0.4582(38)(+12−82)(1)(110) QCDSF/UKQCD 15 [221] A ◦    − − 0.52(5) PACS-CS 12 [190] A      a 2.57(26)(7) 3.68(29)(10) 0.698(51) Laiho 11 [49] C ◦    ◦ − 1.90(8)(21)(10) 4.73(9)(27)(24) 0.401(13)(45) HPQCD 10‡ [11] A ◦     − 2.01(14) 4.77(15) BMW 10A, 10B+ [9,10] A      b 2.15(03)(10) 4.79(07)(12) 0.448(06)(29) Blum 10 [170] A ◦  ◦ ◦  − 2.24(10)(34) 4.65(15)(32) 0.4818(96)(860) MILC 09A [15] C ◦    ◦ − 1.96(0)(6)(10)(12) 4.53(1)(8)(23)(12) 0.432(1)(9)(0)(39) MILC 09 [157] A ◦    ◦ − 1.9(0)(1)(1)(1) 4.6(0)(2)(2)(1) 0.42(0)(1)(0)(4) MILC 04, [197,198] A ◦ ◦ ◦   − 1.7(0)(1)(2)(2) 3.9(0)(1)(4)(2) 0.43(0)(1)(0)(8) HPQCD/ MILC/UKQCD 04 ∗FNAL/MILC/TUMQCD 18 uses  from MILC 18 to produce the individual mu and md masses †MILC 17 additionally quotes an optional 0.0032 uncertainty on mu/md corresponding to QED and QCD separation scheme ambiguities. Because this variation is not per se an error on the determination of mu/md , and because it is generally not included in other results, we choose to omit it here ‡Values obtained by combining the HPQCD 10 result for ms with the MILC 09 results for ms/m+ ud and mu/md The fermion action used is tree-level improved a The masses are renormalized and run nonperturbatively up to a scale of 100 GeV in the N f = 2 SF scheme. In this scheme, nonperturbative and NLO running for the quark masses are shown to agree well from 100 GeV all the way down to 2 GeV [222] b The masses are renormalized and run nonperturbatively up to a scale of 4 GeV in the N f = 3 RI-MOM scheme. In this scheme, nonperturbative and N3LO running for the quark masses are shown to agree from 6 GeV down to 3 GeV to better than 1% [10] Publication status Chiral extrapolation Continuum extrapolation Finite volume Isospin breaking Renormalization Running Eur. Phys. J. C (2022) 82:869 Page 37 of 296 869 mu = 2.27(9)MeV Ref. [20], N f = 2 + 1 : md = 4.67(9)MeV Ref. [20], (43) mu/md = 0.485(19) Ref. [20], with errors of roughly 4%, 2% and 4%, respectively. This estimate is slightly more precise than in the previous edition of FLAG. More importantly, it now comes entirely from a lattice-QCD+QED calculation, whereas phenomenological input was used in previous editions. These numbers result in the following RGI averages MRGIu = 3.15(12)m(4) MeV Ref. [20], N f = 2 + 1 : MRGI = 6.49(12) (7) MeV Ref. [20]. (44)d m  Finally, for N f = 2 + 1 + 1, RM123 17 [19] and FNAL/MILC/TUMQCD 18 [6] enter the average for the individual mu and md masses, and RM123 17 [19] and MILC 18 [21] enter the average for the ratio mu/md , giving mu = 2.14(8)MeV Refs. [6,19], N f = 2 + 1 + 1 : md = 4.70(5)MeV Refs. [6,19], (45) mu/md = 0.465(24) Refs. [19,21.] with errors of roughly 4%, 1% and 5%, respectively. One can observe some marginal discrepancies between results coming from the MILC collaboration and RM123 17 [19]. More specifically, adding all sources of uncertainties in quadrature, one obtains a 1.7σ discrepancy between RM123 17 [19] and MILC 18 [21] for mu/md , and a 2.2σ discrepancy between RM123 17 [19] and FNAL/MILC/TUMQCD 18 [6] for mu . However, the values of md and  are in very good agreement between the two groups. These discrepancies are presently too weak to constitute evidence for concern, and will be monitored as more lattice groups provide results for these quantities. The RGI averages for mu and md are MRGIu = 2.97(11)m(3) MeV Refs. [6,19] , N f = 2 + 1 + 1 : MRGId = 6.53(7)m(8) MeV Refs. [6,19] . (46) Every result for mu and md used here to produce the FLAG averages relies on electro-quenched calculations, so there is some interest to comment on the size of quenching effects. Considering phenomenology and the lattice results presented here, it is reasonable for a rough estimate to use the value (M2 γK ) ∼ 2000 MeV2 for the QED part of the kaon-mass splitting. Using the arguments presented in Sect. 3.1.3, one can assume that the QED sea contribution represents O(10%) of (M2K ) γ . Using SU (3) PQχPT+QED [174,223] gives a ∼ 5% effect. Keeping the more conservative 10% estimate and using the experimental value of the kaon-mass splitting, one finds that the QCD kaon-mass splitting (M2 )SU (2)K suffers from a reduced 3% quenching uncertainty. Considering that this splitting is proportional to mu − md at leading order in SU (3) χPT, we can estimate that a similar error will propagate to the quark masses. So the individual up and down masses look mildly affected by QED quenching. However, one notices that ∼ 3% is the level of error in the new FLAG averages, and increasing significantly this accuracy will require using fully unquenched calculations. In view of the fact that a massless up quark would solve the strong CP problem, many authors have considered this an attractive possibility, but the results presented above exclude this possibility: the value of mu in Eq. (43) differs from zero by 26 standard deviations. We conclude that nature solves the strong CP problem differently. Finally, we conclude this section by giving the FLAG averages for  defined in Eq. (39). For N f = 2 + 1+ 1, we average the results of RM123 17 [19] and MILC 18 [21] with the value of (M2 γK ) from BMW 14 [147] combined with Eq. (40), giving N f = 2 + 1 + 1 :  = 0.79(6) Refs. [19,21,147]. (47) Although BMW 14 [147] focuses on hadron masses and did not extract the light-quark masses, they are the only fully unquenched QCD+QED calculation to date that qualifies to enter a FLAG average. With the exception of renormalization, which is not discussed in the paper, this work has a rating for every FLAG criterion considered for the mu and md quark 123 869 Page 38 of 296 Eur. Phys. J. C (2022) 82:869 masses. For N f = 2 + 1 we use the results from BMW 16 [20], N f = 2 + 1 :  = 0.73(17) Ref. [20]. (48) It is important to notice that the  uncertainties from BMW 16 and RM123 17 are dominated by estimates of the QED quenching effects. Indeed, in contrast with the quark masses,  is expected to be rather sensitive to the sea-quark QED contributions. Using the arguments presented in Sect. 3.1.3, if one conservatively assumes that the QED sea contributions representO(10%) of (M2 γK ) , then Eq. (40) implies that  will have a quenching error of∼ 0.15 for (M2 γ 2∼ K ) ∼ 2000 MeV , representing a large 20% relative error. It is interesting to observe that such a discrepancy does not appear between BMW 15 and RM123 17, although the ∼ 10% accuracy of both results might not be sufficient to resolve these effects. On the other hand, in the context of SU (3) chiral perturbation theory, Bijnens and Danielsson [174] show that the QED quenching effects on  do not depend on unknown LECs at NLO and are therefore computable at that order. In that approach, MILC 18 finds the effect at NLO to be only 5%. To conclude, although the controversy around the value of  has been significantly reduced by lattice-QCD+QED determinations, computing this at few-percent accuracy requires simulations with charged sea quarks. 3.1.7 Estimates for R and Q The quark-mass ratios ≡ ms − m m 2 − m2ud R and Q2 ≡ s ud2 (49)md − mu m 2d − mu compare SU (3) breaking with isospin breaking. Both numbers only depend on the ratios ms/mud and mu/md , ( ) 1 + mu ( )= 1 ms − md 1 mR 1 − m and Q 2 = s + 1 R. (50) 2 m uud 1 m 2 md ud The quantity Q is of particular interest because of a low-energy theorem [224], which relates it to a ratio of meson masses, M̂2 M̂2 − M̂2 Q2 K K π 2 1 2 2 2 M ≡ − , M̂π ≡ (M̂ + + M̂ 0) , M̂K ≡ 1 (M̂2 + + M̂2K K 0) . (51)M̂2 M̂2 M̂2 π π+ 2 2π K 0 K (We remind the reader that the ˆ denotes a quantity evaluated in the α → 0 limit.) Chiral symmetry implies that the expansion of Q2M in powers of the quark masses (i) starts with Q 2 and (ii) does not receive any contributions at NLO: N=LOQM Q . (52) We recall here the N f = 2 estimates for Q and R from FLAG 16, R = 40.7(3.7)(2.2) , Q = 24.3(1.4)(0.6), (53) where the second error comes from the phenomenological inputs that were used. For N f = 2 + 1, we use Eqs. (37) and (43) and obtain R = 38.1(1.5) , Q = 23.3(0.5), (54) where now only lattice results have been used. For N f = 2 + 1 + 1 we obtain R = 35.9(1.7) , Q = 22.5(0.5), (55) which are quite compatible with two- and three-flavour results. It is interesting to notice that the most recent phenomenological determination of R and Q from η → 3π decay [225] gives the values R = 34.4(2.1) and Q = 22.1(7), which are marginally discrepant with some of the averages presented here. The authors of [225,226] point out that this discrepancy is likely due to surprisingly large corrections to the approximation in Eq. (52) used in the phenomenological analysis. Our final results for the masses mu , md , mud , ms and the mass ratios mu/md , ms/mud , R, Q are collected in Tables 10 and 11. 123 Eur. Phys. J. C (2022) 82:869 Page 39 of 296 869 Table 10 Our estimates for the strange-quark and the average up-down-quark masses in the MS scheme at running scale μ = 2 GeV. Mass values are given in MeV. In the results presented here, the error is the one which we obtain by applying the averaging procedure of Sect. 2.3 to the relevant lattice results. N f mud ms ms/mud 2+1+1 3.410(43) 93.44(68) 27.23(10) 2+1 3.364(41) 92.03(88) 27.42(12) Table 11 Our estimates for the masses of the two lightest quarks and related, strong isospin-breaking ratios. Again, the masses refer to the MS scheme at running scale μ = 2 GeV. Mass values are given in MeV N f mu md mu/md R Q 2+1+1 2.14(8) 4.70(5) 0.465(24) 35.9(1.7) 22.5(0.5) 2+1 2.27(9) 4.67(9) 0.485(19) 38.1(1.5) 23.3(0.5) 3.2 Charm-quark mass In the following, we collect and discuss the lattice determinations of the MS charm-quark mass mc. Most of the results have been obtained by analyzing the lattice-QCD simulations of two-point heavy-light- or heavy-heavy-meson correlation functions, using as input the experimental values of the D, Ds , and charmonium mesons. Some groups use the moments method. The latter is based on the lattice calculation of the Euclidean time moments of pseudoscalar-pseudoscalar correlators for heavy- quark currents followed by an OPE expansion dominated by perturbative QCD effects, which provides the determination of both the heavy-quark mass and the strong-coupling constant αs . The heavy-quark actions adopted by various lattice collaborations have been discussed in previous FLAG reviews [2–4], and their descriptions can be found in Sec. A.1.3 of FLAG 19 [4]. While the charm mass determined with the moments method does not need any lattice evaluation of the mass-renormalization constant Zm , the extraction of mc from two-point heavy- meson correlators does require the nonperturbative calculation of Zm . The lattice scale at which Zm is obtained is usually at least of the order 2–3 GeV, and therefore it is natural in this review to provide the values of mc(μ) at the renormalization scale μ = 3 GeV. Since the choice of a renormalization scale equal to mc is still commonly adopted (as by the PDG [165]), we have collected in Table 12 the lattice results for both mc(mc) and mc(3 GeV), obtained for N f = 2 + 1 and 2 + 1 + 1. For N f = 2, interested readers are referred to previous reviews [2,3]. When not directly available in the published work, we apply a conversion factor using perturbative QCD evolution at five loops to run down from μ = 3 GeV to the scales μ = mc and 2 GeV of 0.7739(60) and 0.9026(23), respectively, where the error comes from the uncertainty in QCD. We use QCD = 297(12) MeV for N f = 4 (see Sect. 9). Perturbation theory uncertainties, estimated as the difference between results that use 4- and 5-loop running, are significantly smaller than the parametric uncertainty coming from QCD. For μ = mc, the former is about about 2.5 times smaller. Given the high precision of many of these results, future works should take the uncertainties in QCD and perturbation theory seriously. In the next subsections we review separately the results for mc with three or four flavours of quarks in the sea. 3.2.1 N f = 2 + 1 results Since the last review [4], there are two new results, Petreczky 19 [26] and ALPHA 21 [227], the latter of which was not published at the FLAG deadline. Petreczky 19 employs the HISQ action on ten ensembles with ten lattice spacings down to 0.025 fm, physical strange-quark mass, and two light-quark masses, the lightest corresponding to 161 MeV pions. Their study incorporates lattices with 11 different sizes, ranging from 1.6 to 5.4 fm. The masses are computed from moments of pseudoscalar quarkonium correlation functions, and MS masses are extracted with 4-loop continuum perturbation theory. Thus this work easily rates green stars in all categories. ALPHA 21 uses the O(a)-improved Wilson-clover action with five lattice spacings from 0.087 to 0.039 fm, produced by the CLS collaboration. For each lattice spacing, several light sea-quark masses are used in a global chiral-continuum extrapolation (the lightest pion mass for one ensemble is 198 MeV). The authors also use nonperturbative renormalization and running through application of step-scaling and the Schrödinger functional scheme. Finite-volume effects are investigated at one lattice spacing and only for ∼ 400 MeV pions on the smallest two 123 869 Page 40 of 296 Eur. Phys. J. C (2022) 82:869 Table 12 Lattice results for the MS charm-quark mass mc(mc) and mc(3 GeV) in GeV, together with the colour coding of the calculations used to obtain them Collaboration Refs. N f mc(mc) mc(3 GeV) ETM 21A [204] 2+1+1 P     1.339(22)(+19−10)(10)† 1.036(17)( +15 −8 ) HPQCD 20A [23] 2+1+1 A     1.2719(78) 0.9841(51) HPQCD 18 [13] 2+1+1 A     1.2757(84) 0.9896(61) FNAL/MILC/ [6] 2+1+1 A    − 1.273(4)(1)(10) 0.9837(43)(14)(33)(5) TUMQCD 18 HPQCD 14A [14] 2+1+1 A    − 1.2715(95) 0.9851(63) ETM 14A [22] 2+1+1 A ◦  ◦  1.3478(27)(195) 1.0557(22)(153)∗ ETM 14 [7] 2+1+1 A ◦  ◦  1.348(46) 1.058(35)∗ ALPHA 21 [227] 2+1 A+     1.296(19) 1.007(16) Petreczky 19 [26] 2+1 A     1.265(10) 1.001(16) Maezawa 16 [189] 2+1 A     1.267(12) JLQCD 16 [25] 2+1 A ◦   − 1.2871(123) 1.0033(96) χQCD 14 [24] 2+1 A ◦ ◦ ◦  1.304(5)(20) 1.006(5)(22) HPQCD 10 [11] 2+1 A ◦  ◦ − 1.273(6) 0.986(6) HPQCD 08B [202] 2+1 A ◦  ◦ − 1.268(9) 0.986(10) PDG [165] 1.27(2) †We applied the running factor 0.7739(60) for μ = 3 GeV to mc. The errors are statistical, systematic, and the uncertainty in the running factor∗A running factor equal to 0.900 between the scales μ = 2 GeV and μ = 3 GeV was applied by us +Published after the FLAG deadline volumes where results are compatible within statistical errors. ALPHA 21 satisfies the FLAG criteria for green-star ratings in all of the categories listed in Table 12, but because it is a new result that was unpublished at the deadline, does not enter the average in this review. Descriptions of the other works in this section can be found in the last review [4]. According to our rules on the publication status, the FLAG average for the charm-quark mass at N f = 2 + 1 is obtained by combining the results HPQCD 10, χQCD 14, JLQCD 16, and Petreczky 19, mc(mc) = 1.275(5) GeV Refs. [11,24–26] , (56) N f = 2 + 1 : mc(3 GeV) = 0.992(5) GeV Refs. [11,24–26] , (57) √ where the error on mc(mc) includes a stretching factor χ2/dof 1.16 as discussed in Sect. 2.2. This result corresponds to the following RGI average MRGIc = 1.526(9)m(14) GeV Refs. [11,24–26]. (58) 3.2.2 N f = 2 + 1 + 1 results For a discussion of older results, see the previous FLAG reviews. Since FLAG 19 two groups have produced updated values with charm quarks in the sea. HPQCD 20A [23] is an update of HPQCD 18, including a new finer ensemble (a ≈ 0.045 fm) and EM corrections computed in the quenched approximation of QED for the first time. Besides these new items, the analysis is largely unchanged from 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume Renormalization Eur. Phys. J. C (2022) 82:869 Page 41 of 296 869 HPQCD 18 except for an added α3s correction to the SMOM-to-MS conversion factor and tuning the bare charm mass via the J/ψ mass rather than the ηc. Their new value in pure QCD is mc(3 GeV) = 0.9858(51) GeV which is quite consistent with HPQCD 18 and the FLAG 19 average. The effects of quenched QED in both the bare charm-quark mass and the renormalization constant are small. Both effects are precisely determined, and the overall effect shifts the mass down slightly to mc(3 GeV) = 0.9841(51) where the uncertainty due to QED is invisible in the final error. The shift from their pure QCD value due to quenched QED is about −0.2%. ETM 21A [204] is a new work that follows a similar methodology as ETM 14, but with significant improvements. Notably, a clover-term is added to the twisted mass fermion action which suppressesO(a2) effects between the neutral and charged pions. Additional improvements include new ensembles lying very close to the physical mass point, better control of nonperturbative renormalization systematics, and use of both meson and baryon correlation functions to determine the quark mass. They use the RI-MOM scheme for nonperturbative renormalization. The analysis comprises ten ensembles in total with three lattice spacings (0.095, 0.082, and 0.069 fm), two volumes for the finest lattice spacings and four for the other two, and pion masses down to 134 MeV for the finest ensemble. The values of mπ L range mostly from almost four to greater than five. According to the FLAG criteria, green stars are earned in all categories. The authors find mc(3 GeV) = 1.036(17)(+15−8 ) GeV. In Table 12 we have applied a factor of 0.7739(60) to run from 3 GeV to mc. As in FLAG 19, the new value is consistent with ETM 14 and ETM 14A, but is still high compared to the FLAG average. The authors plan future improvements, including a finer lattice spacing for better control of the continuum limit and a new renormalization scheme, like RI-SMOM. This result has not been published by the deadline, so it does not yet appear in the average. Five results enter the FLAG average for N f = 2 + 1 + 1 quark flavours: ETM 14, ETM 14A, HPQCD 14A, FNAL/MILC/TUMQCD 18, and HPQCD 20A. We note that while the determinations of mc by ETM 14 and 14A agree well with each other, they are incompatible with HPQCD 14A, FNAL/MILC/TUMQCD 18, and HPQCD 20A by several standard deviations. While the latter use the same configurations, the analyses are quite different and independent. As men- tioned earlier, mud and ms values by ETM are also systematically high compared to their respective averages. Combining all four results yields mc(mc) = 1.278(13) GeV Refs. [6,7,14,22,23] , (59) N f = 2 + 1 + 1 : mc(3 GeV) = 0.988(11) GeV Refs. [6,7,14,22,23] , (60) √ where the errors include large stretching factors χ2/dof ≈ 2.0 and 2.5, respectively. We have assumed 100% correlation for statistical errors between ETM results and the same for HPQCD 14A, HPQCD 20A, and FNAL/MILC/TUMQCD 18. These are obviously poor χ2 values, and the stretching factors are quite large. While it may be prudent in such a case to quote a range of values covering the central values of all results that pass the quality criteria, we believe in this case that would obscure rather than clarify the situation. From Fig. 5 we note that not only do ETM 21A, ETM 14A, and ETM 14 lie well above the other 2+1+1 results, but also above all of the 2+1 flavour results. A similar trend is apparent for the light-quark masses (see Figs. 1 and 2) while for mass ratios there is better agreement (Figs. 3, 4 and 6). The latter suggests there may be underestimated systematic uncertainties associated with scale setting and/or renormalization which have not been detected. Finally we note the ETM results are significantly higher than the PDG average. For these reasons, which admittedly are not entirely satisfactory, we continue to quote an average with a stretching factor as in previous reviews. The RGI average reads as follows, MRGIc = 1.520(17)m(14) GeV Refs. [6,7,14,22,23]. (61) Figure 5 presents the values of mc(mc) given in Table 12 along with the FLAG averages obtained for 2 + 1 and 2 + 1 + 1 flavours. 3.2.3 Lattice determinations of the ratio mc/ms Because some of the results for quark masses given in this review are obtained via the quark-mass ratio mc/ms , we review these lattice calculations, which are listed in Table 13, as well. The N f = 2 + 1 results from χQCD 14 and HPQCD 09A [27] are from the same calculations that were described for the charm-quark mass in the previous review. Maezawa 16 does not pass our chiral-limit test (see the previous review), though we note that it is quite consistent with the other values. Combining χQCD 14 and HPQCD 09A, we obtain the same result 123 869 Page 42 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 4 Lattice results and FLAG averages at N f = 2 + 1 and 2 + 1 + 1 for the up-down-quark masses ratio mu/md , together with the current PDG estimate Fig. 5 The charm-quark mass for 2 + 1 and 2 + 1 + 1 flavours. For the latter a large stretching factor is used for the FLAG average due to poor χ2 from our fit reported in FLAG 19, N f = 2 + 1 : mc/ms = 11.82(16) Refs. [24,27], (62) with a χ2/dof 0.85. Turning to N f = 2 + 1 + 1, there is a new result from ETM 21A. The errors have actually increased compared to ETM 14, due to larger uncertainties in the baryon sector which enter their average with the meson sector. Again, ETM 21A does not yet enter the average since it was not published by the deadline for the review. See the earlier reviews for a discussion of previous results. We note that some tension exists between the HPQCD 14A and FNAL/MILC/TUMQCD results. Combining these with ETM 14 yields N f = 2 + 1 + 1 : mc/ms = 11.768(34) Refs. [6,7,14], (63) √ where the error includes the stretching factor χ2/dof 1.5. We have assumed a 100% correlation of statistical errors for FNAL/MILC/TUMQCD 18 and HPQCD 14A. Results for mc/ms are shown in Fig. 6 together with the FLAG averages for N f = 2 + 1 and 2 + 1 + 1 flavours. 123 Eur. Phys. J. C (2022) 82:869 Page 43 of 296 869 Table 13 Lattice results for the quark-mass ratio mc/ms , together with the colour coding of the calculations used to obtain them Collaboration Refs. N f mc/ms ETM 21A [204] 2+1+1 P    11.48(12)(+25−19) FNAL/MILC/TUMQCD 18 [6] 2+1+1 A    11.784(11)(17)(00)(08) HPQCD 14A [14] 2+1+1 A    11.652(35)(55) FNAL/MILC 14A [17] 2+1+1 A    11.747(19)(+59−43) ETM 14 [7] 2+1+1 A ◦  ◦ 11.62(16) Maezawa 16 [189] 2+1 A    11.877(91) χQCD 14 [24] 2+1 A ◦ ◦ ◦ 11.1(8) HPQCD 09A [27] 2+1 A ◦   11.85(16) Fig. 6 Lattice results for the ratio mc/ms listed in Table 13 and the FLAG averages corresponding to 2 + 1 and 2 + 1 + 1 quark flavours. The latter average includes a large stretching factor on the error due a poor χ2 from our fit 3.3 Bottom-quark mass Now we review the lattice results for the MS bottom-quark mass mb. Related heavy-quark actions and observables have been discussed in previous FLAG reviews [2–4], and descriptions can be found in Sec. A.1.3 of FLAG 19 [4]. In Table 14 we collect results for mb(mb) obtained with N f = 2 + 1 and 2 + 1 + 1 sea-quark flavours. Available results for the quark-mass ratio mb/mc are also reported. After discussing the new results we evaluate the corresponding FLAG averages. 3.3.1 N f = 2 + 1 There is one new three-flavour result since the last review, Petreczky 19, which was described already in the charm-quark section. The new result rates green stars, so our new average with HPQCD 10 is (both works quote values in the N f = 5 theory, so we simply use those values), N f = 2 + 1 : mb(mb) = 4.171(20) GeV Refs. [11,26]. (64) 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume 869 Page 44 of 296 Eur. Phys. J. C (2022) 82:869 Table 14 Lattice results for the MS bottom-quark mass mb(mb) in GeV, together with the systematic error ratings for each. Available results for the quark-mass ratio mb/mc are also reported Collaboration Refs. N f mb(mb) mb/mc HPQCD 21 [28] 2+1+1 A    −  4.209(21)++ 4.586(12)∗∗ FNAL/MILC/TUM 18 [6] 2+1+1 A  ◦  −  4.201(12)(1)(8)(1) 4.578(5)(6)(0)(1) Gambino 17 [31] 2+1+1 A ◦  ◦   4.26(18) ETM 16B [30] 2+1+1 A ◦  ◦   4.26(3)(10)+ 4.42(3)(8) HPQCD 14B [29] 2+1+1 A      4.196(0)(23)† ETM 14B [228] 2+1+1 C ◦  ◦   4.26(7)(14) 4.40(6)(5) HPQCD 14A [14] 2+1+1 A    −  4.162(48) 4.528(14)(52) Petreczky19 [26] 2+1 A      4.188(37) 4.586(43) Maezawa 16 [189] 2+1 A      4.184(89) 4.528(57) HPQCD 13B [229] 2+1 A  ◦ − −  4.166(43) HPQCD 10 [11] 2+1 A    −  4.164(23) 4.51(4) ETM 13B [56] 2 A ◦  ◦   4.31(9)(8) ALPHA 13C [230] 2 A      4.21(11) ETM 11A [231] 2 A ◦  ◦   4.29(14) PDG [165] 4.18+0.02−0.03 ++We quote the four-flavour result. For N f = 5, value is 4.202(21) ∗∗The ratio is quoted in the MS scheme for μ = 3 GeV because of the different charges of the bottom and charm quarks +The lattice spacing used in ETM 14B has been updated here †Only two pion points are used for chiral extrapolation The number that is given is mb(10 GeV, N f = 5) = 3.617(25) GeV The corresponding four-flavour RGI average is N f = 2 + 1 : MRGIb = 6.881(33)m(54) GeV Refs. [11,26]. (65) 3.3.2 N f = 2 + 1 + 1 HPQCD 21 [28] is an update of HPQCD 14A (and replaces it in our average), including EM corrections for the first time for the b-quark mass. Four flavours of HISQ quarks are used on MILC ensembles with lattice spacings from about 0.09–0.03 fm. Ensembles with physical and unphysical mass sea-quarks are used. Quenched QED is used to obtain the dominant O(α) effect. The ratio of bottom- to charm-quark masses is computed in a completely nonperturbative formulation, and the b-quark mass is extracted using the value of mc(3 GeV) from HPQCD 20A. Since EM effects are included, the QED renormalization scale enters the ratio which is quoted for 3 GeV and N f = 4. The total error on the new result is more than two times smaller than for HPQCD 14A, but is only slightly smaller compared to the NRQCD result reported in HPQCD 14B. The inclusion of QED shifts the ratio mb/mc up slightly from the pure QCD value by about one standard deviation, and the value of mb(mb) is consistent, within errors, to the other pure QCD results entering our average. Therefore we quote a single average. HPQCD 14B employs the NRQCD action [29] to treat the b quark. The b-quark mass is computed with the moments method, that is, from Euclidean-time moments of two-point, heavy-heavy-meson correlation functions (see also Sect. 9.8 for a description of the method). 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume Renormalization Heavy-quark treatment Eur. Phys. J. C (2022) 82:869 Page 45 of 296 869 In HPQCD 14B the b-quark mass is computed from ratios of the moments Rn of heavy current-current correlation functions, namely, [ ] Rnrn− 1/22 M̄kin = M̄ϒ,ηb , (66) Rn−2rn 2mb 2m̄b(μ) where rn are the perturbative moments calculated at N3LO, M̄kin is the spin-averaged kinetic mass of the heavy-heavy vector and pseudoscalar mesons and M̄ϒ,ηb is the experimental spin average of the ϒ and ηb masses. The average kinetic mass M̄kin is chosen since in the lattice calculation the splitting of the ϒ and ηb states is inverted. In Eq. (66), the bare mass mb appearing on the left-hand side is tuned so that the spin-averaged mass agrees with experiment, while the mass mb at the fixed scale μ = 4.18 GeV is extrapolated to the continuum limit using three HISQ (MILC) ensembles with a ≈ 0.15, 0.12 and 0.09 fm and two pion masses, one of which is the physical one. Their final result is mb(μ = 4.18 GeV) = 4.207(26) GeV, where the error is from adding systematic uncertainties in quadrature only (statistical errors are smaller than 0.1% and ignored). The errors arise from renormalization, perturbation theory, lattice spacing, and NRQCD systematics. The finite-volume uncertainty is not estimated, but at the lowest pion mass they have mπ L 4, which leads to the tag . The next four-flavour result [30] is from the ETM collaboration and updates their preliminary result appearing in a conference proceedings [228]. The calculation is performed on a set of configurations generated with twisted-Wilson fermions with three lattice spacings in the range 0.06 – 0.09 fm and with pion masses in the range 210–440 MeV. The b-quark mass is determined from a ratio of heavy-light pseudoscalar meson masses designed to yield the quark pole mass in the static limit. The pole mass is related to the MS mass through perturbation theory at N3LO. The key idea is that by taking ratios of ratios, the b-quark mass is accessible through fits to heavy-light(strange)-meson correlation functions computed on the lattice in the range ∼ 1–2 × mc and the static limit, the latter being exactly 1. By simulating below mb, taking the continuum limit is easier. They find mb(mb) = 4.26(3)(10) GeV, where the first error is statistical and the second systematic. The dominant errors come from setting the lattice scale and fit systematics. Gambino et al. [31] use twisted-mass-fermion ensembles from the ETM collaboration and the ETM ratio method as in ETM 16. Three values of the lattice spacing are used, ranging from 0.062 to 0.089 fm. Several volumes are also used. The light-quark masses produce pions with masses from 210 to 450 MeV. The main difference with ETM 16 is that the authors use the kinetic mass defined in the heavy-quark expansion (HQE) to extract the b-quark mass instead of the pole mass. The final b-quark mass result is FNAL/MILC/TUM 18 [6]. The mass is extracted from the same fit and analysis done for the charm quark mass. Note that relativistic HISQ valence masses reach the physical b mass on the two finest lattice spacings (a = 0.042 fm, 0.03 fm) at physical and 0.2 ms light-quark mass, respectively. In lattice units the heavy valence masses correspond to aMRGI > 0.90, making the continuum extrapolation challenging, but the authors investigated the effect of leaving out the heaviest points from the fit, and the result did not noticeably change. Their results are also consistent with an analysis dropping the finest lattice from the fit. Since the b-quark mass region is only reached with two lattice spacings, we rate this work with a green circle for the continuum extrapolation. Note however that for other values of the quark masses they use up to five values of the lattice spacing (cf. their charm-quark mass determination). All of the above results enter our average. We note that here the ETM 16 result is consistent with the average and a stretching factor on the error is not used. The average and error is dominated by the very precise FNAL/MILC/TUM 18 value, N f = 2 + 1 + 1 : mb(mb) = 4.203(11) GeV Refs. [6,14,28–31]. (67) We have included a 100% correlation on the statistical errors of ETM 16 and Gambino 17, since the same ensembles are used in both. While FNAL/MILC/TUM 18 and HPQCD 21 also use the same MILC HISQ ensembles, the statistical error in the HPQCD 21 analysis is negligible, so we do not include a correlation between them. The average has χ2/dof = 0.02. The above translates to the RGI average N = 2 + 1 + 1 : MRGIf b = 6.934(18)m(55) GeV Refs. [6,14,28–31]. (68) All the results for mb(mb) discussed above are shown in Fig. 7 together with the FLAG averages corresponding to N f = 2 + 1 and 2 + 1 + 1 quark flavours. 123 869 Page 46 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 7 The b-quark mass for N f = 2 + 1 and 2 + 1 + 1 flavours. The updated PDG value from Ref. [165] is reported for comparison 4 Leptonic and semileptonic kaon and pion decay and |Vud | and |Vus| Authors: T. Kaneko, J. N. Simone, S. Simula, N. Tantalo This section summarizes state-of-the-art lattice calculations of the leptonic kaon and pion decay constants and the kaon semileptonic-decay form factor and provides an analysis in view of the Standard Model. With respect to the previous edition of the FLAG review [4] the data in this section has been updated. As in Ref. [4], when combining lattice data with experimental results, we take into account the strong SU (2) isospin correction, either obtained in lattice calculations or estimated by using chiral perturbation theory (χPT), both for the kaon leptonic decay constant fK± and for the ratio fK±/ fπ± . 4.1 Experimental information concerning |Vud |, |Vus |, f+(0) and fK±/ fπ± The following review relies on the fact that precision experimental data on kaon decays very accurately determine the product |Vus | f+(0) [232] and the ratio |Vus/Vud | fK±/ fπ± [165,232]: ∣∣ ∣| V ∣ f ±Vus | f+(0 us K) = 0.2165(4) , ∣∣ ∣∣ = 0.2760(4) . (69)Vud fπ± Here and in the following, fK± and fπ± are the isospin-broken decay constants, respectively, in QCD. We will refer to the decay constants in the SU (2) isospin-symmetric limit as fK and fπ (the latter at leading order in the mass difference (mu − md ) coincides with fπ± ). The parameters |Vud | and |Vus | are elements of the Cabibbo–Kobayashi–Maskawa matrix and f+(q2) represents one of the form factors relevant for the semileptonic decay K 0 → π− ν, which depends on the 0 − 0 − momentum transfer q between the two mesons. What matters here is the value at q2 = 0: f+(0) ≡ f+K π (0) = f K π0 (0) = qμ〈π−(p′)|s̄γμu|K 0(p)〉/(M2K − M2π ) 2→ . The pion and kaon decay constants are defined by12q 0 〈0| dγμγ5 u|π+(p)〉 = i pμ fπ+ , 〈0| sγμγ +5 u|K (p)〉 = i pμ fK+ . In this normalization, fπ± 130 MeV, fK± 155 MeV. 12 The pion decay constant represents a QCD matrix element – in the full Standard Model, the one-pion state is not a meaningful notion: the correlation function of the charged axial current does not have a pole at p2 = M2+ , but a branch cut extending from M2+ to ∞. The analyticπ π properties of the correlation function and the problems encountered in the determination of fπ are thoroughly discussed in Ref. [233]. The “experimental” value of fπ depends on the convention used when splitting the sum LQCD + LQED into two parts. The lattice determinations of fπ do not yet reach the accuracy where this is of significance, but at the precision claimed by the Particle Data Group [201,234], the numerical value does depend on the convention used [233,235–237]. 123 Eur. Phys. J. C (2022) 82:869 Page 47 of 296 869 In Eq. (69), the electromagnetic effects have already been subtracted in the experimental analysis using χPT. Recently, a new method [238] has been proposed for calculating the leptonic decay rates of hadrons including both QCD and QED on the lattice, and successfully applied to the case of the ratio of the leptonic decay rates of kaons and pions [239,240]. The correction to the tree-level Kμ2/πμ2 decay rate, including both electromagnetic and strong isospin-breaking effects, is found to be equal to −1.26(14)%13 to be compared to the estimate −1.12(21)% based on χPT [164,241]. Using the experimental values of the Kμ2 and πμ2 decay rates the result of Ref. [240] implies ∣∣ ∣∣ Vus ∣∣ fK∣ ∣ = 0.27683 (29)exp (20)th [35] , (70)Vud fπ where the last error in brackets is the sum in quadrature of the experimental and theoretical uncertainties, and the ratio of the decay constants is the one corresponding to isosymmetric QCD. A large part of the theoretical uncertainty comes from the statistics and continuum and chiral extrapolation of lattice data, which can be systematically reduced by a more realistic simulation with high statistics. We also note that an independent study of the electromagnetic effects is in progress [242]. Therefore, it is feasible to more precisely determine |Vus/Vud | using only lattice-QCD+QED for fK±/ fπ± and the ratio of the experimental values of the Kμ2 and πμ2 decay rates. At present, the superallowed nuclear β transitions provide the most precise determination of |Vud |. Its accuracy has been limited by hadronic uncertainties in the universal electroweak radiative correction VR . A recent analysis in terms of a dispersion relation [243,244] found VR larger than the previous estimate [245]. A more straightforward update of Ref. [245] also reported larger VR [246]. In the PDG review, the fourteen precisely measured transitions [247] with the dispersive estimate of VR yield [165] |Vud | = 0.97370(14), (71) which differs by ≈ 3σ from the previous estimate [247]. However, it is not a trivial matter to properly take account of the nuclear corrections at this precision [243,247–255]. For example, the dispersive approach has been applied in a recent update of the so-called inner radiative correction due to quenching of the axial-vector and isoscalar spin-magnetic-moment couplings in nuclei [243], and in a recent estimate of a novel correction due to the distortion of the emitted electron energy spectrum by nuclear polarizabilities [255]. A recent reanalysis of twenty-three β decays [256] obtained |Vud | = 0.97373(31), (72) where the two nuclear corrections tend to cancel with each other and, hence, leave the central value basically unchanged. Their uncertainties, however, doubles that of |Vud |. In Sects. 4.4 and 4.5, we mainly use the PDG value (71) but also test Eq. (72) as an alternative input. The matrix element |Vus | can be determined from semi-inclusive τ decays [257–260]. By separating the inclusive decay τ → hadrons + ν into nonstrange and strange final states, e.g., HFLAV 18 [261] obtains |Vus | = 0.2195(19), (73) and both Maltman et al. [259,262,263] and Gamiz et al. [264,265] arrived at very similar values. Inclusive hadronic τ decay offers an interesting way to measure |Vus |, but the above value of |Vus | differs from the result one obtains from the kaon decays by about three standard deviations (see Sect. 4.5). This apparent tension has been recently solved in Ref. [266] thanks to the use of a different experimental input and to a new treatment of higher orders in the operator product expansion and of violations of quark-hadron duality. A larger value of |Vus | is obtained, namely, |Vus | = 0.2231(27), which is in much better agreement with the results from the kaon decays. This result is also stable against the choice of the upper limit and weight function of the experimental spectral integrals.14 Recently, Ref. [268] proposed a new method to determine |Vus | from inclusive strange τ decays. Through generalized dispersion relations, this method evaluates the spectral integral from lattice-QCD data of the hadronic vacuum polarization 13 This has been updated in Ref. [240] after the previous edition of this review. See also the extended discussion concerning the isospin correction in Sect. 11 on the scale setting. 14 A recent update can be found in Ref. [267]. 123 869 Page 48 of 296 Eur. Phys. J. C (2022) 82:869 function at Euclidean momentum squared in the few-to-several 0.1 GeV2 region. This method, therefore, does not rely on the operator product expansion, and obtained |Vus | consistent with that from the kaon decays. A later analysis yields [267] |Vus | = 0.2240(18), (74) by taking account of updates on experimental strange τ branching fractions in 2018. We quote Eqs. (73) and (74) as |Vus | from the inclusive hadronic τ decays in Sect. 4.5. The experimental results in Eq. (69) are for the semileptonic decay of a neutral kaon into a negatively charged pion and the charged pion and kaon leptonic decays, respectively, in QCD. In the case of the semileptonic decays the corrections for strong and electromagnetic isospin breaking in χPT at NLO have allowed for averaging the different experimentally measured isospin channels [269]. This is quite a convenient procedure as long as lattice-QCD simulations do not include strong or QED isospin-breaking effe(cts. Several lattice results for )fK / fπ are quoted for QCD with (squared) pion and kaon masses of M2 = M2 2 1 2π 0 and MK = 2 MK± + M2 0 − M2± + M20 for which the leading strong and electromagnetic isospinπ K π π violations cancel. For these results, contact with experimental results is made by correcting leading SU (2) isospin breaking guided either by χPT or by lattice calculations. We note, however, that the modern trend for the leptonic decays is to include strong and electromagnetic isospin breaking in the lattice simulations (e.g., Refs. [171,172,193,217,238,239,270–272]). After the previous edition, this trend has been extended to the semileptonic decays. Reference [273] discusses an extension of the method in Refs. [239,240], which led to Eq. (70), for the semileptonic decays. References [274–276] pursue an effective field theory setup supplemented by nonperturbative lattice-QCD inputs to estimate the radiative corrections. 4.2 Lattice results for f+(0) and fK±/ fπ± The traditional way of determining |Vus | relies on using estimates for the value of f+(0), invoking the Ademollo–Gatto theorem [277]. Since this theorem only holds to leading order of the expansion in powers of mu , md , and ms , theoretical models are used to estimate the corrections. Lattice methods have now reached the stage where quantities like f+(0) or fK / fπ can be determined to good accuracy. As a consequence, the uncertainties inherent in the theoretical estimates for the higher order effects in the value of f+(0) do not represent a limiting factor any more and we shall therefore not invoke those estimates. Also, we will use the experimental results based on nuclear β decay and inclusive hadronic τ decay exclusively for comparison – the main aim of the present review is to assess the information gathered with lattice methods and to use it for testing the consistency of the SM and its potential to provide constraints for its extensions. The database underlying the present review of the semileptonic form factor and the ratio of decay constants is listed in Tables 15 and 16. The properties of the lattice data play a crucial role for the conclusions to be drawn from these results: range of Mπ , size of L Mπ , continuum extrapolation, extrapolation in the quark masses, finite-size effects, etc. The key features of the various data sets are characterized by means of the colour code specified in Sect. 2.1. More detailed information on individual computations are compiled in Appendix C.2, which in this edition is limited to new results and to those entering the FLAG averages. For other calculations the reader should refer to the Appendix B.2 of Ref. [3]. The quantity f+(0) represents a matrix element of a strangeness-changing null-plane charge, f+(0) = 〈K |Qūs |π〉 (see Ref. [278]). The vector charges obey the c∑ommutation relation∑s of the Lie algebra of SU (3), in particular [Qūs, Qs̄u] = Qūu−s̄s . This relation implies the sum rule |〈K |Qūs |n〉|2 − |〈K |Qs̄un n |n〉|2 = 1. Since the contribution from the one- pion intermediate state to the first sum is given by f+(0)2, the relation amounts to an exact representation for this quantity [279]: ∑ ∑ f+(0)2 = 1 − |〈K |Qūs |n〉|2 + |〈K |Qs̄u |n〉|2 . (75) n =π n While the first sum on the right extends over nonstrange intermediate states, the second runs over exotic states with strangeness ±2 and is expected to be small compared to the first. The expansion of f+(0) in SU (3) χPT in powers of mu , md , and ms starts with f+(0) = 1+ f2 + f4 +· · · [280]. Since all of the low-energy constants occurring in f2 can be expressed in terms of Mπ , MK , Mη and fπ [278], the NLO correction is known. In the language of the sum rule (75), f2 stems from nonstrange intermediate states with three mesons. Like all other nonexotic intermediate states, it lowers the value of f+(0): f2 = −0.023 when using the experimental value of fπ as input. The corresponding expressions have also been derived in quenched or partially quenched (staggered) χPT [34,281]. At the same order in the SU (2) expansion [282], f+(0) is parameterized in terms of Mπ and two a priori unknown parameters. The latter can be determined from the dependence of the lattice results on the masses of the quarks. Note that any calculation that 123 Eur. Phys. J. C (2022) 82:869 Page 49 of 296 869 Table 15 Colour code for the data on f+(0). In this and previous editions [4], old results with two red tags have been dropped Collaboration Refs. N f f+(0) FNAL/MILC 18 [33] 2+1+1 A    0.9696(15)(12) ETM 16 [32] 2+1+1 A ◦  ◦ 0.9709(45)(9) FNAL/MILC 13E [293] 2+1+1 A    0.9704(24)(22) PACS 19 [294] 2+1 A ◦   0.9603(16)(+50−48) JLQCD 17 [289] 2+1 A ◦  ◦ 0.9636(36)(+57−35) RBC/UKQCD 15A [35] 2+1 A  ◦ ◦ 0.9685(34)(14) RBC/UKQCD 13 [295] 2+1 A  ◦ ◦ 0.9670(20)(+18−46) FNAL/MILC 12I [34] 2+1 A ◦ ◦  0.9667(23)(33) JLQCD 12 [296] 2+1 C ◦   0.959(6)(5) JLQCD 11 [297] 2+1 C ◦   0.964(6) RBC/UKQCD 10 [298] 2+1 A ◦   0.9599(34)(+31−47)(14) RBC/UKQCD 07 [299] 2+1 A ◦   0.9644(33)(34)(14) ETM 10D [300] 2 C ◦  ◦ 0.9544(68)stat ETM 09A [36] 2 A ◦ ◦ ◦ 0.9560(57)(62) relies on the χPT formula for f2 is subject to the uncertainties inherent in NLO results: instead of using the physical value of the pion decay constant fπ , one may, for instance, work with the constant f0 that occurs in the effective Lagrangian and represents the value of fπ in the chiral limit. Although trading fπ for f0 in the expression for the NLO term affects the result only at NNLO, it may make a significant numerical difference in calculations where the latter are not explicitly accounted for. (Lattice results concerning the value of the ratio fπ/ f0 are reviewed in Sect. 5.3.) The lattice results shown in Fig. 8 indicate that the higher order contributions  f ≡ f+(0) − 1 − f2 are negative and thus amplify the effect generated by f2. This confirms the expectation that the exotic contributions are small. The entries in the lower part of the left panel represent various model estimates for f4. In Ref. [283], the symmetry-breaking effects are estimated in the framework of the quark model. The more recent calculations are more sophisticated, as they make use of the known explicit expression for the K3 form factors to NNLO in χPT [284,285]. The corresponding formula for f4 accounts for the chiral logarithms occurring at NNLO and is not subject to the ambiguity mentioned above.15 The numerical result, however, depends on the model used to estimate the low-energy constants occurring in f4 [285–288]. The figure indicates that the most recent numbers obtained in this way correspond to a positive or an almost vanishing rather than a negative value for  f . We note that FNAL/MILC 12I [34], JLQCD 17 [289], FNAL/MILC 18 [33], and Ref. [290] have made an attempt at determining a combination of some of the low-energy constants appearing in f4 from lattice data. 4.3 Direct determination of f+(0) and fK±/ fπ± Many lattice results for the form factor f+(0) and for the ratio of decay constants, which we summarize here in Tables 15 and 16, respectively, have been computed in isospin-symmetric QCD. The reason for this unphysical parameter choice is that there are only a few simulations of isospin-breaking effects in lattice QCD, which is ultimately the cleanest way for predicting these effects [170–172,178,217,238,239,272,291,292]. In the meantime, one relies either on χPT [197,280] to estimate the correction to the isospin limit or one calculates the breaking at leading order in (mu − md) in the valence quark sector by extrapolating the lattice data for the charged kaons to the physical value of the up(down)-quark mass (the result for the 15 Fortran programs for the numerical evaluation of the form factor representation in Ref. [285] are available on request from Johan Bijnens. 123 Publication status Chiral extrapolation Continuum extrapolation Finite-volume errors 869 Page 50 of 296 Eur. Phys. J. C (2022) 82:869 Table 16 Colour code for the data on the ratio of decay constants: fK / fπ is the pure QCD SU (2)-symmetric ratio, while fK±/ fπ± is in pure QCD including the SU (2) isospin-breaking correction. In this and previous editions [4], old results with two red tags have been dropped Collaboration Refs. N f fK / fπ fK±/ fπ± ETM 21 [305] 2+1+1 P    1.1995(44)(7) 1.1957(44)(7) CalLat 20 [39] 2+1+1 A    1.1964(32)(30) 1.1942(32)(31) FNAL/MILC 17 [16] 2+1+1 A    1.1980(12)(+5−15) 1.1950(15)( +6 ◦ ◦ −18 ) ETM 14E [38] 2+1+1 A  1.188(11)(11) 1.184(12)(11) FNAL/MILC 14A [17] 2+1+1 A    1.1956(10)(+26−18) ETM 13F [306] 2+1+1 C ◦  ◦ 1.193(13)(10) 1.183(14)(10) HPQCD 13A [37] 2+1+1 A  ◦  1.1948(15)(18) 1.1916(15)(16) MILC 13A [307] 2+1+1 A    1.1947(26)(37) MILC 11 [308] 2+1+1 C ◦ ◦ ◦ 1.1872(42)†◦ ◦ ◦ stat.ETM 10E [309] 2+1+1 C 1.224(13)stat QCDSF/UKQCD 16 [44] 2+1 A ◦  ◦ 1.192(10)(13) 1.190(10)(13) BMW 16 [43,310] 2+1 A    1.182(10)(26) 1.178(10)(26) RBC/UKQCD 14B [8] 2+1 A    1.1945(45) RBC/UKQCD 12 [188] 2+1 A  ◦  1.199(12)(14) Laiho 11 [49] 2+1 C ◦  ◦ 1.202(11)(9)(2)(5)†† MILC 10 [41] 2+1 C ◦   1.197(2)(+3−7) JLQCD/TWQCD 10 [311] 2+1 C ◦   1.230(19) RBC/UKQCD 10A [117] 2+1 A ◦ ◦  1.204(7)(25) BMW 10 [42] 2+1 A    1.192(7)(6) MILC 09A [15] 2+1 C ◦   1.198(2)(+6) MILC 09 [157] 2+1 A ◦ −8  1.197(3)( +6 ) Aubin 08 [312] 2+1 C ◦ ◦ ◦ −131.191(16)(17) RBC/UKQCD 08 [194] 2+1 A ◦   1.205(18)(62) HPQCD/UKQCD 07 [40] 2+1 A ◦ ◦ 1.189(2)(7) MILC 04 [197] 2+1 A ◦ ◦ ◦◦ 1.210(4)(13) ETM 14D [313] 2 C   ◦ 1.203(5)stat ALPHA 13A [314] 2 C    1.1874(57)(30) ETM 10D [300] 2 C ◦  ◦ 1.190(8)stat ETM 09 [45] 2 A ◦  ◦ 1.210(6)(15)(9) QCDSF/UKQCD 07 [315] 2 C ◦ ◦  1.21(3) †Result with statistical error only from polynomial interpolation to the physical point ††This work is the continuation of Aubin 08 pion decay constant is always extrapolated to the value of the average light-quark mass m̂). This defines the prediction for fK±/ fπ± . Since the majority of results that qualify for inclusion into the FLAG average include the strong SU (2) isospin-breaking correction, we confirm the choice made in the previous edition of the FLAG review [4] and we provide in Fig. 9 the overview of the world data of fK±/ fπ± . For all the results of Table 16 provided only in the isospin-symmetric limit we apply individually an isospin correction that will be described later on (see Eqs. (79)–(80)). 123 Publication status Chiral extrapolation Continuum extrapolation Finite-volume errors Eur. Phys. J. C (2022) 82:869 Page 51 of 296 869 Fig. 8 Comparison of lattice results (squares) for f+(0) with various model estimates based on χPT [283,285–288] (blue circles). The black squares and grey bands indicate our averages (76)–(78). The significance of the colours is explained in Sect. 2 Fig. 9 Comparison of lattice results for fK±/ fπ± . This ratio is obtained in pure QCD including the SU (2) isospin-breaking correction (see Sect. 4.3). The black squares and grey bands indicate our averages in Eqs. (81)–(83) The plots in Figs. 8 and 9 illustrate our compilation of data for f+(0) and fK±/ fπ± . The lattice data for the latter quantity is largely consistent even when comparing simulations with different N f , while in the case of f+(0) a slight tendency to get higher values when increasing N f seems to be visible, even if it does not exceed one standard deviation. We now proceed to form the corresponding averages, separately for the data with N f = 2 + 1 + 1, N f = 2 + 1, and N f = 2 dynamical flavours, and in the following we will refer to these averages as the “direct” determinations. 4.3.1 Results for f+(0) For f+(0) there are currently two computational strategies: FNAL/MILC uses the Ward identity to relate the K → π form factor at zero momentum transfer to the matrix element 〈π |S|K 〉 of the flavour-changing scalar current S = s̄u. 123 869 Page 52 of 296 Eur. Phys. J. C (2022) 82:869 Peculiarities of the staggered fermion discretization used by FNAL/MILC (see Ref. [34]) makes this the favoured choice. The other collaborations are instead computing the vector current matrix element 〈π |s̄γμu|K 〉. Apart from FNAL/MILC 13E, RBC/UKQCD 15A, and FNAL/MILC 18, all simulations in Table 15 involve unphysically heavy quarks and, therefore, the lattice data needs to be extrapolated to the physical pion and kaon masses corresponding to the K 0 → π− channel. We note also that the recent computations of f+(0) obtained by the FNAL/MILC and RBC/UKQCD collaborations make use of the partially-twisted boundary conditions to determine the form-factor results directly at the relevant kinematical point q2 = 0 [301,302], avoiding in this way any uncertainty due to the momentum dependence of the vector and/or scalar form factors. The ETM collaboration uses partially-twisted boundary conditions to compare the momentum dependence of the scalar and vector form factors with the one of the experimental data [32,300], while keeping at the same time the advantage of the high-precision determination of the scalar form factor at the kinematical end-point q2 2max = (MK − Mπ ) [36,303] for the interpolation at q2 = 0. According to the colour codes reported in Table 15 and to the FLAG rules of Sect. 2.2, only the result ETM 09A with N f = 2, the results FNAL/MILC 12I and RBC/UKQCD 15A with N f = 2+1, and the results ETM 16 and FNAL/MILC 18 with N f = 2+ 1+ 1 dynamical flavours of fermions, respectively, can enter the FLAG averages. We note that the new entry in this edition is FNAL/MILC 18 for N f = 2 + 1 + 1, which did not enter the previous FLAG average due to its publication status [4]. At N f = 2+1+1 the result from the FNAL/MILC collaboration, f+(0) = 0.9704(24)(22) (FNAL/MILC 13E), is based on the use of the Highly Improved Staggered Quark (HISQ) action (for both valence and sea quarks), which has been tailored to reduce staggered taste-breaking effects, and includes simulations with three lattice spacings and physical light-quark masses. These features allow to keep the uncertainties due to the chiral extrapolation and to the discretization artifacts well below the statistical error. The remaining largest systematic uncertainty comes from finite-size effects, which have been investigated in Ref. [304] using one-loop χPT (with and without taste-violating effects). In Ref. [33], the FNAL/MILC collaboration presented a more precise determination of f+(0), f+(0) = 0.9696(15)(11) (FNAL/MILC 18). In this update, their analysis is extended to two smaller lattice spacings a = 0.06 and 0.042 fm. The physical light-quark mass is simulated at four lattice spacings. They also added a simulation at a small volume to study the finite-size effects. The improvement of the precision with respect to FNAL/MILC 13E is obtained mainly by an estimate of finite-size effects, which is claimed to be controlled at the level of ∼ 0.05 % by comparing two analyses with and without the one-loop correction. The total uncertainty is largely reduced to ∼ 0.2 %. An independent calculation of such high precision would be highly welcome to solidify the lattice prediction of f+(0), which currently suggests a tension with CKM unitarity with the updated value of |Vud | (see Sect. 4.4). The result from the ETM collaboration, f+(0) = 0.9709(45)(9) (ETM 16), makes use of the twisted-mass discretization adopting three values of the lattice spacing in the range 0.06–0.09 fm and pion masses simulated in the range 210–450 MeV. The chiral and continuum extrapolations are performed in a combined fit together with the momentum dependence, using both a SU (2)-χPT inspired ansatz (following Ref. [300]) and a modified z-expansion fit. The uncertainties coming from the chiral extrapolation, the continuum extrapolation and the finite-volume effects turn out to be well below the dominant statistical error, which includes also the error due to the fitting procedure. A set of synthetic data points, representing both the vector and the scalar semileptonic form factors at the physical point for several selected values of q2, is provided together with the corresponding correlation matrix. ( ) The PACS collaboration obtained a new result for N f = 2 + 1, f+(0) = 0.9603(16) +50−48 , by creating an ensemble with the physical light-quark mass on a large lattice volume of (10.9 fm)4 [294]. Such a large lattice enables them to interpolate f+(q2) to zero momentum transfer and study the momentum-transfer dependence of the form factors without using partially- twisted boundary conditions. Their result, however, does not enter the FLAG average, because they only use a single lattice spacing, which is the source of the largest uncertainty in their calculation. For N f = 2 + 1, the two results eligible to enter the FLAG average are the one from RBC/UKQCD 15A, f+(0) = 0.9685(34)(14) [35], and the one from FNAL/MILC 12I, f+(0) = 0.9667(23)(33) [34]. These results, based on different fermion discretizations (staggered fermions in the case of FNAL/MILC and domain wall fermions in the case of RBC/UKQCD) are in nice agreement. Moreover, in the case of FNAL/MILC the form factor has been determined from the scalar current matrix element, while in the case of RBC/UKQCD it has been determined including also the matrix element of the vector current. To a certain extent both simulations are expected to be affected by different systematic effects. RBC/UKQCD 15A has analyzed results on ensembles with pion masses down to 140 MeV, mapping out the complete range from the SU (3)-symmetric limit to the physical point. No significant cut-off effects (results for two lattice spacings) were observed in the simulation results. Ensembles with unphysical light-quark masses are weighted to work as a guide for small corrections toward the physical point, reducing in this way the model dependence in the fitting ansatz. The systematic 123 Eur. Phys. J. C (2022) 82:869 Page 53 of 296 869 uncertainty turns out to be dominated by finite-volume effects, for which an estimate based on effective theory arguments is provided. The result FNAL/MILC 12I is from simulations reaching down to a lightest RMS pion mass of about 380 MeV (the lightest valence pion mass for one of their ensembles is about 260 MeV). Their combined chiral and continuum extrapolation (results for two lattice spacings) is based on NLO staggered χPT supplemented by the continuum NNLO expression [285] and a phenomenological parameterization of the breaking of the Ademollo–Gatto theorem at finite lattice spacing inherent in their approach. The p4 low-energy constants entering the NNLO expression have been fixed in terms of external input [226]. The ETM collaboration uses the twisted-mass discretization and provides at N f = 2 a comprehensive study of the systematics [36,300], by presenting results for four lattice spacings and by simulating at light pion masses (down to Mπ = 260 MeV). This makes it possible to constrain the chiral extrapolation, using both SU (3) [278] and SU (2) [282] χPT. Moreover, a rough estimate for the size of the effects due to quenching the strange quark is given, based on the comparison of the result for N f = 2 dynamical quark flavours [45] with the one in the quenched approximation, obtained earlier by the SPQcdR collaboration [303]. We now compute the N f = 2 + 1 + 1 FLAG average for f+(0) using the FNAL/MILC 18 and ETM 16 (uncorrelated) results, the N f = 2 + 1 FLAG average based on FNAL/MILC 12I and RBC/UKQCD 15A, which we consider uncorrelated, while for N f = 2 we consider directly the ETM 09A result, respectively: direct, N f = 2 + 1 + 1 : f+(0) = 0.9698(17) Refs. [32,33], (76) direct, N f = 2 + 1 : f+(0) = 0.9677(27) Refs. [34,35], (77) direct, N f = 2 : f+(0) = 0.9560(57)(62) Ref. [36], (78) where the parentheses in the third line indicate the statistical and systematic errors, respectively. We stress that the results (76) and (77), corresponding to N f = 2 + 1 + 1 and N f = 2 + 1, respectively, include already simulations with physical light-quark masses. 4.3.2 Results for fK±/ fπ± In the case of the ratio of decay constants the data sets that meet the criteria formulated in the introduction are HPQCD 13A [37], ETM 14E [38], FNAL/MILC 17 [16] (which updates FNAL/MILC 14A [17]) and CalLat 20 [39] with N f = 2 + 1 + 1, HPQCD/UKQCD 07 [40], MILC 10 [41], BMW 10 [42], RBC/UKQCD 14B [8], BMW 16 [43,310], and QCDSF/UKQCD 16 [44] with N f = 2 + 1 and ETM 09 [45] with N f = 2 dynamical flavours. Note that only CalLat 20 for N f = 2 + 1 + 1 is the new entry for the FLAG average in this edition. CalLat 20 employs a mixed action setup with the Möbius domain-wall valence quarks on gradient-flowed HISQ ensembles at four lattice spacings a = 0.06–0.15 fm. The valence pion mass reaches the physical point at three lattice spacings, and the smallest valence-sea and sea pion masses are below 200 MeV. Finite-volume corrections are studied on three lattice volumes at a = 0.12 fm and Mπ ∼ 220 MeV. Their extrapolation to the continuum limit and the physical point is based on NNLO χPT [316]. A comprehensive study of systematic uncertainties is performed by exploring several options including the use of the mixed-action effective theory expression, and the inclusion of N3LO counter terms. They obtain fK±/ fπ± = 1.1942(32)stat(12)χ (20)a2(1)FV (12)M (7)I B , where the errors are statistical, due to the extrapolation in pion and kaon masses, extrapolation in a2, finite-size effects, choice of the fitting form and isospin breaking corrections. ETM 14E uses the twisted-mass discretization and provides a comprehensive study of the systematics by presenting results for three lattice spacings in the range 0.06–0.09 fm and for pion masses in the range 210–450 MeV. This makes it possible to constrain the chiral extrapolation, using both SU (2) [282] χPT and polynomial fits. The ETM collaboration includes the spread in the central values obtained from different ansätze into the systematic errors. The final result of their analysis is fK±/ fπ± = 1.184(12)stat+fit(3)Chiral(9)a2(1)Z P (3)FV (3)I B where the errors are (statistical + the error due to the fitting procedure), due to the chiral extrapolation, the continuum extrapolation, the mass-renormalization constant, the finite-volume and (strong) isospin-breaking effects. In ETM 21 [305], the ETM collaboration presented an independent estimate of fK / fπ in isosymmetric QCD with 2+1+1 dynamical flavours of the twisted-mass quarks. Their new set of gauge ensembles reaches the physical pion mass. The quark action includes the Sheikoleslami–Wohlert term for a better control of discretization effects. The finite-volume effects are examined by simulating three spatial volumes, and are corrected by SU (2) χPT formulae [146]. Their new estimate fK / fπ = 1.1995(44)stat+fit(7)sys is consistent with ETM 14E with the total uncertainty reduced by a factor of ∼ 3.5. While 123 869 Page 54 of 296 Eur. Phys. J. C (2022) 82:869 ETM 21 satisfies all criteria on simulation parameters, it does not enter the FLAG average in this edition due to the publication status. FNAL/MILC 17 has determined the ratio of the decay constants from a comprehensive set of HISQ ensembles with N f = 2 + 1 + 1 dynamical flavours. They have generated 24 ensembles for six values of the lattice spacing (0.03– 0.15 fm, scale set with fπ+ ) and with both physical and unphysical values of the light sea-quark masses, controlling in this way the systematic uncertainties due to chiral and continuum extrapolations. With respect to FNAL/MILC 14A they have increased the statistics and added three ensembles at very fine lattice spacings, a 0.03 and 0.042 fm, includ- ing for the latter case also a simulation at the physical value of the light-quark mass. The final result of their analysis is fK±/ f +0 π± = 1.1950(14)stat(−17)a2(2)FV (3) fπ ,P DG(3)E M (2)Q2 , where the errors are statistical, due to the continuum extrapolation, finite-volume, pion decay constant from PDG, electromagnetic effects and sampling of the topological charge distribution.16 HPQCD 13A has analyzed ensembles generated by MILC and therefore its study of fK±/ fπ± is based on the same set of ensembles bar the ones at the finest lattice spacings (namely, only a = 0.09−0.15 fm, scale set with fπ+ and relative scale set with the Wilson flow [114,317]) supplemented by some simulation points with heavier quark masses. HPQCD employs a global fit based on continuum NLO SU (3) χPT for the decay constants supplemented by a model for higher- order terms including discretization and finite-volume effects (61 parameters for 39 data points supplemented by Bayesian priors). Their final result is fK±/ fπ± = 1.1916(15)stat(12)a2(1)FV (10), where the errors are statistical, due to the continuum extrapolation, due to finite-volume effects and the last error contains the combined uncertainties from the chiral extrapolation, the scale-setting uncertainty, the experimental input in terms of fπ+ and from the uncertainty in mu/md . Because CalLat 20, FNAL/MILC 17 and HPQCD 13A partly share their gauge ensembles, we assume a 100 % correlation among their statistical errors. A 100 % correlation on the total systematic uncertainty is also assumed between FNAL/MILC 17 and HPQCD 13A with the HISQ valence quarks. For N f = 2 + 1 the results BMW 16 and QCDSF/UKQCD 16 are eligible to enter the FLAG average. BMW 16 has analyzed the decay constants evaluated for 47 gauge ensembles generated using tree-level clover-improved fermions with two HEX-smearings and the tree-level Symanzik-improved gauge action. The ensembles correspond to five values of the lattice spacing (0.05–0.12 fm, scale set by mass), to pion masses in the range 130–680 MeV and to values of the lattice size from 1.7 to 5.6 fm, obtaining a good control over the interpolation to the physical mass point and the extrapolation to the continuum and infinite volume limits. QCDSF/UKQCD 16 has used the nonperturbatively O(a)-improved clover action for the fermions (mildly stout-smeared) and the tree-level Symanzik action for the gluons. Four values of the lattice spacing (0.06–0.08 fm) have been simulated with pion masses down to ∼ 220 MeV and values of the lattice size in the range 2.0–2.8 fm. The decay constants are evaluated using an expansion around the symmetric SU (3) point mu = md = ms = (mu + md + m physs) /3. Note that for N f = 2 + 1 MILC 10 and HPQCD/UKQCD 07 are based on staggered fermions, BMW 10, BMW 16 and QCDSF/UKQCD 16 have used improved Wilson fermions and RBC/UKQCD 14B’s result is based on the domain-wall formulation. In contrast to RBC/UKQCD 14B and BMW 16, the other simulations are for unphysical values of the light-quark masses (corresponding to smallest pion masses in the range 220−260 MeV in the case of MILC 10, HPQCD/UKQCD 07, and QCDSF/UKQCD 16) and therefore slightly more sophisticated extrapolations needed to be controlled. Various ansätze for the mass and cutoff dependence comprising SU (2) and SU (3) χPT or simply polynomials were used and compared in order to estimate the model dependence. While BMW 10, RBC/UKQCD 14B, and QCDSF/UKQCD 16 are entirely independent computations, subsets of the MILC gauge ensembles used by MILC 10 and HPQCD/UKQCD 07 are the same. MILC 10 is certainly based on a larger and more advanced set of gauge configurations than HPQCD/UKQCD 07. This allows them for a more reliable estimation of systematic effects. In this situation we consider both statistical and systematic uncertainties to be correlated. For N f = 2 no new result enters the corresponding FLAG average with respect to the previous edition of the FLAG review [4], which therefore remains the ETM 09 result, which has simulated twisted-mass fermions down to (charged) pion masses equal to 260 MeV. We note that the overall uncertainties quoted by ETM 14E at N f = 2 + 1 + 1 and by BMW 16 and QCDSF/UKQCD 16 at N f = 2 + 1 are much larger than the overall uncertainties obtained with staggered (HPQCD 13A, FNAL/MILC 17 at N f = 2 + 1 + 1, and MILC 10, HPQCD/UKQCD 07 at N f = 2 + 1) and domain-wall fermions (RBC/UKQCD 14B at N f = 2 + 1). 16 To form the average in Eq. (81), we have symmetrized the asymmetric systematic error and shifted the central value by half the difference as will be done throughout this section. 123 Eur. Phys. J. C (2022) 82:869 Page 55 of 296 869 Table 17 Values of the SU (2) isospin-breaking correction δSU (2) applied to the lattice data for fK / fπ , entering the FLAG average at N f = 2+ 1, for obtaining the corrected charged ratio fK±/ fπ± . The last error in the last column is due to a 100 % uncertainty assumed for δSU (2) from SU (3) χPT fK / fπ δSU (2) fK±/ fπ± HPQCD/UKQCD 07 1.189(2)(7) −0.0040(7) 1.187(2)(7)(2) BMW 10 1.192(7)(6) −0.0041(7) 1.190(7)(6)(2) RBC/UKQCD 14B 1.1945(45) −0.0043(9) 1.1919(45)(26) Before determining the average for fK±/ fπ± , which should be used for applications to Standard Model phenomenology, we apply the strong-isospin correction individually to all those results that have been published only in the isospin-symmetric limit, i.e., BMW 10, HPQCD/UKQCD 07 and RBC/UKQCD 14B at N f = 2 + 1 and ETM 09 at N f = 2. To this end, as in the previous editions of the FLAG reviews [2–4], we make use of NLO SU (3) χPT [241,280], which predicts fK± = fK √1 + δ ± SU (2) , (79) fπ fπ where [241] √ [ ( )] δ ≈ 3  − 4 ( f / f − 1)+ 2 M2 − 2M2 − M2 Mln KSU (2) SU (2) 3 K π 2 2 K π π 2 . (80)3(4π) f0 Mπ √ √ We use as input SU (2) = 3/(4R) with the FLAG result for R of Eq. (54), F0 = f0/ 2 = 80 (20) MeV, Mπ = 135 MeV and MK = 495 MeV (we decided to choose a conservative uncertainty on f0 in order to reflect the magnitude of potential higher-order corrections). The results are reported in Table 17, where in the last column the last error is due to the isospin correction (the remaining errors are quoted in the same order as in the original data). For N f = 2 and N f = 2 + 1 + 1 dedicated studies of the strong-isospin correction in lattice QCD do exist. The updated N f = 2 result of the RM123 collaboration [172] amounts to δSU (2) = −0.0080(4) and we use this result for the isospin correction of the ETM 09 result. Note that the above RM123 value for the strong-isospin correction is incompatible with the results based on SU (3) χPT, δSU (2) = −0.004(1) (see Table 17). Moreover, for N f = 2 + 1 + 1 HPQCD [37], FNAL/MILC [16] and ETM [318] estimate a value for δSU (2) equal to−0.0054(14),−0.0052(9) and−0.0073(6), respectively. Note that the RM123 and ETM results are obtained using the insertion of the isovector scalar current according to the expansion method of Ref. [171], while the HPQCD and FNAL/MILC results correspond to the difference between the values of the decay constant ratio extrapolated to the physical u-quark mass mu and to the average (mu + md)/2 light-quark mass. One would not expect the strange and heavier sea-quark contributions to be responsible for such a large effect. Whether higher-order effects in χPT or other sources are responsible still needs to be understood. More lattice-QCD simulations of SU (2) isospin-breaking effects are therefore required. To remain on the conservative side we add a 100% error to the correction based on SU (3) χPT. For further analyses we add (in quadrature) such an uncertainty to the systematic error. Using the results of Table 17 for N f = 2 + 1 we obtain direct, N f = 2 + 1 + 1 : fK±/ fπ± = 1.1932(21) Refs. [16,37–39], (81) direct, N f = 2 + 1 : fK±/ fπ± = 1.1917(37) Refs. [8,40–44], (82) direct, N f = 2 : fK±/ fπ± = 1.205(18) Ref. [45], (83) for QCD with broken isospin. The averages obtained for f+(0) and fK±/ fπ± at N f = 2+1 and N f = 2+1+1 [see Eqs. (76–77) and (81–82)] exhibit a precision better than ∼ 0.3%. At such a level of precision QED effects cannot be ignored and a consistent lattice treatment of both QED and QCD effects in leptonic and semileptonic decays becomes mandatory. 4.3.3 Extraction of |Vud | and |Vus | It is instructive to convert the averages for f+(0) and fK±/ fπ± into a corresponding range for the CKM matrix elements |Vud | and |Vus |, using the relations (69). Consider first the results for N f = 2 + 1 + 1. The average for f+(0) in Eq. (76) is mapped into the interval |Vus | = 0.2232(6), depicted as a horizontal red band in Fig. 10. The one for fK±/ fπ± in Eq. (81) 123 869 Page 56 of 296 Eur. Phys. J. C (2022) 82:869 [165] [255] Fig. 10 The plot compares the information for |Vud |, |Vus | obtained on the lattice for N f = 2 + 1 and N f = 2 + 1 + 1 with |Vud | extracted from nuclear β transitions Eqs. (71) and (72). The dotted line indicates the correlation between |Vud | and |Vus | that follows if the CKM-matrix is unitary. For the N f = 2 results see the 2016 edition [3] and |Vus/Vud |( fK±/ fπ±) in Eq. (69) is converted into |Vus |/|Vud | = 0.2313(5), shown as a tilted red band. The red ellipse is the intersection of these two bands and represents the 68% likelihood contour,17 obtained by treating the above two results as independent measurements. Repeating the exercise for N f = 2 + 1 leads to the green ellipse. The vertical light and dark blue bands show |Vud | from nuclear β decay, Eqs. (71) and (72), respectively. The PDG value (71) indicates a tension with both the N f = 2 + 1 + 1 and N f = 2 + 1 results from lattice QCD. As we mentioned, QED radiative corrections are becoming relevant for the extraction of the CKM elements at the current pre- cision of lattice QCD inputs. We obtain a slightly larger value of |Vus |/|Vud | = 0.2320(5) by inputting |Vus/Vud |( fK±/ fπ±) in Eq. (70) with the QED corrections on the lattice. Figure 11 suggests that the kaon (semi)leptonic decays favour a slightly smaller value of |Vud | than the nuclear transitions. 4.4 Tests of the Standard Model In the Standard Model, the CKM matrix is unitary. In particular, the elements of the first row obey |V |2u ≡ |Vud |2 + |V 2us | + |V 2ub| = 1 . (84) The tiny contribution from |Vub| is known much better than needed in the present context: |V −3ub| = 3.82(24) · 10 [165]. In the following, we test the first row unitarity Eq. (84) by calculating |V 2u | and by analyzing the lattice data within the Standard Model. In Fig. 10, the correlation between |Vud | and |Vus | imposed by the unitarity of the CKM matrix is indicated by a dotted line (more precisely, in view of the uncertainty in |Vub|, the correlation corresponds to a band of finite width, but the effect is too small to be seen here). The plot shows that there is a tension with unitarity in the data for N f = 2 + 1 + 1: Numerically, the outcome for the sum of the squares of the first row of the CKM matrix reads |V |2u = 0.9813(66), which deviates from unity at the level of 2.8 standard deviations. Still, it is fair to say that at this level the Standard Model passes a nontrivial test that exclusively involves lattice data and well-established kaon decay branching ratios. The test sharpens considerably by combining the lattice results for f+(0) with the β decay value of |Vud |: f+(0) in Eq. (76) and the PDG estimate of |Vud | in Eq. (71) lead to |V 2u | = 0.99794(37), which highlights a 5.6 σ deviation with unitarity. A lower tension at the three-σ level is suggested either from fK±/ fπ± in Eq. (81) (|V 2u | = 0.99883(37)) or |Vud | in Eq. (72) with the updated nuclear corrections (|V |2u = 0.99800(65)). Unitarity is fulfilled with fK±/ fπ± and |Vud | (72) (|Vu |2 = 0.99890(68)). Note that, when the PDG value of |Vud | (71) is employed, the uncertainties on |V |2u coming from the errors of |Vud | and |Vus | are of similar magnitude with each other. 17 Note that the ellipses shown in Fig. 5 of both Ref. [1] and Ref. [2] correspond instead to the 39% likelihood contours. Note also that in Ref. [2] the likelihood was erroneously stated to be 68% rather than 39%. 123 Eur. Phys. J. C (2022) 82:869 Page 57 of 296 869 [165] [255] Fig. 11 Same as Fig. 10 but with |Vus |/|Vud | through Eq. (70) The situation is similar for N f = 2 + 1: with the lattice data alone one has |V 2u | = 0.9832(89), which deviates from unity at the level of 1.9 standard deviations. The lattice results for f+(0) in Eq. (77) with the PDG value of |Vud | (71) lead to |Vu |2 = 0.99816(43), implying a 4.3 σ deviation from unitarity, whereas the deviation is reduced to 2.3–2.6σ with fK±/ fπ± in Eq. (82) (|V 2u | = 0.99896(45)) and |Vud | in Eq. (72) (|Vu |2 = 0.99822(69)). For the analysis corresponding to N f = 2 the reader should refer to the 2016 edition [3]. 4.5 Analysis within the Standard Model The Standard Model implies that the CKM matrix is unitary. The precise experimental constraints quoted in Eq. (69) and the unitarity condition Eq. (84) then reduce the four quantities |Vud |, |Vus |, f+(0), fK±/ fπ± to a single unknown: any one of these determines the other three within narrow uncertainties. As Fig. 12 shows, the results obtained for |Vus | and |Vud | from the data on fK±/ fπ± (squares) are consistent with the determinations via f+(0) (triangles), while there is a tendency that |Vus | (|Vud |) from f+(0) is systematically smaller (larger) than that from fK±/ fπ± . In order to calculate the corresponding average values, we restrict ourselves to those determinations that enter the FLAG average in Sect. 4.3. The corresponding results for |Vus | are listed in Table 18 (the error in the experimental numbers used to convert the values of f+(0) and fK±/ fπ± into values for |Vus | is included in the statistical error). For N f = 2 + 1 + 1 we consider the data both for f+(0) and fK±/ fπ± , treating ETM 16 and ETM 14E on the one hand and FNAL/MILC 18, CalLat 20, FNAL/MILC 17, and HPQCD 13A on the other hand, as statistically correlated according √to the prescri√ption of Sect. 2.3. As shown in Table 19, we obtain |Vus | = 0.2248(7), where the error is stretched by a factor χ2/dof ∼ 2.6. This result is indicated on the left hand side of Fig. 12 by the narrow vertical band. In the case N f = 2+1 we consider MILC 10, FNAL/MILC 12I and HPQCD/UKQCD 07 on the one hand and RBC/UKQCD 14B and RBC/UKQCD 15A on the other hand, as mutually statistically correlated, since the analysis in the two cases starts from partly the same set of gauge ensembles. In this way we arrive at |Vus | = 0.2249(5) with χ2/dof 0.8. For N f = 2 we consider ETM 09A and ETM 09 as statistically correlated, obtaining |Vus | = 0.2256(19) with χ2/dof 0.7. The figure shows that the results obtained for the data with N f = 2, N f = 2+√1, and N f = 2+1+1 are consistent with each other. However, the larger error for N f = 2 + 1+ 1 due to the stretch factor χ2/dof suggests a slight tension between the estimates from the semileptonic and leptonic decays. Alternatively, we can solve the relations for |Vud | instead of |Vus |. Again, the result |Vud | = 0.97440(17), which follows from the lattice data with N f = 2+1+1, is perfectly consistent with the values |Vud | = 0.97438(12) and |Vud | = 0.97423(44) obtained from the data with N f = 2+1 and N f = 2, respectively. We observe the difference of about 3 σ from Eq. (71) from the superallowed nuclear transitions. It is, however, reduced to  2 σ with Eq. (72) based on the updated nuclear corrections. As mentioned in Sect. 4.1, the HFLAV value of |Vus | from the inclusive hadronic τ decays differs from those obtained from the kaon decays by about three standard deviations. Assuming the first row unitarity (84) leads to a larger value of |Vud | than those from the kaon and nuclear decays. Such a tension does not appear with |Vus | in Eq. (74) from strange hadronic τ decay data and lattice QCD data of the hadronic vacuum polarization function (Table 19). 123 869 Page 58 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 12 Results for |Vus | and |Vud | that follow from the lattice data for f+(0) (triangles) and fK±/ fπ± (squares), on the basis of the assumption that the CKM matrix is unitary. The black squares and the grey bands represent our averages, obtained by combining these two different ways of measuring |Vus | and |Vud | on a lattice. For comparison, the figure also indicates the results obtained if the data on nuclear β decay and inclusive hadronic τ decay is analyzed within the Standard Model [260] [266] [165] [255] Table 18 Values of |Vus | and | | Collaboration Refs. N From |V | |V |Vud obtained from the lattice f us ud determinations of either f+(0) FNAL/MILC 18 [33] 2 + 1 + 1 f+(0) 0.2233(5)(3) 0.97474(12)(6) or fK±/ fπ± assuming CKM unitarity. The first number in ETM 16 [32] 2 + 1 + 1 f+(0) 0.2230(11)(2) 0.97481(25)(5) brackets represents the statistical CalLat 20 [39] 2 + 1 + 1 fK±/ fπ± 0.2252(7)(6) 0.97431(15)(13) error including the experimental FNAL/MILC 17 [16] 2 + 1 + 1 fK±/ fπ± 0.2251(4)(2) 0.97432(9)(5) uncertainty, whereas the second is the systematic one ETM 14E [38] 2 + 1 + 1 fK±/ fπ± 0.2270(22)(20) 0.97388(51)(47) HPQCD 13A [37] 2 + 1 + 1 fK±/ fπ± 0.2256(4)(3) 0.97420(10)(7) RBC/UKQCD 15A [35] 2 + 1 f+(0) 0.2235(9)(3) 0.97469(20)(7) FNAL/MILC 12I [34] 2 + 1 f+(0) 0.2240(7)(8) 0.97459(16)(18) QCDSF/UKQCD 16 [44] 2 + 1 fK±/ fπ± 0.2259(18)(23) 0.97413(42)(54) BMW 16 [43,310] 2 + 1 fK±/ fπ± 0.2281(19)(48) 0.97363(44)(112) RBC/UKQCD 14B [8] 2 + 1 fK±/ fπ± 0.2256(3)(9) 0.97421(7)(22) MILC 10 [41] 2 + 1 fK±/ fπ± 0.2250(5)(9) 0.97434(11)(21) BMW 10 [42] 2 + 1 fK±/ fπ± 0.2259(13)(11) 0.97413(30)(25) HPQCD/UKQCD 07 [40] 2 + 1 fK±/ fπ± 0.2265(6)(13) 0.97401(14)(29) ETM 09A [36] 2 f+(0) 0.2265(14)(15) 0.97401(33)(34) ETM 09 [45] 2 fK±/ fπ± 0.2233(11)(30) 0.97475(25)(69) 4.6 Direct determination of fK± and fπ± It is useful for flavour-physics studies to provide not only the lattice average of fK±/ fπ± , but also the average of the decay constant fK± . The case of the decay constant fπ± is different, since the the PDG value [234] of this quantity, based on the use of the value of |Vud | obtained from superallowed nuclear β decays [247], is often used for setting the scale in lattice QCD (see Sect. 11 on the scale setting). However, the physical scale can be set in different ways, namely, by using as input the mass of the baryon(m ) or the ϒ-meson spectrum (Mϒ ), which are less sensitive to the uncertainties of the chiral extrapolation 123 Eur. Phys. J. C (2022) 82:869 Page 59 of 296 869 Table 19 The upper half of the table shows our final results for |Vus |, |Vud |, f+(0) and fK±/ fπ± that are obtained by analysing the lattice data within the Standard Model (see text). For comparison, the lower half lists the values that follow if the lattice results are replaced by the experimental results on nuclear β decay and inclusive hadronic τ decay, respectively Refs. |Vus | |Vud | N f = 2 + 1 + 1 0.2248(7) 0.97440(17) N f = 2 + 1 0.2249(5) 0.97438(12) N f = 2 0.2256(19) 0.97423(44) Nuclear β decay [165] 0.2278(6) 0.97370(14) Nuclear β decay [256] 0.2277(13) 0.97373(31) Inclusive τ decay [261] 0.2195(19) 0.97561(43) Inclusive τ decay [267] 0.2240(18) 0.97458(40) in the light-quark mass with respect to fπ± . In such cases the value of the decay constant fπ± becomes a direct prediction of the lattice-QCD simulations. It is therefore interesting to provide also the average of the decay constant fπ± , obtained when the physical scale is set through another hadron observable, in order to check the consistency of different scale-setting procedures. Our compilation of the values of fπ± and fK± with the corresponding colour code is presented in Table 20 and it is unchanged from the corresponding one in the previous FLAG review [4]. In comparison to the case of fK±/ fπ± we have added two columns indicating which quantity is used to set the physical scale and the possible use of a renormalization constant for the axial current. For several lattice formulations the use of the nonsinglet axial-vector Ward identity allows to avoid the use of any renormalization constant. One can see that the determinations of fπ± and fK± suffer from larger uncertainties with respect to the ones of the ratio fK±/ fπ± , which is less sensitive to various systematic effects (including the uncertainty of a possible renormalization constant) and, moreover, is not exposed to the uncertainties of the procedure used to set the physical scale. According to the FLAG rules, for N f = 2 + 1 + 1 three data sets can form the average of fK± only: ETM 14E [38], FNAL/MILC 14A [17], and HPQCD 13A [37]. Following the same procedure already adopted in Sect. 4.3 for the ratio of the decay constants, we assume 100 % statistical and systematic correlation between FNAL/MILC 14A and HPQCD 13A. For N f = 2 + 1 three data sets can form the average of fπ± and fK± : RBC/UKQCD 14B [8] (update of RBC/UKQCD 12), HPQCD/UKQCD 07 [40], and MILC 10 [41], which is the latest update of the MILC program. We consider HPQCD/UKQCD 07 and MILC 10 as statistically correlated and use the prescription of Sect. 2.3 to form an average. For N f = 2 the average cannot be formed for fπ± , and only one data set (ETM 09) satisfies the FLAG rules for fK± . Thus, our averages read N f = 2 + 1 : fπ± = 130.2 (0.8) MeV Refs. [8,40,41], (85) N f = 2 + 1 + 1 : fK± = 155.7 (0.3) MeV Refs. [17,37,38], N f = 2 + 1 : fK± = 155.7 (0.7) MeV Refs. [8,40,41], (86) N f = 2 : fK± = 157.5 (2.4) MeV Ref. [45]. The lattice results of Table 20 and our averages (85–86) are reported in Fig. 13. Note that the FLAG averages of fK± for N f = 2 and N f = 2 + 1 + 1 are based on calculations in which fπ± is used to set the lattice scale, while the N f = 2 + 1 average does not rely on that. 5 Low-energy constants Authors: S. Dürr, H. Fukaya, U. M. Heller 123 869 Page 60 of 296 Eur. Phys. J. C (2022) 82:869 Table 20 Colour code for the lattice data on fπ± and fK± together with information on the way the lattice spacing was converted to physical units and on whether or not an isospin-breaking correction has been applied to the quoted result (see Sect. 4.3). The numerical values are listed in MeV units. In this and previous editions [4], old results with two red tags have been dropped iont s tio n po la ror s r sta tu la tra e ion n p o ex e t le inga tio xt ra m z a k a e uu m lu li sc a ic l in v o a al bre bl ira nt ite - orm sic (2)n y Collaboration Refs. N Pu Ch Co Finf R e Ph SU fπ± fK± ETM 14E [38] 2+1+1 A ◦  ◦ na fπ – 154.4(1.5)(1.3) FNAL/MILC 14A [17] 2+1+1 A    na fπ – 155.92(13)(+34−23) HPQCD 13A [37] 2+1+1 A  ◦  na fπ – 155.37(20)(27) MILC 13A [307] 2+1+1 A  ◦  na fπ – 155.80(34)(54) ETM 10E [309] 2+1+1 C ◦ ◦ ◦ na fπ  – 159.6(2.0) JLQCD 15C [319] 2+1 C ◦   NPR t0 125.7(7.4)stat RBC/UKQCD 14B [8] 2+1 A    NPR m  130.19(89) 155.18(89) RBC/UKQCD 12 [188] 2+1 A  ◦  NPR m  127.1(2.7)(2.7) 152.1(3.0)(1.7) Laiho 11 [49] 2+1 C ◦  ◦ na † 130.53(87)(2.10) 156.8(1.0)(1.7) MILC 10 [41] 2+1 C ◦   na † 129.2(4)(1.4) – MILC 10 [41] 2+1 C ◦   na fπ – 156.1(4)(+6−9) JLQCD/TWQCD 10 [311] 2+1 C ◦   na m  118.5(3.6)stat 145.7(2.7)stat RBC/UKQCD 10A [117] 2+1 A ◦ ◦  NPR m  124(2)(5) 148.8(2.0)(3.0) MILC 09A [15] 2+1 C ◦   na Mϒ 128.0(0.3)(2.9) 153.8(0.3)(3.9) MILC 09A [15] 2+1 C ◦   na fπ – 156.2(0.3)(1.1) MILC 09 [157] 2+1 A ◦   na Mϒ 128.3(0.5)(+2.4−3.5) 154.3(0.4)(+2.1◦ −3.4 ) MILC 09 [157] 2+1 A   na fπ 156.5(0.4)(+1.0−2.7) Aubin 08 [312] 2+1 C ◦ ◦ ◦ na Mϒ 129.1(1.9)(4.0) 153.9(1.7)(4.4) RBC/UKQCD 08 [194] 2+1 A ◦   NPR m  124.1(3.6)(6.9) 149.4(3.6)(6.3) HPQCD/UKQCD 07 [40] 2+1 A ◦ ◦ ◦ na Mϒ  132(2) 156.7(0.7)(1.9) MILC 04 [197] 2+1 A ◦ ◦ ◦ na Mϒ 129.5(0.9)(3.5) 156.6(1.0)(3.6) ETM 14D [313] 2 C   ◦ na fπ  – 153.3(7.5)stat ETM 09 [45] 2 A ◦  ◦ na f  – 157.5(0.8)(2.0)(1.1)††π The label ‘na’ indicates the lattice calculations that do not require the use of any renormalization constant for the axial current, while the label ‘NPR’ (‘1lp’) signals the use of a renormalization constant calculated nonperturbatively (at 1-loop order in perturbation theory) †The ratios of lattice spacings within the ensembles were determined using the quantity r1. The conversion to physical units was made on the basis of Ref. [120] and we note that such a determination depends on the PDG value [234] of the pion decay constant ††Errors are (stat+chiral)(a = 0)(finite size) Fig. 13 Values of fπ and fK . The black squares and grey bands indicate our averages (85) and (86) 123 Eur. Phys. J. C (2022) 82:869 Page 61 of 296 869 5.1 Chiral perturbation theory and lattice QCD In the study of the quark-mass dependence of QCD observables calculated on the lattice, it is beneficial to use chiral perturbation theory (χPT). This framework predicts the nonanalytic quark-mass dependence of hadron masses and matrix elements, and it provides symmetry relations among such observables. These predictions invoke a set of linearly independent and universal (i.e., process-independent) low-energy constants (LECs), defined as coefficients of the polynomial terms (in m or M2q π ) of different observables. χPT is an effective field theory approach to the low-energy properties of QCD based on the spontaneous breaking of chiral symmetry, SU (N f )L × SU (N f )R → SU (N f )V , and its soft explicit breaking by quark-mass terms. In its original implementation (i.e., in infinite volume) it is an expansion in powers of m and p2q with the counting rule M2π ∼ mq ∼ p2. If one expands around the SU (2) chiral limit, two LECs appear at order p2 in the chiral effective Lagrangian, ∣ F ≡ F and B ≡ where ≡ −〈ūu〉 ∣π 2 , ∣ , (87) mu ,md→0 F mu ,md→0 and seven more at order p4, called ̄i with i = 1, . . . , 7. In the analysis of the SU (3) chiral limit there are again18 two LECs at order p2, ∣ F0 ≡ 0Fπ and B0 ≡ 2 , where 0 ≡ −〈 ∣ ūu〉 ∣ mu ,md ,ms→0 F0 mu ,md ,ms→ , (88) 0 but ten more at order p4, indicated by the symbols Li (μ) with i = 1, . . . , 10. These “constants” are independent of the quark masses,19 but they become scale dependent after renormalization (sometimes a superscript r is used). The SU (2) constants ̄i are μ-independent, since they are defined at scale = Mphysμ π (as indicated by the bar). The SU (3) constants Li (μ) are usually quoted at the renormalization scale μ = 770 MeV. For the precise definition of these constants and their scale dependence we refer the reader to Refs. [280,320]. In the previous four versions of the FLAG review, we summarized the χPT formulae for the quark-mass dependence of the pion and kaon mass and decay constant, as well as the scalar and vector pion charge radius. We briefly discussed the different regimes of χPT, touched on partially quenched and mixed action formulations, collected and colour-coded the available lattice results for the LECs considered, and formed FLAG estimates or averages, where possible. Since the fourth edition in 2019 [4] (referred to as FLAG 19 below) only a handful of papers appeared with results on the set of LECs covered in our report, but none that qualifies to be included in an average. We therefore decided to shorten the section on LECs considerably, referring the reader to the 2019 FLAG review for the χPT formulae, description of the results covered there, and the details and explanation of the FLAG estimates and averages. In this edition, we will concentrate on the description of the new results and, for the convenience of our readers, list the FLAG estimates and averages, asking the reader to consult FLAG 19 [4] for the details. In the 2019 edition, we introduced a section on ππ scattering in the context of SU (2) χPT and collected results, from finite-volume lattice calculations, of the isospin I = 0 and I = 2 scattering lengths. In this edition, we will keep this section and describe the new results that appeared since the 2019 FLAG review. We will, further, add a section on πK and K K scattering in the context of SU (3) χPT and collect the available results for the scattering lengths from finite-volume lattice calculations. 5.1.1 ππ scattering The scattering of pseudoscalar octet mesons off each other (mostly ππ and πK scattering) is a useful approach to determine χPT low-energy constants [321–325]. This statement holds true both in experiment and on the lattice. We would like to point out the main difference between these two approaches is not so much the discretization of space-time, but rather the Minkowskian versus Euclidean setup. √ 18 Here and in the following, w√e stick to the notation used in the papers where theχPT formulae were established, i.e., we work with Fπ ≡ fπ/ 2 = 92.2(1)MeV and FK ≡ fK / 2. The occurrence of different normalization conventions is not convenient, but avoiding it by reformulating the formulae in terms of fπ , fK is not a good way out. Since we are using different symbols, confusion cannot arise. 19 More precisely, they are independent of the 2 or 3 light-quark masses that are explicitly considered in the respective framework. However, all low-energy constants depend on the masses of the remaining quarks s, c, b, t or c, b, t in the SU (2) and SU (3) framework, respectively, although the dependence on the masses of the c, b, t quarks is expected to be small [280,320]. 123 869 Page 62 of 296 Eur. Phys. J. C (2022) 82:869 In infinite-volume Minkowski space-time, 4-point Green’s functions can be evaluated (e.g., in experiment) for a continuous range of (on-shell) momenta, as captured, for instance, by the Mandelstam variable s. For a given isospin channel I = 0 or I = 2 the ππ scattering phase shift δ I (s) can be determined for a variety of s values, and by matching to χPT some low-energy constants can be determined (see below). In infinite-volume Euclidean space-time, such 4-point Green’s functions can only be evaluated at kinematic thresholds; this is the content of the so-called Maiani–Testa theorem [326]. However, in the Euclidean case, the finite volume comes to our rescue, as first pointed out by Lüscher [327–330]. By comparing the energy of the (interacting) two-pion system in a box with finite spatial extent L to twice the energy of a pion (with identical bare parameters) in infinite volume information on the scattering length can be obtained. In particular, in the (somewhat idealized) situation where one can “scan” through a narrowly spaced set of box-sizes L such information can be reconstructed in an efficient way. We begin with a brief summary of the relevant formulae in SU (2) χPT terminology. In the x-expansion the formulae for aI with  = 0 and I = 0, 2 are found in Ref. [320] 2 { 2 [ ] } a0 = + 7M 5M 9 210 Mπ 2 1 + ̄1 + 2̄ 22 − ̄3 + +O(x ) , (89)32πF { 84π 2 F2 10 8 2 } 2 = − M − M 2 [ ] a0 Mπ 2 1 2 2 ̄1 + 3 2̄ 22 + +O(x ) , (90) 16πF 12π F 8 where x ≡ M2/(4πF)2 with M2 = (mu +md) /F2 is one possible expansion parameter of χPT. Throughout this report we deviate from the χPT habit of absorbing a factor −Mπ into the scattering length (relative to the convention used in quantum mechanics); we include just a minus sign but not the factor Mπ . Hence, our aI have the dimension of a length so that all quark- or pion-mass dependence is explicit (as is most convenient for the lattice community). But the sign convention is the one of the chiral community (where aI I Mπ > 0 means attraction and a Mπ < 0 indicates repulsion). An important difference between the two S-wave scattering lengths is evident already at tree-level. The isospin-0 scattering length (89) is large and positive at this order, while the isospin-2 counterpart (90) is by a factor∼ 3.5 smaller (in absolute magni- tude) and negative. Hence, in the channel with I = 0 the interaction is attractive, while in the channel with I = 2 the interaction is repulsive and significantly weaker. In this convention, experimental results, evaluated with the unitarity constraint germane to any local quantum field theory, read a00 Mπ = 0.2198(46) 2stat(16)syst(64)theo and a0 Mπ = −0.0445(11)stat(4)syst(8)theo [324,331–333]. The ratio between the two (absolute) central values is about 4.9, i.e., a bit larger than 3.5. This, in turn, suggests that NLO contributions to a00 and a 2 0 are sizeable, but the expansion seems well behaved. Equations (89, 90) may be recast in the ξ -expansion, with ξ ≡ M2π/(4πF )2π , as 7M2 { 1 [ }20 40 18 5] a00 Mπ = + π 232πF2 1 + ξ ̄3 + ξ2̄4 + ξ ̄1 + ̄2 − ̄3 + +O(ξ ) , (91)π { 2 21 21 21 2} 2 M 2 1 [4 8 1] a0 M = − ππ 1 + ξ ̄3 + ξ2̄4 − ξ ̄1 + ̄2 + +O(ξ2) , (92)16πF2π 2 3 3 2 where M2/(4πF)2 = M2π/(4πF 2 1 2π ) {1 + 2ξ ̄3 + 2ξ ̄4 +O(ξ )} has been used. Finally, this expression can be summarized as 2 {7M 9M2 }0 = + π + π (λ 0 2 a M 1 ln 0 ) 0 π 2 2 2 2 +O(ξ2) , (93)32πFπ { 32π Fπ Mπ2 2 2 2 } a20 Mπ = − Mπ 3M1 − π (λ0)ln +O(ξ2) , (94) 16πF2π 32π2 F2π M2π with the abbreviations 9 (λ0 2 ln 0 ) 2 = 20 ̄1 + 40 5 5̄2 − ̄3 + 2̄4 + , (95) 2 Mπ,phys 21 21 14 2 3 (λ20) 2 = 4 8 1 1ln 2 ̄1 + ̄2 − ̄3 − 2̄4 + , (96)2 M phys 3 3 2 2π, 123 Eur. Phys. J. C (2022) 82:869 Page 63 of 296 869 where λI with  = 0 and I = 0, 2 are scales like the i in ̄ = ln(2/M2i i π,phys) for i ∈ {1, 2, 3, 4} (albeit they are not independent from the latter). Here, we made use of the fact that M2π/M 2 π,phys = 1+O(ξ) and thus ξ ln(M2π/M2π,phys) = O(ξ2). In the absence of any knowledge on the ̄i , one would assume λ00 λ20, and with this input Eqs. (93, 94) suggest that the NLO contribution to |a00 | is by a factor ∼ 10.5 larger than the NLO contribution to |a20 |. The experimental numbers quoted before clearly support this view. Given that all of this sounds like a complete success story for the determination of the scattering lengths a00 and a 2 0 , one may wonder whether lattice QCD is helpful at all. It is, because the “experimental” evaluation of these scattering lengths builds on a constraint between these two quantities that, in turn, is based on a (rather nontrivial) dispersive evaluation of scattering phase shifts [324,331–333]. Hence, to overcome this possible loophole, an independent lattice determination of a00 and/or a 2 0 is highly welcome. On the lattice a20 is much easier to determine than a 0 0 , since the former quantity does not involve quark-line disconnected contributions. The main upshot (to be reviewed below) is that the lattice determination of a20 Mπ at the physical mass point is in perfect agreement with the experimental numbers quoted before, thus supporting the view that the scalar condensate is – at least in the SU (2) case – the dominant order parameter, and the original estimate ̄3 = 2.9 ± 2.4 is correct (see below). Still, from a lattice perspective it is natural to see a determination of a00 Mπ and/or a 2 0 Mπ as a means to access the specific linear combinations of ̄i with i ∈ {1, 2, 3, 4} defined in Eqs. (95, 96). In passing, we note that an alternative version of Eqs. (93, 94) is used in the literature, too. For instance, Refs. [94,95,334– 336] give their results in the form 2 { 2 [ 2 ] } a0 M = + 7Mπ + M1 π I=0 + − M 5 9 ln π0 π 2 +O(ξ2) , (97)32πFπ { 32π 2 F2 ππ 2π 2Fπ M2 M2 [ = M2 ] } a2 M = − π π I 2 π 20 π 16πF2 1 − 32π2 F2 ππ + 1 − 3 ln 2 +O(ξ ) , (98)π π 2Fπ where the quantities (used to quote the results of the lattice calculation) 2 I=0 = 40 + 80 − 5 + + Mπ,physππ ̄1 ̄2 ̄3 4̄4 9 ln , (99)21 21 7 2F2π,phys 2 I=2 = 8 + 16 Mπ,physππ ̄1 ̄2 − ̄3 − 4̄4 + 3 ln3 3 2F2 , (100)π,phys amount to linear combination√s of the  ren ren i (μ ) that, due to the explicit logarithms in Eqs. (99, 100), are effectively renormalized at the scale phys physμren = fπ = 2Fπ = 130.41(20)MeV [201]. Note that in these equations the dependence on the physical pion mass in the logarithms cancels the one that comes from the ̄i , so that the right-hand-sides bear no knowledge of M phys π . This alternative form is slightly different from Eqs. (93, 94). Exact equality would be reached upon substituting F2 2π → Fπ,phys in the logarithms of Eqs. (97, 98). Upon expanding F2/F2π π,phys and subsequently the logarithm, one realizes that this difference amounts to a term O(ξ) within the square bracket. It thus makes up for a difference at the NNLO, which is beyond the scope of these formulae. We close by mentioning a few works that elaborate on specific issues inππ scattering relevant to the lattice. Reference [337] does mixed action χPT for 2 and 2+1 flavours of staggered sea quarks and Ginsparg–Wilson valence quarks, Refs. [338,339] work out scattering formulae in Wilson fermion χPT, and Ref. [340] lists connected and disconnected contractions in ππ scattering. 5.1.2 πK and K K scattering The discussion of ππ scattering in the previous subsection carries over, without material changes, to the case of πK and K K scattering. The one (tiny) difference is that results, if contact with χPT is desired, must be matched against the SU (3) 123 869 Page 64 of 296 Eur. Phys. J. C (2022) 82:869 version of this framework. In other words, for ππ scattering there is a choice between SU (2) and SU (3), while for πK and K K scattering matching to the SU (3) version of χPT is mandatory.20 For completeness we also include, below, the SU (3) χPT result for I = 2 ππ scattering. Since, as in the FLAG 19 review, we tabulate the S-wave scattering length with combined isospin I in the dimensionless variable aI0 Mπ , where the physical pion mass is meant, the result can be converted into specific linear combinations of NLO χPT coefficients in either the SU (2) or SU (3) χPT framework. In this conversion, an extra piece to the systematic error is to be included, to account for higher-order terms in the chiral expansion. Below, we continue this tradition by summarizing results in the dimensionless variable aI0μπK for πK scattering and aI0 MK for K K scattering. Throughout this report, μπK ≡ Mπ MK /(Mπ + MK ) is the reduced mass of the kaon-pion system at the physical mass point. Again, these results can be converted into linear combinations of the Li , with proper adjustment of the systematic uncertainty, due to the chiral expansion. In doing so, one should keep in mind that the SU (3) framework does not converge as swiftly as the SU (2) framework, since mud  ms . We basically follow Ref. [344], but we adopt, for masses and decay constants, the conventions of the LEC section in the FLAG 19 report. We consider the χPT formulae at O(p4) in the chiral expansion, as given in Refs. [280,345–349]. The scattering lengths of the ππ(I = 2), K K (I = 1), πK (I = 32 ) and πK (I = 12 ) systems can be written as M2 { [ 2 ]} a2 π0,ππ Mπ = 2 − 1 + 16 2 M 2 Mπ Lscat(μ)− π L5(μ)+ χ216πF F 2 ππ (μ) , (101)π 2 { π [ 2 ]} a1 = MK − 16 M0,K K MK 2 1 + 2 M2 K 116 F F K Lscat(μ)− L5(μ)+ χK K (μ) , (102)π K K 2 μ2 { 16 [ M2 + M2 ]} a3/2 πK π K 3/20,πKμπK = − 1 + Mπ MK Lscat(μ)− L5(μ)+ χπK (μ) , (103)8πFπ FK { Fπ FK 42 [ 2 2 ]} a1/2 μπK 16 M + M 0,πKμπK = 2 + Mπ M π 1/2K Lscat(μ)+ 2 K L5(μ)+ χπK (μ) . (104)8πFπ FK Fπ FK 4 These formulae are written in terms of O(p4) values of the masses and decay constants (Mπ , MK , Fπ and FK ) of the Nambu–Goldstone bosons (which, in turn, depend on the quark masses). We recall that the “Bernese” normalization for the pion decay constant at the physical point is adopted (cf. footnote 18). The constants L5(μ) and 1 Lscat(μ) = 2L1(μ)+ 2L2(μ)+ L3(μ)− 2L4(μ)− L5(μ)+ 2L6(μ)+ L8(μ) (105) 2 are the SU (3) low-energy constants (LECs) at the renormalization scale μ. The objects (I )χP Q(μ) are known functions with chiral logarithmic terms and dependence on the scale μ. In terms of these objects the functions χ IP Q(μ) in Eqs. (101)–(104) read21 20 Note that this could be circumvented if one used a heavy-meson extended version of χPT, in particular SU (2) χPT with an extra (heavy) strange quark [341–343]. However, we have the original Gasser–Leutwyler versions of SU (2) and SU (3) χPT in mind. 2 21 10MThere is a typo in the original version of Ref. [344] which made us mistakenly give the last term in the square bracket of Eq. (107) as K9 7M2 in the arXiv:2111.09849v1 version of this report. The correct expression with the last term K9 agrees with Eq. (32) in [348] which, to the best 2 of our knowledge, is the earliest reference for this quantity. Moreover, in the SU (3) limit (16π)2χ2ππ (μ) → − 14 2 Mπ 4 29 Mπ log( μ2 )+ 9 Mπ , while the 2 2 2 2 Gell–Mann–Oakes–Renner relation and the substitution M2 = M2 +  yield (16 M (Mπ)2χ1 (μ) → π π+) log M( πK π K K 4 2 ) − (M2 M +π + ) log( π ) + μ μ2 (M2π+)(−20M2π−20+11M2 2π ) log Mπ+4/3 7 2 2 36 ( 2 )+ 9 (M2π + ). In this expression the terms O(−1) cancel, and with log Mπ+4/3 M( 2 ) = log( π )+ 4 μ μ μ2 3M2 oneπ 2 obtains (16π)2χ1K K (μ) → − 14 29 Mπ log M( π )+ 4 M22 9 π in the limit  → 0. Hence χ2 1ππ (μ) = χK K (μ) in the SU (3) limit. We are indebted to Andréμ Walker-Loud and Kiyoshi Sasaki for pointing this out to us and for clarifying details, respectively. 123 Eur. Phys. J. C (2022) 82:869 Page 65 of 296 869 [ ( ) ( ) ] 2 = 1 −3M 2 2 π Mπ − M 2 M2η 4M2 χππ(μ) 2 log π 2 log + π2 , (106)(16π) 2 μ 18 μ 9 [ ( ) ( ) 1 = 1 M 2 2 2 2 π M χ (μ) K M log π − M2 Mlog KK K (16π)2 4(M2 − M2 μ2 K) μ2K π ( ) ] + −20M 4 + 11M2 M2 M2 2K π K η 7Mlog + K2 − 2 , (107)36(M 2 [ K M μ 9π ) 3 2 2 4 ( 2 ) 3/2 = 1 22Mπ MK (+ 11Mπ MK − 5Mπ Mχ ππK (μ) )2 log(16π) 8 M2K − M2 μ2π ( ) 9M4 − 134M M3 + 16M3+ K π K π MK − 55M 2 M2 2π K Mlog K 36(M2 2K − M ) μ2π ( ) 36M4 + 48M M3 − 10M3+ K π K π MK + 11M 2 2 π MK − 9M4 M2π η 72(M2K − log M2 ) μ2π ] + 43Mπ MK − 8Mπ MK t1(Mπ , MK ) , (108) 9 9 [ 3 − 2 2 + 4 ( 2 )1/2 1 11Mπ MK 11Mπ M 5Mπ M χπK (μ) = K π(16π)2 4(M2K − log M2 ) μ2π ( ) −9M4 − 67M M3 + 8M3 M + 55M2 M2 M2+ K π K π K π K log K 18(M2 − M2 ) μ2K π ( ) −36M4 + 24M3 M − 5M M3 − 11M2 2 4 2+ K K π K Mπ K Mπ + 9Mπ ηlog 36(M2K − M2 ) μ2π ] + 43Mπ MK + 4Mπ MK 12Mπ MKt1(Mπ , MK )− t2(Mπ , MK ) , (109) 9 9 9 where t1(Mπ , MK ), t2(Mπ , MK ) can be written as √ + ( √ = (MK Mπ )(2MK − Mπ ) 2(MK − Mπ ) MK + ) Mπ t1(Mπ , MK ) − arctan + − , (110)MK Mπ MK 2Mπ 2MK Mπ √ − + ( √ ) = (MK Mπ )(2MK Mπ ) 2(MK + Mπ ) MK − Mπt2(Mπ , MK ) + arctan . (111)MK Mπ MK − 2Mπ 2MK + Mπ In short, these formulae show that – in the SU (3) framework – the four scattering lengths a10 M , a 2 M , a3/2π 0 K 0 μπK , a 1/2 0 μπK determine three linear combinations of L5(μ) and Lscat(μ). Recall that Eq. (105) shows that the latter object is itself a linear combination of the Li (μ). Interestingly, ππ and K K scattering determine the same linear combination Lscat(μ) − 1 2 L5(μ), while a 3/2 1/2 0 μπK and a0 μπK determine two more (ms/mud -dependent) linear combinations. In the last few lines, we established the habit of omitting the particle subscript in aI I0,πK and a0,K K , since the value of I together with the factor Mπ , μπK or MK already tells the particles involved in the scattering process. The remaining zero subscript is meant to indicate the S-wave component. 123 869 Page 66 of 296 Eur. Phys. J. C (2022) 82:869 5.2 Extraction of SU(2) low-energy constants 5.2.1 New results for individual LO SU(2) LECs We are aware of four new papers with results on individual SU (2) LECs plus an additional one which we overlooked in FLAG 19 [4]. They all give results on the LO LECs, B and/or F , where B is frequently traded for the condensate ≡ B F2 (both B and are renormalized at the scale μ = 2 GeV). We start by briefly mentioning their details. The paper ETM 20A [350] presents an N f = 2 calculation with twisted mass fermions, using three pion masses down to the physical value at a single lattice spacing a = 0.0914(15) fm. They report a value of F as given in Table 22 and a value of ̄4 discussed in Sect. 5.2.2 below. The publication status changed from “preprint” to “accepted” after our closing date (as did the quoted uncertainty). In practical terms this change is insignificant, since the quoted number (due to a red tag) would not contribute to the N f = 2 average. The paper χQCD 21 [351] employs N f = 2 + 1 QCD with domain wall fermions and RI/MOM renormalization. They have two ensembles with physical pion mass (139 MeV) at lattice spacings a = 0.114fm and a = 0.084fm, one ensemble with Mπ = 234 MeV at a = 0.071fm, and one with Mπ = 371 MeV at a = 0.063fm that is only used to test the lattice spacing dependence of the scalar renormalization factor. They report the value of 1/3 as listed in Table 21. The paper ETM 21 [305] uses N f = 2 + 1 + 1 flavours of twisted mass fermions, ten ensembles, three lattice spacings (a = 0.092, 0.080, 0.068fm), up to √four pion masses Mπ ∈ [135 MeV, 346 MeV], up to two volumes, and L(Mπ,min) = 5 55 fm. The scale is set by f phys = 2Fphys. π π = 130.4(2)MeV [201]. They analyze the quark mass dependence of both Fπ and the (chiral and finite-volume) log-free quantity X = (F M4 )1/5π π π [352], to determine F and ̄4 in two different ways. The two fitting procedures yield nearly identical results for F . The two central values agree exactly, as do the two systematic uncertainties; only the combined statistical plus fitting uncertainty differs a bit among the two approaches. Since the paper does not give preference to one of the fitting procedures, we take the liberty to condense them, assuming 100% correlation, into the single result F = 87.7(6)(5)MeV as listed in Table 22. They also report a value of ̄4 to be mentioned in Sect. 5.2.2 below. The paper ETM 21A [204] is again based on N f = 2 + 1 + 1 flavours of twisted mass fermions, ten ensembles, three lattice spacings, a = 0.095, 0.082, 0.069fm, up to fo√ur pion masses Mπ ∈ [134 MeV, 346 MeV], up to two volumes, and L(M physπ,min) = 5.52 fm. The scale is set by fπ = 2Fphysπ = 130.4(2)MeV [201], and cross-checked with the nucleon mass. From the analysis of the pion sector they determine values of F and 1/3 as listed in Tables 22 and 21, respectively. Finally, we should mention Ref. [353] which, regrettably, escaped our attention when preparing the last FLAG report [4]. The authors extract the quark condensate from an OPE analysis of the Landau-gauge quark propagator. They use overlap valence quarks on three ensembles with (2+1)-flavor domain-wall fermions with a−1 = 1.75 GeV and sea pion masses of 331, 419 and 557 MeV from the RBC/UKQCD collaboration. Their eight valence pion masses range from 220 to 600 MeV. Their result for 1/3 is listed in Table 21. With only a single lattice spacing, their result does not contribute to the FLAG average. √ Perhaps it is worth comparing the results for f ≡ 2F in Refs. [204,305]. Carrying all errors along, one findsf [MeV] = 124.0(0.9)(0.7)− 122.82(32)(65) = 1.18(1.35), which is less than one standard deviation. Given that the two studies were carried out on largely the same ensemble basis, it is perhaps reasonable to assume the statistical error is ∼ 100% correlated. In this case, the difference would be f [MeV] = 124.0(0.7)− 122.82(65) = 1.18(0.96), which is 1.24σ and thus perfectly acceptable. The chiral analysis in the two papers is treated somewhat differently, which would lead to differences in the neglected NNLO terms, and thus reflects a systematic effect. The new results for 1/3 and Fπ/F , together with the previous ones, are shown in Figs. 14 and 15, respectively. 5.2.2 New results for individual NLO SU(2) LECs Two of the aforementioned papers contain new results on ̄4, i.e., a specific LEC at NLO of the SU (2) framework. ETM 20A [350] quotes ̄4 = 4.31(4)(2)(11)(5) for N f = 2, while ETM 21 [305] finds ̄4 = 3.44(28)(36) for N f = 2 + 1 + 1. These results are listed in Table 23. If one were to ignore N f , the two new results would appear inconsistent. While an implicit dependence on the strange- (and highly suppressed) charm-quark mass in the sea is a logical possibility, it seems to us these results should be considered in conjunction with the FLAG 19 averages for the quantity ̄4. The FLAG 19 average for N f = 2, based on four papers, was 4.40(28), the average for N f = 2 + 1, based on five papers, was 4.02(45), and the estimate for N f = 2 + 1 + 1, 123 Eur. Phys. J. C (2022) 82:869 Page 67 of 296 869 Table 21 Cubic root of the SU (2) quark condensate ≡ − limmu ,md→0〈ūu〉 in MeV units, in the MS-scheme, at the renormalization scale μ = 2 GeV. All ETM values that were available only in r0 units were converted on the basis of r0 = 0.48(2) fm [116,367,368], with this error being added in quadrature to any existing systematic error ∑ Collaboration Refs. N 1/3f ETM 21A [204] 2+1+1 P  ◦   267.6(1.8)(1.1) ETM 17E [81] 2+1+1 A ◦  ◦  318(21)(21) ETM 13 [80] 2+1+1 A ◦    280(8)(15) χQCD 21 [351] 2+1 P     260.3(0.7)(1.7) JLQCD 17A [86] 2+1 A ◦    274(13)(29) Wang 16 [353] 2+1 A ◦    305(15)(21) JLQCD 16B [85] 2+1 A ◦    270.0(1.3)(4.8) RBC/UKQCD 15E [84] 2+1 A     274.2(2.8)(4.0) RBC/UKQCD 14B [8] 2+1 A     275.9(1.9)(1.0) BMW 13 [83] 2+1 A     271(4)(1) Borsanyi 12 [82] 2+1 A ◦ ◦   272.3(1.2)(1.4) JLQCD/TWQCD 10A [354] 2+1 A     234(4)(17) MILC 10A [12] 2+1 C ◦   ◦ ( )281.5(3.4) +2.0−5 9 (4.0). RBC/UKQCD 10A [117] 2+1 A ◦ ◦   256(5)((2)(2)) JLQCD 09 [355] 2+1 A     242(4) +19−18 MILC 09A, SU (3)-fit [15] 2+1 C ◦   ◦ 279(1)((2)()4) MILC 09A, SU (2)-fit [15] 2+1 C ◦   ◦ 280(2) +4 (4) MILC 09 [157] 2+1 A ◦   ◦ (−8)278(1) +2−3 (5) TWQCD 08 [356] 2+1 A     259(6)(9) PACS-CS 08, SU (3)-fit [193] 2+1 A     312(10) PACS-CS 08, SU (2)-fit [193] 2+1 A     309(7) RBC/UKQCD 08 [194] 2+1 A ◦  ◦  255(8)(8)(13) Engel 14 [89] 2 A     263(3)(4) Brandt 13 [88] 2 A ◦  ◦  261(13)(1) ETM 13 [80] 2 A ◦  ◦  283(7)(17) ETM 12 [357] 2 A ◦  ◦  299(26)(29) Bernardoni 11 [358] 2 C ◦    306(11) TWQCD 11 [359] 2 A ◦    230(4)(6) TWQCD 11A [360] 2 A ◦    259(6)(7) JLQCD/TWQCD 10A [354] 2 A     242((5)(2)0() ) Bernardoni 10 [361] 2 A ◦    262 +33 +4−3(4 )−5 ETM 09C [87] 2 A ◦  ◦  270(5) +3−4 ETM 08 [92] 2 A ◦ ◦ ◦  264(3)(5) CERN 08 [362] 2 A ◦  ◦  276(3)(4)(5) Hasenfratz 08 [363] 2 A ◦  ◦  248(6) ( ) JLQCD/TWQCD 08A [364] 2 A ◦    235.7(5.0)(2.0) +12.7 ◦ −0.0JLQCD/TWQCD 07 [365] 2 A    239.8(4.0) JLQCD/TWQCD 07A [366] 2 A     252(5)(10) 123 Publication status Chiral extrapolation Cont. extrapolation Finite volume Renormalization 869 Page 68 of 296 Eur. Phys. J. C (2022) 82:869 Table 22 Results for the SU (2) low-energy constant F (in MeV) and for the ratio Fπ/F . All ETM values that were available only in r0 units were converted on the basis of r0 = 0.48(2) fm [116,367√,368], with this error being added in quadrature to any existing systematic error. Numbers in slanted fonts have been calculated by us, based on 2Fphysπ = 130.41(20)MeV [201], with this error being added in quadrature to any existing systematic error (otherwise to the statistical error). The systematic error in ETM 11 has been carried over from ETM 10 Collaboration Refs. N f F Fπ/F ETM 21A [204] 2+1+1 P  ◦  86.85(23)(46) 1.062(3)(6) ETM 21 [305] 2+1+1 P  ◦  87.7(6)(5) 1.051(7)(6) ETM 11 [90] 2+1+1 C ◦  ◦ 85.60(4)(13) 1.077(2)(2) ETM 10 [218] 2+1+1 A ◦   85.66(6)(13) 1.076(2)(2) RBC/UKQCD 15E [84] 2+1 A    85.8(1.1)(1.5) 1.0641(21)(49) RBC/UKQCD 14B [8] 2+1 A    86.63(12)(13) 1.0645(15)(0) BMW 13 [83] 2+1 A    88.0(1.3)(0.3) 1.055(7)(2) Borsanyi 12 [82] 2+1 A ◦ ◦  86.78(05)((25)) 1.0627(06()(27) NPLQCD 11 [91] 2+1 A ◦ ◦ ◦ 86.8(2.1) +3.3 1.062(26) +42)−3.4 −40 MILC 10 [41] 2+1 C ◦   87.0(4)(5() ) 1.060(5)(6() ) MILC 10A [12] 2+1 C ◦   87.5(1.0) +0.7 1.054(12) +31−2.6 −09 MILC 09A, SU (3)-fit [15] 2+1 C ◦   86.8(2)(4() ) 1.062(1)((3) ) MILC 09A, SU (2)-fit [15] 2+1 C ◦   87.4(0.6) +0.9−1 0 1.054(7) +12◦ ( . ) (−1)1MILC 09 [157] 2+1 A   87.66(17) +28−52 1.052(2) +6−3 PACS-CS 08, SU (3)-fit [193] 2+1 A    90.3(3.6) 1.062(8) PACS-CS 08, SU (2)-fit [193] 2+1 A    89.4(3.3) 1.060(7) RBC/UKQCD 08 [194] 2+1 A ◦  ◦ 81.2(2.9)(5.7) 1.080(8) ETM 20A [350] 2 A   ◦ 86.46(0.06)(2.40) 1.067(1)(30) ETM 15A [368] 2 A   ◦ 86.3(2.8) 1.069(35) Engel 14 [89] 2 A    85.8(0.7)(2.0) 1.075(09)(25) Brandt 13 [88] 2 A ◦  ◦ 84(8)(2) 1.080(16)(6) QCDSF 13 [369] 2 A  ◦ ◦ 86(1) 1.07(1) TWQCD 11 [359] 2 A ◦   83.39(35)((38)) 1.106(5)(5() ) ETM 09C [87] 2 A ◦  ◦ 85.91(07) +78−07 1.0755(6) +08−94 ETM 08 [92] 2 A 86.6(7)(7) 1.067(9)(9) Hasenfratz 08 [363] 2 A ◦◦ ◦ ◦◦ 90(4) ◦ (+4 2) 1.02(5) ( ) JLQCD/TWQCD 08A [364] 2 A   79.0(2.5)(0.7) .−0 0 1.167(37)(10) +02 . −62 JLQCD/TWQCD 07 [365] 2 A ◦   87.3(5.6) 1.06(7) Colangelo 03 [370] 86.2(5) 1.0719(52) based on a single paper, was 4.73(10). In terms of standard deviations the difference “old average minus new result” is 4.40(28)−4.31(13) = 0.09(31) or 0.3σ for N f = 2, while it is 4.73(10)−3.44(46) = 1.29(47) or 2.7σ for N f = 2+1+1. Hence, the new N f = 2 result of ETM 20A [350] is in perfect agreement with the corresponding FLAG 19 average. On the other hand, the new N f = 2+1+1 result of ETM 21 [305] is largely inconsistent with the corresponding FLAG 19 estimate, which was taken from Ref. [90]. Perhaps one should take a step back at this point, and consider the option that the implicit N f -dependence (through a dynamical strange and charm quark) is smaller than some unaccounted-for systematic effects in at least one of the works considered. On the practical side neither one of the new results qualifies for a FLAG average (ETM 20A [350] has a red tag, ETM 21 [305] is still unpublished). In summary, the time is not ripe to give an update on the ̄4 average given in FLAG 19. 123 Publication status Chiral extrapolation Cont. extrapolation Finite volume Eur. Phys. J. C (2022) 82:869 Page 69 of 296 869 Fig. 14 Cubic root of the SU (2) quark condensate ≡ − limmu ,md→0〈ūu〉 in the MS-scheme, at the renormalization scale μ = 2 GeV. Square symbols indicate determinations from correlators in the p-regime, up triangles refer to extractions from the topological susceptibility, diamonds to determinations from the pion form factor, and bullet points refer to the spectral density method Fig. 15 Comparison of the results for the ratio of the physical pion decay constant Fπ and the leading-order SU (2) low-energy constant F . Square symbols indicate determinations from correlators in the p-regime, and diamonds from the pion form factor The two new results on ̄4 in Table 23 are displayed in Fig. 16, along with all previous determinations with systematic error bars. Since there is no new entry in the first column of the table, there is no analogous figure for ̄3. There is also new information on ̄6. It appears in three new papers on the slope of the vector form factor at q2 = 0 (“charge radius”) of the pion. We follow our tradition of quoting and comparing results in terms of 〈r2〉πV rather than ̄6. As mentioned before, we start with a brief discussion of the particulars of these papers. The paper Feng 19 [377] is based on N f = 2 + 1 flavours of domain-wall valence quarks on domain-wall sea. This collaboration uses four ensembles essentially at the physical mass point22 and another one at Mπ = 341 MeV. At the physical mass point they have three lattice spacings in the range a−1 = 1.015−1.73 GeV, i.e., none of them satisfies a < 0.1fm. The respective box sizes are L = [6.22, 4.58, 5.48]fm, hence L(Mπ,min) = 6.22fm. 22 This earns them a green box on “chiral extrapolation”, but the criterion was crafted with the idea of a global fit which takes all available information into account. In the setup of Feng 19 [377] it is barely possible to disentangle a small Mπ dependence in the vicinity of M phys π from cut-off effects. 123 869 Page 70 of 296 Eur. Phys. J. C (2022) 82:869 Table 23 Results for the SU (2) NLO low-energy constants ̄3 and ̄4. For comparison, the last two lines show results from phenomenological analyses. The systematic error in ETM 11 has been carried over from ETM 10 Collaboration Refs. N f ̄3 ̄4 ETM 21 [305] 2+1+1 P  ◦  3.44(28)(36) ETM 11 [90] 2+1+1 C ◦  ◦ 3.53(5)(26) 4.73(2)(10) ETM 10 [218] 2+1+1 A ◦   3.70(7)(26) 4.67(3)(10) RBC/UKQCD 15E [84] 2+1 A    2.81(19)(45) 4.02(8)(24) RBC/UKQCD 14B [8] 2+1 A    2.73(13)(0) 4.113(59)(0) BMW 13 [83] 2+1 A    2.5(5)(4) 3.8(4)(2) RBC/UKQCD 12 [188] 2+1 A  ◦  2.91(23)(07) 3.99(16)(09) Borsanyi 12 [82] 2+1 A ◦ ◦  3.16(10)((29)) 4.03(03)((16)) NPLQCD 11 [91] 2+1 A ◦ ◦ ◦ 4.04(40) +73 4.30(51) +84 ◦ −55 −60MILC 10 [41] 2+1 C   3.18(50)((89)) 4.29(21)((82)◦ +37 +51)MILC 10A [12] 2+1 C   2.85(81) ◦ ◦ −92 3.98(32) −28 RBC/UKQCD 10A [117] 2+1 A  2.57(18) 3.83(9) MILC 09A, SU (3)-fit [15] 2+1 C ◦   3.32(6(4)(4)5) 4.03(16)(17) MILC 09A, SU (2)-fit [15] 2+1 C ◦   3.0(6) +9−6 3.9(2)(3) PACS-CS 08, SU (3)-fit [193] 2+1 A    3.47(11) 4.21(11) PACS-CS 08, SU (2)-fit [193] 2+1 A    3.14(23) 4.04(19) RBC/UKQCD 08 [194] 2+1 A ◦  ◦ 3.13(33)(24) 4.43(14)(77) ETM 20A [350] 2 A   ◦ 4.31(4)(2)(11)(5) ETM 15A [368] 2 A   ◦ 3.3(4) Gülpers 15 [93] 2 A    4.54(30)(0) Gülpers 13 [371] 2 A ◦  ◦ 4.76(13) Brandt 13 [88] 2 A ◦  ◦ 3.0(7)(5) 4.7(4)(1) QCDSF 13 [369] 2 A  ◦ ◦ 4.2(1) Bernardoni 11 [358] 2 C ◦   4.46(30)(14) 4.56(10)(4) TWQCD 11 [359] 2 A ◦   4.149(3(5)(1)4) 4.582(1(7)(2)0) ETM 09C [87] 2 A ◦  ◦ 3.50(9) +09−30 4.66(4) +04−33 JLQCD/TWQCD 09 [372] 2 A ◦   4.09(50)(52) ETM 08 [92] 2 A ◦ ◦ ◦ 3.2(8)(2) ( ) 4.4(2)(1) ( ) JLQCD/TWQCD 08A [364] 2 A ◦   3.38(40)(24) +31 +31 ◦ −00 4.12(35)(30) −00 CERN-TOV 06 [373] 2 A   3.0(5)(1) Colangelo 01 [324] 4.4(2) Gasser 84 [320] 2.9(2.4) 4.3(9) The paperχQCD 20 [376] employs overlap valence quarks on N f = 2+1 ensembles with domain-wall sea quarks. They use a total of seven ensembles, with three of them being at the physical point. They cover five lattice spacings a = 0.083−0.195fm, of which only one is below 0.1fm. The relevant box size is 6.24fm at the physical point, where they have Mπ L = 4.45. Renormalization is done nonperturbatively. The paper Gao 21 [375] is based on N f = 2 + 1 HISQ (staggered) ensembles on which they invert clover valence quarks. They have M 3π,sea = Mπ,val = 140 MeV at a = 0.076fm in a 64 × 64 volume. In addition, they have Mπ,sea = 160 MeV, Mπ,val = 300 MeV at a = 0.06fm (in a 483 × 64 box), and essentially the same sea-valence mass combination at a = 0.04fm (in a 643 × 64 box). The vector form factor is renormalized nonperturbatively. Unfortunately, no continuum 123 Publication status Chiral extrapolation Cont. extrapolation Finite volume Eur. Phys. J. C (2022) 82:869 Page 71 of 296 869 Fig. 16 Effective coupling constant ̄4. Squares indicate determinations from correlators in the p-regime, diamonds refer to determinations from the pion form factor Table 24 Vector form factor of the pion: Lattice results for the charge radius 〈r2〉πV and the chiral coupling constant ̄6 are compared with the experimental value, as obtained by NA7, and some phenomenological estimates. The publication status of χQCD 20 [376] changed from “preprint” to “accepted” after our closing date Collaboration Refs. N 2 πf 〈r 〉V [fm2] ̄6 HPQCD 15B [374] 2+1+1 A  ◦  0.403(18)(6) Gao 21 [375] 2+1 P ◦   0.42(2)tot χQCD 20 [376] 2+1 A  ◦  0.430(5)(13) 17.1(1.4) Feng 19 [377] 2+1 A    0.434(20)(13) JLQCD 15A, SU (2)-fit [378] 2+1 A ◦  ◦ 0.395(26)(32) 13.49(89)(82) JLQCD 14 [379] 2+1 A    0.49(4)(4) 7.5(1.3)(1.5) PACS-CS 11A [380] 2+1 A ◦  ◦◦ 0.441(46) RBC/UKQCD 08A [381] 2+1 A   0.418(31) 12.2(9) LHP 04 [382] 2+1 A    0.310(46) ETM 17F [383] 2 A    0.443(21)(20) 16.21(76)(70) Brandt 13 [88] 2 A ◦  ◦ 0.481(33)(13) 15.5(1.7)(1.3) JLQCD/TWQCD 09 [372] 2 A ◦   0.409(23)(37) 11.9(0.7)(1.0) ETM 08 [92] 2 A ◦ ◦ ◦ 0.456(30)(24) 14.9(1.2)(0.7) QCDSF/UKQCD 06A [384] 2 A ◦   0.441(19)(63) Bijnens 98 [385] 0.437(16) 16.0(0.5)(0.7) NA7 86 [386] 0.439(8) Gasser 84 [320] 16.5(1.1) extrapolation is performed; they quote the result from the a 0.076fm physical pion mass ensemble as listed in Table 24. The error quoted is a total error, comprising systematic uncertainties unrelated to cut-off effects. The available information on 〈r2〉πV is summarized in Fig. 17. It is obvious that the lattice computations for this quantity do not achieve the precision of the experimental result (NA 7) yet. 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume 869 Page 72 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 17 Summary of the pion form factor 〈r2〉πV . The publication status of χQCD 20 [376] changed from “preprint” to “accepted” after our closing date 5.2.3 New results for an SU(2) linear combination linked to ππ scattering We are aware of four new papers on ππ scattering (in the isospin I = 2 and/or I = 0 state). As before, we begin with a brief description of their specifics. Reference [387] by Hörz and Hanlon uses one CLS ensemble of N f = 2+ 1 nonperturbatively improved Wilson (clover) fermions. Since it is away from the physical mass point and no extrapolation to the latter is attempted, we refrain from applying the FLAG criteria, and there will be no listing in tables and/or plots. We add that this procedure is in strict analogy to our treatment of Ref. [388] in FLAG 19. A sequel publication, based on the same data, is Ref. [389]. They find that the ππ (I = 2) spectrum is fit well by an S-wave phase shift that incorporates the expected Adler zero. Obviously, the same comment regarding the applicability of the FLAG criteria applies. The paper Culver 19 [390] uses N f = 2 flavours of nHYP clover fermions at a = 0.12fm, Mπ = 315 MeV on 48 × 242 × {24, 30, 48} and Mπ = 226 MeV on 64 × 242 × {24, 28, 32}. With a conventional analysis technique they find a20 Mπ = −0(.0455(1)6), after extrapolation to physical pion mass. From an inverse amplitude method, they obtain a20 Mπ = −0.0436 +0.0013−0 0012 , again at the physical pion mass. Since the paper does not give preference to one of the analysis. methods, we take the liberty to condense the two numbers into the result a20 Mπ = −0.0445(14)(19), as shown in Table 25. Here, the systematic error reflects the full difference between the two central values given in the paper. The paper Mai 19 [391] employs N f = 2 nHYP clover fermions at a single lattice spacing (a = 0.12fm), with Mπ = 315 MeV on 48×242×{24, 30, 48} lattices(and Mπ) = 224 MeV on 64×242×{24, 28, 32} lattices. They quote, extrapolated to the physical pion mass, a0 M = 0 2132 +0.00080 π . −0 0009 and a20 Mπ = −0.0433± 0.0002 for I = 0 and I = 2, respectively. With. statistical error only, these results go into Table 25, but not into a plot. The paper ETM 20B [392] is based on N f = 2 QCD with twisted mass fermions at a = 0.0914(15)fm, and with cSW = 1.57551. They have three pion masses (Mπ = 340 MeV on 323 × 64 and Mπ = 242 MeV and Mπ = 134 MeV on 483 × 96). They find, for I = 2, at the pion masses considered, a20 Mπ = −0.2061(49),−0.156(15),−0.0481(86), with the last being at physical pion mass, but finite a. Accordingly, we take a20 Mπ = −0.0481(86) with unknown systematic error. With statistical error only, this result goes into Table 25, but not into a plot. These four works, when combined with the information listed in FLAG 19, represent the information from the lattice on the ππ scattering lengths a0 20 and a0 in the isospin channels I = 0 and I = 2, respectively. As can be seen from Eqs. (93, 95), the I = 0 scattering length carries information about 2021 ̄1 + 40 521 ̄2 − 14 ̄3 + 2̄4. And from Eqs. (94, 96) it follows that the I = 2 counterpart carries information about the linear combination 4 83 ̄1 + 3 ̄ − 12 2 ̄3 − 2̄4. Still, we prefer quoting the dimensionless products aI0 Mπ (at the physical mass point) over the aforementioned linear combinations to ease comparison with phenomenology. The updated Table 25 summarizes the present lattice information on aI=00 M and aI=2π 0 Mπ at the physical mass point, and the results are displayed in Fig. 18. We remind the reader that a lattice computation of aI=00 Mπ involves quark-loop 123 Eur. Phys. J. C (2022) 82:869 Page 73 of 296 869 Table 25 Summary of ππ scattering data in the I = 0 (top) and I = 2 (bottom) channels. Some of the results have been adapted to our sign convention. The results of Refs. [324,333] allow for a cross-check with phenomenology Collaboration Refs. N a0 0f 0 Mπ ππ Fu 17 [393] 2+1 A  ◦  0.217(9)(5) 45.6(7.6)(3.8) Fu 13 [335] 2+1 A    0.214(4)(7) 43.2(3.5)(5.6) Fu 11 [394] 2+1 A    0.186(2) 18.7(1.2) Mai 19 [391] 2 P   ◦ 0.2132(9) ETM 16C [336] 2 A    0.198(9)(6) 30(8)(6) Caprini 11 [333] 0.2198(46)(16)(64) Colangelo 01 [324] 0.220(5)tot Collaboration Refs. N 2 2f a0 Mπ ππ ( ) ETM 15E [94] 2+1+1 A ◦   −0 0442 +1.34. (2)(40) 3.79(0.61) −0.11 PACS-CS 13 [344] 2+1 A    −0.04243(22)(43) Fu 13 [335] 2+1 A    −0.04430(25)(40) 3.27(0.77)(1.12) Fu 11 [394] 2+1 A    −0.0416(2) 11.6(9) NPLQCD 11A [395] 2+1 A    −0.0417(07)(02)(16) NPLQCD 07 [334] 2+1 A    −0.04330(42)tot NPLQCD 05 [396] 2+1 A    −0.0426(06)(03)(18) ETM 20B [392] 2 A ◦  ◦ −0.0481(86) Mai 19 [391] 2 P   ◦ −0.0433(2) Culver 19 [390] 2 P   ◦ −0.0445(14)(19) Yagi 11 [397] 2 P ◦   −0.04410(69)(18) ETM 09G [95] 2 A ◦ ◦ ◦ −0.04385(28)(38) 4.65(0.85)(1.07) CP-PACS 04 [398] 2 A    −0.0413(29) Caprini 11 [333] −0.0445(11)(4)(8) Colangelo 01 [324] −0.0444(10)tot disconnected contributions, which tend to be very noisy and thus require large statistics. Compared to the situation in FLAG 19 the number of computations has increased from three to five, but still none of them is free of red tags. The situation is somewhat better for aI=20 Mπ which is computed from quark-line connected contributions only. In this case there is one computation at N f = 2 and one at N f = 2+ 1+ 1 that qualifies for a FLAG average. We quote these numbers in Sect. 5.2.4 below. The available information on aI=0 M and aI=20 π 0 Mπ is summarized in Fig. 18. It is obvious that the former quantity (due to quark-loop disconnected contributions) is much harder to calculate on the lattice than the latter one. Nonetheless, the good news is that in both cases the lattice determinations are in reasonable agreement with EFT results. 5.2.4 LO and NLO SU(2) estimates and averages As promised in an earlier section, here we list our FLAG 19 estimates and averages [4] that all remain unchanged. We refer the reader to that review for details and explanations. 123 Pu Pb ul bic lia ct aio tin onsta stt au ts us Chir Cal he irx at lra ep xo trl aa pti oo ln ation Cont. e Cx ot nra t.p eo xla trti ao pn olation Finite vo Flu inm ite e volume 869 Page 74 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 18 Summary of the ππ scattering lengths a00 Mπ (top) and a 2 0 Mπ (bottom). Results in Table 25 with statistical error only are not shown For the SU (2) LEC , in the MS scheme, at the renormalization scale μ = 2 GeV, we obtained the averages and/or estimate N f = 2 + 1 + 1 : 1/3 = 286(23)MeV Refs. [80,81], N f = 2 + 1 : 1/3 = 272(5)MeV Refs. [12,82–86], (112) N f = 2 : 1/3 = 266(10)MeV Refs. [80,87–89], where the errors include both statistical and systematic uncertainties. For the ratio of the pion decay constant at the physical point, Fπ , to its value in the SU (2) chiral limit (zero up- and down-quark mass but physical strange-quark mass), F , we obtained the averages and/or estimate N f = 2 + 1 + 1 : Fπ/F = 1.077(3) Ref. [90], N f = 2 + 1 : Fπ/F = 1.062(7) Refs. [41,82–84,91], (113) N f = 2 : Fπ/F = 1.073(15) Refs. [87–89,92]. For SU (2) NLO LECs we obtained the averages and/or estimates 123 Eur. Phys. J. C (2022) 82:869 Page 75 of 296 869 N f = 2 + 1 + 1 : ̄3 = 3.53(26) Ref. [90], N f = 2 + 1 : ̄3 = 3.07(64) Refs. [41,82–84,91], (114) N f = 2 : ̄3 = 3.41(82) Refs. [87,88,92], N f = 2 + 1 + 1 : ̄4 = 4.73(10) Ref. [90], N f = 2 + 1 : ̄4 = 4.02(45) Refs. [41,82–84,91], (115) N f = 2 : ̄4 = 4.40(28) Refs. [87,88,92,93], as well as the estimate N f = 2 : ̄6 = 15.1(1.2) Refs. [88,92]. (116) For the scattering length extracted from ππ scattering in the I = 2 channel we quote N 2f = 2 + 1 + 1 : a0 Mπ = −0.0441(4) Ref. [94], N f = 2 : a20 Mπ = − (117) 0.04385(47) Ref. [95], where the errors include both statistical and systematic uncertainties. We remark that our preprocessing procedure23 sym- metrizes the asymmetric errors with a slight adjustment of the central value. In all cases the references shown are the papers with the contributing results, and we ask the readers to cite those papers when quoting these averages. 5.3 Extraction of SU(3) low-energy constants 5.3.1 New results for individual LO SU(3) LECs We are unaware of any new paper that determines a large number of LECs in the SU (3) framework (as was done, in the past, by the MILC collaboration). However, there is one paper, χQCD 21 [351], with a new result on two SU (3) LECs at LO. They find F0 = 67.8(1.2)(3.2) and 0 = 232.6(0.9)(2.7) in the 3-flavour chiral limit.24 They also quote / 0 = 1.40(2)(2) which we consider interesting for reasons detailed in Sect. 5.3.4. These values are listed, together with those of FLAG 19, in Table 26. The paper has been discussed and color coded in Sect. 5.2. As they are not published yet, there is no update to the FLAG averages/estimates here. 5.3.2 New results for individual NLO SU(3) LECs There are a number of new results on L5, for instance in Refs. [349,399,400] to be discussed below in the context of πK scattering. This is not so surprising, since Eqs. (101, 102, 103, 104) indicate that the observables a20 Mπ , a 1 3/2 0 MK , a0 μπK , a1/20 μπK jointly determine the combination Lscat and L5 (both of which are conventionally quoted at the scaleμ = 770 MeV). Determining any of these two LECs is afflicted with an extra uncertainty, compared to the four scattering lengths, due to the convergence of the SU (3) chiral series.25 Therefore we give preference to reviewing the scattering lengths and converting, once they exist, the pertinent FLAG averages into numerical values of Lscat and L5, over collecting values of Lscat and L5 as converted by the individual collaborations. 23 There are two naive procedures to symmetrize an asymmetric systematic error: (i) keep the central value untouched and enlarge the smaller error, (ii) shift the central value by half of the difference between the two origin(al e)rrors and enlarge/shrink both errors by the same amount. Our procedure (iii) is to average the results of (i) and (ii). In other words a result c s +u( ) − with  > u is changed into c + (u − )/4 with statistical error s and a symmetric systematic error (u + 3)/4. The case  < u is handled accordingly. 24 We use = limmu ,md→0 (mu ,md ,ms ,mc, . . .), 0 = limmu ,md ,ms→0 (mu,md ,ms ,mc, . . .), and likewise for B, B0, F and F0. The quantities , 0, B, B0 are renormalized at the scale μ = 2 GeV. 25 One of the issues is whether the convergence in the LECs pertinent to a10 MK , i.e., with two strange quarks involved, is visibly slower than for a3/20 μπK and a 1/2 0 μπK , where only one strange quark appears. 123 869 Page 76 of 296 Eur. Phys. J. C (2022) 82:869 Table 26 Lattice results for the low-energy constants F0, B0 and ≡ F20 0 B0, which specify the effective SU (3) Lagrangian at leading order. The ratios F/F0, B/B0, / 0, which compare these with their SU (2) counterparts, indicate the strength of the Zweig-rule violations in these quantities (in the large-Nc limit, they tend to unity). Numbers in slanted fonts are calculated by us, from the information given in the references Collaboration Refs. N f F0 [MeV] F/F0 B/B0 JLQCD/TWQCD 10A [354] 3 A    71(3)(8) χQCD 21 [351] 2+1 P    67.8(1.2)(3.2) MILC 10 [41] 2+1 C ◦   80.3(2.5)(5.4) ( ) MILC 09A [15] 2+1 C ◦   78.3(1.4)(2.9) 1.104(3)(41) 1.21(4) +5 MILC 09 [157] 2+1 A ◦ ( ) −(6 )  1.15(5) +13−03 1.15(16) +39−13 PACS-CS 08 [193] 2+1 A    83.8(6.4) 1.078(44) 1.089(15) RBC/UKQCD 08 [194] 2+1 A ◦  ◦ 66.1(5.2) 1.229(59) 1.03(05) Collaboration Refs. N 1/3f 0 [MeV] / 0 JLQCD/TWQCD 10A [354] 3 A     214(6)(24) 1.31(13)(52) χQCD 21 [351] 2+1 P     232.6(0.9)(2.7) 1.40(2)(2) MILC 09A [15] 2+1 C ◦   ◦ 245(5)((4)(4)) 1.48(9)(8()(10)) MILC 09 [157] 2+1 A ◦   ◦ 242(9) +05−17 (4) 1.52(17) +38−15 PACS-CS 08 [193] 2+1 A     290(15) 1.245(10) RBC/UKQCD 08 [194] 2+1 A ◦  ◦  1.55(21) On the other hand, there is no new result on those LECs at the NLO in the SU (3) expansion which were covered in previous editions of FLAG (L4, L6, L9, L10). 5.3.3 Results for SU(3) linear combinations linked to πK , K K scattering Since πK , K K scattering were not covered in previous editions of the FLAG report, we list here all works which include such results. Following the example of the section on ππ scattering, where all results were given in the dimensionless variable aI M , we give the results on πK scattering in the form aI0 π 0μπK , where μπK is the pertinent reduced mass, and the results on K K scattering are given in the form aI0 MK . We start with a brief mentioning of all papers we are aware of. The paper NPLQCD 06B [349] uses asqtad (staggered) sea quarks with N f = 2+1 at a single lattice spacing (a = 0.125fm with L 2.5fm) with Mπ = [290, 350, 490, 600]MeV. The domain-wall valence fermions come with quark masses such that the resulting pion masse 1/2 = ( s ma)tch the aforementioned Nambu–(Gol)dstone boson masses. After chiral extrapolation they find a 0 1346 13 +18 3/2 +190 μπK . ( ) −122 and a0 μπK = −0.0448(12) −45 , with L5 pinned down at a value extracted from the analysis of the quark mass dependence of fK / fπ . The color coding in Table 27 is based on Mπ,min(RMS) = 488 MeV. The paper NPLQCD 07B [401] uses asqtad (staggered) sea quarks with N f = 2 + 1 in conjunction with domain-wall valence quarks. They have two lattice spacings (a = 0.125 fm, 0.09 fm) with somewhat unequal span in quark masses. At a = 0.125 fm they cover Mπ 290, 350, 490, 590 MeV with L 2.5fm. At a = 0.09 fm they do not quote Mπ [MeV], but from aMπ = 0.1453 in Tab. II and a 0.09 fm one would conclude Mπ 320 MeV. After chiral extrapolation, they find a10 MK = −0.352(16)tot. The color coding in Table 27 is based on Mπ,min(RMS) = 413 MeV. The paper Fu 11A [399] employs one ensemble of N f = 2 + 1 asqtad (staggered) quarks at a 0.15 fm, ml/ms = 0.2, m physs ms with L = 2.5 fm. It uses six valence pion masses Mπ = 334 − 466 MeV to study S-wave scattering. It quotes, 123 Pu Pb uli bc la icti ao tn ios nta st tu as tus Ch Ci hra irl ae lx et xra trp ao pl oa lt aio tin on C Con ot n. te .x et xr ta rp ao pl oa lt aio tin on F Fin init ie tev vo olu lum me e Renormalization Eur. Phys. J. C (2022) 82:869 Page 77 of 296 869 Table 27 Summary of πK scattering data in the I = 1 , 32 2 channels, and of K K scattering with I = 1. Some of the results have been adapted to our sign convention Collaboration Refs. N a1/2 3/2 1f 0 μπK a0 μπK a0 MK ETM 18B [96] 2+1+1 A ◦  ◦ 0.127(2)tot −0.0463(17)tot ( ) ETM 17G [97] 2+1+1 A ◦  ◦ −0.385(16 +4) −14 PACS-CS 13 [344] 2+1 A    0.150(16)(37) −0.0477(27)(20) −0.312(17)(31) Fu 11A [399] 2+1 A    0.1425(29) −0.0394(15) NPLQCD 07B [401] 2+1 A  ◦ ◦ ( ) ( ) −0.352(16)tot NPLQCD 06B [349] 2+1 A    0.1346 13 +18( ) −122 −0.0448 12 +19( ) −45 after chiral extrapolation, a1/20 μπK = 0.1425(29) and a3/20 μπK = −0.0394(15). The color coding in Table 27 is based on Mπ,min(RMS) = 590 MeV. We are also aware of Ref. [402] which is based on a single ensemble of N f = 2 clover quarks. Since it is away from the physical mass point and no extrapolation to the latter is attempted, we feel it would be unfair (or misleading) to quote its results in Table 27. Reference PACS-CS 13 [344] uses five ensembles of N f = 2 + 1 nonperturbative clover fermions with a = 0.09fm, L = 2.9fm, and Mπ = 166, 297, 414, 575, 707 MeV. They quote, after extrapolation with χPT: a20 Mπ = −0.04243(22)(43) (see Table 25), a1 M 3/2 1/20 K = −0.312(17)(31), a0 μπK = −0.0477(27)(20) and a0 μπK = 0.150(16)(37) (listed in Table 27). These figures reflect the final numbers quoted in the Erratum of Ref. [344]. The reason for the change is the mishap reported in footnote 21; fortunately it turns out that it affected the final analysis only very mildly. We thank the collaboration for keeping us up-to-date with all aspects of the revision. Since there are no FLAG averages for scattering lengths for N f = 2 + 1, these small changes have no impact on the quoted FLAG averages. The paper HS 14A [403] is based on N f = 2 + 1 anisotropic clover fermions at as 0.12 fm, at 0.035 fm, with Mπ = 391 MeV in {163, 203, 243}× 128 boxes, i.e. with L = 1.9, 2.4, 2.9 fm. These parameters yield MK = 549 MeV thus μπK = 228 MeV. They quote various resonance parameters and, in the S-wave I = 3 2 channel, a3/2/ 0 Mπ = −0.278(15) which we convert to a3/2 phys0 μπK = −0.161(9) at the given Mπ . Since this work does not extrapolate to Mπ , we stay away from color coding. The paper ETM 17G [97] uses N f = 2+1+1 twisted mass fermions at three lattice spacings, a = 0.089, 0.082, 0.062 fm, with up to (five)(Mπ) = 230−450 MeV, and L(Mπ,min) 2.8 fm. In the I = 1 channel they find a 1 0 MK = −0.385 16 +0 +0( )(−12) −5 (4). We take the liberty to combine the various non-statistical errors in quadrature, using a 1 0 MK = −0.385(16 +4) −14 as quoted in Table 27. Reference [404] by Brett et al. uses one ensemble of N f = 2 + 1 anisotropic clover fermions with as = 0.115 fm, Mπ = 233 MeV, in a 323 × 256 box, hence L = 3.7fm. These parameters yield MK = 494 MeV and thus μπK = 158 MeV. Their result for I = 1/2 S-wave scattering reads a1/2 1/20 Mπ = −0.353(25), or a0 μπK = −0.240(17) in our notation. Since this work does not extrapolate to Mphysπ , we stay away from color coding. The paper ETM 18B [96] uses N f = 2+1+1 twisted mass fermions at three lattice spacings, a = 0.089, 0.082, 0.062 fm, with up to five pion masses Mπ = 230−450 MeV and up to two volumes. From the tables, one finds Mπ,min = 276, 302, 311 MeV at the three lattice spacings. They find, after chiral extrapolation, a1/20 μπK = 0.127(2)tot and a3/20 μπK = −0.0463(17)tot as quoted in Table 27. An overview of all scattering lengths with at least one kaon involved is shown in Fig. 19. As usual we refrain from displaying data with statistical error only. In passing, we note that there is an additional paper by Z. Fu, Ref. [400], which deals with K K̄ scattering. It employs one ensemble of N f = 2 + 1 asqtad (staggered) quarks at a 0.15 fm, ml/ms = 0 phys.2, ms ms with L = 2.5 fm together with six valence pion masses Mπ = 334 − 466 MeV. Extrapolating to the physical point, the result for K K̄ scattering in 123 Publication status Chiral extrapolation Cont. extrapolation Finite volume 869 Page 78 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 19 Summary of the πK scattering lengths a1/20 μπK (top), a 3/2 0 μπK (middle) and of the K K scattering length a 1 0 MK (bottom). Results in Table 27 with statistical error only are not shown the I = 1 state is a10 MK = 0.211(33). Hence the interaction for K K̄ in the S-wave I = 1 state is found to be attractive, in agreement with LO χPT. In summary, for the quantities a1/2 3/20 μ 1 πK , a0 μπK and a0 MK Refs. [96,97] are the only sources without red tags. Since they appeared in refereed journals and no other works qualify, we take the results quoted in the top two lines of Table 27 as the current FLAG averages. For the reader’s convenience we list them at the end of Sect. 5.3.5. Last but not least we like to remind the reader that K K scattering might be outside the validity of SU (3) χPT, since it involves a scale around 2MK 1 GeV. However, our review focuses on the scattering length a10 MK , where this issue does not 123 Eur. Phys. J. C (2022) 82:869 Page 79 of 296 869 feature prominently. But it is a key topic in the subsequent conversion of such a scattering length to the low-energy constants Li . We hope that forthcoming high-quality data will allow a future edition of FLAG to address this topic. 5.3.4 Implication on Zweig rule violations Let us spend a minute to explain why we consider the result on / 0 of χQCD 21 [351] particularly interesting. The reason is linked to the question of how close real-world QCD with Nc = 3 is to the large-Nc limit of ’t Hooft (see also Ref. [405]). In the large-Nc limit the Zweig rule becomes exact, and the NLO LECs L4 and L6 tend to zero. As discussed in FLAG 19, the available lattice data are consistent with the view that these two couplings approximately satisfy the Zweig rule. Also the ratios F/F0, B/B0 and / 0 (note that they are linearly dependent, since = B F2 and 0 = B0 F20 ) test the validity of this rule. The available data seem to confirm the paramagnetic inequalities of Ref. [406], which require / 0 > 1 and F/F0 > 1. There is much less information concerning B/B0, and this is the point where the new result of χQCD 21 [351] comes in handy. Let us assume, for the sake of an argument, F/F0 = 1.15(5)(5). Together with / 0 = 1.40(2)(2) [351], this would imply B/B0 = 1.06(9)(9). This numerical example illustrates how much precision is lost in forming the ratio ( / )/(F/F )20 0 ; with these numbers it would not be clear whether B/B0 > 1. Therefore we plead with all collaborations to calculate the numbers F/F0, B/B0 and / 0 in their analysis framework to take advantage of correlations. 5.3.5 LO and NLO SU (3) estimates For each of the SU (3) LO and NLO LECs discussed in the 2019 FLAG review [4] exactly one paper contributed and hence constituted the FLAG average. The present status is that this situation is unchanged. For the convenience of the reader, we list the results here but refer to the 2019 FLAG review for the details and explanations. The LO LECs in the SU (3) chiral limit (mu,md ,ms → 0) are denoted by a subscript 0 to distinguish them from their SU (2) chiral limit counterparts. The parameters 0, B0 are in the MS scheme at the renormalization scale μ = 2 GeV. We quote N f = 2 + 1 : 1/3 0 = 245(8)MeV Ref. [15], (118) N f = 2 + 1 : / 0 = 1.48(16) Ref. [15], (119) N f = 2 + 1 : F0 = 80.3(6.0)MeV Ref. [41], (120) N f = 2 + 1 : F/F0 = 1.104(41) Ref. [15], (121) N f = 2 + 1 : B/B0 = 1.21(7) Ref. [15], (122) where the errors include both statistical and systematic uncertainties. The references shown are the papers from which the results are taken. For SU (3) NLO LECs we display the results for individual low-energy constants N f = 2 + 1 + 1 : L4 = +0.09(34)× 10−3 Ref. [37], (123) N f = 2 + 1 : L4 = −0.02(56)× 10−3 Ref. [41], N f = 2 + 1 + 1 : L5 = +1.19(25)× 10−3 Ref. [37], N f = 2 + 1 : L5 = +0.95(41)× 10− (124) 3 Ref. [41], N f = 2 + 1 + 1 : L6 = +0.16(20)× 10−3 Ref. [37], N f = 2 + 1 : (125) L6 = +0.01(34)× 10−3 Ref. [41], N f = 2 + 1 + 1 : L = +0.55(15)× 10−38 Ref. [37], = + (126)N f 2 1 : L = +0.43(28)× 10−38 Ref. [41], at the chiral scaleμ = 770 MeV, where again all errors quoted are total errors. For details of the symmetrization of asymmetric error bars see footnote 23. 123 869 Page 80 of 296 Eur. Phys. J. C (2022) 82:869 For the scattering lengths involving at least one kaon N f = 2 + 1 + 1 : a1/20 μπK = 0.127(2) Ref. [96], (127) N f = 2 + 1 + 1 : a3/20 μπK = −0.0463(17) Ref. [96], (128) N 1f = 2 + 1 + 1 : a0 MK = −0.388(20) Ref. [97], (129) represent the FLAG estimates with all errors added in quadrature. For details of the symmetrization of asymmetric error bars see footnote 23. Throughout we ask the reader to cite the original references when using these values. 6 Kaon mixing Authors: P. Dimopoulos, X. Feng, G. Herdoíza The mixing of neutral pseudoscalar mesons plays an important role in the understanding of the physics of quark-flavour mixing and CP violation. In this section we discuss K 0 − K̄ 0 oscillations, which probe the physics of indirect CP violation. Extensive reviews on this subject can be found in Refs. [407–412]. With respect to the FLAG 19 report, in the new Sect. 6.2 of the present edition the reader will find an updated discussion regarding the lattice determination of the K → ππ decay amplitudes and related quantities. Discussions concerning the kaon mixing within the Standard Model (SM) and Beyond the Standard Model (BSM) are presented in Sects. 6.3 and 6.4, respectively. We note that FLAG averages for SM and BSM bag parameters have not changed with respect to the FLAG 19 report. 6.1 Indirect CP violation and K in the SM Indirect CP violation arises in KL → ππ transitions through the decay of the CP = +1 component of KL into two pions (which are also in a CP = +1 state). Its measure is defined as = A[KL → (ππ)I=0]K A[KS → , (130)(ππ)I=0] with the final state having total isospin zero. The parameter K may also be expressed in terms of K 0 − K̄ 0 oscillations. In the Standard Model, K receives contributions from: (i) short-distance (SD) physics given by S = 2 “box diagrams” involving W± bosons and u, c and t quarks; (ii) the long-distance (LD) physics from light hadrons contributing to the imaginary part of the dispersive amplitude M12 used in the two component description of K 0 − K̄ 0 mixing; (iii) the imaginary part of the absorptive amplitude 12 from K 0 − K̄ 0 mixing; and (iv) Im(A0)/Re(A0), where A0 is the K → (ππ)I=0 decay amplitude. The various factors in this decomposition can vary with phase conventions. In terms of the S = 2 effective Hamiltonian, HS=2eff , it is common to represent contribution (i) by 1 Im(MSD) ≡ Im[〈K̄ 0|HS=2 012 eff |K 〉], (131)2mK and contribution (ii) by Im(MLD12 ). Contribution (iii) can be related to Im(A0)/Re(A0) since (ππ)I=0 states provide the dominant contribution to absorptive part of the integral in 12. Collecting the various pieces yields the following expression for the K factor [411,413–416] [ ] Im(MSD) Im(MLD) Im(A ) K = exp(iφ) sin 0(φ) 12 + 12 + , (132) MK MK Re(A0) where the phase of K is given by = Marctan Kφ . (133) K /2 123 Eur. Phys. J. C (2022) 82:869 Page 81 of 296 869 The quantities MK and K are the mass and decay width differences between long- and short-lived neutral kaons. The experimentally known values of the above quantities read [165]: |K | = 2.228(11)× 10−3, (134) φ = 43.52(5)◦ , (135) MK ≡ MKL − MKS = 3.484(6)× 10−12 MeV, (136)  ≡  −  = 7.3382(33)× 10−12K KS KL MeV, (137) where the latter three measurements have been obtained by imposing CPT symmetry. We will start by discussing the short-distance effects (i) since they provide the dominant contribution to K . To lowest order in the electroweak theory, the contribution to K 0 − K̄ 0 oscillations arises from the so-called box diagrams, in which two W bosons and two “up-type” quarks (i.e., up, charm, top) are exchanged between the constituent down and strange quarks of the K mesons. The loop integration of the box diagrams can be performed exactly. In the limit of vanishing external momenta and external quark masses, the result can be identified with an effective four-fermion interaction, expressed in terms of the effective Hamiltonian 2 2 HS=2 = G F MW F0 QS=2eff 2 + h.c. (138)16π In this expression, G F is the Fermi coupling, MW the W -boson mass, and QS=2 = [ ] [ ]s̄γμ(1 − γ5)d s̄γμ(1 − γ5)d ≡ OVV+AA − OVA+AV, (139) is a dimension-six, four-fermion operator. The subscripts V and A denote vector (s̄γμd) and axial-vector (s̄γμγ5d) bilinears, respectively. The function F0 is given by F0 = λ2c S 20(xc) + λt S0(xt ) + 2λcλt S0(xc, xt ), (140) where λ = V ∗a as Vad , and a = c, t denotes a flavour index. The quantities S0(xc), S0(xt ) and S0(xc, xt ) with xc = m2/M2c W, x 2 2t = mt /MW are the Inami–Lim functions [417], which express the basic electroweak loop contributions without QCD corrections. The contribution of the up quark, which is taken to be massless in this approach, has been taken into account by imposing the unitarity constraint λu + λc + λt = 0. When strong interactions are included, S = 2 transitions can no longer be discussed at the quark level. Instead, the effective Hamiltonian must be considered between mesonic initial and final states. Since the strong coupling is large at typical hadronic scales, the resulting weak matrix element cannot be calculated in perturbation theory. The operator product expansion (OPE) does, however, factorize long- and short- distance effects. For energy scales below the charm threshold, the K 0 − K̄ 0 transition amplitude of the effective Hamiltonian can be expressed as 2 2 [ ] 〈K̄ 0|HS=2 0 G F MW 2 2eff |K 〉 = 2 λc S0(xc)η1 + λt S0(xt )η2 + 2λcλt S0(xc, xt )η31(6π 2 )−γ0/(2β0) {∫ ḡ(μ) ( )} × ḡ(μ) γ (g) γexp 0dg + 〈K̄ 0|QS=2 0 4π β(g) β g R (μ)|K 〉 + h.c., (141) 0 0 where ḡ(μ) and QS=2R (μ) are the renormalized gauge coupling and four-fermion operator in some renormalization scheme. The factors η1, η2 and η3 depend on the renormalized coupling ḡ, evaluated at the various flavour thresholds mt ,mb,mc and MW, as required by the OPE and Renormalization Group (RG) running procedure that separate high- and low-energy contributions. Explicit expressions can be found in Ref. [410] and references therein, except that η1 and η3 have been calculated to NNLO in Refs. [418] and [419], respectively. We follow the same conventions for the RG equations as in Ref. [410]. Thus the Callan–Symanzik function and the anomalous dimension γ (ḡ) of QS=2 are defined by dḡ d QS=2= β(ḡ), R = −γ (ḡ) QS=2, (142) d lnμ d lnμ R 123 869 Page 82 of 296 Eur. Phys. J. C (2022) 82:869 with perturbative expansions 3 5 β(g) = − g gβ0 (4π)2 − β1 4 − · · · ,(4π) g2 g4 γ (g) = γ0 + γ1 + · · · . (143) (4π)2 (4π)4 We stress that β0, β1 and γ0 are universal, i.e., scheme independent. As for K 0 − K̄ 0 mixing, this is usually considered in the naive dimensional regularization (NDR) scheme of MS, and below we specify the perturbative coefficient γ1 in that scheme: { } { ( )} = 11 − 2 = 34 13β0 N N f , β1 N 2 − N f N − 1 , 3 3 3 3 N = 6(N − 1) = N − { } 1 − 57 19 4γ0 , γ1 21 + − N + N f . (144) N 2N N 3 3 Note that for QCD the above expressions must be evaluated for N = 3 colours, while N f denotes the number of active quark flavours. As already stated, Eq. (141) is valid at scales below the charm threshold, after all heavier flavours have been integrated out, i.e., N f = 3. In Eq. (141), the terms proportional to η1, η2 and η3, multiplied by the contributions containing ḡ(μ)2, correspond to the Wilson coefficient of the OPE, computed in perturbation theory. Its dependence on the renormalization scheme and scale μ is canceled by that of the weak matrix element 〈K̄ 0|QS=2R (μ)|K 0〉. The latter corresponds to the long-distance effects of the effective Hamiltonian and must be computed nonperturbatively. For historical, as well as technical reasons, it is convenient to express it in terms of the B-parameter BK , defined as 〈 K̄ 0 ∣∣ ∣ 〉QS=2= R (μ) ∣ K 0 BK (μ) 8 . (145)f 23 K m 2 K The four-quark operator QS=2(μ) is renormalized at scale μ in some regularization scheme, for instance, NDR-MS. Assum- ing that BK (μ) and the anomalous dimension γ (g) are both known in that scheme, the renormalization group independent (RGI) B-parameter B̂K is related to BK (μ) by the exact formula ( 2 )−γ0/(2β0) {∫ ( )} = ḡ(μ) ḡ(μ) γ (g) B̂K exp dg + γ0 BK (μ). (146) 4π 0 β(g) β0g At NLO in perturbation theory the above reduces to ( )− { [ ]}2 γ0/(2β0) = ḡ(μ) + ḡ(μ) 2 β1γ0 − β0γ1 B̂K 1 2 2 BK (μ). (147)4π (4π) 2β0 To this order, this is the scale-independent product of all μ-dependent quantities in Eq. (141). Lattice-QCD calculations provide results for BK (μ). However, these results are usually obtained in intermediate schemes other than the continuum MS scheme used to calculate the Wilson coefficients appearing in Eq. (141). Examples of intermediate schemes are the RI/MOM scheme [420] (also dubbed the “Rome–Southampton method”) and the Schrödinger functional (SF) scheme [421]. These schemes are used as they allow a nonperturbative renormalization of the four-fermion operator, using an auxiliary lattice simulation. This allows BK (μ) to be calculated with percent-level accuracy, as described below. In order to make contact with phenomenology, however, and in particular to use the results presented above, one must convert from the intermediate scheme to the MS scheme or to the RGI quantity B̂K . This conversion relies on 1- or 2-loop perturbative matching calculations, the truncation errors in which are, for many recent calculations, the dominant source of error in B̂K (see, for instance, Refs. [8,49,50,188,422]). While this scheme-conversion error is not, strictly speaking, an error of the lattice calculation itself, it must be included in results for the quantities of phenomenological interest, namely, BK (MS, 2 GeV) and B̂K . Incidentally, we remark that this truncation error is estimated in different ways and that its relative contribution to the total error can considerably differ among the various lattice calculations. We note that this error can be 123 Eur. Phys. J. C (2022) 82:869 Page 83 of 296 869 minimized by matching between the intermediate scheme and MS at as large a scale μ as possible (so that the coupling which determines the rate of convergence is minimized). Recent calculations have pushed the matching μ up to the range 3–3.5 GeV. This is possible because of the use of nonperturbative RG running determined on the lattice [8,48,188]. The Schrödinger functional offers the possibility to run nonperturbatively to scales μ ∼ MW where the truncation error can be safely neglected. However, so far this has been applied only for two flavours for BK in Ref. [423] and for the case of the BSM bag parameters in Ref. [424], see more details in Sect. 6.4. Perturbative truncation errors in Eq. (141) also affect the Wilson coefficients η1, η2 and η3. It turns out that the largest uncertainty arises from the charm quark contribution η1 = 1.87(76) [418]. Although it is now calculated at NNLO, the series shows poor convergence. The net effect from the uncertainty on η1 on the amplitude in Eq. (141) is larger than that of present lattice calculations of BK . Exploiting an idea presented in Ref. [425], it has been recently shown in Ref. [426] that, by using the u − t instead of the usual c − t unitarity in the K computation, the perturbative uncertainties associated with residual short-distance quark contributions can be reduced. We will now proceed to discuss the remaining contributions to K in Eq. (132). An analytical estimate of the leading contribution from Im(MLD12 ) based on χPT, shows that it is approximately proportional to ξ ≡ Im(A0)/Re(A0) so that Eq. (132) can be written as follows [415,416] [ ] = Im(M SD  exp(iφ ) sin(φ ) 12 ) K   + ρ ξ , (148) MK where the deviation of ρ from one parameterizes the long-distance effects in Im(M12). In order to facilitate the subsequent discussions about the status of the lattice studies of K → ππ and of the current estimates of ξ , we proceed by providing a brief account of the parameter ′ that describes direct CP-violation in the kaon sector. The definition of ′ is given by: ′ ≡ √1 A[ ( KS → (ππ)I=2] A[KL → (ππ)I=2] − A[KL → ) (ππ)I=0]  A[ → . (149)2 KS (ππ)I=0] A[KS → (ππ)I=2] A[KS → (ππ)I=0] By selecting appropriate phase conventions for the mixing parameters between K 0 and K̄ 0 CP-eigenstates (see e.g. Ref. [408] for further details), the expression of ′ can be expressed in terms of the real and imaginary parts of the isospin amplitudes, as follows [ ] ′ = iω e i(δ2−δ0) √ Im(A2) − ξ , (150) 2 Re(A2) where ω = Re(A2)/Re(A0), A2 denotes the I = 3/2 K → ππ decay amplitude, and δI denotes the strong scattering phase shifts in the corresponding, I = 0, 2, K → (ππ)I decays. Given that the phase, φ′ = δ2 − δ0 + π/2 = 42.3(1.5)◦ [165] is nearly equal to φ in Eq. (135), the ratio of parameters characterizing the direct and indirect CP-violation in the kaon sector can be approximated in the following way, [ ] ′ ≈ ′ = √ ω Im(A2) / Re( /) | | − ξ , (151)2 K Re(A2) where on the left hand side we have set  ≡ K . The experimentally measured value reads [165], Re(′/) = 16.6(2.3)× 10−4. (152) We remark that isospin breaking and electromagnetic effects (see Refs. [427,428], and the discussion in Ref. [409]) introduce additional correction terms into Eq. (151). 6.2 Lattice-QCD studies of the K → (ππ) ′I decay amplitudes, ξ and  / As a preamble to this section, it should be noted that the study of K → ππ decay amplitudes requires the development of computational strategies that are at the forefront of lattice QCD techniques. These studies represent a significant advance in 123 869 Page 84 of 296 Eur. Phys. J. C (2022) 82:869 Table 28 Results for the real and imaginary parts of the K → ππ decay amplitude A0 from lattice-QCD computations with N f = 2+1 dynamical flavours. Information about the renormalization, running and matching to the MS scheme is indicated in the column “running/matching”, with details given at the bottom of the table. We refer to the text for further details about the main differences between the lattice computations in Refs. [429] and [430] Collaboration Refs. N f Re(A0) [10−7 GeV] Im(A0) [10−11 GeV] RBC/UKQCD 20 [429] 2+1 A  ◦ ◦  a 2.99(0.32)(0.59) −6.98(0.62)(1.44) RBC/UKQCD 15G [430] 2+1 A  ◦ ◦  b 4.66(1.00)(1.26) −1.90(1.23)(1.08) a Nonperturbative renormalization with the RI/SMOM scheme at a scale of 1.53 GeV and running to 4.01 GeV employing a nonperturbatively determined step-scaling function. Conversion to MS at 1-loop order b Nonperturbative renormalization with the RI/SMOM scheme at a scale of 1.53 GeV. Conversion to MS at 1-loop order the study of kaon physics. However, at present, they have not yet reached the same level of maturity of most of the quantities analyzed in the FLAG report, where, for instance, independent results by various lattice collaborations are being compared and averaged. In the present version of this section we will therefore review the current status of K → ππ lattice computations, but we will provide a FLAG average only for the case of the decay amplitude A2. We start by reviewing the determination of the parameter ξ = Im(A0)/Re(A0). An estimate of ξ has been obtained from a direct evaluation of the ratio of amplitudes Im(A0)/Re(A0) where Im(A0) is determined from a lattice-QCD computation by RBC/UKQCD 20 [429] employing N f = 2+1 Möbius domain wall fermions at a single value of the lattice spacing while Re(A0) |A0| and the value |A0| = 3.320(2) × 10−7 GeV are used based on the relevant experimental input [165] from the decay to two pions. This leads to a result for ξ with a rather large relative error, ξ = −2.1(5) · 10−4. (153) Following a similar procedure, an estimate of ξ was obtained through the use of a previous lattice QCD determination of Im(A0) by RBC/UKQCD 15G [430]. We refer to Table 28 for further details about these computations of Im(A0). The comparison of the estimates of ξ based on lattice QCD input are collected in Table 30. Another estimate for ξ can be obtained through a lattice-QCD computation of the ratio of amplitudes Im(A2)/Re(A2) by RBC/UKQCD 15F [46] where the continuum–limit result is based on computations at two values of the lattice spacing employing N f = 2 + 1 Möbius domain wall fermions. Further details about the lattice computations of A2 are collected in Table 29. To obtain the value of ξ , the expression in Eq. (151) together with the experimental values of Re(′/), |K | and ω are used. In this case we obtain ξ = −1.6(2) · 10−4. The use of the updated value of Im(A2) = −8.34(1.03) × 10−13 GeV from Ref. [429],26 in combination with the experimental value of Re(A2) = 1.479(4)×10−8 GeV, introduces a small change with respect to the above result. The value for ξ reads27 ξ = −1.7(2) · 10−4. (154) A phenomenological estimate can also be obtained from the relationship of ξ to Re(′/), using the experimental value of the latter and further assumptions concerning the estimate of hadronic contributions. The corresponding value of ξ reads [415,416] 26 The update in Im(A2) is due to a change in the value of the imaginary part of the ratio of CKM matrix elements, τ = −V ∗ts Vtd/V ∗us Vud , as given in Ref. [431]. The lattice QCD input is therefore the one reported in Ref. [46]. 27 The current estimates for the corrections owing to isospin breaking and electromagnetic effects [428] imply a relative change on the theoretical value for ′/ by about -20% with respect to the determination based on Eq. (151). The size of these isospin breaking and electromagnetic corrections is related to the enhancement of the decay amplitudes between the I = 0 and the I = 2 channels. As a consequence, one obtains a similar reduction on ξ , leading to a value that is close to the result of Eq. (153). 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Running/matching Eur. Phys. J. C (2022) 82:869 Page 85 of 296 869 Table 29 Results for the real and the imaginary parts of the K → ππ decay amplitude A2 from lattice-QCD computations with N f = 2 + 1 dynamical flavours. Information about the renormalization and matching to the MS scheme is indicated in the column “running/matching”, with details given at the bottom of the table Collaboration Refs. N f Re(A2) [10−8 GeV] Im(A2) [10−13 GeV] RBC/UKQCD 15F [46] 2+1 A ◦ ◦   a 1.50(0.04)(0.14) −8.34(1.03) a Nonperturbative renormalization with the RI/SMOM scheme at a scale of 3 GeV. Conversion to MS at 1-loop order  This value of Im(A2) is an update reported in Ref. [429] which is based on the lattice QCD computation in Ref. [46] but where a change in the value of the imaginary part of the ratio of CKM matrix elements τ = −V ∗ts Vtd/V ∗us Vud reported in Ref. [431] has been applied Table 30 Results for the parameter ξ = Im(A0)/Re(A0) obtained through the combination of lattice-QCD determinations of K → ππ decay amplitudes with N f = 2 + 1 dynamical flavours and experimental inputs Collaboration Refs. N f ξ RBC/UKQCD 20† [429] 2+1 −2.1(5) · 10−4 RBC/UKQCD 15G [430] 2+1 −0.6(5) · 10−4 RBC/UKQCD 15F∗ [46] 2+1 −1.7(2) · 10−4 †Estimate for ξ obtained from a direct evaluation of the ratio of amplitudes Im(A0)/Re(A0) where Im(A0) is determined from the lattice-QCD computation of Ref. [429] while for Re(A0) |A | is taken from the experimental value for |A | 0 0Estimate for ξ obtained from a direct evaluation of the ratio of amplitudes Im(A0)/Re(A0) where Im(A0) is determined from the lattice-QCD computation of Ref. [430] while for Re(A ∗ 0 ) |A0| is taken from the experimental value for |A0| Estimate for ξ based on the use of Eq. (151). The new value of Im(A2) reported in Ref. [429] – based on the lattice-QCD computation of Ref. [46] following an update of a nonlattice input – is used in combination with the experimental values for Re(A2), Re(′/), |K | and ω = − · − √ξ 6.0(1.5) 10 2 2 |K | = −1.9(5) · 10−4. (155) We note that the use of the experimental value for Re(′/) is based on the assumption that it is free from New Physics con- tributions. The value of ξ can then be combined with a χPT-based estimate for the long-range contribution, ρ = 0.6(3) [416]. Overall, the combination ρξ appearing in Eq. (148) leads to a suppression of the SM prediction of |K | by about 3(2)% relative to the experimental measurement of |K | given in Eq. (134), regardless of whether the phenomenological estimate of ξ [see Eq. (155)] or the most precise lattice result [see Eq. (153)] are used. The uncertainty in the suppression factor is dominated by the error on ρ. Although this is a small correction, we note that its contribution to the error of K is larger than that arising from the value of BK reported below. Efforts are under way to compute the long-distance contributions to K [432] and to the KL − KS mass difference in lattice QCD [425,433–435]. However, the results are not yet precise enough to improve the accuracy in the determination of the parameter ρ. The lattice-QCD study of K → ππ decays provides crucial input to the SM prediction of K . We now proceed to describe the current status of these computations. In recent years, the RBC/UKQCD collaboration has undertaken a series of lattice-QCD calculations of K → ππ decay amplitudes [46,429,430]. In 2015, the first calculation of the K → (ππ)I=0 decay amplitude A0 was performed using physical kinematics on a 323 × 64 lattice with an inverse lattice spacing of a−1 = 1.3784(68) GeV [430,436]. The main features of the RBC/UKQCD 15G calculation included, fixing the I = 0 ππ energy very close to the kaon mass by imposing G-parity boundary conditions, a continuum-like operator mixing pattern through the use of a domain wall fermion action with accurate chiral symmetry, and the construction of the complete set of correlation functions by computing seventy-five distinct diagrams. Results for the real and the imaginary parts of the decay amplitude A0 from the RBC/UKQCD 15G computation are collected in Table 28, where the first error is statistical and the second one is systematic. The latest 2020 calculation RBC/UKQCD 20 [429] using the same lattice setup has improved the 2015 calculation RBC/UKQCD 15G [430] in three important aspects: (i) an increase by a factor of 3.4 in statistics; (ii) the inclusion of a scalar two-quark operator and the addition of another pion-pion operator to isolate the ground state, and (iii) the use of step 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Running/matching 869 Page 86 of 296 Eur. Phys. J. C (2022) 82:869 scaling techniques to raise the renormalization scale from 1.53 to 4.01 GeV. The updated determinations of the real and the imaginary parts of A0 in Ref. [429] are shown in Table 28. As previously discussed, the determination of Im(A0) from Ref. [429] has been used to obtain the value of the parameter ξ in Eq. (153). A first-principles computation of Re(A0) is essential to address the so-called I = 1/2 puzzle associated to the enhancement of I = 1/2 over I = 3/2 transitions owing, crucially, to long distance effects. Indeed, short- distance enhancements in the Wilson coefficients are not large enough to explain the I = 1/2 rule [437,438]. Lattice-QCD calculations do provide a method to study such a long-distance enhancement. The combination of the result for A0 in Table 28 with the earlier lattice calculation of A2 in Ref. [46] leads to the ratio, Re(A0)/Re(A2) = 19.9(5.0), which agrees with the experimentally measured value, Re(A0)/Re(A2) = 22.45(6). In Ref. [429], the lattice determination of relative size of direct CP violation was updated as follows, Re(′/) = 21.7(2.6)(6.2)(5.0)× 10−4, (156) where the first two errors are statistical and systematic, respectively. The third error arises from the omitted strong and electromagnetic isospin breaking effects. The value of Re(′/) in Eq. (156) uses the experimental values of Re(A0) and Re(A2). The lattice determination of Re(′/) is in good agreement with the experimental result in Eq. (152). However, while the result in Eq. (156) represents a significant step forward, it is important to keep in mind that the calculation of A0 is currently based on a single value of the lattice spacing. It is expected that future work with additional values of the lattice spacing will contribute to improve the precision. For a description of the computation of the ππ scattering phase shifts entering in the determination of Re(′/) in Eq. (156), we refer to Ref. [439]. The real and imaginary values of the amplitude A2 have been determined by RBC/UKQCD 15F [46] employing N f = 2+1 Möbius domain wall fermions at two values of the lattice spacing, namely a = 0.114 fm and 0.083 fm, and performing simulations at the physical pion mass with Mπ L ≈ 3.8. A compilation of lattice results for the real and imaginary parts of the K → ππ decay amplitudes, A0 and A2, with N f = 2 + 1 flavours of dynamical quarks is shown in Tables 28 and 29. In Appendix C.4.1 we collect the corresponding information about the lattice QCD simulations, including the values of some of the most relevant parameters. The results for the parameter ξ , determined through the combined use of K → ππ amplitudes computed on the lattice and experimental inputs, are presented in Table 30. As previously discussed, we remark that the total uncertainty on the reported values of ξ depends on the specific way in which the lattice and experimental inputs are selected. The determination of the real and imaginary parts of A2 by RBC/UKQCD 15F shown in Table 29 is free of red tags. We therefore quote the following FLAG averages: = + : Re(A2) = 1.50(0.04)(0.14)× 10 −8 GeV, N f 2 1 Ref. [46]. (157) Im(A2) = −8.34(1.03)× 10−13 GeV, Besides the RBC/UKQCD collaboration programme [46,429,430] using domain-wall fermions, an approach based on improved Wilson fermions [440,441] has presented a determination of the K → ππ decay amplitudes, A0 and A2, at unphysical quark masses. For an analysis of the scaling with the number of colours of K → ππ decay amplitudes using lattice-QCD computations, we refer to Refs. [442,443]. Recent proposals aiming at the inclusion of electromagnetism in lattice-QCD calculations of K → ππ decays are being explored [444,445] in order to reduce the uncertainties associated with isospin breaking effects. Finally, we notice that K receives a contribution from |Vcb| through the λt parameter in Eq. (140). The present uncertainty on |Vcb| has a significant impact on the error of K (see, e.g., Refs. [446,447] and the recent update in Ref. [448]). 6.3 Lattice computation of BK Lattice calculations of BK are affected by the same type of systematic effects discussed in previous sections of this review. However, the issue of renormalization merits special attention. The reason is that the multiplicative renormalizability of the relevant operator QS=2 is lost once the regularized QCD action ceases to be invariant under chiral transformations. As a result, the renormalization pattern of BK depends on the specific choice of the fermionic discretization. In the case of Wilson fermions, QS=2 mixes with four additional dimension-six operators, which belong to different representations of the chiral group, with mixing coefficients that are finite functions of the gauge coupling. This complicated renormalization pattern was identified as the main source of systematic error in earlier, mostly quenched calculations of BK 123 Eur. Phys. J. C (2022) 82:869 Page 87 of 296 869 with Wilson quarks. It can be bypassed via the implementation of specifically designed methods, which are either based on Ward identities [449] or on a modification of the Wilson quark action, known as twisted-mass QCD [450–452]. An advantage of staggered fermions is the presence of a remnant U (1) chiral symmetry. However, at nonvanishing lattice spacing, the symmetry among the extra unphysical degrees of freedom (tastes) is broken. As a result, mixing with other dimension-six operators cannot be avoided in the staggered formulation, which complicates the determination of the B- parameter. In general, taste conserving mixings are implemented directly in the lattice computation of the matrix element. The effects of the broken taste symmetry are usually treated through an effective field theory, staggered Chiral Perturbation Theory (SχPT) [453,454], parameterizing the quark-mass and lattice-spacing dependences. Fermionic lattice actions based on the Ginsparg–Wilson relation [455] are invariant under the chiral group, and hence four-quark operators such as QS=2 renormalize multiplicatively. However, depending on the particular formulation of Ginsparg–Wilson fermions, residual chiral symmetry breaking effects may be present in actual calculations. For instance, in the case of domain-wall fermions, the finiteness of the extra 5th dimension implies that the decoupling of modes with different chirality is not exact, which produces a residual nonzero quark mass in the chiral limit. The mixing with dimension- six operators of different chirality is expected to be an O(m2res) suppressed effect [456,457] that should be investigated on a case-by-case basis. Before proceeding to the description and compilation of the results of BK , we would like to reiterate a discussion presented in the previous FLAG report about an issue related to the computation of the kaon bag parameters through lattice-QCD simulations with N f = 2+1+1 dynamical quarks. In practice, this only concerns the calculations of the kaon B-parameters including dynamical charm-quark effects in Ref. [47], that were examined in the FLAG 16 report. As described in Sect. 6.1, the effective Hamiltonian in Eq. (138) depends solely on the operator QS=2 in Eq. (139) – which appears in the definition of BK in Eq. (145) – at energy scales below the charm threshold where charm-quark contributions are absent. As a result, a computation of BK based on N f = 2 + 1 + 1 dynamical simulations will include an extra sea-quark contribution from charm-quark loop effects for which there is at present no direct evaluation in the literature. When the matrix element of QS=2 is evaluated in a theory that contains a dynamical charm quark, the resulting estimate for BK must then be matched to the three-flavour theory that underlies the effective four-quark interaction.28 In general, the matching of 2 + 1-flavour QCD with the theory containing 2 + 1 + 1 flavours of sea quarks is performed around the charm threshold. It is usually accomplished by requiring that the coupling and quark masses are equal in the two theories at a renormalization scale μ around mc. In addition, BK should be renormalized and run, in the four-flavour theory, to the value of μ at which the two theories are matched, as described in Sect. 6.1. The corrections associated with this matching are of order (E/mc)2, where E is a typical energy in the process under study, since the subleading operators have dimension eight [458]. When the kaon-mixing amplitude is considered, the matching also involves the relation between the relevant box diagrams and the effective four-quark operator. In this case, corrections of order (E/mc)2 arise not only from the charm quarks in the sea, but also from the valence sector, since the charm quark propagates in the box diagrams. We note that the original derivation of the effective four-quark interaction is valid up to corrections of order (E/mc)2. The kaon-mixing amplitudes evaluated in the N f = 2+1 and 2+1+1 theories are thus subject to corrections of the same order in E/mc as the derivation of the conventional four-quark interaction. Regarding perturbative QCD corrections at the scale of the charm-quark mass on the amplitude in Eq. (141), the uncertainty on η1 and η3 factors is of O(αs(mc)3) [418,419], while that on η2 is of O(αs(m 2c) ) [459].29 On the other hand, the corrections of order (E/m 2c) due to dynamical charm-quark effects in the matching of the amplitudes are further suppressed by powers of αs(mc) and by a factor of 1/Nc, given that they arise from quark-loop diagrams. In order to make progress in resolving this so far uncontrolled systematic uncertainty, it is essential that any future calculation of BK with N f = 2 + 1+ 1 flavours properly addresses the size of these residual dynamical charm effects in a quantitative way. Another issue in this context is how the lattice scale and the physical values of the quark masses are determined in the 2+1 and 2+ 1+ 1 flavour theories. Here it is important to consider in which way the quantities used to fix the bare parameters are affected by a dynamical charm quark. A recent study [167] using three degenerate light quarks, together with a charm quark, indicates that the deviations between the N f = 3 + 1 and the N f = 3 theories are considerably below the 1% level in dimensionless quantities constructed 28 We thank Martin Lüscher for an interesting discussion on this issue. 29 The recent results [426] based on the use of u− t unitarity for the two corresponding perturbative factors, also have an uncertainty of O(αs(m )2c ) and O(αs(m 3c) ). The estimates for the missing higher-order contributions are, however, expected to be reduced with respect to the more traditional case where c − t unitarity is used. 123 869 Page 88 of 296 Eur. Phys. J. C (2022) 82:869 Table 31 Results for the kaon B-parameter in QCD with N f = 2 + 1 + 1 and N f = 2 + 1 dynamical flavours, together with a summary of systematic errors. Information about nonperturbative running is indicated in the column “running”, with details given at the bottom of the table Collaboration Refs. N f BK (MS, 2 GeV) B̂K ETM 15 [47] 2+1+1 A  ◦ ◦  a 0.524(13)(12) 0.717(18)(16)1 RBC/UKQCD 16 [52] 2+1 A ◦ ◦ ◦  b 0.543(9)(13)2 0.744(13)(18)3 SWME 15A [50] 2+1 A  ◦  ◦‡ − 0.537(4)(26) 0.735(5)(36)4 RBC/UKQCD 14B [8] 2+1 A     b 0.5478(18)(110)2 0.7499(24)(150) SWME 14 [422] 2+1 A  ◦  ◦‡ − 0.5388(34)(266) 0.7379(47)(365) SWME 13A [461] 2+1 A  ◦  ◦‡ − 0.537(7)(24) 0.735(10)(33) SWME 13 [462] 2+1 C  ◦  ◦‡ − 0.539(3)(25) 0.738(5)(34) RBC/UKQCD 12A [188] 2+1 A ◦  ◦  b 0.554(8)(14)2 0.758(11)(19) Laiho 11 [49] 2+1 C  ◦ ◦  − 0.5572(28)(150) 0.7628(38)(205)4 SWME 11A [463] 2+1 A  ◦ ◦ ◦‡ − 0.531(3)(27) 0.727(4)(38) BMW 11 [48] 2+1 A     c 0.5644(59)(58) 0.7727(81)(84) RBC/UKQCD 10B [464] 2+1 A ◦ ◦   d 0.549(5)(26) 0.749(7)(26) SWME 10 [465] 2+1 A  ◦ ◦ ◦ − 0.529(9)(32) 0.724(12)(43) Aubin 09 [466] 2+1 A ◦ ◦ ◦  − 0.527(6)(21) 0.724(8)(29) ‡The renormalization is performed using perturbation theory at 1-loop, with a conservative estimate of the uncertainty a BK is renormalized nonperturbatively at scales 1/a ∼ 2.2−3.3 GeV in the N f = 4 RI/MOM scheme using two different lattice momentum scale intervals, the first around 1/a while the second around 3.5 GeV. The impact of the two ways to the final result is taken into account in the error budget. Conversion to MS is at 1-loop at 3 GeV b BK is renormalized nonperturbatively at a scale of 1.4 GeV in two RI/SMOM schemes for N f = 3, and then run to 3 GeV using a nonperturbatively determined step-scaling function. Conversion to MS is at 1-loop order at 3 GeV c BK is renormalized and run nonperturbatively to a scale of 3.5 GeV in the RI/MOM scheme. At the same scale conversion at 1-loop to MS is applied. Nonperturbative and NLO perturbative running agrees down to scales of 1.8 GeV within statistical uncertainties of about 2% d BK is renormalized nonperturbatively at a scale of 2 GeV in two RI/SMOM schemes for N f = 3, and then run to 3 GeV using a nonperturbatively determined step-scaling function. Conversion to MS is at 1-loop order at 3 GeV 1 BK (MS, 2 GeV) and B̂K are related using the conversion factor 1.369, i.e., the one obtained with N f = 2 + 1 2 BK (MS, 2 GeV) is obtained from the estimate for B̂K using the conversion factor 1.369 3 B̂K is obtained from BK (MS, 3 GeV) using the conversion factor employed in Ref. [8] 4 B̂K is obtained from the estimate for BK (MS, 2 GeV) using the conversion factor 1.369 from ratios of gradient flow observables, such as t0 and w0, used for scale setting. This study extends the nonperturbative investigations with two heavy mass-degenerate quarks [158,160] which indicate that dynamical charm-quark effects in low- energy hadronic observables are considerably smaller than the expectation from a naive power counting in terms of αs(mc). For an additional discussion on this point, we refer to Ref. [47]. Given the hierarchy of scales between the charm-quark mass and that of BK , we expect these errors to be modest, but a more quantitative understanding is needed as statistical errors on BK are reduced. Within this review we will not discuss this issue further. However, we wish to point out that the present discussion also applies to N f = 2 + 1 + 1 computations of the kaon BSM B-parameters discussed in Sect. 6.4. A compilation of results for BK with N f = 2, 2 + 1 and 2 + 1 + 1 flavours of dynamical quarks is shown in Tables 31 and 32, as well as Fig. 20. An overview of the quality of systematic error studies is represented by the colour coded entries in Tables 31 and 32. The values of the most relevant lattice parameters, and comparative tables on the various estimates of systematic errors have been collected in the corresponding Appendices of the previous FLAG editions [2–4]. Since the last edition of the FLAG report no new results for BK have appeared in the bibliography. We mention here an ongoing work related to the BK computation where the relevant operators are defined in the gradient flow framework. In a first publication [460] the small flow time expansion method is applied in order to compute, to 1-loop approximation, the finite matching coefficients between the gradient flow and the MS schemes for the operators entering the BK computation. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Running Eur. Phys. J. C (2022) 82:869 Page 89 of 296 869 Table 32 Results for the kaon B-parameter in QCD with N f = 2 dynamical flavours, together with a summary of systematic errors. Information about nonperturbative running is indicated in the column “running”, with details given at the bottom of the table Collaboration Refs. N f BK (MS, 2 GeV) B̂K ETM 12D [51] 2 A  ◦ ◦  e 0.531(16)(9) 0.727(22)(12)1 ETM 10A [467] 2 A  ◦ ◦  f 0.533(18)(12)1 0.729(25)(17) e BK is renormalized nonperturbatively at scales 1/a ∼ 2 − 3.7 GeV in the N f = 2 RI/MOM scheme. In this scheme, nonperturbative and NLO perturbative running are shown to agree from 4 GeV down to 2 GeV to better than 3% [467,468] f BK is renormalized nonperturbatively at scales 1/a ∼ 2 − 3 GeV in the N f = 2 RI/MOM scheme. In this scheme, nonperturbative and NLO perturbative running are shown to agree from 4 GeV down to 2 GeV to better than 3% [467,468] 1 BK (MS, 2 GeV) and B̂K are related using the conversion factor 1.369, i.e., the one obtained with N f = 2 + 1 For a detailed description of previous BK calculations – and in particular those considered in the computation of the average values – we refer the reader to the FLAG 19 [4], FLAG 16 [3] and FLAG 13 [2] reports. We now give the global averages for BK for N f = 2 + 1 + 1, 2 + 1 and 2 dynamical flavours. The details about the calculation of these averages can be found in FLAG 19 [4]. We begin with the N f = 2 + 1 global average since it is estimated by employing four different BK results, namely BMW 11 [48], Laiho 11 [49], RBC/UKQCD 14B [8] and SWME 15A [50]. Note also that the expression of K in terms of BK is obtained in the three-flavour theory (see Sect. 6.1). After constructing the global covariance matrix according to Schmelling [163], we arrive at: N f = 2 + 1 : B̂K = 0.7625(97) Refs. [8,48–50], (158) with χ2/dof = 0.675. After applying the NLO conversion factor B̂ /BMSK K (2 GeV) = 1.369,30 this translates into N f = 2 + 1 : BMSK (2 GeV) = 0.5570(71) Refs. [8,48–50]. (159) Note that the statistical errors of each calculation entering the global average are small enough to make their results statistically incompatible. It is only because of the relatively large systematic errors that the weighted average produces a value of O(1) for the reduced χ2. There is only a single result for N f = 2 + 1 + 1, computed by the ETM collaboration [47]. Since it is free of red tags, it qualifies as the currently best global average, i.e., N f = 2 + 1 + 1 : B̂K = 0.717(18)(16) , BMSK (2 GeV) = 0.524(13)(12) Ref. [47]. (160) For N f = 2 flavours the best global average is given by a single result, that of ETM 12D [51]: N f = 2 : B̂K = 0.727(22)(12), BMSK (2 GeV) = 0.531(16)(19) Ref. [51]. (161) The result in the MS scheme has been obtained by applying the same conversion factor of 1.369 as in the three-flavour theory. 6.4 Kaon BSM B-parameters We now report on lattice results concerning the matrix elements of operators that encode the effects of physics beyond the Standard Model (BSM) to the mixing of neutral kaons. In this theoretical framework both the SM and BSM contributions add 30 We refer to the FLAG 19 report [4] for a discussion about the estimates of these conversion factors. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Running 869 Page 90 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 20 Recent unquenched lattice results for the RGI B-parameter B̂K . The grey bands indicate our global averages described in the text. For N f = 2 + 1 + 1 and N f = 2 the global averages coincide with the results by ETM 15 and ETM 12D, respectively up to reproduce the experimentally observed value of K . Since BSM contributions involve heavy but unobserved particles they are short-distance dominated. The effective Hamiltonian for generic S = 2 processes including BSM contributions reads 5 HS= ∑ 2 eff,BSM = Ci (μ)Qi (μ), (162) i=1 where Q1 is the four-quark operator of Eq. (139) that gives rise to the SM contribution to K . In the so-called SUSY basis introduced by Gabbiani et al. [469] the operators Q 312, . . . , Q5 read ( )( ) Q = a a b b2 = ( s̄ (1 − γ5)d )(s̄ (1 − γ5)d , Q3 (s̄ a(1 − γ5)db)(s̄ b(1 − γ a)5)d , Q = s̄a4 ( (1 − ) (163) γ5)d a b )(s̄ (1 + γ b 5)d = a − ) , Q5 s̄ (1 γ5)d b s̄b(1 + γ5)da , where a and b denote colour indices. In analogy to the case of BK one then defines the B-parameters of Q2, . . . , Q5 according to 〈 | 〉K̄ 0 Q 0= 〈 i (μ)| KBi (μ) 〉 〈 〉 , i = 2, . . . , 5. (164) N 0 0i K̄ |s̄γ5d| 0 0 |s̄γ5d| K The factors {N2, . . . , N5} are given by {−5/3, 1/3, 2, 2/3}, and it is understood that Bi (μ) is specified in some renormalization scheme, such as MS or a variant of the regularization-independent momentum subtraction (RI-MOM) scheme. The SUSY basis has been adopted in Refs. [47,51,52,470]. Alternatively, one can employ the chiral basis of Buras, Misiak and Urban [471]. The SWME collaboration prefers the latter since the anomalous dimension that enters the RG running has been calculated to 2-loops in perturbation theory [471]. Results obtained in the chiral basis can be easily converted to the SUSY basis via ( ) BSUSY = 1 5Bchiral chiral3 2 2 − 3B3 . (165) The remaining B-parameters are the same in both bases. In the following we adopt the SUSY basis and drop the superscript. 31 Thanks to QCD parity invariance lattice computations for three more dimension-six operators, whose parity conserving parts coincide with the corresponding parity conserving contributions of the operators Q1, Q2 and Q3, can be ignored. 123 Eur. Phys. J. C (2022) 82:869 Page 91 of 296 869 Table 33 Results for the BSM B-parameters B2, . . . , B5 in the MS scheme at a reference scale of 3 GeV. Information about nonperturbative running is indicated in the column “running”, with details given at the bottom of the table. Collaboration Refs. N f B2 B3 B4 B5 ETM 15 [47] 2+1+1 A  ◦ ◦  a 0.46(1)(3) 0.79(2)(5) 0.78(2)(4) 0.49(3)(3) RBC/UKQCD [52] 2+1 A ◦ ◦ ◦  b 0.488(7)(17) 0.743(14)(65) 0.920(12)(16) 0.707(8)(44) 16 SWME 15A [50] 2+1 A  ◦  ◦† − 0.525(1)(23) 0.773(6)(35) 0.981(3)(62) 0.751(7)(68) SWME 14C [476] 2+1 C  ◦  ◦† − 0.525(1)(23) 0.774(6)(64) 0.981(3)(61) 0.748(9)(79) SWME 13A‡ [461] 2+1 A  ◦  ◦† − 0.549(3)(28) 0.790(30) 1.033(6)(46) 0.855(6)(43) RBC/ [470] 2+1 A  ◦   b 0.43(1)(5) 0.75(2)(9) 0.69(1)(7) 0.47(1)(6) UKQCD 12E ETM 12D [51] 2 A  ◦ ◦  c 0.47(2)(1) 0.78(4)(2) 0.76(2)(2) 0.58(2)(2) †The renormalization is performed using perturbation theory at 1-loop, with a conservative estimate of the uncertainty a Bi are renormalized nonperturbatively at scales 1/a ∼ 2.2−3.3 GeV in the N f = 4 RI/MOM scheme using two different lattice momentum scale intervals, with values around 1/a for the first and around 3.5 GeV for the second one. The impact of these two ways to the final result is taken into account in the error budget. Conversion to MS is at 1-loop at 3 GeV b The B-parameters are renormalized nonperturbatively at a scale of 3 GeV c Bi are renormalized nonperturbatively at scales 1/a ∼ 2− 3.7 GeV in the N f = 2 RI/MOM scheme using two different lattice momentum scale intervals, with values around 1/a for the first and around 3 GeV for the second one ‡The computation of B4 and B5 has been revised in Refs. [50] and [476] Older quenched results for the BSM B-parameters can be found in Refs. [472–474]. For a nonlattice approach to get estimates for the BSM B-parameters see Ref. [475]. Estimates for B2, . . . , B5 have been reported for QCD with N f = 2 (ETM 12D [51]), N f = 2 + 1 (RBC/UKQCD 12E [470], SWME 13A [461], SWME 14C [476], SWME 15A [50], RBC/UKQCD 16 [52,477]) and N f = 2 + 1 + 1 (ETM 15 [47]) flavours of dynamical quarks. Since the publication of the FLAG 19 report [4] no new results for the BSM B- parameters have appeared in the bibliography. The available results are listed and compared in Table 33 and Fig. 21. In general one finds that the BSM B-parameters computed by different collaborations do not show the same level of consistency as the SM kaon-mixing parameter BK discussed previously. Control over the systematic uncertainties from chiral and continuum extrapolations as well as finite-volume effects in B2, . . . , B5 is expected to be at a commensurate level as for BK , as far as the results by ETM 12D, ETM 15, SWME 15A and RBC/UKQCD 16 are concerned, since the set of gauge ensembles employed in both kinds of computations is the same. The calculation by RBC/UKQCD 12E has been performed at a single value of the lattice spacing and a minimum pion mass of 290 MeV. Let us notice that as reported in RBC/UKQCD 16 [52] the comparison of results obtained in the conventional RI-MOM and two RI-SMOM schemes shows significant discrepancies for B4 and B5 in the MS scheme at the scale of 3 GeV, which amount up to 2.8σ in the case of B5. By contrast, the agreement for B2 and B3 determined for different intermediate scheme is much better. The RBC/UKQCD collaboration has presented an ongoing study [478] in which simulations with two values of the lattice spacing at the physical point and with a third finer lattice spacing at Mπ = 234 MeV are employed in order to obtain the BSM matrix elements in the continuum limit. Results are still preliminary. The findings by RBC/UKQCD 16 [52,477] provide evidence that the nonperturbative determination of the matching factors depends strongly on the details of the implementation of the Rome–Southampton method. The use of nonexceptional momentum configurations in the calculation of the vertex functions produces a significant modification of the renormalization factors, which affects the matching between MS and the intermediate momentum subtraction scheme. This effect is most pronounced in B4 and B5. Furthermore, it can be noticed that the estimates for B4 and B5 from RBC/UKQCD 16 are much closer to those of SWME 15A. At the same time, the results for B2 and B3 obtained by ETM 15, SWME 15A and RBC/UKQCD 16 are in good agreement within errors. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Running 869 Page 92 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 21 Lattice results for the BSM B-parameters defined in the MS scheme at a reference scale of 3 GeV, see Table 33 A nonperturbative computation of the running of the four-fermion operators contributing to the B2, …, B5 parameters has been carried out with two dynamical flavours using the Schrödinger functional renormalization scheme [424]. Renormalization matrices of the operator basis are used to build step-scaling functions governing the continuum-limit running between hadronic and electroweak scales. A comparison to perturbative results using NLO (2-loops) for the four-fermion operator anomalous dimensions indicates that, at scales of about 3 GeV, nonperturbative effects can induce a sizeable contribution to the running. A detailed look at the calculations reported in the works of ETM 15 [47], SWME 15A [50] and RBC/UKQCD 16 [52] reveals that cutoff effects appear to be larger for the BSM B-parameters compared to BK . Depending on the details of the renormalization procedure and/or the fit ansatz for the combined chiral and continuum extrapolation, the results obtained at the coarsest lattice spacing differ by 15–30%. At the same time the available range of lattice spacings is typically much reduced compared to the corresponding calculations of BK , as can be seen by comparing the quality criteria in Tables 31 and 33. Hence, the impact of the renormalization procedure and the continuum limit on the BSM B-parameters certainly requires further investigation. Finally we present our estimates for the BSM B-parameters, quoted in the MS-scheme at scale 3 GeV. For N f = 2 + 1 our estimate is given by the average between the results from SWME 15A and RBC/UKQCD 16, i.e., N f = 2 + 1 : B2 = 0.502(14), B3 = 0.766(32), B4 = 0.926(19), B5 = 0.720(38), Refs. [50,52]. (166) For N f = 2 + 1 + 1 and N f = 2, our estimates coincide with the ones by ETM 15 and ETM 12D, respectively, since there is only one computation for each case. Thus we quote N f = 2 + 1 + 1 : B2 = 0.46(1)(3), B3 = 0.79(2)(5), B4 = 0.78(2)(4), B5 = 0.49(3)(3), Ref. [47], (167) N f = 2 : B2 = 0.47(2)(1), B3 = 0.78(4)(2), B4 = 0.76(2)(2), B5 = 0.58(2)(2), Ref. [51]. (168) Based on the above discussion on the effects of employing different intermediate momentum subtraction schemes in the nonperturbative renormalization of the operators, the discrepancy for B4 and B5 results between N f = 2, 2 + 1 + 1 and N f = 2 + 1 computations should not be considered an effect associated with the number of dynamical flavours. To clarify the present situation, it would be important to perform a direct comparison of results by the ETM collaboration obtained both with RI-MOM and RI-SMOM methods. Furthermore, extending the computation of the BSM-B parameters to include physical point simulations with improved continuum-limit extrapolations would also provide valuable information. As a closing remark, we encourage authors to provide the correlation matrix of the Bi parameters since this information is required in phenomenological studies of New Physics scenarios. 123 Eur. Phys. J. C (2022) 82:869 Page 93 of 296 869 7 Charm hadron decay constants and form factors Authors: Y. Aoki, M. Della Morte, E. Lunghi, S. Meinel, C. Monahan, C. Pena Leptonic and semileptonic decays of charmed D and Ds mesons or c and other charm baryons occur via charged W -boson exchange, and are sensitive probes of c → d and c → s quark flavour-changing transitions. Given experimental measurements of the branching fractions combined with sufficiently precise theoretical calculations of the hadronic matrix elements, they enable the determination of the CKM matrix elements |Vcd | and |Vcs | (within the Standard Model) and a precise test of the unitarity of the second row of the CKM matrix. Here, we summarize the status of lattice-QCD calculations of the charmed leptonic decay constants. Significant progress has been made in charm physics on the lattice in recent years, largely due to the availability of gauge configurations produced using highly-improved lattice-fermion actions that enable treating the c quark with the same action as for the u, d, and s quarks. This section updates the corresponding one in the last FLAG review [4] for results that appeared before April 30, 2021. As already done in Ref. [4], we limit our review to results based on modern simulations with reasonably light pion masses (below approximately 500 MeV). Following our review of lattice-QCD calculations of D(s)-meson leptonic decay constants and charm-hadron semileptonic form factors, we then interpret our results within the context of the Standard Model. We combine our best-determined values of the hadronic matrix elements with the most recent experimentally-measured branching fractions to obtain |Vcd(s)| and test the unitarity of the second row of the CKM matrix. 7.1 Leptonic decay constants fD and fDs In the Standard Model, and up to electromagnetic corrections, the decay constant fD s of a pseudoscalar D or D( ) s meson is related to the branching ratio for leptonic decays mediated by a W boson through the formula | ( )2 2B → = G F V 2 2 cq | τD D (s)( ν ) f 2 m2 m (s)  D m D s 1 − , (169)8π (s) ( ) m2D(s) where q is d or s and Vcd (Vcs) is the appropriate CKM matrix element for a D (Ds) meson. The branching fractions have been experimentally measured by CLEO, Belle, Babar and BES with a precision around 4–5% for both the D and the Ds-meson decay modes [164]. When combined with lattice results for the decay constants, they allow for determinations of |Vcs | and |Vcd |. In lattice-QCD calculations, the decay constants fD s are extracted from Euclidean matrix elements of the axial current( ) 〈0|Aμcq |D p 〉 = i f pμq( ) Dq D , (170)q with q = d, s and Aμcq = c̄γμγ5q. Results for N f = 2, 2 + 1 and 2 + 1 + 1 dynamical flavours are summarized in Table 34 and Fig. 22. Since the publication of the last FLAG review, a handful of results for fD and fDs have appeared, as described below. We consider isospin-averaged quantities, although, in a few cases, results for fD+ are quoted (see, for example, the FNAL/MILC 11,14A and 17 computations, where the difference between fD and fD+ has been estimated to be around 0.5 MeV). Only one new computation appeared for N f = 2. Reference [58], Balasubramanian 19, updates the result for fDs in Blossier 18 [479] (discussed in the previous review) by including in the analysis two additional ensembles at a coarser lattice spacing (a = 0.075 fm, compared to 0.065 fm and 0.048 fm used in Ref. [479]). Pion masses at this coarser resolution reach 282 MeV and Mπ L is always kept larger than 4. The N f = 2 averages for fD and fDs/ fD coincide with those in the previous FLAG review and are given by the values in ETM 13B [56], while the estimate for fDs is the result of the weighted average of the numbers in ETM 13B [56] and Balasubramanian 19 [58]. They read N f = 2 : fD = 208(7) MeV Ref. [56], (171) N f = 2 : fDs = 246(4) MeV Refs. [56,58], (172) = : fDN 2 sf = 1.20(0.02) Ref. [56]. (173) fD 123 869 Page 94 of 296 Eur. Phys. J. C (2022) 82:869 Table 34 Decay constants of the D and Ds mesons (in MeV) and their ratio Collaboration Refs. N f fD fDs fDs / fD FNAL/MILC 17 ∇∇ [16] 2+1+1 A      212.1(0.6) 249.9(0.5) 1.1782(16) ( ) ( ) ( ) FNAL/MILC 14A∗∗ [17] 2+1+1 A      212.6(0.4) +1.0−1 2 249.0(0.3) +1.1 1.1745(10) +29. −1.5 −32 ETM 14E† [38] 2+1+1 A  ◦ ◦   207.4(3.8) 247.2(4.1) 1.192(22) ETM 13F [306] 2+1+1 C ◦ ◦ ◦   202(8) 242(8) 1.199(25) FNAL/MILC 13∇ [480] 2+1+1 C      212.3(0.3)(1.0) 248.7(0.2)(1.0) 1.1714(10)(25) FNAL/MILC 12B [481] 2+1+1 C      209.2(3.0)(3.6) 246.4(0.5)(3.6) 1.175(16)(11) χQCD 20A†† [482] 2+1 A      213(5) 249(7) 1.16(3) ∇ ( )RBC/UKQCD 18A [70] 2+1 P      1.1740(51) +68 ( ) ( ) (−68) RBC/UKQCD 17 [55] 2+1 A   ◦   208.7(2.8) +2.1 +1.3−1 8 246.4(1.3) −1 9 1.1667(77) +57. . −43 χQCD 14† [24] 2+1 A ◦ ◦ ◦   254(2)(4) HPQCD 12A [53] 2+1 A ◦ ◦ ◦   208.3(1.0)(3.3) 246.0(0.7)(3.5) 1.187(4)(12) FNAL/MILC 11 [54] 2+1 A ◦ ◦ ◦ ◦  218.9(11.3) 260.1(10.8) 1.188(25) PACS-CS 11 [483] 2+1 A    ◦  226(6)(1)(5) 257(2)(1)(5) 1.14(3) HPQCD 10A [57] 2+1 A  ◦    213(4)∗ 248.0(2.5) HPQCD/UKQCD 07 [40] 2+1 A ◦ ◦ ◦   207(4) 241 (3) 1.164(11) FNAL/MILC 05 [484] 2+1 A ◦ ◦  ◦  201(3)(17) 249(3)(16) 1.24(1)(7) Balasubramanian 19 [58] 2 A      244(4)(2) Blossier 18 [479] 2 A ◦  ◦   238(5)(2) TWQCD 14 [485] 2 A  ◦    202.3(2.2)(2.6) 258.7(1.1)(2.9) 1.2788(264) ALPHA 13B [486] 2 C ◦  ◦   216(7)(5) 247(5)(5) 1.14(2)(3) ETM 13B [56] 2 A  ◦ ◦   208(7) 250(7) 1.20(2) ETM 11A [231] 2 A  ◦ ◦   212(8) 248(6) 1.17(5) ETM 09 [45] 2 A ◦ ◦ ◦   197(9) 244(8) 1.24(3) †Update of ETM 13F ∇Update of FNAL/MILC 12B ∗This result is obtained by using the central value for fDs / fD from HPQCD/UKQCD 07 and increasing the error to account for the effects from the change in the physical value of r1 Update of ETM 11A and ETM 09 One lattice spacing 0.1 fm only. mπ,min L = 1.93 ∗∗At β = 5.8, mπ,min L = 3.2 but this lattice spacing is not used in the final cont./chiral extrapolations ∇∇Update of FNAL/MILC 14A. The ratio quoted is fDs / fD+ = 1.1749(16). In order to compare with results from other collaborations, we rescale the number by the ratio of central values for fD+ and fD . We use the same rescaling in FNAL/MILC 14A. At the finest lattice spacing the finite-volume criterium would produce an empty green circle, however, as checked by the authors, results would not significantly change by excluding this ensemble, which instead sharpens the continuum limit extrapolation ∇Update of RBC/UKQCD 17 †Two values of sea pion masses ††Four valence pion masses between 208 and 114 MeV have been used at one value of the sea pion mass of 139 MeV Turning to N f = 2 + 1 results, the χQCD collaboration presented in χQCD 20A [482] a calculation of the D(∗)s , D(∗) and φ meson decay constants. The couplings of the vector mesons to the tensor current are also computed. The computation is performed at a single lattice spacing with a−1 ≈ 1.7 GeV on a 2 + 1 domain wall fermion ensemble generated by the RBC/UKQCD Collaboration. The sea pion mass is at its physical value and the spatial extension is 5.5 fm. Overlap valence fermions are used with different values of the light, strange and (quenched) charm quark masses. For the light quarks the corresponding pion masses range between 114 and 208 MeV. The setup follows very closely the one in χQCD 14 [24] (presented in the 2016 FLAG review). The decay constants fD and fDs are obtained from an exactly conserved PCAC Ward identity so they do not depend on renormalization factors. The results, however, do not enter the FLAG average as the simulations do not meet the quality criteria concerning the number of lattice spacings used in the continuum extrapolation. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization/matching Heavy-quark treatment Eur. Phys. J. C (2022) 82:869 Page 95 of 296 869 Fig. 22 Decay constants of the D and Ds mesons [values in Table 34 and Eqs. (171–179)]. As usual, full green squares are used in the averaging procedure, pale green squares have been superseded by later determinations, while pale red squares do not satisfy the criteria. The black squares and grey bands indicate our averages A new result (RBC/UKQCD 18A) for the SU(3)-breaking ratio fDs/ fD has been reported in Ref. [70]. The setup includes 2+1 dynamical flavors of Domain Wall fermions. This new result essentially supersedes RBC/UKQCD 17 [55] (discussed in the previous FLAG review) by implementing a number of improvements. One level of stout smearing for the gauge fields has been introduced before performing the charm-quark inversions, which has allowed them to simulate directly at the physical charm mass. At the same time, the valence the strange-quark mass has been tuned to its physical value in order to eliminate a small correction needed previously. Finally, the number of source positions has been doubled on a few ensembles. As of April 30, 2021 the article has not been published in a journal. Therefore, the result does not contribute to the FLAG estimates. The N f = 2 + 1 FLAG estimates remain unchanged and read N f = 2 + 1 : fD = 209.0(2.4) MeV Refs. [53–55], (174) N f = 2 + 1 : fDs = 248.0(1.6) MeV Refs. [24,54,55,57], (175) fD N sf = 2 + 1 : = 1.174(0.007) Refs. [53–55], (176) fD √ where the error on the N f = 2 + 1 average of fDs has been rescaled by the factor χ2/dof = 1.1. Those come from the results in HPQCD 12A [53], FNAL/MILC 11 [54] as well as RBC/UKQCD 17 [55] concerning fD while for fDs also the χQCD 14 [24] result contributes, and instead of the value in HPQCD 12A [53] the one in HPQCD 10A [57] is used. In addition, the statistical errors between the results of FNAL/MILC and HPQCD have been everywhere treated as 100% correlated since the two collaborations use overlapping sets of configurations. The same procedure had been used in the past reviews. No new result appeared for N f = 2 + 1 + 1 since the last FLAG review. Our estimates, therefore, coincide with those in Ref. [4], namely N f = 2 + 1 + 1 : fD = 212.0(0.7) MeV Refs. [16,38], (177) N f = 2 + 1 + 1 : fDs = 249.9(0.5) MeV Refs. [16,38], (178) N f = 2 + 1 + 1 : fDs = 1.1783(0.0016) Refs. [16,38], (179) fD √ where the error on the average of fD has been rescaled by the factor χ2/dof = 1.22. On a general note, an important recent theoretical development is represented by the nonperturbative calculation of the form factors FA and FV contributing to the radiative leptonic decays of a charged pseudoscalar meson P . As discussed in Ref. [238], those appear in the decomposition of the hadronic matrix element 123 869 Page 96 of 296 Eur. Phys. J. C (2022) 82:869 ∫ Hαr (k,p) = r (k) d4W μ y eiky T〈0| jα μW (0) jem(y)|P(p)〉, (180) with rμ(k) the polarisation vector of the outgoing photon (with momentum k) and j α W and j μ em the weak and electromagnetic currents, respectively. With general kinematics four form factors together with the pseudoscalar decay constant fP are needed; however, for k2 = 0, by choosing in addition a physical basis for the polarisation such that r (k) · k = 0, the decay rate can be calculated once FA, FV , and fP are known. A preliminary study has been presented in Ref. [487] in the theory with 2 + 1 dynamical flavors. While a more complete calculation at three different lattice spacings (in the range 0.09–0.06 fm) and for N f = 2 + 1 + 1 appeared in Ref. [488]. The form factors, once used in combination with the nonperturbative calculation of the corrections to P → ν̄ due to the exchange of a virtual photon, allow for a complete determination of the QED corrections to semileptonic decays of mesons. In Ref. [488] the form factors are defined after removing the point-like, infrared divergent contribution, in order to highlight the interesting structure dependent part. Restricting attention to on-shell photons, the behaviour of discretisation effects is studied in Ref. [488] as the photon momentum is changed and heavy quarks are considered. A prescription is also given to nonperturbatively subtract infrared divergent cutoff effects. Still, for charmed mesons discretization effects turned out to be rather large, relative to the size of the form factors, suggesting that very fine lattice spacings will be needed in the case of B mesons. 7.2 Form factors for D → πν and D → Kν semileptonic decays The SM prediction for the differential decay rate of the semileptonic processes D → πν and D → Kν can be written as √ 2 2 2 2 2 d(D → Pν) G2|V |2 (q − m ) E= F cx  P − m P dq2 2 3[(4π ) q 4m2D ] m2 2× 1 +  m2 (E2 − m2 )| f+(q2 22 D P P )| + 3m 2 2 2 2 2 (m D − m P ) | f0(q )|2 (181)2q 8q where x = d, s is the daughter light quark, P = π, K is the daughter light-pseudoscalar meson, EP is the light-pseudoscalar meson energy in the rest frame of the decaying D, and q = (pD − pP ) is the momentum of the outgoing lepton pair; in this section, the charged lepton  will either be an electron (resp. positron) or (anti)muon. The vector and scalar form factors f+(q2) and f0(q2) parameterize the hadronic matrix element of the heavy-to-light quark flavour-changing vector current Vμ = xγμc, ( ) 2 2 2 2 〈P|Vμ|D〉 = m − mf+(q2) p + p − D P m − mDμ Pμ 2 qμ + f 2 D P0(q ) 2 qμ, (182)q q and satisfy the kinematic constraint f+(0) = f0(0). Because the contribution to the decay width from the scalar form factor is proportional to m2 , within current precision standards it can be neglected for  = e, μ, and Eq. (181) simplifies to d (D → Pν) = G 2 F 2 3 |p |3P |V 2 2cx | | f+(q )|2. (183)dq 24π In models of new physics, decay rates may also receive contributions from matrix elements of other parity-even currents. In the case of the scalar density, partial vector current conservation allows one to write matrix elements of the latter in terms of f+ and f0, while for tensor currents Tμν = x̄σμνc a new form factor has to be introduced, viz., 〈P| 2 [ ]T |D〉 = p p − p 2μν + Pμ Dν Pν pDμ fT (q ). (184)m D m P Recall that, unlike the Noether current Vμ, the operator Tμν requires a scale-dependent renormalization. Lattice-QCD computations of f+,0 allow for comparisons to experiment to ascertain whether the SM provides the correct prediction for the q2-dependence of d(D → Pν)/dq2; and, subsequently, to determine the CKM matrix elements |Vcd | and |Vcs | from Eq. (181). The inclusion of fT allows for analyses to constrain new physics. Currently, state-of-the-art experimental results by CLEO-c [489] and BESIII [490,491] provide data for the differential rates in the whole q2 range available, with a precision of order 2–3% for the total branching fractions in both the electron and muon final channels. 123 Eur. Phys. J. C (2022) 82:869 Page 97 of 296 869 Calculations of the D → πν and D → Kν form factors typically use the same light-quark and charm-quark actions as those of the leptonic decay constants fD and fDs . Therefore, many of the same issues arise; in particular, considerations about cutoff effects coming from the large charm-quark mass, or the normalization of weak currents, apply. Additional complications arise, however, due to the necessity of covering a sizeable range of values in q2: • Lattice kinematics imposes restrictions on the values of the hadron momenta. Because lattice calculations are performed in a finite spatial volume, the pion or kaon three-momentum can only take discrete values in units of 2π/L when periodic boundary conditions are used. For typical box sizes in recent lattice D- and B-meson form-factor calculations, L ∼ 2.5– 3 fm; thus, the smallest nonzero momentum in most of these analyses lies in the range |pP | ∼ 400–500 MeV. The largest momentum in lattice heavy-light form-factor calculations is typically restricted to |pP | ≤ 4π/L . For D → πν and D → Kν, q2 = 0 corresponds to |pπ | ∼ 940 MeV and |pK | ∼ 1 GeV, respectively, and the full recoil-momentum region is within the range of accessible lattice momenta. This has implications for both the accuracy of the study of the q2-dependence, and the precision of the computation, since statistical errors and cutoff effects tend to increase at larger meson momenta. As a consequence, many recent studies have incorporated the use of nonperiodic (“twisted”) boundary conditions (tbc) [492,493] in the valence fields used for the computation of observables, as a means to alleviate some of these difficulties. In particular, while they will not necessarily lead to a decrease of numerical noise or cutoff effects, the use of tbc allows not only for a better momentum resolution, but also to better control the q2 = 0 endpoint [59,494–498]. • Final-state pions and kaons can have energies  1 GeV, given the available kinematical range 0  q2 ≤ q2max = (m D − m P )2. This makes the use of (heavy-meson) chiral perturbation theory to extrapolate to physical light-quark masses potentially problematic. • Accurate comparisons to experiment, including the determination of CKM parameters, requires good control of systematic uncertainties in the parameterization of the q2-dependence of form factors. While this issue is far more important for semileptonic B decays, where existing lattice computations cover just a fraction of the kinematic range, the increase in experimental precision requires accurate work in the charm sector as well. The parameterization of semileptonic form factors is discussed in detail in Appendix B.1. The most advanced N f = 2 lattice-QCD calculation of the D → πν and D → Kν form factors is by the ETM collaboration [494]. This work, which did not proceed beyond the preliminary stage, uses the twisted-mass Wilson action for both the light and charm quarks, with three lattice spacings down to a ≈ 0.068 fm and (charged) pion masses down to mπ ≈ 270 MeV. The calculation employs the method of Ref. [499] to avoid the need to renormalize the vector current, by introducing double-ratios of lattice three-point correlation functions in which the vector current renormalization cancels. Discretization errors in the double ratio are of O((amc)2), due to the automatic O(a) improvement at maximal twist. The vector and scalar form factors f+(q2) and f0(q2) are obtained by taking suitable linear combinations of these double ratios. Extrapolation to physical light-quark masses is performed using SU (2) heavy-light meson χPT. The ETM collaboration simulates with twisted boundary conditions for the valence quarks to access arbitrary momentum values over the full physical q2 range, and interpolate to q2 = 0 using the Bečirević–Kaidalov ansatz [500]. The statistical errors in f+Dπ (0) and f+DK (0) are 9% and 7%, respectively, and lead to rather large systematic uncertainties in the fits to the light-quark mass and energy dependence (7% and 5%, respectively). Another significant source of uncertainty is from discretization errors (5% and 3%, respectively). On the finest lattice spacing used in this analysis amc ∼ 0.17, so O((amc)2) cutoff errors are expected to be about 5%. This can be reduced by including the existing N f = 2 twisted-mass ensembles with a ≈ 0.051 fm discussed in Ref. [87]. The first published N f = 2 + 1 lattice-QCD calculation of the D → πν and D → Kν form factors came from the Fermilab Lattice, MILC, and HPQCD collaborations [501].32 This work uses asqtad-improved staggered sea quarks and light (u, d, s) valence quarks and the Fermilab action for the charm quarks, with a single lattice spacing of a ≈ 0.12 fm, and a minimum RMS-pion mass of ≈ 510 MeV, dictated by the presence of fairly large staggered taste splittings. The vector current is normalized using a mostly nonperturbative approach, such that the perturbative truncation error is expected to be negligible compared to other systematics. Results for the form factors are provided over the full kinematic range, rather than focusing just at q2 = 0 as was customary in previous work, and fitted to a Bečirević–Kaidalov ansatz. In fact, the publication of this result predated the precise measurements of the D → Kν decay width by the FOCUS [502] and Belle experiments [503], and showed good agreement with the experimental determination of the shape of f+DK (q2). Progress on extending this work 32 Because only two of the authors of this work are members of HPQCD, and to distinguish it from other more recent works on the same topic by HPQCD, we hereafter refer to this work as “FNAL/MILC.” 123 869 Page 98 of 296 Eur. Phys. J. C (2022) 82:869 was reported in [504]; efforts are aimed at reducing both the statistical and systematic errors in f+Dπ (q2) and f+DK (q2) by increasing the number of configurations analyzed, simulating with lighter pions, and adding lattice spacings as fine as a ≈ 0.045 fm. The most precise published calculations of the D → πν [60] and D → Kν [62] form factors in N f = 2 + 1 QCD are by the HPQCD collaboration. They are also based on N f = 2 + 1 asqtad-improved staggered MILC configurations, but use two lattice spacings a ≈ 0.09 and 0.12 fm, and a HISQ action for the valence u, d, s, and c quarks. In these mixed-action calculations, the HISQ valence light-quark masses are tuned so that the ratio ml/ms is approximately the same as for the sea quarks; the minimum RMS sea-pion mass≈ 390 MeV. Form factors are determined only at q2 = 0, by using a Ward identity to relate matrix elements of vector currents to matrix elements of the absolutely normalized quantity (mc−mx )〈P|x̄c|D〉 (where x = u, d, s), and exploiting the kinematic identity f+(0) = f0(0) to yield f+(q2 = 0) = (m −m )〈P|x̄c|D〉/(m2 −m2c x D P ). A modified z-expansion (cf. Appendix B.1) is employed to simultaneously extrapolate to the physical light-quark masses and the continuum and to interpolate to q2 = 0, and allow the coefficients of the series expansion to vary with the light- and charm-quark masses. The form of the light-quark dependence is inspired by χPT, and includes logarithms of the form m2 2π log(mπ ) as well as polynomials in the valence-, sea-, and charm-quark masses. Polynomials in Eπ(K ) are also included to parameterize momentum-dependent discretization errors. The number of terms is increased until the result for f+(0) stabilizes, such that the quoted fit error for f+(0) not only contains statistical uncertainties, but also reflects relevant systematics. The largest quoted uncertainties in these calculations are from statistics and charm-quark discretization errors. Progress towards extending the computation to the full q2 range have been reported in Refs. [495,496]; however, the information contained in these conference proceedings is not enough to establish an updated value of f+(0) with respect to the previous journal publications. The most recent N f = 2+1 computation of D semileptonic form factors has been carried out by the JLQCD collaboration, and so far only published in conference proceedings; most recently in Ref. [505]. They use their own Möbius domain-wall configurations at three values of the lattice spacing a = 0.080, 0.055, 0.044 fm, with several pion masses ranging from 226 to 501 MeV (though there is so far only one ensemble, with mπ = 284 MeV, at the finest lattice spacing). The vector and scalar form factors are computed at four values of the momentum transfer for each ensemble. The computed form factors are observed to depend mildly on both the lattice spacing and the pion mass. The momentum dependence of the form factors is fitted to a BCL z-parameterization (see Appendix B.1) with a Blaschke factor that contains the measured value of the D∗(s) mass in the vector channel, and a trivial Blaschke factor in the scalar channel. The systematics of this latter fit is assessed by a BCL fit with the experimental value of the scalar resonance mass in the Blaschke factor. Continuum and chiral extrapolations are carried out through a linear fit in the squared lattice spacing and the squared pion and ηc masses. A global fit that uses hard-pion HMχPT to model the mass dependence is furthermore used for a comparison of the form factor shapes with experimental data.33 Since the computation is only published in proceedings so far, it will not enter our N f = 2 + 1 average.34 The first full computation of both the vector and scalar form factors in N f = 2 + 1 + 1 QCD was achieved by the ETM collaboration [59]. Furthermore, they have provided a separate determination of the tensor form factor, relevant for new physics analyses [498]. Both works use the available N f = 2 + 1 + 1 twisted-mass Wilson lattices [218], totaling three lattice spacings down to a ≈ 0.06 fm, and a minimal pion mass of 220 MeV. Matrix elements are extracted from suitable double ratios of correlation functions that avoid the need of nontrivial current normalizations. The use of twisted boundary conditions allows both for imposing several kinematical conditions, and considering arbitrary frames that include moving initial mesons. After interpolation to the physical strange- and charm-quark masses, the results for form factors are fitted to a modified z-expansion that takes into account both the light-quark mass dependence through hard-pion SU (2) χPT [507], and the lattice-spacing dependence. In the latter case, a detailed study of Lorentz-breaking effects due to the breaking of rotational invariance down to the hypercubic subgroup is performed, leading to a nontrivial momentum-dependent parameterization of cutoff effects. The z-parameterization (see Appendix B.1) itself includes a single-pole Blaschke factor (save for the scalar channel in D → K , where the Blaschke factor is trivial), with pole masses treated as free parameters. The final quoted uncertainty on the form factors is about 5–6% for D → π , and 4% for D → K . The dominant source of uncertainty is quoted as statistical+fitting procedure+input parameters – the latter referring to the values of quark masses, the lattice spacing (i.e., scale setting), and the LO SU (2) LECs. 33 It is important to stress the finding in Ref. [506] that the factorization of chiral logs in hard-pion χPT breaks down, implying that it does not fulfill the expected requisites for a proper effective field theory. Its use to model the mass dependence of form factors can thus be questioned. 34 The ensemble parameters quoted in Ref. [505] appear to show that the volumes employed at the lightest pion masses are insufficient to meet our criteria for finite-volume effects. There is, however, a typo in the table which results in a wrong assignment of lattice sizes, whereupon the criteria are indeed met. We thank T. Kaneko for correspondence on this issue. 123 Eur. Phys. J. C (2022) 82:869 Page 99 of 296 869 Table 35 Summary of computations of charmed-meson semileptonic form factors. Note that HPQCD 20 (discussed in Sect. 7.4) addresses the Bc → Bs and Bc → Bd transitions – hence the absence of quoted values for f+Dπ (0) and f+DK (0) – while ETM 18 provides a computation of tensor form factors Collaboration Refs. N f f+Dπ (0) f+DK (0) HPQCD 21A [61] 2+1+1 P   ◦†   n/a 0.7380(44) HPQCD 20 [510] 2+1+1 A  ◦    n/a n/a ETM 17D, 18 [59,498] 2+1+1 A  ◦ ◦   0.612(35) 0.765(31) JLQCD 17B [505] 2+1 C   ◦   0.615(31)(+17)(+28)∗ 0.698(29)(18)(+32)∗ HPQCD 11 [60] 2+1 A ◦ ◦ ◦ −16 −7 −12  0.666(29) HPQCD 10B [62] 2+1 A ◦ ◦ ◦   0.747(19) FNAL/MILC 04 [501] 2+1 A   ◦ ◦  0.64(3)(6) 0.73(3)(7) ETM 11B [494] 2 C ◦ ◦    0.65(6)(6) 0.76(5)(5) ∗The first error is statistical, the second from the q2 → 0 extrapolation, the third from the chiral-continuum extrapolation †The volumes used in the computation satisfy the◦nominal criterion for finite-volume effects. However, the impact of the topologically frozen ensemble ata 0.044 fm is neglected. We therefore assign a rating here, as a mark of caution Another N f = 2 + 1 + 1 computation of f+ and f0 in the full kinematical range for the D → Klν mode, performed by HPQCD, has recently been published – HPQCD 21A (Ref. [61]). This work uses MILC’s HISQ ensembles at five values of the lattice spacing, and pion masses reaching to the physical point for the three coarsest values of a. Vector currents are normalized nonperturbatively by imposing that form factors satisfy Ward identities exactly at zero recoil. Results for the form factors are fitted to a modified z-expansion ansatz, with all sub-threshold poles removed by using the experimental value of the mass shifted by a factor that matches the corresponding result at finite lattice spacing. The accuracy of the description of the q2 dependence is crosschecked by comparing to a fit based on cubic splines. Finite-volume effects are expected to be small, and chiral-perturbation-theory-based estimates for them are included in the chiral fit. However, the impact of frozen topology at the finest lattice spacing is neglected. The final uncertainty from the form factors in the determination of |Vcs | quoted in HPQCD 21A is at the 0.5% level, and comparable to the rest of the uncertainty (due to the experimental error, as well as weak and electromagnetic corrections); in particular, the precision of the form factors is around seven times higher than that of the other existing N f = 2+1+1 determination by ETMC. The work also provides an accurate prediction for the lepton flavour universality ratio between the muon and electron modes, where the uncertainty is overwhelmingly dominated by the electromagnetic corrections. The FNAL/MILC collaboration has also reported ongoing work on extending their computation to N f = 2+ 1+ 1, using MILC HISQ ensembles at four values of the lattice spacing down to a = 0.042 fm and pion masses down to the physical point. The latest updates on this computation, focusing on the form factors at q2 = 0, but without explicit values of the latter yet, can be found in Refs. [508,509]. Table 35 contains our summary of the existing calculations of the D → πν and D → Kν semileptonic form factors. Additional tables in Appendix C.5.1 provide further details on the simulation parameters and comparisons of the error estimates. Recall that only calculations without red tags that are published in a refereed journal are included in the FLAG average. We will quote no FLAG estimate for N f = 2, since the results by ETM have only appeared in conference proceedings. For N f = 2+1, only HPQCD 10B,11 qualify, which provides our estimate for f+(q2 = 0) = f (q20 = 0). For N f = 2+1+1, we quote as the FLAG estimate for f+Dπ (0) the only published result by ETM 17D, while for f+DK (0) we quote the weighted average of the values published by ETM 17D and HPQCD 21A: 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment 869 Page 100 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 23 D → πν and D → Kν semileptonic form factors at q2 = 0. The N f = 2 + 1 HPQCD result for f+Dπ (0) is from HPQCD 11, the one for f+DK (0) represents HPQCD 10B (see Table 35) = + : f+ Dπ (0) = 0.666(29) Ref. [60], N f 2 1 (185) f+DK (0) = 0.747(19) Ref. [62], = + + : f+ Dπ (0) = 0.612(35) Ref. [59], N f 2 1 1 = (186)f+DK (0) 0.7385(44) Refs. [59,61]. It is worth noting that, at the current level of precision, no significant effect of the dynamical charm quark is observed. However, given the paucity of results, it is premature to infer strong conclusions on this point. In Fig. 23, we display the existing N f = 2, N f = 2 + 1, and N f = 2 + 1 + 1 results for f+Dπ (0) and f+DK (0); the grey bands show our estimates of these quantities. Section 7.5 discusses the implications of these results for determinations of the CKM matrix elements |Vcd | and |Vcs | and tests of unitarity of the second row of the CKM matrix. In the case of N f = 2 + 1 + 1, we can also provide a complete result for the q2 dependence of f+ and f0. In the case of the D → πν channel, the latter is provided by the fit given in ETM 17D (Ref. [59]), to which we refer the reader. For D → Kν, we can average the results in ETM 17D (Ref. [59]), and HPQCD 21A (Ref. [61]). To that purpose, we use the parameterizations provided in the papers to produce synthetic data for both f+(q2) and f (q20 ) at a number of values of q2. The large correlations involved make covariance matrices ill-behaved as the number of values of q2 considered increases; we have settled for two q2 values for ETM 17D and three q2 values for HPQCD 21A, in both cases including the kinematical endpoints q2 = 0 and q2 = (m D−m 2K ) of the semileptonic interval. This choice allows us to obtain well-behaved covariance matrices. We fit the resulting dataset to a BCL ansatz (cf. Eqs. (532, 533)) for a number of combinations of the highest orders N+ and N0 considered for either form factor; the constraint f+(0) = f0(0) is used to rewrite the highest-order coefficient a0N −1 in f0 in terms of the other N+ + N0 − 1 coefficients. In both form factors, we include non-trivial Blaschke factors,0 with pole masses set to the experimental values of the D∗s (for the vector channel) and Ds0 (scalar channel) masses found in the PDG [165]. We take flavour averages of charged and neutral states for the D and K masses. Our external input is thus m D = 1.87265 GeV, mK = 495.644 MeV, m D∗ = 2.1122 GeV, and m Ds0 = 2.317 GeV. With this setup, we observe stables fits beyond the linear approximation in z for the form factors, although precision is rapidly lost for coefficients of terms of O(z3) and higher. We quote as our preferred fit, and, therefore, FLAG average, the N+ = N0 = 3 result, quoted in full in Table 36, and illustrated in Fig. 24. As clearly shown in the figure, there is some tension between the two datasets, that grows with q2 to reach the ∼ 2σ level. This results in a relatively poor χ2/d.o.f. = 9.17/3, which has resulted in our rescaling the errors of our average fit accordingly. 123 Eur. Phys. J. C (2022) 82:869 Page 101 of 296 869 Table 36 Coefficients for the + = D → Kν (N = 2 + 1 + 1)N 3, N 0 = 3 z-expansion f of the N f = 2 + 1 + 1 FLAG Values Correlation matrix average for the D → K form + factors f+ and f0, and their a0 0.7877(87) 1.000000 −0.498440 0.073805 0.687417 0.363513 correlation matrix a+1 −0.97(18) −0.498440 1.000000 −0.609159 −0.063023 0.309377 a+2 −0.3(2.0) 0.073805 −0.609159 1.000000 0.020575 0.007175 a00 0.6959(47) 0.687417 −0.063023 0.020575 1.000000 0.273019 a01 0.775(69) 0.363513 0.309377 0.007175 0.273019 1.000000 Fig. 24 The form factors f+(q2) and f (q20 ) for D → Kν plotted versus z (left panel) and q2 (right panel). In the left plot, we removed the Blaschke factors. See text for a discussion of the data set. The grey and salmon bands display our preferred N+ = N 0 = 3 BCL fit (five parameters) 7.3 Form factors for c and c semileptonic decays The motivation for studying charm-baryon semileptonic decays is two-fold. First, these decays allow for independent deter- minations of |Vcs |. Second, given that possible new-physics contributions to the c → sν weak effective Hamiltonian are already constrained to be much smaller compared to b → uν̄ and b → s, charm-baryon semileptonic decays allow testing the lattice techniques for baryons that are also employed for bottom-baryon semileptonic decays (see Sect. 8.6) in a better-controlled environment. The amplitudes of the decays c → ν receive contributions from both the vector and the axial components of the current in the matrix element 〈|s̄γ μ(1 − γ5)c|c〉, and can be parameterized in terms of six different form factors f+, f0, f⊥, g+, g0, g⊥ – see, e.g., Ref. [511] for a complete description. The computation in Meinel 16 [512] uses RBC/UKQCD N f = 2 + 1 DWF ensembles, and treats the c quarks within the Columbia RHQ approach. Two values of the lattice spacing (a ≈ 0.11, 0.085 fm) are considered, with the absolute scale set from the ϒ(2S)–ϒ(1S) splitting. In one ensemble, the pion mass mπ ≈ 139 MeV is at the physical point, while for other ensembles it ranges from 295 to 352 MeV. Results for the form factors are obtained from suitable three-point functions, and fitted to a modified z-expansion ansatz that combines the q2-dependence with the chiral and continuum extrapolations. The paper predicts for the total rates in the e and μ channels (c → e+νe) | |2 = 0.2007(71)(74) ps −1, Vcs (187) (c → μ+νμ) | |2 = 0.1945(69)(72) ps −1, Vcs where the uncertainties are statistical and systematic, respectively. In combination with the recent experimental determination of the total branching fractions by BESIII [513,514], it is possible to extract |Vcs | as discussed in Sect. 7.5 below. Lattice results are also available for the c → N form factors, where N is a neutron or proton [515]. This calculation uses the same lattice actions but a different set of ensembles with parameters matching those used in the 2015 calculation of the b → p form factors in Ref. [516] (cf. Sect. 8.6). Predictions are given for the rates of the c → d semileptonic 123 869 Page 102 of 296 Eur. Phys. J. C (2022) 82:869 Table 37 Summary of computations of charmed-baryon semileptonic form factors Process Collaboration Refs. N f c → ν Zhang 21 [517] 2+1 P ◦  ◦   c → nν Meinel 17 [515] 2+1 A ◦ ◦  ◦  c → ν Meinel 16 [512] 2+1 A ◦   ◦  decays c → n+ν; these modes have not yet been observed. Reference [515] also studies the phenomenology of the flavour-changing neutral-current decay c → pμ+μ−. As is typical for rare charm decays to charged leptons, this mode is dominated by long-distance effects that have not yet been calculated on the lattice and whose description is model-dependent. Recently, the authors of Zhang 21 [517] also performed a first lattice calculation of the c → form factors and extracted |Vcs |, with still large uncertainties, from the recent Belle measurement of the c → +ν branching fractions [518]. This calculation uses only two ensembles with 2 + 1 flavours of clover fermions, with lattice spacings of 0.108 and 0.080 fm and nearly identical pion masses of 290 and 300 MeV. The results are extrapolated to the continuum limit but are not extrapolated to the physical pion mass. No systematic uncertainty is estimated for the effect of the missing chiral extrapolation. A summary of the lattice calculations of charm-baryon semileptonic decay form factors is given in Table 37. 7.4 Form factors for charm semileptonic decays with heavy spectator quarks Two other decays mediated by the c → sν and c → dν transitions are Bc → Bsν and B 0c → B ν, respectively. At present, there are no experimental results for these processes, but it may be possible to produce them at LHCb in the future. The HPQCD Collaboration has recently computed the form factors for both of these Bc decay modes with N f = 2+ 1+ 1 [510]. The calculation uses six different MILC ensembles with HISQ light, strange, and charm quarks, and employs the PCAC Ward identity to nonperturbatively renormalize the c → s and c → d currents. Data were generated for two different choices of lattice action for the spectator b quark: lattice NRQCD on five of the six ensembles, and HISQ on three of the six ensembles (cf. Sect. 8 for a discussion of different lattice approaches used for the b quark). For the NRQCD calculation, two of the ensembles have a physical light-quark mass, and the lattice spacings are 0.15 fm, 0.12 fm, and 0.09 fm. The heavy-HISQ calculation is performed only at ml/ms = 0.2, and at lattice spacings of 0.12 fm, 0.09 fm, and 0.06 fm. The largest value of the heavy-HISQ mass used is 0.8 in lattice units on all three ensembles, which does not reach the physical b-quark mass even at the finest lattice spacing. Form-factor fits are performed using z-expansions (see Appendix B.1) modified to include dependence on the lattice spacing and quark masses, including an expansion in the inverse heavy quark mass in the case of the heavy-HISQ approach. The parameters t+ are set to (m Bc +m 2B s ) even though the branch cuts start at (m D +m )2K or (m D +m )2π , as also noted by( ) the authors. The variable z is rescaled by a constant. The lowest charmed-meson poles are removed before the z-expansion, but this still leaves the branch cuts and higher poles below t+. As a consequence of this structure, the good convergence properties of the z-expansion are not necessarily expected to apply. Fits are performed (i) using the NRQCD data only, (ii) using the HISQ data only, and (iii) using the NRQCD data, but with priors on the continuum-limit form-factor parameters equal to the results of the HISQ fit. The results from fits (i) and (ii) are mostly consistent, with the NRQCD fit having smaller uncertainties than the HISQ fit. Case (iii) then results in the smallest uncertainties and gives the predictions (for massless leptons) (Bc → B +s ν) = 1.738(55)× 10−11| 2 MeV,Vcs | → 0 + (188)(Bc B  ν) | |2 = 2.29(12)× 10 −11 MeV. Vcd 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment Eur. Phys. J. C (2022) 82:869 Page 103 of 296 869 Table 38 Determinations of | | | | Collaboration Refs. N f From |Vcd | or |V |Vcd (upper panel) and V cscs (lower panel) obtained from FNAL/MILC 17 [16] 2+1+1 fD 0.2179(6)(57) lattice calculations of D-meson leptonic decay constants and ETM 17D/Riggio 17 [59,521] 2+1+1 D → πν 0.2341(74) semileptonic form factors. The ETM 14E [38] 2+1+1 fD 0.2228(41)(57) errors shown are from the lattice RBC/UKQCD 17 [55] 2+1 fD 0.2214(36)(57) calculation and experiment (plus nonlattice theory), respectively, HPQCD 12A [53] 2+1 fD 0.2218(36)(57) save for ETM 17D/Riggio 17, HPQCD 11 [60] 2+1 D → πν 0.2140(93)(29) where the joint fit to lattice and FNAL/MILC 11 [54] 2+1 fD 0.2110(108)(55) experimental data does not ETM 13B [56] 2 fD 0.2221(74)(57)provide a separation of the two sources of error (although the HPQCD 21A [61] 2+1+1 D → Kν 0.9750(54)(45)† latter is largely theory FNAL/MILC 17 [16] 2+1+1 f 0.983(2)(18) dominated, like other results Ds using D → π and D → K ETM 17D/Riggio 17 [59,521] 2+1+1 D → Kν 0.970(33) decays) ETM 17D (q2 = 0) [59] 2+1+1 D → Kν 0.939(38) ETM 14E [38] 2+1+1 fDs 0.994(17)(19) RBC/UKQCD 17 [55] 2+1 fDs 0.997(9)(19) Meinel 16 [512] 2+1 c → ν 0.949(24)(51) χQCD 14 [24] 2+1 fDs 0.968(17)(19) FNAL/MILC 11 [54] 2+1 fDs 0.945(40)(19) HPQCD 10A [57] 2+1 fDs 0.991(10)(19) HPQCD 10B [62] 2+1 D → Kν 0.975(25)(7) Balasubramanian 19 [58] 2 fDs 1.007(18)(19) ETM 13B [56] 2 fDs 0.983(28)(19) †The value quoted in HPQCD 21A is actually |Vcs | = 0.9663(53)latt(39)exp(19)ηEW (40)EM, and takes into account an electroweak correction ηEW = 1.009(2) that we have eliminated to allow for a straight comparison with the other results. The three remaining errors have been combined in quadrature. Note also that the other computations in the table do not incorporate estimates of electroweak and soft electromagnetic corrections. HPQCD 21A also quotes a value for |Vcs | obtained from the total branching fraction that results in a very small decrease in the total error due to a reduction in the estimate of electromagnetic corrections We note that there is a discrepancy between the NRQCD and HISQ results in the case of f0(Bc → B0), and the uncertainty quoted for method (iii) does not cover this discrepancy. However, this form factor does not enter in the decay rate for massless leptons. 7.5 Determinations of |Vcd | and |Vcs | and test of second-row CKM unitarity We now interpret the lattice-QCD results for the D(s) meson decays as determinations of the CKM matrix elements |Vcd | and |Vcs | in the Standard Model. For the leptonic decays, we use the latest experimental averages from the Particle Data Group [165, see Sec. 71.3.1] fD|Vcd | = 46.2(1.2) MeV, fDs |Vcs | = 245.7(4.6) MeV, (189) where the errors include those from nonlattice theory, e.g., estimates of radiative corrections to lifetimes [519]. By combining these with the average values of fD and fDs from the individual N f = 2, N f = 2 + 1 and N f = 2 + 1 + 1 lattice-QCD calculations that satisfy the FLAG criteria, we obtain the results for the CKM matrix elements |Vcd | and |Vcs | in Table 38. For our preferred values we use the averaged N f = 2, 2 + 1, and 2 + 1 + 1 results for fD and fDs in Eqs. (171–179). We obtain leptonic decays, N f = 2 + 1 + 1 : |Vcd | = 0.2179(7)(57), |Vcs | = 0.983(2)(18), (190) Refs. [16,38], 123 869 Page 104 of 296 Eur. Phys. J. C (2022) 82:869 leptonic decays, N f = 2 + 1 : |Vcd | = 0.2211(25)(57), |Vcs | = 0.991(7)(19), (191) Refs. [24,53–55,57], leptonic decays, N f = 2 : |Vcd | = 0.2221(74)(57), |Vcs | = 0.998(16)(19), (192) Refs. [56,58], where the errors shown are from the lattice calculation and experiment (plus nonlattice theory), respectively. For the N f = 2+1 and the N f = 2 + 1 + 1 determinations, the uncertainties from the lattice-QCD calculations of the decay constants are significantly smaller than the experimental uncertainties in the branching fractions. The leptonic determinations of these CKM matrix elements have uncertainties that are reaching the few-percent level. However, higher-order electroweak and hadronic-structure dependent corrections to the rate have not been computed for the case of D(s) mesons, whereas they have been estimated to be around 1–2% for pion and kaon decays [520]. Therefore, it is important that such theoretical calculations are tackled soon, perhaps directly on the lattice, as proposed in Ref. [238]. For D meson semileptonic decays, there are still no N f = 2 results, and for N f = 2 + 1 the only works entering the FLAG averages are still HPQCD 10B/11 [60,62]. For N f = 2 + 1 + 1, on the other hand, there is a new work that enters FLAG averages, HPQCD 21A (Ref. [61]). There is also a new experimental result by BESIII [522], in which the muon mode D0 → K−μ+νμ has been measured for the first time. This has two consequences. First, HFLAV has updated their averages for the combinations f+(0)|Vcx | [261]. They now find f+Dπ (0)|Vcd | = 0.1426(18), f+DK (0)|Vcs | = 0.7180(33) (193) The previous HFLAV average f+DK (0)|Vcs | = 0.7226(34) differed from the new one by 1.4 standard deviations. Second, we now determine |Vcs | using the full q2 dependence of the form factors provided by both HPQCD 21A and ETM 17D (Ref. [59]). Using both the new lattice and new experimental input, we perform a joint lattice+experimental fit to determine the CKM matrix elements. This reduces the error on the CKM matrix elements significantly compared with just using the form factor at q2 = 0, especially for |Vcd | (cf. Fig. 26). This was, indeed, the strategy to extract |Vcd | and |Vcs | pursued in a companion paper to ETM 17D, Ref. [521], as as well as in HPQCD 21A (for |Vcs | only).35 The result for |Vcd | in Ref. [521] is still state-of-the-art, and we will quote it as the FLAG estimate. In the case of |Vcs |, we have performed joint lattice+experiment fits using the same ansatz as described for the lattice average of form factors in Sect. 7.2, including |V 2cs | as an additional coefficient that provides the normalization of the experimental data. The experimental datasets we include are three different measurements of the D0 → K−e+νe mode by BaBar (BaBar 07, Ref. [523]), CLEO-c (CLEO 09/0, Ref. [489]), and BESIII (BESIII 15, Ref. [524]); CLEO-c (CLEO 09/+, Ref. [489]) and BESIII measurements of the D+ → K̄ 0e+νe mode (BESIII 17, Ref. [525]); and the recent first measurement of the D0 → K−μ+νμ mode by BESIII, Ref. [522]. There is also a Belle dataset available in Ref. [526], but it provides results for parameterized form factors rather than partial widths, which implies that reverse modeling of the q2 dependence of the form factor would be needed to add them to the fit, which involves an extra source of systematic uncertainty; it is, furthermore, the measurement with the largest error. Thus, we will drop it. The CLEO collaboration provides correlation matrices for the systematic uncertainties across the channels in their two measurements; the latter are, however, not available for BESIII, and, therefore, we will conservatively treat their systematics with a 100% correlation, following the same prescription as in the HFLAV review [261]. Since all lattice results have been obtained in the isospin limit, we will average over the D0 and D+ electronic modes. We observe that the error of the final result for |Vcs | is independent of the specific ansatz, while the central values differ by at most one standard deviation. From the lattice point of view, HPQCD 21A dominates the result completely, because of its much smaller uncertainties than in ETM 17D. The precision of the data does not allow us to consistently resolve the higher-order coefficients of the z-expansion beyond N+ = N0 = 3, at which point the result for |Vcs | becomes insensitive to increasing the order. Thus, we quote the result from the latter fit, provided in full detail in Table 39 and illustrated in Fig. 25, as the N 2f = 2 + 1 + 1 FLAG average. The χ /d.o.f. of our preferred fit is 1.46, and we have rescaled the full covariance matrix with that value to obtain conservative error estimates. Notice that, notwithstanding the fact that HPQCD 21A dominates the fit, our final value |Vcs | = 0.9714(69) is slightly higher than their quoted value |Vcs | = 0.9663(66) (where for the error we have combined in quadrature their lattice and 35 Notice that the estimate for |Vcs | in Ref. [521] does not include the later experimental result in Ref. [522]. The value obtained in Ref. [521] is however completely dominated by the uncertainty of the lattice form factors, and changes very little once the full experimental information is incorporated into the determination. 123 Eur. Phys. J. C (2022) 82:869 Page 105 of 296 869 Table 39 Coefficients for the N+ = N 0 = 3 z-expansion of the D → K form factors f+ and f0, |Vcs |, and their correlation matrix D → Kν (N f = 2 + 1 + 1) Values Correlation matrix a+0 0.7864(54) 1 −0.282248 −0.052775 0.760032 0.631483 −0.899274 a+1 −0.849(68) −0.282248 1 −0.640953 −0.088377 0.041977 0.128087 a+2 −1.5(1.1) −0.052775 −0.640953 1 0.018139 0.115382 0.020790 a00 0.6958(32) 0.760032 −0.088377 0.018139 1 0.300343 −0.734376 a01 0.781(45) 0.631483 0.041977 0.115382 0.300343 1 −0.664113 |Vcs | 0.9714(69) −0.899274 0.128087 0.020790 −0.734376 −0.664113 1 Fig. 25 The D → Kν differential decay rates experiment error, in order to allow for a direct comparison, and dropped the estimated systematic uncertainties due to electroweak and electromagnetic corrections also provided in HPQCD 21A). This is due to the fact that HPQCD 21A has applied the structure-independent electroweak correction factor ηEW = 1.009(2) in their analysis, which we are not doing for consistency with other determinations in this review; if we had applied the same procedure, our final result would be |Vcs | = 0.9628(68). Meinel 16 has also determined the form factors for c → ν decays for N f = 2 + 1, which results in a determination of |Vcs | in combination with the experimental measurement of the branching fractions for the e+ and μ+ channels in Refs. [513,514]. In Ref. [512] the value |Vcs | = 0.949(24)(14)(49) is quoted, where the first error comes from the lattice computation, the second from the c lifetime, and the third from the branching fraction of the decay. While the lattice uncertainty is competitive with meson channels (for N f = 2 + 1), the experimental uncertainty is far larger. Our estimates for |Vcd | and |Vcs | from semileptonic decay are |Vcd | = 0.2141(93)(29) Ref. [60], SL averages for N f = 2 + 1 : |Vcs | = 0.967(25)(5) Ref. [62], (194) |Vcs |(c) = 0.949(24)(51) Ref. [512], |Vcd | = 0.2341(74) Refs. [59,521], SL averages for N = 2 + 1 + 1 : | | = (195)f Vcs 0.9714(69) Refs. [59,61], where the errors for N f = 2 + 1 are lattice and experimental (plus nonlattice theory), respectively. It has to be stressed that for meson decay errors are largely theory-dominated, save for the D → K mode for N f = 2 + 1 + 1 where the lattice contribution to the error is only slightly larger than the experimental one; while in the baryon mode for |Vcs | the dominant error is experimental. The above values are compared with individual leptonic determinations in Table 38. 123 869 Page 106 of 296 Eur. Phys. J. C (2022) 82:869 Table 40 Comparison of From Refs. |V determinations of |V | and cd | |Vcs |cd |Vcs | obtained from lattice N f = 2 + 1 + 1 fD & fDs [16,38] 0.2179(57) 0.983(18)methods with nonlattice determinations and the Standard N f = 2 + 1 fD & fDs [24,53–55,57] 0.2211(62) 0.991(20) Model prediction from global N f = 2 fD & fDs [56,58] 0.2220(93) 0.999(25) fits assuming CKM unitarity. Experimental and lattice errors N f = 2 + 1 + 1 D → πν and D → Kν [59,61,521] 0.2341(74) 0.9714(69) have been combined in N f = 2 + 1 D → πν and D → Kν [60,62] 0.2141(97) 0.967(25) quadrature. The PDG figures N f = 2 + 1 c → ν [512] n/a 0.949(56) quoted are taken from the “CKM Quark-Mixing Matrix” review PDG Neutrino scattering [165] 0.230(11) PDG CKM unitarity [165] 0.2265(5) 0.9732(1) Fig. 26 Comparison of determinations of |Vcd | and |Vcs | obtained from lattice methods with nonlattice determinations and the Standard Model prediction based on CKM unitarity. When two references are listed on a single row, the first corresponds to the lattice input for |Vcd | and the second to that for |Vcs |. The results denoted by squares are from leptonic decays, while those denoted by triangles are from semileptonic decays. The points indicated as (q2 = 0) do not contribute to the average, and are shown to stress the decrease in the final uncertainty obtained by considering the full q2 dependence. Notice that the HPQCD 21A point includes estimates of the electroweak and soft electromagnetic uncertainties that we have not incorporated into our average In Table 40, we summarize the results for |Vcd | and |Vcs | from leptonic and semileptonic decays, and compare them to determinations from neutrino scattering (for |Vcd | only) and global fits assuming CKM unitarity. These results are also plotted in Fig. 26. For both |Vcd | and |Vcs |, the errors in the direct determinations from leptonic and semileptonic decays are approximately one order of magnitude larger than the indirect determination from CKM unitarity. The direct and indirect determinations are still always compatible within at most 1.2σ , save for the leptonic determinations of |Vcs | – that show a ∼ 2σ deviation for all values of N f – and |Vcd | using the N f = 2 + 1 + 1 lattice result, where the difference is 1.8σ . In order to provide final estimates, we average all the available results separately for each value of N f . Whenever two results share ensembles, we have conservatively fully correlated their statistical uncertainties. This is a particularly sensitive issue in the average for |Vcs |, that is dominated by the FNAL/MILC 17 and HPQCD 21A results, and for which precision has been greatly improved by the latter; however, the uncertainty of the leptonic determination is completely dominated by the experimental uncertainty, and therefore the impact of the statistical correlation is all but negligible. We have also 100% correlated the errors from the heavy-quark discretization and scale setting in HPQCD’s N f = 2 + 1 results. Finally, we include a 100% correlation in the fraction of the error of |Vcd(s)| leptonic determinations that comes from the experimental input, to avoid an artificial reduction of the experimental uncertainty in the averages. Our results thus are our average, N f = 2 + 1 + 1 |Vcd | = 0.2236(37), |Vcs | = 0.9741(65), (196) Refs. [16,38,59,61,521], 123 Eur. Phys. J. C (2022) 82:869 Page 107 of 296 869 our average, N f = 2 + 1 : |Vcd | = 0.2192(54), |Vcs | = 0.982(16), (197) Refs. [24,53–55,57,60,62,512] , our average, N f = 2 : |Vcd | = 0.2221(93), |Vcs | = 0.998(24), (198) Refs. [56,58], where the errors include both theoretical and experimental uncertainties. These averages also appear in Fig. 26. The mutual consistency between the various lattice results is good except for the case of |Vcd | with N f = 2+1+1, where a ∼ 2σ tension between the leptonic and semileptonic determinations is observed. Currently, the leptonic and semileptonic determinations of Vcd are controlled by experimental and lattice uncertainties, respectively. The leptonic error will be reduced by Belle II and BES III. It would be valuable to have other lattice calculations of the semileptonic form factors. Using the lattice determinations of |Vcd | and |Vcs | in Table 40, we can test the unitarity of the second row of the CKM matrix. We obtain N = 2 + 1 + 1 : |V |2f cd + |Vcs |2 + |V 2cb| − 1 = −0.001(8), (199) N f = 2 + 1 : |V 2 2 2cd | + |Vcs | + |Vcb| − 1 = 0.01(3), (200) N f = 2 : |Vcd |2 + |Vcs |2 + |Vcb|2 − 1 = 0.05(6). (201) The much-improved precision in |Vcs | – cf. the value 0.025(22) quoted in the latest PDG review, Ref. [165] – has thus not resulted in any tension with CKM unitarity. Note that, given the current level of precision, this result does not depend on |V |, which is of O(10−2cb ). Notice, on the other hand, that the final quoted precision of 0.7% makes the incorporation of electromagnetic corrections from first principles a necessary step for the near future, similarly to the ongoing developments in the light-meson sector. 8 Bottom hadron decays and mixings Authors: Y. Aoki, M. Della Morte, E. Lunghi, S. Meinel, C. Monahan, C. Pena The (semi)leptonic decay and mixing processes of B(s) mesons have been playing a crucial role in flavour physics. In particular, they contain important information for the investigation of the b−d unitarity triangle in the Cabibbo–Kobayashi– Maskawa (CKM) matrix, and can be ideal probes of physics beyond the Standard Model. The charged-current decay channels B+ → l+νl and B0 → π−l+νl , where l+ is a charged lepton with νl being the corresponding neutrino, are essential in extracting the CKM matrix element |Vub|. Similarly, the B to D(∗) semileptonic transitions can be used to determine |Vcb|. The flavour-changing neutral current (FCNC) processes, such as B → K (∗)+− and Bd(s) → +−, occur only beyond the tree level in weak interactions and are suppressed in the Standard Model. Therefore, these processes can be sensitive to new physics, since heavy particles can contribute to the loop diagrams. They are also suitable channels for the extraction of the CKM matrix elements involving the top quark that can appear in the loop. The decays B → D(∗)ν and B → K (∗) can also be used to test lepton flavour universality by comparing results for  = e,μ and τ . In particular, anomalies have been seen in the ratios R(D(∗)) = B(B → D(∗)τν)/B(B → D(∗)ν)=e,μ and R(K (∗)) = B(B → K (∗)μμ)/B(B → K (∗)ee). In addition, the neutral Bd(s)-meson mixings are FCNC processes and are dominated by the 1-loop “box” diagrams containing the top quark and the W bosons. Thus, using the experimentally measured neutral B0d(s)-meson oscillation frequencies, Md(s), and the theoretical calculations for the relevant hadronic mixing matrix elements, one can obtain |Vtd | and |Vts | in the Standard Model. At the Large Hadron Collider, decays of b quarks can also be probed with b and other bottom baryons, which can provide complementary constraints on physics beyond the Standard Model. The most important processes are the charged-current decays b → pν̄ and b → cν̄, and the neutral-current decay b → +−. Accommodating the light quarks and the b quark simultaneously in lattice-QCD computations is a challenging endeavour. To incorporate the pion and the b hadrons with their physical masses, the simulations have to be performed using the lattice size L̂ = L/a ∼ O(102), where a is the lattice spacing and L is the physical (dimensionful) box size. The most ambitious calculations are now using such volumes; however, many ensembles are smaller. Therefore, in addition to employing Chiral Perturbation Theory for the extrapolations in the light-quark mass, current lattice calculations for quantities involving b hadrons often make use of effective theories that allow one to expand in inverse powers of mb. In this regard, two general 123 869 Page 108 of 296 Eur. Phys. J. C (2022) 82:869 approaches are widely adopted. On the one hand, effective field theories such as Heavy-Quark Effective Theory (HQET) and Nonrelativistic QCD (NRQCD) can be directly implemented in numerical computations. On the other hand, a relativistic quark action can be improved à la Symanzik to suppress cutoff errors, and then re-interpreted in a manner that is suitable for heavy-quark physics calculations. This latter strategy is often referred to as the method of the Relativistic Heavy-Quark Action (RHQA). The utilization of such effective theories inevitably introduces systematic uncertainties that are not present in light- quark calculations. These uncertainties can arise from the truncation of the expansion in constructing the effective theories (as in HQET and NRQCD), or from more intricate cutoff effects (as in NRQCD and RHQA). They can also be introduced through more complicated renormalization procedures which often lead to significant systematic effects in matching the lattice operators to their continuum counterparts. For instance, due to the use of different actions for the heavy and the light quarks, it is more difficult to construct absolutely normalized bottom-light currents. Complementary to the above “effective theory approaches”, another popular method is to simulate the heavy and the light quarks using the same (normally improved) lattice action at several values of the heavy-quark mass mh with amh < 1 and mh < mb. This enables one to employ HQET-inspired relations to extrapolate the computed quantities to the physical b mass. When combined with results obtained in the static heavy-quark limit, this approach can be rendered into an interpolation, instead of extrapolation, in mh . The discretization errors are the main source of the systematic effects in this method, and very small lattice spacings are needed to keep such errors under control. In recent years, it has also been possible to perform lattice simulations at very fine lattice spacings and treat heavy quarks as fully relativistic fermions without resorting to effective field theories. Such simulations are, of course, very demanding in computing resources. Because of the challenge described above, the efforts that have been made to obtain reliable, accurate lattice-QCD results for physics of the b quark have been enormous. These efforts include significant theoretical progress in formulating QCD with heavy quarks on the lattice. This aspect is briefly reviewed in Appendix A.1.3 of FLAG 19 [4]. In this section, we summarize the results of the B-meson leptonic decay constants, the neutral B-mixing parameters, and the semileptonic form factors of B mesons and b baryons, from lattice QCD. To focus on the calculations that have strong phenomenological impact, we limit the review to results based on modern simulations containing dynamical fermions with reasonably light pion masses (below approximately 500 MeV). Following our review of B(s)-meson leptonic decay constants, the neutral B-meson mixing parameters, and semileptonic form factors, we then interpret our results within the context of the Standard Model. We combine our best-determined values of the hadronic matrix elements with the most recent experimentally-measured branching fractions to obtain |Vub| and |Vcb|, and compare these results to those obtained from inclusive semileptonic B decays. 8.1 Leptonic decay constants fB and fBs The B- and Bs-meson decay constants are crucial inputs for extracting information from leptonic B decays. Charged B mesons can decay to a lepton-neutrino final state through the charged-current weak interaction. On the other hand, neutral Bd(s) mesons can decay to a charged-lepton pair via a flavour-changing neutral current (FCNC) process. In the Standard Model, the decay rate for B+ → +ν is described by a formula identical to Eq. (169), with D(s) replaced by B, and the relevant CKM matrix element Vcq replaced by Vub, ( ) 2 2m B (B → ν ) = G2 2 F fB |Vub|2m2 1 − m 2 . (202)8π m B The only two-body charged-current B-meson decay that has been observed so far is B+ → τ+ντ , which has been measured by the Belle and Babar collaborations [527,528]. Both collaborations have reported results with errors around 20%. These measurements can be used to determine |Vub| when combined with lattice-QCD predictions of the corresponding decay constant. Neutral B + −d(s)-meson decays to a charged-lepton pair Bd(s) → l l is a FCNC process, and can only occur at one loop in the Standard Model. Hence these processes are expected to be rare, and are sensitive to physics beyond the Standard Model. The corresponding expression for the branching fraction has the form √ G2 ( )2 + α m2B(Bq →  −) = τ F Y m 2Bq 2 Bq fB |V ∗tbV |2tq m2 1 − 4 π 4π sin  q m2 , (203)W B 123 Eur. Phys. J. C (2022) 82:869 Page 109 of 296 869 where the light quark q = s or d, and the function Y includes NLO QCD and electro-weak corrections [417,529]. Evidence for the Bs → μ+μ− decay was first observed by the CMS and the LHCb collaborations, and a combined analysis was presented in 2014 in Ref. [530]. In 2020, the ATLAS, CMS and LHCb collaborations reported their measurements from a preliminary combined analysis as [531] B(B → μ+μ−d ) < (1.9)× 10−10 at 95% CL, B(B → μ+μ−) = (2.69+0.37s −0.35)× 10−9, (204) which are compatible with the Standard Model predictions within approximately 2 standard deviations [532]. We note that the errors of these results are currently too large to enable a precise determination of |Vtd | and |Vts |. The decay constants fBq (with q = u, d, s) parameterize the matrix elements of the corresponding axial-vector currents Aμbq = b̄γ μγ 5q analogously to the definition of fDq in Sect. 7.1: 〈0|Aμ|Bq(p)〉 = i pμB fBq . (205) For heavy-light mesons, it is convenient to define and analyse the quantity √ Bq ≡ fBq m Bq , (206) which approaches a constant (up to logarithmic corrections) in the m B → ∞ limit, because of heavy-quark symmetry. In the following discussion, we denote lattice data for , and the corresponding decay constant f , obtained at a heavy-quark mass mh and light valence-quark mass m as h and fhl , to differentiate them from the corresponding quantities at the physical b- and light-quark masses. The SU (3)-breaking ratio fBs/ fB is of phenomenological interest, because many systematic effects can be partially reduced in lattice-QCD calculations of this ratio. The discretization errors, heavy-quark mass tuning effects, and renor- malization/matching errors may all be partially reduced. This SU (3)-breaking ratio is, however, still sensitive to the chi- ral extrapolation. Provided the chiral extrapolation is under control, one can then adopt fBs/ fB as an input in extracting phenomenologically-interesting quantities. In addition, it often happens to be easier to obtain lattice results for fBs with smaller errors than direct calculations of fB . Therefore, one can combine the Bs-meson decay constant with the SU (3)- breaking ratio to calculate fB . Such a strategy can lead to better precision in the computation of the B-meson decay constant, and has been adopted by the ETM [30,56] and the HPQCD collaborations [66]. An alternative strategy, used in Ref. [58], is to obtain the Bs-meson decay constant by combining the Ds-meson decay constant with the ratio fBs/ fDs . It is clear that the decay constants for charged and neutral B mesons play different roles in flavour-physics phenomenology. Knowledge of the B+-meson decay constant fB+ is essential for extracting |Vub| from leptonic B+ decays. The neutral B- meson decay constants fB0 and fBs are inputs to searches for new physics in rare leptonic B 0 decays. In view of this, it is desirable to include isospin-breaking effects in lattice computations for these quantities, and have results for fB+ and fB0 . With the increasing precision of recent lattice calculations, isospin splittings for B-meson decay constants can be significant, and will play an important role in the foreseeable future. A few collaborations have reported fB+ and fB0 separately by taking into account strong isospin effects in the valence sector, and estimated the corrections from electromagnetism. The N f = 2+ 1+ 1 strong isospin-breaking effect was computed in HPQCD 13 [63] (see Table 41 in this subsection). However, since only unitary points (with equal sea- and valence-quark masses) were considered in HPQCD 13 [63], this procedure only correctly accounts for the effect from the valence-quark masses, while introducing a spurious sea-quark contribution. The decay constants fB+ and fB0 are also separately reported in FNAL/MILC 17 [16] by taking into account the strong-isospin effect. The new FNAL/MILC results were obtained by keeping the averaged light sea-quark mass fixed when varying the quark masses in their analysis procedure. Their finding indicates that the strong isospin-breaking effects, fB+ − fB ∼ 0.5 MeV, could be smaller than those suggested by previous computations. One would have to take into account QED effects in the B-meson leptonic decay rates to properly use these results for extracting phenomenologically relevant information.36 Currently, errors on the experimental measurements on these decay rates are still very large. In this review, we will therefore concentrate on the isospin-averaged result fB and the Bs-meson decay constant, as well as the SU (3)-breaking ratio fBs/ fB . The status of lattice-QCD computations for B-meson decay constants and the SU (3)-breaking ratio, using gauge-field ensembles with light dynamical fermions, is summarized in Tables 41 and 42, while Figs. 27 and 28 contain the graphical 36 See Ref. [238] for a strategy that has been proposed to account for QED effects. 123 869 Page 110 of 296 Eur. Phys. J. C (2022) 82:869 Table 41 Decay constants of the B, B+, B0 and Bs mesons (in MeV). Here fB stands for the mean value of fB+ and fB0 , extrapolated (or interpolated) in the mass of the light valence-quark to the physical value of mud Collaboration Refs. N f fB+ fB0 fB fBs FNAL/MILC 17 [16] 2+1+1 A      189.4(1.4) 190.5(1.3) 189.9(1.4) 230.7(1.2) HPQCD 17A [64] 2+1+1 A ◦   ◦  − − 196(6) 236(7) ETM 16B [30] 2+1+1 A  ◦ ◦ ◦  − − 193(6) 229(5) ETM 13E [533] 2+1+1 C  ◦ ◦ ◦  − − 196(9) 235(9) HPQCD 13 [63] 2+1+1 A ◦   ◦  184(4) 188(4) 186(4) 224(5) RBC/UKQCD 14 [68] 2+1 A ◦ ◦ ◦ ◦  195.6(14.9) 199.5(12.6) − 235.4(12.2) RBC/UKQCD 14A [67] 2+1 A ◦ ◦ ◦ ◦  − − 219(31) 264(37) RBC/UKQCD 13A [534] 2+1 C ◦ ◦ ◦ ◦  − − 191(6) stat 233(5)stat HPQCD 12 [66] 2+1 A ◦ ◦ ◦ ◦  − − 191(9) 228(10) HPQCD 12 [66] 2+1 A ◦ ◦ ◦ ◦  − − 189(4) − HPQCD 11A [65] 2+1 A  ◦    − − − 225(4)∇ FNAL/MILC 11 [54] 2+1 A ◦ ◦  ◦  197(9) − − 242(10) HPQCD 09 [72] 2+1 A ◦ ◦ ◦ ◦  − − 190(13)• 231(15)• Balasubramamian 19† [58] 2 A    ◦  − − − 215(10)(2)(+−25) ALPHA 14 [69] 2 A      − − 186(13) 224(14) ALPHA 13 [535] 2 C      − − 187(12)(2) 224(13) ETM 13B, 13C‡ [56,536] 2 A  ◦  ◦  − − 189(8) 228(8) ALPHA 12A [537] 2 C      − − 193(9)(4) 219(12) ETM 12B [538] 2 C  ◦  ◦  − − 197(10) 234(6) ALPHA 11 [539] 2 C  ◦    − − 174(11)(2) − ETM 11A [231] 2 A  ◦  ◦  − − 195(12) 232(10) ETM 09D [540] 2 A  ◦ ◦ ◦  − − 194(16) 235(12) Statistical errors only Obtained by combining fBs from HPQCD 11A with fBs / fB calculated in this work∇This result uses one ensemble per lattice spacing with light to strange sea-quark mass ratio m/ms ≈ 0.2 •This result uses an old determination of r1 = 0.321(5) fm from Ref. [118] that has since been superseded ‡Obtained by combining fDs , updated in this work, with fBs / fDs , calculated in this work ‡Update of ETM 11A and 12B presentation of the collected results and our averages. Most results in these tables and plots have been reviewed in detail in FLAG 19 [4]. Here, we only describe the new results published after January 2019. One new N f = 2 calculation of fBs has appeared after the publication of the previous FLAG review [4]. In Table 41, this result is labelled Balasubramamian 19 [58]. In Balasubramamian 19 [58], simulations at three values of the lattice spacing, a = 0.0751, 0.0653 and 0.0483 fm were performed with nonperturbatively O(a)-improved Wilson-clover fermions and the Wilson plaquette gauge action. The pion masses in this work range from 194 to 439 MeV, and the lattice sizes are between 2.09 and 4.18 fm. A key feature of this calculation is the use of a variant of the ratio method [540], applied for the first time to Wilson-clover fermions. This variant is required because, in contrast to twisted-mass Wilson fermions, there is no simple relationship between the heavy quark pole mass and the bare quark mass. In the application of this approach to the Bs-decay constant, one first computes the quantity Fhq ≡ fhq/Mhq , where fhq and Mhq are the decay constant and mass of the pseudoscalar meson composed of valence (relativistic) heavy quark h and light (or strange) quark q. The matching between the lattice and the continuum heavy-light currents for extracting the above fhq is straightforward because the valence heavy quark is also described by 123 Publication status Continuum extrap C olh ai tr ia ol nextrapolatio F ninite volume Renormalization/m H ae ta cv hy in-q guark treatment Eur. Phys. J. C (2022) 82:869 Page 111 of 296 869 Table 42 Ratios of decay constants of the B and Bs mesons (for details see Table 41) Collaboration Refs. N f fBs / fB+ fBs / fB0 fBs / fB FNAL/MILC 17 [16] 2+1+1 A      1.2180(49) 1.2109(41) − HPQCD 17A [64] 2+1+1 A ◦   ◦  − − 1.207(7) ETM 16B [30] 2+1+1 A  ◦ ◦ ◦  − − 1.184(25) ETM 13E [533] 2+1+1 C  ◦ ◦ ◦  − − 1.201(25) HPQCD 13 [63] 2+1+1 A ◦   ◦  1.217(8) 1.194(7) 1.205(7) RBC/UKQCD 18A [70] 2+1 P      − − 1.1949(60)( ◦ ◦ ◦ ◦ − +95 175) RBC/UKQCD 14 [68] 2+1 A  1.223(71) 1.197(50) − RBC/UKQCD 14A [67] 2+1 A ◦ ◦ ◦ ◦  − − 1.193(48) RBC/UKQCD 13A [534] 2+1 C ◦ ◦ ◦ ◦  − − 1.20(2) ◦ ◦ ◦ ◦ statHPQCD 12 [66] 2+1 A  − − 1.188(18) FNAL/MILC 11 [54] 2+1 A ◦ ◦  ◦  1.229(26) − − RBC/UKQCD 10C [541] 2+1 A    ◦  − − 1.15(12) HPQCD 09 [72] 2+1 A ◦ ◦ ◦ ◦  − − 1.226(26) ALPHA 14 [69] 2 A      − − 1.203(65) ALPHA 13 [535] 2 C      − − 1.195(61)(20) ETM 13B, 13C† [56,536] 2 A  ◦  ◦  − − 1.206(24) ALPHA 12A [537] 2 C      − − 1.13(6) ETM 12B [538] 2 C  ◦  ◦  − − 1.19(5) ETM 11A [231] 2 A ◦ ◦  ◦  − − 1.19(5) Statistical errors only †Update of ETM 11A and 12B Fig. 27 Decay constants of the B and Bs mesons. The values are taken from Table 41 (the fB entry for FNAL/MILC 11 represents fB+ ). The significance of the colours is explained in Sect. 2. The black squares and grey bands indicate our averages in Eqs. (207), (210), (213), (208), (211) and (214) 123 Publication statu C sontinuum extra C poh li ara til oe nxtrapolat F ioin nite volume Renormalization H /me aa tv cy h- iq nu gark treatment 869 Page 112 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 28 Ratio of the decay constants of the B and Bs mesons. The values are taken from Table 42. Results labelled as FNAL/MILC 17 1 and FNAL/MILC 17 2 correspond to those for fBs / fB0 and fBs / fB+ reported in FNAL/MILC 17. The significance of the colours is explained in Sect. 2. The black squares and grey bands indicate our averages in Eqs. (209), (212), and (215) Wilson-clover fermions. In the second step, the ratio z M 3/2 3/2q( hq , λ) ≡ [F stathqCA (Mh′q)Mhq ]/[F ′ Cstath q A (Mhq)Mh′q ] is calcu- lated, where CstatA (Mhq) is the matching coefficient for the (hq)-meson decay constant in QCD and its counterpart in HQET, and Mhq = λMh′q . The authors of Balasubramamian 19 [58] use the NNLO perturbative result of CstatA (Mhq) [542–544] and λ = 1.18. By starting from a “triggering” point with the heavy-meson mass around that of the Ds meson, one can proceed with the calculations in steps, such that Mhq is increased by a factor of λ at each step. The authors simulate up to heavy-quark mass around 4.5 GeV, but observed significant (aM 2Hs ) cutoff effects on ensembles with lattice spacings a = 0.0751 and 0.0653 fm an√d so simulate up to 3.2 GeV on these lattices. In this formulation of the ratio method, the ratio obeys zq(Mhq , λ) → 1/ λ in the limit Mhq → ∞. Designing the computations in such a way that in the last step Mhq is equal to the physical Bs mass, one obtains fB s / fD s . Combining this ratio with results for fD s , updated with( ) ( ) ( ) a third lattice spacing, the decay constant of the Bs meson can be extracted. The authors estimated the systematic uncer- tainty associated with their generic fit form, which combines chiral-continuum extrapolation with heavy quark discretization effects, and quote a single systematic uncertainty. The systematic uncertainty associated with scale-setting is estimated from fDs . There have been no new N f = 2 calculations of fB or fBs/ fB . Therefore, our averages for these two cases stay the same as those in Ref. [4]. We update our average of fBs to include the new calculation of Balasubramamian 19 [58]: N f = 2 : fB = 188(7) MeV Refs. [56,69], (207) N f = 2 : fBs = 225.3(6.6) MeV Refs. [56,58,69], (208) N f = 2 : fBs = 1.206(0.023) Refs. [56,69]. (209) fB One new N f = 2 + 1 calculation of fBs/ fB was completed after the publication of the previous FLAG review [4]. In Table 42, this result is labelled RBC/UKQCD 18A [70]. The RBC/UKQCD collaboration presented in RBC/UKQCD 18A [70] the ratio of decay constants, fBs/ fB , using N f = 2 + 1 dynamical ensembles generated using Domain Wall Fermions (DWF). Three lattice spacings, of a = 0.114, 0.0835 and 0.0727 fm, were used, with pion masses ranging from 139 to 431 MeV, and lattice sizes between 2.65 and 5.47 fm. Two different Domain Wall discretizations (Möbius and Shamir) have been used for both valence and sea quarks. These discretizations correspond to two different choices for the DWF kernel. The Möbius DWF are loosely equivalent to Shamir DWF at twice the extension in the fifth dimension [8]. The bare parameters for these discretizations were cho- sen to lie on the same scaling trajectory, to enable a combined continuum extrapolation. Heavy quark masses between the charm and approximately half the bottom quark mass were used, with a linear extrapolation in 1/m H applied to reach the physical Bs mass, where m H is the mass of the heavy meson used to set the heavy quark mass. For the central fit, the 123 Eur. Phys. J. C (2022) 82:869 Page 113 of 296 869 authors set the heavy quark mass through the pseudoscalar heavy-strange meson Hs , and estimate systematic uncertain- ties by comparing these results to those obtained with H a heavy-light meson or a heavy-heavy meson. For the quenched heavy quark Möbius DWF are always used, with a domain-wall height slightly different from the one adopted for light valence quarks. The choice helps to keep cutoff effects under control, according to the study in Ref. [545]. The chiral- continuum extrapolations are performed with a Taylor expansion in a2 and m2π − mphys( 2π ) and the associated systematic error is estimated by varying the fit function to apply cuts in the pion mass. The corresponding systematic error is estimated as approximately 0.5%, which is roughly equal to the statistical uncertainty and to the systematic uncertainties associated with extrapolation to the physical m Bs mass and with higher-order corrections to the static limit. These latter corrections take the form O(2/m2B ). The error estimate comes from assuming the coefficient of such terms is up to five times largers than the fitted O(/m Bs ) coefficient. Isospin corrections and heavy-quark discretization effects are estimated to be less than 0.1%. At time of writing, RBC/UKQCD 18A [70] has not been published and therefore is not included in our average. Thus, our averages for these quantities remain the same as in Ref. [4], N f = 2 + 1 : fB = 192.0(4.3) MeV Refs. [54,65–68], (210) N f = 2 + 1 : fBs = 228.4(3.7) MeV Refs. [54,65–68], (211) f N f = 2 + 1 : Bs = 1.201(0.016) Refs. [54,66–68,70]. (212) fB No new N f = 2 + 1 + 1 calculations of fB , fBs/ fB or fB s have appeared since the last FLAG review. Therefore, our( ) averages for these quantities remain the same as in Ref. [4], N f = 2 + 1 + 1 : fB = 190.0(1.3) MeV Refs. [16,30,63,64], (213) N f = 2 + 1 + 1 : fBs = 230.3(1.3) MeV Refs. [16,30,63,64], (214) N f = 2 + 1 + 1 : fBs = 1.209(0.005) Refs. [16,30,63,64]. (215) fB The PDG presented averages for the N f = 2+ 1 and N f = 2+ 1+ 1 lattice-QCD determinations of the isospin-averaged fB , fBs and fBs/ fB in 2020 [165]. The N f = 2 + 1 and N f = 2 + 1 + 1 lattice-computation results used in Ref. [165] are identical to those included in our current work, and the averages quoted in Ref. [165] are those determined in [4]. 8.2 Neutral B-meson mixing matrix elements Neutral B-meson mixing is induced in the Standard Model through 1-loop box diagrams to lowest order in the electroweak theory, similar to those for short-distance effects in neutral kaon mixing. The effective Hamiltonian is given by HB= 2 2 2,SM = G F MW 0 d 0 seff 16π2 (FdQ1 + Fs Q1)+ h.c., (216) with Qq = [1 b̄γμ(1 − ] [ ] γ5)q b̄γμ(1 − γ5)q , (217) where q = d or s. The short-distance function F0q in Eq. (216) is much simpler compared to the kaon mixing case due to the hierarchy in the CKM matrix elements. Here, only one term is relevant, F0 2q = λtq S0(xt ) (218) where λ ∗tq = Vtq Vtb, (219) 123 869 Page 114 of 296 Eur. Phys. J. C (2022) 82:869 and where S0(xt ) is an Inami–Lim function with xt = m2t /M2W , which describes the basic electroweak loop contributions without QCD [417]. The transition amplitude for B0q with q = d or s can be written as 2 2 [ ] 〈B̄0|HB=2 0 G F MW 2q eff |Bq 〉 = 2 λtq S0(xt )η2B1(6π 2 )− { }γ0/(2βḡ 0) ∫(μ) ḡ(μ) ( )× γ (g)exp dg + γ0 4π 0 β(g) β0g × 〈B̄0q |QqR(μ)|B0q 〉 + h.c., (220) where QqR(μ) is the renormalized four-fermion operator (usually in the NDR scheme of MS). The running coupling ḡ, the β-function β(g), and the anomalous dimension of the four-quark operator γ (g) are defined in Eqs. (142) and (143). The product of μ-dependent terms on the second line of Eq. (220) is, of course, μ-independent (up to truncation errors arising from the use of perturbation theory). The explicit expression for the short-distance QCD correction factor η2B (calculated to NLO) can be found in Ref. [410]. For historical reasons the B-meson mixing matrix elements are often parameterized in terms of bag parameters defined as 〈 ∣ ∣ 〉 B̄0 ∣q QqR(μ)∣ B0q BBq (μ) = 8 . (221) 3 f 2 2 B mq B The renormalization group independent (RGI) B parameter B̂ is defined as in the case of the kaon, and expressed to 2-loop order as ( 2 )− { [ ]}γ0/(2βḡ 0) 2= (μ) + ḡ(μ) β1γ0 − β0γ1B̂Bq 1 BB4π (4π)2 2 2 q (μ), (222)β0 with β0, β1, γ0, and γ1 defined in Eq. (144). Note, as Eq. (220) is evaluated above the bottom threshold (mb < μ < mt ), the active number of flavours here is N f = 5. Nonzero transition amplitudes result in a mass difference between the CP eigenstates of the neutral B-meson system. Writing the mass difference for a B0q meson as mq , its Standard Model prediction is G2= F m 2 W m Bm q 2 2q 6π2 |λtq | S0(xt )η2B fB B̂Bq . (223)q Experimentally, the mass difference is determined from the oscillation frequency of the CP eigenstates. The frequencies are measured precisely with an error of less than a percent. Many different experiments have measured md , but the current average [165] is dominated by the LHCb experiment. For ms the experimental average is again dominated by results from LHCb [165] and the precision reached is about one per mille. With these experimental results and lattice-QCD calculations of f 2B B̂Bq , λtq can be determined. In lattice-QCD calculations the flavour SU (3)-breaking ratioq 2 2 = fB Bs Bsξ 2 (224)fB Bd Bd can be obtained more precisely than the individual Bq -mixing matrix elements because statistical and systematic errors cancel in part. From ξ2, the ratio |Vtd/Vts | can be determined and used to constrain the apex of the CKM triangle. Neutral B-meson mixing, being loop-induced in the Standard Model, is also a sensitive probe of new physics. The most general B = 2 effective Hamiltonian that describes contributions to B-meson mixing in the Standard Model and beyond is given in terms of five local four-fermion operators: ∑ ∑5 HB=2eff,BSM = C qiQi , (225) q=d,s i=1 123 Eur. Phys. J. C (2022) 82:869 Page 115 of 296 869 Table 43 Neutral B- and Bs-meson mixing matrix elements (in MeV) and bag parameters √ √ Collaboration Refs. N f fBd B̂Bd fBs B̂Bs B̂Bd B̂Bs HPQCD 19A [71] 2+1+1 A ◦ ◦  ◦  210.6(5.5) 256.1(5.7) 1.222(61) 1.232(53) FNAL/MILC 16 [73] 2+1 A  ◦  ◦  227.7(9.5) 274.6(8.4) 1.38(12)(6) 1.443(88)(48) RBC/UKQCD 14A [67] 2+1 A ◦ ◦ ◦ ◦  240(15)(33) 290(09)(40) 1.17(11)(24) 1.22(06)(19) FNAL/MILC 11A [548] 2+1 C  ◦  ◦  250(23)† 291(18)† − − HPQCD 09 [72] 2+1 A ◦ ◦∇ ◦ ◦  216(15)∗ 266(18)∗ 1.27(10)∗ 1.33(6)∗ HPQCD 06A [549] 2+1 A    ◦  − 281(21) − 1.17(17) ETM 13B [56] 2 A  ◦ ◦   216(6)(8) 262(6)(8) 1.30(5)(3) 1.32(5)(2) ETM 12A, 12B [538,550] 2 C  ◦ ◦   − − 1.32(8) 1.36(8) PDG averages of decay constant fB0 and fBs [164] are used to obtain these values †Reported f 2B B at μ = mb is converted to RGI by multiplying the 2-loop factor 1.517∇While wrong-spin contributions are not included in the HMrSχPT fits, the effect is expected to be small for these quantities (see description in FLAG 13 [2]) ∗This result uses an old determination of r1 = 0.321(5) fm from Ref. [118] that has since been superseded, which however has only a small effect in the total error budget (see description in FLAG 13 [2]) Reported B at μ = mb = 4.35 GeV is converted to RGI by multiplying the 2-loop factor 1.521 where Q1 is defined in Eq. (217) and where Qq = [ − ] [b̄(1 γ )q b̄(1 − ] [ ] [ ]γ )q Qq, = b̄α(1 − γ )qβ β α2 5 Qq = [ ] [ 5 ] 3 [ 5 ] [b̄ (1 − γ5)q ] , b̄ q4 (1 − γ5)q b̄(1 + γ5)q , Q5 = b̄α(1 − γ )qβ b̄β5 (1 + γ )qα5 , (226) with the superscripts α, β denoting colour indices, which are shown only when they are contracted across the two bilinears. There are three other basis operators in the B = 2 effective Hamiltonian. When evaluated in QCD, however, they give identical matrix elements to the ones already listed due to parity invariance in QCD. The short-distance Wilson coefficients Ci depend on the underlying theory and can be calculated perturbatively. In the Standard Model only matrix elements of Qq1 contribute to mq , while all operators do, for example, for general SUSY extensions of the Standard Model [469]. The matrix elements or bag parameters for the non-SM operators are also useful to estimate the width difference q between the CP eigenstates of the neutral B meson in the Standard Model, where combinations of matrix elements of Qq1 , Qqq 2 , andQ3 contribute to q at O(1/mb) [546,547]. In√this sectio√n, we report on results from lattice-QCD calculations for the neutral B-meson mixing parameters B̂Bd , B̂Bs , fBd B̂Bd , fBs B̂Bs and the SU (3)-breaking ratios BBs/BBd and ξ defined in Eqs. (221), (222), and (224). The results are summarized in Tables 43 and 44 and in Figs. 29 and 30. Additional details about the underlying simulations and systematic error estimates are given in Appendix C.6.2. Some collaborations do not provide the RGI quantities B̂Bq , but quote instead BB(μ)M S,N DR . In such cases, we convert the results using Eq. (222) to the RGI quantities quoted in Table 43 with a brief description for each case. More detailed descriptions for these cases are provided in FLAG13 [2]. We do not provide the B-meson matrix elements of the other operators Q2−5 in this report. They have been calculated in Ref. [56] for the N f = 2 case and in Refs. [73,548] for N f = 2 + 1. There are no new results for N f = 2 reported after the previous FLAG review. In this category, one work (ETM 13B) [56] passes the quality criteria. A description of this work can be found in the FLAG 13 review [2] where it did not enter the average as it had not appeared in a journal. Because this is the only result available for N f = 2, we quote their values as our estimates √ √ fBd B̂bd = 216(10) MeV fBs B̂Bs = 262(10) MeV Ref. [56], (227) 123 Publication status Continuum extrap C oh lai tr ia ol nextrapolatio F ninite volume Renormalization/m H ae ta cv hy i- nq guark treatment 869 Page 116 of 296 Eur. Phys. J. C (2022) 82:869 Table 44 Results for SU (3)-breaking ratios of neutral Bd - and Bs -meson mixing matrix elements and bag parameters Collaboration Refs. N f ξ BBs/BBd HPQCD 19A [71] 2+1+1 A ◦ ◦  ◦  1.216(16) 1.008(25) RBC/UKQCD 18A [70] 2+1 P      1.1939(67)(+95−177) 0.9984(45)(+80−63) FNAL/MILC 16 [73] 2+1 A  ◦  ◦  1.206(18) 1.033(31)(26) RBC/UKQCD 14A [67] 2+1 A ◦ ◦ ◦ ◦  1.208(41)(52) 1.028(60)(49) FNAL/MILC 12 [551] 2+1 A ◦ ◦  ◦  1.268(63) 1.06(11) RBC/UKQCD 10C [541] 2+1 A    ◦  1.13(12) − HPQCD 09 [72] 2+1 A ◦ ◦∇ ◦ ◦  1.258(33) 1.05(7) ETM 13B [56] 2 A  ◦ ◦   1.225(16)(14)(22) 1.007(15)(14) ETM 12A, 12B [538,550] 2 C  ◦ ◦   1.21(6) 1.03(2) PDG average of the ratio of decay constants fBs / fB0 [164] is used to obtain the value∇Wrong-spin contributions are not included in the HMrSχPT fits. As the effect may not be negligible, these results are excluded from the average (see description in FLAG 13 [2]) Fig. 29 Neutral B- and Bs-meson mixing matrix elements and bag parameters [values in Table 43 and Eqs. (227), (230), (233), (228), (231), (234)] N f = 2 : B̂Bd = 1.30(6) B̂Bs = 1.32(5) Ref. [56], (228) ξ = 1.225(31) BBs/BBd = 1.007(21) Ref. [56]. (229) For the N f = 2+ 1 case the RBC/UKQCD collaboration reported their new results on the flavour SU(3) breaking ratio of neutral B-meson mixing parameters in 2018. Their paper [70] has not been published yet, thus the results will not be included in our averages presented here. Their computation uses ensembles generated by the 2+1 flavour domain-wall fermion (DWF) formulation. The use of the DWFs also for the heavy quarks makes the renormalization structure simple. Because of the chiral symmetry, the mixing is the same as in the continuum theory. The operators for standard model mixing matrix elements are multiplicatively renormalized. Since they only report the SU(3) breaking ratio, the renormalization of the operators is not needed. The lattice spacings employed are not as fine as some of the recent results reported here. But, by applying successive 123 Publication status Continuum extrap C olh ai tr ia ol nextrapolatio F ninite volume Renormalization/m H ae ta cv hy in-q guark treatment Eur. Phys. J. C (2022) 82:869 Page 117 of 296 869 Fig. 30 The SU (3)-breaking quantities ξ and BBs /BBd [values in Table 44 and Eqs. (229), (232), (235)] stout link smearings in the heavy DWF, the reach to heavy mass is improved, which allows them to simulate up to half of the physical bottom mass. Two ensembles are of physical ud quark mass at a = 0.11 and 0.09 fm, and there is yet another ensemble off the physical point but with finer lattice spacing, a = 0.07 fm. This is the first computation using physical light-quark mass for these quantities, which yields a drastic reduction of the chiral extrapolation error. The results that enter our averages for N f = 2 + 1 are FNAL/MILC 16 [73], which had been included in the averages at FLAG 19 [4], RBC/UKQCD 14A [67], included in the averages at FLAG 16 [3], and HPQCD 09 [72] for which a description is available in FLAG 13 [2]. Thus, the averages for N f = 2 + 1 are unchanged: N f = 2 + 1 : √ √ fBd B̂Bd = 225(9)MeV fBs B̂Bs = 274(8)MeV Refs. [67,72,73], (230) B̂Bd = 1.30(10) B̂Bs = 1.35(6) Refs. [67,72,73], (231) ξ = 1.206(17) BBs/BBd = 1.032(38) Refs. [67,73]. (232) Here all the above equations have not been changed from the FLAG 19. The averages were obtained using the nested averaging scheme described in Sect. 2.3.2, due to a nested correlation structure among the results. Details are discussed in the FLAG 19 report [4]. We have the first N f = 2 + 1 + 1 calculation for these quantities by the HPQCD collaboration HPQCD 19A [71], using the MILC collaboration’s HISQ ensembles. The lattice spacings used are 0.15, 0.12 and 0.09 fm, among which the mass of the Nambu–Goldstone pion (lightest in the staggered taste multiplets) is as small as 130 MeV for two coarser lattices. However, the smallest root-mean-squared pion mass through all taste multiplets is 241 MeV, which is a similar size as the FNAL/MILC 16 result [73] with N f = 2 + 1 and makes the rating on the chiral extrapolation a green circle. The heavy quark formulation used is non-relativistic QCD (NRQCD). The NRQCD action employed is improved from that used in older calculations, especially by including one-loop radiative corrections to most of the coefficients of the O(v4b) terms [552]. The b-quark mass is pre-tuned with the spin-averaged kinetic mass of the ϒ and ηb states. Therefore, there is no need for extrapolation or interpolation on the b-quark mass. The HISQ-NRQCD four-quark operators are matched through O(1/M) and renormalized to one-loop, which includes the effects of O(αs), O(QCD/M), O(αs/aM), and O(αs QCD/M). The remaining error is dominated by O(αsQCD/M) 2.9% and O(α2s ) 2.1% for individual bag parameters. The bag parameters are the primary quantities calculated in this work. The mixing matrix elements are obtained by combining the so-obtained bag parameters with the B-meson decay constants calculated by Fermilab-MILC collaboration (FNAL/MILC 17 [16]). Because this is the only result available for N f = 2 + 1 + 1, we quote their values as the FLAG estimates 123 869 Page 118 of 296 Eur. Phys. J. C (2022) 82:869 N f = 2 + 1 + 1: √ √ fBd B̂bd = 210.6(5.5) MeV fBs B̂Bs = 256.1(5.7) MeV Ref. [71], (233) B̂Bd = 1.222(61) B̂Bs = 1.232(53) Ref. [71], (234) ξ = 1.216(16) BBs/BBd = 1.008(25) Ref. [71]. (235) We note that the above results within same N f (e.g., those in Eqs. (233–235)) are all correlated with each other, due to the use of the same gauge field ensembles for different quantities. The results are also correlated with the averages obtained in Sect. 8.1 and shown in Eqs. (207)–(209) for N f = 2, Eqs. (210)–(212) for N f = 2 + 1 and Eqs. (213)–(215) for N f = 2 + 1 + 1. This is because the calculations of B-meson decay constants and mixing quantities are performed on the same (or on similar) sets of ensembles, and results obtained by a given collaboration use the same actions and setups. These correlations must be considered√when using our averages as inputs to unitarity triangle (UT) fits. For this reason, if one were for example to estimate fBs B̂s from the separate averages of fBs (Eq. (211)) and B̂s (Eq. (231)) for N f = 2 + 1, one would obtain a value about one standard deviation below the one quoted above in Eq. (230). While these two estimates lead to compatible results, giving us confidence that all uncertainties have been properly addressed, we do not recommend combining averages this way, as many correlations would have to be taken into account to properly assess the errors. We recommend instead using the numbers quoted above. In the future, as more independent calculations enter the averages, correlations between the lattice-QCD inputs to UT fits will become less significant. 8.3 Semileptonic form factors for B decays to light flavours The Standard Model differential rate for the decay B(s) → Pν involving a quark-level b → u transition is given, at leading order in the weak interaction, by a formula analogous to the one for D decays in Eq. (181), but with D → B(s) and the relevant CKM matrix element |Vcq | → |Vub|: √ → 2 2 (q2 − m2)2 E2 − m2d(B(s) Pν) 2 = G F |Vub|  P P dq 24π3 q4m2 [( ) B(s) ] m2 3m2× 1 +  m22 B (E2 − m2 )| f+(q2)|2 +  (m2 2 2 2 22q (s) P P 8q2 B − m P ) | f0(q )| . (236)(s) Again, for  = e, μ the contribution from the scalar form factor f0 can be neglected, and one has a similar expression to Eq. (183), which, in principle, allows for a direct extraction of |Vub| by matching theoretical predictions to experimental data. However, while for D (or K ) decays the entire physical range 0 ≤ q2 ≤ q2max can be covered with moderate momenta accessible to lattice simulations, in B → πν decays one has q2max ∼ 26 GeV2 and only part of the full kinematic range is reachable. As a consequence, obtaining |Vub| from B → πν is more complicated than obtaining |Vcd(s)| from semileptonic D-meson decays. In practice, lattice computations are restricted to large values of the momentum transfer q2 (see Sect. 7.2) where statistical and momentum-dependent discretization errors can be controlled,37 which in existing calculations roughly cover the upper third of the kinematically allowed q2 range. Since, on the other hand, the decay rate is suppressed by phase space at large q2, most of the semileptonic B → π events are observed in experiment at lower values of q2, leading to more accurate experimental results for the binned differential rate in that region.38 It is, therefore, a challenge to find a window of intermediate values of q2 at which both the experimental and lattice results can be reliably evaluated. State-of-the-art determinations of CKM matrix elements, say, e.g., |Vub|, are obtained from joint fits to lattice and experi- mental results, keeping the relative normalization |Vub|2 as a free parameter. This requires, in particular, that both experimental and lattice data for the q2-dependence be parameterized by fitting data to specific ansätze, with the ultimate aim of minimizing the systematic uncertainties involved. This plays a key role in assessing the systematic uncertainties of CKM determinations, 37 The variance of hadron correlation functions at nonzero three-momentum is dominated at large Euclidean times by zero-momentum multiparticle states [553]; therefore the noise-to-signal grows more rapidly than for the vanishing three-momentum case. 38 Upcoming data from Belle II are expected to significantly improve the precision of experimental results, in particular, for larger values of q2. 123 Eur. Phys. J. C (2022) 82:869 Page 119 of 296 869 and will be discussed extensively in this section. A detailed discussion of the parameterization of form factors as a function of q2 can be found in Appendix B.1. 8.3.1 Form factors for B → πν The semileptonic decay process B → πν enables determination of the CKM matrix element |Vub|within the Standard Model via Eq. (236). Early results for B → πν form factors came from the HPQCD [554] and FNAL/MILC [555] collaborations. Only HPQCD provided results for the scalar form factor f0. Our 2016 review featured a significantly extended calculation of B → πν from FNAL/MILC [556] and a new computation from RBC/UKQCD [557]. All the above computations employ N f = 2 + 1 dynamical configurations, and provide values for both form factors f+ and f0. In addition, HPQCD using MILC ensembles had published the first N f = 2 + 1 + 1 results for the B → πν scalar form factor, working at zero recoil (q2 = q2max) and pion masses down to the physical value [558]; this adds to previous reports on ongoing work to upgrade their 2006 computation [559,560]. Since this latter result has no immediate impact on current |Vub| determinations, which come from the vector-form-factor-dominated decay channels into light leptons, we will from now on concentrate on the N f = 2+1 determinations of the q2-dependence of B → π form factors. Several groups are working on new calculations of the B → π form factors and have reported on their progress at the annual Lattice conferences and the 2020 Asia-Pacific Symposium for Lattice Field Theory. The results are preliminary or blinded, so not yet ready for inclusion in this review. The JLQCD collaboration is using Möbius Domain Wall fermions (including for the heavy quark) with a ≈ 0.08, 0.055, and 0.044 fm and pion masses down to 225 MeV to study this process [561–563]. FNAL/MILC is using N f = 2 + 1 + 1 HISQ ensembles with a ≈ 0.15, 0.12, 0.088 fm, 0.057 fm, with Goldstone pion mass down to its physical value [564,565]. The RBC/UKQCD Collaborations have added a new Möbius-domain-wall-fermion ensemble with a ≈ 0.07 fm and mπ ≈ 230 MeV to their analysis [566]. Returning to the calculations that contribute to our averages (with no new results since FLAG 19), both the HPQCD and the FNAL/MILC computations of B → πν amplitudes use ensembles of gauge configurations with N f = 2 + 1 flavours of rooted staggered quarks produced by the MILC collaboration; however, the latest FNAL/MILC work makes a much more extensive use of the currently available ensembles, both in terms of lattice spacings and light-quark masses. HPQCD have results at two values of the lattice spacing (a ≈ 0.12, 0.09 fm), while FNAL/MILC employs four values (a ≈ 0.12, 0.09, 0.06, 0.045 fm). Lattice-discretization effects are estimated within heavy-meson rooted staggered chiral perturbation theory (HMrSχPT) in the FNAL/MILC computation, while HPQCD quotes the results at a ≈ 0.12 fm as central values and uses the a ≈ 0.09 fm results to quote an uncertainty. The relative scale is fixed in both cases through the quark– antiquark potential-derived ratio r1/a. HPQCD set the absolute scale through the ϒ 2S–1S splitting, while FNAL/MILC uses a combination of fπ and the same ϒ splitting, as described in Ref. [54]. The spatial extent of the lattices employed by HPQCD is L 2.4 fm, save for the lightest mass point (at a ≈ 0.09 fm) for which L 2.9 fm. FNAL/MILC, on the other hand, uses extents up to L 5.8 fm, in order to allow for light-pion masses while keeping finite-volume effects under control. Indeed, while in the 2006 HPQCD work the lightest RMS pion mass is 400 MeV, the latest FNAL/MILC work includes pions as light as 165 MeV – in both cases the bound mπ L  3.8 is kept. Other than the qualitatively different range of MILC ensembles used in the two computations, the main difference between HPQCD and FNAL/MILC lies in the treatment of heavy quarks. HPQCD uses the NRQCD formalism, with a 1-loop matching of the relevant currents to the ones in the relativistic theory. FNAL/MILC employs the clover action with the Fermilab interpretation, with a mostly nonperturbative renormalization of the relevant currents, within which the overall renormalization factor of the heavy-light current is written as a product of the square roots of the renormalization factors of the light-light and heavy-heavy temporal vector currents (which are determined nonperturbatively) and a residual factor that is computed using 1-loop perturbation theory. (See Table 45; full details about the computations are provided in tables in Appendix C.6.3.) The RBC/UKQCD computation is based on N f = 2 + 1 DWF ensembles at two values of the lattice spacing (a ≈ 0.12, 0.09 fm), and pion masses in a narrow interval ranging from slightly above 400 MeV to slightly below 300 MeV, keeping mπ L  4. The scale is set using the − baryon mass. Discretization effects coming from the light sector are estimated in the 1% ballpark using HMχPT supplemented with effective higher-order interactions to describe cutoff effects. The b quark is treated using the Columbia RHQ action, with a mostly nonperturbative renormalization of the relevant currents. Discretization effects coming from the heavy sector are estimated with power-counting arguments to be below 2%. Given the large kinematical range available in the B → π transition, chiral extrapolations are an important source of systematic uncertainty: apart from the eventual need to reach physical pion masses in the extrapolation, the applicability of χPT is not guaranteed for large values of the pion energy Eπ . Indeed, in all computations Eπ reaches values in the 1 GeV ballpark, and chiral extrapolation systematics is the dominant source of errors. FNAL/MILC uses SU (2) NLO HMrSχPT 123 869 Page 120 of 296 Eur. Phys. J. C (2022) 82:869 Table 45 Results for the B → πν semileptonic form factor Collaboration Refs. N f FNAL/MILC 15 [556] 2+1 A  ◦  ◦  BCL RBC/UKQCD 15 [557] 2+1 A ◦ ◦ ◦ ◦  BCL HPQCD 06 [554] 2+1 A ◦ ◦ ◦ ◦  n/a for the continuum-chiral extrapolation, supplemented by NNLO analytic terms and hard-pion χPT terms [507];39 systematic uncertainties are estimated through an extensive study of the effects of varying the specific fit ansatz and/or data range. RBC/UKQCD uses SU (2) hard-pion HMχPT to perform its combined continuum-chiral extrapolation, and obtains estimates for systematic uncertainties by varying the ansätze and ranges used in fits. HPQCD performs chiral extrapolations using HMrSχPT formulae, and estimates systematic uncertainties by comparing the result with the ones from fits to a linear behaviour in the light-quark mass, continuum HMχPT, and partially quenched HMrSχPT formulae (including also data with different sea and valence light-quark masses). FNAL/MILC and RBC/UKQCD describe the q2-dependence of f+ and f0 by applying a BCL parameterization to the form factors extrapolated to the continuum limit, within the range of values of q2 covered by data. (A discussion of the various parameterizations can be found in Appendix B.1.) RBC/UKQCD generate synthetic data for the form factors at some values of q2 (evenly spaced in z) from the continuous function of q2 obtained from the joint chiral-continuum extrapolation, which are then used as input for the fits. After having checked that the kinematical constraint f+(0) = f0(0) is satisfied within errors by the extrapolation to q2 = 0 of the results of separate fits, this constraint is imposed to improve fit quality. In the case of FNAL/MILC, rather than producing synthetic data a functional method is used to extract the z-parameterization directly from the fit functions employed in the continuum-chiral extrapolation. In the case of HPQCD, the parameterization of the q2-dependence of form factors is somewhat intertwined with chiral extrapolations: a set of fiducial values {E (n)π } is fixed for each value of the light-quark mass, and f+,0 are interpolated to each of the E (n)π ; chiral extrapolations are then performed at fixed Eπ (i.e., m 2π and q are varied subject to Eπ = constant). The interpolation is performed using a Ball–Zwicky (BZ) ansatz [567]. The q2-dependence of the resulting form factors in the chiral limit is then described by means of a BZ ansatz, which is cross-checked against Becirevic–Kaidalov (BK) [500], Richard Hill (RH) [568], and Boyd–Grinstein–Lebed (BGL) [569] parameterizations (see Appendix B.1), finding agreement within the quoted uncertainties. Unfortunately, the correlation matrix for the values of the form factors at different q2 is not provided, which severely limits the possibilities of combining them with other computations into a global z-parameterization. The different ways in which the current results are presented do not allow a straightforward averaging procedure. RBC/UKQCD only provides synthetic values of f+ and f0 at a few values of q2 as an illustration of their results, and FNAL/MILC does not quote synthetic values at all. In both cases, full results for BCL z-parameterizations defined by Eq. (532) are quoted. In the case of HPQCD 06, unfortunately, a fit to a BCL z-parameterization is not possible, as discussed above. In order to combine these form factor calculations, we start from sets of synthetic data for several q2 values. HPQCD and RBC/UKQCD directly provide this information; FNAL/MILC present only fits to a BCL z-parameterization from which we can easily generate an equivalent set of form factor values. It is important to note that in both the RBC/UKQCD synthetic data and the FNAL/MILC z-parameterization fits the kinematic constraint at q2 = 0 is automatically included (in the FNAL/MILC case the constraint is manifest in an exact degeneracy of the (a+n , a0n) covariance matrix). Due to these considerations, in our opinion, the most accurate procedure is to perform a simultaneous fit to all synthetic data for the vector and scalar form factors. Unfortunately, the absence of information on the correlation in the HPQCD result between the vector and scalar form factors even at a single q2 point makes it impossible to include consistently this calculation in the overall fit. 39 It is important to stress the finding in Ref. [506] that the factorization of chiral logs in hard-pion χPT breaks down, implying that it does not fulfill the expected requisites for a proper effective field theory. Its use to model the mass dependence of form factors can thus be questioned. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment z-parameterization Eur. Phys. J. C (2022) 82:869 Page 121 of 296 869 Fig. 31 The form factors f+(q2) and f0(q2) for B → πν plotted versus z (left panel) and q2 (right panel). In the left plot, we removed the Blaschke factors. See text for a discussion of the data set. The grey and salmon bands display our preferred N+ = N 0 = 3 BCL fit (five parameters) Table 46 Coefficients and correlation matrix for the N+ = N 0 = 3 z-expansion fit of the B → π form factors f+ and f0. The coefficient a02 is fixed by the f+(q2 = 0) = f0(q2 = 0) constraint. The chi-square per degree of freedom is χ2/dof = 0.82. The lattice calculations that enter this fit are taken from FNAL/MILC 15 [556] and RBC/UKQCD 15 [557]. The parameterizations are defined in Eqs. (532) and (533) B → π (N f = 2 + 1) Central values Correlation matrix a+0 0.404 (13) 1 0.404 0.118 0.327 0.344 a+1 −0.68 (13) 0.404 1 0.741 0.310 0.900 a+2 −0.86 (61) 0.118 0.741 1 0.363 0.886 a00 0.490 (21) 0.327 0.310 0.363 1 0.233 a01 −1.61 (16) 0.344 0.900 0.886 0.233 1 In fact, the HPQCD and FNAL/MILC statistical uncertainties are highly correlated (because they are based on overlapping subsets of MILC N f = 2 + 1 ensembles) and, without knowledge of the f+ − f0 correlation we are unable to construct the HPQCD-FNAL/MILC off-diagonal entries of the overall covariance matrix. In conclusion, we will present as our best result a combined vector and scalar form factor fit to the FNAL/MILC and RBC/UKQCD results that we treat as completely uncorrelated. For sake of completeness, we will also show the results of a vector form factor fit alone in which we include one HPQCD datum at q2 = 17.34 GeV2 assuming conservatively a 100% correlation between the statistical error of this point and of all FNAL/MILC synthetic data. In spite of contributing just one point, the HPQCD datum has a significant weight in the fit due to its small overall uncertainty. We stress again that this procedure is slightly inconsistent because FNAL/MILC and RBC/UKQCD include information on the kinematic constraint at q2 = 0 in their f+ results. The resulting data set is then fitted to the BCL parameterization in Eqs. (532) and (533). We assess the systematic uncertainty due to truncating the series expansion by considering fits to different orders in z. In Fig. 31, we show the FNAL/MILC, RBC/UKQCD, and HPQCD data points for (1 − q2/m2 2 2B∗) f+(q ) and f0(q ) versus z. The data is highly linear and we get χ2/dof = 0.82 with N+ = N 0 = 3. Note that this implies three independent parameters for f+ corresponding to a polynomial through O(z3) and two independent parameters for f0 corresponding to a polynomial through O(z2) (the coefficient a02 is fixed using the q2 = 0 kinematic constraint). We cannot constrain the coefficients of the z-expansion beyond this order; for instance, including a fourth parameter in f+ results in 100% uncertainties on a+2 and a + 3 . The outcome of the five-parameter N+ = N 0 = 3 BCL fit to the FNAL/MILC and RBC/UKQCD calculations is shown in Table 46. The uncertainties on a+,00 , a+,01 and a + 2 encompass the central values obtained from N + = 2, 4 and N 0 = 2, 4, 5 fits and thus adequately reflect the systematic uncertainty on those series coefficients. The fit shown in Table 46 can therefore be used as the averaged FLAG result for the lattice-computed form factor f+(q2). The coefficient a+3 can be obtained from the values for a + 0 –a + 2 using Eq. (531). The coefficient a02 can be obtained from all other coefficients imposing the f+(q2 = 0) = f (q20 = 0) constraint. The fit is illustrated in Fig. 31. 123 869 Page 122 of 296 Eur. Phys. J. C (2022) 82:869 We emphasize that future lattice-QCD calculations of semileptonic form factors should publish their full statistical and systematic correlation matrices to enable others to use the data. It is also preferable to present a set of synthetic form factors data equivalent to the z-fit results, since this allows for an independent analysis that avoids further assumptions about the compatibility of the procedures to arrive at a given z-parameterization.40 It is also preferable to present covariance/correlation matrices with enough significant digits to calculate correctly all their eigenvalues. For the sake of completeness, we present also a standalone z-fit to the vector form factor. In this fit, we are able to include the single f+ point at q2 = 17.34 GeV2 that we mentioned above. This fit uses the FNAL/MILC and RBC/UKQCD results that do make use of the kinematic constraint at q2 = 0, but is otherwise unbiased. The results of the three-parameter BCL fit to the HPQCD, FNAL/MILC and RBC/UKQCD calculations of the vector form factor are: N f = 2 + 1 : a+0 =⎛0.421(13), a + 1 = −0.⎞35(10), a + 2 = −0.41(64) ; 1.000 0.306 0.084 corr(ai , a j ) = ⎝ 0.306 1.000 0.856⎠ . (237) 0.084 0.856 1.000 Note that the a+0 coefficient, that is the most relevant for input to the extraction of Vub from semileptonic B → πν( = e, μ) decays, shifts by about a standard deviation. 8.3.2 Form factors for Bs → Kν Similar to B → πν, measurements of Bs → Kν decay rates enable determinations of the CKM matrix element |Vub| within the Standard Model via Eq. (236). From the lattice point of view, the two channels are very similar. As a matter of fact, Bs → Kν is actually somewhat simpler, in that the kaon mass region is easily accessed by all simulations making the systematic uncertainties related to chiral extrapolation smaller. At the time of our FLAG 19 review [4], results for Bs → Kν form factors were provided by HPQCD [570] and RBC/UKQCD [557] for both form factors f+ and f0, in both cases using N f = 2 + 1 dynamical configurations. HPQCD has recently emphasized the value of using ratios of form factors for the processes Bs → Kν and Bs → Dsν for the determination of |Vub/Vcb| [571]. In the FLAG Review 19 [4], FNAL/MILC preliminary results had been reported for both N f = 2+1 [572] and N f = 2+1+1 [564], but were not included in the average due to their non-final status. The N f = 2+1 results have since been published [573]; we will, therefore, include them in the average here. The RBC/UKQCD computation has been published together with the B → πν computation discussed in Sect. 8.3.1, all technical details being practically identical. The main difference is that errors are significantly smaller, mostly due to the reduction of systematic uncertainties due to the chiral extrapolation. Detailed information is provided in Appendix C.6.3. The RBC/UKQCD collaboration is also working on an improved determination of Bs → K form factors that includes a finer lattice spacing, with preliminary results shown in Ref. [566], but these results cannot yet be included in the average. The HPQCD computation uses ensembles of gauge configurations with N f = 2 + 1 flavours of asqtad rooted staggered quarks produced by the MILC collaboration at two values of the lattice spacing (a ≈ 0.12, 0.09 fm), for three and two different sea-pion masses, respectively, down to a value of 260 MeV. The b quark is treated within the NRQCD formalism, with a 1-loop matching of the relevant currents to the ones in the relativistic theory, omitting terms of O(αsQCD/mb). The HISQ action is used for the valence s quark. The continuum-chiral extrapolation is combined with the description of the q2-dependence of the form factors into a modified z-expansion (cf. Appendix B.1) that formally coincides in the continuum with the BCL ansatz. The dependence of form factors on the pion energy and quark masses is fitted to a 1-loop ansatz inspired by hard-pion χPT [507], that factorizes out the chiral logarithms describing soft physics. The FNAL/MILC computation coincides with HPQCD’s in using ensembles of gauge configurations with N f = 2+ 1 flavours of asqtad rooted staggered quarks produced by the MILC collaboration, but only one ensemble is shared, and a different valence regularization is employed; we will thus treat the two results as fully independent from the statistics point of view. FNAL/MILC uses three values of the lattice spacing (a ≈ 0.12, 0.09, 0.06 fm); only one value of the sea pion mass and the volume is available at the extreme values of the lattice spacing, while four different masses and volumes are considered at a = 0.09 fm. Heavy quarks are treated within the Fermilab approach. HMrSχPT expansion is used at next-to-leading order in SU (2) and leading order in 1/MB , including next-to-next-to-leading-order (NNLO) analytic and generic discretization terms, to perform continuum-chiral extrapolations. 40 Note that generating synthetic data is a trivial task, but less so is choosing the number of required points and the q2 values that lead to an optimal description of the form factors. 123 Eur. Phys. J. C (2022) 82:869 Page 123 of 296 869 Table 47 Summary of lattice calculations of the Bs → Kν semileptonic form factors Collaboration Refs. N f FNAL/MILC 19 [573] 2+1 A  ◦  ◦  BCL RBC/UKQCD 15 [557] 2+1 A ◦ ◦ ◦ ◦  BCL HPQCD 14 [570] 2+1 A ◦ ◦ ◦ ◦  BCL† †Results from modified z-expansion Table 48 Coefficients and Bs → K (N f = 2 + 1)correlation matrix for the N+ = N 0 = 4 z-expansion of Central values Correlation matrix the Bs → K form factors f+ and f0. The coefficient a0 is a + 0 0.374 (12) 1 0.2471 −0.1715 −0.2396 0.6445 0.3791 0.28573 fixed by the a+1 −0.672 (64) 0.2471 1 0.4198 0.1724 0.4626 0.8183 0.7948 f+(q2 = 0) = f (q20 = 0) constrain. The chi-square per a + 2 0.07 (31) −0.1715 0.4198 1 0.8136 0.3804 0.7293 0.7481 degree of freedom is a+3 1.34 (52) −0.2396 0.1724 0.8136 1 0.2823 0.5120 0.5529 χ2/dof = 1.54 and the errors on a00 0.2203 (68) 0.6445 0.4626 0.3804 0.2823 1 0.6570 0.4837 the z-parame√ters have been a02 = 1 0.089 (57) 0.3791 0.8183 0.7293 0.5120 0.6570 1 0.9220rescaled by χ /dof 1.24 a02 0.24 (23) 0.2857 0.7948 0.7481 0.5529 0.4837 0.9220 1 Fig. 32 The form factors f+(q2) and f (q20 ) for Bs → Kν plotted versus z (left panel) and q2 (right panel). In the left plot, we remove the Blaschke factors. See text for a discussion of the data sets. The grey and salmon bands display our preferred N+ = N 0 = 4 BCL fit (seven parameters) Hard kaons are assumed to decouple, i.e., their effect is reabsorbed in the SU (2) LECs. Continuum- and chiral-extrapolated values of the form factors are fitted to a z-parametrization imposing the kinematical constraint f+(0) = f0(0). See Table 47 and the tables in Appendix C.6.3 for full details. In order to combine the results for the q2 dependence of the form factors from the three collaborations, we will follow a similar approach to the one adopted above for B → πν, and produce synthetic data from the preferred fits quoted in the papers, to obtain a dataset to which a joint fit can be performed. Note that the kinematic constraint at q2 = 0 is included in all three cases; we will impose it in our fit as well, since the synthetic data will implicitly depend on that fitting choice. However, it is worth mentioning that the systematic uncertainty of the resulting extrapolated value f+(0) = f0(0) can be fairly large, the main reason being the required long extrapolation from the high-q2 region covered by lattice data. While we stress that the average far away from the high-q2 region has to be used carefully, it is possible that increasing the number of 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment z-parameterization 869 Page 124 of 296 Eur. Phys. J. C (2022) 82:869 z coefficients beyond what is sufficient for a good description of the lattice data and using unitarity constraints to control the size of additional terms, might yield fits with a more stable extrapolation at very low q2. We plan to include said unitarity analysis into the next edition of the FLAG review. It is, however, important to emphasize that joint fits with experimental data, where the latter accurately map the q2 region, are expected to be safe. √ Our fits employ a BCL ansatz with t+ = (M + M )2 and t = t+ − t+(t+ − t−), with t− = (M − M )2B π 0 Bs K . Our pole factors will contain a single pole in both the vector and scalar channels, for which we take the mass values MB∗ = 5.32465 GeV and MB∗(0+) = 5.68 GeV.41 The constraint f+(0) = f0(0) is imposed by expressing the coefficient b0 0− in terms of allN 1 others. The outcome of the seven-parameter N+ = N 0 = 4 BCL fit, which we quote as our preferred result, is shown in Table 48. The fit has a chi-square per degree of freedom χ2/dof = 1.54. Following th√e PDG recommendation, we rescale the whole covariance matrix by χ2/dof: the errors on the z-parameters are increased by χ2/dof = 1.24 and the correlation matrix is unaffected. The parameters shown in Table 48 provide the averaged FLAG results for the lattice-computed form factors f+(q2) and f (q20 ). The coefficient a+4 can be obtained from the values for a + + 0 –a3 using Eq. (531). The fit is illustrated in Fig. 32.42 We will conclude by pointing out progress in the application of the npHQET method to the extraction of semileptonic form factors, reported for Bs → K transitions in Ref. [574], which extends the work of Ref. [575]. This is a methodological study based on CLS N f = 2 ensembles at two different values of the lattice spacing and pion masses, and full 1/mb corrections are incorporated within the npHQET framework. Emphasis is on the role of excited states in the extraction of the bare form factors, which are shown to pose an impediment to reaching precisions better than a few percent. 8.3.3 Form factors for rare and radiative B-semileptonic decays to light flavours Lattice-QCD input is also available for some exclusive semileptonic decay channels involving neutral-current b → q transi- tions at the quark level, where q = d, s. Being forbidden at tree level in the SM, these processes allow for stringent tests of potential new physics; simple examples are B → K ∗γ , B → K (∗)+−, or B → π+− where the B meson (and therefore the light meson in the final state) can be either neutral or charged. The corresponding SM effective weak Hamiltonian is considerably more complicated than the one for the tree-level processes discussed above: after integrating out the top quark and the W boson, as many as ten dimension-six operators formed by the product of two hadronic currents or one hadronic and one leptonic current appear.43 Three of the latter, coming from penguin and box diagrams, dominate at short distances and have matrix elements that, up to small QED corrections, are given entirely in terms of B → (π, K , K ∗) form factors. The matrix elements of the remaining seven operators can be expressed, up to power corrections whose size is still unclear, in terms of form factors, decay constants and light-cone distribution amplitudes (for the π , K , K ∗ and B mesons) by employing OPE arguments (at large di-lepton invariant mass) and results from Soft Collinear Effective Theory (at small di-lepton invariant mass). In conclusion, the most important contributions to all of these decays are expected to come from matrix elements of current operators (vector, tensor, and axial-vector) between one-hadron states, which in turn can be parameterized in terms of a number of form factors (see Ref. [577] for a complete description). In channels with pseudoscalar mesons in the final state, the level of sophistication of lattice calculations is similar to the B → π case and there are results for the vector, scalar, and tensor form factors for B → K+− decays by HPQCD [579], and more recent results for both B → π+− [580] and B → K+− [578] from FNAL/MILC. Full details about these two calculations are provided in Table 49 and in Appendix C.6.4. Both computations employ MILC N f = 2 + 1 asqtad ensembles. HPQCD [581] and FNAL/MILC [582] have also companion papers in which they calculate the Standard Model predictions for the differential branching fractions and other observables and compare to experiment. The HPQCD computation employs NRQCD b quarks and HISQ valence light quarks, and parameterizes the form factors over the full kinematic range using a model-independent z-expansion as in Appendix B.1, including the covariance matrix of the fit coefficients. In the case of the (separate) FNAL/MILC computations, both of them use Fermilab b quarks and asqtad light quarks, and a BCL z-parameterization of the form factors. 41 These are the values used in the FNAL/MILC determination, while HPQCD and RBC/UKQCD use MB∗(0+) = 5.6794(10) GeV and MB∗(0+) = 5.63 GeV, respectively. They also employ different values of t+ and t0 than employed here, which again coincide with FNAL/MILC’s choice. 42 Note that in FLAG 19 [4] we had adopted the threshold t+ = (MBs + MK )2 rather than t+ = (MB + M 2π ) . This change impacted the z-range which the physical q2 interval maps onto. We also point out that, in the FLAG 19 version of Fig. 32, the three synthetic f0 data points from HPQCD were plotted incorrectly, but this did not affect the fit. 43 See, e.g., Ref. [576] and references therein. 123 Eur. Phys. J. C (2022) 82:869 Page 125 of 296 869 Table 49 Summary of lattice calculations of the B → K semileptonic form factors Collaboration Refs. N f FNAL/MILC 15D [578] 2+1 A  ◦  ◦  BCL HPQCD 13E [579] 2+1 A ◦ ◦ ◦ ◦  BCL Table 50 Coefficients and B → π (N = 2 + 1) correlation matrix for the f N+ = N 0 = 3 z-expansion of Central values Correlation matrix the B → π form factor fT aT0 0.393(17) 1.000 0.400 0.204 0.166 aT1 −0.65(23) 0.400 1.000 0.862 0.806 aT2 −0.6(1.5) 0.204 0.862 1.000 0.989 aT3 0.1(2.8) 0.166 0.806 0.989 1.000 Reference [580] includes results for the tensor form factor for B → π+− not included in previous publications on the vector and scalar form factors [556]. Nineteen ensembles from four lattice spacings are used to control continuum and chiral extrapolations. The results for Nz = 4 z-expansion of the tensor form factor and its correlations with the expansions for the vector and scalar form factors, which we consider the FLAG estimate, are shown in Table 50. Partial decay widths for decay into light leptons or τ+τ− are presented as a function of q2. The former is compared with results from LHCb [583], while the latter is a prediction. The averaging of the HPQCD and FNAL/MILC results for the B → K form factors is similar to our treatment of the B → π and Bs → K form factors. In this case, even though the statistical uncertainties are partially correlated because of some overlap between the adopted sets of MILC ensembles, we choose to treat the two calculations as independent. The reason is that, in B → K , statistical uncertainties are subdominant and cannot be easily extracted from the results presented by HPQCD and FNAL/MILC. Both collaborations provide only the outcome of a simultaneous z-fit to the vector, scalar and tensor form factors, that we use to generate appropriate synthetic data. We then impose the kinematic constraint f+(q2 = 0) = f (q20 = 0) and fit to (N+ = N 0 = N T = 3) BCL parameterization. The functional forms of the form factors that we use are identical to those adopted in Ref. [582].44 The results of the fit are presented in Table 51. The fit is illustrated in Fig. 33. Note that the average for the fT form factor appears to prefer the FNAL/MILC synthetic data. This happens because we perform a correlated fit of the three form factors simultaneously (both FNAL/MILC and HPQCD present covariance matrices that include correlations between all form factors). We checked that the average for the fT form factor, obtained neglecting correlations with f0 and f+, is a little lower and lies in between the two data sets. There is still a noticeable tension between the FNAL/MILC and HPQCD data for the tensor form factor; indeed, a standalone fit to these data results in χ2red = 7.2/3 = 2.4, while a similar standalone joint fit to f+ and f0 has χ2red = 9.2/7 = 1.3. Finally, the global fit that is shown in the figure has χ2red = 18.6/10 = 1.86. Lattice computations of form factors in channels with a vector meson in the final state face extra challenges with respect to the case of a pseudoscalar meson: the state is unstable, and the extraction of the relevant matrix element from correlation functions is significantly more complicated; χPT cannot be used as a guide to extrapolate results at unphysically heavy pion masses to the chiral limit. While field-theory procedures to take resonance effects into account are available [328–330,585–592], they have not yet been implemented in the existing preliminary computations, which therefore suffer from uncontrolled systematic errors in calculations of weak decay form factors into unstable vector meson final states, such as the K ∗ or ρ mesons.45 44 Note in particular that not much is known about the sub-threshold poles for the scalar form factor. FNAL/MILC includes one pole at the B∗s0 mass as taken from the calculation in Ref. [584]. 45 In cases such as B → D∗ transitions, that will be discussed below, this is much less of a practical problem due to the very narrow nature of the resonance. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment z-parameterization 869 Page 126 of 296 Eur. Phys. J. C (2022) 82:869 Table 51 Coefficients and B → K (N = 2 + 1) correlation matrix for the f N+ = N 0 = N T = 3 Central values Correlation matrix z-expansion of the B → K form + factors f+, f0 and fT . The a0 0.471 (14) 1 0.513 0.128 0.773 0.594 0.613 0.267 0.118 coefficient a02 is fixed by the a+ −0.74 (16) 0.513 1 0.668 0.795 0.966 0.212 0.396 0.263 f+(q2 = = 2 = 10) f0(q 0) a+2 0.32 (71) 0.128 0.668 1 0.632 0.768 −0.104 0.0440 0.187constraint. The chi-square per 0 degree of freedom is a0 0.301 (10) 0.773 0.795 0.632 1 0.864 0.393 0.244 0.200 χ2/dof = 1.86 and the errors on a01 0.40 (15) 0.594 0.966 0.768 0.864 1 0.235 0.333 0.253 the z-parame√ters have been= a T 0 0.455 (21) 0.613 0.212 −0.104 0.393 0.235 1 0.711 0.608rescaled by χ2/dof 1.36 aT1 −1.00 (31) 0.267 0.396 0.0440 0.244 0.333 0.711 1 0.903 aT2 −0.9 (1.3) 0.118 0.263 0.187 0.200 0.253 0.608 0.903 1 Fig. 33 The B → K form factors f+(q2), f (q2) and f (q20 T ) plotted versus z (left panels) and q2 (right panels). In the plots as a function of z, we remove the Blaschke factors. See text for a discussion of the data sets. The grey, salmon and blue bands display our preferred N+ = N 0 = N T = 3 BCL fit (eight parameters) As a consequence of the complexity of the problem, the level of maturity of these computations is significantly below the one present for pseudoscalar form factors. Therefore, we only provide a short guide to the existing results. Horgan et al. have obtained the seven form factors governing B → K ∗+− (as well as those for Bs → φ +− and for the charged-current decay Bs → K ∗ν) in Ref. [593] using NRQCD b quarks and asqtad staggered light quarks. In this work, they use a modified z-expansion to simultaneously extrapolate to the physical light-quark masses and fit the q2-dependence. As discussed above, the unstable nature of the vector mesons was not taken into account. Horgan et al. use their form-factor results to calculate the differential branching fractions and angular distributions and discuss the implications for phenomenology in a companion paper [594]. An update of the form factor fits that enforces endpoint relations and also provides the full correlation matrices can be found in Ref. [595]. Finally, preliminary results on B → K ∗+− and Bs → φ+− by RBC/UKQCD have been reported in Refs. [596–598]. 123 Eur. Phys. J. C (2022) 82:869 Page 127 of 296 869 8.4 Semileptonic form factors for B(s) → D(s)ν and B ∗(s) → D(s)ν The semileptonic processes B(s) → D(s)ν and B(s) → D∗(s)ν have been studied extensively by experimentalists and theorists over the years. They allow for the determination of the CKM matrix element |Vcb|, an extremely important parameter of the Standard Model. The matrix element Vcb appears in many quantities that serve as inputs to CKM unitarity triangle analyses and reducing its uncertainties is of paramount importance. For example, when K , the measure of indirect CP violation in the neutral kaon system, is written in terms of the parameters ρ and η that specify the apex of the unitarity triangle, a factor of |Vcb|4 multiplies the dominant term. As a result, the errors coming from |Vcb| (and not those from BK ) are now the dominant uncertainty in the Standard Model (SM) prediction for this quantity. The decay rate for B → Dν can be parameterized in terms of vector and scalar form factors in the same way as, e.g., B → πν (see Sect. 8.3). The decay rate for B → D∗ν is different because the final-state hadron is spin-1. There are four form factors used to describe the vector and axial-vector current matrix elements that are needed to calculate this decay. We define the 4-velocity of the meson P as vP = p ∗P/m P and the polarization vector of the D as . When the light lepton  = e, or μ, it is traditional to use w = v · v ∗ rather than q2B D( ) as the variable upon which the form factors depend. Then, the form factors hV and h Ai , with i = 1, 2 or 3 are defined by 〈D∗| √Vμ|B〉 = m βBm D∗hV (w)ε ∗ν αμναβ vD∗vB, (238) 〈 ∗| | 〉 = √ [ ]D Aμ B i m Bm ∗ h ∗μD A1(w)(1 + w) − h A2(w)∗ · vBvBμ − h A3(w)∗ · vBvD∗μ . (239) The differential decay rates can then be written as46 dB−→D0− 2 3ν̄ =GFm D 2 2 3/2 2 2 2 dw 48π3 (m B + m D) (w − 1) |ηEW| |Vcb| |G(w)| , (240) dB−→D0∗ − G2 3 ν̄ = Fm D∗3 (m B − m 2 2 1/2 2 2 2D∗) (w − 1) |ηEW| |Vcb| χ(w)|F(w)| , (241)dw 4π where w = vB · vD(∗) (depending on whether the final-state meson is D or D∗) and ηEW = 1.0066 is the 1-loop electroweak correction [599]. The function χ(w) in Eq. (241) depends on the recoil w and the meson masses, and reduces to unity at zero recoil [576].47 These formulas do not include terms that are proportional to the lepton mass squared, which can be neglected for  = e, μ. Further details of the definitions of F and G (which can be expressed in terms of the form factors hV and h Ai ) may be found, e.g., in Ref. [576]. Until recently, most unquenched lattice calculations for B → D∗ν and B → Dν decays ∗ focused on the form factors at zero recoil F B→D (1) and GB→D(1); these can then be combined with experimental input to extract |Vcb|. The main reasons for concentrating on the zero recoil point are that (i) the decay rate then depends on a single form factor, and (ii) for B → D∗ν, there are no O(QC D/m Q) contributions due to Luke’s theorem [600]. Further, the zero recoil form factor can be computed via a double ratio in which most of the current renormalization cancels and heavy-quark discretization errors are suppressed by an additional power of  /m . Recent work on B → D(∗)QC D Q ν transitions has started to explore the dependence of the relevant form factors on the momentum transfer, using a similar methodology to the one employed in B → πν transitions; see Sect. 8.3 for a detailed discussion. Early computations of the form factors for B → Dν decays include N f = 2 + 1 results by FNAL/MILC [601,602] for GB→D(1) and the N f = 2 study by Atoui et al. [603], that in addition to providing GB→D(1) explored the w > 1 region. This latter work also provided the first results for Bs → Dsν amplitudes, again including information about the∗ momentum-transfer dependence. The first published unquenched results for F B→D (1), obtained by FNAL/MILC, date from 2008 [604]. In 2014 and 2015, significant progress was achieved in N f = 2 + 1 computations: the FNAL/MILC value∗ for F B→D (1) was updated in Ref. [605], and full results for B → Dν at w ≥ 1 were published by FNAL/MILC [606] 46 These are the only meson decay channels dealt with in this review where we apply the Sirlin correction factor ηEW , that incorporates leading- order, structure-independent corrections. This is in keeping with common practice. While including ηEW in the analysis of b → c transitions is nearly universal in the literature, this is not so in other flavour-changing decays. It is worth stressing that this is just part of the expected corrections – cf. the discussion of QED corrections in the sections of this review dealing with light meson decay – and therefore its inclusion is largely arbitrary, insofar as a precise control of the full corrections, including the structure-dependent ones, is unavailable for a given channel. It is also necessary to remark, on the other hand, that different practices contribute to a small ambiguity in the comparison of CKM matrix elements determined from different decays, precisely of the order of the typically neglected electromagnetic corrections. 47 The reason to keep the factor χ(w) outside the combination of form factors that defines F(w) is conventional, and inspired by the heavy-quark limit. One particular consequence of this notation is that at zero recoil F(1) = h A1 (1). 123 869 Page 128 of 296 Eur. Phys. J. C (2022) 82:869 and HPQCD [607]. These works also provided full results for the scalar form factor, allowing analysis of the decay with a final-state τ . In the FLAG 19 review [4], we included new results for Bs → Dsν form factors over the full kinematic range for N f = 2 + 1 from HPQCD [608,609], and for B(s) → D∗(s)ν form factors at zero recoil with N f = 2 + 1 + 1 also from HPQCD [610,611]. Most recently, HPQCD published further new calculations of the B → D∗s s form factor at zero recoil [612] and of the Bs → Ds form factors in the full kinematic range [613], now using MILC’s HISQ N f = 2+1+1 ensembles and using the HISQ action also for the b quark. Both of these calculations have recently been used by LHCb to determine |Vcb| [614,615], as discussed further in Sect. 8.9. Improved calculations of the B → D and Bs → Ds form factors are also underway by RBC/UKQCD [566], and the Fermilab/MILC computation of the B → D∗ form factors at nonzero recoil is nearing completion [616]. The JLQCD collaboration also presented preliminary results for the B → D and B → D∗ form factors, both at nonzero recoil [617]. In the discussion below, we mainly concentrate on the latest generation of results, which supersedes previous N f = 2 + 1 determinations and allows for an extraction of |Vcb| that incorporates information about the q2-dependence of the decay rate (cf. Sect. 8.9). 8.4.1 B(s) → D(s) decays We will first discuss the N f = 2+1 computations of B → Dν by FNAL/MILC and HPQCD mentioned above, both based on MILC asqtad ensembles. Full details about all the computations are provided in Table 54 and in the tables in Appendix C.6.5. The FNAL/MILC study [606] employs ensembles at four values of the lattice spacing ranging between approximately 0.045 fm and 0.12 fm, and several values of the light-quark mass corresponding to pions with RMS masses ranging between 260 MeV and 670 MeV (with just one ensemble with MRMSπ 330 MeV at the finest lattice spacing). The b and c quarks are treated using the Fermilab approach. The quantities directly studied are the form factors h± defined by 〈D(pD√)|i c̄γμb|B(pB)〉 = h+(w)(vB + vD)μ + h−(w)(vB − vD)μ, (242) m Dm B which are related to the standard vector and scalar form factors by 1 [ ] f+(q2) = √ (1 + r)h+(w)− (1 − r)h−(w) , (243) 2 r √ [ + − ]2 = 1 w 1 wf0(q ) r + h+(w)+ − h−(w) , (244)1 r 1 r with r = m /m . (Recall that q2 = (p − p )2 = m2 2D B B D B + √m D − 2wm Bm D .) The hadronic form factor relevant for experiment, G(w), is then obtained from the relation G(w) = 4r f+(q2)/(1 + r). The form factors are obtained from double ratios of three-point functions in which the flavour-conserving current renormalization factors cancel. The remaining matching factor to the flavour-changing normalized current is estimated with 1-loop lattice perturbation theory. In order to obtain h±(w), a joint continuum-chiral fit is performed to an ansatz that contains the light-quark mass and lattice-spacing dependence predicted by next-to-leading order HMrSχPT, and the leading dependence on mc predicted by the heavy-quark expansion (1/m2c for h+ and 1/mc for h−). The w-dependence, which allows for an interpolation in w, is given by analytic terms up to (1−w)2, as well as a contribution from the logarithm proportional to g2D∗Dπ . The total resulting systematic error, determined as a function of w and quoted at the representative point w = 1.16 as 1.2% for f+ and 1.1% for f0, dominates the final error budget for the form factors. After f+ and f0 have been determined as functions of w within the interval of values of q2 covered by the computation, synthetic data points are generated to be subsequently fitted to a z-expansion of the BGL form, cf. Sect. 8.3, with pole factors set to unity. This in turn enables one to determine |Vcb| from a joint fit of this z-expansion and experimental data. The value of the zero-recoil form factor resulting from the z-expansion is GB→D(1) = 1.054(4)stat(8)sys. (245) The HPQCD computations [607,609] use ensembles at two values of the lattice spacing, a = 0.09, 0.12 fm, and two and three values of light-quark masses, respectively. The b quark is treated using NRQCD, while for the c quark the HISQ action is used. The form factors studied, extracted from suitable three-point functions, are 123 Eur. Phys. J. C (2022) 82:869 Page 129 of 296 869 √ √ 〈D 0(s)(pD s )|V |B 〉 = 2M f (s) k k (s)( ) (s) B(s) ‖ , 〈D(s)(pD s )|V |B( ) (s)〉 = 2MB s p( ) D f(s) ⊥ , (246) where Vμ is the relevant vector current and the B(s) rest frame is chosen. The standard vector and scalar form factors are retrieved as [ ] f (s) = √ 1 f (s)+ + (M − E f (s)B D ) , (247)2M ‖ (s) (s) ⊥√ B(s) (s) = 2M [ ] B f (s) (s)(M − E ) f + (M2 − E2 f (s)0 2 − 2 B s D s ‖ B D ) ⊥ . (248)M M ( ) ( ) (s) (s)B(s) D(s) The currents in the effective theory are matched at 1-loop to their continuum counterparts. Results for the form factors are then fitted to a modified BCL z-expansion ansatz, that takes into account simultaneously the lattice spacing, light-quark masses, and q2-dependence. For the mass dependence, NLO chiral logarithms are included, in the form obtained in hard-pion χPT (see footnote 33). As in the case of the FNAL/MILC computation, once f+ and f0 have been determined as functions of q2, |Vcb| can be determined from a joint fit of this z-expansion and experimental data. The papers quote for the zero-recoil vector form factor the result GB→D(1) = 1.035(40) GBs→Ds (1) = 1.068(40). (249) The HPQCD and FNAL/MILC results for B → D differ by less than half a standard deviation (assuming they are uncorrelated, which they are not as some of the ensembles are common) primarily because of lower precision of the former result. The HPQCD central value is smaller by 1.8 of the FNAL/MILC standard deviations than the FNAL/MILC value. The dominant source of errors in the |Vcb| determination by HPQCD are discretization effects and the systematic uncertainty associated with the perturbative matching. In order to combine the form factor determinations of HPQCD and FNAL/MILC into a lattice average, we proceed in a similar way as with B → πν and Bs → Kν above. FNAL/MILC quotes synthetic values for each form factor at three values of w (or, alternatively, q2) with a full correlation matrix, which we take directly as input. In the case of HPQCD, we use their preferred modified z-expansion parameterization to produce synthetic values of the form factors at five different values of q2 (three for f+ and two for f0). This leaves us with a total of six (five) data points in the kinematical range w ∈ [1.00, 1.11] for the form factor f+ ( f0). As in the case of B → πν, we conservatively assume a 100% correlation of statistical uncertainties between HPQCD and FNA√L/MILC. We then fit this data set to a BCL ansatz, using t+ = (MB0 + MD±)2 51.12 GeV2 and t0 = (MB0 + MD±)( MB0 − √ MD±) 2 6.19 GeV2. In our fits, pole factors have been set to unity, i.e., we do not take into account the effect of sub-threshold poles, which is then implicitly absorbed into the series coefficients. The reason for this is our imperfect knowledge of the relevant resonance spectrum in this channel, which does not allow us to decide the precise number of poles needed.48 This, in turn, implies that unitarity bounds do not rigorously apply, which has to be taken into account when interpreting the results (cf. Appendix B.1). With a procedure similar to what we adopted for the B → π and Bs → K cases, we impose the kinematic constraint at q2 = 0 by expressing the a0 0− coefficient in the z-expansion of f0 in terms of all the other coefficients. As mentioned above,N 1 FNAL/MILC provides synthetic data for f+ and f0 including correlations; HPQCD presents the result of simultaneous z-fits to the two form factors including all correlations, thus enabling us to generate a complete set of synthetic data for f+ and f0. Since both calculations are based on MILC ensembles, we then reconstruct the off-diagonal HPQCD-FNAL/MILC entries of the covariance matrix by conservatively assuming that statistical uncertainties are 100% correlated. The Fermilab/MILC (HPQCD) statistical error is 58% (31%) of the total error for every f+ value, and 64% (49%) for every f0 one. Using this information we can easily build the off-diagonal block of the overall covariance matrix (e.g., the covariance between [ f+(q2)] and [ f (q2)] is (δ[ f+(q2)] × 0.58) (δ[ f (q21 FNAL 0 2 HPQCD 1 FNAL 0 2 )]HPQCD × 0.49), where δ f is the total error). For our central value, we choose an N+ = N 0 = 3 BCL fit, shown in Table 52. The coefficient a+3 can be obtained from the values for a+0 –a + 2 using Eq. (531). We find χ 2/dof = 4.6/6 = 0.77. The fit, which is dominated by the FNAL/MILC calculation, is illustrated in Fig. 34. Reference [603] is the only existing N f = 2 work on B → Dν transitions, that furthermore provided the first available results for Bs → Dsν. This computation uses the publicly available ETM configurations obtained with the twisted-mass QCD action at maximal twist. Four values of the lattice spacing, ranging between 0.054 and 0.098 fm, are considered, with 48 As noted above, this is the same approach adopted by FNAL/MILC in their fits to a BGL ansatz. HPQCD, meanwhile, uses one single pole in the pole factors that enter their modified z-expansion, using their spectral studies to fix the value of the relevant resonance masses. 123 869 Page 130 of 296 Eur. Phys. J. C (2022) 82:869 Table 52 Coefficients and correlation matrix for the N+ = N 0 = 3 z-expansion of the B → D form factors f+ and f0. The chi-square per degree of freedom is χ2/dof = 4.6/6 = 0.77. The lattice calculations that enter this fit are taken from FNAL/MILC [606] and HPQCD [607] B → D (N f = 2 + 1) ain Central values Correlation matrix a+0 0.896 (10) 1 0.423 −0.231 0.958 0.596 a+1 −7.94 (20) 0.423 1 0.325 0.498 0.919 a+2 51.4 (3.2) −0.231 0.325 1 −0.146 0.317 a00 0.7821 (81) 0.958 0.498 −0.146 1 0.593 a01 −3.28 (20) 0.596 0.919 0.317 0.593 1 Fig. 34 The form factors f+(q2) and f0(q2) for B → Dν plotted versus z (left panel) and q2 (right panel). See text for a discussion of the data sets. The grey and salmon bands display our preferred N+ = N 0 = 3 BCL fit (five parameters) physical box lengths ranging between 1.7 and 2.7 fm. At two values of the lattice spacing two different physical volumes are available. Charged-pion masses range between ≈ 270 and ≈ 490 MeV, with two or three masses available per lattice spacing and volume, save for the a ≈ 0.054 fm point at which only one light mass is available for each of the two volumes. The strange- and heavy-valence quarks are also treated with maximally twisted-mass QCD. The quantities of interest are again the form factors h± defined above. In order to control discretization effects from the heavy quarks, a strategy similar to the one employed by the ETM collaboration in their studies of B-meson decay constants (cf. Sect. 8.1) is employed: the value of G(w) is computed at a fixed value of mc and several values of a heavier quark mass m(k)h = λkmc, where λ is a fixed scaling parameter, and step-scaling functions are built as = G(w, λ k+1mc,mc, a2) k(w) G 2 . (250)(w, λkmc,mc, a ) Each ratio is extrapolated to the continuum limit, σk(w) = lima→0 k(w). One then exploits the fact that the mh → ∞ limit of the step-scaling is fixed. In particular, it is easy to find from the heavy-quark expansion that limmh→∞ σ(1) = 1. In this way, the physical result at the b-quark mass can be reached by interpolating σ(w) between the charm region (where the computation can be carried out with controlled systematics) and the known static limit value. In practice, the values of mc and ms are fixed at each value of the lattice spacing such that the experimental kaon and Ds masses are reached at the physical point, as determined in Ref. [618]. For the scaling parameter, λ = 1.176 is chosen, and eight scaling steps are performed, reaching mh/mc = 1.1769 4.30, approximately corresponding to the ratio of the physical b- and c-masses in the MS scheme at 2 GeV. All observables are obtained from ratios that do not require (re)normalization. The ansatz for the continuum and chiral extrapolation of k contains a constant and linear terms in msea and a2. Twisted boundary conditions in space are used for valence-quark fields for better momentum resolution. Applying this strategy, the form factors are finally obtained at four reference values of w between 1.004 and 1.062, and, after a slight extrapolation to w = 1, the result is GBs→Ds (1) = 1.052(46). (251) 123 Eur. Phys. J. C (2022) 82:869 Page 131 of 296 869 The authors also provide values for the form factor relevant for the meson states with light-valence quarks, obtained from a similar analysis to the one described above for the Bs → Ds case. Values are quoted from fits with and without a linear msea/ms term in the chiral extrapolation. The result in the former case, which safely covers systematic uncertainties, is GB→D(1) = 1.033(95). (252) Given the identical strategy, and the small sensitivity of the ratios used in their method to the light valence- and sea-quark masses, we assign this result the same ratings in Table 54 as those for their calculation of GBs→Ds (1). Currently, the precision of this calculation is not competitive with that of N f = 2+1 works, but this is due largely to the small number of configurations analyzed by Atoui et al. The viability of their method has been clearly demonstrated, however, which leaves significant room for improvement on the errors of both the B → D and Bs → Ds form factors with this approach by including either additional two-flavour data or analysing more recent ensembles with N f > 2. Atoui et al. also study the scalar and tensor form factors, as well as the momentum-transfer dependence of f+,0. The value of the ratio f0(q2)/ f+(q2) is provided at a reference value of q2 as a proxy for the slope of G(w) around the zero-recoil limit. Let us finally discuss the most recent results for Bs → Ds form factors, obtained by the HPQCD collaboration using MILC’s N f = 2+ 1+ 1 ensembles in Ref. [613]. Three values of the lattice spacing are used, including a very fine ensemble at a 0.044 fm; the pion mass is kept fixed at around 300 MeV, and in addition at the coarser a 0.09 fm lattice an ensemble with the physical pion mass is included. The scalar current needs no renormalization because of the PCVC relation, while the vector current is nonperturbatively normalized by imposing a condition based on the PCVC relation at zero recoil. Heavy quarks are treated in a fully relativistic fashion through the use of the HISQ regularization, employing bare values of the quark mass up to amh = 0.8 for the extrapolation to the physical b point. Results for the form factors are fitted to a modified z-expansion ansatz, based on a BCL ansatz with a Blaschke factor containing one sub-threshold pole, tuned to reproduce the lattice-spacing and heavy-quark-mass-dependent mass of the corresponding resonance. The final error budget is equally dominated by statistics and the combined effect of the continuum and heavy quark mass extrapolations, which correspond to 1.1% and 1.2% uncertainties, respectively, for the scalar form factor at zero recoil. The total uncertainty of the latter is thus below 2%, which remains true in the whole q2 range. The uncertainty of f+ is somewhat larger, starting at around 2% at q2 = 0 and increasing up to around 3.5% at zero recoil. One important matter of concern with this computation is the use of the a 0.044 fm ensemble with periodic boundary conditions, which suffers from severe topology freezing. Other than possible implications for statistical uncertainties, the lack of topology fluctuations are expected to significantly enhance finite-volume effects, which are no longer exponential in mπ L , but become power-like in the spatial volume. The authors neglect the impact of finite-volume effects in the computation, with a twofold argument: for the two coarser lattice spacings, the impact of pion-mass-related corrections on the heavy-meson states involved is presumably negligible; and, for the finest ensemble, the estimate of finite-volume effects on the Ds decay constant obtained in Ref. [142] turns out to be very small, a result which is presumed to extend to form factors. It is however unclear whether the latter argument would really hold, since the computation in Ref. [142] does show that the expected effect is heavily observable-dependent, reaching, e.g., more than 1% for fD . We have, therefore, concluded that our standard criteria for finite-volume effects cannot be applied at the finest lattice spacing, and opted to assign ◦ rating to them. We thus proceed to quote the final result of HPQCD 19 as the FLAG estimate for the N f = 2 + 1 + 1 Bs → Ds form factors. The preferred fit is a constrained BCL form with the imposition of the kinematical constraint f+(0) = f0(0), carried through z2 for f0 and z3 for f+. Both form factors contain just one sub-threshold pole, to which the masses MB∗ = 6.329 GeVc and MBc0 = 6.704 GeV, respectively, have been assigned. The fit parameters and covariance matrix, quoted in Table VIII of Ref. [613], are reproduced in Table 53. 8.4.2 Lepton-flavour-universality ratios R(D) and R(Ds) The availability of results for the scalar form factor f0 for B → Dν amplitudes allows us to study interesting observables that involve the decay in the τ channel. One such quantity is the ratio = B(B → Dτν)R(D) B → with  = e, μ, (253)(B Dν) which, in the Standard Model, depends only on the form factors and hadron and lepton masses. Indeed, the recent availability of experimental results for R(D) has made this quantity particularly relevant in the search for possible physics beyond the 123 869 Page 132 of 296 Eur. Phys. J. C (2022) 82:869 Table 53 Coefficients and B → D (N = 2 + 1 + 1) correlation matrix for the s s f z-expansion of the B is → Ds an Central values Correlation matrix form factors f+ and f0 a00 0.666(12) 1 0.62004 0.03149 1 0.03973 0.00122 a01 −0.26(25) 0.62004 1 0.36842 0.62004 0.12945 0.00002 a02 −0.1(1.8) 0.03149 0.36842 1 0.03149 0.22854 -0.00168 a+0 −0.075(12) 1 0.62004 0.03149 1 0.03973 0.00122 a+1 −3.24(45) 0.03973 0.12945 0.22854 0.03973 1 0.11086 a+2 0.7(2.0) 0.00122 0.00002 −0.00168 0.00122 0.11086 1 Standard Model. The most recent HFLAV average reads [619]: R(D)exp = 0.340(27)(13). (254) HPQCD provides a Standard-Model prediction for R(D) using the form factors from their 2015 lattice computation [607], R(D)lat = 0.300(8) Ref. [607]. (255) The FNAL/MILC collaboration computed R(D) using form factors from a combined fit of their lattice data and B → Dν experimental data: R(D)lat+exp = 0.299(11) Ref. [606]. (256) Note that other authors have obtained even smaller uncertainties from fits including B → Dν experimental data, cf. the value R(D)lat+exp = 0.299(3) quoted in Ref. [620]. One important reason for this is that, while Refs. [606,607] only use the 2009 BaBar data in Refs. [620,621] also incorporates the more precise Belle 2015 data in Ref. [622]. If instead we take the lattice-only form factors from FNAL/MILC, we obtain R(D)lat = 0.285(15) Ref. [606]. (257) (We average over electrons and muons in the denominator, which only affects the last digit.) Finally, using the FLAG average of the B → D form factors discussed above, we find R(D)FLAGlat = 0.2934(38). The ratio R(D) requires the integral of the branching ratios for  = e, μ, τ over the whole phase space. Since lattice simulations are sensitive mostly to relatively large q2 values, lattice-only calculations of R(D) rely on the extrapolation of the form factors to low q2 and are especially sensitive to the choice of parameterization. In order to estimate this source of systematics, we repeated the fit using the parameterization adopted by HPQCD in Ref. [607]. The main difference with respect to our default paremeterization is the inclusion of Blaschke factors for the form factors f+ and f0 located at M+ = MB∗ = 6.330(9) GeVc and M0 = 6.420(9) GeV; additionally, the parameter t0 is set to (m B − m D)2. Using five coefficients (a+ 01,2,3 and a1,2 with a03 fixed by the f+(q2 = 0) = f0(q2 = 0) condition) we find R D HPQCD( )lat = 0.3009(38) which deviates from R(D)FLAGlat by 1.4 σ . To take this potential source of systematic uncertainty into account we rescale accordingly the uncertainty of our default fit and obtain: R(D)lat = 0.2934(53), our average. (258) This result is about 1.5σ lower than the current experimental average [261] for this quantity. It has to be stressed that achieving this level of precision critically depends on the reliability with which the low-q2 region is controlled by the parameterizations of the form factors. After including the B → Dν ( = e, μ) data in the fit, as discussed at the end of Sect. 8.9, we obtain the following combined lattice plus experiment result: R(D)lat+exp = 0.2951(31), our average. (259) 123 Eur. Phys. J. C (2022) 82:869 Page 133 of 296 869 HPQCD also computes values for R(Ds), the analog of R(D) with both heavy-light mesons containing a strange quark. The earlier calculation using NRQCD b quarks gives R(Ds)lat = 0.301(6), N f = 2 + 1 [609]. (260) The newer calculation with HISQ b quarks yields the somewhat more precise value R(Ds)lat = 0.2987(46), N f = 2 + 1 + 1 [613]. (261) A similar ratio R(D∗) can be considered for B → D∗ transitions. As a matter of fact, the experimental value of R(D∗) is significantly more accurate than the one of R(D). A preliminary lattice-QCD-only prediction of R(D∗) was shown by A. Vaquero [623]. 8.4.3 Fragmentation fraction ratio fs/ fd Another area of immediate interest in searches for physics beyond the Standard Model is the measurement of Bs → μ+μ− decays, recently studied at the LHC. One of the inputs required by the LHCb analysis is the ratio of Bq meson (q = d, s) fragmentation fractions fs/ fd , where fq is the probability that a q quark hadronizes into a Bq . This ratio can be measured by writing it as a product of ratios that involve experimentally measurable quantities, cf. Refs. [624,625]. One of the factors is the ratio f (s) 2 (d) 20 (Mπ )/ f0 (MK ) of scalar form factors for the corresponding semileptonic meson decay, which is where lattice input becomes useful. A dedicated N = 2 + 1 study by FNAL/MILC49f [626] addresses the ratios of scalar form factors f (q)0 (q2), and quotes: f (s) M2 f (d)( )/ 20 π 0 (MK ) = 1.046(44)(15 f (s)), (M2 f (d)0 π )/ 0 (M2π ) = 1.054(47)(17), (262) where the first error is statistical and the second systematic. The more recent results from HPQCD [609] are: f (s) M2 f (d)0 ( π )/ 0 (M 2 K ) = 1.000 62 f (s) M2 f (d)( ), 0 ( π )/ 0 (M2π ) = 1.006(62). (263) Results from both groups lead to fragmentation fraction ratios fs/ fd that are consistent with LHCb’s measurements via other methods [625]. 8.4.4 B ∗(s) → D(s) decays The most precise computation of the zero-recoil form factors needed for the determination of |Vcb| from exclusive B semilep-∗ tonic decays comes from the B → D∗ν form factor at zero recoil F B→D (1), calculated by the FNAL/MILC collaboration. The original computation, published in Ref. [604], has been updated [605] by employing a much more extensive set of gauge ensembles and increasing the statistics of the ensembles originally considered, while preserving the analysis strategy. There are currently no final unquenched results for the relevant form factors at nonzero recoil, but work is in progress by FNAL/MILC [616] and JLQCD [617]. Reference [605] uses the MILC N f = 2 + 1 ensembles. The bottom and charm quarks are simulated using the clover action with the Fermilab interpretation and light quarks are treated via the asqtad staggered fermion action. Recalling the ∗ definition of the form factors in Eq. (239), at zero recoil F B→D (1) reduces to a single form factor h A1(1) coming from the axial-vector current 〈 ∗ ′ | | 〉 = √D ∗(v,  ) Aμ B(v) i 2m ′B2m D∗ μ h A1(1), (264) where ′ is the polarization of the D∗. The form factor is accessed through a ratio of three-point correlators, viz., R = 〈D ∗|c̄γ jγ5b|B〉 〈B|b̄γ ∗jγ5c|D 〉 2 A1 〈D∗| = |h A (1)| . (265)c̄γ4c|D∗〉 〈B|b̄γ4b|B〉 1 49 This work also provided a value for R(D), now superseded by Ref. [606]. 123 869 Page 134 of 296 Eur. Phys. J. C (2022) 82:869 Simulation data is obtained on MILC ensembles with five lattice spacings, ranging from a ≈ 0.15 fm to a ≈ 0.045 fm, and as many as five values of the light-quark masses per ensemble (though just one at the finest lattice spacing). Results are then extrapolated to the physical, continuum/chiral, limit employing staggered χPT. The D∗ meson is not a stable particle in QCD and decays predominantly into a D plus a pion. Nevertheless, heavy-light meson χPT can be applied to extrapolate lattice simulation results for the B → D∗ν form factor to the physical light-quark mass. The D∗ width is quite narrow, 0.096 MeV for the D∗±(2010) and less than 2.1 MeV for the D∗0(2007), making this system much more stable and long lived than the ρ or the K ∗ systems. The fact that the D∗ − D mass difference is close to the pion mass leads to the well-known “cusp” in RA1 just above the physical pion mass [627–629]. This cusp makes the chiral extrapolation sensitive to values used in the χPT formulas for the D∗Dπ coupling gD∗Dπ . The error budget in Ref. [605] includes a separate error of 0.3% coming from the uncertainty in gD∗Dπ in addition to general chiral extrapolation errors in order to take this sensitivity into account. The final value presented in Ref. [605] is N f = 2 + 1 : F B→D∗(1) = 0.906(4)(12), (266) where the first error is statistical, and the second the sum of systematic errors added in quadrature, making up a total error of 1.4% (down from the original 2.6% of Ref. [604]). The largest systematic uncertainty comes from discretization errors followed by effects of higher-order corrections in the chiral perturbation theory ansatz. In 2017, the HPQCD collaboration has published the first study of B(s) → D∗(s)ν form factors at zero recoil for N f = 2+1+1 using eight MILC ensembles with lattice spacing a ≈ 0.15 fm, 0.12 fm, and 0.09 fm [611]. There are three ensembles with varying light-quark masses for the two coarser lattice spacings and two choices of light-quark mass for the finest lattice spacing. In each case, there is one ensemble for which the light-quark mass is very close to the physical value. The b quark is treated using NRQCD and the light quarks are treated using the HISQ action. The resulting zero-recoil form factors are: N f = 2 + + ∗ ∗1 1 : F B→D (1) = 0.895(10)(24), F Bs→Ds (1) = 0.883(12)(28). (267) In 2019, the HPQCD collaboration published a new N f = 2 + 1 + 1 calculation of the Bs → D∗s form factor at zero recoil, now using the HISQ action also for the b quark [612]. The lattice methodology and ensembles used are the same as in their 2019 calculation of the Bs → Ds form factors [613], which was discussed in detail in Sect. 8.4.1. The resulting form factor is: N = 2 + 1 + 1 : F Bs→D∗f s (1) = 0.9020(96)(90). (268) The calculations in Refs. [611,612] use different b-quark actions and share only two ensambles at a = 0.09 fm and can be considered essentially independent, yielding the average: N f = 2 + 1 + 1 : F B →D∗s s (1) = 0.899(12), our average. (269) In recent years, the FNAL/MILC, HPQCD, and JLQCD collaborations have periodically reported about their efforts to determine the full momentum dependence of B ∗(s) → D(s)ν form factors at Lattice conferences. JLQCD efforts are based on N f = 2 + 1 Möbius domain-wall ensembles and a relativistic heavy quark action. The latest status update published in conference proceedings can be found in Ref. [617]. At the time of finalizing this review, FNAL/MILC and HPQCD have produced preprints with their final results for B → D∗ and B ∗s → Ds form factors, respectively. The FNAL/MILC computation, Ref. [630] is based on N f = 2 + 1 asqtad sea-quark ensembles with lattice spacing between approximately 0.15 and 0.045 fm, and uses a relativistic heavy-quark action with the Fermilab interpretation. The HPQCD computation, Ref. [631], is based on N f = 2+1+1 HISQ ensembles, and uses the same regularization for heavy quarks. Upon publication of both works, we intend to include full details for them in an upcoming intermediate update of this section (Table 54). 8.5 Semileptonic form factors for Bc → (ηc, J/ψ)ν decays In a recent publication, HPQCD 20B [632] provided the first full determination of Bc → J/ψ form factors, extending earlier preliminary work that also covered Bc → ηc, Refs. [633,634]. While the latter employed both NRQCD and HISQ actions for the valence b quark, and the HISQ action for the c quark, in HPQCD 20B the HISQ action is used throughout for all flavors. The setup is the same as for the Bs → Ds computation discussed above, HPQCD 19; we refer to the entries for the 123 Eur. Phys. J. C (2022) 82:869 Page 135 of 296 869 Table 54 Lattice results for mesonic processes involving b → c transitions Collaboration Refs. N f w = 1 form factor/ratio HPQCD 15, HPQCD 17 [607,609] 2+1 A ◦ ◦ ◦ ◦  GB→D(1) 1.035(40) FNAL/MILC 15C [606] 2+1 A  ◦  ◦  GB→D(1) 1.054(4)(8) Atoui 13 [603] 2 A  ◦  –  GB→D(1) 1.033(95) HPQCD 19 [613] 2+1+1 A  ◦ ◦∗   GBs→Ds (1) 1.071(37) HPQCD 15, HPQCD 17 [607,609] 2+1 A ◦ ◦ ◦ ◦  GBs→Ds (1) 1.068(40) Atoui 13 [603] 2 A  ◦  –  GBs→Ds (1) 1.052(46) HPQCD 17B [611] 2+1+1 A ◦   ◦  F B→D∗ (1) 0.895(10)(24) ∗ FNAL/MILC 14 [605] 2+1 A  ◦  ◦  F B→D (1) 0.906(4)(12) HPQCD 17B [611] 2+1+1 A ◦   ◦  F B →D∗s s (1) 0.883(12)(28)  ◦ ◦∗   F B →D∗HPQCD 19B [612] 2+1+1 A s s (1) 0.9020(96)(90) HPQCD 15, HPQCD 17 [607,609] 2+1 A ◦ ◦ ◦ ◦  GBs→Ds (1) 1.068(40) HPQCD 20B [632] 2+1+1 A  ◦ ◦∗   n/a n/a HPQCD 15, HPQCD 17 [607,609] 2+1 A ◦ ◦ ◦ ◦  R(D) 0.300(8) FNAL/MILC 15C [606] 2+1 A  ◦  ◦  R(D) 0.299(11) ∗The rationale for assigning a◦ rating is discussed in the text latter paper in summary tables for details. The flavor singlet nature of the final state means that there are contributions to the relevant three-point functions from disconnected Wick contractions, which are not discussed in the paper. There are however some relevant differences with Bs → Ds decays. In the J/ψ case, since the hadron in the final state has vector quantum numbers, the description of the hadronic amplitude requires four independent form factors V , A0, A1, A2. Specifically, 〈J/ψ(p′ 2iV (q 2) , λ)|c̄γ μb|B− μνρσ ∗ ′ ′c (p)〉 = ε  (p , λ)p pσ ,M ν ρBc + MJ/ψ ′ − ∗ ′ [ ∗ ′ ] 〈J/ψ(p , λ)|c̄γ μγ 5b|Bc (p)〉 =  (p , λ) · q  (p , λ) · q 2MJ/ψ A0(q 2) qμ + (M + M )A 2 ∗μ ′ μB J/ψ 1(q )  (p , λ)− q q2 c 2 ∗ ′ [ ] q · M2 2− 2  (p , λ) q B − MJ/ψA2(q ) + p μ + p′μ − c μ2 q , (270)MBc MJ/ψ q where μ is the polarization vector of the J/ψ state. The computed form factors are fitted to a z-parameterization-inspired ansatz, where coefficients are modified to model the lattice-spacing and the heavy- and light-mass dependences, for a total of 280 fit parameters. In the continuum and at physical kinematics only 16 parameters survive, as each form factor is parameterized by an expression of the form 3 2 = 1 ∑ F(q ) 2 a n nz , (271) P(q ) n=0 where the pole factor is given by ∏ P(q2) = z(q2, M2k ) (272) k 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment 869 Page 136 of 296 Eur. Phys. J. C (2022) 82:869 with {Mk} a different set of pole energies below the B D∗ threshold for each set of J P quantum numbers, taken from a mixture of experimental results, lattice determinations, and model estimates. The values used (in GeV) are 0− : 6.275, 6.872, 7.25; 1− : 6.335, 6.926, 7.02, 7.28; (273) 1+ : 6.745, 6.75, 7.15, 7.15. The outcome of the fit, that we quote as a FLAG estimate, is a0 a1 a2 a3 V 0.1057(55) −0.746(92) 0.10(98) 0.006(1.000) A0 0.1006(37) −0.731(72) 0.30(90) −0.02(1.00) A1 0.0553(19) −0.266(40) 0.31(70) 0.11(99) A2 0.0511(91) −0.22(19) −0.36(82) −0.05(1.00) The correlation matrix for the coefficients is provided in Tables XIX–XXVII of Ref. [632] 8.6 Semileptonic form factors for b → (p (∗), c )ν̄ decays The b → cν̄ and b → uν̄ transitions can also be probed in decays of b baryons. With the LHCb experiment, the final state of b → pμν̄ is easier to identify than that of B → πμν̄ [635], and the first determination of |Vub|/|Vcb| at the Large Hadron Collider was performed using a ratio of b → pμν̄ and b → cμν̄ decay rates [636] (cf. Sect. 8.10). The amplitudes of the decays b → pν̄ and b → cν̄ receive contributions from both the vector and the axial-vector components of the current in the matrix elements 〈p|ūγ μ(1 − γ5)b|b〉 and 〈c|c̄γ μ(1 − γ5)b|b〉. The matrix elements split into three form factors f+, f0, f⊥ mediated by the vector component of the current, and another three form factors g+, g0, g⊥ mediated by the axial-vector component – see, e.g., Ref. [511] for a complete description. Given the sensitivity to all Dirac structures, measurements of the baryonic decay rates also provides useful complementary constraints on right-handed couplings beyond the Standard Model [636]. To date, only one unquenched lattice-QCD computation of the b → p and b → c form factors with physical heavy- quark masses has been published: Detmold 15 [516]. This computation uses RBC/UKQCD N f = 2 + 1 DWF ensembles, and treats the b and c quarks within the Columbia RHQ approach. The renormalization of the currents is carried out using a mostly nonperturbative method, with residual matching factors computed at one loop. Two values of the lattice spacing (a ≈ 0.11, 0.085 fm) are considered, with the absolute scale set from the ϒ(2S)–ϒ(1S) splitting. Sea pion masses lie in a narrow interval ranging from slightly above 400 MeV to slightly below 300 MeV, keeping mπ L  4; however, lighter pion masses are considered in the valence DWF action for the u, d quarks. The lowest valence-valence pion mass is 227(3) MeV, which leads to a  rating of finite-volume effects. Results for the form factors are obtained from suitable three-point functions, and fitted to a modified z-expansion ansatz that combines the q2-dependence with the chiral and continuum extrapolations. The main results of the paper are the predictions (errors are statistical and systematic, respectively) ∫ 1 q 2 max 2 d(b → pμ−ν̄μ)ζ 2 −1pμν̄(15GeV ) ≡ | |2 2 dq = 12.31(76)(77) ps , (274)Vub 2 dq∫15 GeVq21 max d( →  μ−ν̄ ) ζcμν̄(7GeV 2 ≡ b c μ) dq2| |2 2 = 8.37(16)(34) ps −1, (275) Vcb 7 GeV2 dq ζ 2pμν̄(15GeV ) 2 = 1.471(95)(109), (276)ζcμν̄(7GeV ) which are the input for the LHCb analysis. Predictions for the total rates in all possible lepton channels, as well as for ratios similar to R(D) (cf. Sect. 8.4) between the τ and light-lepton channels are also available, in particular, = (b → c τ −ν̄τ ) R(c) → − = 0.3328(74)(70). (277)(b c μ ν̄μ) 123 Eur. Phys. J. C (2022) 82:869 Page 137 of 296 869 Datta 2017 [637] additionally includes results for the b → c tensor form factors h+, h⊥, h̃+, h̃⊥, based on the same lattice computation as Detmold 15 [516]. The main focus of Datta 2017 is the phenomenology of the b → cτντ decay and how it can be used to constrain contributions from beyond the Standard Model physics. Unlike in the case of the vector and axial-vector currents, the residual matching factors of the tensor currents are set to their tree-level value. While the matching systematic uncertainty is augmented to take this fact into account, the procedure implies that the tensor current retains an uncanceled logarithmic divergence at O(αs). Recently, first lattice calculations have also been completed for b semileptonic decays to negative-parity baryons in the final state. Such calculations are substantially more challenging and have not yet reached the same level of precision. Meinel 21 [638] considers the decaysb → ∗c(2595)ν̄ andb → ∗c(2625)ν̄, where the∗c(2595) and∗c(2625) are the lightest− − charm baryons with isospin 0 and J P = 12 and J P = 32 , respectively. These decay modes may eventually provide new opportunities to test lepton-flavor universality at the LHC, but are also very interesting from a theoretical point of view. The lattice results for the form factors may help tighten dispersive constraints in global analyses of b → c semileptonic decays [639], and may provide new insights into the internal structure of the negative-parity heavy baryons and their description in heavy-quark-effective-theory. The ∗c(2595) and ∗c(2625) are very narrow resonances decaying through the strong interaction into cππ . The strong decays are neglected in Meinel 21 [638]. The calculation was performed using the same lattice actions as previously for b → c, albeit with newly tuned RHQ parameters. Only three ensembles are used, with a ≈ 0.11, 0.08 fm and pion masses in the range from approximately 300–430 MeV, with valence-quark masses equal to the sea-quark masses. Chiral-continuum extrapolations linear in m2π and a 2 are performed, with systematic uncertainties estimated using higher-order fits. Finite-volume effects and effects associated with the strong decays of the ∗c ’s are not quantified. The calculation is done in the ∗c rest frame, where the cubic symmetry is sufficient to avoid mixing with unwanted lower-mass states. As a consequence, the calculation is limited to a small kinematic region near the zero-recoil point w = 1. On each ensemble, lattice data were produced for two values of w − 1 of approximately 0.01 and 0.03. The final results for the form factors are parameterized as linear functions of w− 1 and can be found in Meinel 21 [638] and associated supplemental files. 8.7 Semileptonic form factors for b → (∗) The decays b → +− are mediated by the same underlying b → s+− FCNC transition as, for example, B → K+− and B → K ∗+−, and can therefore provide additional information on the hints for physics beyond the Standard Model seen in the meson decays. The  baryon in the final state decays through the weak interaction into pπ− (or nπ0), leading to a wealth of angular observables even for unpolarized b. When including the effects of a nonzero b polarization,  → (→ pπ−b )+− decays are characterized by five angles leading to 34 angular observables [640], which have been measured by LHCb in the bin q2 ∈ [15, 20]GeV2 [641]. Given that the  is stable under the strong interactions, the b →  form factors parametrizing the matrix elements of local s̄b currents can be calculated on the lattice with high precision using standard methods. Of course, the process  → +−b also receives contributions from nonlocal matrix elements of four-quark and quark-gluon operators in the weak effective Hamiltonian combined with the electromagnetic current. As with the mesonic b → s+− decays, these contributions cannot easily be calculated on the lattice and one relies on other theoretical tools for them, including the local OPE at high q2 and a light-cone OPE/QCD factorization at low q2. Following an early calculation with static b quarks [642], Detmold 16 [643] provides results for all ten relativistic b →  form factors parametrizing the matrix elements of the local vector, axial-vector and tensor b → s currents. The lattice setup is identical to that used in the 2015 calculation of the b → p form factors in Detmold 15 [516], and similar considerations as in the previous section thus apply. The lattice data cover the upper 60% of the q2 range, and the form factors are extrapolated to the full q2 range using BCL z-expansion fits. This extrapolation is done simultaneously with the chiral and continuum extrapolations. The caveat regarding the renormalization of the tensor currents also applies here. Reference [644] uses the lattice results for the b →  form factors together with the experimental results for b → (→ pπ−)μ+μ− from LHCb [641,645] to perform fits of the b → sμ+μ− Wilson coefficients and of the b polarization parameter. Given the uncertainties (which are still dominated by experiment), the results for the Wilson coefficients are presently consistent both with the Standard-Model values and with the deviations seen in global fits that include all mesonic decays [646,647]. As with the b → c semileptonic form factors, a first lattice calculation, Meinel 2020 [648], was also recently completed − for a b → s transition to a negative-parity baryon in the final state, in this case the ∗(1520) with J P = 32 (no calculation− has yet been published for the strange J P = 12 final states, which would be the broader and even more challenging ∗(1405)/∗(1380) [165]). The ∗(1520) decays primarily to pK−/nK̄ 0, π , and ππ with a total width of 15.6 ± 1.0 123 869 Page 138 of 296 Eur. Phys. J. C (2022) 82:869 Table 55 Summary of computations of bottom baryon semileptonic form factors (see also Refs. [642,649] for calculations with static b quarks). The rationale for the  rating of finite-volume effects in Meinel 20 and 21 (despite meeting the ◦ criterion based on the minimum pion mass) is that the unstable nature of the final-state baryons was neglected in the analysis Process Collaboration Refs. N f  ∗b → c (2625) −ν̄ Meinel 21 [638] 2+1 A ◦ ◦  ◦   → ∗b c (2595) −ν̄ Meinel 21 [638] 2+1 A ◦ ◦  ◦   ∗b →  (1520) +− Meinel 20 [648] 2+1 A ◦ ◦  ◦  b → +− Detmold 16 [643] 2+1 A ◦ ◦  ◦   → p −b ν̄ Detmold 15 [516] 2+1 A ◦ ◦  ◦  b →  −c  ν̄ Detmold 15, Datta 17 [516,637] 2+1 A ◦ ◦  ◦  Table 56 Experimental Collaboration Tagging method B(B− → τ−ν̄)× 104 measurements for B(B− → τ−ν̄). The first error Belle [650] Hadronic 0.72+0.27 ± 0.11 on each result is statistical, while −0.25 the second error is systematic Belle [528] Semileptonic 1.25 ± 0.28 ± 0.27 BaBar [527] Hadronic 1.83+0.53−0.49 ± 0.24 BaBar [651] Semileptonic 1.7 ± 0.8 ± 0.2 MeV [165] . The analysis of the lattice data again neglects the strong decays and does not quantify finite-volume effects, and is again limited to a small kinematic region near q2max (Table 55). 8.8 Determination of |Vub| We now use the lattice-determined Standard Model transition amplitudes for leptonic (Sect. 8.1) and semileptonic (Sect. 8.3) B-meson decays to obtain exclusive determinations of the CKM matrix element |Vub|. In this section, we describe the aspect of our work that involves experimental input for the relevant charged-current exclusive decay processes. The relevant formulae are Eqs. (202) and (236). Among leptonic channels the only input comes from B → τντ , since the rates for decays to e and μ have not yet been measured. In the semileptonic case, we only consider B → πν transitions (experimentally measured for  = e, μ). We first investigate the determination of |Vub| through the B → τντ transition. This is the only experimentally measured leptonic decay channel of the charged B meson. The experimental measurements of the branching fraction of this channel, B(B− → τ−ν̄), have not been updated since the publication of the FLAG Review in 2016 [3]. The status of the experimental results for this branching fraction, summarized in Table 56, is unchanged from FLAG Review 16 [3]. Our corresponding values of |Vub| are unchanged from FLAG Review 19 [4]. It is obvious that all the measurements listed in Table 56 have significance smaller than 5σ , and the large uncertainties are dominated by statistical errors. These measurements lead to the averages of experimental measurements for B(B− → τ ν̄) [527,528], B(B− → τ ν̄)× 104 = 0.91 ± 0.22 from Belle, (278) = 1.79 ± 0.48 from BaBar, (279) = 1.06 ± 0.33 average, (280) 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Heavy-quark treatment Eur. Phys. J. C (2022) 82:869 Page 139 of 296 869 where, following our standard procedure we perform a weighted average and rescale the uncertainty by the square root of the reduced chi-squared. Note that the Particle Data Group [164] did not inflate the uncertainty in the calculation of the averaged branching ratio. Combining the results in Eqs. (278–280) with the experimental measurements of the mass of the τ -lepton and the B-meson lifetime and mass we get |Vub| fB = 0.72 ± 0.09 MeV from Belle, (281) = 1.01 ± 0.14 MeV from BaBar, (282) = 0.77 ± 0.12 MeV average, (283) which can be used to extract |Vub|, viz., N = 2 Belle B → τν : |V | = 3.83(14)(48)× 10−3f τ ub , (284) N f = 2 + 1 Belle B → τντ : |Vub| = 3.75(8)(47)× 10−3, (285) N f = 2 + 1 + 1 Belle B → τντ : |Vub| = 3.79(3)(47)× 10−3; (286) N f = 2 Babar B → τντ : |Vub| = 5.37(20)(74)× 10−3, (287) N f = 2 + 1 Babar B → τντ : |Vub| = 5.26(12)(73)× 10−3, (288) N f = 2 + 1 + 1 Babar B → τν : |V | = 5.32(4)(74)× 10−3τ ub , (289) N f = 2 average B → τντ : |Vub| = 4.10(15)(64)× 10−3, (290) N f = 2 + 1 average B → τντ : |Vub| = 4.01(9)(63)× 10−3, (291) N f = 2 + 1 + 1 average B → τντ : |Vub| = 4.05(3)(64)× 10−3, (292) where the first error comes from the uncertainty in fB and the second comes from experiment. Let us now turn our attention to semileptonic decays. The experimental value of |V 2ub| f+(q ) can be extracted from the measured branching fractions for B0 → π±ν and/or B± → π0ν applying Eq. (236);50 |Vub| can then be determined by performing fits to the constrained BCL z-parameterization of the form factor f+(q2) given in Eq. (532). This can be done in two ways: one option is to perform separate fits to lattice and experimental results, and extract the value of |Vub| from the ratio of the respective a0 coefficients; a second option is to perform a simultaneous fit to lattice and experimental data, leaving their relative normalization |Vub| as a free parameter. We adopt the second strategy, because it combines the lattice and experimental input in a more efficient way, leading to a smaller uncertainty on |Vub|. The available state-of-the-art experimental input consists of five data sets: three untagged measurements by BaBar (6- bin [652] and 12-bin [653]) and Belle [654], all of which assume isospin symmetry and provide combined B0 → π− and B+ → π0 data; and the two tagged Belle measurements of B̄0 → π+ (13-bin) and B− → π0 (7-bin) [655]. Including all of them, along with the available information about cross-correlations, will allow us to obtain a meaningful final error estimate.51 The lattice input data set will be the same discussed in Sect. 8.3. We perform a constrained BCL fit of the vector and scalar form factors (this is necessary in order to take into account the f+(q2 = 0) = f (q20 = 0) constraint) together with the combined experimental data sets. We find that the error on |Vub| stabilizes for N+ = N 0 = 3. The result of the combined fit is presented in Table 57. The fit has a chi-square per degree of freedom χ2/dof = 78.7/56 = 1.41. Following the√PDG recommendation we rescale the whole covariance matrix by χ2/dof: the errors on the z-parameters are increased by χ2/dof = 1.19 and the correlation matrix is unaffected. In Fig. 35, we show both the lattice and experimental data for (1 − q2/m2 ∗) f+(q2B ) as a function of z(q2), together with our preferred fit; experimental data has been rescaled by the resulting value for |Vub|2. It is worth noting the good consistency between the form factor shapes from lattice and experimental data. This can be quantified, e.g., by computing the ratio of the two leading coefficients in the constrained BCL parameterization: the fit to lattice form factors yields a+1 /a + 0 = −1.67(35) (cf. the results presented in Sect. 8.3.1), while the above lattice+experiment fit yields a+1 /a + 0 = −1.19(13). 50 Since  = e, μ the contribution from the scalar form factor in Eq. (236) is negligible. 51 See, e.g., Sec. V.D of Ref. [556] for a detailed discussion. 123 869 Page 140 of 296 Eur. Phys. J. C (2022) 82:869 Table 57 |Vub|, coefficients for the N+ = N 0 = N T = 3 z-expansion of the B → π form factors f+ and f0, and their c√orrelation matrix. The chi-square per degree of freedom is χ2/dof = 78.7/56 = 1.41 and the errorson the fit parameters have been rescaled by χ2/dof = 1.19. The lattice calculations that enter this fit are taken from FNAL/MILC [556] and RBC/UKQCD [557]. The experimental inputs are taken from BaBar [652,653] and Belle [654,655] B → πν (N f = 2 + 1) Central values Correlation matrix Vub × 103 3.74 (17) 1 −0.851 −0.349 0.375 −0.211 −0.246 a+0 0.415 (14) −0.851 1 0.155 −0.454 0.260 0.144 a+1 −0.488 (53) −0.349 0.155 1 −0.802 −0.0962 0.220 a+2 −0.31 (18) 0.375 −0.454 −0.802 1 0.0131 −0.100 a00 0.500 (23) −0.211 0.260 −0.0962 0.0131 1 −0.453 a01 −1.424 (54) −0.246 0.144 0.220 −0.100 −0.453 1 Fig. 35 Lattice and experimental data for f+B→π (q2) and f B→π 20 (q ) versus z (left panel) and q2 (right panel). Experimental data has been rescaled by the value for |Vub| found from the joint fit. Green symbols denote lattice-QCD points included in the fit, while blue and indigo points show experimental data divided by the value of |Vub| obtained from the fit. The grey and orange bands display the preferred N+ = N 0 = 3 BCL fit (five z-parameters and |Vub|) We plot the values of |Vub| we have obtained in Fig. 37, where the GGOU [656] determination through inclusive decays, |Vub|incl = (4.32±0.12 −3exp ±0.13theo ±0.23B F )×10 [165,261] (the B F error has been added in Ref. [165] to account for the spread in results obtained using different theoretical models), is also shown for comparison.52 In this plot the tension between the BaBar and the Belle measurements of B(B− → τ−ν̄) is manifest. As discussed above, it is for this reason that we do not extract |Vub| through the average of results for this branching fraction from these two collaborations. In fact this means that a reliable determination of |Vub| using information from leptonic B-meson decays is still absent; the situation will only clearly improve with the more precise experimental data expected from Belle II [657,658]. The value for |Vub| obtained from semileptonic B decays for N f = 2 + 1, on the other hand, is significantly more precise than both the leptonic and the inclusive determinations, and exhibits a ∼ 1.7σ tension with the latter. 8.9 Determination of |Vcb| We will now use the lattice-QCD results for the B → D(∗)ν form factors in order to obtain determinations of the CKM matrix element |Vcb| in the Standard Model. The relevant formulae are given in Eq. (241). Let us summarize the lattice input that satisfies FLAG requirements for the control of systematic uncertainties, discussed in Sect. 8.4. In the (experimentally more precise) B → D∗ν channel, there is only one N f = 2 + 1 lattice computation of∗ the relevant form factor F B→D at zero recoil. Concerning the B → Dν channel, for N f = 2 there is one determination of the relevant form factor GB→D at zero recoil, while for N f = 2 + 1 there are two determinations of the B → D form 52 Note that a recent Belle measurement of partial B → X +u ν branching fractions which superseeds their previous result and which yields the somewhat lower value |Vub| = 4.10(9)(22)(15)× 10−3, has not been included in the HFLAV average yet. 123 Eur. Phys. J. C (2022) 82:869 Page 141 of 296 869 factor as a function of the recoil parameter in roughly the lowest third of the kinematically allowed region. In this latter case, it is possible to replicate the analysis carried out for |Vub| in Sect. 8.8, and perform a joint fit to lattice and experimental∗ data; in the former, the value of |Vcb| has to be extracted by matching to the experimental value for F B→D (1)ηEW|Vcb| and GB→D(1)ηEW|Vcb|. The latest experimental average by HFLAV [619] for the B → D∗ form factor at zero recoil makes use of the CLN [659] parameterization of the B → D∗ form factor and is [F B→D∗(1)ηEW|Vcb|]CLN,HFLAV = 35.61(43)× 10−3. (293) Recently the Belle collaboration presented an updated measurement of the B → D∗ν branching ratio [660] in which, as suggested in Refs. [661–663], the impact of the form factor parameterization has been studied by comparing the CLN [659] and BGL [569,664] ansätze. The fit results using the two parameterizations are now consistent. In light of the fact that the BGL parameterization has a much stronger theoretical standing than the CLN one and that it imposes less stringent constraints on the shape of the form factors, we do not consider the CLN determination any further and focus on the BGL fit: [F B→D∗(1)ηEW|Vcb|]BGL, Belle = 35.44(23)(60)× 10−3, (294) where the first error is statistical and the second is systematic.53 Given the fact that the two determinations in Eqs. (293) and (294) are quite compatible and that the BGL parameterization is on firmer theoretical ground, in the following we present the determination of |Vcb| obtained from Eq. (294). We refer to the discussion presented at the end of Sec. 8.8 of the previous edition of this review [4] for further comments on the CLN and BGL parameterizations. ∗ By using ηEW = 1.0066254 and the N f = 2 + 1 lattice value for F B→D (1) in Eq. (266)55, we thus extract the average N f = 2 + 1[B → D∗ν] −3BGL,Belle : |Vcb| = 38.86(54)(70)× 10 , (295) where the first uncertainty comes from the lattice computation and the second from the experimental input. For the zero-recoil B → D form factor, HFLAV [619] quotes HFLAV: GB→D(1)ηEW|Vcb| = 41.57(45)(89)× 10−3, (296) yielding the following average for N f = 2: N f = 2 B → Dν : |Vcb| = 40.0(3.7)(1.0)× 10−3, (297) where the first uncertainty comes from the lattice computation and the second from the experimental input. Finally, for N f = 2 + 1 we perform, as discussed above, a joint fit to the available lattice data, discussed in Sect. 8.4, and state-of-the-art experimental determinations. In this case, we will combine the aforementioned Belle measurement [666], which provides partial integrated decay rates in 10 bins in the recoil parameter w, with the 2010 BaBar data set in Ref. [667], which quotes the value of GB→D(w)ηEW|Vcb| for ten values of w.56 The fit is dominated by the more precise Belle data; given this, and the fact that only partial correlations among systematic uncertainties are to be expected, we will treat both data sets as uncorrelated.57 A constrained (N+ = N 0 = 3) BCL fit using the same ansatz as for lattice-only data in Sect. 8.4, yields our average, which we present in Table 58. The chi-square per degree of freedom is χ2/dof = 20.0/25 = 0.80. The fit is illustrated in 53 Note∗ that the BGL fit employed by Belle uses very few z parameters and that this could lead to an underestimation of the error on[F B→D (1)ηEW|Vcb|. See Ref. [665] for a through review of this point. 54 Note that this determination does not include the electromagnetic Coulomb correction roughly estimated in Ref. [605]. Currently the numerical impact of this correction is negligible. 55 In light of our policy not to average N f = 2 + 1 and N f = 2 + 1 + 1 calculations and of the controversy over the use of the CLN vs. BGL∗ parameterizations, we prefer to simply use only the more precise N = 2 + 1 determination of F B→Df (1) in Eq. (266) for the extraction of Vcb. 56 We thank Marcello Rotondo for providing the ten bins result of the BaBar analysis. 57 We have checked that results using just one experimental data set are compatible within 1σ . In the case of BaBar, we have taken into account the introduction of some EW corrections in the data. 123 869 Page 142 of 296 Eur. Phys. J. C (2022) 82:869 Table 58 |Vcb|, coefficients for the N+ = N 0 z-expansion of the B → D form factors f+ and f0, and their correlation matrix. The coefficient a02 is fixed by the f+(q2 = 0) = f0(q2 = 0) constrain. The chi-square per degree offreedom is χ2/dof = 20.0/25 = 0.80. The lattice calculations that enter this fit are taken from FNAL/MILC [606] and HPQCD [607]. The experimental inputs are taken from BaBar [667] and Belle [666] B → Dν (N f = 2 + 1) Central values Correlation matrix |V | × 103cb 40.0 (1.0) 1.00 −0.525 −0.339 0.0487 −0.521 −0.433 a+0 0.8946 (94) −0.525 1.00 0.303 −0.351 0.953 0.529 a+1 −8.03 (16) −0.339 0.303 1.00 0.203 0.375 0.876 a+2 50.1 (3.1) 0.0487 −0.351 0.203 1.00 −0.276 0.196 a00 0.7804 (75) −0.521 0.953 0.375 −0.276 1.0 0.502 a01 −3.38 (16) −0.433 0.529 0.876 0.196 0.502 1.0 Fig. 36 Lattice and experimental data for f+B→D(q2) and f B→D0 (q2) versus z (left panel) and q2 (right panel). Green symbols denote lattice-QCD points included in the fit, while blue and indigo points show experimental data divided by the value of |Vcb| obtained from the fit. The grey and orange bands display the preferred N+ = N 0 = 3 BCL fit (five z-parameters and |Vcb|) Fig. 36. In passing, we note that, if correlations between the FNAL/MILC and HPQCD calculations are neglected, the |Vcb| central value rises to 40.3 × 10−3 in nice agreement with the results presented in Ref. [620]. Before discussing the combination of the above |Vcb| results, we note that the LHCb Collaboration recently reported the first determination of |Vcb| at the Large Hadron Collider using B → D−s s μ+νμ and B → D∗−s s μ+νμ decays [614,615]. The differential decay rates, in combination with the N f = 2 + 1+ 1 HPQCD 19 [613] and HPQCD 19B [612] lattice results for f B →D+ s s and F Bs→D ∗ s (1), were analyzed using either the CLN or BGL form-factor parameterizations. The result for |Vcb| from the BGL fit is [615] |Vcb| × 103 = (41.7 ± 0.8 ± 0.9 ± 1.1) B → D(∗)− +s s μ νμ, BGL,LHCb. (298) The LHCb analysis used ratios to the reference decay modes B0 → D−μ+ν and B0 → D∗−μ+μ νμ, whose branching fractions are used as input in the form of the Particle Data Group averages of measurements by other experiments [431]. The result (298) is therefore correlated with the determinations of |V | from B → D and B → D∗cb semileptonic decays. Given the challenges involved in performing our own fit to the LHCb data, we do not, at present, include the LHCb results for B → D−μ+s s νμ and B ∗− +s → Ds μ νμ in our combination of |Vcb|. We now proceed to combine the determinations of |Vcb| from exclusive B → D and B → D∗ semileptonic decays. To this end, we need to estimate the correlation between the lattice uncertainties in the two modes. We assume conservatively that the statistical component of the lattice error in both determinations are 100% correlated because they are based on the same MILC configurations (albeit on different subsets). We obtain: |Vcb| × 103 = 39.36(68) BGL,Belle. (299) 123 Eur. Phys. J. C (2022) 82:869 Page 143 of 296 869 Table 59 Results for |Vcb|. When two errors are quoted in our averages, the first one comes from the lattice form factor, and the second from the experimental measurement. The LHCb result using B → D(∗)s s ν decays [612–615], as well as the inclusive average obtained in the kinetic scheme from Ref. [668] are shown for comparison From |Vcb| × 103 Our average for N f = 2 + 1 (BGL) B → D∗ν 38.86(54)(70) Our average for N f = 2 + 1 B → Dν 40.0(1.0) Our average for N f = 2 + 1 (BGL) B → (D, D∗)ν 39.36(68) Our average for N f = 2 B → Dν 40.0(3.7)(1.0) LHCb result for N f = 2 + 1 + 1 (BGL) B → D(∗)s s ν 41.7(0.8)(0.9)(1.1) Gambino et al. B → Xcν 42.00(64) Fig. 37 Left: Summary of |Vub| determined using: i) the B-meson leptonic decay branching fraction, B(B− → τ−ν̄), measured at the Belle and BaBar experiments, and our averages for fB from lattice QCD; and ii) the various measurements of the B → πν decay rates by Belle and BaBar, and our averages for lattice determinations of the relevant vector form factor f+(q2). Right: Same for determinations of |Vcb| using semileptonic decays. The inclusive results are taken from Refs. [619,668] Our results are summarized in Table 59, which also shows the HFLAV inclusive determination of |Vcb| = 42.00(64) × 10−3 [668] for comparison, and illustrated in Fig. 37. Finally, using the fit results in Table 59, we extract a value for R(D) which includes both lattice and experimental information: R(D)lat+exp = 0.2951(31), our average. (300) Note that we do not need to rescale the uncertainty on R(D)lat+exp because, after the inclusion of experimental B → Dν ( = e, μ) results, the shift in central value caused by using a different parameterization is negligible (see the discussion above Eq. (258)). 8.10 Determination of |Vub/Vcb| from b decays In 2015, the LHCb Collaboration reported a measurement of the ratio [636] ∫ q2max dB(b → pμ−ν̄μ) 2 dq 2 R ( ) = ∫15 GeV 2 dq BF b , (301) q2max dB( →  −b cμ ν̄μ) dq22 7 GeV2 dq 123 869 Page 144 of 296 Eur. Phys. J. C (2022) 82:869 which, combined with the lattice QCD prediction [516] discussed in Sect. 8.6 yields a determination of |Vub/Vcb|. The LHCb analysis uses the decay c → pKπ to reconstruct the c and requires the branching fraction B(c → pKπ) of this decay as an external input. Using the latest world average of B(c → pKπ) = (6.28 ± 0.32)% [165] to update the LHCb measurement gives [261] RBF(b) = (0.92 ± 0.04 ± 0.07)× 10−2, (302) ζ (15GeV2) and, combined with the lattice QCD prediction for pμν̄ 2 discussed in Sect. 8.6,ζcμν̄ (7GeV ) |Vub/Vcb| = 0.079 ± 0.004 lat. ± 0.004 exp.. (303) 8.11 Determination of |Vub/Vcb| from Bs decays More recently, LHCb reported the measurements [669] ∫ 7 GeV2 dB(B → K−s μ+νμ) 2 dq 2 q2 2 dq R B min =mμ ( −3BF s, low) = B → − + = (1.66 ± 0.12)× 10 , (304)(Bs Ds μ νμ) ∫ q2max=(m 2Bs −mK ) dB(B → K−μ+s νμ) dq2 2 dq2 RBF(Bs, high) = 7 GeV B → − + = (3.25 ± 0.28)× 10 −3, (305) (Bs Ds μ νμ) B(Bs → K−μ+νμ) RBF(Bs, all) = B → − + = (4.89 ± 0.33)× 10 −3. (306) (Bs Ds μ νμ) Using our average of the Bs → K form factors from lattice QCD as discussed in Sect. 8.3.2, we obtain the Standard-Model predictions ∫ 1 7 GeV 2 d(B → K−s μ+νμ) | |2 = (2.26 ± 0.38) ps −1, (307) Vub q2min= 2 m2 dq ∫ μ2 1 qmax=(m Bs −m )2K d(Bs → K−μ+νμ) | |2 2 = (4.02 ± 0.31) ps −1, (308) Vub 7 GeV2 dq 1 − + | |2 (Bs → K μ νμ) = (6.28 ± 0.67) ps −1. (309) Vub For the denominator, we use the Bs → Ds form factors from Ref. [613], which yields 1 | |2 (B → D − s s μ +νμ) = (9.15 ± 0.37) ps−1. (310) Vcb Combined with the LHCb measurements we obtain |Vub| | | (low) = 0.0819 ± 0.0072 lat. ± 0.0029 exp., (311)Vcb |Vub| | | (high) = 0.0860 ± 0.0037 lat. ± 0.0038 exp., (312)Vcb |Vub| | | (all) = 0.0844 ± 0.0048 lat. ± 0.0028 exp.. (313)Vcb We note the excellent compatibility of the results in the high and low q2 regions. Nevertheless, we will use the result from the high-q2 region in our combination in Sect. 8.12, as this is the region in which the form factor shape is most reliably constrained by the lattice data. 123 Eur. Phys. J. C (2022) 82:869 Page 145 of 296 869 Fig. 38 Summary of |Vub| and |Vcb| determin∣ations. Th∣e black solid and dashed lines correspond to 68% and 95% C.L. contours, respectively. The result of the global fit (which does not include ∣V /V ∣ub cb from baryon modes or |Vcb| from Bs → D(∗)s ν) is (|Vcb|, |Vub|) = (39.48±0.68, 3.63± 0.14)×10−3 with a p-value of 0.39. The lattice and experimental results that contribute to the various contours are the following. B → πν: lattice (FNAL/MILC [556] and RBC/UKQCD [557]) and experiment (BaBar [652,653] and Belle [654,655]). B → Dν: lattice (FNAL/MILC [606] and HPQCD [607]) and experiment (BaBar [667] and Belle [666]). B → D∗ν: lattice (FNAL/MILC [605]) and experiment (Belle [660]). B → τν: lattice ( fB determinations in Fig 27) and experiment (BaBar [528] and Belle [527]). Bs → Kν/Bs → Dsν: lattice (HPQCD [570], RBC/UKQCD [556], FNAL/MILC [573], HPQCD [613]) and experiment (LHCb [669]). b → pν/b → cν: lattice (Detmold 15 [516]) and experiment (LHCb [636]). Bs → D∗s ν/Bs → Dsν: lattice (HPQCD 19 [613] and HPQCD 19B [612]) and experiment (LHCb [614,615]). The inclusive determinations are taken from Refs. [165,261,668] and read (|Vcb|, |Vub|)incl = (42.00 ± 0.64, 4.32 ± 0.29)× 10−3 8.12 Summary: |Vub| and |Vcb| In Fig. 38, we present a summary of determinations of |Vub| and |Vcb| from B → (π, D(∗))ν, Bs → (K , Ds)ν (high q2 o∣nly), B → τν and  → (p, )ν, as well as the results from inclusive B → X ν decays. Note that constraints on ∣ ∣∣ b c u,c Vub/Vcb from baryon modes are displayed but, in view of the rating in Table 55, are not included in the global fit. As discussed in Sect. 8.9, experimental inputs used in the extraction of |V | from B → D(∗)cb s s ν decays [614,615] given in Eq. (298) are highly correlated with those entering the global (|Vub|, |Vcb|)fit described in this section. Given these correlations and the challenges in reproducing the LHCb analysis, for the time being we do not include the result Eq. (298) into the global fit. Currently, the determinations of Vcb from B → D∗ and B → D decays are quite compatible; however, a sizeable tension involving the extraction of Vcb from inclusive decays remains. In the determination of the 1σ and 2σ contours for our average, we have included an estimate of the correlation between |Vub| and |Vcb| from semileptonic B decays: the lattice inputs to these quantities are dominated by results from the Fermilab/MILC and HPQCD collaborations that are both based on MILC N f = 2 + 1 ensembles, leading to our conservatively introducing a 100% correlation between the lattice statistical uncertainties of the three computations involved. The results of the fit are |V | = 39.48(68)× 10−3cb , (314) |Vub| = 3.63(14)× 10−3, (315) p−value = 0.39. (316) For reference, the inclusive determinations read |Vcb|incl = (42.00 ± 0.64)× 10−3 [668] and |Vub|incl = (4.32 ± 0.12exp ± 0.13theo ± 0.23B F ) × 10−3 [165,261] (the B F error has been added in Ref. [165] to account for the spread in results obtained using different theoretical models). Note that a recent Belle analysis [670] of partial B → X +u ν branching fractions finds a slightly lower central value |Vub|incl,Belle = (4.10 ± 0.09 −3stat ± 0.22syst ± 0.15theo)× 10 . 123 869 Page 146 of 296 Eur. Phys. J. C (2022) 82:869 9 The strong coupling αs Authors58 : R. Horsley, P. Petreczky, S. Sint 9.1 Introduction The strong coupling ḡs(μ) defined at scale μ, plays a key role in the understanding of QCD and in its application to collider physics. For example, the parametric uncertainty from αs is one of the dominant sources of uncertainty in the Standard- Model prediction for the H → bb̄ partial width, and the largest source of uncertainty for H → gg. Thus higher precision determinations of αs are needed to maximize the potential of experimental measurements at the LHC, and for high-precision Higgs studies at future colliders and the study of the stability of the vacuum [671–678]. The value of αs also yields one of the essential boundary conditions for completions of the Standard Model at high energies. In order to determine the running coupling at scale μ ḡ2= s (μ)αs(μ) , (317) 4π we should first “measure” a short-distance quantity Q at scale μ either experimentally or by lattice calculations, and then match it to a perturbative expansion in terms of a running coupling, conventionally taken as αMS(μ), Q(μ) = c1αMS(μ)+ c2αMS(μ)2 + · · · . (318) The essential difference between continuum determinations of αs and lattice determinations is the origin of the values of Q in Eq. (318). The basis of continuum determinations are experimentally measurable cross sections or decay widths from which Q is defined. These cross sections have to be sufficiently inclusive and at sufficiently high scales such that perturbation theory can be applied. Often hadronization corrections have to be used to connect the observed hadronic cross sections to the perturbative ones. Experimental data at highμ, where perturbation theory is progressively more precise, usually have increasing experimental errors, and it is not easy to find processes that allow one to follow the μ-dependence of a single Q(μ) over a range where αs(μ) changes significantly and precision is maintained. In contrast, in lattice gauge theory, one can design Q(μ) as Euclidean short-distance quantities that are not directly related to experimental observables. This allows us to follow the μ-dependence until the perturbative regime is reached and nonperturbative “corrections” are negligible. The only experimental input for lattice computations of αs is the hadron spectrum which fixes the overall energy scale of the theory and the quark masses. Therefore experimental errors are completely negligible and issues such as hadronization do not occur. We can construct many short-distance quantities that are easy to calculate nonperturbatively in lattice simulations with small statistical uncertainties. We can also simulate at parameter values that do not exist in nature (for example, with unphysical quark masses between bottom and charm) to help control systematic uncertainties. These features mean that precise results for αs can be achieved with lattice-gauge-theory computations. Further, as in the continuum, the different methods available to determine αs in lattice calculations with different associated systematic uncertainties enable valuable cross-checks. Practical limitations are discussed in the next section, but a simple one is worth mentioning here. Experimental results (and therefore the continuum determinations) of course have all quarks present, while in lattice gauge theories in practice only the lighter ones are included and one is then forced to use the matching at thresholds, as discussed in the following subsection. It is important to keep in mind that the dominant source of uncertainty in most present day lattice-QCD calculations of αs are from the truncation of continuum/lattice perturbation theory and from discretization errors. Perturbative truncation errors are of particular concern because they often cannot easily be estimated from studying the data itself. Further, the size of higher-order coefficients in the perturbative series can sometimes turn out to be larger than naive expectations based on power counting from the behaviour of lower-order terms. We note that perturbative truncation errors are also the dominant source of uncertainty in several of the phenomenological determinations of αs . The various phenomenological approaches to determining the running coupling constant, (5)α (MZ ) are summarized byMS the Particle Data Group [165]. The PDG review lists five categories of phenomenological results used to obtain the running 58 There is a strong overlap with the FLAG 19 report’s section on αs , authored by Horsley, Onogi and Sommer [4]. In particular the introduction, and the description of methods without new data have been taken over almost unchanged. 123 Eur. Phys. J. C (2022) 82:869 Page 147 of 296 869 coupling: using hadronic τ decays, hadronic final states of e+e− annihilation, deep inelastic lepton–nucleon scattering, electroweak precision data, and high energy hadron collider data. Excluding lattice results, the PDG quotes the weighted average as (5) α (MZ ) = 0.1176(11), PDG 20 [165] (319)MS compared to (5)α (MZ ) = 0.1174(16) of the older PDG 2018 [431]. For a general overview of the various phenomenologicalMS and lattice approaches see, e.g., Ref. [679]. The extraction of αs from τ data, which is one of the most precise and thus has a large impact on the nonlattice average in Eq. (319), is especially sensitive to the treatment of higher-order perturbative terms as well as the treatment of nonperturbative effects. This is important to keep in mind when comparing our chosen range for (5) α (MZ ) from lattice determinations in Eq. (396) with the nonlattice average from the PDG.MS 9.1.1 Scheme and scale dependence of αs and QCD Despite the fact that the notion of the QCD coupling is initially a perturbative concept, the associated  parameter is nonperturbatively defined  ≡ μϕs(ḡs(μ)), [ ∫ ( )] − − ḡs= 2 b /(2b2) 1/(2b ḡ2) 1 1 bϕ 1 0 1s(ḡs) (b0 ḡ ) 0 e ss exp − dx + − , (320) 0 β(x) b 3 20x b0x where ∂ ḡ (μ)β(ḡs) = μ s is the full renormalization group function in the scheme which defines ḡs , and b0 and b1 are the first∂μ two scheme-independent coefficients of the perturbative expansion β(x) ∼ −b 3 50x − b1x + · · · , (321) with ( ) ( ) 1 2 1 38 b0 = 2 11 − N f , b1 = 4 102 − N f . (322)(4π) 3 (4π) 3 Thus the  parameter is renormalization-scheme-dependent but in an exactly computable way, and lattice gauge theory is an ideal method to relate it to the low-energy properties of QCD. In the MS scheme presently bnl up to nl = 4 are known [199,680–683]. The change in the coupling from one scheme S to another (taken here to be the MS scheme) is perturbative, g2 (μ) = g2S(μ)(1 + c(1)g2g S(μ)+ · · · ), (323)MS where c(i)g , i ≥ 1 are finite renormalization coefficients. The scale μ must be taken high enough for the error in keeping only the first few terms in the expansion to be small. On the other hand, the conversion to the  parameter in the MS scheme is given exactly by [ ] MS = S exp c(1)g /(2b0) . (324) The fact that MS can be obtained exactly from S in any scheme S where c (1) g is known together with the high-order knowledge (5-loop by now) of βMS means that the errors in αMS(mZ) are dominantly due to the errors ofS . We(will)therefore mostly discuss them in that way. Starting fr(om E)q. (320), we have to consider (i) the error of ḡ 2(μ) (denoted as S ) and αS (ii) the truncation error in βS (denoted as  (nl )  trunc). Concerning (ii), note that knowledge of cg for the scheme S means that βS is known to nl + 1 loop order; bnl is known. We thus see that in the region where perturbation theory can be applied, the following errors of S (or consequently MS) have to be considered 123 869 Page 148 of 296 Eur. Phys. J. C (2022) 82:869 ( )  = αS(μ)2 × [1 +O(αS(μ))] , (325) (  )αS 8πb0αS(μ)  = k n nα l(μ)+O(α l+1S S (μ)), (326) trunc where k depends on bnl+1 and in typical good schemes such as MS it is numerically of order one. Statistical and systematic errors such as discretization effects contribute to αS(μ). In the above we dropped a scheme subscript for the -parameters because of Eq. (324). By convention αMS is usually quoted at a scale μ = MZ where the appropriate effective coupling is the one in the 5-flavour theory: (5)α (M MS Z ). In order to obtain it from a result with fewer flavours, one connects effective theories with different number of flavours as discussed by Bernreuther and Wetzel [684]. For example, one considers the MS scheme, matches the 3-flavour theory to the 4-flavour theory at a scale given by the charm-quark mass [685–687], runs with the 5-loop β-function [199,680– 683] of the 4-flavour theory to a scale given by the b-quark mass, and there matches to the 5-flavour theory, after which one runs up to μ = MZ with the 5-loop β function. For the matching relation at a given quark threshold we use the mass m which satisfies m = mMS(m), where m is the running mass (analogous to the running coupling). Then ∑ ḡ2 − (m ) = ḡ2N 1  N (m)× [1 + 0 × ḡ2N (m)+ t ḡ2nn N (m)] (327)f f f f n≥2 with [685,687,688] 1 11 t2 = 2 2 , (328)(4π ) 7[2 ] = 1 −82043 + 564731 − 2633t3 2 3 ζ3 (N f − 1) , (329)(4π ) 27648 124416 31104 t4 = 1 [ ] 2 4 5.170347 − 1.009932(N 2f − 1)− 0.021978 (N f − 1) , (330)(4π ) (where ζ3 is the Riemann zeta-function) provides the matching at the thresholds in the MS scheme. Often the packageRunDec is used for quark-threshold matching and running in the MS-scheme [689,690]. While t2, t3, t4 are numerically small coefficients, the charm-threshold scale is also relatively low and so there are nonper- turbative uncertainties in the matching procedure, which are difficult to estimate but which we assume here to be negligible. Obviously there is no perturbative matching formula across the strange “threshold”; here matching is entirely nonperturbative. Model dependent extrapolations of ḡ2N from N f = 0, 2 to N f = 3 were done in the early days of lattice gauge theory.f We will include these in our listings of results but not in our estimates, since such extrapolations are based on untestable assumptions. 9.1.2 Overview of the review of αs We begin by explaining lattice-specific difficulties in Sect. 9.2.1 and the FLAG criteria designed to assess whether the associated systematic uncertainties can be controlled and estimated in a reasonable manner. These criteria are taken over unchanged from the FLAG 19 report, as there has not yet been sufficiently broad progress to make these criteria more stringent. We would also like to point to a recent review [691] of lattice methodology and systematic uncertainties for αs . There, a systematic scale variation is advocated to assess systematic errors due to the truncation of the perturbative series and such a procedure may indeed be incorporated into future FLAG criteria, as it can be applied without change to most lattice approaches. We then discuss, in Sects. 9.3–9.9, the various lattice approaches and results from calculations with N f = 0, 2, 2+1, and 2+1+1 flavours. Besides new results and upgrades of previous works, a new strategy of nonperturbative renormalization by decoupling has been proposed by the ALPHA collaboration [692], which shifts the perspective on results with unphysical flavour numbers, in particular for N f = 0. As these can be nonperturbatively related to N f > 0 results by a nonperturbative matching calculation, 123 Eur. Phys. J. C (2022) 82:869 Page 149 of 296 869 it becomes very important to obtain precise and controlled N f = 0 results, with obvious implications for this and future FLAG reports. A short account of the decoupling strategy is given in Sect. 9.4. In Sect. 9.11, we present averages together with our best estimates for (5)α . These are currently determined from 3- MS and 4-flavour QCD simulations only, however, in the near future the decoupling strategy is expected to link e.g. 3-flavour simulations with the pure gauge theory simulations. Therefore, for the  parameter, we also give results for other numbers of flavours, including N f = 0 and N f = 2. 9.1.3 Additions with respect to the FLAG 19 report The additional papers since the FLAG 19 report are: Dalla Brida 19 [693] and Nada 20 [694] from step-scaling methods (Sect. 9.3). ALPHA 19A [692] from the decoupling method (Sect. 9.4). TUMQCD 19 [75] and Ayala 20 [74] and Husung 20 [695] from the static quark potential (Sect. 9.5). Cali 20 [76] from (light-quark) vacuum polarization in position space (Sect. 9.6). Petreczky 20 [696], Petreczky 19 [26], and Boito 20 [697,698] from heavy-quark current two-point functions (Sect. 9.8). Zafeiropoulos 19 [699] from QCD vertices (Sect. 9.9). 9.2 General issues 9.2.1 Discussion of criteria for computations entering the averages As in the PDG review, we only use calculations of αs published in peer-reviewed journals, and that use NNLO or higher-order perturbative expansions, to obtain our final range in Sect. 9.11. We also, however, introduce further criteria designed to assess the ability to control important systematics, which we describe here. Some of these criteria, e.g., that for the continuum extrapolation, are associated with lattice-specific systematics and have no continuum analogue. Other criteria, e.g., that for the renormalization scale, could in principle be applied to nonlattice determinations. Expecting that lattice calculations will continue to improve significantly in the near future, our goal in reviewing the state-of-the-art here is to be conservative and avoid prematurely choosing an overly small range. In lattice calculations, we generally take Q to be some combination of physical amplitudes or Euclidean correlation functions which are free from UV and IR divergences and have a well-defined continuum limit. Examples include the force between static quarks and two-point functions of quark-bilinear currents. In comparison to values of observables Q determined experimentally, those from lattice calculations require two more steps. The first step concerns setting the scale μ in GeV, where one needs to use some experimentally measur√able low-energy scale as input. Ideally one employs a hadron mass. Alternatively convenient intermediate scales such as t0, w0, r0, r1, [114,317,700,701] can be used if their relation to an experimental dimensionful observable is established. The low-energy scale needs to be computed at the same bare parameters where Q is determined, at least as long as one does not use the step-scaling method (see below). This induces a practical difficulty given present computing resources. In the determination of the low-energy reference scale the volume needs to be large enough to avoid finite-size effects. On the other hand, in order for the perturbative expansion of Eq. (318) to be reliable, one has to reach sufficiently high values of μ, i.e., short enough distances. To avoid uncontrollable discretization effects the lattice spacing a has to be accordingly small. This means L  hadron size ∼ −1QCD and 1/a  μ, (331) (where L is the box size) and therefore L/a ≫ μ/QCD. (332) The currently available computer power, however, limits L/a, typically to L/a = 32−96. Unless one accepts compromises in controlling discretization errors or finite-size effects, this means one needs to set the scale μ according to μ ≪ L/a ×QCD ∼ 10−30 GeV. (333) 123 869 Page 150 of 296 Eur. Phys. J. C (2022) 82:869 (Here ≪ or ≫ means at least one order of magnitude smaller or larger.) Therefore, μ can be 1−3 GeV at most. This raises the concern whether the asymptotic perturbative expansion truncated at 1-loop, 2-loop, or 3-loop in Eq. (318) is sufficiently accurate. There is a finite-size scaling method, usually called step-scaling method, which solves this problem by identifying μ = 1/L in the definition of Q(μ), see Sect. 9.3. For the second step after setting the scale μ in physical units (GeV), one should compute Q on the lattice, Qlat(a, μ) for several lattice spacings and take the continuum limit to obtain the left hand side of Eq. (318) as Q(μ) ≡ l→im Qlat(a, μ) with μ fixed. (334)a 0 This is necessary to remove the discretization error. Here it is assumed that the quantity Q has a continuum limit, which is regularization-independent. The method discussed in Sect. 9.7, which is based on the perturbative expansion of a lattice-regulated, divergent short-distance quantity Wlat(a) differs in this respect and must be treated separately. In summary, a controlled determination of αs needs to satisfy the following: 1. The determination of αs is based on a comparison of a short-distance quantity Q at scale μ with a well-defined continuum limit without UV and IR divergences to a perturbative expansion formula in Eq. (318). 2. The scale μ is large enough so that the perturbative expansion in Eq. (318) is precise to the order at which it is truncated, i.e., it has good asymptotic convergence. 3. If Q is defined by physical quantities in infinite volume, one needs to satisfy Eq. (332). Nonuniversal quantities need a separate discussion, see Sect. 9.7. Conditions 2. and 3. give approximate lower and upper bounds for μ respectively. It is important to see whether there is a window to satisfy 2. and 3. at the same time. If it exists, it remains to examine whether a particular lattice calculation is done inside the window or not. Obviously, an important issue for the reliability of a calculation is whether the scale μ that can be reached lies in a regime where perturbation theory can be applied with confidence. However, the value ofμ does not provide an unambiguous criterion. For instance, the Schrödinger Functional, or SF-coupling (Sect. 9.3) is conventionally taken at the scale μ = 1/L , but one could also choose μ = 2/L . Instead of μ we therefore define an effective αeff . For schemes such as SF (see Sect. 9.3) or qq (see Sect. 9.5) this is directly the coupling of the scheme. For other schemes such as the vacuum polarization we use the perturbative expansion Eq. (318) for the observable Q to define αeff = Q/c1. (335) If there is an αs-independent term it should first be subtracted. Note that this is nothing but defining an effective, regularization- independent coupling, a physical renormalization scheme. Let us now c∑omment further on the use of the perturbative series. Since it is only an asymptotic expansion, the remainder R (Q) = Q − ≤ c αi ∑ n i n i s of a truncated perturbative expression Q ∼ i≤n ciαis cannot just be estimated as a perturbative error k αn+1s . The error is nonperturbative. Often one speaks of “nonperturbative contributions”, but nonperturbative and perturbative cannot be strictly separated due to the asymptotic nature of the series (see, e.g., Ref. [702]). Still, we do have some general ideas concerning the size of nonperturbative effects. The known ones such as instantons or renormalons decay for large μ like inverse powers of μ and are thus roughly of the form exp(−γ /αs), (336) with some positive constant γ . Thus we have, loosely speaking, Q = c α 2 n1 s + c2αs + · · · + cnαs +O(αn+1s )+O(exp(−γ /αs)). (337) For small αs , the exp(−γ /αs) is negligible. Similarly the perturbative estimate for the magnitude of relative errors in Eq. (337) is small; as an illustration for n = 3 and αs = 0.2 the relative error is ∼ 0.8% (assuming coefficients |cn+1/c1| ∼ 1). For larger values of αs nonperturbative effects can become significant in Eq. (337). An instructive example comes from the values obtained from τ decays, for which αs ≈ 0.3. Here, different applications of perturbation theory (fixed order and 123 Eur. Phys. J. C (2022) 82:869 Page 151 of 296 869 contour improved) each look reasonably asymptotically convergent59 but the difference does not seem to decrease much with the order (see, e.g., the contribution of Pich in Ref. [704]). In addition nonperturbative terms in the spectral function may be nonnegligible even after the integration up to mτ (see, e.g., Refs. [705,706]). All of this is because αs is not really small. Since the size of the nonperturbative effects is very hard to estimate one should try to avoid such regions of the coupling. In a fully controlled computation one would like to verify the perturbative behaviour by changing αs over a significant range instead of estimating the errors as ∼ αn+1s . Some computations try to take nonperturbative power ‘corrections’ to the perturbative series into account by including such terms in a fit to the μ-dependence. We note that this is a delicate procedure, both because the separation of nonperturbative and perturbative is theoretically not well defined and because in practice a term like, e.g., αs(μ)3 is hard to distinguish from a 1/μ2 term when the μ-range is restricted and statistical and systematic errors are present. We consider it safer to restrict the fit range to the region where the power corrections are negligible compared to the estimated perturbative error. The above considerations lead us to the following special criteria for the determination of αs : • Renormalization scale  all points relevant in the analysis have α < 0.2 ◦ effall points have αeff < 0.4 and at least one αeff ≤ 0.25  otherwise • Perturbative behaviour  verified over a range of a factor 4 change in nα leff without power corrections or alternatively n α l 1 2eff ≤ 2αeff/(8πb0αeff) is reached ◦ agreement with perturbation theory over a range of a factor (3/2)2 in nα leff possibly fitting with power corrections or alternatively nα leff ≤ αeff/(8πb 20αeff) is reached  otherwise Here αeff is the accuracy cited for the determination of αeff and nl is the loop order to which the connection of αeff to the MS scheme is known. Recall the discussion around Eqs. (325,326); the β-function of αeff is then known to nl + 1 loop order.60 • Continuum extrapolation At a reference point of αeff = 0.3 (or less) we require  three lattice spacings with μa < 1/2 and full O(a) improvement, or three lattice spacings with μa ≤ 1/4 and 2-loop O(a) improvement, or μa ≤ 1/8 and 1-loop O(a) improvement ◦ three lattice spacings with μa < 3/2 reaching down to μa = 1 and full O(a) improvement, or three lattice spacings with μa ≤ 1/4 and 1-loop O(a) improvement  otherwise We also need to specify what is meant by μ. Here are our choices: step-scaling : μ = 1/L , heavy quark–antiquark potential : μ = 2/r, observables in position space : μ = 1/|x |, observables in momentum space : μ = q, moments of heavy-quark currents : μ = 2m̄c, eigenvalues of the Dirac operator : μ = λMS (338) 59 See, however, the recent discussion in [703]. 60 Once one is in the perturbative region with αeff , the error in extracting the  parameter due to the truncation of perturbation theory scales like n α leff , as discussed around Eq. (326). In order to detect/control such corrections properly, one needs to change the correction term significantly; we require a factor of four for a and a factor (3/2)2 for a◦. An exception to the above is the situation where the correction terms are small anyway, i.e., nα l 2eff ≈ (/)trunc < (/)α ≈ αeff/(8πb0αeff ) is reached. 123 869 Page 152 of 296 Eur. Phys. J. C (2022) 82:869 where |x | is the Euclidean norm of the 4-vector x , q is the magnitude of the momentum, m̄c is the heavy-quark mass (in the MS scheme) and usually taken around the charm-quark mass and λMS is the eigenvalue of the Dirac operator, see Sect. 9.10. We note again that the above criteria cannot be applied when regularization dependent quantities Wlat(a) are used instead of Q(μ). These cases are specifically discussed in Sect. 9.7. In principle one should also account for electro-weak radiative corrections. However, both in the determination of αs at intermediate scales μ and in the running to high scales, we expect electro-weak effects to be much smaller than the presently reached precision. Such effects are therefore not further discussed. The attentive reader will have noticed that bounds such as μa < 3/2 or at least one value of αeff ≤ 0.25 which we require for a ◦ are not very stringent. There is a considerable difference between ◦ and . We have chosen the above bounds, unchanged as compared to FLAG 16 and FLAG 19, since not too many computations would satisfy more stringent ones at present. Nevertheless, we believe that the ◦ criteria already give reasonable bases for estimates of systematic errors. An exception may be Cali 20, which is discussed in detail in Sect. 9.6. In the future, we expect that we will be able to tighten our criteria for inclusion in the average, and that many more computations will reach the present  rating in one or more categories. In addition to our explicit criteria, the following effects may influence the precision of results: Topology sampling: In principle a good way to improve the quality of determinations of αs is to push to very small lattice spacings thus enabling largeμ. It is known that the sampling of field space becomes very difficult for the HMC algorithm when the lattice spacing is small and one has the standard periodic boundary conditions. In practice, for all known discretizations the topological charge slows down dramatically for a ≈ 0.05 fm and smaller [116,125,128–132]. Open boundary conditions solve the problem [133] but are not frequently used. Since the effect of the freezing on short distance observables is not known, we also do need to pay attention to this issue. Remarks are added in the text when appropriate. Quark-mass effects: We assume that effects of the finite masses of the light quarks (including strange) are negligible in the effective coupling itself where large, perturbative, μ is considered. Scale setting: The scale does not need to be very precise, since using the lowest-order β-function shows that a 3% error in the scale determination corresponds to a ∼ 0.5% error in αs(MZ ). As long as systematic errors from chiral extrapolation and finite-volume effects are well below 3% we do not need to be concerned about those at the present level of precision in αs(MZ ). This may change in the future. 9.2.2 Physical scale Since FLAG 19, a new FLAG working group on scale setting has been established. We refer to Sect. 11 for definitions and the current status. Note that the error from scale setting is sub-dominant for current αs determinations. A popular scale choice has been the intermediate r0 scale, and its variant r1, which both derive from the force between static quarks, see Eq. (358). One should bear in mind that their determination from physical observables also has to be taken into account. The phenomenological value of r0 was originally determined as r0 ≈ 0.49 fm through potential models describing quarkonia [700]. Of course the quantity is precisely defined, independently of such model considerations. But a lattice computation with the correct sea-quark content is needed to determine a completely sharp value. When the quark content is not quite realistic, the value of r0 may depend to some extent on which experimental input is used to determine (actually define) it. The latest determinations from two-flavour QCD are r0 = 0.420(14)–0.450(14) fm by the ETM collaboration [45,87], using as input fπ and fK and carrying out various continuum extrapolations. On the other hand, the ALPHA collaboration [707] determined r0 = 0.503(10) fm with input from fK , and the QCDSF collaboration [107] cites 0.501(10)(11) fm from the mass of the nucleon (no continuum limit). Recent determinations from three-flavour QCD are consistent with r1 = 0.313(3) fm and r0 = 0.472(5) fm [41,120,708]. Due to the uncertainty in these estimates, and as many results are based directly on r0 to set the scale, we shall often give both the dimensionless number r0MS, as well as MS. In the cases where no physical r0 scale is given in the original papers or we convert to the r0 scale, we use the value r0 = 0.472 fm. In case r1MS is given in the publications, we use r0/r1 = 1.508 [708], to convert, which remains√well consistent with the update [116] neglecting the error on this ratio. In some, mostly early, computations the string tension, σ was used. We convert to r0 using r20σ = 1.65−π/12, which has been shown to be an excellent approximation in the relevant pure gauge theory [709,710]. The new scales t0, w0 based on the gradient flow are very attractive alternatives to r0 but their discretization errors are still under discussion [112,711–713] and their values at the physical point are not yet determined with great precision. We 123 Eur. Phys. J. C (2022) 82:869 Page 153 of 296 869 remain with r0 as our main reference scale for now. A general discussion of the various scales is given in [714] and in the scale-setting section of this FLAG report, cf. Sect. 11. 9.2.3 Studies of truncation errors of perturbation theory As discussed previously, we have to determine αs in a region where the perturbative expansion for the β-function, Eq. (321) in the integral Eq. (320), is reliable. In principle this must be checked, however, this is difficult to achieve as we need to reach up to a sufficiently high scale. A frequently used recipe to estimate the size of truncation errors of the perturbative series is to vary the renormalization-scale dependence around the chosen ‘optimal’ scale μ∗, of an observable evaluated at a fixed order in the coupling from μ = μ∗/2 to 2μ∗. For examples, see Ref. [691]. Alternatively, or in addition, the renormalization scheme chosen can be varied, which investigates the perturbative con- version of the chosen scheme to the perturbatively defined MS scheme and in particular ‘fastest apparent convergence’ when the ‘optimal’ scale is chosen so that the O(α2s ) coefficient vanishes. The ALPHA collaboration in Ref. [715] and ALPHA 17 [716], within the SF approach defined a set of ν-schemes for which the 3-loop (scheme-dependent) coefficient of the β-function for N f = 2 + 1 flavours was computed to be bν2 = −(0.064(27) + 1.259(1)ν)/(4π)3. The standard SF scheme has ν = 0. For comparison, bMS2 = 0.324/(4π)3. A range of scales from about 4 GeV to 128 GeV was investigated. It was found that while the procedure of varying the scale by a factor 2 up and down gave a correct estimate of the residual perturbative error for ν ≈ 0 . . . 0.3, for negative values, e.g., ν = −0.5, the estimated perturbative error is much too small to account for the mismatch in the -parameter of ≈ 8% at αs = 0.15. This mismatch, however, did, as expected, still scale with nα ls with nl = 2. In the schemes with negative ν, the coupling αs has to be quite small for scale-variations of a factor 2 to correctly signal the perturbative errors. For a systematic study of renormalization scale variations as a measure of perturbative truncation errors in various lattice determinations of αs we refer to the recent review by Del Debbio and Ramos [691]. 9.3 αs from step-scaling methods 9.3.1 General considerations The method of step-scaling functions avoids the scale problem, Eq. (331). It is in principle independent of the particular boundary conditions used and was first developed with periodic boundary conditions in a two-dimensional model [717]. The essential idea of the step-scaling strategy is to split the determination of the running coupling at large μ and of a hadronic scale into two lattice calculations and connect them by ‘step-scaling’. In the former part, we determine the running coupling constant in a finite-volume scheme in which the renormalization scale is set by the inverse lattice size μ = 1/L . In this calculation, one takes a high renormalization scale while keeping the lattice spacing sufficiently small as μ ≡ 1/L ∼ 10 . . . 100 GeV, a/L  1. (339) In the latter part, one chooses a certain ḡ2 2max = ḡ (1/Lmax), typically such that Lmax is around 0.5–1 fm. With a co√mmon discretization, one then determines Lmax/a and (in a large volume L ≥ 2–3 fm) a hadronic scale such as a hadron mass, t0/a or r0/a at the same bare parameters. In this way one gets numbers for, e.g., Lmax/r0 and by changing the lattice spacing a carries out a continuum limit extrapolation of that ratio. In order to connect ḡ2(1/L 2max) to ḡ (μ) at high μ, one determines the change of the coupling in the continuum limit when the scale changes from L to L/s, starting from L = L kmax and arriving at μ = s /Lmax. This part of the strategy is called step-scaling. Combining these results yields ḡ2(μ) at μ = sk (r0/L −1max) r0 , where r0 stands for the particular chosen hadronic scale. Most applications use a scale factor s = 2. At present most applications in QCD use Schrödinger functional boundary conditions [421,718] and we discuss this below in a little more detail. (However, other boundary conditions are also possible, such as twisted boundary conditions and the discussion also applies to them.) An important reason is that these boundary conditions avoid zero modes for the quark fields and quartic modes [719] in the perturbative expansion in the gauge fields. Furthermore the corresponding renormalization scheme is well studied in perturbation theory [720–722] with the 3-loop β-function and 2-loop cutoff effects (for the standard Wilson regularization) known. In order to have a perturbatively well-defined scheme, the SF scheme uses Dirichlet boundary conditions at time t = 0 and t = T . These break translation invariance and permit O(a) counter terms at the boundary through quantum corrections. 123 869 Page 154 of 296 Eur. Phys. J. C (2022) 82:869 Table 60 Results for the  parameter from computations using step-scaling of the SF-coupling. Entries without values for  computed the running and established perturbative behaviour at large μ Collaboration Refs. N f Scale MS [MeV] r0MS ALPHA 10A [735] 4 A    Only running of αs in Fig. 4 Perez 10 [736] 4 C   ◦ Only step-scaling function in Fig. 4 √ ALPHA 17 [77] 2+1 A    8t0 = 0.415 fm 341(12) 0.816(29) PACS-CS 09A [78] 2+1 A   ◦ mρ 371(13)(8)(+0−27)# 0.888(30)(18)(+0−65)† A   ◦ mρ 345(59)## 0.824(141)† ALPHA 12∗ [707] 2 A    fK 310(20) 0.789(52) ALPHA 04 [737] 2 A    r0 = 0.5 fm§ 245(16)(16)§ 0.62(2)(2)§ ALPHA 01A [738] 2 A    Only running of αs in Fig. 5 Nada 20 [694] 0 A    Consistency checks for [693], same gauge configurations Dalla Brida 19 [693] 0 A    r0 = 0.5fm 260.5(4.4) 0.660(11) Ishikawa 17 [732] 0 A    r0, [√σ ] 253(4)(+13−2 )† 0.606(9)(+31−5 )+ CP-PACS 04& [723] 0 A   ◦ Only tables of g2 ◦ SFALPHA 98†† [739] 0 A   r0 = 0.5fm 238(19) 0.602(48) Lüscher 93 [720] 0 A  ◦ ◦ r0 = 0.5fm 233(23) 0.590(60)§§ #Result with a constant (in a) continuum extrapolation of the combination Lmaxmρ †In conversion from MS to r0MS and vice versa, r0 is taken to be 0.472 fm ##Result with a linear continuum extrapolation in a of the combination L ∗ max mρ Supersedes ALPHA 04 §The N f = 2 results were based on values for r0/a which have later been found to be too small by [707]. The effect will be of the order of 10–15%, presumably an increase in r0. We have taken this into account by a  in the renormalization scale &This investigation was a precursor for PACS-CS 09A and confirmed two step-scaling functions as well as the scale setting of ALPHA 98 ††Uses data of Lüscher 93 and therefore supersedes it §§Converted f√rom αMS(37r −1 0 ) = 0.1108(25)+Also MS/ σ = 0.532(8)(+27−5 ) is quoted Therefore, the leading discretization error is O(a). Improving the lattice action is achieved by adding counter terms at the boundaries whose coefficients are denoted as ct , c̃t . In practice, these coefficients are computed with 1-loop or 2-loop perturbative accuracy. A better precision in this step yields a better control over discretization errors, which is important, as can be seen, e.g., in Refs. [709,723]. Also computations with Dirichlet boundary conditions do in principle suffer from the insufficient change of topology in the HMC algorithm at small lattice spacing. However, in a small volume the weight of nonzero charge sectors in the path integral is exponentially suppressed [724]61 and in a Monte Carlo run of typical length very few configurations with nontrivial topology should appear. Considering the issue quantitatively Ref. [725] finds a strong suppression below L ≈ 0.8 fm. Therefore the lack of topology change of the HMC is not a serious issue for the high energy regime in step-scaling studies. However, the matching to hadronic observables requires volumes where the problem cannot be ignored. Therefore, Ref. [726] includes a projection to zero topology into the definition of the coupling. We note also that a mix of Dirichlet and open boundary conditions is expected to remove the topology issue entirely [727] and may be considered in the future. Apart from the boundary conditions, the very definition of the coupling needs to be chosen. We briefly discuss in turn, the two schemes used at present, namely, the ‘Schrödinger Functional’ (SF) and ‘Gradient Flow’ (GF) schemes. 61 We simplify here and assume that the classical solution associated with the used boundary conditions has charge zero. In practice this is the case. 123 Publication status Renormalization scale Perturbative behaviour Continuum extrapolation Eur. Phys. J. C (2022) 82:869 Page 155 of 296 869 The SF scheme is the first one, which was used in step-scaling studies in gauge theories [421]. Inhomogeneous Dirichlet boundary conditions are imposed in time, Ak(x)|x0=0 = C ′k, Ak(x)|x0=L = Ck, (340) for k = 1, 2, 3. Periodic boundary conditions (up to a phase for the fermion fields) with period L are imposed in space. The matrices ( ) LCk = i diag(η − π/3,−η/2,−η/2 + π/3 , ) LC ′k = i diag − (η + π), η/2 + π/3, η/2 + 2π/3 , just depend on the dimensionless parameter η. The coupling ḡSF is obtained from the η-derivative of the effective action, 〈 | 〉 = 12π∂ηS η=0 . (341) ḡ2SF For this scheme, the finite c(i)g , Eq. (323), are known for i = 1, 2 [721,722]. More recently, gradient-flow couplings have been used frequently because of their small statistical errors at large couplings (in contrast to ḡSF, which has small statistical errors at small couplings). The gradient flow is introduced as follows [317,728]. Consider the flow gauge field Bμ(t, x) with the flow time t , which is a one parameter deformation of the bare gauge field Aμ(x), where Bμ(t, x) is the solution to the gradient-flow equation ∂t Bμ(t, x) = DνGνμ(t, x), Gμν = ∂μBν − ∂ν Bμ + [Bμ, Bν], (342) with initial condition Bμ(0, x) = Aμ(x). The renormalized coupling is defined by [317] ∣ ḡ2GF(μ) = N t2〈 ∣E(t, x)〉∣ = √ , (343)μ 1/ 8t with N = 16π2/3 +O((a/L)2) and where E(t, x) is the action density given by 1 E(t, x) = Gaμν(t, x)Gaμν(t, x). (344)4 In a finite volume, one needs to specify additional conditions. In order not to introduce two independent scales one sets √ 8t = cL , (345) for some fixed number c [729]. Schrödinger functional boundary conditions [730] or twisted boundary conditions [731,732] have been employed. Matching of the GF coupling to the MS-scheme coupling is known to 1-loop for twisted boundary conditions with zero quark flavours and SU (3) group [732] and to 2-loop with SF boundary conditions with zero quark flavours [733]. The former is based on a MC evaluation at small couplings62 and the latter on numerical stochastic perturbation theory. 9.3.2 Discussion of computations In Table 60 we give results from various determinations of the  parameter. For a clear assessment of the N f -dependence, the last column also shows results that refer to a common hadronic scale, r0. As discussed above, the renormalization scale can be chosen large enough such that αs < 0.2 and the perturbative behaviour can be verified. Consequently only is present for these criteria except for early work where the nl = 2 loop correction to MS was not yet known and we assigned a  62 For a variant of the twisted periodic finite volume scheme the 1-loop matching has been computed analytically [734]. 123 869 Page 156 of 296 Eur. Phys. J. C (2022) 82:869 concerning the renormalization scale. With dynamical fermions, results for the step-scaling functions are always available for at least a/L = μa = 1/4, 1/6, 1/8. All calculations have a nonperturbatively O(a) improved action in the bulk. For the discussed boundary O(a) terms this is not so. In most recent calculations 2-loop O(a) improvement is employed together with at least three lattice spacings.63 This means a for the continuum extrapolation. In other computations only 1-loop c ◦ twas available and we arrive at . We note that the discretization errors in the step-scaling functions of the SF coupling are usually found to be very small, at the percent level or below. However, the overall desired precision is very high as well, and the results in CP-PACS 04 [723] show that discretization errors at the below percent level cannot be taken for granted. In particular with staggered fermions (unimproved except for boundary terms) few percent effects are seen in Perez 10 [736]. In the work by PACS-CS 09A [78], the continuum extrapolation in the scale setting is performed using a constant function in a and with a linear function. Potentially the former leaves a considerable residual discretization error. We here use, as discussed with the collaboration, the continuum extrapolation linear in a, as given in the second line of PACS-CS 09A [78] results in Table 60. After perturbative conversion from a three-flavour result to five flavours (see Sect. 9.2.1), they obtain (5) α (MZ ) = 0.118(3). (346)MS In Ref. [77], the ALPHA collaboration determined (3) combining step-scaling in ḡ2GF in the lower scale region μMS had ≤ μ ≤ μ0, and step-scaling in ḡ2SF for higher scales μ0 ≤ μ ≤ μPT. Both schemes are defined with SF boundary conditions. For ḡ2GF a projection to the sector of zero topological charge is included, Eq. (344) is restricted to the magnetic components, and c = 0.3. The scales μhad, μ0, and μPT are defined by ḡ2 2GF(μhad) = 11.3, ḡSF(μ0) = 2.012, and μPT = 16μ0 which are roughly estimated as 1/Lmax ≡ μhad ≈ 0.2 GeV, μ0 ≈ 4 GeV, μPT ≈ 70 GeV. (347) Step-scaling is carried out with an O(a)-improved Wilson quark action [740] and Lüscher–Weisz gauge action [741] in the low-scale region and an O(a)-improved Wilson quark action [742] and Wilson gauge action in the high-energy part. For the step-scaling using steps of L/a → 2L/a, three lattice sizes L/a = 8, 12, 16 were simulated for ḡ2GF and four lattice sizes L/a = (4, ) 6, 8, 12 for ḡ2SF. The final results do not use the small lattices given in parenthesis. The parameter (3) is thenMS obtained via (3) (3 ) = MS × μPT × μhad × f MS πK , (348) ︸μ︷P︷T︸ ︸μ︷h︷ad︸ ︸f︷π︷K︸ ︸︷︷︸experimental data perturbation theory step-scaling large volume simulation where the hadronic scale fπK is f = 1πK 3 (2 fK + fπ ) = 147.6(5) MeV. The first factor on the right hand side of Eq. (348) is obtained from αSF(μPT) which is the output from SF step-scaling using Eq. (320) with αSF(μPT) ≈ 0.1 and the 3-loop β-function and the exact conversion to the MS-scheme. The second factor is essentially obtained from step-scaling in the GF scheme and the measurement of ḡ2SF(μ0) (except for the trivial scaling factor of 16 in the SF running). The third factor is obtained from a measurement of the hadronic quantity at large volume. A large-volume simulation is done for three lattice spacings with sufficiently large volume and reasonable control over the chiral extrapolation so that the scale determination is precise enough. The step-scaling results in both schemes satisfy renormalization criteria, perturbation theory criteria, and continuum limit criteria just as previous studies using step-scaling. So we assign green stars for these criteria. The dependence of , Eq. (320) with 3-loop β-function, on αs and on the chosen scheme is discussed in [715]. This investigation provides a warning on estimating the truncation error of perturbative series. Details are explained in Sect. 9.2.3. The result for the parameter is (3)  = 341(12) MeV, where the dominant error comes from the error of αSF(μPT) afterMS step-scaling in the SF scheme. Using 4-loop matching at the charm and bottom thresholds and 5-loop running one finally obtains (5) α (M ) = 0.11852(84). (349) MS Z 63 With 2-loop O(a) improvement we here mean ct including the g40 term and c̃t with the g20 term. For gluonic observables such as the running coupling this is sufficient for cutoff effects being suppressed to O(g6a). 123 Eur. Phys. J. C (2022) 82:869 Page 157 of 296 869 Several other results do not have a sufficient number of quark flavours or do not yet contain the conversion of the scale to physical units (ALPHA 10A [735], Perez 10 [736]). Thus no value for (5)α (MZ ) is quoted.MS The computation of Ishikawa et al. [732] is based on the gradient flow coupling with twisted boundary conditions [731] (TGF coupling) in the pure gauge theory. Again they use c = 0.3. Step-scaling with a scale factor s = 3/2 is employed, covering a large range of couplings from αs ≈ 0.5 to αs ≈ 0.1 and taking the continuum limit through global fits to the step-scaling function on L/a = 12, 16, 18 lattices with between 6 and 8 parameters. Systematic errors due to variations of the fit functions are estimated. Two physical scales are considered: r 20/a is taken from [709] and σa from [194] and [743]. As the ratio TGF/MS has not yet been computed analytically, Ref. [732] determines the 1-loop relation between ḡSF and ḡTGF from MC simulations performed in the weak coupling region and then uses the known SF/MS. Systematic errors due to variations of the fit functions dominate the overall uncertainty. Since FLAG 19 two new and quite extensive N f = 0 step-scaling studies have been carried out in Dalla Brida 19 [693] and by Nada and Ramos [694]. They use different strategies for the running from mid to high energies, but use the same gauge configurations and share the running at low energies and matching to the hadronic scales. These results are therefore correlated. However, given the comparatively high value for r0MS, it is re-assuring that the√se conceptually different approaches yield perfectly compatible results within errors of similar size of around 1.5% for 8t0MS = 0.6227(98), or, alternatively r0MS = 0.660(11). In Dalla Brida 19 [693] two GF-coupling definitions with SF-boundary conditions are considered, corresponding to (colour-) magnetic and electric components of the action density respectively. The coupling definitions include the projection to Q = 0, as was also done in [77]. The flow time parameter is set to c = 0.3, and both Zeuthen and Wilson flow are measured. Lattice sizes range from L/a = 8 to L/a = 48, covering up to a factor of 3 in lattice spacings for the step-scaling function, where both L/a and 2L/a are needed. Lattice effects in the step-scaling function are visible but can be extrapolated using global fits with a2 errors. Some remnant O(a) effects from the boundaries are expected, as their perturbative cancellation is incomplete. These O(a) contaminations are treated as a systematic error on the data, following [77] and are found to be subdominant. An intermediate reference scale μref is defined where α = 0.2, and the scales above and below are analyzed separately. Again this is similar to [77], except that here GF coupling data is available also at high energy scales. The GF β-functions are then obtained by fitting to the continuum extrapolated data for the step-scaling functions. In addition, a nonperturbative matching to the standard SF coupling is performed above μref for a range of couplings covering a factor 2. The nonperturbative β-function for the SF scheme can thus be inferred from the GF β-function. It turns out that GF schemes are very slow to reach the perturbative regime. Particularly the -parameter for the magnetic GF coupling shows a large slope in α2, which is the parametric uncertainty with known 3-loop β-function. Also, convincing contact with the 3-loop β-function is barely seen down to α = 0.08. This is likely to be related to the rather large 3-loop β-function coefficients, especially for the magnetic GF scheme [733]. In contrast, once the GF couplings are matched nonperturbatively to the SF scheme the contact to perturbative running can be safely made. It is also re-assuring that in all cases the extrapolations (linear in α2) to α = 0 for the -parameters agree very well, and the authors argue in favour of such extrapolations. Their data confirms that this procedure yields consistent results with the SF scheme for ν = 0, where such an extrapolation is not required. The low energy regime between μref and a hadronic scale μhad is covered again using the nonperturbative step-scaling function and the derived β-function. Finally, contact between μhad and hadronic scales t0 and r0 is established using 5 lattice spacings covering a factor up to 2.7. The multitude of cross checks of both continuum limit and perturbative truncation errors make this a study which passes all current FLAG criteria by some margin. The comparatively high value for r0MS found in this study must therefore be taken very seriously. In Nada 20 [694], Nada and Ramos provide further consistency checks of [693] for scales larger than μref . The step scaling function for c = 0.2 is constructed in 2 steps, by determining first the relation between couplings for c = 0.2 and c = 0.4 at the same L and then increasing L to 2L keeping the flow time fixed (in units of the lattice spacing), so that one arrives again at c = 0.2 on the 2L volume. The authors demonstrate that the direct construction of the step-scaling function for c = 0.2 would require much larger lattices in order to control the continuum limit at the same level of precision. The consistency with [693] for the -parameter is therefore a highly non-trivial check on the systematic effects of the continuum extrapolations. The study obtains results for the -parameter (again extrapolating to α = 0) with a similar error as in [693]. using the low-energy running and matching to the hadronic scale from that reference. For this reason and since gauge configurations are shared between both papers, these results are not independent of [693], so Dalla Brida 19 will be taken as representative for both works. 123 869 Page 158 of 296 Eur. Phys. J. C (2022) 82:869 9.4 The decoupling method The ALPHA collaboration has proposed a new strategy to compute the  parameter in QCD with N f ≥ 3 flavours based on simultaneous decoupling of N f ≥ 3 heavy quarks with RGI mass M [692]. We refer to [744] for a pedagogical introduction and to [745] for recent results. Generically, a running coupling in a mass-dependent renormalization scheme ḡ2(μ, M)(N f ) = ḡ2(μ)(N f =0) + O(M−k) (350) can be represented by the corresponding N f = 0 coupling, up to power corrections in 1/M . The leading power is usually k = 2, however renormalization schemes in finite volume may have k = 1, depending on the set-up. For example, this is the case with standard SF or open boundary conditions in combination with a standard mass term. In practice one may try to render such boundary contributions numerically small by a careful choice of the scheme’s parameters. In principle, power corrections can be either (μ/M)k or (/M)k . Fixing μ = μdec, e.g. by prescribing a value for the mass-independent coupling, such that μdec/ = O(1) thus helps to reduce the need for very large M . Defining ḡ2(μdec, M) = uM at fixed ḡ2(μdec, M = 0), Eq. (350) translates to a relation between -parameters, which can be cast in the form, ⎛ ⎞ (N f ) (0)  MS ⎝ M μdec ⎠ = MS (N f =0) (√ )P ϕs uM +O(M−k), (351) μ μ (N f ) (0)dec dec  s MS with the function ϕs as defined in Eq. (320), for scheme s and N f = 0. A crucial observation is that the function P , which (N ) gives the ratios of -parameters (0) f  / , can be evaluated perturbatively to a very good approximation [158,160]. MS MS Eq. (350) also implies a relation between the couplings in mass-independent schemes, in the theories with N f and zero flavours, respectively. In the MS scheme this relation is analogous to Eq. (327), ( ) ḡ2 (m )(N f =0) = ḡ2 (m )(N f ) × C ḡ (m )(N f )   (352)MS MS MS (N ) and the function C(g) is also known up to to 4-loop order [685–688,746]. The function P f(y), with y ≡ M/ can MS therefore be evaluated perturbatively in the MS scheme, as the ratio (N f =0) ( √ ) ϕ g= (y) C(g (y)) MS  = (NP y g y ḡ f )( ) , ( ) (m ). (353) (N f ) MS  ϕ (g(y)) MS Hence, perturbation theory is only required at the scale set by the heavy-quark mass, which works the better the larger M can be chosen. Once P is known, the LHS of (351) can be inferred from a N f = 0 computation of the RHS in the scheme s, assuming the ratio MS/s is known from a 1-loop calculation. To put the decoupling strategy into practice, the ALPHA collaboration uses N f = 3, so that information from [77] can be used. Using the massless GF coupling in finite volume from this project, μdec is defined through ḡ2GF(μdec) = 3.95, and thus known in physical units, μdec = 789(15)MeV. Varying L/a between 12 and 32 (five lattice spacings) defines a range of values for the bare coupling along a line of constant μdec and for vanishing quark mass. Next, a mass-dependent GF coupling is defined at constant μdec, using the available information on nonperturbative mass renormalization [203] and O(a) improvement. In order to obtain a larger suppression of the leading 1/M boundary correction term, the time extent T is here set to 2L , so as to maximize the distance to the time boundaries. Choosing 4 values of z = M/μdec within the range from 2 to (N =3) 8, with up to 5 lattice spacings64 and using precision results for N f = 0 from [693] then leads to the result for f , upMS to power corrections in 1/z, expected to be predominantly of order 1/z2. Figure 39, taken from [692] shows the continuum extrapolated results obtained for (3) /μdec at different values of z, together with the FLAG 19 average for three-flavourMS QCD. While the authors of [692] stopped short of quoting an extrapolated value for the three-flavour -parameter, the result (3)  = 332(10)(2) MeV is now given in the 2021 lattice conference proceedings [745], compatible with ALPHA 17 albeit MS with a somewhat smaller error. Despite some common elements with ALPHA 17, the authors emphasize that the decoupling 64 At the largest mass, z = 8, only the 2-3 finest lattice spacings are useful in a linear extrapolation in a2. 123 Eur. Phys. J. C (2022) 82:869 Page 159 of 296 869 Fig. 39 Illustration of the decoupling method, taken from Ref. [692] method is largely independent, with the overlap in squared error amounting to ca. 40 percent. This is due to the fact that the error in ALPHA 17 is dominated by the N f = 3 step scaling procedure at high energy, and this part is completely replaced by the N f = 0 result by Dalla Brida 19 [693]. The decoupling method thus seems to offer scope for a further error reduction, the major challenges being the continuum extrapolation for the GF coupling at fixed and large RGI masses, followed by the large M limit. It is important to note that this new method relies on new precision results for N f = 0 which have appeared in the last two years [693,694]. Therefore, the pure gauge theory acquires new relevance for αs results, beyond its traditional rôle as a test bed for the study of systematic errors. FLAG will take account of this development by continuing to carefully monitor N f = 0 results. It is hoped that this will encourage more groups to undertake precision studies with N f = 0. 9.5 αs from the potential at short distances 9.5.1 General considerations The basic method was introduced in Ref. [747] and developed in Ref. [748]. The force or potential between an infinitely massive quark and antiquark pair defines an effective coupling constant via = dV (r) = αqq(r)F(r) CF 2 . (354)dr r The coupling can be evaluated nonperturbatively from the potential through a numerical differentiation, see below. In pertur- bation theory one also defines couplings in different schemes αV̄ , αV via = − αV̄ (r) αV (Q)V (r) CF , or Ṽ (Q) = −CF 2 , (355)r Q where one fixes the unphysical constant in the potential by limr→∞ V (r) = 0 and Ṽ (Q) is the Fourier transform of V (r). Nonperturbatively, the subtraction of a constant in the potential introduces an additional renormalization constant, the value of V (rref) at some distance rref . Perturbatively, it is believed to entail a renormalon ambiguity. In perturbation theory, the different definitions are all simply related to each other, and their perturbative expansions are known including the α4s , α 4 s logαs and α5s logαs, α 5 s (logαs) 2 terms [749–756]. The potential V (r) is determined from ratios of Wilson loops, W (r, t), which behave as ∑ 〈W (r, t)〉 = |c |2e−V (r)t + |c 2 −Vn(r)t0 n| e , (356) n =0 where t is taken as the temporal extension of the loop, r is the spatial one and Vn are excited-state potentials. To improve the overlap with the ground state, and to suppress the effects of excited states, t is taken large. Also various additional techniques are used, such as a variational basis of operators (spatial paths) to help in projecting out the ground state. Furthermore some lattice-discretization effects can be reduced by averaging over Wilson loops related by rotational symmetry in the continuum. In order to reduce discretization errors it is of advantage to define the numerical derivative giving the force as = V (r)− V (r − a)F(rI) , (357) a 123 869 Page 160 of 296 Eur. Phys. J. C (2022) 82:869 Table 61 Short-distance potential results Collaboration Refs. N f Scale MS [MeV] r0MS Ayala 20 [74] 2+1 A ◦  ◦ r1 = 0.3106(17) fmc 338(13) 0.802(31) TUMQCD 19 [75] 2+1 A ◦  ◦ r1 = 0.3106(17) fmc 314+16 0.745(+38√ −8 −19) Takaura 18 [758,759] 2+1 A  ◦ ◦ t = 0.1465(25)fma 334(10)(+200 −18)b 0.799(51)+ Bazavov 14 [760] 2+1 A ◦  ◦ r1 = 0.3106(17) fmc 315(+18)d 0.746(+42−12 −27) Bazavov 12 [761] 2+1 A ◦† ◦ ◦# r0 = 0.468 fm 295(30)  0.70(7) Karbstein 18 [762] 2 A ◦ ◦ ◦ r0 = 0.420(14) fme 302(16) 0.643(34) Karbstein 14 [763] 2 A ◦ ◦ ◦ r0 = 0.42 fm 331(21) 0.692(31) ETM 11C [764] 2 A ◦ ◦ ◦ r0 = 0.42 fm 315(30)§ 0.658(55) Husung 20 [695] 0 C ◦   No quoted value for MS Husung 17 [765] 0 C ◦   r0 = 0.50 fm 232(6) 0.590(16) Brambilla 10 [766] 0 A ◦ ◦ ◦ †† √ 266(13) + 0.637(+32− )††30 UKQCD 92 [748] 0 A  ++  √σ = 0.44 GeV 256(20) 0.686(54) Bali 92 [767] 0 A  ◦++  σ = 0.44 GeV 247(10) 0.661(27) aScale determined from t0 in Ref. [114] b (5)α (MZ ) = 0.1179(7)(+13MS −12) cDetermination on lattices with mπ L = 2.2 − 2.6. Scale from r1 [116] as determined from fπ in Ref. [41] d (3)α (1.5 GeV) = 0 336 (5). (+12−8 ), α (MZ ) = 0.1166(+12)MS MS −8 eScale determined from fπ , see [87] †Since values of αeff within our designated range are used, we assign a◦◦despite values of αeff up to αeff = 0.5 being used#Since values of 2a/r within our designated range are used, we assign a although only values of 2a/r ≥ 1.14 are used at αeff = 0.3 Using results from Ref. [708]  (3) 1 5 GeV = 0 326 19 , (5)α ( . ) . ( ) α (MZ ) = 0.1156(+21)MS MS −22 §Both potential and r0/a are determined on a small (L = 3.2r0) lattice ††Uses lattice results of Ref. [709], some of which have very small lattice spacings where according to more recent investigations a bias due to the freezing of topology may be present +Our conversion using r = 0.472 fm ++ 0We give a◦because only a NLO formula is used and the error bars are very large; our criterion does not apply well to these very early calculations where rI is chosen so that at tree level the force is the continuum force. F(rI) is then a ‘tree-level improved’ quantity and similarly the tree-level improved potential can be defined [757]. Lattice potential results are in position space, while perturbation theory is naturally computed in momentum space at large momentum. Usually, the Fourier transform of the perturbative expansion is then matched to lattice data. Finally, as was noted in Sect. 9.2.1, a determination of the force can also be used to determine the scales r0, r1, by defining them from the static force by r2 20 F(r0) = 1.65, r1 F(r1) = 1. (358) 9.5.2 Discussion of computations In Table 61, we list results of determinations of r0MS (together with MS using the scale determination of the authors). Since the last review, FLAG 19, there have been three new publications, namely, TUMQCD 19 [75], Ayala 20 [74] and Husung 20 [695]. The first determinations in the three-colour Yang Mills theory are by UKQCD 92 [748] and Bali 92 [767] who used αqq as explained above, but not in the tree-level improved form. Rather a phenomenologically determined lattice-artifact correction 123 Publication status Renormalization scale Perturbative behaviour Continuum extrapolation Eur. Phys. J. C (2022) 82:869 Page 161 of 296 869 was subtracted from the lattice potentials. The comparison with perturbation theory was on a more qualitative level on the basis of a 2-loop β-function (nl = 1) and a continuum extrapolation could not be performed as yet. A much more precise computation of αqq with continuum extrapolation was performed in Refs. [709,757]. Satisfactory agreement with perturbation theory was found [757] but the stability of the perturbative prediction was not considered sufficient to be able to extract a  parameter. In Brambilla 10 [766] the same quenched lattice results of Ref. [757] were used and a fit was performed to the continuum potential, instead of the force. Perturbation theory to nl = 3 loop was used including a resummation of terms α3s (αs ln αs)n and α4s (αs ln αs) n . Close agreement with perturbation theory was found when a renormalon subtraction was performed. Note that the renormalon subtraction introduces a second scale into the perturbative formula which is absent when the force is considered. Bazavov 14 [760] updates Bazavov 12 [761] and modifies this procedure somewhat. They consider the perturbative expansion for the force. They set μ = 1/r to eliminate logarithms and then integrate the force to obtain an expression for the potential. The resulting integration constant is fixed by requiring the perturbative potential to be equal to the nonperturbative one exactly at a reference distance rref and the two are then compared at other values of r . As a further check, the force is also used directly. For the quenched calculation of Brambilla 10 [766] very small lattice spacings, a ∼ 0.025 fm, were available from Ref. [757]. For ETM 11C [764], Bazavov 12 [761], Karbstein 14 [763] and Bazavov 14 [760] using dynamical fermions such small lattice spacings are not yet realized (Bazavov 14 reaches down to a ∼ 0.041 fm). They all use the tree-level improved potential as described above. We note that the value of MS in physical units by ETM 11C [764] is based on a value of r0 = 0.42 fm. This is at least 10% smaller than the large majority of other values of r0. Also the values of r0/a on the finest lattices in ETM 11C [764] and r1/a for Bazavov 14 [760] come from rather small lattices with mπ L ≈ 2.4, 2.2 respectively. Instead of the procedure discussed previously, Karbstein 14 [763] reanalyzes the data of ETM 11C [764] by first estimating the Fourier transform Ṽ (p) of V (r) and then fitting the perturbative expansion of Ṽ (p) in terms of αMS(p). Of course, the Fourier transform requires some modelling of the r -dependence of V (r) at short and at large distances. The authors fit a linearly rising potential at large distances together with string-like corrections of order r−n and define the potential at large distances by this fit.65 Recall that for observables in momentum space we take the renormalization scale entering our criteria as μ = q, Eq. (338). The analysis (as in ETM 11C [764]) is dominated by the data at the smallest lattice spacing, where a controlled determination of the overall scale is difficult due to possible finite-size effects. Karbstein 18 [762] is a reanalysis of Karbstein 14 and supersedes it. Some data with a different discretization of the static quark is added (on the same configurations) and the discrete lattice results for the static potential in position space are first parameterized by a continuous function, which then allows for an analytical Fourier transformation to momentum space. Similarly also for Takaura 18 [758,759] the momentum space potential Ṽ (Q) is the central object. Namely, they assume that renormalon/power-law effects are absent in Ṽ (Q) and only come in through the Fourier transformation. They provide evidence that renormalon effects (both u = 1/2 and u = 3/2) can be subtracted and arrive at a nonperturbative term k 3 r2. MS Two different analyses are carried out with the final result taken from “Analysis II”. Our numbers including the evaluation of the criteria refer to it. Together with the perturbative 3-loop (including the α4s logαs term) expression, this term is fitted to the nonperturbative results for the potential in the region 0.04 fm ≤ r ≤ 0.35 fm, where 0.04 fm is r = a on the finest lattice. The nonperturbative potential data originates from JLQCD ensembles (Symanzik-improved gauge action and Möbius domain-wall quarks) at three lattice spacings with a pion mass around 300 MeV. Since at the maximal distance in the analysis we find αMS(2/r) = 0.43, the renormalization scale criterion yields a  . The perturbative behaviour is ◦ because of the high orders in perturbation theory known. The continuum-limit criterion yields a ◦. One of the main issues for all these computations is whether the perturbative running of the coupling constant has been reached. While for N f = 0 fermions Brambilla 10 [766] reports agreement with perturbative behaviour at the smallest distances, Husung 17 (which goes to shorter distances) finds relatively large corrections beyond the 3-loop αqq. For dynamical fermions, Bazavov 12 [761] and Bazavov 14 [760] report good agreement with perturbation theory after the renormalon is subtracted or eliminated. A second issue is the coverage of configuration space in some of the simulations, which use very small lattice spacings with periodic boundary conditions. Affected are the smallest two lattice spacings of Bazavov 14 [760] where very few tunnelings of the topological charge occur [116]. With present knowledge, it also seems possible that the older data by Refs. [709,757] used by Brambilla 10 [766] are partially obtained with (close to) frozen topology. 65 Note that at large distances, where string breaking is known to occur, this is not any more the ground state potential defined by Eq. (356). 123 869 Page 162 of 296 Eur. Phys. J. C (2022) 82:869 The computation in Husung 17 [765], for N f = 0 flavours, first determines the coupling ḡ2qq(r, a) from the force and then performs a continuum extrapolation on lattices down to a ≈ 0.015 fm, using a step-scaling method at short distances, r/r <0 ∼ 0.5. Using the 4-loop βqq function this allows r0qq to be estimated, which is then converted to the MS scheme. αeff = αqq ranges from ∼ 0.17 to large values; we give ◦ for renormalization scale and  for perturbative behaviour. The range aμ = 2a/r ≈ 0.37–0.14 leads to a in the continuum extrapolation. Recently these calculations have been extended in Husung 20 [695]. A finer lattice spacing of a = 0.01 fm (scale from r0 = 0.5 fm) is reached and lattice volumes up to L/a = 192 are simulated (in Ref. [765] the smallest lattice spacing is 0.015 fm). The Wilson action is used despite its significantly larger cutoff effects compared to Symanzik-improved actions; this avoids unitarity violations, thus allowing for a clean ground state extraction via a generalized eigenvalue problem. Open boundary conditions are used to avoid the topology-freezing problem. Furthermore, new results for the continuum approach are employed, which determine the cutoff dependence at O(a2) including the exact coupling-dependent terms, in the asymptotic region where the Symanzik effective theory is applicable [768]. An ansatz for the remaining higher order cutoff effects at O(a4) is propagated as a systematic error to the data, which effectively discards data for r/a < 3.5. The large volume step-scaling function with step factor 3/4 is computed and compared to perturbation theory. For αqq > 0.2 there is a noticeable difference between the 2-loop and 3-loop results. Furthermore, the ultra-soft contributions at 4-loop level give a significant contribution to the static Q Q̄ force. While this study is for N f = 0 flavours it does raise the question whether the weak coupling expansion for the range of r -values used in present analyses of αs is sufficiently reliable. Around αqq ≈ 0.21 the differenc√es get smaller but the errorn =0 increases significantly, mainly due to the propagated lattice artifacts. The dependence of f 8t on α3 is very similar MS 0 qq to the one observed in the previous study but no value for its αqq → 0 limit is quoted. Husung 20 [695] is more pessimistic about the error on the  parameter stating the relative error has to be 5% or larger, while Husung 17 quotes a relative error of 3%. In 2+1-flavor QCD two new papers appeared on the determination of the strong coupling constant from the static quark anti-quark potential after the FLAG 19 report [74,75]. In TUMQCD 19 [75]66 the 2014 analysis of Bazavov 14 [760] has been extended by including three finer lattices with lattice spacing a = 0.035, 0.030 and 0.025 fm as well as lattice results on the free energy of static quark anti-quark pair at non-zero temperature. On the new fine lattices the effect of freezing topology has been observed, however, it was verified that this does not affect the potential within the estimated errors [769,770]. The comparison of the lattice result on the static potential has been performed in the interval r = [rmin, rmax], with rmax = 0.131, 0.121, 0.098, 0.073 and 0.055 fm. The main result quoted in the paper is based on the analysis with rmax = 0.073 fm [75]. Since the new study employs a much wider range in r than the previous one [760] we give it a for the perturbative behaviour. Since αeff = αqq varies in the range 0.2–0.4 for the r values used in the main analysi√s we give ◦ for the renormalization scale. Several values of rmin have been used in the analysis, the largest being r /a = 8 2.82, which corresponds to aμ 0.71. Therefore, we give a ◦ minfor continuum extrapolation in this case. An important difference compared to the previous √study [760] is the variation of the renormalization scale. In Ref. [760] the renormalization scale was varied by a factor of 2 around the nominal value of μ = 1/r , in order to exclude very low scales, for which the running of the strong coupling constant is no longer perturbative. In the new analysis the renormalization scale was varied by a factor of two. As the result, despite the extended data set and shorter distances used in the new study the perturbative error did not decrease [75]. We also note that the scale dependence turned out to be non-monotonic in the range μ = 1/(2r)–2/r [75]. The final result reads (“us” stands for “ultra-soft”), N f =3  = 314.0 ± 5.8(stat)± 3.0(lat)± 1.7(scale)+13.4−1.8 (pert)± 4.0(pert. us) MeV = 314+16−08 MeV, (359)MS where all errors were combined in quadrature. This is in very good agreement with the previous determination [760]. The analysis was also applied to the singlet static quark anti-quark free energy at short distances. At short distances the free energy is expected to be the same as the static potential. This is verified numerically in the lattice calculations TUMQCD 19 [75] for rT < 1/4 with T being the temperature. Furthermore, this is confirmed by the perturbative calculations at T > 0 at NLO [771]. The advantage of using the free energy is that it gives access to much shorter distances. On the other hand, one has fewer data points because the condition rT < 1/4 has to be satisfied. The analysis based on the free energy gives N f =3  = 310.9 ± 11.3(stat)± 3.0(lat)± 1.7(scale)+5.6−0.8(pert)± 2.1(pert. us) MeV = 311(13) MeV, (360)MS in good agreement with the above result and thus, providing additional confirmation of it. 66 The majority of authors are the same as in [760]. 123 Eur. Phys. J. C (2022) 82:869 Page 163 of 296 869 The analysis of Ayala 20 [74] uses a subset of data presented in TUMQCD 19 [75] with the same correction of the lattice effects. For this reason the continuum extrapolation gets ◦, too. They match to perturbation theory for 1/r > 2 GeV, which corresponds to αeff = αqq = 0.2–0.4. Therefore, we give ◦ for the renormalization scale. They verify the perturbative behaviour in the region 1 GeV < 1/r < 2.9 GeV, which corresponds to variation of α3eff by a factor of 3.34. However, the relative error on the final result has δ/ 0.035 which is larger than α3eff = 0.011. Therefore, we give a  for the perturbative behaviour in this case. The final result for the -parameter reads: N f =3  = 338 ± 2(stat)± 8(matching)± 10(pert) MeV = 338(13)MeV. (361) MS This is quite different from the above result. This difference is mostly due to the organization of the perturbative series. The authors use ultra-soft (log) resummation, i.e. they resum the terms α3+ns lnn αs to all orders instead of using fixed- order perturbation theory. They also include what is called the terminant of the perturbative series associated to the leading renormalon of the force [74]. When they use fixed order perturbation theory they obtain very similar results to Refs. [75,760]. It has been argued that log resummation cannot be justified since for the distance range available in the lattice studies αs is not small enough and the logarithmic and non-logarithmic higher-order terms are of a similar size [760]. On the other hand, the resummation of ultra-soft logs does not lead to any anomalous behaviour of the perturbative expansion like large scale dependence or bad convergence [74]. N =3 To obtain the value of f from the static potential we combine the results in Eqs. (359) and (361) using the weighted MS average with the weight given by the perturbative error and using the difference in the central value as the error estimate. This leads to N f =3  = 330(24) MeV. (362) MS from the static potential determination. In the case of TUMQCD 19, where the perturbative error is very asymmetric we used the larger upper error for the calculation of the corresponding weight. 9.6 αs from the light-quark vacuum polarization in momentum/position space 9.6.1 General considerations Except for the new calculation Cali 20 [76], where position space is used (see below), the light-flavour-current 2-point function is usually evaluated in momentum space, in terms of the vacuum-polarization function. For the flavour-nonsinglet currents J aμ (a = 1, 2, 3) in the momentum representation this is parametrized as 〈J a J b〉 = δab[ (1) (0)μ ν (δ 2μν Q − QμQν)#J (Q)− QμQν#J (Q)], (363) where Qμ is a space-like momentum and Jμ ≡ Vμ for a vector current and Jμ ≡ Aμ for an axial-vector current. Defining Q ≡ (0) (1)#J ( ) #J (Q)+#J (Q), the operator product expansion (OPE) of #V/A(Q) is given by 2 | 2 = + V/A 2 + V/A 2 m̄ (Q) ∑ #V/A OPE(Q , αs) c C1 (Q ) Cm (Q ) 2 + V/A 2 〈mqq̄q〉C Q q̄q (Q ) Q4 q=u,d,s + V/A 2 〈αs GG〉CGG (Q ) −64 +O(Q ), (364)Q for larg(e Q 2. T)he perturbative coefficient functions C V/A X (Q 2) for the operators X (X = 1, q̄q, GG) are given as CV/A 2 ∑ X (Q ) = V/A (i) i 2 i≥0 CX αs(Q ) and m̄ is the running mass of the mass-degenerate up and down quarks. C V/A 1 is known including α4s in a continuum renormalization scheme such as the MS scheme [772–775]. Nonperturbatively, there are terms in C V/A X that do not have a series expansion in αs . For an example for the unit operator see Ref. [776]. The term c is Q-independent and divergent in the limit of infinite ultraviolet cutoff. However the Adler function defined as 2 D(Q2 d#(Q ) ) ≡ −Q2 2 , (365)d Q 123 869 Page 164 of 296 Eur. Phys. J. C (2022) 82:869 is a scheme-independent finite quantity. Therefore one can determine the running-coupling constant in the MS scheme from the vacuum-polarization function computed by a lattice-QCD simulation. Of course, there is the choice whether to use the vector or the axial vector channel, or both, the canonical choice being #V+A = #V +#A. While perturbation theory does not distinguish between these channels, the nonperturbative contributions are different, and the quality of lattice data may differ, too. For a given choice, the lattice data of the vacuum polarization is fitted with the perturbative formula Eq. (364) with fit parameter MS parameterizing the running coupling αMS(Q 2). While there is no problem in discussing the OPE at the nonperturbative level, the ‘condensates’ such as 〈αs GG〉 are ambiguous, since they mix with lower-dimensional operators including the unity operator. Therefore one should work in the high-√Q2 regime where power corrections are negligible within the given accuracy. Thus setting the renormalization scale as μ ≡ Q2, one should seek, as always, the window QCD  μ  a−1. 9.6.2 Definitions in position space The 2-point current correlation functions in position space contain the same physical information as in momentum space, but the technical details are sufficiently different to warrant a separate discussion. The (Euclidean) current-current correlation function for Jμf f ′ (with flavour indices f, f ′) is taken to be either the flavour non-diagonal vector or axial vector current, with the Lorentz indices contracted, ∑〈 〉 ( ) CA,V(x) = − Jμ x Jμ 6 αsf f ′A,V( ) f ′ 2f A,V(0) = 4 2 3 1 + +O(α ) . (366)π (x ) π μ In the chiral limit, the perturbative expansion is known to α4s [777], and is identical for vector and axial vector correlators. The only scale is set by the Euclidean distance μ = 1/|x | and the effective coupling can thus be defined as [ ] αeff(μ = 1/|x |) = π (x2)3(π4/6)CA,V(x)− 1 . (367) As communicated to us by the authors of [76], there is a typo in Eq. (35) of [777]. For future reference, the numerical coefficients for the 3-loop conversion αeff(μ) = αMS(μ)+ c α2 (μ)+ c α31 2 (μ)+ c α4 (μ), (368)MS MS 3 MS should read c1 = −1.4346, c2 = 0.16979, c3 = 3.21120. (369) 9.6.3 Discussion of computations Results using this method in momentum space are, to date, only available using overlap fermions or domain-wall fermions. Since the last review, FLAG 19, there has been one new computation, Cali 20 [76], which uses the vacuum polarization in position space, using O(a) improved Wilson fermions. The results are collected in Table 62 for N f = 2, JLQCD/TWQCD 08C [778] and for N f = 2 + 1, JLQCD 10 [779], Hudspith 18 [780] and Cali 20 [76]. We first discuss the results of JLQCD/TWQCD 08C [778] and JLQCD 10 [779]. The fit to Eq. (364) is done with the 4-loop relation between the running coupling and MS. It is found that without introducing condensate contributions, the momentum scale where the perturbative formula gives good agreement with the lattice results is very narrow, aQ 0.8–1.0. When a condensate contribution is included the perturbative formula gives good agreement with the lattice results for the extended range aQ 0.6–1.0. Since there is only a single lattice spacing a ≈ 0.11 fm there is a  for the continuum limit. The renormalization scale μ is in the range of Q = 1.6–2 GeV. Approximating αeff ≈ αMS(Q), we estimate that αeff = 0.25–0.30 for N f = 2 and αeff = 0.29–0.33 for N f = 2 + 1. Thus we give a ◦ and  for N f = 2 and N f = 2 + 1, respectively, for the renormalization scale and a  for the perturbative behaviour. A further investigation of this method was initiated in Hudspith 15 [781] and completed by Hudspith 18 [780] (see also [782]) based on domain-wall fermion configurations at three lattice spacings, a−1 = 1.78, 2.38, 3.15 GeV, with three different light-quark masses on the two coarser lattices and one on the fine lattice. An extensive discussion of condensates, using continuum finite-energy sum rules was employed to estimate where their contributions might be negligible. It was found that 123 Eur. Phys. J. C (2022) 82:869 Page 165 of 296 869 Table 62 Results from the vacuum polarization in both momentum and position space Collaboration Ref. N f Scale MS [MeV] r0MS Cali 20 [76] 2+1 A ◦   m§ϒ 342(17) 0.818(41)a Hudspith 18 [780] 2+1 P ◦ ◦  m 337(40) 0.806(96)b Hudspith 15 [781] 2+1 C ◦ ◦  m 300(24)+ 0.717(58) JLQCD 10 [779] 2+1 A  ◦  r0 = 0.472 fm 247(5)† 0.591(12) JLQCD/TWQCD 08C [778] 2 A ◦ ◦  r0 = 0.49 fm 234(9)(+16 +40−0 ) 0.581(22)(−0 ) §Via t /a20 , still unpublished. We use r0 = 0.472 fm Determined in [8] aEvaluates to (5)α (MZ ) = 0.11864(114)MS In conversion to r0 we used r0 = 0.472 fm. b (5)α (M ) = 0.1181(27)(+8Z −22). MS determined by us from (3)α (2 GeV) = 0.2961(185). In conversion to r0 we used r0 = 0.472 fmMS MS +Determined by us from (3)α (2 GeV (5)) = 0.279(11). Evaluates to α (MZ ) = 0.1155(18)MS MS † (5)α (M ) = 0.1118(3)(+16 MS Z −17) even up to terms of O((1/Q2)8) (a higher order than depicted in Eq. (364) but with constant coefficients) no single condensate dominates and apparent convergence was poor for low Q2 due to cancellations between contributions of similar size with alternating signs. (See, e.g., the list given by Hudspith 15 [781].) Choosing Q2 to be at least∼ 3.8 GeV2 mitigated the problem, but then the coarsest lattice had to be discarded, due to large lattice artefacts. So this gives a  for continuum extrapolation. With the higher Q2 the quark-mass dependence of the results was negligible, so ensembles with different quark masses were averaged over. A range of Q2 from 3.8–16 GeV2 gives αeff = 0.31–0.22, so there is a ◦ for the renormalization scale. The value of α3eff reaches αeff/(8πb0αeff) and thus gives a ◦ for perturbative behaviour. In Hudspith 15 [781] (superseded by Hudspith 18 [780]) about a 20% difference in #V (Q2) was seen between the two lattice spacings and a result is quoted only for the smaller a. 9.6.4 Vacuum polarization in position space Cali 20 [76] evaluate the light-current 2-point function in position space. The 2-point functions for the nonperturbatively renormalized (non-singlet) flavour currents is computed for distances |x | between 0.1 and 0.25 fm and extrapolated to the chiral limit. The available CLS configurations are used for this work, with lattice spacings between 0.039 and 0.086 fm. Despite fully nonperturbative renormalization and O(a) improvement, the remaining O(a2) effects, as measured by O(4) symmetry violations, are very large, even after subtraction of tree-level lattice effects. Therefore the authors performed a numerical stochastic perturbation theory (NSPT) simulation in order to determine the lattice artifacts at O(g2). Only after subtraction of these effects the constrained continuum extrapolations from 3 different lattice directions to the same continuum limit are characterized by reasonable χ2-values, so the feasibility of the study crucially depends on this step. Interestingly, there is no subtraction performed of nonperturbative effects. For instance, chiral-symmetry breaking would manifest itself in a difference between the vector and the axial vector 2-point functions, and is invisible to perturbation theory, where these 2-point functions are known to α4s [777]. According to the authors, phenomenological estimates suggest that a difference of 1.5% between the continuum correlators would occur around 0.3 fm and this difference would not be resolvable by their lattice data. Equality within their errors is confirmed for shorter distances. We note, however, that chiral symmetry breaking effects are but one class of nonperturbative effects, and their smallness does not allow for the conclusion that such effects are generally small. In fact, the need for explicit subtractions in momentum space analyses may lead one to suspect that such effects are N =3 not negligible at the available distance scales. For the determination of f the authors limit the range of distances to MS 0.13–0.19 fm, where αeff ∈ [0.2354, 0.3075] (private communication by the authors). These effective couplings are converted 123 Publication status Renormalization scale Perturbative behaviour Continuum extrapolation 869 Page 166 of 296 Eur. Phys. J. C (2022) 82:869 to MS couplings at the same scales μ = 1/|x | by solving Eq. (368) numerically. Central values for the -parameter thus obtained are in the range 325–370 MeV (using the β-function at 5-loop order) and a weighted average yields the quoted result 342(17) MeV, where the average emphasizes the data around |x | = 0.16 fm, or μ = 1.3 GeV. Applying the FLAG criteria the range of lattice spacings yields  for the continuum extrapolation. However, the FLAG criterion implicitly assumes that the remaining cutoff effects after non-perturbative O(a) improvement are small, which is not the case here. Some hypercubic lattice artefacts are still rather large even after 1-loop subtraction, but these are not used for the analysis. As for the renormalization scale, the lowest effective coupling entering the analysis is 0.235 < 0.25, so we give◦. As for perturbative behaviour, for the range of couplings in the above interval α3eff changes by (0.308/0.235) 3 ≈ 2.2, marginally reaching (3/2)2 = 2.25. The errors αeff after continuum and chiral extrapolations are 4–6% (private communication by the authors) and the induced uncertainty in  is comfortably above 2α3eff, which gives a according to FLAG criteria. Although the current FLAG criteria are formally passed by this result, the quoted error of 5% for  seems very optimistic. We have performed a simple test, converting to the MS scheme by inverting Eq. (368) perturbatively (instead of solving the fixed-order equation numerically). The differences between the couplings are of order α5s and thus indicative of the sensitivity to perturbative truncation errors. The resulting -parameter estimates are now in the range 409–468 MeV, i.e. ca. 15–30% larger than before. While the difference between both estimates decreases proportionally to the expected α3eff, an extraction of the -parameter in this energy range is a priori affected by systematic uncertainties corresponding to such differences. The FLAG criterion might fail to capture this e.g. if the assumption of an O(1) coefficient for the asymptotic α3eff behaviour is not correct. Some indication for a problematic behaviour is indeed seen when perturbatively inverting Eq. (368) to order α3s . The resulting MS couplings are then closer to the values used in Cali 20, although the difference is formally O(α4s ) rather than O(α5s ). 9.7 αs from observables at the lattice spacing scale 9.7.1 General considerations The general method is to evaluate a short-distance quantity Q at the scale of the lattice spacing ∼ 1/a and then determine its relationship to αMS via a perturbative expansion. This is epitomized by the strategy of the HPQCD collaboration [783,784], discussed here for illustration, which computes and then fits to a variety of short-distance quantities n∑max Y = c n ∗nαV′(q ). (370) n=1 The quantity Y is taken as the logarithm of small Wilson loops (including some nonplanar ones), Creutz ratios, ‘tadpole- improved’ Wilson loops and the tadpole-improved or ‘boosted’ bare coupling (O(20) quantities in total). The perturbative coefficients cn (each depending on the choice of Y ) are known to n = 3 with additional coefficients up to nmax being fitted numerically. The running coupling αV′ is related to αV from the static-quark potential (see Sect. 9.5).67 The coupling constant is fixed at a scale q∗ = d/a. The latter is chosen as the mean value of ln q with the one-gluon loop as measure [785,786]. (Thus a different result for d is found for every short-distance quantity.) A rough estimate yields d ≈ π , and in general the renormalization scale is always found to lie in this region. For example, for the Wilson loop Wmn ≡ 〈W (ma, na)〉 we have ( ) W ln mn+ = c ∗ 2 ∗ 3 ∗1αV′(q )+ c2αV′(q )+ c3αV′(q )+ · · · , (371)u2(m n)0 for the tadpole-improved version, where c1, c2, . . . are the appropriate perturbative coefficients and u0 = W 1/411 . Substituting the nonperturbative simulation value in the left hand side, we can determine αV′(q∗), at the scale q∗. Note that one finds empirically that perturbation theory for these tadpole-improved quantities have smaller cn coefficients and so the series has a faster apparent convergence compared to the case without tadpole improvement. 67 ′ ′αV′ is defined by V′ = V and bVi = bVi for i = 0, 1, 2 but bVi = 0 for i ≥ 3. 123 Eur. Phys. J. C (2022) 82:869 Page 167 of 296 869 Using the β-function in the V′ scheme, results can be run to a reference value, chosen as α0 ≡ αV′(q0), q0 = 7.5 GeV. This is then converted perturbatively to the continuum MS scheme αMS(q0) = α0 + d1α20 + d2α30 + · · · , (372) where d1, d2 are known 1-and 2-loop coefficients. Other collaborations have focused more on the bare ‘boosted’ coupling constant and directly determined its relationship to αMS. Specifically, the boosted coupling is defined by 2 αP(1/a) = 1 g04 , (373)4π u0 again determined at a scale ∼ 1/a. As discussed previously, since the plaquette expectation value in the boosted coupling contains the tadpole-diagram contributions to all orders, which are dominant contributions in perturbation theory, there is an expectation that the perturbation theory using the boosted coupling has smaller perturbative coefficients [785], and hence smaller perturbative errors. 9.7.2 Continuum limit Lattice results always come along with discretization errors, which one needs to remove by a continuum extrapolation. As mentioned previously, in this respect the present method differs in principle from those in whichαs is determined from physical observables. In the general case, the numerical results of the lattice simulations at a value of μ fixed in physical units can be extrapolated to the continuum limit, and the result can be analyzed as to whether it shows perturbative running as a function of μ in the continuum. For observables at the cutoff-scale (q∗ = d/a), discretization effects cannot easily be separated out from perturbation theory, as the scale for the coupling comes from the lattice spacing. Therefore the restriction aμ  1 (the ‘continuum-extrapolation’ criterion) is not applicable here. Discretization errors of order a2 are, however, present. Since a ∼ exp(−1/(2b0g20)) ∼ exp(−1/(8πb α(q∗0 )), these errors now appear as power corrections to the perturbative running, and have to be taken into account in the study of the perturbative behaviour, which is to be verified by changing a. One thus usually fits with power corrections in this method. In order to keep a symmetry with the ‘continuum-extrapolation’ criterion for physical observables and to remember that discretization errors are, of course, relevant, we replace it here by one for the lattice spacings used: • Lattice spacings  3 or more lattice spacings, at least 2 points below a = 0.1 fm ◦ 2 lattice spacings, at least 1 point below a = 0.1 fm  otherwise 9.7.3 Discussion of computations Note that due to μ ∼ 1/a being relatively large the results easily have a or ◦ in the rating on renormalization scale. The work of El-Khadra 92 [794] employs a 1-loop formula to relate (0)α (π/a) to the boosted coupling for three lattice MS spacings a−1 = 1.15, 1.78, 2.43 GeV. (The lattice spacing is determined from the charmonium 1S-1P splitting.) They obtain (0)  = 234 MeV, corresponding to (0)αeff = α (π/a) ≈ 0.15–0.2. The work of Aoki 94 [791] calculates (2)αV and (2)α for aMS MS MS single lattice spacing a−1 ∼ 2 GeV, again determined from charmonium 1S-1P splitting in two-flavour QCD. Using 1-loop perturbation theory with boosted coupling, they obtain (2)αV = 0 169 and (2). α = 0.142. Davies 94 [790] gives a determinationMS of αV from the expansion − ≡ 4π (N f ) (N )ln W f11 αV (3.41/a)× [1 − (1.185 + 0.070N f )α ], (374)3 V neglecting higher-order terms. They compute the ϒ spectrum in N f = 0, 2 QCD for single lattice spacings at a−1 = 2.57, 2.47 GeV and obtain αV(3.41/a) 0.15, 0.18, respectively. Extrapolating the inverse coupling linearly in N f , a value of 123 869 Page 168 of 296 Eur. Phys. J. C (2022) 82:869 (3) αV (8.3 GeV) = 0.196(3) is obtained. SESAM 99 [788] follows a similar strategy, again for a single lattice spacing. They linearly extrapolated results for 1 (0) (2) (3)/αV , 1/αV at a fixed scale of 9 GeV to give αV , which is then perturbatively converted to (3)α . This finally gave (5)α (MZ ) = 0.1118(17). Wingate 95 [789] also follows this method. With the scale determinedMS MS from the charmonium 1S-1P splitting for single lattice spacings in N = 0, 2 giving a−1f 1.80 GeV for N f = 0 and a−1 1.66 GeV for N f = 2, they obtain (0) (2)αV (3.41/a) 0.15 and αV 0.18, respectively. Extrapolating the inverse coupling linearly in N f , they obtain (3) αV (6.48 GeV) = 0.194(17). The QCDSF/UKQCD collaboration, QCDSF/UKQCD 05 [787,795–797], use the 2-loop relation (re-written here in terms of α) 1 = 1 + 4π(2b0 ln aμ− tP)+ (4π)2(2b P1 ln aμ− t )αP(1/a), (375) αMS(μ) α (1/a) 1 2 P where tP1 and t P 2 are known. (A 2-loop relation corresponds to a 3-loop lattice β-function.) This was used to directly compute αMS, and the scale was chosen so that the O(α0P) term vanishes, i.e., { ∗ = 1 2.63/a N = 0μ exp [tP 2b ] ≈ f1 /( 0) . (376)a 1.4/a N f = 2 The method is to first compute αP(1/a) and from this, using Eq. (375) to find α ∗MS(μ ). The RG equation, Eq. (320), then determines μ∗/MS and hence using Eq. (376) leads to the result for r0MS. This avoids giving the scale in MeV until the end. In the N f = 0 case seven lattice spacings were used [709], giving a range μ∗/MS ≈ 24–72 (or a−1 ≈ 2–7 GeV) and α ∗eff = αMS(μ ) ≈ 0.15–0.10. Neglecting higher-order perturbative terms (see discussion after Eq. (377) below) in Eq. (375) this is sufficient to allow a continuum extrapolation of r0MS. A similar computation for N f = 2 by QCDSF/UKQCD 05 [787] gave μ∗/MS ≈ 12–17 (or roughly a−1 ≈ 2–3 GeV) and α ∗eff = αMS(μ ) ≈ 0.20–0.18. The N f = 2 results of QCDSF/UKQCD 05 [787] are affected by an uncertainty which was not known at the time of publication: It has been realized that the values of r0/a of Ref. [787] were significantly too low [707]. As this effect is expected to depend on a, it influences the perturbative behaviour leading us to assign a  for that criterion. Since FLAG 13, there has been one new result for N f = 0 by FlowQCD 15 [793], later updated and published in Kitazawa 16 [792]. They also use the techniques as described in Eqs. (375), (376), but together with the gradient flow scale w0 (rather than the r0 scale) leading to a determination of w0MS. The continuum limit is estimated by extrapolating the data at 6 lattice spacings linearly in a2. The data range used is μ∗/MS ≈ 50–120 (or a−1 ≈ 5–11 GeV) and α ∗MS(μ ) ≈ 0.12–0.095. Since a very small value of αMS is reached, there is a in the perturbative behaviour. Note that our conversion to the common r0 scale unfortunately leads to a significant increase of the error of the  parameter compared to using w0 directly [714]. Again we note that the results of QCDSF/UKQCD 05 [787] (N f = 0) and Kitazawa 16 [792] may be affected by frozen topology as they have lattice spacings significantly below a = 0.05 fm. Kitazawa 16 [792] investigate this by evaluating w0/a in a fixed topology and estimate any effect at about ∼ 1%. The work of HPQCD 05A [783] (which supersedes the original work [798]) uses three lattice spacings a−1 ≈ 1.2, 1.6, 2.3 GeV for 2 + 1 flavour QCD. Typically the renormalization scale q ≈ π/a ≈ 3.50–7.10 GeV, corresponding to αV′ ≈ 0.22–0.28. In the later update HPQCD 08A [784] twelve data sets (with six lattice spacings) are now used reaching up to a−1 ≈ 4.4 GeV, corresponding to αV′ ≈ 0.18. The values used for the scale r1 were further updated in HPQCD 10 [11]. Maltman 08 [79] uses most of the same lattice ensembles as HPQCD 08A [784], but not the one at the smallest lattice spacing, a ≈ 0.045 fm. Maltman 08 [79] also considers a much smaller set of quantities (three versus 22) that are less sensitive to condensates. They also use different strategies for evaluating the condensates and for the perturbative expansion, and a slightly different value for the scale r1. The central values of the final results from Maltman 08 [79] and HPQCD 08A [784] differ by 0.0009 (which would be decreased to 0.0007 taking into account a reduction of 0.0002 in the value of the r1 scale used by Maltman 08 [79]). As mentioned before, the perturbative coefficients are computed through 3-loop order [799], while the higher-order per- turbative coefficients cn with nmax ≥ n > 3 (with nmax = 10) are numerically fitted using the lattice-simulation data for the lattice spacings with the help of Bayesian methods. It turns out that corrections in Eq. (371) are of order |ci/c1|αi = 5–15% and 3–10% for i = 2, 3, respectively. The inclusion of a fourth-order term is necessary to obtain a good fit to the data, and leads to a shift of the result by 1 – 2 sigma. For all but one of the 22 quantities, central values of |c4/c1| ≈ 2–4 were found, with errors from the fits of ≈ 2. It should be pointed out that the description of lattice results for the short distance quantities 123 Eur. Phys. J. C (2022) 82:869 Page 169 of 296 869 does not require Bayesian priors, once the term proportional to c4 is included [79]. We also stress that different short distance quantities have quite different nonperturbative contributions [800]. Hence the fact that different observables lead to consistent αs values is a nontrivial check of the approach. An important source of uncertainty is the truncation of perturbation theory. In HPQCD 08A [784], 10 [11] it is estimated to be about 0.4% of αMS(MZ ). In FLAG 13 we included a rather detailed discussion of the issue with the result that we prefer for the time being a more conservative error based on the above estimate |c4/c1| = 2. From Eq. (370) this gives an estimate of the uncertainty in αeff of ∣∣ ∣c4 ∣ αeff(μ1) = ∣∣ ∣∣α4eff(μ1), (377)c1 at the scale μ1 where αeff is computed from the Wilson loops. This can be used with a variation in  at lowest order of perturbation theory and also applied to αs evolved to a different scale μ2,68  = 1 αs αs(μ ) α 2 2 = s (μ2), . (378)  8πb0αs αs αs(μ1) α2s (μ1) With μ2 = MZ and αs(μ1) = 0.2 (a typical value extracted from Wilson loops in HPQCD 10 [11], HPQCD 08A [784] at μ = 5 GeV) we have αMS(m Z ) = 0.0012, (379) which we shall later use as the typical perturbative uncertainty of the method with 2 + 1 fermions. Table 63 summarizes the results. Within the errors of 3–5% N f = 3 determinations of r0 nicely agree. 9.8 αs from heavy-quark current two-point functions 9.8.1 General considerations The method has been introduced in HPQCD 08, Ref. [202], and updated in HPQCD 10, Ref. [11], see also Ref. [801]. In addition there is a 2+1+1-flavour result, HPQCD 14A [14]. The basic observable is constructed from a current, J (x) = iamcψc(x)γ5ψc′(x), (380) of two mass-degenerate heavy-valence quarks, c, c′, usually taken to be at or around the charm-quark mass. The pre-factor mc denotes the bare mass of the quark. When the lattice discretization respects chiral symmetry, J (x) is a renormalization group invariant local field, i.e., it requires no renormalization. Staggered fermions and twisted-mass fermions have such a residual chiral symmetry. The (Euclidean) time-slice correlation function ∑ G(x ) = a6 †0 〈J (x)J (0)〉, (381) x (J †(x) = iamcψc′(x)γ5ψc(x)) has a ∼ x−30 singularity at short distances and moments T∑/2−a G = a xnn 0 G(x0) (382) x0=−(T/2−a) are nonvanishing for even n and furthermore finite for n ≥ 4. Here T is the time extent of the lattice. The moments are dominated by contributions at x0 of order 1/mc. For large mass mc these are short distances and the moments become 68 From Eq. (327) we see that at low order in PT the coupling αs is continuous and differentiable across the mass thresholds (at the same scale). Therefore to leading order αs and αs are independent of N f . 123 869 Page 170 of 296 Eur. Phys. J. C (2022) 82:869 Table 63 Wilson loop results. Some early results for N f = 0, 2 did not determine MS Collaboration Refs. N f Scale MS [MeV] r0MS § HPQCD 10a [11] 2+1 A ◦   r1 = 0.3133(23) fm 340(9) 0.812(22) HPQCD 08Aa [784] 2+1 A ◦   r1 = 0.321(5) fm†† 338(12) 0.809(29) Maltman 08a [79] 2+1 A ◦ ◦  r1 = 0.318 fm 352(17)† 0.841(40) HPQCD 05Aa [783] 2+1 A ◦ ◦ ◦ r††1 319(17) 0.763(42) QCDSF/UKQCD 05 [787] 2 A    r0 = 0.467(33) fm 261(17)(26) 0.617(40)(21)b SESAM 99c [788] 2 A ◦   cc̄(1S-1P) Wingate 95d [789] 2 A    cc̄(1S-1P) Davies 94e [790] 2 A    ϒ Aoki 94 f [791] 2 A    cc̄(1S-1P) Kitazawa 16 [792] 0 A    w0 260(5) j 0.621(11) j FlowQCD 15 [793] 0 P    wi0.4 258(6)i 0.618(11)i QCDSF/UKQCD 05 [787] 0 A  ◦  r0 = 0.467(33) fm 259(1)(20) 0.614(2)(5)b SESAM 99c [788] 0 A    cc̄(1S-1P) Wingate 95d [789] 0 A    cc̄(1S-1P) Davies 94e [790] 0 A    ϒ El-Khadra 92g [794] 0 A   ◦ cc̄(1S-1P) 234(10) 0.560(24)h aThe numbers for  have been converted from the values for (5)αs (MZ ) § (3)α (5 GeV) = 0.2034 21 , (5)( ) α (M ) = 0.1184(6), only update of intermediate scale and c-, b-quark masses, supersedes HPQCD 08A MS MS Z † (5)α (M MS Z ) = 0.1192(11)  (3)αV (7.5 GeV) = 0.2120 28 , (5)( ) α (MZ ) = 0.1183(8), supersedes HPQCD 05MS ††Scale is originally determined from ϒ mass splitting. r1 is used as an intermediate scale. In conversion to r0MS, r0 is taken to be 0.472 fm  (3) (5)αV (7.5 GeV) = 0.2082(40), α (MZ ) = 0.1170(12)MS bThis supersedes Refs. [795–797]. (5)α (MZ ) = 0.112(1)(2). The N f = 2 results were based on values for r0/a which have later been found to beMS too small [707]. The effect will be of the order of 10–15%, presumably an increase in r0 c (5)α (MZ ) = 0.1118(17)MS d (3)αV (6.48 GeV) = 0.194(7) extrapolated from N = 0 2. (5)f , α (M ) = 0.107(5)MS Z e (3)αP (8.2 GeV) = 0.1959(34) extrapolated from N f = 0 (5), 2. α (MZ ) = 0.115(2)MS f Estimated (5)α (MZ ) = 0.108(5)(4)MS gThis early computation violates our requirement that scheme conversions are done at the 2-loop level. (4) = 160 +47 (4)(−37)MeV, α (5GeV) =MS MS 0.174(12). We converted this number to give (5)α (MZ ) = 0.106(4)MS hWe used r0 = 0.472 fm to convert to r0MS ∣ i Reference scale w0.4 where wx is defined by t∂ [t2t 〈E(t)〉]∣t= 2 = x in terms of the action density E(t) at positive flow time t [793]. Ourw conversion to r0 scale using [793] r0/w0.4 = 2.587(45) and r0 = x 0.472 fm j Our conversion fromw0MS = 0.2154(12) to r0 scale using r0/w0 = (r0/w0.4)·(w0.4/w0) = 2.885(50)with the factors cited by the collaboration [793] and with r0 = 0.472 fm increasingly perturbative for decreasing n. Denoting the lowest-order perturbation theory moments by G(0)n , one defines the normalized moments ⎧ ⎪⎨ G G(0)4/ for n = 4, R = ( ) 4 n ⎪ 1/(n−4) (383)⎩ amηc Gn 2am (0) for n ≥ 6,c Gn 123 Publication status Renormalization scale Perturbative behaviour Lattice spacings Eur. Phys. J. C (2022) 82:869 Page 171 of 296 869 of even order n. Note that Eq. (380) contains the variable (bare) heavy-quark mass mc. The normalization G (0) n is introduced to help in reducing lattice artifacts. In addition, one can also define moments with different normalizations, R̃n = 2Rn/mηc for n ≥ 6. (384) While R̃n also remains renormalization-group invariant, it now also has a scale which might introduce an additional ambiguity [25]. The normalized moments can then be parameterized in terms of functions { ≡ r4(αs(μ)) for n = 4,Rn m (385)ηc 2m̄ rn(αs(μ)) for n ≥ 6,c(μm ) with m̄c(μm) being the renormalized heavy-quark mass. The scale μm at which the heavy-quark mass is defined could be different from the scale μ at which αs is defined [802]. The HPQCD collaboration, however, used the choice μ = μm = 3mc(μ). This ensures that the renormalization scale is never too small. The reduced moments rn have a perturbative expansion rn = 1 + r 2 3n,1αs + rn,2αs + rn,3αs + . . . , (386) where the written terms rn,i (μ/m̄c(μ)), i ≤ 3 are known for low n from Refs. [803–807]. In practice, the expansion is performed in the MS scheme. Matching nonperturbative lattice results for the moments to the perturbative expansion, one determines an approximation to αMS(μ) as well as m̄c(μ). With the lattice spacing (scale) determined from some extra physical input, this calibrates μ. As usual suitable pseudoscalar masses determine the bare-quark masses, here in particular the charm mass, and then through Eq. (385) the renormalized charm-quark mass. A difficulty with this approach is that large masses are needed to enter the perturbative domain. Lattice artifacts can then be sizeable and have a complicated form. The ratios in Eq. (383) use the tree-level lattice results in the usual way for normalization. This results in unity as the leading term in Eq. (386), suppressing some of the kinematical lattice artifacts. We note that in contrast to, e.g., the definition of αqq, here the cutoff effects are of order akαs , while there the tree-level term defines αs and therefore the cutoff effects after tree-level improvement are of order akα2s . To obtain the continuum results for the moments it is important to perform fits with high powers of a. This implies many fit parameters. To deal with this problem the HPQCD collaboration used Bayesian fits of their lattice results. More recent analyses of the moments, however, did not rely on Bayesian fits [25,26,189,696]. Finite-size effects (FSE) due to the omission of |x0| > T/2 in Eq. (382) grow with n as (m nηc T/2) exp (−mηc T/2). In practice, however, since the (lower) moments are short-distance dominated, the FSE are expected to be small at the present level of precision. Possible exception could be the ratio R8/R10, where the finite-volume effects could be significant as discussed below. Moments of correlation functions of the quark’s electromagnetic current can also be obtained from experimental data for e+e− annihilation [808,809]. This enables a nonlattice determination of αs using a similar analysis method. In particular, the same continuum perturbation-theory computation enters both the lattice and the phenomenological determinations. 9.8.2 Discussion of computations The determination of the strong-coupling constant from the moments of quarkonium correlators by HPQCD collaboration have been discussed in detail in the FLAG 2016 and 2019 reports. Therefore, we only give the summary of these determinations in Table 64. Two additional computations have appeared between the FLAG 16 and the FLAG 19 reports. We re-discuss them here (see also the summary section), as the assessment in FLAG 19 was partially based on an inconsistent use of the FLAG criteria and has now been changed. Maezawa and Petreczky, [189] computed the two-point functions of the cc̄ pseudoscalar operator and obtained R4, R6/R8 and R8/R10 based on the HotQCD collaboration HISQ staggered ensembles, [116]. The scale is set by measuring r1 = 0.3106(18) fm. Continuum limits are taken fitting the lattice-spacing dependence with a2 +a4 form as the best fit. For R , they also employ other forms for fit functions such as a2, αboosted 24 s a + a4, etc., the results agreeing within errors. Matching R4 with the 3-loop formula Eq. (386) through order α3 [803], where μ is fixed to mMS c, they obtain (3)α (μ = mc) = 0.3697(54)(64)(15). The first error is statistical, the second is the uncertainty in the continuumMS 123 869 Page 172 of 296 Eur. Phys. J. C (2022) 82:869 Table 64 Heavy-quark current two-point function results. Note that all analysis using 2 + 1 flavour simulations perturbatively add a dynamical charm quark. Partially they then quote results in N f = 4-flavour QCD, which we converted back to N f = 3, corresponding to the nonperturbative sea quark content Collaboration Refs. N f Scale MS [MeV] r0MS HPQCD 14A [14] 2+1+1 A ◦  ◦ w0 = 0.1715(9) fma 294(11)bc 0.703(26) Petreczky 20 [696] 2+1 P ◦ ◦  r1 = 0.3106(18) fm 332(17)h 0.792(41)g Boito 20 [698] 2+1 A   ◦ m hc(mc) = 1.28(2) GeV 328(30) 0.785(72) Petrezcky 19, mh=mc [26] 2+1 A    r = 0.3106(18) fmg 314(10) 0.751(24)g Petrezcky 19, mhm =1.5 [26] 2+1 A  c ◦ 1 r g g1 = 0.3106(18) fm 310(10) 0.742(24) Maezawa 16 [189] 2+1 A   ◦ r√1 = 0.3106(18) fmd 309(10)e 0.739(24)e JLQCD 16 [25] 2+1 A  ◦ ◦ t0 = 0.1465(25) fm 331(38) f 0.792(89) f HPQCD 10 [11] 2+1 A ◦  ◦ r † 1 = 0.3133(23) fm 338(10) 0.809(25) HPQCD 08B [202] 2+1 A    r1 = 0.321(5) fm† 325(18)+ 0.777(42) aScale determined in [37] using fπ b (4)α (5 GeV (5)) = 0.2128(25), α (MZ ) = 0.11822(74)MS MS cWe evaluated (4) from (4)α . We also used r0 = 0.472 fmMS MS d Scale is determined from fπ e (3) (5)α (mc = 1.267 GeV) = 0.3697(85), α (MZ ) = 0.11622(84). Our conversion with r = 0.472 fmMS MS 0 f We evaluated (3) from the given (4)α (3 GeV) = 0.2528(127). (5)α (MZ ) = 0.1177(26). We also used r0 = 0.472 fm to convertMS MS MS gWe used r0 = 0.472 fm to convert hWe back-engineered from (5)α (MZ ) = 0.1177(20). We used r0 = 0.472 fm to convertMS  (3)α (5 GeV) = 0.2034(21 , (5)) α (M ) = 0.1183(7) MS MS Z †Scale is determined from ϒ mass splitting +We evaluated (3) from the given (4)α (3 GeV (5)) = 0.251(6). α (MZ ) = 0.1174(12)MS MS MS extrapolation, and the third is the truncation error in the perturbative approximation of r4. This last error is estimated by the “typical size” of the missing 4-loop contribution, which they assume to be α4 (μ) multiplied by 2 times the 3-loop coefficient MS 2 × r 44,3 × α (μ) = 0.2364 × α4 (μ). The result is converted toMS MS (5) α (MZ ) = 0.11622(84). (387)MS Since αeff = 0.38 we assign  for the criterion of the renormalization scale. As / < α2eff , we assign  for the criterion of perturbative behaviour. The lattice cutoff ranges as a−1 = 1.42–4.89 GeV with μ = 2mc ∼ 2.6 GeV so that we assign ◦ for continuum extrapolation. JLQCD 16 [25] also computed the two-point functions of the cc̄ pseudoscalar operator and obtained R6, R8, R10 and their ratios bas√ed on 2+1-flavour QCD with Möbius domain-wall quark for three lattice cutoff a−1 = 2.5, 3.6, 4.5 GeV. The scale is set by t0 = 0.1465(21)(13) fm. The continuum limit is taken assuming linear dependence on a2. They find a sizeable lattice-spacing dependence of R4, which is therefore not used in their analysis, but for R6, R8, R10 the dependence is mild giving reasonable control over the continuum limit. They use the perturbative formulae for the vacuum polarization in the pseudoscalar channel #P S through order α3 in the MS scheme [805,806] to obtain (4) α . Combining the matching of lattice MS MS results with continuum perturbation theory for R6, R (4) 6/R8 and R10, they obtain α (μ = 3 GeV) = 0.2528(127), where theMS error is dominated by the perturbative truncation error. To estimate the truncation error they study the dependence of the final result on the choice of the renormalization scales μ, μm which are used as renormalization scales for αs and the quark mass. Independently [802] the two scales are varied in the range of 2 GeV to 4 GeV. The above result is converted to (5)α (M ) as MS Z 123 Publication status Renormalization scale Perturbative behaviour Continuum extrapolation Eur. Phys. J. C (2022) 82:869 Page 173 of 296 869 (5) α (MZ ) = 0.1177(26). (388)MS Since αeff 0.37, they have  for the renormalization scale criterion. Since / α2eff , we also assign ◦ for the criterion of perturbative behaviour. The lattice cutoff ranges over a−1 = 2.5–4.5 GeV with μ = 3 GeV so we also give them a ◦ for continuum extrapolation. We note, however, that the χ2/dof of the a2 extrapolation was quite bad, namely between 2.1 and 5.1 [25]. Please note that the 2019 FLAG review mistakenly took αMS(2mc) for αeff . This resulted in a ◦ rating for the renormalization scale for both Maezawa 16 and JLQCD 16. With the consistent definition of αeff both determinations now have  for the renormalization scale. Three new determinations of αs from the moments of quarkonium correlators appeared since the 2019 FLAG review [26,696,698]. Petreczky 19 [26] extended the calculation of [189] by considering heavy-quark masses larger than the charm- quark mass, namely, mh = 1.5mc, 2mc and 3mc. Also three additional lattice spacings, a = 0.025, 0.03 and 0.035 fm have been added to the analysis. Another improvement compared to Maezawa 16 was the use of random-colour wall sources which greatly reduced the statistical errors. In fact, the statistical errors on the moments were completely negligible compared to other sources of errors. The lattices corresponding to the three smallest lattice spacings have been generated for the calculations of the QCD equation of state at high temperature [769] at light sea-quark masses corresponding to the pion mass of 300 MeV in the continuum limit, instead of the pion mass of 160 MeV as in the previous calculations. However, it has been checked that the effect of the larger light sea-quark masses is very small, about the size of the statistical errors [26]. Therefore, the calculations at the two light sea-quark masses have been combined into a single analysis [26]. For each value of the heavy-quark mass the continuum extrapolations have been performed using various fit ansätze, some of which included high powers of a. Due to availability of many lattice spacing it was possible to perform such fits without using Bayesian priors. The variation of the continuum-extrapolated values with the variation of the fit range in a2 and the fit forms has been investigated and included as the systematic error of the continuum results. The renormalization scale μ was fixed to the heavy-quark N =3 mass, and αs(μ = mh) and the corresponding f has been determined for each value of mh using continuum resultsM S for R4, R6/R8 and R8/R10. The perturbative error was estimated as in Maezawa 16 but with the coefficient of the 4-loop N =3 term being 1.6 times the coefficient of the 3-loop term. The values of f obtained for m = m M S h c and mh = 1.5mc were N =3 N =3 consistent with each other, f = 314(10) MeV for mh = mc and f = 310(10) MeV for mh = 1.5mc. However, N = M S M S3 the f values turned out to be significantly lower for mh = 2mc and 3mc. In Petreczky 20 [696], it has been arguedM S that reliable continuum extrapolations of R4, R6/R8 and R8/R10 are not possible for mh ≥ 2mc. Therefore, we only review the results obtained for mh = mc and mh = 1.5mc. There are many lattice spacings available for analysis, including three lattice spacings a ≤ 0.035 fm, implying that aμ < 0.5. Therefore, we assign for the continuum extrapolation. The value of αeff is 0.38 and 0.31 for mh = mc and mh = 1.5mc, respectively. So we assign  for the renormalization scale. Since (/) 2α < αeff we assign  for the perturbative behaviour. Petreczky 20 [696] used the same raw lattice data as Petreczky 19 but a different strategy for continuum extrapolation and αs extraction. The lattice spacing dependence of the results of R4 at different quark masses was fitted simultaneously in a similar manner as in the HPQCD 10 and HPQCD 14 analyses, but without using Bayesian priors. In extracting αs several choices of the renormalization scale μ in the range 2/3mh–3mh have been considered. The perturbative error was estimated as in Petreczky 19 but the variation of the results due to the scale variation was larger than the estimated perturbative error. N =3 The final error of the result f = 331(17) MeV comes mostly from the scale variation [696]. Since there are three M S lattice spacing available with aμ < 0.5 we give  for continuum extrapolation. Because αeff = 0.22 − 0.38 we give ◦ for the renormalization scale. Finally, since (/) 2α > αeff for the smallest αeff value we give ◦ for the perturbative behaviour. In addition to R4 Petreczky 20 also considered using R6/R8 and R8/R10 for the αs determination. It was pointed out that the lattice spacing dependence of R6/R8 is quite subtle and therefore reliable continuum extrapolations for this ratio are not possible for mh ≥ 2mc [696]. For mh = mc and 1.5mc the ratio R6/R8 leads to αs values that are consistent with the ones from R4. Furthermore, it was argued that finite-volume effects in the case of R8/R10 are large for mh = mc and therefore the corresponding data are not suitable for extracting αs . This observation may explain why the central values of αs extracted from R8/R10 in some previous studies were systematically lower [26,189,202]. On the other hand for mh ≥ 1.5mc the finite-volume effects are sufficiently small in the continuum extrapolated results if some small-volume lattice data are excluded from the analysis [696]. The αs obtained from R8/R10 with mh ≥ 1.5mc were consistent with the ones obtained from R4. Boito 20 [698] use published continuum extrapolated lattice results on R4, R6/R8 and R8/R10 from various groups combined with experimental results on e+e− annihilation. They quote a separate result for each lattice determinations of R4, 123 869 Page 174 of 296 Eur. Phys. J. C (2022) 82:869 Table 65 Moments and the HPQCD 08 HPQCD 10 Maezawa 16 JLQCD 16 Petreczky 19 Petreczky 20 ratios of the moments from N f = 3 simulations at the R4 1.272(5) 1.282(4) 1.265(7) – 1.279(4) 1.278(2) charm mass R6 1.528(11) 1.527(4) 1.520(4) 1.509(7) 1.521(3) 1.522(2) R8 1.370(10) 1.373(3) 1.367(8) 1.359(4) 1.369(3) 1.368(3) R10 1.304(9) 1.304(2) 1.302(8) 1.297(4) 1.311(7) 1.301(3) R6/R8 1.113(2) – 1.114(2) 1.111(2) 1.1092(6) 1.10895(32) R8/R10 1.049(2) – 1.0495(7) 1.0481(9) 1.0485(8) – R6/R8 and R8/R10 for mh = mc from different lattice groups. They vary the scale μ and μm independently in the region between mc and 4 GeV. As the typical value they quote αs(MZ ) = 0.1177(20). The error is dominated by the perturbative uncertainty. Since the effective coupling is around 0.38 we give  for the renormalization scale. Because (/) < α2 we give this determination  for perturbative behaviour. The continuum results used in the analysis were rated as ◦ ◦ α eff with the exception of HPQCD 08B, which however, does not affect the quoted αs value. Therefore we give them for the continuum extrapolation. An interesting point of the Boito 20 analysis is that the αs values extracted from R8/R10 are systematically lower than the ones extracted from R4. This confirms the above assertion that finite volume effects are significant for R8/R10 at mh = mc. Aside from the final results forαs(m Z ) obtained by matching with perturbation theory, it is interesting to make a comparison of the short distance quantities in the continuum limit Rn which are available from HPQCD 08 [202], JLQCD 16 [25], Maezawa 16 [189], Petreczky 19 [26] and Petreczky 20 [696] (all using 2 + 1 flavours). This comparison is shown in Table 65. The results are in quite good agreement with each other. For future studies it is of course interesting to check agreement of these numbers before turning to the more involved determination of αs . 9.9 αs from QCD vertices 9.9.1 General considerations The most intuitive and in principle direct way to determine the coupling constant in QCD is to compute the appropriate three- or four-point gluon vertices or alternatively the quark-quark-gluon vertex or ghost-ghost-gluon vertex (i.e., qq A or ccA vertex, respectively). A suitable combination of renormalization constants then leads to the relation between the bare (lattice) and renormalized coupling constant. This procedure requires the implementation of a nonperturbative renormalization condition and the fixing of the gauge. For the study of nonperturbative gauge fixing and the associated Gribov ambiguity, we refer to Refs. [810–812] and references therein. In practice the Landau gauge is used and the renormalization constants are defined by requiring that the vertex is equal to the tree-level value at a certain momentum configuration. The resulting renormalization schemes are called ‘MOM’ scheme (symmetric momentum configuration) or ‘M˜OM’ (one momentum vanishes), which are then converted perturbatively to the MS scheme. A pioneering work to determine the three-gluon vertex in the N f = 0 theory is Alles 96 [813] (which was followed by Ref. [814] for two flavour QCD); a more recent N f = 0 computation was Ref. [815] in which the three-gluon vertex as well as the ghost-ghost-gluon vertex was considered. (This requires a computation of the propagator of the Faddeev–Popov ghost on the lattice.) The latter paper concluded that the resulting MS depended strongly on the scheme used, the order of perturbation theory used in the matching and also on nonperturbative corrections [816]. Subsequently in Refs. [817,818] a specific M˜OM scheme with zero ghost momentum for the ghost-ghost-gluon vertex was used. In this scheme, dubbed the ‘MM’ (Minimal MOM) or ‘Taylor’ (T) scheme, the vertex is not renormalized, and so the renormalized coupling reduces to 2 = Dgluon ghostα (μ) (μ, a)D (μ, a)2 g0T lat lat , (389)4π where Dghostlat and D gluon lat are the (bare lattice) dressed ghost and gluon ‘form factors’ of these propagator functions in the Landau gauge, 123 Eur. Phys. J. C (2022) 82:869 Page 175 of 296 869 ghost ( ) gluon Dab(p) = −δab D (p) pμ pν D (p)2 , Dabμν(p) = δab δμν −p p2 p2 , (390) and we have written the formula in the continuum with Dghost/gluon p = Dghost/gluon( ) lat (p, 0). Thus there is now no need to compute the ghost-ghost-gluon vertex, just the ghost and gluon propagators. 9.9.2 Discussion of computations For the calculations considered here, to match to perturbative scaling, it was first necessary to reduce lattice artifacts by an H(4) extrapolation procedure (addressing O(4) rotational invariance), e.g., ETM 10F [824] or by lattice perturbation theory, e.g., Sternbeck 12 [822]. To match to perturbation theory, collaborations vary in their approach. In ETM 10F [824], it was necessary to include the operator A2 in the OPE of the ghost and gluon propagators, while in Sternbeck 12 [822] very large momenta are used and a2 p2 and a4 p4 terms are included in their fit to the momentum dependence. A further later refinement was the introduction of higher nonperturbative OPE power corrections in ETM 11D [821] and ETM 12C [820]. Although the expected leading power correction, 1/p4, was tried, ETM finds good agreement with their data only when they fit with the next-to-leading-order term, 1/p6. The update ETM 13D [819] investigates this point in more detail, using better data with reduced statistical errors. They find that after again including the 1/p6 term they can describe their data over a large momentum range from about 1.75 GeV to 7 GeV. In all calculations except for Sternbeck 10 [823], Sternbeck 12 [822], the matching with the perturbative formula is performed including power corrections in the form of condensates, in particular 〈A2〉. Three lattice spacings are present in almost all calculations with N f = 0, 2, but the scales ap are rather large. This mostly results in a  on the continuum extrapolation (Sternbeck 10 [823], Boucaud 01B [814] for N f = 2. Ilgenfritz 10 [825], Boucaud 08 [818], Boucaud 05 [815], Becirevic 99B [830], Becirevic 99A [831], Boucaud 98B [832], Boucaud 98A [833], Alles 96 [813] for N f = 0). A ◦ is reached in the N f = 0 computations Boucaud 00A [829], 00B [828], 01A [827], Soto 01 [826] due to a rather small lattice spacing, but this is done on a lattice of a small physical size. The N f = 2 + 1 + 1 calculation, fitting with condensates, is carried out for two lattice spacings and with ap > 1.5, giving  for the continuum extrapolation as well. In ETM 10F [824] we have 0.25 < αeff < 0.4, while in ETM 11D [821], ETM 12C [820] (and ETM 13 [80]) we find 0.24 < αeff < 0.38, which gives a ◦ in these cases for the renormalization scale. In ETM 10F [824] the values of ap violate our criterion for a continuum limit only slightly, and we give a ◦. In Sternbeck 10 [823], the coupling ranges over 0.07 ≤ αeff ≤ 0.32 for N◦ f = 0 and 0.19 ≤ αeff ≤ 0.38 for N f = 2 giving  and for the renormalization scale, respectively. The fit with the perturbative formula is carried out without condensates, giving a satisfactory description of the data. In Boucaud 01A [827], depending on a, a large range of αeff is used which goes down to 0.2 giving a ◦ for the renormalization scale and perturbative behaviour, and several lattice spacings are used leading to◦ in the continuum extrapolation. The N f = 2 computation Boucaud 01B [827], fails the continuum limit criterion because both aμ is too large and an unimproved Wilson fermion action is used. Finally in the conference proceedings Sternbeck 12 [822], the N f = 0, 2, 3 coupling αT is studied. Subtracting 1-loop lattice artifacts and subsequently fitting with a2 p2 and a4 p4 additional lattice artifacts, agreement with the perturbative running is found for large momenta (r20 p 2 > 600) without the need for power corrections. In these comparisons, the values of r0MS from other collaborations are used. As no numbers are given, we have not introduced ratings for this study. Since the previous FLAG review, there has been one new result, Zafeiropoulos 19 [699], again based on the method described in ETM 10F, [824] but now for N f = 3 flavours rather than two. Again an 〈A2〉 condensate is included, but cannot be determined; an estimate is used from ETM 10F (N f = 2) and ETM12C (N f = 4). The scale  is determined from the largest momenta available (when a plateau appears), and the error is estimated from the larger range p ∼ 3.0–3.7 GeV. This is used to determine αM S . In this work there is also some emphasis on being close to the physical-quark masses, using three domain-wall fermion data sets and careful consideration of discretization effects following [834]. The disadvantage is that a lower upper bound on the momenta is now reached. The range of effective couplings is 0.35<∼ α 1), the errors of the input quantities are scaled by χ /dof. In the following step, correlations among different calculations are taken into account in the error estimate by applying Schmelling’s procedure [163]. 10.3 Isovector charges The axial, scalar and tensor isovector charges are needed to interpret the results of many experiments and phenomena mediated by weak interactions, including probes of new physics. The most natural process from which isovector charges can be measured is neutron beta decay (n → p+e−νe). At the quark level, this process occurs when a down quark in a neutron transforms into an up quark due to weak interactions, in particular due to the axial current interaction. While scalar and tensor currents have not been observed in nature, effective scalar and tensor interactions arise in the SM due to loop effects. At the TeV and higher scales, contributions to these three currents could arise due to new interactions and/or loop effects in BSM theories. These super-weak corrections to standard weak decays can be probed through high precision measurements of the neutron decay distribution by examining deviations from SM predictions as described in Ref. [957]. The lattice-QCD methodology for the calculation of isovector charges is well-established, and the control over statistical and systematic uncertainties is becoming robust. The axial charge gu−dA is an important parameter that encapsulates the strength of weak interactions of nucleons. It enters in many analyses of nucleon structure and of SM and BSM physics. For example, it enters in (i) the extraction of Vud and tests of the unitarity of the Cabibbo–Kobayashi–Maskawa (CKM) matrix; (ii) the analysis of neutrinoless double-beta decay, (iii) neutrino-nucleus quasi-elastic scattering cross-section; (iv) the rate of proton-proton fusion, the first step in the thermonuclear reaction chains that power low-mass hydrogen-burning stars like the Sun; (v) solar and reactor neutrino fluxes; (vi) muon capture rates, etc. The current best determination of the ratio of the axial to the vector charge, gA/gV , comes from measurement of neutron beta decay using polarized ultracold neutrons by the UCNA collaboration, 1.2772(20) [958,959], and by PERKEO II, 1.2761+14−17 [960]. Note that, in the SM, gV = 1 up to second-order corrections in isospin breaking [961,962] as a result of the conservation of the vector current. Given the accuracy with which gu−dA has been measured in experiments, the goal of lattice-QCD calculations is to calculate it directly with O(1%) accuracy. Isovector scalar or tensor interactions contribute to the helicity-flip parameters, called b and B, in the neutron decay distribution. By combining the calculation of the scalar and tensor charges with the measurements of b and B, one can put constraints on novel scalar and tensor interactions at the TeV scale as described in Ref. [957]. To optimally bound such scalar and tensor interactions using measurements of b and B parameters in planned experiments targeting 10−3 precision [963–965], we need to determine gu−dS and g u−d T at the 10% level as explained in Refs. [881,957]. Future higher-precision measurements of b and B would require correspondingly higher-precision calculations of the matrix elements to place even more stringent bounds on these couplings at the TeV-scale. One can estimate gu−d QCDS using the conserved vector current (CVC) relation, gS/gV = (Mneutron − Mproton) /(md − mu)QCD, as done by Gonzalez-Alonso et al. [966]. In their analysis, they took estimates of the two mass differences on the right-hand side from the global lattice-QCD data [2] and obtained gu−dS = 1.02(8)(7). The tensor charge gu−dT can be extracted experimentally from semi-inclusive deep-inelastic scattering (SIDIS) data [967– 970]. A sample of these phenomenological estimates is shown in Fig. 45, and the noteworthy feature is that the current uncertainty in these phenomenological estimates is large. 10.3.1 Results for gu−dA Calculations of the isovector axial charge have a long history, as can be seen from the compilation given in Table 69 and plotted in Fig. 43. The issue of excited-state contamination received little if any attention before 2010. As a consequence, the range of source-sink separations employed in many of the early calculations prior to that year was rather limited, offering little control over this important systematic effect. This concerns, in particular, the calculations by LHPC 05 [974], LHPC 10 [897], RBC 08 [975], RBC/UKQCD 08B [883], RBC/UKQCD 09B [884] and QCDSF 06 [867]. Since the last edition of the FLAG report, no new results in two-flavour QCD have been published. An exception is the calculation ETM 19 [971], which reanalyzed two ensembles with N f = 2 around the physical pion mass to study finite-volume effects, while the main result is quoted from a calculation with N f = 2 + 1 + 1. These two-flavour calculations still do not qualify for inclusion in the FLAG average (see Table 69). We thus refrain from providing a detailed discussion of the results in Refs. 123 869 Page 196 of 296 Eur. Phys. J. C (2022) 82:869 Table 69 Overview of results for gu−dA Collaboration Refs. N gu−df A CalLat 19 [100] 2+1+1 C ◦    ◦ 1.2642(93) ETM 19 [971] 2+1+1 A  ◦   ◦ 1.286(23) PNDME 18a [98] 2+1+1 A ‡    ◦ 1.218(25)(30) CalLat 18 [99] 2+1+1 A ◦   ◦ ◦ 1.271(10)(7) CalLat 17 [882] 2+1+1 P    ◦ 1.278(21)(26) PNDME 16a [881] 2+1+1 A ◦‡    ◦ 1.195(33)(20) NME 21a [972] 2+1 P ◦‡    ◦ 1.31(6)(5) LHPC 19 [851] 2+1 A ‡    ◦ 1.265(49) Mainz 19 [102] 2+1 A  ◦   ◦ 1.242(25)(+0−0.030) PACS 18A [849] 2+1 A     ◦ 1.273(24)(5)(9) PACS 18 [847] 2+1 A      1.163(75)(14) χQCD 18 [101] 2+1 A ◦    ◦ 1.254(16)(30)$ JLQCD 18 [890] 2+1 A  ◦ ◦  ◦ 1.123(28)(29)(90) LHPC 12Ab [973] 2+1 A ‡    ◦ 0.97(8) LHPC 10 [897] 2+1 A  ◦    1.21(17) RBC/UKQCD 09B [884] 2+1 A   ◦   1.19(6)(4) RBC/UKQCD 08B [883] 2+1 A   ◦   1.20(6)(4) LHPC 05 [974] 2+1 A      1.226(84) Mainz 17 [871] 2 A      1.278(68)(+0 ◦ ◦ ◦ −0.087 ) ETM 17B [875] 2 A   1.212(33)(22) ETM 15D [873] 2 A  ◦ ◦  ◦ 1.242(57) RQCD 14 [869] 2 A ◦     1.280(44)(46) QCDSF 13 [369] 2 A ◦     1.29(5)(3) Mainz 12 [868] 2 A  ◦ ◦   1.233(63)(+0.035−0.060) RBC 08 [975] 2 A      1.23(12) QCDSF 06 [867] 2 A ◦     1.31(9)(7) aThe improvement coefficient in the valence quark action is set to its tadpole-improved tree-level value bThe quark action is tree-level improved ‡The rating takes into account that the action is not fully O(a) improved by requiring an additional lattice spacing $For this partially quenched analysis the criteria are applied to the unitary points [369,867–869,871,873,875,975] and refer the reader to the corresponding chapter in the previous edition of the FLAG report. Estimates for the axial charge with N f = 2+1 have been published by many collaborations, i.e., LHPC [851,897,973,974], RBC/UKQCD [883,884], JLQCD 18 [890], χQCD 18 [101], PACS 18/PACS 18A [847,849], Mainz 19 [102] (superseding the previously listed result in [976]) and NME 21 [972]. The calculations in LHPC 05 [974] and LHPC 10 [897] were based on a mixed-action setup, combining domain-wall fermions in the valence sector with staggered (asqtad) gauge ensembles generated by MILC. Although the dependence of the results on the source-sink separation was studied to some extent in LHPC 10, excited-state effects are not sufficiently controlled according to our quality criteria described in Sect. 10.2. A different discretization of the quark action was used in their later studies (LHPC 12A [973] and LHPC 19 [851]), employing tree-level improved Wilson fermions with smeared gauge links, both in the sea and valence sectors. While this setup does not realize full O(a) improvement, it was found 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Excited states Eur. Phys. J. C (2022) 82:869 Page 197 of 296 869 Fig. 43 Lattice results and FLAG averages for the isovector axial charge gu−dA for N f = 2, 2 + 1 and 2 + 1 + 1 flavour calculations. Also shown is the experimental result as quoted in the PDG [165] that smeared gauge links reduce the leading discretization effects of O(a) substantially. The most recent publication (LHPC 19) is based on two ensembles within 1.5% of the physical pion, at two different values of the lattice spacing. Results for gu−dA were determined using the summation and ratio methods, with and without including the first excitation in the fit. LHPC quotes the result from the finer lattice spacing, with an error that covers the spread of uncertainties on both ensembles. The RBC/UKQCD collaboration has employed N f = 2 + 1 flavours of domain-wall fermions. The results quoted in RBC/UKQCD 08B [883] and RBC/UKQCD 09B [884] were obtained at relatively heavy pion masses at a single value of the lattice spacing, with only limited control over excited-state effects. While systematic investigation of different source-sink separations has been recently performed on two ensembles at the same lattice spacing and pion masses of 250 and 170 MeV, respectively [977], an estimate for gu−dA at the physical point has not been quoted. The JLQCD collaboration (JLQCD 18 [890]) has performed a calculation using N f = 2 + 1 flavours of overlap fermions and the Iwasaki gauge action. Owing to the large numerical cost of overlap fermions, which preserve exact chiral symmetry at nonzero lattice spacing, they have only simulated four light quark masses with 290 < Mπ < 540 MeV and at a single lattice spacing so far. Their simultaneous fit to the data for the correlator ratio RA(t, τ ) computed at six values of τ to a constant, gives a low value for gu−dA at the physical point. Overlap valence quarks were also used by the χQCD collaboration in their study of various nucleon matrix elements (χQCD 18 [101]), utilizing the gauge ensembles generated by RBC/UKQCD with domain-wall fermions. The quoted estimate for the axial charge was obtained from a combination of two-state fits and the summation method, applied over a range of source-sink separations. Calculations with N f = 2 + 1 flavours of O(a) improved Wilson fermions have been performed by PACS, the Mainz group and NME. The calculations by the PACS collaboration (PACS 18 [847] and PACS 18A [849]) were performed on very large volumes (8.2 fm and 10.8 fm, respectively) at or near the physical pion mass. In PACS 18A, the ratio method without including excited states was used to determine the isovector axial charge, which was found to be in good agreement with the experimental value. However, only a single lattice spacing was used in PACS 18 and PACS 18A, so that these calculations lack control over discretization effects. The Mainz group (Mainz 19 [102]) has presented results for the axial charge, obtained by performing two-state fits to six different nucleon matrix elements (including the scalar and tensor charges), assuming that the mass gap to the excited state can be more reliably constrained in this way. Up to six source-sink separations per ensemble have been studied. The final results are obtained from a combined chiral, continuum and finite-volume extrapolation. The NME collaboration (NME 21 [972]) has recently published the results from a calculation of various nucleon form factors and charges. Results were obtained from multi-state fits, using up to four (three) states in the two-point (three-point) correlation functions. In order to describe and control excited-state effects, Nπ and Nππ states with different relative momenta were included in the analysis. The preferred result for gu−dA was obtained from the axial form factor G (Q2A ) extrapolated to Q2 = 0. 123 869 Page 198 of 296 Eur. Phys. J. C (2022) 82:869 Three groups, PNDME, CalLat and ETMC, have published results for N f = 2+1+1, i.e. PNDME 16 [881], PNDME 18 [98], CalLat 17 [882] CalLat 18 [99], CalLat 19 [100]. PNDME and CalLat share the staggered (HISQ) gauge ensem- bles generated by the MILC collaboration, but employ different discretizations in the valence quark sector: PNDME use O(a) improved Wilson fermions with the improvement coefficient csw set to its tadpole-improved tree-level value. By contrast, CalLat use the Möbius variant of domain-wall fermions, which are fully O(a) improved. The CalLat set of ensem- bles includes three values of the lattice spacing, i.e. a = 0.09, 0.12, and 0.15 fm, while PNDME added another set of ensembles at the finer lattice spacing of 0.06 fm to this collection. Both groups have included physical pion mass ensem- bles in their calculations. The operator matrix elements are renormalized nonperturbatively, using the Rome–Southampton method. In order to control excited-state contamination, PNDME perform multi-state fits, including up to four (three) energy levels in the two-point (three-point) correlation functions. By contrast, CalLat have employed the Feynman–Hellmann-inspired implementation of summed operator insertions described in Sect. 10.1.2. Plotting the summed correlator SA(τ ) as a function of the source-sink separation, they find that excited-state effects cannot be detected for τ  1.0 fm at their level of statistics. After subtracting the leading contributions from excited states determined from two-state fits, they argue that the data for SA(τ ) can be described consistently down to τ 0.3 fm. The recent calculation by ETMC (ETM 19 [971]) with N f = 2 + 1 + 1 was performed using a single twisted-mass QCD ensemble with mπ ≈ 139 MeV. In order to control excited-state effects, the summation method and multi-state fits were used. No significant finite-volume effects were expected based on a similar analysis of two N f = 2 ensembles with different spatial extents. The quoted estimate is identified with the result obtained from a two-state fit on the single N f = 2+ 1+ 1 ensemble, which agrees with the value determined from the summation method. We now proceed to discuss global averages for the axial charge, in accordance with the procedures in Sect. 10.2. For QCD with N f = 2 + 1 + 1, the calculations of PNDME and CalLat pass all our quality criteria, while the result of ETM 19 is excluded due to the fact that it was performed at a single value of the lattice spacing. Hence the results from PNDME 18 [98] and CalLat 19 [100], which is an update of CalLat 18 [99], qualify for being included in a global average. Since both PNDME and CalLat use gauge ensembles produced by MILC, we assume that the quoted errors are 100% correlated, even though the range of pion masses and lattice spacings explored in Refs. [98] and [99,100] is not exactly identical. Performing a weighted average yields gu−dA = 1.2617(126) with χ2/dof = 1.33, where the error has been scaled by about 15% because of the large χ2/dof. The result by CalLat dominates the weighted average due to its smaller error. Given that the calculations of PNDME 18 and CalLat 19 are correlated, the large value of χ2/dof indicates a slight tension between the two results. In this situation we adopt a more conservative approach, by requiring that the uncertainty assigned to the FLAG estimate encompasses the central value of PNDME 18. As a result, we choose to represent the axial charge by the interval 1.218 ≤ gu−dA ≤ 1.274, where the lower bound is identified with the result of PNDME 18, while the upper bound is the weighted average plus the scaled 1σ uncertainty. Hence, for N f = 2+1+1 we quote gu−dA = 1.246(28) as the FLAG estimate, where the central value marks the mid-point of the interval, and half the width is taken to be the error. For QCD with N f = 2 + 1 dynamical quarks, the calculations of χQCD 18 [101], Mainz 19 [102] and NME 21 [972] are free of red tags, while the calculation by PACS 18A [849] and LHPC 19 [851] do not offer enough control over lattice artefacts according to the FLAG criteria. Since the result by NME 21 was published only as a preprint by the FLAG deadline, it does not qualify for being included in a global average. Hence, for N f = 2 + 1 we compute a weighted average from χQCD 18 [101] and Mainz 19 [102], assuming no correlations between the two calculations. This yields gu−dA = 1.248(23) with χ2/dof = 0.07. Due to the modified criteria for excited-state contamination, none of the results obtained in two-flavour QCD qualify for a global average. Nonetheless, we find it instructive to show the results for N f = 2 together with the calculations with N f = 2 + 1 and 2 + 1 + 1 and the respective FLAG estimates in Fig. 43. To summarize, the FLAG averages for the axial charge read N f = 2 + 1 + 1 : gu−dA = 1.246(28) Refs. [98–100], (433) N = 2 + 1 : gu−df A = 1.248(23) Refs. [101,102], (434) Within errors, these averages are compatible with the result of gu−dA = 1.2724(23) quoted by the PDG. While the most recent lattice calculations reproduce the axial charge at the level of a few percent or even better, the experimental result is more precise by an order of magnitude. 123 Eur. Phys. J. C (2022) 82:869 Page 199 of 296 869 Table 70 Overview of results for gu−dS Collaboration Refs. N f g u−d S ETM 19 [971] 2+1+1 A  ◦   ◦ 1.35(17) PNDME 18 [98] 2+1+1 A ‡    ◦ 1.022(80)(60) PNDME 16 [881] 2+1+1 A ◦‡    ◦ 0.97(12)(6) PNDME 13 [878] 2+1+1 A ‡    ◦ 0.72(32) NME 21 [972] 2+1 P ◦‡    ◦ 1.06(10)(6) χQCD 21A [978] 2+1 P     ◦ 0.94(10)(6) RBC/UKQCD 19 [977] 2+1 A  ◦    0.9(3) Mainz 19 [102] 2+1 A  ◦   ◦ 1.13(11)(7) LHPC 19 [851] 2+1 A ‡    ◦ ◦ ◦ 6 0.927(303) JLQCD 18 [890] 2+1 A   ◦ 0.88(8)(3)(7) LHPC 12 [979] 2+1 A ‡    ◦ 1.08(28)(16) ETM 17 [877] 2 A  ◦ ◦  ◦ 0.930(252)(48)(204) RQCD 14 [869] 2 A ◦     1.02(18)(30) ‡The rating takes into account that the action is not fully O(a) improved by requiring an additional lattice spacing Fig. 44 Lattice results and FLAG averages for the isovector scalar charge gu−dS for N f = 2, 2+ 1, and 2+ 1+ 1 flavour calculations. Also shown is a phenomenological result obtained using the conserved vector current (CVC) relation [966] (circle) 10.3.2 Results for gu−dS Calculations of the isovector scalar charge have, in general, larger errors than the isovector axial charge as can be seen from the compilation given in Table 70 and plotted in Fig. 44. The isovector scalar charge can also be determined indirectly via the conserved vector current (CVC) relation from results for the neutron-proton mass difference [147,172,221,980–985] and the down and up quark mass difference (see Sect. 3.1.6). For comparison, Fig. 44 also shows an indirect determination obtained using lattice and phenomenological input [966]. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Excited states 869 Page 200 of 296 Eur. Phys. J. C (2022) 82:869 As in FLAG 19, for 2+1+1 flavours, only PNDME 18 [98], which supersedes PNDME 16 [881] and PNDME 13 [878], meets all the criteria for inclusion in the average. The discussions for this and other past calculations are repeated from FLAG 19 for completion. This mixed-action calculation was performed using the MILC HISQ ensembles, with a clover valence action. The 11 ensembles used include three pion mass values, Mπ ∼ 135, 225, 320 MeV, and four lattice spacings, a ∼ 0.06, 0.09, 0.12, 0.15 fm. Note that four lattice spacings are required to meet the green star criteria, as this calculation is not fully O(a) improved. Lattice size ranges between 3.3  Mπ L  5.5. Physical point extrapolations were performed simultaneously, keeping only the leading-order terms in the various expansion parameters. For the finite-volume extrapolation, the asymptotic limit of the χPT prediction, Eq. (432), was used. Excited-state contamination is controlled using two-state fits to between three and five source-sink time separations between 0.72  τ  1.68 fm. Renormalization was performed nonperturbatively using the RI-SMOM scheme and converted to MS at 2 GeV using 2-loop perturbation theory. The calculation performed in ETM 19 [971] was generated using twisted-mass fermions with a clover term. The calculation utilized a single 2+1+1-flavour gauge configuration, with a pion mass near the physical point, mπ ∼ 139 MeV, lattice spacing of a ∼ 0.08 fm, and volume corresponding to mπ L = 3.86. Seven source-sink separations were used in the analysis, ranging from t = 0.64–1.6 fm. Two further two-flavour ensembles were also explored, having the same pion mass, mπ ∼ 130 MeV and lattice spacing a ∼ 0.09 fm, but with different volumes corresponding to mπ L ∼ 3 and mπ L ∼ 4. The final result is quoted from the single 2+1+1 flavour ensemble and does not include an assessment of discretization systematics, and therefore does not meet the continuum quality criterion for inclusion in the average. Regarding 2+1-flavour calculations, a single calculation meets all criteria necessary for inclusion in the average. The Mainz 19 [102] calculation was performed on the Wilson CLS ensembles, using four lattice spacings (a ∼ 0.05 fm to 0.086 fm), several pion masses ranging from ∼ 200 to ∼ 350 MeV, and volumes corresponding to mπ L ∼ 3 to ∼ 5.4. Physical point extrapolations were performed simultaneously in the lattice spacing, pion mass, and volume. Excited states were controlled using two-state simultaneous fits to multiple observables, and included several source-sink separations typically in the range 1-1.5 fm. Renormalization was performed nonperturbatively using the RI-SMOM scheme and converted to MS at 2 GeV using 2-loop perturbation theory. The 2+1-flavour calculation of χQCD 21A [978] was performed using a mixed-action approach with domain-wall fermion gauge configurations generated by the RBC/UKQCD collaboration and overlap valence quarks. They include five pion masses ranging from mπ ∼ 140 to 370 MeV, four lattice spacings (a ∼ 0.06, 0.08, 0.11, and 0.14 fm). Three to six different valence- quark masses are computed on each ensemble. The extrapolation to the physical pion mass, continuum and infinite-volume limits is obtained by a global fit of all data to a partially quenched chiral perturbation theory ansatz. Excited-state contamination is assessed using three to five sink-source separations and multi-state fits. Renormalization is performed using RI/MOM and the final result quoted in MS at 2 GeV. At the time of writing of this review, this calculation was unpublished and the results are therefore not included in the average. The NME 21 [972] 2+1-flavour calculation utilized seven ensembles of Wilson-clover fermions. Three lattice spacings, ranging from a ∼ 0.07 to 0.13 fm, several pion masses, mπ ∼ 165 to 285 MeV, and volumes corresponding to mπ L ∼ 3.75 to 6.15 were used. Combined continuum, chiral, and infinite-volume extrapolations are performed to the physical point using leading-order fit functions. Several fitting strategies are explored using four to six source-sink separations ranging from 0.7–1.8 fm. Final results are quoted by averaging results from two of these fitting strategies, in which the excited-state energy for the three-point function is fixed using two different prior strategies. Renormalization is non-perturbative (RI-SMOM) using two strategies, and quoted in MS at 2 GeV. This work was also unpublished at the time of writing of this review and is not included in the average. The RBC/UKQCD 19 [977] calculation employed 2+1 flavours of domain-wall fermions using an Iwasaki and dislocation- suppressing-determinant-ratio gauge action. They utilized two values of the pion mass, mπ ∼ 250 and 170 MeV with volumes corresponding to mπ L ∼ 5.8 and 4.0, respectively. The results are quoted using only one lattice spacing of 0.14 fm, and a single source-sink separation of 1.3 fm and therefore do not meet the criteria for continuum or excited-state contamination. The LHPC 19 [851] calculation used a 2+1 flavour 2-HEX-smeared Wilson-clover action with two ensembles near the physical pion mass, mπ ∼ 133 and 137 MeV. The lattice spacings corresponded to a ∼ 0.09 and 0.12 fm and volumes mπ L ∼ 4. They used 3 and 8 different time separations for the two ensembles and compare ratio, summation, and multi-state methods to assess excited-state contamination. Because the calculation is not fully O(a) improved, an additional lattice spacing would be necessary to meet the continuum criterion for inclusion in the average. The JLQCD 18 [890] calculation, performed using overlap fermions on the Iwasaki gauge action, covered four pion masses down to 290 MeV. The lattice size was adjusted to keep Mπ L ≥ 4 in all four cases. However, the single lattice spacing of a = 0.11 fm does not meet the criteria for continuum extrapolation. The calculations presented in LHPC 12A used three 123 Eur. Phys. J. C (2022) 82:869 Page 201 of 296 869 Table 71 Overview of results for gu−dT Collaboration Refs. N gu−df T ETM 19 [971] 2+1+1 A  ◦   ◦ 0.936(25) PNDME 18 [98] 2+1+1 A ‡    ◦ 0.989(32)(10) PNDME 16 [881] 2+1+1 A ◦‡    ◦ 0.987(51)(20) PNDME 15 [879,880] 2+1+1 A ◦‡    ◦ 1.020(76) PNDME 13 [878] 2+1+1 A ‡    ◦ 1.047(61) NME 21 [972] 2+1 P ◦‡    ◦ 0.95(5)(2) RBC/UKQCD 19 [977] 2+1 A  ◦    1.04(5) Mainz 19 [102] 2+1 A  ◦   ◦ 0.965(38)(1341) LHPC 19 [851] 2+1 A ‡    ◦ 0.972(41) JLQCD 18 [890] 2+1 A  ◦ ◦  ◦ 1.08(3)(3)(9) LHPC 12 [979] 2+1 A ‡    ◦ 1.038(11)(12) RBC/UKQCD 10D [885] 2+1 A   ◦   0.9(2) ETM 17 [877] 2 A  ◦◦ ◦◦  ◦ 1.004(21)(2)(19) ETM 15D [873] 2 A   ◦ 1.027(62) RQCD 14 [869] 2 A ◦     1.005(17)(29) RBC 08 [975] 2 A      0.93(6) ‡The rating takes into account that the action is not fully O(a) improved by requiring an additional lattice spacing different lattice actions, Wilson-clover, domain-wall, and mixed action. Pion masses ranged down to near the physical pion mass. Data at two lattice spacings were produced with the domain-wall and Wilson actions, however, the final result utilized only the single lattice spacing of a = 0.116 fm from the Wilson action. Because the action is not fully O(a) improved, two lattice spacings are not sufficient for meeting the quality criteria for the continuum extrapolation. The two-flavour calculations in Table 70 include ETM 17, which employed twisted-mass fermions on the Iwasaki gauge action.76 This work utilized a single physical pion mass ensemble with lattice spacing a ∼ 0.09 fm, and therefore does not meet the criteria for continuum extrapolation. The RQCD 14 calculation included three lattice spacings down to 0.06 fm and several pion masses down to near the physical point. While a study of excited-state contamination was performed on some ensembles using multiple source-sink separations, many ensembles included only a single time separation, so it does not meet the criteria for excited states. The final FLAG value for gu−dS is N = 2 + 1 + 1 : gu−df S = 1.02(10) Ref. [98], (435) N u−df = 2 + 1 : gS = 1.13(14) Ref. [102]. (436) 10.3.3 Results for gu−dT Estimates of the isovector tensor charge are currently the most precise of the isovector charges with values that are stable over time, as can be seen from the compilation given in Table 71 and plotted in Fig. 45. This is a consequence of the smaller statistical fluctuations in the raw data and the very mild dependence on a, Mπ , and the lattice size Mπ L . As a result, the 76 The earlier work, ETM 15D [873], did not give a final value for gu−dS and is therefore not included in the tables. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Excited states 869 Page 202 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 45 Lattice results and FLAG averages for the isovector tensor charge gu−dT for N f = 2, 2+1, and 2+1+1 flavour calculations. Also shown are phenomenological results using measures of transversity [986–990] (circles) uncertainty due to the various extrapolations is small. Also shown for comparison in Fig. 45 are phenomenological results using measures of transversity [986–990]. As in FLAG 19, for 2+1+1 flavours, only PNDME 18 [98], which supersedes PNDME 16 [881], PNDME 15 [879] and PNDME 13 [878], meets all the criteria for inclusion in the average. The details for this calculation are the same as those for gu−dS described in the previous section (Sect. 10.3.2), except that three-state fits were used to remove excited-state effects. The details of the 2+1+1 flavour calculation by ETM 19, which does not meet the criteria for averaging, are also the same as those described in the previous section for gu−dS . For 2+1-flavour calculations, only Mainz 19 [102] meets all criteria for inclusion in the averages. Details of this calculation are the same as for gu−dS , described in the previous section. Details for the 2+1-flavour NME 21, RBC/UKQCD 19, LHPC 19, Mainz 18, JLQCD 18, and LHPC 12A, calculations are identical to those presented previously in Sect. 10.3.2. The earlier RBC/UKQCD 10 calculation was performed using domain-wall fermions on the Iwasaki gauge action, with two volumes and several pion masses. The lowest pion mass used was Mπ ∼ 330 MeV and does not meet the criteria for chiral extrapolation. In addition, the single lattice spacing and single source-sink separation do not meet the criteria for continuum extrapolation and excited states. Two-flavour calculations include RQCD 14, with details identical to those described in Sect. 10.3.2. There are two calcu- lations, ETM 15D [873] and ETM 17 [877], which employed twisted-mass fermions on the Iwasaki gauge action. The earlier work utilized three ensembles, with three volumes and two pion masses down to the physical point. The more recent work used only the physical pion mass ensemble. Both works used only a single lattice spacing a ∼ 0.09 fm, and therefore do not meet the criteria for continuum extrapolation. The early work by RBC 08 with domain-wall fermions used three heavy values for the pion mass, and a single value for the lattice spacing, volume, and source-sink separation, and therefore do not meet many of the criteria. The final FLAG value for gu−dT is N f = 2 + 1 + 1 : gu−dT = 0.989(34) Ref. [98], (437) N = 2 + 1 : gu−df T = 0.965(61) Ref. [102]. (438) 10.4 Flavour diagonal charges Three examples of interactions for which matrix elements of flavour-diagonal operators (qq where  defines the Lorentz structure of the bilinear quark operator) are needed are the neutral current interactions of neutrinos, elastic scattering of electrons off nuclei, and the scattering of dark matter off nuclei. In addition, these matrix elements also probe intrinsic properties of nucleons (the spin, the nucleon sigma term and strangeness content, and the contribution of the electric dipole moment 123 Eur. Phys. J. C (2022) 82:869 Page 203 of 296 869 (EDM) of the quarks to the nucleon EDM) as explained below. For brevity, all operators are assumed to be appropriately renormalized as discussed in Sect. 10.1.3. The matrix elements of the scalar operator qq with flavour q give the rate of change in the nucleon mass due to nonzero values of the corresponding quark mass. This relationship is given by the Feynman–Hellmann theorem. The quantities of interest are the nucleon σ -term, σπN , and the strange and charm content of the nucleon, σs and σc, σπN = mud〈N |uu + dd|N 〉, (439) σs = ms〈N |ss|N 〉, (440) σc = mc〈N |cc|N 〉. (441) Here mud is the average of the up and down quark masses and ms (mc) is the strange (charm) quark mass. The σπN ,s,c give the shift in MN due to nonzero light-, strange- and charm-quark masses. The same matrix elements are also needed to quantify the spin independent interaction of dark matter with nucleons. Note that, while σb and σt are also phenomenologically interesting, they are unlikely to be calculated on the lattice due to the expected tiny signal in the matrix elements. In principle, the heavy sigma terms can be estimated using σu,d,s by exploiting the heavy-quark limit [991–993]. The matrix elements of the axial operator qγμγ5q give the contribution q of quarks of flavour q to the spin of the nucleon: 〈N |q qγμγ5q|N 〉 = gAuNγμγ∫ 5 uN , 1 gqA ≡ q = dx(q(x)+q(x)). (442) 0 The charge gqA is thus the contribution of the spin of a quark of flavour q to the spin of the nucleon. It is also related to the first Mellin moment of the polarized parton distribution function (PDF) q as shown in the second line in Eq. (442). Measurements by the European Muon collaboration in 1987 of the spin asymmetry in polarized deep inelastic scattering showed that the sum of the spins of the quarks contributes less than half of the total spin of the proton [994]. To understand this unexpected result, called the “proton spin crisis”, it is common to start with Ji’s sum rule [995], which provides a gauge invariant decomposition of the nucleon’s total spin, as ( ) 1 ∑= 1q + Lq + Jg, (443) 2 = 2q u,d,s,c,· where q/2 ≡ gqA/2 is the contribution of the intrinsic spin of a quark with flavour q; Lq is the orbital angular momentum of that quark; and Jg is the total angular momentum of the gluons. Thus, to obtain the spin of the proton starting from QCD requires calculating the contributions of the three terms: the spin and orbital angular momentum of the quarks, and the angular momentum of the gluons. Lattice-QCD calculations of the various matrix elements needed to extract the three contributions are underway. An alternate decomposition of the spin of the proton has been provided by Jaffe and Manohar [996]. The two formulations differ in the decomposition of the contributions of the quark orbital angular momentum and of the gluons. The contribution of the quark spin, which is the subject of this review and given in Eq. (442), is the same in both formulations. The tensor charges are defined as the matrix elements of the tensor operator qσμνq with σμν = {γμ, γν}/2: gqT uNσμνuN = 〈N |qσμνq|N 〉. (444) These flavour-diagonal tensor charges gu,d,s,cT quantify the contributions of the u, d, s, c quark EDM to the neutron electric dipole moment (nEDM) [879,997]. Since particles can have an EDM only due to P and T (or CP assuming CPT is a good symmetry) violating interactions, the nEDM is a very sensitive probe of new sources of CP violation that arise in most extensions of the SM designed to explain nature at the TeV scale. The current experimental bound on the nEDM is d < 2.9 × 10−26n e cm [998], while the known CP violation in the SM implies dn < 10−31 e cm [999]. A nonzero result over the intervening five orders of magnitude would signal new physics. Planned experiments aim to reduce the bound to around 10−28 e cm. A discovery or reduction in the bound from these experiments will put stringent constraints on many BSM theories, provided the matrix elements of novel CP-violating interactions, of which the quark EDM is one, are calculated with the required precision. 123 869 Page 204 of 296 Eur. Phys. J. C (2022) 82:869 Table 72 Overview of results for gqA Collaboration Refs. N f u d PNDME 20 [1000] 2+1+1 C ‡    ◦ 0.790(23)(30) −0.425(15)(30) ETM 19 [971] 2+1+1 A  ◦   ◦ 0.862(17) −0.424(16) PNDME 18A [103] 2+1+1 A ‡    ◦ 0.777(25)(30)# −0.438(18)(30)# Mainz 19A [1001] 2+1 C  ◦   ◦ 0.84(3)(4) −0.40(3)(4) χQCD 18 [101] 2+1 A ◦    ◦ 0.847(18)(32)$ −0.407(16)(18)$ ETM 17C [876] 2 A  ◦ ◦  ◦ 0.830(26)(4) −0.386(16)(6) s PNDME 20 [1000] 2+1+1 C ‡    ◦ −0.053(7) ETM 19 [971] 2+1+1 A  ◦   ◦ −0.0458(73) PNDME 18A [103] 2+1+1 A ‡    ◦ −0.053(8)# Mainz 19A [1001] 2+1 C  ◦   ◦ −0.044(4)(5) χQCD 18 [101] 2+1 A ◦    ◦ −0.035(6)(7)$ JLQCD 18 [890] 2+1 A  ◦ ◦  ◦ −0.046(26)(9)# χQCD 15 [887] 2+1 A  ◦   ◦ −0.0403(44)(78)# Engelhardt 12 [1002] 2+1 A  ◦   ◦ −0.031(17)# ETM 17C [876] 2 A  ◦ ◦  ◦ −0.042(10)(2) #Assumed that Zn.s. = ZsA A ‡The rating takes into account that the action is not fully O(a) improved by requiring an additional lattice spacing $For this partially quenched analysis the criteria are applied to the unitary points One can also extract these tensor charges from the zeroth moment of the transversity distributions that are measured in many experiments including Drell–Yan and semi-inclusive deep inelastic scattering (SIDIS). Of particular importance is the active program at Jefferson Lab (JLab) to measure them [967,968]. Transversity distributions describe the net transverse polarization of quarks in a transversely polarized nucleon. Their extraction from the data taken over a limited range of Q2 and Bjorken x is, however, not straightforward and requires additional phenomenological modeling. At present, lattice-QCD estimates of gu,d,sT are the most accurate [879,969,970] as can be deduced from Fig. 45. Future experiments will significantly improve the extraction of the transversity distributions. Thus, accurate calculations of the tensor charges using lattice QCD will continue to help elucidate the structure of the nucleon in terms of quarks and gluons and provide a benchmark against which phenomenological estimates utilizing measurements at JLab and other experimental facilities worldwide can be compared. The methodology for the calculation of flavour-diagonal charges is also well-established. The major challenges are the much larger statistical errors in the disconnected contributions for the same computational cost and the need for the additional calculations of the isosinglet renormalization factors. 10.4.1 Results for gu,d,sA A compilation of results for the flavour-diagonal axial charges for the proton is given in Table 72 and plotted in Fig. 46. Results for the neutron can be obtained by interchanging the u and d flavour indices. Only two calculations already discussed in FLAG 19 [4] qualify for global averages: the PNDME 18A [103] for 2+1+1 flavours and the χQCD 18 [101] for 2+1 flavours. The PNDME 18A [103] results were obtained using the 2+1+1 flavour clover-on-HISQ formulation. The connected contributions were obtained on 11 HISQ ensembles generated by the MILC collaboration with a ≈ 0.057, 0.87, 0.12 and 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Excited states Eur. Phys. J. C (2022) 82:869 Page 205 of 296 869 Fig. 46 Lattice results and FLAG averages for gu,d,sA for the N f = 2, 2 + 1, and 2 + 1 + 1 flavour calculations 0.15 fm, Mπ ≈ 135, 220 and 320 MeV, and 3.3 < Mπ L < 5.5. The light disconnected contributions were obtained on six of these ensembles with the lowest pion mass Mπ ≈ 220 MeV, while the strange disconnected contributions were obtained on seven ensembles, i.e., including an additional one at a ≈ 0.087 fm and Mπ ≈ 135 MeV. The excited state and the chiral-continuum fits were done separately for the connected and disconnected contributions, which introduces a systematic that is hypothesised to be small as explained in Ref. [103]. The analysis of the excited-state contamination, discussed in Sect. 10.1.2, was done using three-state fits for the connected contribution and two-state fits for the disconnected contributions. The chiral-continuum extrapolation was done keeping the leading correction terms proportional to M2π and a in both cases, and the leading finite-volume correction in Mπ L was included in the analysis of the connected contributions. Isovector renormalization constants, calculated on the lattice in the RI-SMOM scheme and converted to MS, are used for all three flavour diagonal operators. The PNDME 20 [1000] provided a status update to PNDME 18A [103] and presented results showing that flavour mixing in the calculation of renormalization constants is small, and the isovector renormalization factor is a good approximation for renormalizing flavour diagonal axial charges as discussed in Sect. 10.1.3. It is not considered for the average as it is a conference proceeding. The ETM 19 [971] presented new results for gu,d,s,cA from a single ensemble with 2+1+1-flavour twisted-mass fermions with a clover term at a = 0.0801(4) fm and Mπ = 139.3(7) MeV. These are not considered for the averages as they do not satisfy the criteria for the continuum extrapolation. 123 869 Page 206 of 296 Eur. Phys. J. C (2022) 82:869 The 2+1+1 flavour FLAG values for the axial charges gu,d,sA of the proton are, therefore, the same as the corresponding results given in Table 72 and unchanged from FLAG 19 [4]: N f = 2 + 1 + 1 : guA = 0.777(25)(30) Ref. [103], (445) N f = 2 + 1 + 1 : gdA = −0.438(18)(30) Ref. [103], (446) N f = 2 + 1 + 1 : gsA = −0.053(8) Ref. [103]. (447) There are also new results for gu,d,sA from Mainz 19A [1001] with 2+1-flavour ensembles. While they satisfy all the criteria, they are not included in the averages as [1001] is a conference proceeding. The 2+1 flavour FLAG results from χQCD 18 [101] were obtained using the overlap-on-domain-wall formalism. Three domain-wall ensembles with lattice spacings 0.143, 0.11 and 0.083 fm and sea-quark pion masses Mπ = 171, 337 and 302 MeV, respectively, were analyzed. In addition to the three approximately unitary points, the paper presents data for an additional 4–5 valence quark masses on each ensemble, i.e., partially quenched data. Separate excited-state fits were done for the connected and disconnected contributions. The continuum, chiral and volume extrapolation to the combined unitary and nonunitary data is made including terms proportional to both M2π,valence and M 2 π,sea, and two O(a2) discretization terms for the two different domain-wall actions. With just three unitary points, not all the coefficients are well constrained. The Mπ,sea dependence is omitted and considered as a systematic, and a prior is used for the coefficients of the a2 terms to stabilize the fit. These χQCD 18 2+1 flavour results for the proton, which supersede the χQCD 15 [887] analysis, are N f = 2 + 1 : guA = 0.847(18)(32) Ref. [101], (448) N f = 2 + 1 : gdA = −0.407(16)(18) Ref. [101], (449) N f = 2 + 1 : gsA = −0.035(6)(7) Ref. [101]. (450) The JLQCD 18 [890], ETM 17C [876] and Engelhardt 12 [1002] calculations were not considered for the averages as they did not satisfy the criteria for the continuum extrapolation. All three calculations were done at a single lattice spacing. The JLQCD 18 calculation used overlap fermions and the Iwasaki gauge action. They perform a chiral fit using data at four pion masses in the range 290–540 MeV. Finite volume corrections are assumed to be negligible since each of the two pairs of points on different lattice volumes satisfy Mπ L ≥ 4. The ETM 17C calculation is based on a single twisted-mass ensemble with Mπ = 130 MeV, a = 0.094 and a relatively small Mπ L = 2.98. Engelhardt 12 [1002] calculation was done on three asqtad ensembles with Mπ = 293, 356 and 495 MeV, but all at a single lattice spacing a = 0.124 fm. Results for gsA were also presented recently by LHPC in Ref. [843]. However, this calculation is not included in Table 72 as it has been performed on a single ensemble with a = 0.114 fm and a heavy pion mass value of Mπ ≈ 317 MeV. 10.4.2 Results for gu,d,sS from direct and hybrid calculations of the matrix elements The sigma terms σq = mq〈N |q̄q|N 〉 = mq gqS or the quark mass fractions fTq = σq/MN are normally computed rather than gqS . These combinations have the advantage of being renormalization group invariant in the continuum, and this holds on the lattice for actions with good chiral properties, see Sect. 10.1.3 for a discussion. In order to aid comparison with phenomenological estimates, e.g. from π -N scattering [1003–1005], the light quark sigma terms are usually added to give the πN sigma term, σπN = σu + σd . The direct evaluation of the sigma terms involves the calculation of the corresponding three-point correlation functions for different source-sink separations τ . For σπN there are both connected and disconnected contributions, while for most lattice fermion formulations only disconnected contributions are needed for σs . The techniques typically employed lead to the availability of a wider range of τ for the disconnected contributions compared to the connected ones (both, however, suffer from signal-to-noise problems for large τ , as discussed in Sect. 10.1) and we only comment on the range of τ computed for the latter in the following. Recent results for σπN and for σs from the direct approach are compiled in Table 73. ETM 19 [971] (discussed below) is the only new study included in this table since the last FLAG report. Ref. [1006] is also new, however, it was submitted to the arXiv after the deadline and will be reviewed in the next edition. For completeness, the descriptions of other works are reproduced from FLAG 19. For both σπN and for σs , only the results from χQCD 15A [106] qualify for global averaging. In this mixed-action study, three RBC/UKQCD N f = 2 + 1 domain-wall ensembles are analyzed comprising two lattice spacings, a = 0.08 fm with 123 Eur. Phys. J. C (2022) 82:869 Page 207 of 296 869 Table 73 Overview of results for σπN and σs from the direct approach (above) and σs from the hybrid approach (below) Collaboration Refs. N f σπN [MeV] σs [MeV] ETM 19 [971] 2+1+1 A  ◦  na/na ◦ 41.6(3.8) 45.6(6.2) JLQCD 18 [890] 2+1 A  ◦ ◦ na/na ◦ 26(3)(5)(2) 17(18)(9) χQCD 15A [106] 2+1 A ◦   na/na ◦ 45.9(7.4)(2.8)$ 40.2(11.7)(3.5)$ χQCD 13A [886] 2+1 A   ◦ −/na ◦ − 33.3(6.2)$ JLQCD 12A [889] 2+1 A  ◦ ◦ −/na ◦ − 0.009(15)(16)×m†N Engelhardt 12 [1002] 2+1 A  ◦  −/na ◦ − 0.046(11)×m†N ETM 16A [874] 2 A  ◦ ◦ ◦ na/na ◦ 37.2(2.6)(4.72.9) 41.1(8.2)(7.85.8) RQCD 16 [870] 2 A   na/  35(6) 35(12) MILC 12C [108] 2+1+1 A    −/◦ ◦ − 0.44(8)(5)×m¶§s MILC 12C [108] 2+1 A  ◦  −/◦ ◦ − 0.637(55)(74)×m¶§s MILC 09D [1007] 2+1 A  ◦  −/na ◦ − 59(6)(8)§ The renormalization criteria is given for σπN (first) and σs (second). The label ‘na’ indicates that no renormalization is required $For this partially quenched analysis the criteria are applied to the unitary points †This study computes the strange quark fraction fTs /m N §This study employs a hybrid method, see Ref. [1007] ¶The matrix element 〈N |s̄s|N 〉 at the scale μ = 2 GeV in the MS scheme is computed Mπ,sea = 300 MeV and a = 0.11 fm with Mπ,sea = 330 MeV and 139 MeV. Overlap fermions are employed with a number of nonunitary valence quark masses. The connected three-point functions are measured with three values of τ in the range 0.9–1.4 fm. A combined chiral, continuum and volume extrapolation is performed for all data with Mπ < 350 MeV. The leading order expressions are taken for the lattice-spacing and volume dependence while partially quenched SU (2) HBχPT up to M3π terms models the chiral behaviour for σπN . The strange-quark sigma term has a milder dependence on the pion mass and only the leading-order quadratic terms are included in this case. The lack of other qualifying studies is an indication of the difficulty and computational expense of performing these calcula- tions. Nonetheless, this situation is likely to improve in the future. We note that although the recent analyses, ETM 16A [874], the new study ETM 19 [971] and JLQCD 18 [890], are at a single lattice spacing (a = 0.09 fm, 0.08 fm and 0.11 fm, respectively), they satisfy the criteria for chiral extrapolation, finite volume and excited states. ETM 16A is a single ensemble study with N f = 2 twisted-mass fermions with a pion mass close to the physical point and Mπ L = 3.0. Excited states are investigated utilizing τ = 0.9 fm up to τ = 1.7 fm for the connected three-point functions. In ETM 19 a high statistics analysis was carried out employing a N f = 2 + 1 + 1 physical point ensemble and seven source-sink separations in the range τ = 0.6–1.6 fm, improving the precision they obtain for both σπN and σs compared to their N f = 2 results. JLQCD in JLQCD 18 utilize N f = 2 + 1 overlap fermion ensembles with pion masses reaching down to 293 MeV (Mπ L = 4.0) and apply techniques which give a wide range of τ for the connected contribution, with the final results extracted from τ ≥ 1.2 fm. RQCD (RQCD 16 [870]) investigates the continuum, physical quark mass and infinite-volume limits, where the lattice spacing spans the range 0.06–0.08 fm, the minimum Mπ is 150 MeV and Mπ L is varied between 3.4 to 6.7 at Mπ = 290 MeV. This N f = 2 study has a red tag for the excited state criterion as multiple source-sink separations for the connected three- point functions are only computed on a subset of the ensembles. Clover fermions are employed and the lack of good chiral properties for this action means that there is mixing between quark flavours under renormalization when determining σs and a gluonic term needs to be considered for full O(a) improvement (which has not been included, see Sect. 10.1.3 for a discussion). 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Excited states 869 Page 208 of 296 Eur. Phys. J. C (2022) 82:869 Earlier work focuses only on σs . The analysis of JLQCD 12A [889], is performed on the same set of ensembles as the JLQCD 18 study discussed above and in addition includes smaller volumes for the lightest two pion masses.77 No significant finite-volume effects are observed. Engelhardt 12 [1002] and χQCD 13A [886] have less control over the systematics. The former is a single lattice spacing analysis restricted to small spatial volumes while the latter is a partially quenched study on a single ensemble with unitary Mπ > 300 MeV. MILC has also computed σs using a hybrid method [∫1007] which makes use of the Feynman–Hellmann (FH) theorem and involves evaluating the nucleon matrix element 〈N | d4x s̄s|N 〉.78 This method is applied in MILC 09D [1007] to the N f = 2 + 1 asqtad ensembles with lattice spacings a = 0.06, 0.09, 0.12 fm and values of Mπ ranging down to 224 MeV. A continuum and chiral extrapolation is performed including terms linear in the light-quark mass and quadratic in a. As the coefficient of the discretization term is poorly determined, a Bayesian prior is used, with a width corresponding to a 10% discretization effect between the continuum limit and the∫coarsest lattice spacing.79 A similar updated analysis is presented in MILC 12C [108], with an improved evaluation of 〈N | d4x s̄s|N 〉 on a subset of the N f = 2 + 1 asqtad ensembles. The study is also extended to HISQ N f = 2 + 1 + 1 ensembles comprising four lattice spacings with a = 0.06–0.15 fm and a minimum pion mass of 131 MeV. Results are presented for gsS = 〈N |s̄s|N 〉 (in the MS scheme at 2 GeV) rather than for σs . The scalar matrix element is renormalized for both three and four flavours using the 2-loop factor for the asqtad action [196]. The error incurred by applying the same factor to the HISQ results is expected to be small.80 Both MILC 09D and MILC 12C achieve green tags for all the criteria, see Table 73. As the same set of asqtad ensembles is utilized in both studies we take MILC 12C as superseding MILC 09D for the three-flavour case. The global averaging is discussed in Sect. 10.4.4. 10.4.3 Results for gu,d,sS using the Feynman–Hellmann theorem An alternative approach for accessing the sigma terms is to determine the slope of the nucleon mass as a function of the quark masses, or equivalently, the squared pseudoscalar meson masses. The Feynman–Hellman (FH) theorem gives = ∂MN + ∂MN ≈ 2 ∂MN = ∂MN ≈ 2 ∂MNσπN mu md M ∂m ∂m π ∂M2 , σs ms M , (451) u d π ∂m s̄s 2 s ∂Ms̄s where the fictitious s̄s meson has a mass squared M2 = 2M2s̄s K − M2π . In principle this is a straightforward method as the nucleon mass can be extracted from fits to two-point correlation functions, and a further fit to MN as a function of Mπ (and also MK for σs) provides the slope. Nonetheless, this approach presents its own challenges: a functional form for the chiral behaviour of the nucleon mass is needed, and while baryonic χPT (BχPT) is the natural choice, the convergence properties of the different formulations are not well established. Results are sensitive to the formulation chosen and the order of the expansion employed. If there is an insufficient number of data points when implementing higher order terms, the coefficients are sometimes fixed using additional input, e.g. from analyses of experimental data. This may influence the slope extracted. Simulations with pion masses close to or bracketing the physical point can alleviate these difficulties. In some studies the nucleon mass is used to set the lattice spacing. This naturally forces the fit to reproduce the physical nucleon mass at the physical point and may affect the extracted slope. Note that, if the nucleon mass is fitted as a function of the pion and kaon masses, the dependence of the meson masses on the quark masses also, in principle, needs to be considered in order to extract the sigma terms. An overview of recent determinations of σπN and σs is given in Table 74. BMW 20A [1010] (discussed below) is the only new study since the last FLAG report. For completeness, the descriptions of other works are reproduced from FLAG 19. Note that the renormalization criterion is not included in Table 74 as renormalization is not normally required when computing the sigma terms in the Feynman–Hellmann approach.81 At present, a rating indicating control over excited state contamination is also not considered since a wide range of source-sink separations are available for nucleon two-point functions and ground 77 JLQCD also determine fTs in Ref. [1008] in a single la∫ttice spacing study on small volumes with heavy pion masses. 78 Note that in the direct method the matrix element 〈N | d3x s̄s|N 〉, involving the spatial volume sum, is evaluated for a fixed timeslice. 79 This is consistent with discretization effects observed in other quantities at a = 0.12 fm. 80 At least at 1-loop the Z factors for HISQ and asqtad are very similar, cf. Ref. [1009]. 81 An exception to this is when clover fermions are employed. In this case one must take care of the mixing between quark flavours when renormalizing the quark masses that appear in Eq. (451). 123 Eur. Phys. J. C (2022) 82:869 Page 209 of 296 869 Table 74 Overview of results for σπN and σs from the Feynman–Hellmann approach Collaboration Refs. N f σπN [MeV] σs [MeV] BMW 20A [1010] 1+1+1+1 P ‡   0.0398(32)(44)×m†N 0.0577(46)(33)×m†N ETM 14A [22] 2+1+1 A  ◦ ◦ 64.9(1.5)(13.2) − BMW 15 [105] 2+1 A ‡   38(3)(3) 105(41)(37) Junnarkar 13 [109] 2+1 A ◦ ◦ ◦ − 48(10)(15) Shanahan 12 [1011] 2+1 A  ◦ ◦ 45(6)/51(7) 21(6)/59(6) JLQCD 12A [889] 2+1 A  ◦ ◦ − 0.023(29)(28)×m†N QCDSF 11 [1012] 2+1 A   ◦ 31(3)(4) 71(34)(59) BMW 11A [104] 2+1 A ◦‡  ◦ 39(4)(187 ) 67(27)(5547) Martin Camalich 10 [1013] 2+1 A    59(2)(17) −4(23)(25) PACS-CS 09 [872] 2+1 A    75(15) − Walker-Loud 08 [1014] 2+1 A  ◦  84(17)(20)/42(14)(9) − QCDSF 12 [107] 2 A ◦  ◦ 37(8)(6) − JLQCD 08B [888] 2 A  ◦  53(2)(+21−7 ) − Two results for σπN are quoted arising from different fit ansätze to the nucleon mass. The systematic error is the same as in Ref. [1015] for a combined N f = 2 and N f = 2 + 1 + 1 analysis [1016] ‡The rating takes into account that the action is not fully O(a) improved by requiring an additional lattice spacing Two results are quoted †The quark fractions fTud = fTu + fTd = σπN /m N and/or fTs = σs/m N are computed state dominance is normally achieved. This issue may be revisited in the future as statistical precision improves and this systematic is further investigated. There are several results for σπN that can be included in a global average. For N f = 2, one study meets the selection crite- ria.82 The analysis of QCDSF 12 [107] employs nonperturbatively improved clover fermions over three lattice spacings (a = 0.06–0.08 fm) with pion masses reaching down to around 160 MeV. Finite volume corrected nucleon masses are extrapolated via O(p4) covariant BχPT with three free parameters. The other coefficients are taken from experiment, phenomenology or FLAG, with the corresponding uncertainties accounted for in the fit for those coefficients that are not well known. The nucleon mass is used to set the scale. A novel feature of this study is that a direct determination of σπN at around Mπ = 290 MeV was used as an additional constraint on the slope. Turning to N f = 2 + 1, two studies performed by the BMW collaboration and one by χQCD are relevant. In BMW 11A [104], stout-smeared tree-level clover fermions are employed on 15 ensembles with simulation parameters encompassing a = 0.06–0.12 fm, Mπ ∼ 190–550 MeV and M >π L ∼ 4. Taylor, Padé and covariant SU (3) BχPT fit forms are considered. Due to the use of smeared gauge links, discretization effects are found to be mild even though the fermion action is not fully O(a) improved. Fits are performed including an O(a) or O(a2) term and also without a lattice-spacing dependent term. Finite volume effects were assessed to be small in an earlier work [1018]. The final results are computed considering all combinations of the fit ansatz weighted by the quality of the fit. In BMW 15 [105], a more extensive analysis on 47 ensembles is presented for HEX-smeared clover fermions involving five lattice spacings and pion masses reaching down to 120 MeV. Bracketing the physical point reduces the reliance on a chiral extrapolation. Joint continuum, chiral and infinite-volume extrapolations are carried out for a number of fit parameterisations with the final results determined via the Akaike information criterion procedure [956]. Although only σπN is accessible in the FH approach in the isospin limit, the 82 The ETM collaboration also determine σπN in Ref. [1017] as part of an N f = 2 analysis to determine the lattice spacing from the nucleon mass. However, no final result is given. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume 869 Page 210 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 47 Lattice results and FLAG averages for the nucleon sigma term, σπN , for the N f = 2, 2 + 1, and 2 + 1 + 1 flavour calculations. Determinations via the direct approach are indicated by squares and the Feynman–Hellmann method by triangles. Results from calculations which analyze more than one lattice data set within the Feynman–Hellmann approach [1015,1021–1029] are shown for comparison (pentagons) along with those from recent analyses of π -N scattering [1003–1005,1030] (circles) individual quark fractions fTq = σq/MN for q = u, d for the proton and the neutron are also quoted in BMW 15, using isospin relations.83 Regarding N f = 2+1+1, there is only one recent study. In ETM 14A [22], fits are performed to the nucleon mass utilizing SU (2) χPT for data with Mπ ≥ 213 MeV as part of an analysis to set the lattice spacing. The expansion is considered to O(p3) and O(p4), with two and three of the coefficients as free parameters, respectively. The difference between the two fits is taken as the systematic error. No discernable discretization or finite-volume effects are observed where the lattice spacing is varied over the range a = 0.06–0.09 fm and the spatial volumes cover Mπ L = 3.4 up to Mπ L > 5. The results are unchanged when a near physical point N f = 2 ensemble is added to the analysis in Ref. [1015]. Since FLAG 19, BMW have performed a new N f = 1+1+1+1 study BMW 20A [1010]. A two step analysis is followed: the dependence of the nucleon mass on the pion and kaon masses is determined on HEX-smeared clover ensembles with a = 0.06–0.1 fm and pion masses in the range Mπ = 195–420 MeV. The meson masses as a function of the quark masses are evaluated on stout-staggered ensembles with a similar range in a and quark masses which bracket their physical values. As [1010] is a preprint, the result is not considered for the average. We note that the N f = 2 + 1 study by χQCD [1020] based on overlap valence fermions on four domain-wall fermion ensembles with a = 0.08–0.14 fm and Mπ down to the physical point is also new. However, since σπN is determined from a single fit and the systematic uncertainties are not estimated, we do not present the result in the table. Other determinations of σπN in Table 74 receive one or more red tags. Walker-Loud 08 [1014], JLQCD 08B [888], PACS- CS 09 [872] and QCDSF 11 [1012] are single lattice spacing studies. In addition, the volume for the minimum pion mass is rather small for Walker-Loud 08, JLQCD 08B and PACS-CS 09, while QCDSF 11 is restricted to heavier pion masses. We also consider publications that are based on results for baryon masses found in the literature. As different lattice setups (in terms of N f , lattice actions, etc.) will lead to different systematics, we only include works in Table 74 which utilize a single setup. These correspond to Shanahan 12 [1011] and Martin Camalich 10 [1013], which fit PACS-CS data [193] (the PACS-CS 09 study is also based on these results). Note that Shanahan 12 avoids a red tag for the volume criterion as the lightest pion mass ensemble is omitted. Recent studies which combine data from different setups/collaborations are displayed for comparison in Figs. 47 and 48 in the next section. Several of the above studies have also determined the strange quark sigma term. This quantity is difficult to access via the Feynman–Hellmann method since in most simulations the physical point is approached by varying the light-quark mass, keeping ms approximately constant. While additional ensembles can be generated, it is hard to resolve a small slope with respect to ms . Such problems are illustrated by the large uncertainties in the results from BMW 11A and BMW 15. Alternative approaches have been pursued in QCDSF 11, where the physical point is approached along a trajectory keeping the average 83 These isospin relations were also derived in Ref. [1019]. 123 Eur. Phys. J. C (2022) 82:869 Page 211 of 296 869 Fig. 48 Lattice results and FLAG averages for σs for the N f = 2, 2+1, and 2+1+1 flavour calculations. Determinations via the direct approach are indicated by squares, the Feynman–Hellmann method by triangles and the hybrid approach by circles. Results from calculations which analyze more than one lattice data set within the Feynman–Hellmann approach [1022,1023,1025,1026,1029] are shown for comparison (pentagons) of the light- and strange-quark masses fixed, and JLQCD 12A [889], where quark mass reweighting is applied. The latter is a single lattice spacing study. One can also fit to the whole baryon octet and apply SU (3) flavour symmetry constraints as investigated in, e.g. Martin Camalich 10, Shanahan 12, QCDSF 11 and BMW 11A. The determinations of σs in BMW 11A and BMW 15 qualify for averaging. The mixed action study of Junnarkar 13 [109] with domain-wall valence fermions on MILC N f = 2 + 1 asqtad ensembles also passes the FLAG criteria. The derivative ∂MN/∂ms is determined from simulations above and below the physical strange quark mass for Mπ around 240–675 MeV. The resulting values of σs are extrapolated quadratically in Mπ . The quark fraction fTs = σs/MN exhibits a milder pion-mass dependence and extrapolations of this quantity were also performed using ansätze linear and quadratic in Mπ . A weighted average of all three fits was used to form the final result. Two lattice spacings were analyzed, with a around 0.09 fm and 0.12 fm, however, discretization effects could not be resolved. We note that BMW in their N f = 1 + 1 + 1 + 1 study [1010] significantly improve the precision of their estimate of σs . Even though all the criteria are satisfied, it is not considered for the average as Ref. [1010] is a preprint. The global averaging of all calculations that qualify is discussed in the next section. 10.4.4 Summary of results for gu,d,sS We consider computing global averages of results determined via the direct, hybrid and Feynman–Hellmann (FH) methods. These are unchanged from FLAG 19. Beginning with σπN , Tables 73 and 74 show that for N f = 2+1+1 only ETM 14A (FH) satisfies the selection criteria. We take this value as our FLAG result for the four-flavour case. N f = 2 + 1 + 1 : σπN = 64.9(1.5)(13.2) MeV Ref. [22]. (452) We remark that although the N f = 1 + 1 + 1 + 1 BMW 20A study also satisfies the criteria, as Ref. [1010] is a preprint this work is not considered for averaging. For N f = 2 + 1 we form an average from the BMW 11A (FH), BMW 15 (FH) and χQCD 15A (direct) results, yielding N f = 2 + 1 : σπN = 39.7(3.6) MeV Refs. [104–106]. (453) Note that both BMW results are included as they were obtained on independent sets of ensembles (employing different fermion actions). The average is dominated by the BMW 15 calculation, which has much smaller overall errors compared to the other two studies. 123 869 Page 212 of 296 Eur. Phys. J. C (2022) 82:869 Turning to the results for N f = 2, only QCDSF 12 (FH) qualifies. This is taken as the FLAG result N f = 2 : σπN = 37(8)(6) MeV Ref. [107]. (454) Moving on to σs and the calculations detailed in Table 73, for N f = 2+ 1+ 1 MILC 12C (hybrid) and BMW 20A satisfy the quality criteria, however, the latter is a preprint and is not considered for averaging. In order to convert the result for 〈N |s̄s|N 〉 given in MILC 12C to a value for σs , we multiply by the appropriate FLAG average for ms given in Eq. (35) of FLAG 19. This gives our result for four flavours. N f = 2 + 1 + 1 : σs = 41.0(8.8) MeV Ref. [108]. (455) For N f = 2+ 1 we perform a weighted average of BMW 11A (FH), MILC 12C (hybrid), Junnarkar 13 (FH), BMW 15 (FH) and χQCD 15A (direct). MILC 09D [1007] also passes the FLAG selection rules, however, this calculation is superseded by MILC 12C. As for Eq. (455), the strangeness scalar matrix element determined in the latter study is multiplied by the three flavour FLAG average for ms given in Eq. (33) of FLAG 19. There are correlations between the MILC 12C and Junnarkar 13 results as there is some overlap between the sets of asqtad ensembles used in both cases. To be conservative we take the statistical errors for these two studies to be 100% correlated. The global average is N f = 2 + 1 : σs = 52.9(7.0) MeV Refs. [104–106,108,109]. (456) Given that all of the N f = 2 studies have at least one red tag we are not able to give an average in this case. All the results for σπN and σs are displayed in Figs. 47 and 48 along with the averages given above. Note that where fTud = fTu + fTd or fTs is quoted in Tables 73 and 74, we multiply by the experimental proton mass in order to include the results in the figures. Those results which pass the FLAG criteria, shown in green, are reasonably consistent. However, there is some fluctuation in the central values, in particular, when taking the lattice results as a whole into account, and we caution the reader that the averages may change as new results become available. Also shown for comparison in the figures are determinations from the FH method which utilize more than one lattice data set [1015,1021–1029] as well as results for σπN obtained from recent analyses of π -N scattering [1003–1005,1030]. There is some tension, at the level of three to four standard deviations, between the lattice average for N f = 2 + 1 and Hoferichter et al. [1005] (Hoferichter 15 in Fig. 47), who quote a precision similar to that of the average. Finally we remark that, by exploiting the heavy-quark limit, the light- and strange-quark sigma terms can be used to estimate σq for the charm, bottom and top quarks [991–993]. The resulting estimate for the charm quark, see, e.g. the RQCD 16 N f = 2 analysis of Ref. [870] that reports fTc = 0.075(4) or σc = 70(4) MeV is consistent with the direct determinations of ETM 19 [971] for N f = 2+1+1 of σc = 107(22) MeV, ETM 16A [874] for N f = 2 of σ 12c = 79(21)(8 ) MeV and χQCD 13A [886] for N f = 2 + 1 of σc = 94(31) MeV. BMW in BMW 20A [1010] employing the Feynman–Hellmann approach obtain fTc = σc/m N = 0.0734(45)(55) for N f = 1 + 1 + 1 + 1. MILC in MILC 12C [108] find 〈N |c̄c|N 〉 = 0.056(27) in the MS scheme at a scale of 2 GeV for N f = 2 + 1 + 1 via the hybrid method. Considering the large uncertainty, this is consistent with the other results once multiplied by the charm quark mass. 10.4.5 Results for gu,d,sT A compilation of recent results for the flavour-diagonal tensor charges gu,d,sT for the proton in the MS scheme at 2 GeV is given in Table 75 and plotted in Fig. 49. Results for the neutron can be obtained by interchanging the u and d flavour indices. Only the PNDME 2+1+1 flavour calculations qualify for the global average. The FLAG values remain the same as in FLAG 19, i.e., the PNDME 18B [110] results, which supersede the PNDME 16 [881] and the PNDME 15 [879] results: N f = 2 + 1 + 1 : guT = 0.784(28)(10) Ref. [110], (457) N f = 2 + 1 + 1 : gdT = −0.204(11)(10) Ref. [110], (458) N f = 2 + 1 + 1 : gsT = −0.0027(16) Ref. [110]. (459) 123 Eur. Phys. J. C (2022) 82:869 Page 213 of 296 869 Table 75 Overview of results for gqT Collaboration Refs. N f gu gdT T PNDME 20 [1000] 2+1+1 C ‡    ◦ 0.783(27)(10) −0.205(10)(10) ETM 19 [971] 2+1+1 A  ◦   ◦ 0.729(22) −0.2075(75) PNDME 18B [110] 2+1+1 A ‡    ◦ 0.784(28)(10)# −0.204(11)(10)# PNDME 16 [881] 2+1+1 A ◦‡    ◦ 0.792(42)#& −0.194(14)#& PNDME 15 [879,880] 2+1+1 A ◦‡    ◦ 0.774(66)# −0.233(28)# Mainz 19A [1001] 2+1 C  ◦   ◦ 0.77(4)(6) −0.19(4)(6) JLQCD 18 [890] 2+1 A  ◦ ◦  ◦ 0.85(3)(2)(7) −0.24(2)(0)(2) ETM 17 [877] 2 A  ◦ ◦  ◦ 0.782(16)(2)(13) −0.219(10)(2)(13) gsT PNDME 20 [1000] 2+1+1 C ‡    ◦ −0.0022(12) ETM 19 [971] 2+1+1 A  ◦   ◦ −0.00268(58) PNDME 18B [110] 2+1+1 A ‡    ◦ −0.0027(16)# PNDME 15 [879,880] 2+1+1 A ◦‡    ◦ 0.008(9)# Mainz 19A [1001] 2+1 C  ◦   ◦ −0.0026(73)(42) JLQCD 18 [890] 2+1 A  ◦ ◦  ◦ −0.012(16)(8) ETM 17 [877] 2 A  ◦ ◦  ◦ −0.00319(69)(2)(22) ‡The rating takes into account that the action is not fully O(a) improved by requiring an additional lattice spacing #Assumed that Zn.s. sT = ZT &Disconnected terms omitted The ensembles and the analysis strategy used in PNDME 18B is the same as described in Sect. 10.4.1 for gu,d,sA . The only difference for the tensor charges was that a one-state (constant) fit was used for the disconnected contributions as the data did not show significant excited-state contamination. The isovector renormalization constant, used for all three flavour- diagonal tensor operators, was calculated on the lattice in the RI-SMOM scheme and converted to MS at 2 GeV using 2-loop perturbation theory. The PNDME 20 [1000] provided a status update on gu,d,sT to PNDME 18B [103] but is not considered for the average as it is a conference proceeding. It also presented results showing that flavour mixing in the calculation of tensor renormalization constants is small, and the isovector renormalization factor is a good approximation for renormalizing flavour-diagonal tensor charges as discussed in Sect. 10.1.3. The ETM 19 [971] presented new results for gu,d,s,cT from a single ensemble with 2+1+1-flavour twisted-mass fermions with a clover term at a = 0.0801(4) fm and Mπ = 139.3(7) MeV. It was not considered for the final averages because it did not satisfy the criteria for the continuum extrapolation as already discussed in Sect. 10.4.1. The same applies to the JLQCD 18 [890] and ETM 17 [877] calculations. The Mainz 19A [1001] results with 2+1-flavour ensembles of clover fermions are not included in the averages as Ref. [1001] is a conference proceeding. 123 Publication status Continuum extrapolation Chiral extrapolation Finite volume Renormalization Excited states 869 Page 214 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 49 Lattice results and FLAG averages for gu,d,sT for the N f = 2, 2 + 1, and 2 + 1 + 1 flavour calculations 11 Scale setting Authors: R. Sommer, N. Tantalo, U. Wenger Matching QCD to Nature requires fixing the quark masses and matching an overall scale to experiment. That overall energy scale S may be taken, for example, as the nucleon mass. This process is referred to as scale setting. 11.1 Impact The scale setting procedure, described in some detail below, is a rather technical step necessary to obtain predictions from QCD. What may easily be overlooked is that the exact predictions obtained may depend rather sensitively on the scale. As long as the theory is incomplete, e.g., because we have predictions from N f = 2 + 1 QCD, results will depend on which physics scale is used. Whenever a theory scale (see Sect. 11.5) is used, it matters which value one imposes. Thus, to know whether computations of a particular quantity agree or not, one should check which (value for a) scale was used. The sensitivity of predictions to the scale vary with the observable. For example, the  parameter of the theory has a linear dependence, δ ≈ δSS , (460) 123 Eur. Phys. J. C (2022) 82:869 Page 215 of 296 869 because  has mass dimension one and other hidden dependences on the scale are (usually) suppressed.√Let us preview the results. The present precision on the most popular theory scale, w0 in Eq. (510) is about 0.4% and for t0 it is 0.6%. On the  parameter it is about 3%. Thus, we would think that the scale un√certainty is irrelevant. However, in Sect. 11.8 we will discuss that differences between N f = 2 + 1 and 2+1+1 numbers for t0 are at around 2% which does matter. Also, light-quark masses have an approximatively linear dependence on the scale (roughly speaking one determines, e.g., m = 1 × [m2 ] × [mud Sud S π exp 2 ]lat) and scale uncertainties may play an important rôle in the discussion of agreementmπ vs. disagreement of computations within their error budget. The list of quantities where scale setting is very important may be continued; we just want to mention an observable very much discussed at present, the hadronic vacuum polarisation contribution to the anomalous magnetic moment of the muon [1031]. It is easily seen that the dependence on the scale is about quadratic in that case [1032], δaHVPμ δS aHVP ≈ 2 S . (461)μ This fact means that scale setting has to be precise at the few per-mille precision to have an impact [115] on the discussion whether or not aμ computed in the standard model shows a deviation from experiment. 11.2 Scale setting as part of hadronic renormalization schemes We consider QCD with N f quarks and without a θ -parameter. This theory is completely defined by its coupling constant as well as N f quark masses. After these parameters are specified all other properties of the theory are predictions. Coupling and quark masses depend on a renormalization scale μ as well as on a renormalization scheme. The most popular scheme in the framework of perturbative computations is the MS scheme, but one may also define nonperturbative renormalization schemes, see Sects. 3 and 9. In principle, a lattice computation may, therefore, use these N f + 1 parameters as input together with the renormalization scale μ to fix the bare quark masses and coupling of the discretized Lagrangian, perform continuum and infinite volume limit and obtain desired results, e.g., for decay rates.84 However, there are various reasons why this strategy is inefficient. The most relevant one is that coupling and quark masses cannot be obtained from experiments without invoking perturbation theory and thus necessarily truncation errors. Moreover, these parameters are naturally short distance quantities, since this is where perturbation theory applies. Lattice QCD on the other hand is most effective at long distances, where the lattice spacing plays a minor role. Therefore, it is more natural to proceed differently. Namely, we may fix N f + 1 nonperturbative, long-distance observables to have the values found in Nature. An obvious choice are N f + 1 hadron masses that are stable in the absence of weak interactions. This hadronic renormalization scheme is defined by exp Mi (g0, {am0,j}) { } = Mi exp , i = 2 . . . N f + 1, j = 1 . . . N f . (462)M1(g0, am0,j ) M1 Here, Mi are the chosen hadron masses, g0 is the bare coupling, and am0, j are the bare quark masses in lattice units. The ratio Mi/M1 is, precisely speaking, defined through the hadron masses in lattice units, but in infinite volume. In QCD (without QED), all particles are massive. Therefore, the infinite volume limit of the properties of stable particles is approached with exponentially small corrections which are assumed to be estimated reliably. The power-like finite-volume corrections in QCD + QED are discussed in subsection 11.3. For fixed g0, Eq. (462) needs to be solved for the bare quark masses, am0,j = μ j (g0). (463) The functions μ j define a line in the bare parameter space, called the line of constant physics. Its dependence on the set of masses {Mi } is suppressed. The continuum limit is obtained as g0 → 0 with the lattice spacing shrinking roughly as 2 aM −1/(2b g )1 ∼ e 0 0 . More precisely, consider observables O with mass dimension dO. One defines their dimensionless ratio 84 At first sight this seems like too many inputs, but note that it is the scale μ, at which α(μ) has a particular value, which is the input. The coupling α by itself can have any (small) value as it runs. 123 869 Page 216 of 296 Eur. Phys. J. C (2022) 82:869 Fig. 50 Effective masses for Mproton [1034], M [1035], V (≈ r0), V (≈ r1) [707] and fπ [314] on N f = 2 CLS ensemble N6 with a = 0.045 fm, Mπ = 340 MeV on a 483 96 lattice [314]. All effective “masses” have been scaled such that the errors in the graph reflect directly the errors of the determined scales. They are shifted vertically by arbitrary amounts. Figure from Ref. [714]. Note that this example is at still favourably large quark masses. The situation for Mproton becomes worse closer to the physical point, but may be changed by algorithmic improvements ∣ Ô = O ∣ (aM1) dO ∣ ∣ ∣ , (464)M1 am0, j=μ j (g0) and obtains the continuum prediction as Ocont = ( exp)dM O1 lim→ Ô(aM1) (465)aM1 0 which explains why the determination and use of aM1 is referred to as scale setting. Equation (463) has to be obtained from numerical results. Therefore, it is easiest and most transparent if the i-th mass ratio depends predominantly on the i-th quark mass. Remaining for a while in the isospin-symmetric theory with m0,1 = m0,2 (we enumerate the quark masses in the order up, down, strange, charm, bottom and ignore the top quark), we have natural candidates for the numerators as the pseudoscalar masses in the associated flavour sectors, i.e., π, K , D, B. The desired strong dependence on light- (strange-)quark masses of π - (K -)meson masses derives from their pseudo-Goldstone nature of the approximate SU (3)L×SU (3)R symmetry of the massless QCD Lagrangian which predicts that M2π is roughly proportional to the light-quark mass and M2K to the sum of light- and strange-quark masses. For D and B mesons approximate heavy-quark symmetry predicts MD and MB to be proportional to charm- and bottom-quark masses. Also other heavy-light bound states have this property. There is another important feature which singles out pseudoscalar masses. Because they are the lightest particles with the given flavour quantum numbers, their correlation functions have the least signal/noise problem in the Monte Carlo evaluation of the path integral [866,1033]. Still restricting ourselves to isospin-symmetric QCD (isoQCD), we thus take it for granted that the choice Mi , i ≥ 2 is easy, and we do not need to discuss it in detail: the pseudoscalar meson masses are very good choices, and some variations for heavy quarks may provide further improvements. The choice of M1 is more difficult. From the point of view of physics, a natural choice is the nucleon mass, M1 = Mnucl. Unfortunately it has a rather bad signal/noise problem when quark masses are close to their physical values. The ratio of signal to noise of the correlation function at time x0 from N measurements behaves as [866] ( ( ) ) nucl x0 large √ 3 √ RS/N ∼ N exp − mnucl − mπ x0 ≈ N exp(−x0/0.27 fm), (466)2 where the numerical value of 0.27 fm uses the experimental masses. The behaviour in practice, but at still favourably large quark masses, is illustrated in Fig. 50. 123 Eur. Phys. J. C (2022) 82:869 Page 217 of 296 869 Because this property leads to large statistical errors and it is further difficult to control excited-state contaminations when statistical errors are large, it is useful to search for alternative physics scales. The community has gone this way, and we discuss some of them below. For illustration, here we just give one example: the decay constants of leptonic π or K decays have mass dimension one and can directly replace M1 above. Figure 50 demonstrates their long and precise plateaux as a function of the Euclidean time. Advantages and disadvantages of this choice and others are discussed more systematically in Sect. 11.4. 11.2.1 Theory scales Since the signal/noise problem of physics scales is rather severe, they were already replaced by theory scales in the very first days of lattice QCD. These scales cannot be determined from experiment alone. Rather, their values have to be computed by lattice QCD using a physics scale as input. Creutz already used the string tension in his seminal paper on SU (2) Yang Mills theory [1036], because it is by far easier to determine than glueball masses. A further step was made by the potential scale r0, defined in terms of the static force F(r) as [700] r20 F(r0) = 1.65. (467) Even though r0 can vaguely be related to the phenomenology of charmonium and bottomonium states, its precise definition is in terms of F(r) which can be obtained accurately from Monte Carlo lattice computations with (improvable) control over the uncertainties, but not from experiment. In that sense, it is a prototype of a theory scale. Useful properties of a good theory scale are high statistical accuracy, easy to control systematics (also large volume), quark mass dependence only due to the fermion determinant, and low numerical cost for its evaluation. These properties are realized to varying degrees by the different theory scales covered in this section and, in this respect, they are much preferred compared to physics scales. Consequently, the physics scale M has often been replaced by a theory scale as, e.g., S = r−11 0 in the form ( ) O dcont = Sphys O [ ] lim ÔS(aS) with ÔS(aS) = S−dO O , (468) aS→0 am0, j=μ j (g0) and Sphys = (Mexp)1 lim→ ŜM1(aM1). (469)aM1 0 In this section, we review the determination of numerical results for the values of various theory scales in physical units, Eq. (469). The main difficulty is that a physics scale M1 has to be determined first in order to connect to Nature and, in particular, that the continuum limit of the theory scale in units of the physics scale has to be taken. 11.3 Isospin breaking, electromagnetism, and definition of hadronic schemes 11.3.1 The approximate nature of QCD For simplicity and because it is a very good approximation, we have assumed above that all other interactions except for QCD can be ignored when hadron masses and many other properties of hadrons are considered. This is a natural point of view because QCD is a renormalizable field theory and thus provides unique results. However, we must be aware that while it is true that the predictions (e.g., for hadron masses Mi , i > N f + 1) are unique once Eq. (462) is specified, they will change when we change the inputs Mexpi . These ambiguities are due to the neglected electroweak and gravitational interactions, namely because QCD is only an approximate – even if precise – theory of hadrons. At the sub-percent level, QED effects and isospin violations due to mu = md must be included. At that level one has a very precise description of Nature, where weak decays or weak effects, in general, can be included perturbatively and systematically in an effective field theory description through the weak effective interaction Hamiltonian, while gravity may be ignored. We now discuss how to handle the scale setting as part of the renormalization of QCD+QED. Note that a similar discussion with emphasis on quark masses can be found in Sect. 3. In the following discussion, we focus more on the issues related to the scale setting (see also Ref. [240]). In this connection, triviality of QED does not play a rôle at small enough α: we may think of replacing the continuum limit a → 0 by a limit a → aw with aw nonzero but very far below all QCD+QED scales. 123 869 Page 218 of 296 Eur. Phys. J. C (2022) 82:869 11.3.2 Hadronic renormalization of QCD + QED The definition and implementation of a hadronic renormalization scheme of QCD+QED defined on the lattice needs some additions to Sect. 11.2 which we now discuss. In addition to the N f + 1 parameters of the QCD action (without isospin symmetry), one now also has the elementary electric charge e. This requires N f + 2 experimentally measurable observables to fix the bare parameters of the theory. A natural choice for the experimental inputs are again hadron masses. Indeed, hadron masses are infrared safe quantities also in QCD + QED, while in the cases of cross sections and decay rates, infrared divergences appear at intermediate stages of the calculations (see below). Therefore, we consider the generalization exp Mi (g0, e0, {am0,j}) Mi { } = exp , i = 2 . . . N f + 2, j = 1 . . . N f (470)M1(g0, e0, am0,j ) M1 of Eq. (462). Here, Mi are the chosen hadron masses, g0 the bare strong coupling, e0 the bare electric charge, and am0, j are the bare quark masses in lattice units. For fixed g0, the system of equations (470) now needs to be solved for the bare quark masses and the bare electric charge, am0,j = μ j (g0), e0 = e(g0), (471) to obtain the line of constant physics of the theory. Some observations are in order. So far, we have assumed that QCD+QED is simulated nonperturbatively in the electromagnetic coupling constant αem. In this case, the bare electric charge can be conveniently fixed by considering among the experimental inputs both the charged and neutral pion masses. Indeed, by neglecting terms of O((m 2 2 2u −md) ) [216] one has that m −m ∼ α . If the theory isπ+ π0 em instead treated neglecting O(α2em) contributions, the electric charge does not renormalize and it is consistent and convenient to fix it by the condition [165] 4π 1 2 = Thomson = 137.035999084(21) . (472)e0 αem Another important difference from pure QCD concerns finite-volume effects. In contrast to the exponentially suppressed finite-volume effects of stable hadron masses at αem = 0, in QCD + QED with αem > 0 finite-volume effects are M (g , e , am ; L) α q2ξ(1) α q2i 0 0 0,j = + em i + em i ξ(2) ( − ) ;∞ 1 ;∞ [ ]2 + O L n, α2em , (473)Mi (g0, e0, am0,j ) L Mi (g0, e0, am0,j ) L Mi (g0, e0, am0,j;∞) where qi is the electric charge of the hadron in units of the charge of the positron, the ξ(i) are known numerical constants that depend on the spatial boundary conditions, and the remainder terms start with a power n = 3 in the QEDL formulation[147,148] and with n = 4 in QEDC [177]. These two definitions of QED in a finite volume are discussed in Refs. [170,179,180] and Refs. [177,185,186], respectively. For other formulations [178,183,184], we refer to Sect. 3 and for a discussion on open problems to Ref. [1037]. Since the bare parameters need to be fixed through experimental observables, finite volume effects have to be removed from the Mi and the behaviour Eq. (473) is crucial in this respect. Another important observation concerns the use of observables associated with decay rates or cross sections in setting the scale. The issue is particularly subtle in QCD + QED because of the well known problem associated with the appearance of infrared divergences at intermediate stages of the calculations. The solution requires a proper definition of infrared-safe observables, according to the Bloch–Nordsieck mechanism [1038]. These measurable observables are obtained by including in the final state of a process any number of soft photons with total energy up to a given physical threshold. Once the infrared- safe measurable observable has been constructed, it can be used in the scale setting as any other measurable quantity. A particularly relevant example is the leptonic decay rate of the pion, QCD+QED[π− !→ μν̄μ(γ ), Eγ ]. (474) Here, phase space is integrated over with the constraint that the total energy of the photons is below Eγ . The feasibility of using such an observable in place of a stable hadron mass has to be judged on the basis of the overall precision, statistical plus systematics, that is achievable in the lattice calculation (see Refs. [239,240]). 123 Eur. Phys. J. C (2022) 82:869 Page 219 of 296 869 11.3.3 Hadronic definition of QCD and of QED corrections Under the assumption of negligible weak (and gravity) corrections QCD + QED is the complete theory and, therefore, the predictions obtained from lattice simulations for any observable OQCD+QED, that has not already been used in the scale setting, are unambiguous. On the contrary, what we call the QCD contribution OQCD and the associated radiative corrections, OQCD+QED δOQCD = OQCD − 1, (475) do depend upon the inputs used to define QCD. Going back to Eq. (462), different hadronic definitions of QCD can be obtained by choosing different hadron masses and/or different values for the “physical” inputs. Once we have chosen which hadron masses to use, the different hadronic schemes can be identified by writing M QCDi (g0, {am0,j}) = Mi , i = 2 . . . N f + 1, j = 1 . . . N f , (476) M1(g QCD0, {am0,j}) M1 and by specifying the values of the external inputs, for example parameterized by QCDεi in 85 ( ) MQCD = Mexp 1 + QCDi i εi . (477) A “natural” choice is to set QCDεi = 0, i.e., to define QCD by using exactly the experimental values for the stable hadron masses entering the calibration procedure. In this case, if the same hadron mass is used in the definition of the full theory, Eq. (470), and in the definition of QCD, then the radiative corrections on these quantities are zero by construction. Radiative corrections on any other predictable quantity are well defined and nonvanishing. In light of this observation, the introduction of the QCDεi parameters might appear unnecessary. However, this is not the case for the following reasons. Isosymmetric QCD (isoQCD), already introduced in Sect. 11.2, is another good approximation of the real world. Due to mu = md = mud , the theory only depends on N f parameters. In order to set the masses of the light and strange quarks in isoQCD the options of using the charged or the neutral pion and kaon masses are equally valid from the physical point of view. If one picks, e.g., the neutral meson masses, then one has nonzero εi when the right hand side of Eq. (477) is written in terms of the charged ones. Furthermore, on the basis of symmetry arguments and/or (chiral) effective theory calculations one may argue that certain linear combinations of charged and neutral meson masses are more “natural” than others (see the discussion in the quark-mass section, Sect. 3) because the resulting radiative corrections are smaller. As a matter of fact, many of the existing lattice calculations have been performed in the isospin-symmetric limit, but not all the results considered in this review correspond to the very same definition of QCD. The commonly adopted values for the pion and masses in isoQCD are M isoQCD = Mexp M isoQCD expπ 0 , = M − in Refs. [8,111,193].π For the kaon mass in isoQCD different collaborations made different choices, e.g., the values M isoQCDK = 494.2(4) MeV in Refs. [7,15,111], M isoQCDK = 495.7 MeV in Ref. [8], M isoQCDK = 497.6 MeV in Ref. [193]. The different choices of experimental inputs are perfectly legitimate if QED radiative corrections are neglected, but in principle predictions of isoQCD do depend on these choices, and it is not meaningful to average numbers obtained with different inputs. 85 After having calibrated the full theory (QCD + QED) with physical hadronic inputs, one can compute the strong coupling constant and the quark masses in a given renormalization scheme. These can then be used to define QCD by matching the corresponding renormalized quantities. This is the so-called GRS approach originally introduced in Ref. [235]. We refer to Ref. [240] for a discussion concerning the connection of the ε-language used here and the GRS scheme, and to Sect. 3 for a detailed discussion of the different schemes that have been used in the literature to define (iso)QCD including the original references on the subject. 123 869 Page 220 of 296 Eur. Phys. J. C (2022) 82:869 However, at the present level of precision the sub-percent differences in the inputs are most likely not relevant, and we will average and compare isoQCD results irrespective of these differences. The issue will become important when results become significantly more precise. Of course, it may not be ignored, when radiative corrections, Eq. (475), are directly compared between collaborations. In this case, we strongly suggest to compare results for the unambiguous full theory observable or to stick to a standard. Indeed, for the future, it is highly desirable to define such a standard for the parameters used to define (iso)QCD. We suggest using86 M isoQCD = Mexpπ π0 , M isoQCD = MexpK 0 , (478)K while it is difficult to define a standard scale M1 right now. Going by the majority of the large-scale computations, the two options m and pion leptonic decay rate are equally popular at the moment (see Sect. 11.4). Since leptonic decay rates of pion and kaon play a prominent rôle in scale setting, we discuss the (pure) QCD definition of these quantities and of the associated radiative corrections in some detail. There are no ambiguities in the definition of the physical observable in QCD + QED that, in this case, is the decay rate introduced in Eq. (474) above. We now assume that (iso)QCD has been already defined by using as hadronic inputs hadron masses. It is then possible to compute the leptonic decay rate in QCD, [ ( ) ] 2 exp 2 ( ) QCD[π !→ ] = G F |V |2 Mexp (mexp)2 m 2μν̄μ ud π− μ 1 −8 ( μ ) f QCD π exp 2 π (479) M π− where the so-called decay constant of the pion is given by 0 5 QCD = 〈0| ūγ γ d |π〉 QCD fπ MQCD . (480) π Radiative corrections to f QCDπ are then defined by √ QCD+QED −QCD = [π !→ μν̄μ(γ ), Eγ ]δ fπ (Eγ ) QCD[π !→ μν̄μ] − 1, (481) such that [ ]2 QCD+QED[π− !→ μν̄μ(γ ), E QCDγ ] =  [π !→ μν̄ ] 1 + δ f QCDμ π (Eγ ) . (482) We want to stress once again that the definition of f QCDδ π (Eγ ) is not unique. As in the case of any other observable, different values of δ f QCDπ (Eγ ) are obtained if one changes the prescription used to define QCD. In this case, in addition, one has to specify the photon energy threshold Eγ and, moreover, the exact expression used to defineQCD. Indeed, it would be perfectly legitimate to replace Mexp− appearing in the kinematical factors of Eq. (479) with M QCD π . The effect of such a different definitionπ of QCD would be compensated by a change in f QCDδ (E ) with no ambiguities in the full theory observable QCD+QEDπ γ . We mentioned in Sect. 11.2 that there are advantages from the numerical point of view in using the leptonic decay constants of the mesons in the QCD scale setting procedure. That observation can now be made more precise in light of the discussion of the previous paragraph. When we say that we use fπ to calibrate QCD, we mean that we choose a value for the QCD δ fπ (Eγ ) and we define 86 We note that the π0 is unstable in QCD + QED and, therefore, it is much more convenient to use Mexp+ to calibrate the full theory. Althoughπ it is perfectly consistent to use different observables in the calibration of the full theory and of isoQCD, in the specific case one can write M isoQCD = Mexp 1 + isoQCD with isoQCDπ + ( επ ) επ = Mexp exp exp( 0 − M + )/M + . In that language, the same observable is used in both theories but with aπ π π π nonvanishing ε. 123 Eur. Phys. J. C (2022) 82:869 Page 221 of 296 869 √√ exp − QCD = 1 √  [π !→ μν̄μ([γ ), Eγ ]fπ ] 1 + f QCD √√ ( . (483)δ (E )√ G2 exp)2π γ F |V |2 Mexp ( exp)28π ud − mμ 1 − ( mμ ) π Mexp 2 π− In the notation of the ε parameters introduced above, one has 1 + QCD = 1ε fπ 1 + f QCD . (484)δ π (Eγ ) Again, a possible choice would be to set QCDε f to zero and to use directly the experimentally measured decay rate at a givenπ value of E .87γ Common practice among the different lattice collaborations is to set exp [ ( exp)2 ] E maxγ = Eγ = M π− 1 − m 2 ( μ exp)2 , (485)M π− the maximum energy allowed to a single photon in the case of negligible O(α2em) corrections, and to use the value δ f isoQCDπ (E max γ ) = 0.0088(11) (486) obtained in Refs. [241,520,1039] in chiral perturbation theory and using the standard definition Eq. (478). The corresponding number for kaon decays is f isoQCDδ K (E max γ ) = 0.0053(11). (487) A recent lattice determination in the electro-quenched approximation [240] δ f isoQCDπ (E max γ ) = 0.0076(9), (488) agrees well with Eq. (486), while the number for Kaon decays, f isoQCDδ K (E max γ ) = 0.0012(5), (489) differs by more than three (quadratically combined) error bars from Eq. (487). The scheme dependence can be neglected at the present level of accuracy. 11.4 Physical scales The purpose of this short section is to summarize the most popular scales and give a short discussion of their advantages and disadvantages. We restrict ourselves to those used in more recent computations and thus have a rather short list. 11.4.1 The mass of the baryon As already discussed, masses of hadrons that are stable in QCD + QED and have a small width, in general, are very good candidates for physical scales since there are no QED infrared divergences to be discussed. Furthermore, remaining within this class, the radiative corrections QCDδMi , Eq. (475), are expected to be small. Furthermore, the baryon has a significantly better noise/signal ratio than the nucleon (see Fig. 50). It also has little dependence on up- and down-quark masses, since it is composed entirely of strange valence quarks. Still, one has to be aware that the mass is not extracted from the plateau region but from a modelling of the approach to a plateau in the form of fits [8,111,114,115,117,193]. In this sense, the noise/signal ratio problem may persist. The use of various interpolating fields for the helps in constraining such analyses, but it would be desirable to have a theoretical 87 This procedure unavoidably requires that one provides a value for the CKM matrix element Vud that has then to be considered an input of the lattice calculation and not a predictable quantity. 123 869 Page 222 of 296 Eur. Phys. J. C (2022) 82:869 understanding of multi-hadron (or in QCD + QED multi-hadron + photon) contributions as it exists for the nucleon [1040] as discussed in Sect. 10. In the present review, we take the estimates of the collaborations at face value and do not try to apply a rating or an estimate of systematic error due to excited-state contributions. 11.4.2 Pion and kaon leptonic decay rates These decay rates have been discussed above. Here, we just summarize the main issues. In QCD+QED there is so far only one computation of the decay rate in the electro-quenched approximation [240]. The derived estimate for the radiative corrections agrees with the estimates from chiral perturbation theory (see Eqs. (486) and (487)). The quoted uncertainties are at the level of 0.001. This directly sets a limit to the achievable precision on the scale in isoQCD. At present, this limit is not yet relevant. A second source of uncertainty is due to the knowledge of Vud and Vus . For convenience, we summarize the isoQCD values f isoQCDπ |Vud | = 127.13(2)exp(13)QED MeV, (490) f isoQCDπ = 130.56(2)exp(13)QED(2)Vud MeV, (491) f isoQCDK |Vus | = 35.09(4)exp(4)QED MeV, (492) f isoQCDK = 157.2(2)exp(2)QED(4)Vus MeV, (493) where we have used the PDG values [165] for fx |Vy | (equivalent to Eqs. (486) and (487)), and the values Vud = 0.97370(14), Vus = 0.2232(6). Here, Vud is from the PDG [165] (beta decays) and the latter from Sect. 4 ( f+(0) for N f = 2 + 1 + 1). Of course, the information on pion and kaon leptonic decays do not enter the determinations of Vud and Vus used here. The uncertainties in the above values are in the following assumed to have been considered in the estimates of the scale given by the collaborations. This is analogous to the systematics due to excited-state contaminations in hadron masses, an issue which is irrelevant in the pseudoscalar channel (see Fig. 50). Depending on the lattice formulation, there is also a nontrivial renormalization of the axial current. Since it is easily determined from a chiral Ward identity, it does not play an important rôle. When it is present, it is assumed to be accounted for in the statistical errors. 11.4.3 Other physics scales Scales derived from bottomonium have been used in the past, in particular, the splitting mϒ = mϒ(2s)−mϒ(1s). They have very little dependence on the light-quark masses, but need an input for the b-quark mass. In all relevant cases, the b quark is treated by NRQCD. 11.5 Theory scales In the following, we consider in more detail the two classes of theory scales that are most commonly used in typical lattice computations. The first class consists of scales related to the static quark–antiquark potential [700]. The second class is related to the action density renormalized through the gradient flow [317]. 11.5.1 Potential scales In this approach, lattice scales are derived from the properties of the static quark–antiquark potential. In particular, a scale can be defined by fixing the force F(r) between a static quark and antiquark separated by the distance r in physical units [700]. Advantages of using the potential include the ease and accuracy of its computation, and its mild dependence on the valence-quark mass. In general, a potential scale rc can be fixed through the condition that the static force takes a predescribed value, i.e., r2c F(rc) = Xc (494) 123 Eur. Phys. J. C (2022) 82:869 Page 223 of 296 869 where Xc is a suitably chosen number. Phenomenological and computational considerations suggest that the optimal choice for Xc is in the region where the static force turns over from Coulomb-like to linear behaviour and before string breaking occurs. In the original work [700], it was suggested to use X0 = 1.65 leading to the condition r20 F(r0) = 1.65. (495) In Ref. [701], the value X1 = 1.0 was proposed yielding the scale r1. The static force is the derivative of the static quark–antiquark potential V (r)which can be determined through the calculation of Wilson loops. More specifically, the potential at distance r is extracted from the asymptotic time dependence of the r×t-sized Wilson loops W (r, t), V (r) = − dl→im∞ log〈W (r, t)〉. (496)t dt The derivative of the potential needed for the force is then determined through the derivative of a suitable local parameterization of the potential as a function of r , e.g., 1 V (r) = C− + C0 + C+r, (497) r estimating uncertainties due to the parameterization. In some calculations, the gauge field is fixed to Coulomb or temporal gauge in order to ease the computation of the potential at arbitrary distances. In order to optimize the overlap of the Wilson loops with the ground state of the potential, one can use different types and levels of spatial gauge field smearing and extract the ground state energy from the corresponding correlation matrix by solving a generalized eigenvalue problem [925,926,1041]. Finally, one can also make use of the noise reduction proposed in Refs. [1042,1043]. It changes the definition of the discretized loops by a smearing of the temporal parallel transporter [1044] and thus yields a different discretization of the continuum force. 11.5.2 Gradient flow scales The gradient flow Bμ(t, x) of gauge fields is defined in the continuum by the flow equation ∣ Ḃμ = DνGνμ, B ∣μ t=0 = Aμ , (498) Gμν = ∂μBν − ∂ν Bμ + [Bμ, Bν], Dμ = ∂μ + [Bμ, · ], (499) where Aμ is the fundamental gauge field, Gμν the field strength tensor, and Dμ the covariant derivative [317]. At finite lattice spacing, a possible form of Eqs. (498) and (499) is d a2 Vt (x, μ) = −g20 · ∂x,μSG(Vt ) · Vt (x, μ), (500)dt where Vt (x, μ) is the flow of the original gauge field U (x, μ) at flow time t , SG is an arbitrary lattice discretization of the gauge action, and ∂x,μ denotes the su(3)-valued differential operator with respect to Vt (x, μ). An important point to note is that the flow time t has the dimension of a length squared, i.e., t ∼ a2, and hence provides a means for setting the scale. One crucial property of the gradient flow is that any function of the gauge fields evaluated at flow times t > 0 is renormalized [1045] by just renormalizing the gauge coupling. Therefore, one can define a scale by keeping a suitable gluonic observable defined at constant flow time t , e.g., the action density E = − 12 Tr GμνGμν [317], fixed in physical units. This can, for example, be achieved through the condition 1 t2c 〈E(tc, x)〉 = c, E(t, x) = − Tr Gμν(t, x)Gμν(t, x) (501)2 where Gμν(t, x) is the field strength tensor evaluated on the flown gauge field Vt . Then, the lattice scale a can be determined from the dimensionless flow time in lattice units, t̂ 2c = a tc. The original proposal in [317] was to use c = 0.3 yielding the scale t0, t2 √ 0 〈E(t0)〉 = 0.3 . (502) For convenience one sometimes also defines s0 = t0. 123 869 Page 224 of 296 Eur. Phys. J. C (2022) 82:869 An alternative scale w0 has been introduced in Ref. [114]. It is defined by fixing a suitable derivative of the action density, ( ) W (tc) = tc · ∂ t2t 〈E(t)〉 = = c . (503)t tc Setting c = 0.3 yields the scale w0 through W (w20) = 0.3 . (504) In addition to the lattice scales from t0 and w0, one can also consider the scale from the dimensionful combination t0/w0. This combination has been found to have a very weak dependence on the quark mass [305,368,1046]. A useful property of the gradient flow scales is the fact that their quark-mass dependence is known from χPT [1047]. Since the action density at t ∼ t0 ∼ w20 usually suffers from large autocorrelation [1046,1048], the calculation of the statistical error needs special care. Lattice artefacts in the gradient flow scales originate from different sources [1049], which are systematically discussed by considering t as a coordinate in a fifth dimension. First, there is the choice of the action SG for t > 0. Second, there is the discretization of E(t, x). Third, there is the discretization of the 4-dimensional quantum action, which is always there, and fourth, there are also terms localized at the boundary t = 0+. The interplay between the different sources of lattice artefacts turns out to be rather subtle [1049]. Removing discretization errors due to the first two sources requires only classical (g0-independent) improvement. Those due to the quantum action are common to all t = 0 observables, but the effects of the boundary terms are not easily removed in practice. At tree level, the Zeuthen flow [1049] does the complete job, but none of the computations reviewed here have used it. Discretization effects due to SG can be removed by using an improved action such as the tree-level Symanzik-improved gauge action [114,1050]. More phenomenological attempts of improving the gradient flow scales consist of applying a t-shift [1051], or tree-level improvement [712]. 11.5.3 Other theory scales The MILC collaboration has been using another set of scales, the partially quenched pseudoscalar decay constant f p4s with degenerate valence quarks with a mass mq = 0.4 ·mstrange, and the corresponding partially quenched pseudoscalar mass Mp4s . So far it has been a quantity only used by the MILC collaboration [16,17,145]. We do not perform an in-depth discussion or an average but will list numbers in the results section. Yet another scale that has been used is the leptonic decay constant of the ηs . This fictitious particle is a pseudoscalar made of a valence quark–antiquark pair with different (fictitious) flavours which are mass-degenerate with the strange quark [118,120,552]. 11.6 List of computations and results 11.6.1 Gradient flow scales √ We now turn to a review of the calculations of the gradient flow scales t0 and w0. The results are compiled in Table 76 and shown in Fig. 51. In the following, we briefly discuss the calculations in the order that they appear in√the table and figure. ETM 21 [305] finalizes an√d supersedes ETM 20 discussed below. It determines the scales t0, w0, also t0/w0 = 0.11969(62) fm, and the ratio t0/w0 = 0.82930(65), cf. also HPQCD 13A [37]. CalLat 20A [111] use Möbius Domain-Wall valence fermions on HISQ ensembles generated by the MILC and CalLat collaborations. The gauge fields entering the Möbius Domain-Wall operator are gradient-√flow smeared with t = a2. They compute the mass and the scales w0, t0 and perform global fits to determine w0 M and t0 M at the physical point. The flow is discretized with the Symanzik tree-level improved action and the clover discretization of E(t) is used. A global fit with Bayesian priors is performed including terms derived from χPT for finite volume and quark-mass dependences, as well as a2 and a2αs(1.5/a) terms for discretization errors. Also, a tree-level improved definition of the GF scales is used where the leading-in-g2 cutoff effects are removed up to and including O(a8/t4). BMW 20 [115] presents a result for w0 in the context of their staggered fermion calculation of the muon anomalous magnetic moment. It is the first computation that takes QED and isospin-breaking corrections into account. The simulations are performed by using staggered fermions with stout gauge field smearing with six lattice spacings and several pion masses around the physical point with Mπ between 110 and 140 MeV. Volumes are around L = 6 fm. At the largest lattice spacing, 123 Eur. Phys. J. C (2022) 82:869 Page 225 of 296 869 Table 76 Results for gradient flow scales at the physical point, cf. Eq. (469). Note that BMW 20 [115] take IB and QED corrections into account. Some additional results for ratios of scales are: ETM 21 [305]: t0/w0 = 0.11969(62) fm √ Collaboration Refs. N f t0 [fm] w0 [fm] ETM 21 [305] 2+1+1 P    fπ 0.14436(61) 0.17383(63) CalLat 20A [111] 2+1+1 A    m 0.1422(14) 0.1709(11) BMW 20 [115] 1+1+1+1 A    m 0.17236(29)(63)[70] ETM 20 [1052] 2+1+1 C    fπ 0.1706(18) MILC 15 [112] 2+1+1 A    F ( f )#p4s π 0.1416(+8/-5) 0.1714(+15/-12) HPQCD 13A [37] 2+1+1 A  ◦  fπ 0.1420(8) 0.1715(9) CLS 16 [113] 2+1 A ◦   fπ , fK 0.1467(14)(7) QCDSF/UKQCD 15B [713] 2+1 P ◦ ◦ ◦ mSU (3)P 0.1511(22)(6)(5)(3) 0.1808(23)(5)(6)(4) RBC/UKQCD 14B [8] 2+1 A    m 0.14389(81) 0.17250(91) HotQCD 14 [116] 2+1 A    r1( fπ )# 0.1749(14) BMW 12A [114] 2+1 A    m 0.1465(21)(13) 0.1755(18)(4) # These scales are not physical scales and have been determined from fπ Fig. 51 Results for gradient flow scales it is demonstrated how the effective masses of the correlator almost reach the plateau value extracted from a four-state fit (two states per parity). Within the range where the data is fitted, the deviation of data points from the estimated plateau is less than a percent. Isospin-breaking corrections are computed by Taylor expansion around isoQCD with QED treated as QEDL. Finite volume effects in QED are taken from the 1/L , 1/L2 universal corrections and O(1/L3) effects are neglected. The results for M 2 4 w0 are extrapolated to the continuum by a fit with a and a terms. ETM 20 [1052] presents in their proceedings contribution a preliminary analysis of their N f = 2 + 1+ 1 Wilson twisted- mass fermion simulations at maximal twist (i.e., automatic O(a) improved), at three lattice spacings and pion masses at the physical point. Their determination of w0 = 0.1706(18) fm from fπ using an analysis in terms of Mπ is the value quoted above. They obtain the consistent value w0 = 0.1703(18) fm from an analysis in terms of the renormalized light quark mass. 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume Physical scale 869 Page 226 of 296 Eur. Phys. J. C (2022) 82:869 MILC 15 [112] sets the physical scale using the fictitious pseudoscalar decay constant Fp4s=153.90(9)(+21/− 28) MeV with degenerate valence quarks of mass mv = 0.4ms and physical sea-quark masses [145]. (Fp4s has strong dependence on the valence-quark mass and is determined from fπ .) They use a definition of the flow scales where the tree-level lattice artefacts up to O(a4/t2) are divided out. Charm-quark mass mistunings are between 1 and 11%. They are taken into account at leading order in 1 m through (3)/ c QCD applied directly to Fp4s and 1/mc corrections are included as terms in the fits. They use elaborate variations of fits in order to estimate extrapolation errors (both in GF scales and Fp4s). They include errors from FV effects and experimental errors in fπ in Fp4s . HPQCD 13A [37] uses eight MILC-HISQ ensembles with lattice spacings a = 0.088, 0.121, 0.151 fm. Values of L are between 2.5 fm and 5.8 fm with Mπ L = 3.3–4.6. Pion masses range between 128 and 306 MeV. QCD is defined by using the inputs Mπ = 134.98(32) MeV√, MK = 494.6(3) MeV, fπ+ = 130.4(2) MeV derived by model subtractions of IB effects. Additional scale ratios are given: t0/w0 = 0.835(8), r1/w0 = 1.789(26). CLS 13 [113] uses CLS configurations of 2+1 nonperturbatively O(a)-improved Wilson fermions. There are a few pion masses with the strange mass adjusted along a line of mu +md +ms = const. Three different lattice spacings are used. They determine t0 at the physical point defined by π and K masses and the linear combination f 1K + 2 fπ . They use the Wilson flow with the clover definition of E(t). QCDSF 15B [713,1053] results, unpublished, are obtained by simulating N f = 2 + 1 QCD with the tree-level Symanzik improved gauge action and clover Wilson fermions with single level stout smearing for the hopping terms together with unsmeared links for the clover term (SLiNC action). Simulations are performed at four different lattice spacings, in the range [0.06, 0.08] fm, with Mπ,min = 228 MeV and Mπ,minL = 4.1. The results for the gradient flow scales have been obtained by relying on the observation that flavour-symmetric quantities get corrections of O((m )2q ) where mq is the difference of the quark mass from the SU (3)-symmetric valu√e. The O(m2q) terms are not detected in the data and subsequently neglected. RBC/UKQCD 14B [8] presents results for t0 and w0 obtained in QCD with 2 + 1 dynamical flavours. The simulations are performed by using domain-wall fermions on six ensembles with lattice spacing a−1 = 1.38, 1.73, 1.78, 2.36, 2.38, and 3.15 GeV, pion masses in the range Munitar yπ ∈ [139, 360] MeV. The simulated volumes are such that Mπ L > 3.9. The effective masses of the correlator are extracted with two-state fits and it is shown, by using √two different nonlocal interpolating operators at the source, that the co√rrelators almost reach a plateau. In the calculation of t0 and w0, the clover definition of E(t) is used. The values given are t0 = 0.7292(41)GeV−1 and w0 = 0.8742(46)GeV−1 which we converted to the values in Table 76. HotQCD 14 [116] determines the equation of state with N f = 2 + 1 flavours using highly improved staggered quarks (HISQ/tree). As a byproduct, they update the results of HotQCD 11 [708] by adding simulations at four new values of β, for a total of 24 ensembles, with lattice spacings in the range [0.04, 0.25] fm and volumes in the range [2.6, 6.1] fm with Mπ = 160 MeV. They obtain values for the scale parameters r0 and w0, via the ratios r0/r1, w0/r1 and using r1 = 0.3106(14)(8)(4) fm from MILC 10 [41]. They obtain for the ratios (r0/r1)cont = 1.5092(39) and (w0/r1)cont = 0.5619(21) in the continuum. They crosscheck their determination of the scale r1 using the hadronic quantities fK , fη from HPQCD 09B [120] and the experimental value of Mϕ , and find good agreement. BMW 12A [114] is the work in which w0 was introduced. Simulations with 2HEX smeared Wilson fermions and two-level stout-smeared rooted staggered fermions are done. The Wilson flow with clover E(t) is used, and a test of the Symanzik flow is carried out. They take the results with Wilson fermions as their central value, because those “do not rely on the ‘rooting’ of the fermion determinant”. Staggered fermion results agree within uncertainties. 11.6.2 Potential scales We now turn to a review of the calculations of the potential scales r0 and r1. The results are compiled in Table 77 and shown in Fig. 52. The most recent calculations date back to 2014, and we discuss them in the order that they appear in the table and the figure. ETM 14 [7] uses N f = 2 + 1 + 1 Wilson twisted-mass fermions at maximal twist (i.e., automatic O(a)-improved), three lattice spacings and pion masses reaching down to Mπ = 211 MeV. They determine the scale r0 through fπ = fπ+ = 130.41 MeV. A crosscheck of the so obtained lattice spacings with the ones obtained via the fictitious pseudoscalar meson Ms′s′ made of two strange-like quarks gives consistent results. The crosscheck is done using the dimensionless combinations r0 Ms′s′ (with r0 in the chiral limit) and fπ/Ms′s′ determined in the continuum, and then using r0/a and the value of Ms′s′ obtained from the experimental value of fπ . We also note that in Ref. [1046] using the same ensembles the preliminary value w0 = 0.1782 fm is determined, however, without error due to the missing or incomplete investigation of the systematic effects. HPQCD 13A [37] was already discussed above in connection with the gradient flow scales. 123 Eur. Phys. J. C (2022) 82:869 Page 227 of 296 869 Table 77 Results for potential scales at the physical point, cf. Eq. (469). Mϒ = Mϒ(2s) − Mϒ(1s) Collaboration Ref. N f r0 [fm] r1 [fm] ETM 14 [7] 2+1+1 A ◦   fπ 0.474(14) HPQCD 13A [37] 2+1+1 A  ◦  fπ 0.3112(30) HPQCD 11B [552] 2+1+1 A ◦ ◦ ◦ Mϒ , fηs 0.3209(26) HotQCD 14 [116] 2+1 A    r1([41])# 0.4688(41) χQCD 14 [24] 2+1 A ◦ ◦ ◦ Three inputsa 0.465(4)(9) HotQCD 11 [708] 2+1 A    fπ 0.468(4) RBC/UKQCD 10A [117] 2+1 A ◦ ◦ ◦ M 0.487(9) 0.333(9) MILC 10 [41] 2+1 C ◦   fπ 0.3106(8)(14)(4) MILC 09 [157] 2+1 A ◦   fπ 0.3108(15)(+26−79) MILC 09A [15] 2+1 C ◦   fπ 0.3117(6)(+12−31) HPQCD 09B [120] 2+1 A ◦  ◦ Three inputs 0.3133(23)(3) PACS-CS 08 [193] 2+1 A    M 0.4921(64)(+74−2 ) HPQCD 05B [118] 2+1 A ◦ ◦ ◦ Mϒ 0.469(7) 0.321(5) Aubin 04 [119] 2+1 A ◦ ◦ ◦ Mϒ 0.462(11)(4) 0.317(7)(3) #This theory scale was determined in turn from r1 [41] a MD∗ , MD∗ − MDs , Ms s J/ψ Fig. 52 Results for potential scales HPQCD 11B [552] uses five MILC-HISQ ensembles and determines r1 from Mϒ(2s)− Mϒ(1s) and the decay constant fηs (see HPQCD 09B). The valence b quark is treated by NRQCD, while the light valence quarks have the HISQ discretization, identical to the sea quarks. HotQCD 14 [116] was already discussed in connection with the gradient flow scales. χQCD 14 [24] uses overlap fermions as valence quarks on N f = 2 + 1 domain-wall fermion gauge configurations generated by the RBC/UKQCD collaboration [117]. Using the physical masses of D ∗s, Ds and J/ψ as inputs, the strange and charm quark masses and the decay constant fDs are determined as well as the scale r0. 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume Physical scale 869 Page 228 of 296 Eur. Phys. J. C (2022) 82:869 HotQCD 11 [708] uses configurations with tree-level improved Symanzik gauge action and HISQ staggered quarks in addition to previously generated ensembles with p4 and asqtad staggered quarks. In this calculation, QCD is defined by generating lines of constant physics with ml/ms =√ {0.2, 0.1, 0.05, 0.025} and setting the strange quark mass by requiring that the mass of a fictitious ηss̄ meson is M 2ηss̄ = 2MK − M2π . The physical point is taken to be at ml/ms = 0.037. The physical scale is set by using the value r1 = 0.3106(8)(18)(4) fm obtained in Ref. [41] by using fπ as physical input. In the paper, this result is shown to be consistent within the statistical and systematic errors with the choice of fK as physical input. The result r0/r1 = 1.508(5) is obtained by averaging over 12 ensembles at ml/ms = 0.05 with lattice spacings in the range [0.066, 0.14] fm. This result is then used to get r0 = 0.468(4) fm. Finite volume effects have been monitored with 20 ensembles in the range [3.2, 6.1]fm with Mπ L > 2.6. RBC/UKQCD 10A [117] uses N f = 2 + 1 flavours of domain-wall quarks and the Iwasaki gauge action at two values of the lattice spacing with unitary pion masses in the approximate range [290, 420] MeV. They use the masses of π and K meson and of the baryon to determine the physical quark masses and the lattice spacings, and so obtain estimates of the scales r0, r1 and the ratio r1/r0 from a combined chiral and continuum extrapolation. MILC 10 [41] presents a further update of r1 with asqtad staggered quark ensembles with a ∈ {0.045, 0.06, 0.09} fm. It supersedes MILC 09 [15,157,1054]. MILC 09 [157] presents an N f = 2 + 1 calculation of the potential scales on asqtad staggered quark ensembles with a ∈ {0.045, 0.06, 0.09, 0.12, 0.15, 0.18} fm. The continuum extrapolation is performed by using Goldstone boson pions as light as Mπ = 224 MeV (RMS pion mass of 258 MeV). The physical scale is set from fπ . The result for r1 obtained in the published paper [157] is then updated and, therefore, superseded by the conference proceedings MILC 09A and 09B [15,1054]. HPQCD 09B [120] is an extension of HPQCD 05B [118] and uses HISQ valence quarks instead of asqtad quarks. The scale r1 is obtained from three different inputs. First r1 = 0.309(4) fm from the splitting of 2S and 1S ϒ states as in Ref. [118], second r1 = 0.316(5) fm from MDs − Mηs/2 and third r1 = 0.315(3) fm from the decay constant of the ηs . The fictitious ηs state is operationally defined by setting quark masses to the s-quark mass and dropping disconnected diagrams. Its mass and decay constant are obtained from a partially quenched chiral perturbation theory analysis using the pion and kaon states from experiment together with various partially quenched lattice data. The three results are combined to r1 = 0.3133(23)(3) fm. PACS-CS 08 [193] presents a calculation of r0 in N f = 2 + 1 QCD by using NP O(a)-improved clover Wilson quarks and Iwasaki gauge action. The calculation is done at fixed lattice spacing a = 0.09 fm and is extrapolated to the physical point from (unitary) pion masses in the range [156, 702] MeV. The N f = 2 + 1 theory is defined by fixing Mπ , MK , and M to 135.0, 497.6, and 1672.25 MeV, respectively. The effective masses of smeared-local correlators averaged over the four spin polarizations show quite good plateaux. HPQCD 05B [118] performed the first bottomonium spectrum calculation in full QCD with N f = 2 + 1 on MILC asqtad configurations and the b quark treated by NRQCD. They find agreement of the low lying ϒ states with experiment and also compare to quenched and N f = 2 results. They determined r0 and r1 from the splitting of 2S and 1S states. Aubin 04 [119] presents an N f = 2+1 calculation of the potential scales by using asqtad staggered quark ensembles with a = 0.09 and 0.12} fm. The continuum extrapolation is performed by using Goldstone boson pions as light as mπ = 250 MeV. The physical scale is set from the ϒ 2S-1S and 1P-1S splittings computed with NRQCD by HPQCD [1055]. 11.6.3 Ratios of scales It is convenient in many cases to also have ratios of scales at hand. In addition to translating from one scale to another, the ratios provide important crosschecks between different determinations. Results on ratios provided by the collaborations are compiled in Table 78 and Fig. 53. The details of the computations were already discussed in the previous sections. 11.7 Averages √ Gradient flow scale t0 For N f = 2 + 1 + 1, we have two very recent calculations from ETM 21 [305], CalLat 20A [111], and two less recent ones from MILC 15 [112], and HPQCD 13A [37] fulfilling the FLAG criteria to enter the average. The latter two and CalLat 20A are based on the same MILC-HISQ gauge field ensembles, hence we consider their statistical errors to be 100% correlated. ETM 21 is not published, and, hence, does not enter the FLAG average. 123 Eur. Phys. J. C (2022) 82:869 Page 229 of 296 869 Table 78 Results for dimensionless ratios of scales √ Collaboration Refs. N f t0/w0 r0/r1 r1/w0 ETM 21 [305] 2+1+1 P    0.82930(65) HPQCD 13A [37] 2+1+1 A  ◦  0.835(8) 1.789(26) HotQCD 14 [116] 2+1 A    1.7797(67) HotQCD 11 [708] 2+1 A    1.508(5) RBC/UKQCD 10A [117] 2+1 A ◦ ◦ ◦ 1.462(32)# Aubin 04 [119] 2+1 A ◦ ◦ ◦ 1.474(7)(18) #This value is obtained from r1/r0 = 0.684(15)(0)(0) Fig. 53 Results for dimensionless ratios of scales 123 Publication status Chiral extrapolation Continuum extrapolation Finite volume 869 Page 230 of 296 Eur. Phys. J. C (2022) 82:869 For N f = 2 + 1, we have three calculations from CLS 16 [113], RBC/UKQCD 14B [8], and BMW 12A [114] which all enter the FLAG average. They are all independent computations, so there is no correlation to be taken into account. QCDSF/UKQCD 15B [713] does not contribute to the average, because it is not published. Performing the weighted and correlated average we obtain N f = 2 + 1 + 1 : √√t0 = 0.14186(88) fm Refs. [37,111,112], (505) N f = 2 + 1 : t0 = 0.14464(87) fm Refs. [8,113,114]. (506) We note that for the N f = 2 + 1 average the stretching factor based on the χ2-value from the weighted average is 1.25. Gradient flow scale w0 For N f = 1 + 1 + 1 + 1, including QED, there is a single calculation, BMW 20 [115] with the result N f = 1 + 1 + 1 + 1 + QED : w0 = 0.17236(70) fm Ref. [115]. (507) For N f = 2 + 1 + 1, we have four calculations ETM 21 [305], CalLat 20A [111], MILC 15 [112], and HPQCD 13A [37] fulfilling the FLAG criteria to enter the average. However, ETM 20 is a proceedings contribution, w√hile ETM 21 is not published, hence, they do not contribute to the FLAG average. As discussed above in connection with t0 we correlate the statistical errors of CalLat 20A, MILC 15, and HPQCD 13A. For N f = 2 + 1, we have three calculations RBC/UKQCD 14B [8], HotQCD 14 [116], and BMW 12A [114] that enter the FLAG average. These calculations are independent, and no correlation needs to be taken into account. QCDSF/UKQCD 15B [713] does not contribute to the average, because it is not published. Performing the weighted and correlated average, we obtain N f = 2 + 1 + 1 : w0 = 0.17128(107) fm Refs. [37,111,112], (508) N f = 2 + 1 : w0 = 0.17355(92) fm Refs. [8,114,116]. (509) We note that for the N f = 2 + 1 average, the stretching factor based on the χ2-value from the weighted average is 1.23. Isospin-breaking and electromagnetic corrections are expected to be small at the level of present uncertainties. This is also confirmed by the explicit computation by BMW 12A. Therefore, we also perform an average over all N f > 2 + 1 computations and obtain N f > 2 + 1 : w0 = 0.17177(67) fm Refs. [37,111,112,115]. (510) Potential scale r0 For N f = 2 + 1 + 1, there is currently only one determination of r0 from ETM 14 [7], namely r0 = 0.474(14) fm, which, therefore, represents the FLAG average. For N f = 2+1, all but one calculation fulfill all the criteria to enter the FLAG average. HotQCD 14 [116] is essentially an update of HotQCD 11 [708] by enlarging the set of ensembles used in the computation. Therefore, the result from HotQCD 14 supersedes the one from HotQCD 11 and, hence, we only use the former in the average. The computation of χQCD [24] is based on the configurations produced by RBC/UKQCD 10A [117], and we, therefore, assume a 100% correlation between the statistical errors of the two calculations. HPQCD 05B [118] enhances the calculation of Aubin 04 [119] by adding ensembles at a coarser lattice spacing and using the same discretization for the valence fermion. Therefore, we consider the full errors (statistical and systematic) on the results from Aubin 04 and HPQCD 05B to be 100% correlated. Performing the weighted and correlated average, we obtain N f = 2 + 1 + 1 : r0 = 0.474(14) fm Ref. [7], (511) N f = 2 + 1 : r0 = 0.4701(36) fm Refs. [24,116–119]. (512) We note that for the N f = 2 + 1 average, the stretching factor based on the χ2-value from the weighted average is 1.14. 123 Eur. Phys. J. C (2022) 82:869 Page 231 of 296 869 Potential scale r1 For N f = 2 + 1 + 1, there are two works that fulfill the criteria to enter the FLAG average, namely HPQCD 13A [37] and HPQCD 11B [552]. Both are based on MILC-HISQ ensembles, the former uses eight, the latter only five. The result from HPQCD 13A supersedes the result from HPQCD 11B (in line with a corresponding statement in HPQCD 13A) and forms the FLAG average. For N f = 2 + 1, all the results quoted in Table 77 fulfill the FLAG criteria, but not all of them enter the average. The published result from MILC 09 [157] is superseded by the result in the proceedings MILC 10 [41], while MILC 09A [15] is a proceedings contribution and does not enter the average. HPQCD 09B [120] uses HISQ valence quarks instead of asqtad valence quarks as in HPQCD 05B [118]. Therefore, we have RBC/UKQCD 10A [117], MILC 10, HPQCD 09B, HPQCD 05B, and Aubin 04 entering the average. However, since the latter four calculations are based on the aqtad MILC ensembles, we attribute 100% correlation on the statistical error between them and 100% correlation on the systematic error between HPQCD 05B and Aubin 04 as discussed above in connection with r0. Performing the weighted and correlated average, we obtain N f = 2 + 1 + 1 : r1 = 0.3112(30) fm Ref. [37], (513) N f = 2 + 1 : r1 = 0.3127(30) fm Refs. [41,117–120]. (514) We note that for the N f = 2 + 1 average the stretching factor based on the χ2-value from the weighted average is 1.57. The scales Mp4s and f p4s As mentioned in Sect. 11.5.3, these scales have been used only by the MILC and FNAL/MILC collaborations [16,17,145]. The latest numbers from Ref. [16] are f = 153.98(11)(+24ps −12)(12)[4] MeV and M +17p4s = 433.12(14)(−6 )(4)[40] MeV and, hence, we have N f = 2 + 1 + 1 : f4ps = 153.98(20) MeV Ref. [16], (515) N f = 2 + 1 + 1 : M4ps = 433.12(30) MeV Ref. [16]. (516) Dimensionless ratios of scales √ We start with the ratio t0/w0 for which two N f = 2 + 1 + 1 calculations from ETM 21 [305] and HPQCD 13A [37] are available. Since the former is not published, HPQCD 13A forms the FLAG value, N f = 2 + 1 + √1 : t0/w0 = 0.835(8) Ref. [37]. (517) For the ratio r0/r1 there are three calculations from HotQCD 11 [708], RBC/UKQCD 10A [117], and Aubin 04 [119] available. They all fulfill the FLAG criteria and enter the FLAG average of this ratio, N f = 2 + 1 : r0/r1 = 1.5049(74) Refs. [117,119,708]. (518) We note that the stretching factor based on the χ2-value from the weighted average is 1.54. Finally, for the ratio r1/w0 there is one computation from HotQCD 14 [116] for N f = 2 + 1 + 1, and one from HPQCD 13A [37] for N f = 2 + 1 fulfilling the FLAG criteria, and, hence, forming the FLAG values N f = 2 + 1 + 1 : r1/w0 = 1.789(26) Ref. [37], (519) N f = 2 + 1 : r1/w0 = 1.7797(67) Ref. [116]. (520) 11.8 Observations and conclusions The different computations for theory scales reported here are generally in good agreement within each set of 2+1+1 and 2+1 flavour content. Quantitatively, the computed stretching factors are almost all below 1.3, a value which is insignificant considering the small number of computations that enter the averages. The only exception are the potential scales, where the domain-wall fermion results of RBC/UKQCD 10A are above the staggered fermion results, and also the ratio r0/r1 needs a stretching factor of 1.54. Of course, the limited number of large-scale QCD simulations that are available means that we 123 869 Page 232 of 296 Eur. Phys. J. C (2022) 82:869 have only a small number of truly independent determinations of the scales. For example, three out of the four computations entering our average for w0 are based on the same HISQ rooted staggered fermion configurations and thus those differences are only due to the choice of the physical scale (m vs. fπ ), the valence quark action (Möbius domain-wall valence fermions vs. staggered fermions) employed to compute it and different analysis of continuum limit, etc. Differences between N f = 2 + 1 and 2+1+1 QCD are small but still larger than expectations [15√8,159]. A marginally significant difference is indicated by the figure and√numbers on w0 and a larger one by the ones for t0. The effect of the charm quark is −1.0(7)% on w0 and −1.9(9)% on t0 as computed from the FLAG averages. In contrast, precision studies of the decoupling of charm quarks predict generic effects of a magnitude of only ≈ 0.2% [158,159] for low energy quantities. We are looking forward to further and more precise results for clarifying whether th√ese −1.9(9)% hold up over time. In this respect, it is highly desirable for future computations to also publish ratios such as t0/w0 where numbers are rare so far. Such ratios of gradient flow scales are also of high interest in order to better understand the specific discretization errors of gradient flow observables. So far, systematic studies and information on the different contributions (see Sect. 11.5.2 and Ref. [1049]) are missing. A worrying result is, for example, the scale-setting study of Ref. [167] on ratios of scales. The authors find indications that the asymptotic ∼ a2 scaling does not set in before a ≈ 0.05 fm and the a = 0.04 fm data has a relevant influence on their continuum extrapolations. A final word concerns the physics scales that all results depend on. While the mass of the baryon is more popular than the leptonic decay rate of the pion, both have systematics which are difficult to estimate. For the baryon it is the contaminations by excited states and for the decay rates it is the QED effects f isoQCDδ π . The uncertainty in Vud is not relevant at this stage, but means that one is relying more on the standard model being an accurate low energy theory than in the case of the mass. In principle, excited state effects are controlled by just going to large Euclidean time, but, in practice, this yields errors that are too large. One, therefore, performs fits with a very small number of excitations while theoretically there is a multitude of multi-hadron states that are expected to contribute. For the leptonic decay rate of the pion, the situation is quite reversed, namely, the problematic QED contributions have a well-motivated theory: chiral perturbation theory. The needed combination of low-energy constants is not accessible from experiment but its large-N estimate [520] has been (indirectly) confirmed by the recent computation of f isoQCDδ π [240]. Unfortunately the same comparison is not so favourable for the leptonic Kaon decay. Acknowledgements We are very grateful to the external reviewers for providing detailed comments and suggestions on the draft of this review. These reviewers were Gunnar Bali, Claude Bernard, Johann Bijnens, Jérôme Charles, Luigi Del Debbio, Paolo Gambino, Elvira Gámiz, Christian Hoelbling, Marc Knecht, Bira van Kolck, Laurent Lellouch, Zoltan Ligeti, Kim Maltman, Emilie Passemar, Chris Sachrajda, Tilo Wettig, and Takeshi Yamazaki. The kick-off meeting for the present review was held in February 2020 at the University of Bern and was supported by the Albert Einstein Center for fundamental physics. The mid-review meeting was held virtually in December 2020. We thank our hosts for their hospitality and financial support is gratefully acknowledged. Members of FLAG were supported by funding agencies; in particular: • Y.A. was supported by JSPS KAKENHI Grant No. 16K05320. • S.C. acknowledges the support of the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 813942 (ITN EuroPLEx). • M.D.M. was supported by DFF Research project 1. Grant n. 8021-00122B. • P.D. acknowledges support from the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 813942 (EuroPLEx) and from INFN under the research project INFN-QCDLAT. • X.F. is supported in part by NSFC of China under Grant No. 11775002, 12070131001, 12125501 and National Key Research and Development Program of China under Contracts No. 2020YFA0406400. • S.G. acknowledges support from the U.S. Department of Energy through grant DE-SC0010120. •G.H. and C.P. acknowledge support from the EU H2020-MSCA-ITN-2018-813942 (EuroPLEx), Spanish MICINN grant PGC2018-094857-B-I00, Spanish Agencia Estatal de Investigación through the grant ”IFT Centro de Excelencia Severo Ochoa” SEV-2016-0597 and CEX2020-001007-S. • S.H. is supported in part by JSPS KAKENHI Grant Number 18H03710. S.H. is supported in part by MEXT as “Program for Promoting Researches on the Supercomputer Fugaku” (Simulation for basic science: from fundamental laws of particles to creation of nuclei, JPMXP1020200105) through the Joint Institute for Computational Fundamental Science (JICFuS). • A.J. received funding from the STFC consolidated grant ST/P000711/1 and ST/ T000775/1. • T.K. is supported in part by JSPS KAKENHI Grant Number 21H01085. T.K. is supported in part by MEXT as “Program for Promoting Researches on the Supercomputer Fugaku” (Simulation for basic science: from fundamental laws of particles to creation of nuclei, JPMXP1020200105) through the Joint Institute for Computational Fundamental Science (JICFuS). • S.M. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0009913. • C.J.M. is supported in part by USDOE grant No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. • A.P. is supported in part by UK STFC grant ST/P000630/1 and also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreements No 757646 & 813942. • A.R. is supported by the Generalitat Valenciana through the plan GenT program (CIDEGENT/2019/040) and the Ministerio de Ciencia e Innovacion (PID2020-113644GB-I00). 123 Eur. Phys. J. C (2022) 82:869 Page 233 of 296 869 • S.R.S. is supported in part by the U.S. Department of Energy (grant DE-SC0011637). • S.S. acknowledges support from the EU H2020-MSCA-ITN-2018-813942 (EuroPLEx). Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There is no data.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adapta- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this arti- cle are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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A List of acronyms BχPT Baryonic chiral perturbation theory BCL Bourrely–Caprini–Lellouch BGL Boyd–Grinstein–Lebed BK Becirevic–Kaidalov BSM Beyond standard model BZ Ball–Zwicky χPT Chiral perturbation theory CKM Cabibbo–Kobayashi–Maskawa CLN Caprini–Lellouch–Neubert CP Charge-parity CPT Charge-parity-time reversal CVC Conserved vector current DSDR Dislocation suppressing determinant ratio DW Domain wall DWF Domain wall fermion EDM Electric dipole moment EFT Effective field theory EM Electromagnetic ESC Excited state contributions EW Electroweak FCNC Flavor-changing neutral current FH Feynman–Hellman FSE Finite-size effects FV Finite volume GF Gradient flow GGOU Gambino–Giordano–Ossola–Uraltsev GRS Gasser–Rusetsky–Scimemi HEX Hypercubic stout HISQ Highly-improved staggered quarks HMχPT Heavy-meson chiral perturbation theory HMC Hybrid Monte Carlo HMrSχPT Heavy-meson rooted staggered chiral perturbation theory HQET Heavy-quark effective theory IR Infrared isoQCD Isospin-symmetric QCD LD Long distance LEC Low-energy constant LO Leading order 123 869 Page 234 of 296 Eur. Phys. J. C (2022) 82:869 LW Lüscher–Weisz MC Monte Carlo MM Minimal MOM MOM Momentum subtraction MS Modified minimal substraction scheme NDR Naive dimensional regularization nEDM Nucleon electric dipole moment NGB Nambu–Goldstone bosons NLO Next-to-leading order NME Nucleon matrix elements NNLO Next-to-next-to-leading order NP Nonperturbative npHQET Nonperturbative heavy-quark effective theory NRQCD Nonrelativistic QCD NSPT Numerical stochastic perturbation theory OPE Operator product expansion PCAC Partially-conserved axial current PDF Parton distribution function PDG Particle data group QCD Quantum chromodynamics QED Quantum electrodynamics QEDL Formulation of QED in finite volume (see [179]) QEDTL Formulation of QED in finite volume (see [1056]) RG Renormalization group RGI Renormalization group invariant RH R. Hill RHQ Relativistic heavy-quark RHQA Relativistic heavy-quark action RI-MOM Regularization-independent momentum subtraction (also RI/MOM) RI-SMOM Regularization-independent symmetric momentum (also RI/SMOM) RMS Root mean square SχPT Staggered chiral perturbation theory SD Short distance SF Schrödinger functional SIDIS Semi-inclusive deep-inelastic scattering SM Standard model SSF Step-scaling function SUSY Supersymmetric SW Sheikholeslami–Wohlert UT Unitarity triangle UV Ultraviolet B Appendix B.1 Parameterizations of semileptonic form factors In this section, we discuss the description of the q2-dependence of form factors, using the vector form factor f+ of B → πν decays as a benchmark case. Since in this channel the parameterization of the q2-dependence is crucial for the extraction of |Vub| from the existing measurements (involving decays to light leptons), as explained above, it has been studied in great detail in the literature. Some comments about the generalization of the techniques involved will follow. 123 Eur. Phys. J. C (2022) 82:869 Page 235 of 296 869 The vector form factor for B → πν All form factors are analytic functions of q2 outside physical poles and inelastic threshold branch points; in the case of B → πν, the only pole expected below the Bπ production region, starting at q2 = t+ = (m + m )2B π , is the B∗. A simple ansatz for the q2-dependence of the B → πν semileptonic form factors that incorporates vector-meson dominance is the Bečirević–Kaidalov (BK) parameterization [500], which for the vector form factor reads: f (0) f+(q2) = − 2 − 2 . (521)(1 q2/m B∗)(1 αq2/m B∗) Because the BK ansatz has few free parameters, it has been used extensively to parameterize the shape of experimental branching-fraction measurements and theoretical form-factor calculations. A variant of this parameterization proposed by Ball and Zwicky (BZ) adds extra pole factors to the expressions in Eq. (521) in order to mimic the effect of multiparticle states [1057]. A similar idea, extending the use of effective poles also to D → πν decays, is explored in Ref. [1058]. Finally, yet another variant (RH) has been proposed by Hill in Ref. [568]. Although all of these parameterizations capture some known properties of form factors, they do not manifestly satisfy others. For example, perturbative QCD scaling constrains the behaviour of f+ in the deep Euclidean region [1059–1061], and angular momentum conservation constrains the asymptotic behaviour near thresholds – e.g., Im f+(q2) ∼ (q2− t+)3/2 (see, e.g., Ref. [1062]). Most importantly, these parameterizations do not allow for an easy quantification of systematic uncertainties. A more systematic approach that improves upon the use of simple models for the q2 behaviour exploits the positivity and analyticity properties of two-point functions of vector currents to obtain optimal parameterizations of form factors [569,664, 1061,1063–1066]. Any form factor f can be shown to admit a series expansion of the form ∑∞1 f (q2) = 2 2 an(t0) z(q2, t0)n, (522)B(q )φ(q , t0) n=0 where the squared momentum transfer is replaced by the variable √ 2 = √t+ − √ q2 −√t+ − t0z(q , t0) . (523) t+ − q2 + t+ − t0 This is a conformal transformation, depending on an arbitrary real parameter t0 < t+, that maps the q2 plane cut for q2 ≥ t+ onto the disk |z(q2, t0)| < 1 in the z complex plane. The function B(q2) is called the Blaschke factor, and contains poles and cuts below t+ – for instance, in the case of B → π decays, 2 2 2 = z(q , t0)− z(m B∗ , t0)B(q ) − = z(q 2,m2B∗). (524)1 z(q2, t0)z(m2B∗, t0) Finally, the quantity φ(q2, t0), called the outer function, is some otherwise arbitrary function that does not introduce further poles or branch cuts. The crucial property of this series expansion is that the sum of the squares of the coefficients ∑∞ ∮ 2 1 dza 2n = |B(z)φ(z) f (z)| , (525) = 2π i zn 0 is a finite quantity. Therefore, by using this parameterization an absolute bound to the uncertainty induced by truncating the series can be obtained. The aim in choosing φ is to obtain a bound that is useful in practice, while (ideally) preserving the correct behaviour of the form factor at high q2 and aroun∑d thresholds. The simplest form of the bound would correspond to ∞ 2n=0 an = 1. Imposing this bound yields the following “standard” choice for the outer function √ ( 1 √ √ ) φ(q2, t0) = t+ − q2 + t− + − t032πχ1 (0) (√ √ )3/2 (√ √ )−5× + − 2 + t+ − − + − 2 + + + − q 2 t q t t t q t , (526) (t+ − t 1/40) 123 869 Page 236 of 296 Eur. Phys. J. C (2022) 82:869 where t− = (m B−mπ )2, and χ1−(0) is the derivative of the transverse component of the polarization function (i.e., the Fourier transform of the vector two-point function) #μν(q) at Euclidean momentum Q2 = −q2 = 0. It is computed perturbatively, using operator product expansion techniques, by relating the B → πν decay amplitude to ν → Bπ inelastic scattering via crossing symmetry and reproducing the correct value of the inclusive ν → Xb amplitude. We will refer to the series parameterization with the outer function in Eq. (526) as Boyd, Grinstein, and Lebed (BGL). The perturbative and OPE truncations imply that the bound is not strict, and one should take it as ∑N a2n  1, (527) n=0 where this holds for any choice of N . Since the values of |z| in the kinematical region of interest are well below 1 for judicious choices of t0, this provides a very stringent bound on systematic uncertainties related to truncation for N ≥ 2. On the other hand, the outer function in Eq. (526) is somewhat unwieldy and, more relevantly, spoils the correct large q2 behaviour and induces an unphysical singularity at the Bπ threshold. A simpler choice of outer function has been proposed by Bourrely, Caprini and Lellouch (BCL) in Ref. [1062], which leads to a parameterization of the form 1 ∑N f+(q2) = + 2 n 1 − a (t )z(q , t ) . (528)q2 n 0 0/m2B∗ n=0 This satisfies all the basic properties of the form factor, at the price of changing the expression for the bound to ∑N B (t )a+(t )a+jk 0 j 0 k (t0) ≤ 1. (529) j,k=0 The constants B jk can be computed and shown to be |B |  O(10−2jk ) for judicious choices of t0; therefore, one again finds that truncating at N ≥ 2 provides sufficiently stringent bounds for the current level of experimental and theoretical precision. It is actually possible to optimize the properties of the expansion by taking t0 = √ √topt = (m B + mπ )( m − m 2B π ) , (530) which for physical values of the masses results in the semileptonic domain being mapped onto the symmetric interval |z| <∼ 0.279 (where this range differs slightly for the B± and B0 decay channels), minimizing the maximum truncation error. If one also imposes that the asymptotic behaviour Im f+(q2) ∼ (q2 − t+)3/2 near threshold is satisfied, then the highest-order coefficient is further constrained as − N N∑−1+ = − ( 1)aN (−1)n n a+N n . (531) n=0 Substituting the above constraint on a+N into Eq. (528) leads to the constrained BCL parameterization N∑−11 + [ ]f+(q2) = a zn − (−1)n−N n− 2 n z N , (532) 1 q2/m B∗ Nn=0 which is the standard implementation of the BCL parameterization used in the literature. Parameterizations of the BGL and BCL kind, to which we will refer collectively as “z-parameterizations”, have already been adopted by the BaBar and Belle collaborations to report their results, and also by the Heavy Flavour Averaging Group (HFAG, later renamed HFLAV). Some lattice collaborations, such as FNAL/MILC and ALPHA, have already started to report their results for form factors in this way. The emerging trend is to use the BCL parameterization as a standard way of presenting results for the q2-dependence of semileptonic form factors. Our policy will be to quote results for z-parameterizations when the latter are provided in the paper (including the covariance matrix of the fits); when this is not the case, but the published form factors include the full correlation matrix for values at different q2, we will perform our own fit to the constrained BCL ansatz in Eq. (532); otherwise no fit will be quoted. We however stress the importance of providing, apart from parameterization 123 Eur. Phys. J. C (2022) 82:869 Page 237 of 296 869 coefficients, values for the form factors themselves (in the continuum limit and at physical quark masses) for a number of values of q2, so that the results can be independently parameterized by the readers if so wished. The scalar form factor for B → πν The discussion of the scalar B → π form factor is very similar. The main differences are the absence of a constraint analogue to Eq. (531) and the choice of the overall pole function. In our fits we adopt the simple expansion: N∑−1 f (q2) = a0 zn0 n . (533) n=0 We do impose the exact kinematical constraint f+(0) = f0(0) by expressing the a0N−1 coefficient in terms of all remaining a+n and a0n coefficients. This constraint introduces important correlations between the a+ and a0n n coefficients; thus only lattice calculations that present the correlations between the vector and scalar form factors can be used in an average that takes into account the constraint at q2 = 0. Finally we point out that we do not need to use the same number of parameters for the vector and scalar form factors. For instance, with (N+ = 3, N 0 = 3) we have a+ 00,1,2 and a0,1, while with (N+ = 3, N 0 = 4) we have a+ 00,1,2 and a0,1,2 as independent fit parameters. In our average we will choose the combination that optimizes uncertainties. Extension to other form factors The discussion above largely extends to form factors for other semileptonic transitions (e.g., Bs → K and B → D(∗)(s) (s) , and semileptonic D and K decays). Details are discussed in the relevant sections. A general discussion of semileptonic meson decay in this context can be found, e.g., in Ref. [1067]. Extending what has been discussed above for B → π , the form factors for a generic H → L transition will display√a cut starting at the production threshold t+, and the optimal value of t0 required in z-parameterizations is t0 = t+(1 − 1 − t−/t+) (where t± = (m ±m )2H L ). For unitarity bounds to apply, the Blaschke factor has to include all sub-threshold poles with the quantum numbers of the hadronic current – i.e., vector (resp. scalar) resonances in Bπ scattering for the vector (resp. scalar) form factors of B → π , Bs → K , or b → p; and vector (resp. scalar) resonances in Bcπ scattering for the vector (resp. scalar) form factors of B → D or b → c.88 Thus, as emphasized above, the control over systematic uncertainties brought in by using z-parameterizations strongly depends on implementation details. This has practical consequences, in particular, when the resonance spectrum in a given channel is not sufficiently well-known. Caveats may also apply for channels where resonances with a nonnegligible width appear. A further issue is whether t+ = (m H +m 2L) is the proper choice for the start of the cut in cases such as Bs → Kν and B → Dν, where there are lighter two-particle states that project on the current (B,π and Bc,π for the two processes, respectively).89 In any such situation, it is not clear a priori that a given z-parameterization will satisfy strict bounds, as has been seen, e.g., in determinations of the proton charge radius from electron-proton scattering [1068–1070]. The HPQCD collaboration pioneered a variation on the z-parameterization approach, which they refer to as a “modified z-expansion,” that is used to simultaneously extrapolate their lattice simulation data to the physical light-quark masses and the continuum limit, and to interpolate/extrapolate their lattice data in q2. This entails allowing the coefficients an to depend on the light-quark masses, squared lattice spacing, and, in some cases the charm-quark mass and pion or kaon energy. Because the modified z-expansion is not derived from an underlying effective field theory, there are several potential concerns with this approach that have yet to be studied. The most significant is that there is no theoretical derivation relating the coefficients of the modified z-expansion to those of the physical coefficients measured in experiment; it therefore introduces an unquantified model dependence in the form-factor shape. As a result, the applicability of unitarity bounds has to be examined carefully. Related to this, z-parameterization coefficients implicitly depend on quark masses, and particular care should be taken in the event that some state can move across the inelastic threshold as quark masses are changed (which would in turn also affect the form of the Blaschke factor). Also, the lattice-spacing dependence of form factors provided by Symanzik effective theory techniques may not extend trivially to z-parameterization coefficients. The modified z-expansion is now being utilized by collaborations other than HPQCD and for quantities other than D → πν and D → Kν, where it was originally employed. We advise treating results that utilize the modified z-expansion to obtain form-factor shapes and CKM matrix elements with caution, however, since the systematics of this approach warrant further study. 88 A more complicated analytic structure may arise in other cases, such as channels with vector mesons in the final state. We will however not discuss form-factor parameterizations for any such process. 89 We are grateful to G. Herdoíza, R.J. Hill, A. Kronfeld and A. Szczepaniak for illuminating discussions on this issue. 123 869 Page 238 of 296 Eur. Phys. J. C (2022) 82:869 C Notes In the following Appendices we provide more detailed information on the simulations used to calculate the quantities discussed in Sects. 3–10. We present this information only for results that are new w.r.t. FLAG 19. For all other results the information is available in the corresponding Appendices B.1–8 in FLAG 19 [4] and B.1–7 of FLAG 16 [3]. The complete information is available on the FLAG website http://flag.unibe.ch [5]. C.1 Notes to Sect. 3 on quark masses See Tables 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102. Table 79 Continuum extrapolations/estimation of lattice artifacts in determinations of mud , ms and, in some cases mu and md , with N f = 2+1+1 quark flavours Collab. Ref. N f a [fm] Description ETM 21A [204] 2+1+1 0.07, 0.08, 0.09 Scale set from w0 Table 80 Continuum extrapolations/estimation of lattice artifacts in determinations of mud , ms and, in some cases mu and md , with N f = 2 + 1 quark flavours Collab. Ref. N f a [fm] Description ALPHA 19 [18] 2+1 0.05, 0.064, 0.076, 0.086 Table 81 Chiral extrapolation/minimum pion mass in determinations of mud , ms and, in some cases, mu and md , with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f Mπ,min [MeV] Description ETM 21A [204] 2+1+1 Physical Table 82 Chiral extrapolation/minimum pion mass in determinations of mud , ms and, in some cases mu and md , with N f = 2 + 1 quark flavours Collab. Ref. N f Mπ,min [MeV] Description ALPHA 19 [18] 2+1 198 Table 83 Finite-volume effects in determinations of mud , ms and, in some cases mu and md , with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f L [fm] Mπ,min L Description ETM 21A [204] 2+1+1 2.5–5.6 3.6 Table 84 Finite-volume effects in determinations of mud , ms and, in some cases mu and md , with N f = 2 + 1 quark flavours Collab. Ref. N f L [fm] Mπ,min L Description ALPHA 19 [18] 2+1 2.4–4.1  4.0 Table 85 Renormalization in determinations of mud , ms and, in some cases mu and md , with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f Description ETM 21A [204] 2+1+1 Nonperturbative (RI/MOM) 123 Eur. Phys. J. C (2022) 82:869 Page 239 of 296 869 Table 86 Renormalization in determinations of mud , ms and, in some cases mu and md , with N f = 2 + 1 quark flavours Collab. Ref. N f Description ALPHA 19 [18] 2+1 Schrödinger functional. Table 87 Continuum extrapolations/estimation of lattice artifacts in the determinations of mc with N f = 2 + 1 + 1 quark flavours Collab. Refs. N f a [fm] Description ETM 21A [204] 2+1+1 0.07, 0.08, 0.09 Scale set from w0. Twisted mass action for charm quarks HPQCD 20A [23] 2+1+1 0.03, 0.042,0.06, 0.09, 0.12 Scale set from w0 and fπ . HISQ action for charm quarks Table 88 Continuum extrapolations/estimation of lattice artifacts in the determinations of mc with N f = 2 + 1 quark flavours Collab. Ref. N f a [fm] Description ALPHA 21 [227] 2+1 0.039–0.087 fm Wilson-clover action for the charm quark Petreczky 19 [26] 2+1 0.025–0.11 fm HISQ action for the charm quark. Scale set from r1 parameter of heavy quark potential and fπ Table 89 Chiral extrapolation/minimum pion mass in the determinations of mc with N f = 2 + 1 + 1 quark flavours Collab. Refs. N f Mπ,min [MeV] Description ETM 21A [204] 2+1+1 Physical HPQCD 20A [23] 2+1+1 Physical Table 90 Chiral extrapolation/minimum pion mass in the determinations of mc with N f = 2 + 1 quark flavours Collab. Refs. N f Mπ,min [MeV] Description ALPHA 21 [227] 2+1 198 Petreczky 19 [26] 2+1 161 Table 91 Finite-volume effects in the determinations of mc with N f = 2 + 1 + 1 quark flavours Collab. Refs. N f L [fm] Mπ,min L Description ETM 21A [204] 2+1+1 2.5–5.6 3.6 HPQCD 20A [23] 2+1+1 1.9–5.76 3.8 Table 92 Finite-volume effects in the determinations of mc with N f = 2 + 1 quark flavours Collab. Refs. N f L [fm] Mπ,min L Description ALPHA 21 [227] 2+1 2.5–4.0 4.2 Petreczky 19 [26] 2+1 1.6–5.2 4.2 Table 93 Renormalization in the determinations of mc with N f = 2 + 1 + 1 quark flavours Collab. Refs. N f Description ETM 21A [204] 2+1+1 Nonperturbative (RI/MOM) HPQCD 20A [23] 2+1+1 Nonperturbative (RI/SMOM) 123 869 Page 240 of 296 Eur. Phys. J. C (2022) 82:869 Table 94 Renormalization in the determinations of mc with N f = 2 + 1 quark flavours Collab. Refs. N f Description ALPHA 21 [227] 2+1 Schrödinger functional Petreczky 19 [26] 2+1 Not required Table 95 Continuum extrapolations/estimation of lattice artifacts in the determinations of mb with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f a [fm] Description HPQCD 21 [28] 2+1+1 0.03, 0.042, 0.06, 0.09 Scale set from w0 and fπ . HISQ action for charm quarks Table 96 Continuum extrapolations/estimation of lattice artifacts in the determinations of mb with N f = 2 + 1 quark flavours Collab. Ref. N f a [fm] Description Petreczky 19 [26] 2+1 0.025–0.11 fm HISQ action for the charm quark. Scale set from r1 parameter of heavy quark potential and fπ Table 97 Chiral extrapolation/minimum pion mass in the determinations of mb with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f Mπ,min [MeV] HPQCD 21 [28] 2+1+1 Physical Table 98 Chiral extrapolation/minimum pion mass in the determinations of mb with N f = 2 + 1 quark flavours Collab. Ref. N f Mπ,min [MeV] Petreczky 19 [26] 2+1 161 Table 99 Finite-volume effects in the determinations of mb with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f L [fm] Mπ,min L HPQCD 21 [28] 2+1+1 1.9–5.76 3.8 Table 100 Finite-volume effects in the determinations of mb with N f = 2 + 1 quark flavours Collab. Ref. N f L [fm] Mπ,min L Petreczky 19 [26] 2+1 1.6–5.2 4.2 Table 101 Lattice renormalization in the determinations of mb with N f = 2 + 1 + 1 flavours Collab. Ref. N f Description ETM 21A [204] 2+1+1 Nonperturbative (RI/MOM) Table 102 Lattice renormalization in the determinations of mb with N f = 2 + 1 flavours Collab. Ref. N f Description Petreczky 19 [26] 2+1 Not required 123 Eur. Phys. J. C (2022) 82:869 Page 241 of 296 869 C.2 Notes to Sect. 4 on |Vud | and |Vus | See Tables 103, 104, 105, 106, 107, 108. Table 103 Continuum extrapolations/estimation of lattice artifacts in the determinations of f+(0) Collab. Refs. N f a [fm] Description FNAL/MILC 18 [33] 2+1+1 0.042, 0.06, 0.09, 0.12, 0.15 HISQ quark action. Relative scale through r1 PACS 19 [294] 2+1 0.085 Nonperturbative O(a) clover quark action. Scale set from -baryon mass Table 104 Chiral extrapolation/minimum pion mass in determinations of f+(0). The subscripts RMS and π, 5 in the case of staggered fermions indicate the root-mean-square mass and the Nambu–Goldstone boson mass, respectively Collab. Refs. N f Mπ,min [MeV] Description FNAL/MILC 18 [33] 2+1+1 144RMS(135π,5) Chiral interpolation through NLO SU (3) PQ staggered χPT with continuum χPT at NNLO. Lightest Nambu–Goldstone mass is 135 MeV and lightest RMS mass is 144 MeV at the same gauge ensemble with a 0.06 fm PACS 19 [294] 2+1 135 Physical point simulation at a single pion mass 135 MeV Table 105 Finite-volume effects in determinations of f+(0). The subscripts RMS and π, 5 in the case of staggered fermions indicate the root- mean-square mass and the Nambu–Goldstone boson mass, respectively Collab. Refs. N f L [fm] Mπ,min L Description FNAL/MILC 18 [33] 2+1+1 2.6–5.8 4.2RMS(3.9π,5) The values correspond to Mπ,RMS = 144 MeV and Mπ,5 = 135 MeV, respectively PACS 19 [294] 2+1 10.9 7.5 Table 106 Continuum extrapolations/estimation of lattice artifacts in determinations of fK / fπ for N f = 2 + 1 + 1 simulations Collab. Refs. N f a [fm] Description ETM 21 [305] 2+1+1 0.07, 0.08, 0.09 Wilson-clover twisted mass quark action. Relative scale through gradient flow scale w0 and absolute scale trough fπ CalLat 20 [39] 2+1+1 0.06, 0.09, 0.12, 0.15 Möbius domain-wall valence quarks on gradient-flowed HISQ ensembles. Relative scale through the gradient flow scale w0 Table 107 Chiral extrapolation/minimum pion mass in determinations of fK / fπ for N f = 2 + 1 + 1 simulations Collab. Refs. N f Mπ,min [MeV] Description ETM 21 [305] 2+1+1 134 Chiral extrapolation based on NLO SU (2) χPT CalLat 20 [39] 2+1+1 157 Chiral extrapolation based on NNLO SU (3) χPT. We quote the root-mean-square (RMS) mass of the valence and valence-sea pions as Mπ,min. The smallest mass is 176 MeV for the HISQ sea pions, which do not enter until NNLO in the χPT expression Table 108 Finite-volume effects in determinations of fK / fπ for N f = 2 + 1 + 1 Collab. Refs. N f L [fm] Mπ,min L Description ETM 21 [305] 2+1+1 2.0–5.6 3.8 Three different volumes at Mπ = 253 MeV and a = 0.08 fm CalLat 20 [39] 2+1+1 2.4–7.2 3.8 Three different volumes at Mπ = 220 MeV and a = 0.12 fm 123 869 Page 242 of 296 Eur. Phys. J. C (2022) 82:869 C.3 Notes to Sect. 5 on low-energy constants See Tables 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120. Table 109 Continuum extrapolations/estimation of lattice artifacts in determinations of the SU (2) low-energy constants , F, ̄4, ̄6, and SU (3) low-energy constants 0, F0 Collab. Refs. N f a [fm] Description ETM 21A [204] 2+1+1 0.095, 0.082, 0.069 Scale set by fπ = 130.4(2) MeV ETM 21 [305] 2+1+1 0.092, 0.080, 0.068 Scale set by fπ = 130.4(2) MeV χQCD 21 [351] 2+1 0.063, 0.071, 0.084, 0.114 Same configs. as RBC/UKQCD 15E Wang 16 [353] 2+1 0.113 Same configs. as RBC/UKQCD 08A ETM 20A [350] 2 0.0914(15) Single lattice spacing Table 110 Continuum extrapolations/estimation of lattice artifacts in determinations of the low-energy constants related to the vector form factor of the pion Collab. Refs. N f a [fm] or a−1 [GeV] Description Gao 21 [375] 2+1 0.04, 0.06, 0.076 One lattice spacing at phys. pt χQCD 20 [376] 2+1 0.083–0.195 One lattice spacing below 0.1 fm Feng 19 [377] 2+1 1.015, 1.378, 1.730 a > 0.1 fm Table 111 Continuum extrapolations/estimation of lattice artifacts in determinations of the low-energy constants related to ππ scattering Collab. Refs. N f a [fm] Description ETM 20B [392] 2 0.0914(15) Single lattice spacing Mai 19 [391] 2 0.12 Single lattice spacing Culver 19 [390] 2 0.12 Single lattice spacing Table 112 Continuum extrapolations/estimation of lattice artifacts in determinations of the low-energy constants related to πK scattering Collab. Refs. N f a [fm] Description ETM 18B [96] 2+1+1 0.089, 0.082, 0.062 Same configuration with ETM 17G ETM 17G [97] 2+1+1 0.089, 0.082, 0.062 Scale set by the Sommer parameter r0 PACS-CS 13 [344] 2+1 0.09 Single lattice spacing Fu 11A [399] 2+1 0.15 Single lattice spacing NPLQCD 07B [401] 2+1 0.09, 0.125 Configurations generated by MILC NPLQCD 06B [349] 2+1 0.125 Single lattice spacing Table 113 Chiral extrapolation/minimum pion mass in determinations of the SU (2) low-energy constants , F, ̄4, ̄6, and SU (3) low-energy constants 0, F0 Collab. Refs. N f Mπ,min [MeV] Description ETM 21A [204] 2+1+1 134 4 pion masses in [134, 346] MeV ETM 21 [305] 2+1+1 135 4 pion masses in [134, 346] MeV χQCD 21 [351] 2+1 139 3 pion masses with different a Wang 16 [353] 2+1 220 8 (3) pion masses in val (sea) sector ETM 20A [350] 2 132 123 Eur. Phys. J. C (2022) 82:869 Page 243 of 296 869 Table 114 Chiral extrapolation/minimum pion mass in determinations of the low-energy constants related to the vector form factor of the pion Collab. Refs. N f Mπ,min [MeV] Description Gao 21 [375] 2+1 140 χQCD 20 [376] 2+1 139 Feng 19 [377] 2+1 ∼ 135 Table 115 Chiral extrapolation/minimum pion mass in determinations of the low-energy constants related to ππ scattering Collab. Refs. N f Mπ,min [MeV] Description ETM 20B [392] 2 134 2 pion masses Mai 19 [391] 2 224 Culver 19 [390] 2 226 2 pion masses Table 116 Chiral extrapolation/minimum pion mass in determinations of the low-energy constants related to πK scattering Collab. Refs. N f Mπ,min [MeV] Description ETM 18B [96] 2+1+1 276 5 pion masses in [230, 450] MeV ETM 17G [97] 2+1+1 276 5 pion masses in [230, 450] MeV PACS-CS 13 [344] 2+1 166 5 pion masses in [166, 707] MeV Fu 11A [399] 2+1 590 (RMS) 6 valence pion masses NPLQCD 07B [401] 2+1 413 (RMS) 4 pion masses NPLQCD 06B [349] 2+1 488 (RMS) 4 pion masses Table 117 Finite-volume effects in determinations of the SU (2) low-energy constants , F, ̄4, ̄6, and SU (3) low-energy constants 0, F0 Collab. Refs. N f L [fm] #V ETM 21A [204] 2+1+1 5.52 2 ETM 21 [305] 2+1+1 5.55 2 χQCD 21 [351] 2+1 5.4 2 at physical point. Wang 16 [353] 2+1 2.7 1 ETM 20A [350] 2 4.39 2 Table 118 Finite-volume effects in determinations of the low-energy constants related to the vector form factor of the pion Collab. Refs. N f L [fm] #V Gao 21 [375] 2+1 4.86 1 χQCD 20 [376] 2+1 6.24 3 Feng 19 [377] 2+1 6.22 3 Table 119 Finite-volume effects in determinations of the low-energy constants related to ππ scattering Collab. Refs. N f L [fm] #V ETM 20B [392] 2 2.92 2 Mai 19 [391] 2 2.88 1 Culver 19 [390] 2 2.88 3 123 869 Page 244 of 296 Eur. Phys. J. C (2022) 82:869 Table 120 Finite-volume effects in determinations of the low-energy constants related to πK scattering Collab. Refs. N f L [fm] #V ETM 18B [96] 2+1+1 2.832 2 ETM 17G [97] 2+1+1 2.832 2 PACS-CS 13 [344] 2+1 2.9 1 Fu 11A [399] 2+1 2.4 1 NPLQCD 07B [401] 2+1 2.52 2 NPLQCD 06B [349] 2+1 2.5 2 C.4 Notes to Sect. 6 on kaon mixing C.4.1 K → ππ decay amplitudes See Tables 121, 122, 123, 124. Table 121 Continuum extrapolations/estimation of lattice artifacts in determinations of the K → ππ decay amplitudes Collab. Refs. N f a [fm] Description RBC/UKQCD 20 [429] 2+1 0.143 Single lattice spacing RBC/UKQCD 15G [430] 2+1 0.143 Single lattice spacing RBC/UKQCD 15F [46] 2+1 0.114, 0.084 Combined chiral-continuum fit based on two values of the lattice spacing. Systematic error associated with the extrapolation to the continuum limit is negligible with respect to the statistical errors Table 122 Chiral extrapolation/minimum pion mass in determinations of the K → ππ decay amplitudes Collab. Refs. N f Mπ,min [MeV] Description RBC/UKQCD 20 [429] 2+1 142.3 Single pion mass value, close to the physical point RBC/UKQCD 15G [430] 2+1 143.1 Single pion mass value, close to the physical point RBC/UKQCD 15F [46] 2+1 139.1, 139.2 Single pion mass value – close to the physical point – at each of the two values of the lattice spacing Table 123 Finite volume effects in determinations of the K → ππ decay amplitudes Collab. Refs. N f L [fm] Mπ,min L Description RBC/UKQCD 20 [429] 2+1 4.6 3.3 Finite volume effects amount to a 7% systematic error contribution to the final error budget of A0 RBC/UKQCD 15G [430] 2+1 4.6 3.3 Finite volume effects amount to a 7% systematic error contribution to the final error budget of A0 RBC/UKQCD 15F [46] 2+1 5.5, 5.4 3.8 Finite volume effects amount to a 3.5% systematic error contribution to the error budget of Im(A2)/Re(A2) 123 Eur. Phys. J. C (2022) 82:869 Page 245 of 296 869 Table 124 Running and matching in determinations of the K → ππ decay amplitudes Collab. Refs. N f Ren. Running match Description RBC/UKQCD 20 [429] 2+1 RI PT1 Two different RI-SMOM schemes are used. The relative systematic errors due to the renormalisation of the relevant operators amount to 4%, while those arising from the computation of the Wilson coefficients in the MS scheme correspond to 12% RBC/UKQCD 15G [430] 2+1 RI PT1 Two different RI-SMOM schemes are used. The relative systematic errors due to the renormalisation of the relevant operators amount to 15%, while those arising from the computation of the Wilson coefficients in the MS scheme correspond to 12% RBC/UKQCD 15F [46] 2+1 RI PT1 Two different RI-SMOM schemes are used. The relative systematic errors for the conversion to MS are 2.9% for ReA0 and 7% for ImA0 C.4.2 Kaon B-parameter BK No new calculations w.r.t. the previous FLAG report. C.4.3 Kaon BSM B-parameters No new calculations w.r.t. the previous FLAG report. C.5 Notes to Sect. 7 on D-meson decay constants and form factors See Tables 125, 126, 127, 128, 129, 130, 131, 132, 133, 134. Table 125 Chiral extrapolation/minimum pion mass in N f = 2 + 1 determinations of the D- and Ds -meson decay constants. For actions with multiple species of pions, masses quoted are the RMS pion masses. The different Mπ,min entries correspond to the different lattice spacings Collab. Refs. N f Mπ,min [MeV] Description χQCD 20A [482] 2+1 114 Simulations are performed very close to the physical point and linear interpolations/extrapolations are used to correct for mismatches RBC/UKQCD 17, [55,70] 2+1 139, 139, 232 The lattice spacing, pion-mass and charm-quark RBC/UKQCD 18A mass dependences are fit simultaneously through a Taylor expansion in a2, m2 − m2phys( π π ) and 1/m H − 1/m D(s) Table 126 Chiral extrapolation/minimum pion mass in N f = 2 determinations of the D- and Ds -meson decay constants. For actions with multiple species of pions, masses quoted are the RMS pion masses. The different Mπ,min entries correspond to the different lattice spacings Collab. Refs. N f Mπ,min [MeV] Description Blossier 18, [58,479] 2 282, 194, 269 Linear fits (in m2π and in a 2) are used in the Balasubramanian 19 combined chiral/continuum extrapolation. NLO HMχPT expressions are used for a cross-check, concluding however that there are not enough data points to be sensitive to the NLO terms 123 869 Page 246 of 296 Eur. Phys. J. C (2022) 82:869 Table 127 Finite-volume effects in N f = 2 + 1 determinations of the D- and Ds -meson decay constants. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, the lightest masses are quoted Collab. Refs. N f L [fm] Mπ,min L Description χQCD 20A [482] 2+1 5.5 3.2 No explicit discussion of FSE RBC/UKQCD 17, [55,70] 2+1 5.5/2.7, 5.4/2.7, 3.4 3.86, 3.78, 4.11 FV errors estimated to be at the permille level by RBC/UKQCD 18A either comparing values of mπ L to the study of FSE by MILC in [17] or by 1-loop HMχPT Table 128 Finite-volume effects in N f = 2 determinations of the D- and Ds -meson decay constants. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, the lightest masses are quoted Collab. Refs. N f L [fm] Mπ,min L Description Blossier 18, [58,479] 2 2.4/3.6, 2.1/3.1/4.2, 2.3/3.1 5.2, 4.1, 4.2 No explicit discussion of FV effects, but Balasubramanian 19 mπ L > 4 always Table 129 Lattice spacings and description of actions used in N f = 2 + 1 determinations of the D- and Ds -meson decay constants Collab. Refs. N f a [fm] Continuum extrapolation Scale Setting χQCD 20A [482] 2+1 0.115 Cutoff effects are estimated to be m used for scale setting by 2% by comparing the result for RBC/UKQCD, which produced fDs with the one obtained in [24] the ensemble RBC/UKQCD 17, [55,70] 2+1 0.11, 0.08, 0.07 The lattice spacing, pion-mass and The lattice scale and physical RBC/UKQCD 18A charm-quark mass dependences light-quark masses have been are fit simultaneously through a determined using mπ , mK and Taylor expansion in a2, m as inputs m2 − m2phys( π π ) and 1/m H − 1/m D(s) Table 130 Lattice spacings and description of actions used in N f = 2 determinations of the D- and Ds -meson decay constants Collab. Refs. N f a [fm] Continuum extrapolation Scale Setting Blossier 18, [58,479] 2 0.075, 0.065, 0.048 Linear fits (in m2π and in a 2) are Scale set through fK Balasubramanian 19 used in the combined chiral/continuum extrapolation Table 131 Operator renormalization in N f = 2 + 1 determinations of the D- and Ds -meson decay constants Collab. Refs. N f Ren. Description χQCD 20A [482] 2+1 RI The decay constants are extracted from an exact lattice Ward identity or from NP renormalized operators RBC/UKQCD 17, [55,70] 2+1 mNPR The local current is renormalized nonperturbatively RBC/UKQCD 18A for the case of the unmixed action; however in the actual computation the domain wall height is chosen differently in the valence than in the sea and the effect of that on the renormalization constant is estimated to be 0.4% through a study in the RI/SMOM scheme 123 Eur. Phys. J. C (2022) 82:869 Page 247 of 296 869 Table 132 Operator renormalization in N f = 2 determinations of the D- and Ds -meson decay constants Collab. Refs. N f Ren. Description Blossier 18, Balasubramanian 19 [58,479] 2 SF NP renormalization and improvement of the axial current (am terms included at 1-loop) Table 133 Heavy-quark treatment in N f = 2 + 1 determinations of the D- and Ds -meson decay constants Collab. Refs. N f Action Description χQCD 20A [482] 2+1 Overlap on DW amc ≈ 0.73 set by using m Ds RBC/UKQCD 17, RBC/UKQCD 18A [55,70] 2+1 Möbius-DWF on Shamir- 0.18 < amh < 0.4. Charm discretization DWF or Möbius-DWF errors estimated using different ways to define the charm quark mass (through D, Ds or ηc) in the global fits Table 134 Heavy-quark treatment in N f = 2 determinations of the D- and Ds -meson decay constants Collab. Refs. N f Action Description Blossier 18, Balasubramanian 19 [58,479] 2 npSW amc ≤ 0.32. Axial current nonperturbatively improved (O(am) at 1-loop) C.5.1 Form factors for semileptonic decays of charmed hadrons See Tables 135, 136, 137, 138, 139. Table 135 Continuum extrapolations/estimation of lattice artifacts in N f = 2 + 1 + 1 determinations of form factors for semileptonic decays of charmed hadrons Collab. Refs. N f a [fm] Continuum extrapolation Scale setting HPQCD 21A [61] 2+1+1 0.042, 0.06, 0.09, 0.12, 0.15 Modified z-expansion fit combining Scale setting from fπ the continuum and chiral via the flow quantity extrapolations and the w0 [14,37,1071] momentum-transfer dependence. Discretization effects assumed dominated by the charm scale. Discretization errors on form factors between 0.4 and 1.2% as a function of the momentum transfer Zhang 21 [517] 2+1 0.080, 0.11 Continuum extrapolation combined Set from Wilson-flow with fit to q2-dependence of form quantity w0 factors in a “modified” z-expansion. Systematics estimated from difference between extrapolated results and results at smallest lattice spacing, and difference between two current renormalization methods HPQCD 20 [510] 2+1+1 0.06, 0.09, 0.12, 0.15 Modified z-expansion fit combining Scale setting from fπ the continuum and chiral via the flow quantity extrapolations and the w0 [14,37,1071] momentum-transfer dependence, and, for the heavy-HISQ spectator b quark, the dependence on 1/m Q . The analysis combines data with NRQCD b quarks and data with HISQ heavy quarks 123 869 Page 248 of 296 Eur. Phys. J. C (2022) 82:869 Table 136 Chiral extrapolation/minimum pion mass in determinations of form factors for semileptonic decays of charmed hadrons. For actions with multiple species of pions, masses quoted are the RMS pion masses for N f = 2 + 1 and the Goldstone mode mass for N f = 2 + 1 + 1. The different Mπ,min entries correspond to the different lattice spacings Collab. Refs. N f Mπ,min [MeV] Description HPQCD 21A [61] 2+1+1 315, 329, 129, Modified z-expansion fit combining the continuum and chiral 132, 131 extrapolations and the momentum-transfer dependence. Polynomial dependence on quark masses, supplemented by a pion chiral logarithm. Fit result compared with alternative approach based on cubic splines in q2 Zhang 21 [517] 2+1 300, 290 Dependence on pion mass neglected. No estimate of resulting systematic uncertainty HPQCD 20 [510] 2+1+1 329, 316, Modified z-expansion fit combining the continuum and chiral 132/305, 131/305 extrapolations and the momentum-transfer dependence, and, for the heavy-HISQ spectator b quark, the dependence on 1/m Q . The analysis combines data with NRQCD b quarks and data with HISQ heavy quarks Table 137 Finite-volume effects in determinations of form factors for semileptonic decays of charmed hadrons. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, the lightest pion masses are quoted Collab. Refs. N f L [fm] Mπ,min L Description HPQCD 21A [61] 2+1+1 2.73, 2.72,  3.7 Finite volume correction included in chiral fit, 2.81/5.62, claimed to be a negligible effect. Effect of 2.93/5.87, frozen topology in finest ensemble not 2.45/4.89 discussed Zhang 21 [517] 2+1+1 2.6, 2.6  3.8 No discussion of finite-volume effects HPQCD 20 [510] 2+1+1 2.72, 2.81,  3.8 Physical point ensemble at a 0.15 fm has 2.93/5.87, mπ L = 3.3; the statement mπ L  3.8 2.45/4.89 applies to the other five ensembles Table 138 Operator renormalization in determinations of form factors for semileptonic decays of charmed hadrons Collab. Refs. N f Ren. Description HPQCD 21A [61] 2+1+1 NP Vector current normalized by imposing Ward identity at zero recoil Zhang 21 [517] 2+1 NP Local vector current renormalized using ratio to conserved vector current. Axial current renormalized using ratio of off-shell quark matrix elements HPQCD 20 [510] 2+1+1 NP Vector current normalized by imposing Ward identity at zero recoil Table 139 Heavy-quark treatment in determinations of form factors for semileptonic decays of charmed hadrons Collab. Refs. N f Action Description HPQCD 21A [61] 2+1+1 HISQ Bare charm-quark mass 0.194  amc  0.8605 Zhang 21 [517] 2+1+1 SW Bare charm-quark mass 0.235  amc  0.485. No O(a) improvement of currents Charm: HISQ Bottom Bare charm-quark HIQS mass 0.274  amc  0.827HPQCD 20 [510] 2+1+1 (spectator): Bare bottom-quark HIQS mass 0.274  amb  0.8 HISQ and NRQCD 123 Eur. Phys. J. C (2022) 82:869 Page 249 of 296 869 C.6 Notes to Sect. 8 on B-meson decay constants, mixing parameters and form factors C.6.1 B(s)-meson decay constants See Tables 140, 141, 142, 143, 144, 145, 146, 147, 148. Table 140 Chiral extrapolation/minimum pion mass in determinations of the B- and Bs -meson decay constants for N f = 2 + 1 simulations. For actions with multiple species of pions, masses quoted are the RMS pion masses. The different Mπ,min entries correspond to the different lattice spacings Collab. Ref. N f Mπ,min [MeV] Description RBC/UKQCD 18A [70] 2+1 139, 139, 232 Three or four light-quark masses per lattice spacing except for the finest lattice spacing. Generic fits to m2 − mphys( )2 and a2π π in the combined chiral-continuum extrapolation, with systematic errors estimated to be from 0.3% to 0.5% in fBs / fB Table 141 Chiral extrapolation/minimum pion mass in determinations of the B- and Bs -meson decay constants for N f = 2 simulations. For actions with multiple species of pions, masses quoted are the RMS pion masses (where available). The different Mπ,min entries correspond to the different lattice spacings Collab. Ref. N f Mπ,min [MeV] Description Balasubramamian 19 [58] 2 282, 194, 269 Two pion masses for the coarsest and finest lattice spacings, and four pion masses for the intermediate lattice spacing. Generic fits to m2π and a2 in the combined continuum-chiral extrapolation, with systematic effects obtained by including a generic NLO chiral term m2π log m 2 π . Detailed error budget not provided, but total systematic uncertainty estimated as 1.1–1.2% Table 142 Finite-volume effects in determinations of the B- and Bs -meson decay constants for N f = 2+1 simulations. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings Collab. Ref. N f L [fm] Mπ,min L Description RBC/UKQCD 18A [70] 2+1 2.65/5.47, 3.86, 3.78, 4.11 Finite-volume effects are estimated 2.65/5.35,3.49 to be 0.18% in fBs / fB Table 143 Finite-volume effects in determinations of the B- and Bs -meson decay constants for N f = 2 simulations. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings Collab. Ref. N f L [fm] Mπ,min L Description Balasubramamian 19 [58] 2 2.4/3.6, 4.1, 4.1, 4.0 No explicit estimate of FV errors, but expected to be 2.1/3.1/4.2, much smaller than other uncertainties 2.3/3.1 Table 144 Continuum extrapolations/estimation of lattice artifacts in determinations of the B and Bs meson decay constants for N f = 2 + 1 simulations Collab. Ref. N f a [fm] Continuum extrapolation Scale setting RBC/UKQCD 18A [70] 2+1 0.11, 0.084, Combined continuum and chiral Scale set by the baryon mass. No 0.073 extrapolation with linear in a2 estimate for scale uncertainty, but term. Systematic errors estimated expected to be negligible in fBs / fB to be from 0.3% to 0.5% in fBs / fB 123 869 Page 250 of 296 Eur. Phys. J. C (2022) 82:869 Table 145 Description of the renormalization/matching procedure adopted in the determinations of the B- and Bs -meson decay constants for N f = 2 + 1 simulations Collab. Ref. N f Ren. Description [70] 2+1 − Mass-independent operator renormalization cancels in fBs / fB Table 146 Description of the renormalization/matching procedure adopted in the determinations of the B- and Bs -meson decay constants for N f = 2 simulations Collab. Ref. N f Ren. Description Balasubramamian 19 [58] 2 PT2 Perturbative coefficient relating HQET and QCD currents included at NNLO in continuum perturbation theory. Systematic uncertainties estimated using fits including only NLO coefficients Table 147 Heavy-quark treatment in N f = 2 + 1 determinations of the B-and Bs -meson decay constants Collab. Ref. N f Action Description RBC/UKQCD 18A [70] 2+1 DWF Linear extrapolation in inverse heavy quark mass 1/MH , with systematic errors estimated, by excluding heaviest and lightest valence quark masses, to be 0.47% in fBs / fB . HQ discretization effects are estimated to be 0.01% in fBs / fB Table 148 Heavy-quark treatment in N f = 2 determinations of the B-and Bs -meson decay constants Collab. Ref. N f Action Description Balasubramamian 19 [58] 2 Wilson Expansion in inverse of pseudoscalar heavy-strange meson mass, 1/MHs . HQ discretization effects included in generic fit linear in a2(aM 2Hs ) . Detailed error budget not provided, but systematic uncertainty estimated as 1.1–2.3% in fBs C.6.2 B(s)-meson mixing matrix elements See Tables 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159. Table 149 Continuum extrapolations/estimation of lattice artifacts in determinations of the neutral B-meson mixing matrix elements for N f = 2 + 1 + 1 simulations Collab. Ref. N f a [fm] Continuum extrapolation Scale setting HPQCD 19A [71] 2+1+1 0.15, 0.12, 0.09 Discretization errors start from Scale setting done using ϒ and ϒ ′ mass α a2s and are included in the splitting [552] systematic error. It is estimated as 1.8% for individual bag parameters. Residual α 2sa and a4 errors from wrong-spin contributions are subtracted by including them in the chiral fit 123 Eur. Phys. J. C (2022) 82:869 Page 251 of 296 869 Table 150 Continuum extrapolations/estimation of lattice artifacts in determinations of the neutral B-meson mixing matrix elements for N f = 2+1 simulations Collab. Ref. N f a [fm] Continuum extrapolation Scale setting RBC/UKQCD 18A [70] 2+1 0.11, 0.08, 0.07 Combined continuum (a2) and heavy Lattice scale and target quark masses quark (1/m H ) extrapolation with the are set using , K and π masses LO pion mass dependence (m2π ) in [8,55,117] the global fit Table 151 Chiral extrapolation/minimum pion mass in determinations of the neutral B-meson mixing matrix elements. For actions with multiple species of pions, masses quoted are the RMS pion masses (where available). The different Mπ,min entries correspond to the different lattice spacings Collab. Refs. N f Mπ,min [MeV] Description HPQCD 19A [71] 2+1+1 311, 241, – Pion mass in the Goldston channel is as small as 130 MeV for two coarser lattices. NLO HMrSχPT is used with NNLO analytic terms and other discretization errors. Staggered wrong-spin contributions are included RBC/UKQCD 18A [70] 2+1 139, 139, 234 Combined continuum (a2) and heavy quark (1/m H ) extrapolation with the LO pion mass dependence (m2π ) in the global fit Table 152 Finite-volume effects in determinations of the neutral B-meson mixing matrix elements. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, masses quoted are the RMS pion masses (where available) Collab. Refs. N f L [fm] Mπ,min L Description HPQCD 19A [71] 2+1+1 2.4/3.5/4.6, 2.9/3.8/5.7, 2.8 7.3, 7.0, – FV error is estimated to be negligible from FV HMχPT RBC/UKQCD 18A [70] 2+1 2.7/5.5, 2.6/5.3, 3.5 3.9, 3.8, 4.0 FV error is estimated to be less than 0.18% for SU (3)-breaking ratios from FV HMχPT Table 153 Operator renormalization in determinations of the neutral B-meson mixing matrix elements Collab. Refs. N f Ren. Description HPQCD 19A [71] 2+1+1 PT1 HISQ-NRQCD 4-quark operators are matched through O(1/M) and renormalized to 1-loop: included are those of O(αs), O(QCD/M), O(αs/aM), O(αs QCD/M). Remnant error is dominated by O(αsQCD/M) 2.9% and O(α2s ) 2.1% for individual bag parameters. Associated error for their SU(3) breaking ratio are negligible RBC/UKQCD 18A [70] 2+1 – Operators are renormalized multiplicatively due to chiral symmetry of DWF. No need to calculate the renormalization factor since only the SU (3) breaking ratios are examined 123 869 Page 252 of 296 Eur. Phys. J. C (2022) 82:869 Table 154 Heavy-quark treatment in determinations of the neutral B-meson mixing matrix elements Collab. Refs. N f Action Description HPQCD 19A [71] 2+1+1 NRQCD See the entry in Table 153 RBC/UKQCD 18A [70] 2+1 DWF Domain-wall fermion with 3 stout-smearing extends the reach to heavy mass, allowing to simulate up to half of the b-quark mass. Heavy mass errors on ξ are estimated as 0.8% from fitting range and 0.4% from higher order (1/M2) by power counting C.6.3 Form factors entering determinations of |Vub| (B → πν, Bs → Kν, b → pν̄) Table 155 Continuum extrapolations/estimation of lattice artifacts in determinations of B → πν, Bs → Kν, and b → pν̄ form factors Collab. Ref. N f a [fm] Continuum extrapolation Scale setting FNAL/MILC 19 [573] 2+1 0.06, 0.09, 0.12 HMrSχPT expansion used at next-to-leading Relative scale r1/a set from order in SU (2) and leading order in 1/MB , the static-quark potential. including next-to-next-to-leading-order Absolute scale r1, (NNLO) analytic and generic discretization including related terms. Hard kaons assumed to decouple. uncertainty estimates, Systematic uncertainties estimated by taken from [54] varying fit ansatz and data range. The (stat + chiral extrap + HQ discretization + gπ ) uncertainty dominates the error budget, ranging from 2–3% at q2  21 GeV2 to up to 8–10% in the lower end of the accessed q2 interval Table 156 Chiral extrapolation/minimum pion mass in determinations of B → πν, Bs → Kν, and b → pν̄ form factors. For actions with multiple species of pions, masses quoted are the RMS pion masses. The different Mπ,min entries correspond to the different lattice spacings Collab. Ref. N f Mπ,min [MeV] Description FNAL/MILC 19 [573] 2+1 255, 277, 456 HMrSχPT expansion used at next-to-leading order in SU (2) and leading order in 1/MB , including next-to-next-to-leading-order (NNLO) analytic and generic discretization terms. Hard kaons assumed to decouple. Systematic uncertainties estimated by varying fit ansatz and data range Table 157 Finite-volume effects in determinations of B → πν, Bs → Kν, and b → pν̄ form factors. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, the lightest masses are quoted Collab. Ref. N f L [fm] Mπ,min L Description FNAL/MILC 19 [573] 2+1 3.8, 2.5/2.9/3.6/5.8, 2.9  3.8 FV effects estimated by comparing infinite volume integrals with finite sums in HMrSχPT, found to be negligible Table 158 Operator renormalization in determinations of B → πν, Bs → Kν, and b → pν̄ form factors Collab. Ref. N f Ren. Description FNAL/MILC 19 [573] 2+1 mNPR Perturbative truncation error estimated at 1% with size of 1-loop correction on next-to-finest ensemble 123 Eur. Phys. J. C (2022) 82:869 Page 253 of 296 869 Table 159 Heavy-quark treatment in determinations of B → πν, Bs → Kν, and b → pν̄ form factors Collab. Ref. N f Action Description FNAL/MILC 19 [573] 2+1 Fermilab (See comments for continuum limit extrapolation.) C.6.4 Form factors for rare decays of beauty hadrons See Tables 160, 161, 162, 163, 164. Table 160 Continuum extrapolations/estimation of lattice artifacts in determinations of form factors for rare decays of beauty hadrons Collab. Ref. N f a [fm] Continuum extrapolation Scale setting Meinel 20 [648] 2+1 0.0828(3), 0.1106(3) Combined chiral-continuum Scale setting using mass in Ref. [8] extrapolation as part of the expansion of form factor shape in powers of w − 1. Systematic uncertainty estimated by repeating fit with added higher-order terms Table 161 Chiral extrapolation/minimum pion mass in determinations of form factors for rare decays of beauty hadrons. For actions with multiple species of pions, masses quoted are the RMS pion masses. The different Mπ,min entries correspond to the different lattice spacings Collab. Ref. N f Mπ,min [MeV] Description Meinel 20 [648] 2+1 303, 340 Combined chiral-continuum extrapolation as part of the expansion of form factor shape in powers of w − 1. Systematic uncertainty estimated by repeating fit with added higher-order terms Table 162 Finite-volume effects in determinations of form factors for rare decays of beauty hadrons. Each L-entry corresponds to a different lattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, the lightest masses are quoted Collab. Ref. N f L [fm] Mπ,min L Description Meinel 20 [648] 2+1 2.7, 2.7 4.1, 4.6 FV effects not quantified. Effects from unstable ∗(1520) not quantified Table 163 Operator renormalization in determinations of form factors for rare decays of beauty hadrons Collab. Ref. N f Ren. Description Meinel 20 [648] 2+1 mNPR Residual matching factors ρ computed at one-loop for vector and axial-vector currents, but at tree-level only for tensor currents. A systematic uncertainty is assigned to ρTμν as the double of max(|ρAμ − 1|, |ρVμ − 1|) Table 164 Heavy-quark treatment in determinations of form factors for rare decays of beauty hadrons Collab. Ref. N f Action Description Meinel 20 [648] 2+1 Columbia RHQ Discretization errors discussed as part of combined chiral-continuum-w fit. Higher-order fit also includes O(αsa|p|) terms to account for missing radiative corrections to O(a) improvement of the currents 123 869 Page 254 of 296 Eur. Phys. J. C (2022) 82:869 C.6.5 Form factors entering determinations of |Vcb| (B (∗) (∗)(s) → D(s) ν, b → c ν̄) and R(D(s)) See Tables 165, 166, 167, 168, 169. Table 165 Continuum extrapolations/estimation of lattice artifacts in N f = 2 + 1 determinations of B (∗) (∗)(s) → D(s) ν and b → c ν̄ form factors, and of R(D(s)) Collab. Refs. N f a [fm] Continuum extrapolation Scale setting Meinel 21 [638] 2+1 0.0828(3), Combined chiral-continuum extrapolation as part of the Scale setting using mass 0.1106(3) expansion of form factor shape in powers of w − 1. in Ref. [8] Systematics estimated by varying fit form HPQCD 19 [613] 2+1+1 0.044, Combined chiral-continuum extrapolation. Fractional Scale setting from fπ via 0.059, error from continuum limit and extrapolation to the flow quantity ≈ 0.09 physical b mass at zero recoil quoted as 1.20% w0 [14,37,1071] HPQCD 19B [612] 2+1+1 0.044, Combined chiral-continuum extrapolation. Fractional Scale setting from fπ via 0.059, error from continuum limit and extrapolation to the flow quantity ≈0.09 physical b mass at zero recoil quoted as 0.73% and w0 [14,37,1071] 0.69%, respectively Table 166 Chiral extrapolation/minimum pion mass in N f = 2 + 1 determinations of B(s) → D(∗) (∗)(s) ν and b → c ν̄ form factors, and of R(D(s)). For actions with multiple species of pions, masses quoted are the RMS pion masses for N f = 2 + 1 and the Goldstone mode mass for N f = 2 + 1 + 1. The different Mπ,min entries correspond to the different lattice spacings Collab. Refs. N f Mπ,min [MeV] Description Meinel 21 [638] 2+1 303, 340 Combined chiral-continuum extrapolation as part of the expansion of form factor shape in powers of w − 1. Systematic uncertainty estimated by repeating fit with added higher-order terms HPQCD 19 [613] 2+1+1 315, 329, 129 [16] Combined chiral-continuum extrapolation using rSχPT. No specific uncertainty coming from chiral extrapolation quoted HPQCD 19B [612] 2+1+1 315, 329, 129 [16] Combined chiral-continuum extrapolation using rSχPT. No specific uncertainty coming from chiral extrapolation quoted Table 167 Finite-volume effects in determinations of B → D(∗)(s) (s) ν and b → (∗)c ν̄ form factors, and of R(D(s)). Each L-entry corresponds to a differentlattice spacing, with multiple spatial volumes at some lattice spacings. For actions with multiple species of pions, the lightest pion masses are quoted Collab. Refs. N f L [fm] Mπ,min L Description Meinel 21 [638] 2+1 2.7, 2.7 4.1, 4.6 FV effects not quantified. Effects from unstable ∗c not quantified HPQCD 19 [613] 2+1+1 2.73, 2.72,2.81/5.62 4.3, 4.5, FV effects neglected. This includes the effect of frozen 3.7/4.5, topology in the finest ensemble 3.9/4.5, 3.3/3.8 HPQCD 19B [612] 2+1+1 2.73, 2.72,2.81/5.62 4.3, 4.5, FV effects neglected. This includes the effect of frozen 3.7/4.5, topology in the finest ensemble 3.9/4.5, 3.3/3.8 123 Eur. Phys. J. C (2022) 82:869 Page 255 of 296 869 Table 168 Operator renormalization in determinations of B(s) → D(∗) (∗)(s) ν and b → c ν̄ form factors, and of R(D(s)) Collab. Refs. N f Ren. Description Meinel 21 [638] 2+1 mNPR Residual matching factors ρ computed at one-loop for vector and axial-vector currents, but at tree-level only for tensor currents. A systematic uncertainty is assigned to ρTμν as the double of max(|ρAμ − 1|, |ρVμ − 1|) HPQCD 19 [613] 2+1+1 NP Currents normalized non-perturbatively by imposing that f+(0) = f0(0) = 1 for transitions involving initial and final mesons with identical masses HPQCD 19B [612] 2+1+1 NP Currents renormalized nonperturbatively using PCAC relation Table 169 Heavy-quark treatment in determinations of B(s) → D(∗) (∗)(s) ν and b → c ν̄ form factors, and of R(D(s)) Collab. Refs. N f Action Description Meinel 21 [638] 2+1 Columbia RHQ for both the Discretization errors discussed as part of combined b and c quarks chiral-continuum-w fit. Higher-order fit also includes O(αsa|p|) terms to account for missing radiative corrections to O(a) improvement of the currents HPQCD 19 [613] 2+1+1 HISQ for both the b and c Values of bare heavy quark masses up to amh = 0.8. quarks Fractional error from continuum limit and extrapolation to physical b mass at zero recoil quoted as 1.20% HPQCD 19B [612] 2+1+1 HISQ for both the b and c Values of bare heavy-quark masses up to amh = 0.8. quarks Fractional error from extrapolation to physical b mass quoted as 0.69% C.7 Notes to Sect. 9 on the strong coupling αs In this section we provide more detailed information on the simulations used to calculate the strong coupling αs . We present this information only for results that have appeared since FLAG 19. For information on previous calculations not listed here we refer the previous reports FLAG 19 [4] and FLAG 16 [3]. C.7.1 Renormalization scale and perturbative behaviour See Tables 170, 171. Table 170 Renormalization scale and perturbative behaviour of αs determinations for N f = 0 Collab. Refs. N f αeff nl Description Nada 20 [694] 0 SF: 0.08–0.15 2 Step-scaling with GF, non-pert matching to SF, GF: 0.08–0.24 Same gauge configurations as in [693] Husung 20 [695] 0 0.17−0.36 3 Estimated from αqq Dalla Brida 19 [693] 0 SF: 0.08–0.15 2 Step-scaling with GF, non-pert matching to SF GF: 0.08–0.95 123 869 Page 256 of 296 Eur. Phys. J. C (2022) 82:869 Table 171 Renormalization scale and perturbative behaviour of αs determinations for N f = 3 Collab. Refs. N f αeff nl Description Petreczky 20 [696] 2+1 0.22−0.38 2 mh = mc − 4mc Ayala 20 [74] 2+1 0.2−0.4 3 1/r > 2 GeV Cali 20 [76] 2+1 0.235−0.308 3 |x | = 0.13−0.19 fm Boito 20 [698] 2+1 0.38 2 Only mh = mc is used TUMQCD 19 [75] 2+1 0.2−0.4 3 r < 0.073 fm Zafeiropoulos 19 [699] 2+1 0.35–0.42 3 αT for p ∼ 3.0−3.7 GeV Petreczky 19 [26] 2+1 0.31, 0.38 2 Only results for mh = mc, 1.5mc are reviewed C.7.2 Continuum limit See Tables 172, 173. Table 172 Continuum limit for αs determinations with N f = 0 Collab. Refs. N f a μ Description GF: 0.125 < aμ < 0.042 Step scaling GF scheme, Nada 20 [694] 0 SF: 0.17 < aμ < 0.063 Non-pert matching to SF scheme Husung 20 [695] 0 aμ < 0.57 Only r/a > 3/5 are used in the analysis GF: 0.125 < aμ < 0.042 Step scaling GF scheme, Dalla Brida 19 [693] 0 SF: 0.17 < aμ < 0.063 Non-pert matching to SF scheme Table 173 Continuum limit for αs determinations with N f = 3 Collab. Refs. N f a μ Description Petreczky 20 [696] 2+1 aμ = 0.32−5.40 a = 0.025−0.104 fm, mh = mc − 4mc √ Ayala 20 [74] 2+1 aμ < 0.71 Only r/a > 8 are considered Cali 20 [76] 2+1 aμ = 0.21−0.58 a = 0.039−0.076 fm Boito 20 [698] 2+1 NA Previously published continuum results were used TUMQCD 19 [75] 2+1 aμ < 2 r/a values down to 1 are used but it is c√hecked that the result does not change if only r/a > 8 are considered Zafeiropoulos 19 [699] 2+1 aμ ∼ 1.3−1.6 Two lattice spacings a = 0.11, 0.08 fm αeff = 0.3 not reached Petreczky 19 [26] 2+1 aμ < 0.49 Only results for mh = mc, 1.5mc are reviewed C.8 Notes to Sect. 10 on nucleon matrix elements See Tables 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200. 123 Eur. Phys. J. C (2022) 82:869 Page 257 of 296 869 Table 174 Continuum extrapolations/estimation of lattice artifacts in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 + 1 quark flavours Collab. Refs. N f a [fm] Description CalLat 19 [100] 2+1+1 0.15, 0.12, 0.09 Extrapolation to the physical point via simultaneous fit in the lattice spacing, Mπ and Mπ L , including terms of order a2 and a4 ETM 19 [971] 2+1+1 0.08 Single lattice spacing Table 175 Continuum extrapolations/estimation of lattice artifacts in determinations of the isovector axial, scalar and tensor charges with N f = 2+1 quark flavours Collab. Refs. N f a [fm] Description χQCD 21 [978] 2+1 0.06, 0.08, 0.11, 0.14 Extrapolation performed using a quadratic term in a as part of a simultaneous fit in a, Mπ and Mπ L NME 21 [972] 2+1 0.07, 0.09, 0.13 Extrapolation performed using a linear term in a as part of a simultaneous fit in a, Mπ and Mπ L RBC/UKQCD 19 [977] 2+1 0.14 Single lattice spacing LHPC 19 [851] 2+1 0.09, 0.12 No extrapolation performed; final result taken from finer lattice with an enlarged error bar to account for result at coarser a Mainz 19 [102] 2+1 0.05, 0.06, 0.08, 0.09 Extrapolation performed as part of a simultaneous fit in a, Mπ and Mπ L PACS 18A [849] 2+1 0.085 Single lattice spacing Table 176 Chiral extrapolation/minimum pion mass in determinations of the isovector axial, scalar and tensor charges with N f = 2+ 1+ 1 quark flavours Collab. Ref. N f Mπ,min [MeV] Description CalLat 19 [100] 2+1+1 135, 130, 220 Fit performed including analytic and non-analytic terms in Mπ up to order M4π ETM 19 [971] 2+1+1 139 Single pion mass within 3% of the physical value Table 177 Chiral extrapolation/minimum pion mass in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 quark flavours Collab. Ref. N f Mπ,min [MeV] Description χQCD 21 [978] 2+1 139, 171, 302, 337, 371 Extrapolation performed using quadratic terms in Mπ from partially quenched chiral perturbation theory as part of a simultaneous fit in a, Mπ and Mπ L NME 21 [972] 2+1 170, 170, 270, 285 Extrapolation performed using a quadratic term in Mπ as part of a simultaneous fit in a, Mπ and Mπ L RBC/UKQCD 19 [977] 2+1 170, 250 Does not quote extrapolated value, but shows low mass dependence LHPC 19 [851] 2+1 133, 137 Does not perform extrapolation; comparison of results at two near-physical pion masses Mainz 19 [102] 2+1 220, 290, 200, 260 Extrapolation performed using logarithmic and quadratic terms in Mπ as part of a simultaneous fit in a, Mπ and Mπ L PACS 18A [849] 2+1 135 Single, physical pion mass 123 869 Page 258 of 296 Eur. Phys. J. C (2022) 82:869 Table 178 Finite-volume effects in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f L [fm] Mπ,min L Description CalLat 19 [100] 2+1+1 2.4–7.2, 4.9, 3.8, 4.7 Fit performed√including a term of the form 2.9–5.8, M2 e−Mπ Lπ / Mπ L as part of a 2.9–4.2 simultaneous fit in a2, Mπ and Mπ L ETM 19 [971] 2+1+1 5.12, 4.5–6.0 3.62, 2.98 Final result quoted from the Mπ L = 3.62 ensemble only Table 179 Finite-volume effects in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 quark flavours Collab. Refs. N f L [fm] Mπ,min L Description χQCD 21 [978] 2+1 4.5, 3.7, Extrapolation performed including a term of the 5.3, 3.9, form e−Mπ L as part of a simultaneous fit in a, 2.6, 3.6, Mπ and Mπ L 2.6, 3.8, 1.92 3.6 NME 21 [972] 2+1 4.2, 5.87, Extrapolation perfo√rmed including a term of the 2.9–4.3, 4.09, form M2 −Mπ Lπ e / Mπ L as part of a 4.3–5.8, 3.75, simultaneous fit in a2, Mπ and Mπ L 3.4–5.0 4.28 RBC/UKQCD 19 [977] 2+1 4.5 4.0 Does not quote extrapolated value LHPC 19 [851] 2+1 5.9, 4.0, Does not quote extrapolated value 5.6 3.9 Mainz 19 [102] 2+1 2.8–4.1, 4.7, Extrapolation perfo√rmed including a term of the 2.4–3.6, 5.3, form M2 −Mπ Lπ e / Mπ L as part of a 2.1–4.1, 4.2, simultaneous fit in a2, Mπ and Mπ L 2.4–3.2 4.3 PACS 18A [849] 2+1 10.8 7.4 Single spatial volume of 10.8 fm at physical pion mass Mπ L = 7.4 Table 180 Renormalization in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 + 1 quark flavours Collab. Ref. N f Ren. CalLat 19 [100] 2+1+1 RI-MOM ETM 19 [971] 2+1+1 RI-MOM Table 181 Renormalization in determinations of the isovector axial, scalar and tensor charges with 2 + 1 quark flavours Collab. Ref. N f Ren. χQCD 21 [978] 2+1 RI-MOM NME 21 [972] 2+1 RI-SMOM RBC/UKQCD 19 [977] 2+1 RI-SMOM LHPC 19 [851] 2+1 RI-MOM, RI-SMOM Mainz 19 [102] 2+1 RI-MOM, SF PACS 18A [849] 2+1 SF 123 Eur. Phys. J. C (2022) 82:869 Page 259 of 296 869 Table 182 Control of excited state contamination in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 + 1 quark flavours. The comma-separated list of numbers in square brackets denote the range of source-sink separations τ (in fermi) at each value of the bare coupling Collab. Ref. N f τ [fm] Description CalLat 19 [100] 2+1+1 All Two-state fits to the τ -dependence of summed operator insertion for τ ≥ 0.3 fm ETM 19 [971] 2+1+1 [0.64−1.6] Fits to the τ - and t-dependence of three-point correlators using multiple methods (plateau, two- and three-state fits, summation), with the two-state fits quoted for the final result Table 183 Control of excited state contamination in determinations of the isovector axial, scalar and tensor charges with N f = 2 + 1 quark flavours. The comma-separated list of numbers in square brackets denote the range of source-sink separations τ (in fermi) at each value of the bare coupling Collab. Refs. N f τ [fm] Description χQCD 21 [978] 2+1 [0.9–1.1] Fits to the τ - and t-dependence of three-point [1–1.2] correlators using up to three lowest-lying [0.9–1.3,0.9–1.3] states [1–1.5] NME 21 [972] 2+1 [1.0–1.8] Fits to the τ - and t-dependence of three-point [0.75–1.3,0.75–1.7] correlators using three lowest-lying states. [0.72–1.46,0.72–1.46] [0.8–1.4,0.9–1.5] RBC/UKQCD 19 [977] 2+1 [1.0,1.3] Result quoted from t=1.3 fm point only; consistency in results from t = 1.0 fm is demonstrated LHPC 19 [851] 2+1 [0.4–1.4] Multiple analysis methods used (plateau, [0.9–1.5] summation, and two-state fits to each), and combined to produce final result Mainz 19 [102] 2+1 [1.0–1.4,1.0–1.4,1.0–1.4] Fits to the τ - and t-dependence of correlator [1.0–1.5,1.0–1.5] ratios using the two lowest-lying states [1.0–1.4,1.0–1.4,1.0–1.4,1.0–1.4] [1.0–1.4,1.0–1.3] PACS 18A [849] 2+1 [0.9−1.4] Average of plateau values at the three largest source-sink separations Table 184 Continuum extrapolations/estimation of lattice artifacts in determinations of gq qA and gT Collab. Ref. N f a [fm] Description ETM 19 [971] 2+1+1 0.08 Not estimated Table 185 Chiral extrapolation/minimum pion mass in determinations of gq and gqA T Collab. Ref. N f Mπ,min [MeV] Description ETM 19 [971] 2+1+1 139 Simulate close to M physπ Table 186 Finite-volume effects in determinations of gqA and g q T Collab. Ref. N f L [fm] Mπ,min L Description ETM 19 [971] 2+1+1 5.1 3.6 FVE are anticipated to be small based on an investigation using two near physical point N f = 2 ensembles with Mπ L = 3.0 and 4.0 123 869 Page 260 of 296 Eur. Phys. J. C (2022) 82:869 Table 187 Renormalization in determinations of gqA and g q T Collab. Ref. N f Ren. ETM 19 [971] 2+1+1 RI-SMOM Table 188 Control of excited state contamination in determinations of gqA and g q T . The comma-separated list of numbers in square brackets denote the range of source-sink separations τ (in fermi) at each value of the bare coupling Collab. Ref. N f τ [fm] Description ETM 19 [971] 2+1+1 [0.6−1.6]/all Two-state fit to all τ . A comparison is made with three-state fits, plateau fits and the summation method Table 189 Continuum extrapolation/estimation of lattice artifacts in direct determinations of σπN and σs Collab. Ref. N f a [fm] Description ETM 19 [971] 2+1+1 0.08 Not estimated Table 190 Chiral extrapolation/minimum pion mass in direct determinations of σπN and σs Collab. Ref. N f Mπ,min [MeV] Description ETM 19 [971] 2+1+1 139 Simulate close to M physπ Table 191 Finite-volume effects in direct determinations of σπN and σs Collab. Ref. N f L [fm] Mπ,min L Description ETM 19 [971] 2+1+1 5.1 3.6 FVE are found to be small from an investigation using two near physical point N f = 2 ensembles with Mπ L = 3.0 and 4.0 Table 192 Renormalization for direct determinations of σπN and σs . The type of renormalization (Ren.) is given for σπN first and σs second. The label ’na’ indicates that no renormalization is required Collab. Ref. N f Ren. Description ETM 19 [971] 2+1+1 na/na 6.3 cm Table 193 Control of excited state contamination in direct determinations of σπN and σs . The comma-separated list of numbers in square brackets denote the range of source-sink separations τ (in fermi) at each value of the bare coupling. The range of τ for the connected (disconnected) contributions to the three-point correlation functions is given first (second). If a wide range of τ values is available this is indicated by “all” in the table Collab. Ref. N f τ [fm] Description ETM 19 [971] 2+1+1 [0.6−1.6]/all Two-state fit to all τ . A comparison is made with three-state fits, plateau fits and the summation method Table 194 Continuum extrapolations/estimation of lattice artifacts in determinations of σπN and σs from the Feynman–Hellmann method Collab. Ref. N f a [fm] Description BMW 20A [1010] 1+1+1+1 0.10, 0.09, 0.08, 0.06 Combined continuum, chiral and volume fit within an extended frequentist method. O(αa) or O(a2) terms are included. MN used to fix the lattice spacing 123 Eur. Phys. J. C (2022) 82:869 Page 261 of 296 869 Table 195 Chiral extrapolation/minimum pion mass in determinations of σπN and σs from the Feynman–Hellmann method Collab. Ref. N f Mπ,min [MeV] Description BMW 20A [1010] 1+1+1+1 238, 200, 219, 199 Combined continuum, chiral and volume fit within an extended frequentist method. Terms with M2,3,4π and M2K are included and cuts of Mπ < 420 MeV and Mπ < 360 MeV are made Walker-Loud 08 [1014] 2+1 294 Fit using NNLO SU (2) HBχPT [1072] with a zero and non-zero value for gN . Errors on the LECs that are not free parameters determine the systematic uncertainty Table 196 Finite-volume effects in determinations of σπN and σs from the Feynman–Hellmann method Collab. Ref. N f L [fm] Mπ,min L Description BMW 20A [1010] 1+1+1+1 3.2, 4.0, 4.3, 4.1, 4.1 Combined continuum, chiral and volume fit within 2.9–4.3, an extended frequentist method. FV term 2.6–3.8, M1/2 −3/2 −Mπ Lπ L e added 1.9–3.8 Walker-Loud 08 [1014] 2+1 2.5 3.7 FV effects expected to be 1% or less for the lightest pion mass based on [1073] C.9 Notes to Sect. 11 on scale setting Table 197 Continuum extrapolations/estimation of lattice artifacts in scale determinations with N f = 2 + 1 + 1 quark flavours Collab. Refs. N f a [fm] Description ETM 21 [305] 2+1+1 0.069, 0.079, 0.097 CalLat 20A [111] 2+1+1 [0.056, 0.082, 0.11, 0.13], Möbius Domain-wall valence quarks on [0.057, 0.087, 0.12, 0.15] HISQ/MILC sea; lattice spacings depending on scheme BMW 20 [115] 1+1+1+1 0.0640, 0.0787, 0.0952, Staggered fermion computation with 0.1116, 0.1191, 0.1315 isospin breaking and QED ETM 20 [1052] 2+1+1 0.069, 0.079, 0.097 Wilson TM fermions at maximal twist including clover term. Note that lattice spacings are not explicitly given in [1052] ETM 18A [1074] 2+1+1 0.08 Wilson TM fermions at maximal twist including clover term FNAL/MILC 17 [16] 2+1+1 0.03, 0.042, 0.06, 0.09, Determination of f p4s 0.12, 0.15 MILC 15 [112] 2+1+1 0.06, 0.09, 0.12, 0.15 Highly improved staggered quarks ETM 14 [7] 2+1+1 0.089, 0.082, 0.062 Wilson TM fermions at maximal twist FNAL/MILC 14A [17] 2+1+1 0.06, 0.09, 0.12, 0.15 HPQCD 13A [37] 2+1+1 0.15, 0.12, 0.09 HPQCD 11B [552] 2+1+1 0.09, 0.12, 0.15 NRQCD for b quark and HISQ for valence light quarks and MILC sea 123 869 Page 262 of 296 Eur. Phys. J. C (2022) 82:869 Table 198 Continuum extrapolations/estimation of lattice artifacts in scale determinations with N f = 2 + 1 quark flavours Collab. Refs. N f a [fm] Description CLS 16 [113] 2+1 0.085, 0.065, NP O(a)-improved Wilson fermions with LW 0.05 gauge action QCDSF/UKQCD 15B [713] 2+1 0.059, 0.068, (amax/a 2min) = 1.94 0.074, 0.082 χQCD 14 [24] 2+1 0.084, 0.112 Valence overlap fermions on domain wall fermion gauge configurations HotQCD 14 [116] 2+1 [0.04, 0.25] HISQ and tree-level improved Symanzik gauge action. Several ensembles with a in the quoted range and with a single Mπ ≈ 160 MeV RBC/UKQCD 14 [8] 2+1 0.06, 0.08, 0.11, Two versions of Domain Wall Fermions 0.14 combined each with the Iwasaki gauge action BMW 12A [114] 2+1 0.053, 0.065, Wilson fermion computation 0.077, 0.093 HotQCD 11 [708] 2+1 [0.066, 0.25] HISQ and tree-level improved Symanzik gauge action. Several ensembles with a in the quoted range RBC/UKQCD 10A [117] 2+1 0.087, 0.114 MILC 10 [41] 2+1 0.045, 0.06, 0.09 Asqtad staggered quarks HPQCD 09B [120] 2+1 0.044, 0.059, NRQCD for b quark and HISQ for valence light 0.085, 0.12, quarks on MILC asqtad sea 0.15 MILC 09 [15,157,1054] 2+1 [0.045, 0.18] Asqtad staggered quarks, 6 different lattice spacings in the quoted range PACS-CS 08 [193] 2+1 0.09 Several ensembles of Wilson clover quarks and Iwasaki gauge action at a single lattice spacing HPQCD 05B [118] 2+1 0.09, 0.12, 0.17 NRQCD with tree level tadpole improved couplings on MILC asqtad sea Aubin 04 [119] 2+1 0.09, 0.12 Asqtad staggered quarks 123 Eur. Phys. J. C (2022) 82:869 Page 263 of 296 869 Table 199 Chiral extrapolation and finite-volume effects in scale determinations with N f = 2 + 1 + 1 quark flavours. We list the minimum pion mass Mπ,min and Mπ L ≡ Mπ,min[L(Mπ,min)]max is evaluated at the maximum value of L available at Mπ = Mπ,min Collab. Refs. N f Mπ,min [MeV] Mπ L Description ETM 21 [305] 2+1+1 CalLat 20A [111] 2+1+1 130 3.9 BMW 20 [115] 1+1+1+1 Several pion masses 3.0 within ±3% of physical value ETM 20 [1052] 2+1+1 135 3.5 Only indirectly inferred ETM 18A [1074] 2+1+1 139.8 3.6 One ensemble only FNAL/MILC 17 [17] 2+1+1 129 3.7 Determination of f4ps , M4ps MILC 15 [112] 2+1+1 125 3.7 Four ensembles at physical point ETM 14 [7] 2+1+1 211 3.19 FNAL/MILC 14A [17] 2+1+1 130 3.7 Determination of f4ps , M4ps HPQCD 13A [37] 2+1+1 128 3.7 HPQCD 11B [552] 2+1+1 211 4.0 NRQCD ϒ 2s-1s splitting and ηs as input 123 869 Page 264 of 296 Eur. Phys. J. C (2022) 82:869 Table 200 Chiral extrapolation and finite-volume effects in scale determinations with N f = 2+ 1 quark flavours. We list the minimum pion mass Mπ,min and Mπ L ≡ Mπ,min[L(Mπ,min)]max is evaluated at the maximum value of L available at Mπ = Mπ,min Collab. Refs. N f Mπ,min [MeV] Mπ L Description CLS 16 [113] 2+1 200 4.2 Extrapolation along line mu + md + ms = const. QCDSF/UKQCD 15B [713] 2+1 228 4.1 χQCD 14 [24] 2+1 290 4.0 HotQCD 14 [116] 2+1 160 4.8 HISQ staggered quarks, the pion mass quoted is Mπ . Results on 20 ensembles with volumes in the range [2.6, 6.1] fm, we quote Mmin maxπ L RBC/UKQCD 14 [8] 2+1 139 3.9 BMW 12A [114] 2+1 131 and 120 3.92 and 3.0 HotQCD 11 [708] 2+1 160 4.8 HISQ staggered quarks, the pion mass quoted is Mπ . Results on 20 ensembles with volumes in the range [3.2, 6.1] fm, we quote Mminπ L max RBC/UKQCD 10A [117] 2+1 290 4.0 MILC 10 [41] 2+1 255 4.8 Determination from global fit, scale from fπ HPQCD 09B [120] 2+1 211 4.0 Extension of HPQCD 05B [118] MILC 09 [15,157,1054] 2+1 258 4.3 Asqtad staggered quarks, the pion mass quoted is M RM Sπ PACS-CS 08 [193] 2+1 156 2.3 Clover quarks, several pion masses in the range [156, 702] MeV, single lattice spacing HPQCD 05B [118] 2+1 270 3.7 NRQCD ϒ 2s-1s splitting as input Aubin 04 [119] 2+1 253 3.8 Asqtad staggered quarks, the pion mass quoted is the Goldstone mass References 1. 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