A Lattice QCD study of the rho resonance and the timelike pion form factor
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Abstract
The $rho$ resonance with its principal decay into two pions is the subject of many experimental and theoretical investigations. The $rho$ is the lightest of all vector mesons and has some interesting applications, like giving access to the timelike pion form factor or a contribution to the hadronic vacuum polarisation of the anomalous magnetic moment of the muon, $(g-2)_mu$.
In this work we study the $rho$ resonance using the Lattice formulation of Quantum Chromodynamics (Lattice QCD). Lattice QCD allows us to compute observables at low energies, where perturbation theory does not work due to the large coupling of QCD in this regime. The $rho$ has become a test piece for the study of resonances in Lattice QCD, which is always a difficult task because in Euclidean space, where Lattice QCD is formulated, we can directly extract only QCD-stable states. Special formalisms are therefore needed which study the impact of a resonance on stable states.
We extract correlator data using state-of-the-art formalisms like Distillation with stochastic Laplacian-Heaviside (LapH) smearing to extract single- and two-meson correlators from the lattice, in a centre-of-mass frame and moving frames with three different total momenta. We then use the variational method in 8 different irreducible representations of those frames and obtain phase-shift information from this spectrum with a L"uscher-type analysis.
Furthermore, we are computing the timelike pion form factor based on the phase-shift information as well as correlators from current operators. We parametrise the form factor using a fit of our data to an Omnes representation of the pion form factor and find that our data is well described both by the $2$-subtracted as well as the $3$-subtracted version of this representation. This fit allows us to extract the square radius of the pion which we compare to an independent lattice calculation of the spacelike pion form factor, performed on the same lattice ensembles.
We use both the parametrised version of the pion form factor as well as a reconstruction of the light-quark correlator to get two seperate estimates of the vector-vector correlator, which we use to constrain the long-time behaviour of $(g-2)_mu$. Both methods give consistent results and lower the uncertainty estimate substantially compared to the simpler procedures. Because they do not rely on a fit to the low-energy data they are an independent check of the vector-vector correlator signal.