Phenomenology of
New Physics Models
at Colliders and in
Gravitational Waves
Dissertation submitted for the award of the title
“Doctor of Natural Sciences”
to the Faculty of Physics, Mathematics and Computer Science
of Johannes Gutenberg University Mainz in Mainz
Eric Madge Pimentel
born in Ratingen, Germany
Mainz, June 1, 2020
date of submission: June 2, 2020
date of defense: October 14, 2020
Abstract
The existence of physics beyond the Standard Model of particle physics is very well
motivated. This dissertation studies the phenomenology of models that accommodate
such new physics. It mainly covers two aspects: collider phenomenology and gravitational
waves.
We first present a search for Higgs-portal dark matter at the LHC and its prospective
high-luminosity and high-energy upgrades, entertaining the vector-boson fusion channel.
We derive the limits on the portal coupling as a function of the dark matter mass, in
particular also for masses close to the transition between the on- and off-shell Higgs
regime. Subsequently, a study of the h→ Zγ decay in top-pair associated production is
considered. We evaluate the observational prospects at future proton colliders and derive
the corresponding indirect constraints that can be put on the new physics’ contribution to
the decay rate. Our exploration of collider probes of physics beyond the Standard Model is
then concluded with a comprehensive analysis of the phenomenology of a model in which
lepton number is gauged. The model automatically provides a candidate for particle dark
matter. We investigate the parameter space in which the measured relic abundance is
reproduced, impose constraints from direct and indirect dark matter searches, and assess
the limits from collider experiments.
We then move on to study the gravitational wave phenomenology of new physics, fo-
cusing on stochastic gravitational wave backgrounds generated in cosmological first-order
phase transitions. After an introduction to the topic, we return to the gauged-lepton-
number-model and investigate the lepton number breaking phase transition. We identify
the parameter regions in which the transition is of first order and which are consistent
with the collider and dark matter constraints. We then calculate the respective gravita-
tional wave spectrum and evaluate its detectability at LISA and other future gravitational
wave observatories. Finally, we consider phase transitions occurring in decoupled dark
sectors, particularly focusing on sub-MeV hidden sectors. We investigate the interplay
between cosmological constraints on the number of relativistic degrees of freedom and
the detectability of the gravitational wave background generated by a phase transition in
such a sector.
i

List of Publications
References to collaborators by name have been anonymized in the electronic version.
This thesis is based on the publications and preprints [1–4]. In the following, a brief
summary of these works is provided, highlighting the respective contributions of the
author.
[1] E. Madge and P. Schwaller, Leptophilic dark matter from gauged lepton number: Phe-
nomenology and gravitational wave signatures, JHEP 02 (2019) 048, [1809.09110]
We perform a comprehensive study of an extension of the Standard Model in which
lepton number is promoted to a gauge group. Additional fermions required to cancel
anomalies provide a dark matter candidate. The lepton number gauge group is spon-
taneously broken using the Higgs mechanism. We assess the collider and dark matter
phenomenology of the model and investigate the possibility of observing the lepton
number breaking phase transition through gravitational waves.
The paper updates and extends the discussion of the model in ref. [5]. All calcu-
lations, simulations and parameter scans, the derivation of the resulting constraints
and experimental sensitivity, the creation of the corresponding graphical presenta-
tions, as well as the composition of the publication text (except for the introduction)
were performed by the author with advise and corrections from the collaborators.
Correspondingly, the majority of the contents of chapters 5 and 7 of this dissertation
has been copied literally (partially with minor modifications to the text) from the
publication, which is subject to the creative commons license CC-BY 4.0 [6].
[2] M. Breitbach, J. Kopp, E. Madge, T. Opferkuch and P. Schwaller, Dark, Cold, and
Noisy: Constraining Secluded Hidden Sectors with Gravitational Waves, JCAP 1907
(2019) 007, [1811.11175]
This work investigates the detectability of stochastic gravitational wave backgrounds
generated in cosmological phase transitions in decoupled hidden sectors, particularly
focusing on sub-MeV sectors and the interplay with constraints from the effective
number of neutrino species.
The parameter scans calculating the phase transition properties were performed inde-
pendently by a collaborator and the author. All authors contributed to the derivation
of the dependence of the gravitational wave spectrum on the temperature ratio be-
tween the hidden sector and photon bath, and to the text in the published manuscript.
The figures in the publication were created by a collaborator and cross-checked by
the author. The author further particularly contributed by the extraction of noise
and sensitivity curves of pulsar timing arrays from the literature, as well as by an
approximate analytic analysis of the phase transition of the singlet scalars toy model
with one or two scalar fields.
iii
List of Publications
[3] F. Goertz, E. Madge, P. Schwaller and V. T. Tenorth, Discovering the h→ Zγ Decay
in tt̄ Associated Production, Phys. Rev. D 102 (2020) 053004, [1909.07390]
We propose a collider search for the so-far unobserved decay of the Higgs boson
into a photon and a Z boson, entertaining the top-pair associated Higgs production
channel. The discovery prospects in this channel, as well as the respective bounds on
new physics, are investigated for the high-luminosity and high-energy upgrades of the
LHC , as well as for a 100TeV pp collider (FCC), performing a Monte Carlo study.
The Monte Carlo simulations of the signal and irreducible background process as
well as the analysis of the corresponding events were performed independently by
a collaborator and the author. Additional sub-leading backgrounds were divided up
amongst these two and simulated only once. All authors contributed to the derivation
of the significance and the corresponding limits on new physics contributions, as well
as to the text.
[4] J. Heisig, M. Krämer, E. Madge and A. Mück, Probing Higgs-portal dark matter with
vector-boson fusion, JHEP 03 (2020) 183, [1912.08472]
In this work, a study of Higgs-portal dark matter produced in vector-boson fusion is
presented. Constraints from the current LHC as well as projections for the HL-LHC
and HE-LHC updates are derived, including an estimate of the systematic uncertain-
ties in the case of data-driven background determination. Particular care is taken to
obtain consistent limits in the dark matter mass region close to the Higgs resonance.
The author performed all Monte Carlo simulations of the signal and background
processes and their successive analysis, with advice from the collaborators. The plots
depicting the respective limits on the portal couping are due to a collaborator and were
cross-checked by the author. The author further derived the perturbative unitarity
bound on the coupling in the tensor dark matter case, as well as the limits on the
portal coupling for different cut-offs on the integral in eq. (3.13). The published text
contains contributions from all authors.
All figures depicted in this dissertation are due to the author. Figures from the publi-
cations listed above that were produced by collaborators, as well as some of the figures
produced by the author himself, have been recreated for this thesis to obtain a (mostly)
uniform plot layout.
iv
Contents
Abstract i
List of Publications iii
Prologue
1. Introduction 3
2. The Standard Model and Beyond 6
2.1. The Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . 6
2.2. New Physics in the Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1. WIMP Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2. Axion-Like Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.3. Sterile Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Part I: New Physics at Colliders
Prelude 19
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion 21
3.1. The Scalar Singlet Higgs-Portal Model . . . . . . . . . . . . . . . . . . . . 22
3.2. Threshold at Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3. Current LHC Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1. Reinterpretation of Upper Limits . . . . . . . . . . . . . . . . . . . 28
3.3.2. Recasting of the Cut-and-Count Analysis . . . . . . . . . . . . . . 30
3.4. HL-LHC and HE-LHC Projections . . . . . . . . . . . . . . . . . . . . . . 32
3.4.1. Background Predictions and Systematic Uncertainties . . . . . . . 33
3.4.2. Sensitivity Projections . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Appendix 3.A. Reinterpretation for Other Higgs-Portal Models . . . . . . . . 38
4. Discovering the h → Zγ Decay in tt̄ Associated Production 41
4.1. HL-LHC Sensitivity Estimate . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1. Semi-Leptonic Channel . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2. All-Hadronic and All-Leptonic Channel . . . . . . . . . . . . . . . 45
4.2.3. Predictions for 27 and 100 TeV Colliders . . . . . . . . . . . . . . . 46
v
Contents
4.3. Constraints on New Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5. Leptophilic Dark Matter from Gauged Lepton Number 51
5.1. The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.1. Gauge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.2. Scalar Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.3. Fermion Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.1.4. RG Running . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2. Leptophilic Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.1. Relic Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.2. Direct and Indirect Detection . . . . . . . . . . . . . . . . . . . . . 61
5.3. Collider Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.1. Z ′ Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2. Higgs Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.3. Constraints on Heavy Leptons . . . . . . . . . . . . . . . . . . . . 69
5.4. Intermediate Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Part II: Gravitational Waves from Cosmological Phase Transitions
Prelude 75
6. Stochastic Gravitational Wave Backgrounds 77
6.1. Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2. Gravitational Wave Experiments . . . . . . . . . . . . . . . . . . . . . . . 78
6.3. Detection and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4. Cosmological Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4.1. Bubble Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.4.2. Phase Transition Parameters . . . . . . . . . . . . . . . . . . . . . 84
6.4.3. Generation of a Stochastic Gravitational Wave Background . . . . 85
6.5. The Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Appendix 6.A. Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . 90
Appendix 6.B. Further Details on the Effective Potential . . . . . . . . . . . . 92
6.B.1. Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.B.2. The One-Loop Effective Potential at Zero-Temperature . . . . . . 94
6.B.3. The One-Loop Effective Potential at Finite-Temperature . . . . . . 95
7. Gravitational Wave Signatures from Lepton Number Breaking 99
7.1. The Lepton Number Breaking Phase Transition . . . . . . . . . . . . . . . 99
7.1.1. Finite-Temperature Effective Potential . . . . . . . . . . . . . . . . 100
7.1.2. A First-Order Lepton-Number-Breaking Phase Transition . . . . . 101
7.2. Gravitational Waves Signature . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
vi
Contents
Appendix 7.A. Goldstone Divergences . . . . . . . . . . . . . . . . . . . . . . 111
7.A.1. Scalar Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.A.2. On-Shell Renormalization of the Effective Potential . . . . . . . . . 113
Appendix 7.B. Bubble Wall Velocity . . . . . . . . . . . . . . . . . . . . . . . 114
8. Constraining Secluded Hidden Sectors with Gravitational Waves 116
8.1. Decoupled Hidden Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.1.1. Neutrino Decoupling and Electron-Positron Annihilation . . . . . . 117
8.1.2. Effective Number of Neutrino Species . . . . . . . . . . . . . . . . 119
8.1.3. Hidden Sector Cosmology . . . . . . . . . . . . . . . . . . . . . . . 120
8.2. Gravitational Waves from Decoupled Hidden Sectors . . . . . . . . . . . . 123
8.2.1. Temperature Ratio Dependence . . . . . . . . . . . . . . . . . . . . 124
8.2.2. Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3. Toy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.3.1. Singlet Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.3.2. Dark Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3.3. Parameter Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Epilogue
9. Conclusion and Summary 143
Acknowledgements 146
Bibliography 147
List of Abbreviations 173
List of Experiments 174
vii

Prologue

1. Introduction
The Standard Model (SM) of particle physics is one of the most successful theories ever
developed in the field of physics. Since its formulation in the late 1960s [7–9], it correctly
describes the known particles as well as their strong, weak and electromagnetic interac-
tions. Undoubtedly, one of its most notable triumphs was the correct prediction of the
existence of a massive, neutral scalar particle, the Higgs boson [10–15], which was finally
discovered at the LHC in 2012 [16, 17]. Over the past decades, an extensive program
for experimental tests of the SM has been carried out, which impressively confirmed the
validity of its predictions over a variety of processes.
Despite its tremendous success in providing accurate predictions for the LHC and
other colliders, the SM suffers from various short-comings that require the introduction
of new physics beyond the Standard Model (BSM). These open problems include questions
motivated from a theoretical point of view, partially based on arguments of philosophical
nature, such as the hierarchy or naturalness problem and the strong CP problem, but
also puzzles related to experimental observations, mostly astrophysical or cosmological,
that are not accommodated within the SM. Examples of the latter category are neutrino
masses and oscillations, the generation of a baryon asymmetry in the early Universe, and
the existence of dark matter (DM) and dark energy.
A plethora of models has been proposed to potentially solve one or several of the open
problems described above, ranging from simple extensions of the SM in which only a
single new field is added, over models with rather complicated sectors of new physics,
to novel frameworks extending the symmetries of space-time, such as supersymmetry
(SUSY), or models of extra dimensions. This dissertation is dedicated to the study of
the phenomenology of such models. Particular focus is put on models of DM, which are
discussed in chapters 3, 5 and 7, whereas chapters 4 and 8 consider new physics in a
more general context. We here choose two different, complementary paths to constrain
BSM models. Part I is devoted to collider studies of new physics, mostly focusing on pp
machines. In part II we then investigate the gravitational wave (GW) phenomenology of
BSM models, in particular regarding cosmological phase transitions (PTs) in the early
Universe.
Particle accelerators and colliders have proven to be invaluable discovery tools in ele-
mentary particle physics. Since the early scattering experiments by Rutherford, shooting
helium nuclei on gold targets in 1911 [18], the procedure of smashing particles into one
another and inferring elementary physics from the respective outcome has been refined
immensely, leading to the development of giant machines that reach unprecedented en-
ergies and precision. Much of the observational confirmation of the SM was provided
by collider experiments, as for instance the discovery of various particles predicted on
theoretical grounds such as the W and Z bosons, the top quark, and, last but not least,
3
1. Introduction
the Higgs boson. Colliders are therefore commonly regarded promising facilities to unveil
the nature of new physics.
Since, despite the ample efforts to unravel new physics taken at colliders and other
types of experiments, no definite BSM signatures have been found so far, we may have to
face the possibility that whatever new physics cures the open problems could interact only
very weaky (or maybe even not at all) with the particles of the SM. We however know
that, at least in the case of DM, the new physics should feature gravitational interactions.
As a consequence, even in this very pessimistic scenario, gravity may provide a handle
to probe BSM physics. This is particularly promising in the light of the recent direct
observation of GWs by LIGO and Virgo in 2015 [19], which led to the proposal and
elaboration of concrete realizations for various future GW observatories. Most notably,
the first-ever space-based GW interferometer, LISA [20], will prospectively be launched in
the mid 2030’s. These future experimental facilities pave the way for a potential detection
of new physics via GWs.
In the context of GWs, new physics may be observable in deviations from predictions
for astrophysical events such as mergers of black holes (BHs) or neutron stars (NS), or in
the form a stochastic gravitational wave background (SGWB) of cosmological origin. A
cosmological SGWB can for instance be generated in the era of inflation, from the decays
of cosmic strings, or in cosmological first-order PTs. In this thesis, we will focus on the
latter case.
According to our current understanding of the history of our Universe, it started with an
epoch of exponential expansion, the epoch of inflation, from which the Universe emerged
flat, homogeneous, isotropic, and basically empty (except for the inflaton field). During
the subsequent epoch of reheating, the inflaton then decayed, repopulating the Universe
with a thermal plasma of elementary particles. Due to the expansion of the Universe
driven by the energy in the plasma, its temperature dropped during the further evolution.
In the course of this cooling process, the Universe may have undergone one or several PTs,
which, if these were of first order, could have generated a stochastic background of GWs
detectable by future observatories. The SM predicts two transitions: the electroweak PT
(EWPT), in which the electroweak (EW) gauge symmetry is broken to electromagnetism
(EM), and the confining PT of quantum chromodynamics (QCD), which breaks chiral
symmetry. However, in the SM both of these transitions are cross-overs.1 The observation
of the SGWB generated by a cosmological PT would therefore be a clear indication of
new physics, potentially providing insight into the nature of the underlying theory.
Laboratory and cosmological probes of new physics provide complementary ways to
asses BSM models. Collider experiments may directly detect new particles, e.g. in the
form of resonances, or observe them indirectly via their effects on SM observables. Cos-
mological observations, on the other hand, can for instance provide limits on the number
of relativistic degrees of freedom (DOFs) in the early Universe, as we will discuss in
chapter 8, or on the mass of DM if it is thermal (we briefly touch upon this bound in
section 2.3). While all evidence for DM is of astrophysical and cosmological nature, there
is still a good chance that it can be produced at collider experiments. Cosmology and
colliders further exhibit a particular interplay in the context of spontaneous symmetry
1In a misuse of language, we will nonetheless refer to them as PTs.
4
1. Introduction
breaking (SSB). As pointed out above, the corresponding cosmological PT may be ob-
servable in GWs if it is of first order, while indications for the order of the PT can be
obtained at colliders, e.g. probing the potential of the Higgs boson in the context of the
EWPT, or by the creation of a quark-gluon plasma in the case of the chiral PT. Fur-
thermore, as all elementary SM particles obtain their masses from the Higgs mechanism,
it is suggestive to assume that the masses of BSM particles are generated in a similar
manner. If the new particles are much heavier than the weak scale, they cannot obtain
their full masses via electroweak symmetry breaking (EWSB), potentially indicating the
spontaneous breaking of additional symmetries, which may in turn be associated with an
observable PT.
This thesis is organized as follows. We first briefly recapitulate the SM and its open
problems in chapter 2. Next, in part I, collider studies searching for new physics are
considered. We start with a DM search at proton colliders in chapter 3, focusing on
Higgs-portal DM in the vector-boson fusion (VBF) channel. In chapter 4 we then assess
the prospects of detecting the decay of the Higgs boson into a photon and a Z boson at
future collider experiments, and evaluate how this can be used to indirectly constrain the
impact of new physics on the decay process. We conclude the part in chapter 5 with a
comprehensive study of the DM and collider phenomenology of a model in which lepton
number is promoted to a gauge symmetry. Subsequently, part II is devoted to probing
BSM physics via GWs. Chapter 6 provides an introduction to SGWBs, their detection,
and how they are generated in cosmological first-order PTs. As an example, chapter 7
investigates the lepton number breaking PT of the model we considered in chapter 5.
Chapter 8 then considers PTs in general hidden sectors, addressing the question how the
corresponding SGWB is affected when the dark sector is sequestered. Finally, conclusions
of the thesis are presented in chapter 9.
5
2. The Standard Model and Beyond
Before diving into the phenomenology of new physics beyond the Standard Model (BSM),
let us first very briefly recapitulate the Standard Model (SM) itself, assuming that the
reader is mostly familiar with this subject. For a more detailed review, the reader shall
be referred to the usual text books, such as refs. [21, 22]. We then proceed in this chapter
by pointing out some of the open questions of the SM with emphasis on those related to
the models studied in this dissertation.
2.1. The Standard Model of Particle Physics
The SM is a gauge theory describing the strong, weak and electromagnetic forces. It is
based on the symmetry group SU(3)c × SU(2)L × U(1)Y , corresponding to the SU(3)c
color group of quantum chromodynamics (QCD) [23–28] as well as the weak isospin
SU(2)L and hypercharge U(1)Y gauge groups uniting the electroweak forces [7–9]. The
structure of the SM is completely fixed by this gauge symmetry, its particle content, and
by requiring renormalizability [29].
Its matter content consists mostly of fermionic fields. The quark fields QL, uR, and dR
transform as triplets upon the SU(3)c group of QCD, whereas the lepton fields `L and
eR are QCD singlets. The left-handed fields (with index L) are doublets under the weak
gauge group SU(2)L, while the right-handed fields (with index R) are SU(2)L-singlets.
Each of these five types of fermions comes in three generations. Hence, the representations
and charges of th(e SM fer)mions under th(e SU(3)c)× SU(2)L × U((1)Y gauge)group are
Qi
1 2 1
L ∼ (3, 2, i6 , uR ∼ 3, 1, 3 , diR ∼ 3, 1, −3 ,
i ∼ 1 2 −1
) (2.1)
`L , , , e
i
2 R ∼ (1, 1, −1) ,
where i = 1, 2, 3 is a generation index.
In addition to the fermions, the SM features an SU(2)L doublet scalar field H trans-
forming as H ∼ (1, 2, 1/2), the Higgs doublet. It spontaneously breaks the electroweak
(EW) gauge symmetry to electromagnetism (EM), SU(2)L × U(1)Y → U(1)EM, via the
Higgs mechanism [10–15], acquiring a vacuum expectation value (VEV). Rotating the
VEV into the lower real component of the√doublet using a global transformation, it can
be expanded as H = (G±, (v + h+ iG0)/ 2)T , where h is the physical Higgs mode, v its
(space-time independent) VEV, and Gi are the would-be Goldstone bosons that provide
the longitudinal degrees of freedom (DOFs) to the massive gauge bosons. The Higgs VEV
induces mass terms for the EW gauge bosons from the Higgs’ covariant derivative term.
The resulting mass eigenstates are the charged W± and neutral Z bosons with masses
mW and mZ , and the massless photon γ.
6
2.1. The Standard Model of Particle Physics
Apart from the kinetic and potential terms (as well as gauge-fixing and ghost terms),
the SM symmetries allow Yukawa interactions of the form
Lyuk = −(Y ) Q̄iu ij L H̃ u
j
R − (Y )
i j ¯̀i j
d ij Q̄LH dR − (Ye)ij LH eR + h.c. , (2.2)
where H̃ ≡ iσ2H∗ with σ2 denoting the second Pauli matrix. Upon spontaneous sym-
metry breaking (SSB), the Yukawa interactions generate mass terms for the quarks and
leptons.1 Writing the doublet fields in their SU(2) components Qi = (uiL L L, di TL) and
`i = (νiL L, eiL), these mass terms are
Lyuk ⊃ −(Mu) i jij ūL H̃ uR − (M ) d̄
i
d ij LH d
j i
R − (Me)ij ēLH e
j
R + h.c. , (2.3)√
where Mf = Yf v/ 2 for f = u, d, e. The mass matrices Mf can be diagonalized via
singular value decomposition, rotating the quark and lepton fields by unitary 3 × 3 ma-
trices UfC , i.e. f i → (U
f
C C)ijf
j
C , where C = L,R, with left and right-handed fields rotated
independently. The resulting mass eigenstates are six massive quarks, to wit, in order
of increasing mass, the up (u), down (d), strange (s), charm (c), bottom (b) and top (t)
quarks, three massive charged leptons, namely the electron (e), muon (µ) and tau lep-
ton (τ), as well as the corresponding three massless neutrinos νe, νµ and ντ .
The unitary transformations that diagonalize the fermion masses reappear in the ki-
netic terms ψ̄iD/ψi with ψi = QiL, uiR, diR, `i iL, eR. They however mostly combine to unity,
except in the charged current interactions of the left-handed doublets, such as for instance
ūi γµL d
i
L. In the quark sector, the rotations of the left-handed fields combine to the unitary
Cabibbo-Kobayashi-Maskawa (CKM) matrix [30, 31] V ≡ (Uu)† UdL L, which describes the
quark-flavor-changing charged-current interactions with the W boson. These then take
the form ūiLγµV
j + i
ijdLWµ + h.c., where u and di now denote the up- and down-type mass
eigenstate quarks.
As a unitary matrix, the CKM matrix has 9 DOFs, viz. three rotation angles and six
phases. The mass terms are however invariant under phase changes of each of the six
quark fields, whereas the CKM matrix is only invariant under a simultaneous change of
all six phases. We can therefore absorb five phases in the quark fields, resulting in four
physical, real parameters: three mixing angles, and one (CP -violating) phase.
In the lepton sector, on the other hand, no such matrix appears in the SM. As neu-
trinos remain massless, we can simply transform νL in the same way as eL, so that the
matrix disappears. However, when right-handed neutrinos are added, as we will discuss
in section 2.3.3, the neutrino mass terms force us to rotate νL differently from eL. The
resulting mixing matrix of the leptons is called the Pontecorvo-Maki-Nakagawa-Sakata
(PMNS) matrix [32, 33].
As already mentioned, the SM features two potential phase transitions (PTs). The first
one comes from the spontaneous breaking of the EW gauge symmetry in which the Higgs
field acquires its VEV. Lattice simulations indicate that the electroweak PT (EWPT)
of the SM proceeds as a weak cross-over at a temperature around T ' 160GeV [34].
However, even simple extensions of the SM by a single scalar field may render the tran-
sition first-order [35–37]. The second PT is the chiral PT of QCD, in which the quark-
gluon plasma of the early Universe confines into hadrons, with the quark condensate
1Note that explicit mass terms are inconsistent with the weak gauge symmetry, as left-handed fields are
doublets while the right-handed ones are singlets.
7
2. The Standard Model and Beyond
breaking chiral symmetry. This is also a cross-over, occurring at a temperature around
T ' 160MeV [38]. Due to this low temperature, the chiral PT may also be probed di-
rectly in heavy ion collisions at the LHC or RHIC [39]. Furthermore, the corresponding
transition in QCD-like sectors of BSM models may very well be a first-order PT [40, 41].
2.2. New Physics in the Higgs Sector
Many of the motivations to consider BSM physics are to some degree related to the Higgs
boson or the spontaneous breaking of the EW gauge symmetry. One particular example
which has inspired numerous models of new physics is the EW hierarchy problem. It can
be boiled down to the question why the observed mass of the Higgs boson or the EW
scale are so much lighter than the scale of gravity, the Planck scale. The extremely tiny
ratio m2/M2 ∼ 10−34h P violates ’t Hooft’s principle of technical naturalness [42], which
states that dimensionless parameters of fundamental theories should be O (1) numbers
unless setting them to zero enhances the symmetry of the theory.
The hierarchy problem stems from the observation that the Higgs mass in the SM is
additively renormalized, i.e. quantum corrections to the Higgs mass are independent of
the Higgs mass itself.2 This can be realized noting that the loop corrections to the Higgs
mass exhibit a quadratic di[vergence. In the SM, these corr]ectio(ns are [43]∆m2h = 3 Λ2 m2 2 2 )24 t2 8 2 2 2 − mZ mW2 − 2 2 − 1 + . . . ≈ Λ500GeV , (2.4)mh π v mh mh mh
where the loop integrals have been regularized imposing a cut-off Λ on the virtual mo-
mentum, and v ' 246GeV is the Higgs’ VEV. Considering the SM as an effective field
theory (EFT),3 this means that the Higgs mass is quadratically sensitive to the scale of
new physics, i.e. it is ultraviolet (UV) sensitive. As a result, loop corrections to the Higgs
mass exceed its physical value when considering scales above Λ ∼ 500GeV.
Note that the cut-off scale Λ should be interpreted as a placeholder for the mass scale of
whatever new physics appears in the UV. If we extend the SM adding a new particle with
mass M that couples to the Higgs boson, this particle will induce mass corrections that
go like ∆m2h ∼ M2, assuming that we now employ dimensional regularization. Hence,
any new massive particle coupled to the Higgs will in general generate corrections on
the order of the new physics’ scale; heavy particles do not decouple. As a result, we
encounter a fine-tuning problem. Suppose that there is new physics UV-completing the
SM at some high scale Λ  mh. The Higgs mass parameter of the theory then needs
to be tuned at the level m2 2h/Λ to cancel the quantum corrections and give the observed
mass of mh = 125GeV. The higher Λ, the worse the tuning.
Although the fine-tuning problem is in principle mostly an aesthetic problem counter-
acting our intuition, it is still an inherently unsatisfying feature of the SM, and therefore
served as a guideline for the construction of numerous models of new physics. Possible
2In contrast, multiplicatively renormalized parameters, such as for instance the electron mass, receive
corrections that are proportional to the parameter itself.
3 Note that the running of the hypercharge gauge coupling in the SM exhibits a Landau pole around
Λ ∼ 1041GeV, so that the SM has to be considered an EFT with a cut-off around or below that scale.
8
2.2. New Physics in the Higgs Sector
solutions to the hierarchy problem include little Higgs [44, 45] or composite Higgs [46–48]
models, in which the Higgs boson arises as a pseudo-Goldstone boson from the sponta-
neous breaking or confinement of a gauge or global symmetry, and where that symmetry
protects the Higgs mass from large corrections, or models of supersymmetry (SUSY) [49–
51], where bosonic corrections are canceled by the corresponding contributions from their
fermionic super-partners and vice-versa.
A further open question directly related to the Higgs potential regards the stability
of the EW vacuum. At large field values, the potential of the Higgs field h can be
approximated by the quartic term V (h) ' λ(h)h4/4, including the renormalization group
(RG) evolution of the quartic coupling λ(µ). The latter is governed by the β function [52,
53]
≡ dλ 1
[ ]
β = 12λ2 + 6λy2 4λ d log 16 2 t − 3yt + . . . , (2.5)µ π
where λ is the Higgs quartic and yt is the top Yukawa coupling. If the top contribution
dominates, it drives the quartic coupling negative at high field values. As a consequence,
the Higgs potential is not bounded from below and the EW vacuum is not stable. This is
indeed the case for the measured values of the Higgs and top masses. The corresponding
tunneling rate is however very low, such that the lifetime of the vacuum exceeds the age
of the Universe by orders of magnitude, and we live in a meta-stable vacuum very close
to the border of stability [52]. New physics contributions to the running of λ may render
the EW vacuum absolutely stable, or conversely spoil its (meta-)stability.
Another important motivation for BSM physics is the failure of the SM to provide
suitable conditions for the generation of the baryon asymmetry of the Universe. At some
point during the history of the Universe, a tiny excess of baryons over anti-baryons of [54]
= nB − nη B̄ ' 10−9 (2.6)
nγ
must have been generated. The anti-baryons subsequently annihilated with the baryons,
leaving the Universe dominated by the excess baryons, eventually leading to our mere
existence. Although the baryon asymmetry is not necessarily related to electroweak
symmetry breaking (EWSB), one of the standard scenarios for its generation relies on a
first-order EWPT.
The generation of a baryon asymmetry from an initially baryon-symmetric Universe
requires that the so-called Sakharov conditions [55] are satisfied. These are the non-
conservation of baryon number, violation of C and CP invariance, and deviation from
thermal equilibrium. The first requirement is obvious, C and CP violation are needed to
produce matter and anti-matter at different rates, and non-equilibrium is required as the
equilibrium number densities of particles and anti-particles are the same. In principle,
the SM could establish all these conditions at the EWPT, generating the asymmetry in a
scenario called electroweak baryogenesis (EWBG) [56–58]. The weak interactions of the
SM exhibit C and CP violation, baryon number (or B+L, to be more precise) is violated
by non-perturbative sphalerons, and a first-order EWPT can provide the required out-of-
equilibrium conditions. However, the amount of CP violation in the SM is not sufficient,
and the EWPT is a cross-over [34]. EWBG therefore requires BSM physics.
9
2. The Standard Model and Beyond
All these open problems and their potential solutions more or less directly relate to
the Higgs potential and the EWPT. Collider and gravitational wave (GW) experiments
therefore provide complementary ways to probe the corresponding BSM extensions. At
colliders, we can directly search for additional particles coupled to the Higgs boson or
other SM particles. Furthermore, the Higgs potential can be probed directly, e.g. measur-
ing a cubic term via double Higgs production. GW observatories, on the other hand, may
detect the stochastic gravitational wave background (SGWB) produced in the EWPT if
it is of first order, as for instance required in EWBG, or from additional PTs such as a
confining PT in a composite Higgs model.
2.3. Dark Matter
Among the various open problems of the SM, a large fraction of the work presented in this
dissertation particularly regards the existence of dark matter (DM). DM is a mysterious
form of matter that is non-luminous but interacts gravitationally, and whose existence
can be inferred from astrophysical and cosmological observations (see e.g. ref. [59] for
a review). Today, it is well established that only roughly 5% of the energy content
of our Universe consists of the baryonic matter we know from the SM, whereas about
27% is constituted of DM [60]. The remaining 68% are dark energy, an even more
mysterious form of energy, required to accommodate the observed accelerated expansion
of our Universe. Both of these non-baryonic components, DM and dark energy, are now
incorporated into the ΛCDM Standard Model of cosmology (including cold dark matter
(CDM) and a cosmological constant Λ). The SM of particle physics, however, lacks a
suitable explanation for these phenomena.
The question about the nature of DM is a long-standing open problem in particle
physics, with a history of about 100 years (see ref. [61] for a review). Nowadays, we have
ample observational evidence for its existence over a large range of scales [59]. On galactic
scales, observations of the rotation curves of spiral galaxies, as for instance the Andromeda
galaxy analyzed by Rubin and Ford in√1970 [62], show a flat velocity distribution of the
outer stars deviating from the ∝ 1/ r expectation, which may be explained by the
presence of a DM halo. Similar conclusions can be drawn on the scale of galaxy clusters,
for example based on the velocity dispersion in the Coma cluster studied by Zwicky
in 1933 [63]. Furthermore, one of the most compelling hints for DM was observed in
the Bullet cluster [64], where the collision of two clusters revealed that the bulk mass
was mostly unaffected by the collision, while it was clearly visible in the hot baryonic
gas component. Finally, measurements of the power spectrum of the cosmic microwave
background (CMB) allow for a precise determination of the total energy density of DM
of h2ΩDM = 0.120± 0.001 [60].
Although the observations indicating the existence of DM may in principle be explained
by a population of non-radiating but baryonic astrophysical objects, such as primordial
black holes (PBHs) or other types of massive astrophysical compact halo objects (MA-
CHOs),4 the power spectrum of the CMB [60] as well as the light element abundances
4These are however rather strongly constrained and basically ruled out as a main component of DM,
see e.g. ref. [65].
10
2.3. Dark Matter
predicted by Big Bang Nucleosynthesis (BBN) [66] indicate that the majority of the
matter in the Universe is non-baryonic. We therefore here assume that DM consists of
elementary or composite particles, potentially having various sub-components.5 The for-
mation of cosmological structure and the matter power spectrum further imply that the
DM mostly consists of cold dark matter (CDM). “Cold” here signifies that the DM is
non-relativistic at the times of matter-radiation equality and the onset of structure for-
mation. While density perturbations in the baryonic component cannot collapse to form
structures due to radiation pressure until the time of photon decoupling, perturbations
in the CDM component can collapse as soon as the Universe becomes matter-dominated,
allowing for the early formation of structure at small scales. Hot DM on the other hand is
relativistic at matter-radiation equality and has a non-negligible free-streaming length, so
that it does not form structures until it becomes non-relativistic. The observed small-scale
structure requires that the gross of matter is non-baryonic CDM [68].
Apart from its total abundance, as well as the fact that it is non-luminous (i.e. electro-
magnetically neutral6) and gravitationally interacting, little is known about the nature of
DM. The DM particles of course need to be stable, or at least have a lifetime exceeding
the age of the Universe [70]. Furthermore, observations of collisions of galaxy clusters,
as for instance the Bullet cluster, impose an upper bound on the self-interactions of
DM [71]. The possible range of DM masses is almost unconstrained, spanning many or-
ders of magnitude. For bosonic DM, a lower bound on the mass is given by its de Broglie
wavelength λ ∼ 1/mDM: it has to be smaller than the size of the smallest observed
structures, i.e. dwarf galaxies. Observations of the Lyman-α forest put a lower bound
of mDM & 10−21 eV [72]. For fermionic DM on the other hand, the Pauli’s exclusion
principle sets a much more stringent bound from the mass and size of dwarf galaxies
of roughly mDM & 100 eV [73]. Thermal DM, i.e. DM that was in thermal equilibrium
with the SM plasma in the early Universe, is furthermore required to be heavier than
mDM & 5.3 keV [74], as it would otherwise erase small-scale structures due to its large
free-streaming length at matter-radiation equality. Finally, an upper bound on the DM
mass can be obtained from the fact that extremely heavy DM would disrupt star clusters
and similar structures when passing through them, imposing a limit ofmDM . 5M [65],
where M = 2× 1030 kg is the solar mass.
The genesis of the abundance of DM particles can be roughly divided into two cate-
gories: thermal and non-thermal production [75]. In thermal production, the DM is pro-
duced from particles that are in thermal equilibrium, resulting in an energy spectrum that
is proportional to that of an equilibrium species. The standard scenario is the so-called
thermal freeze-out, in which the DM itself is initially in thermal equilibrium, with an
abundance determined by the temperature at which the DM decouples from the plasma.
We will discuss this scenario in slightly more detail in section 2.3.1. Commonly employed
modifications of or alternatives to thermal freeze-out include co-annihilation with part-
ner particles [76], as well as freeze-in production of a non-equilibrium DM species from
5Some of the observations may also be explained by modifications of gravity. These models are however
mostly ruled out, in particular in the light of constraints on the deviation of the speed of gravity from
the speed of light, derived from the observation of a neutron star (NS) binary merger in GWs and EM
radiation [67].
6Or at least very close to neutral. A small EM charge (milli-charge) may still be allowed, see e.g. ref. [69].
11
2. The Standard Model and Beyond
decays or collisions of thermal-bath particles [77]. Another variation that is particularly
interesting in the light of GW signatures is the possibility to produce DM in processes
that are only temporarily active due to kinematic thresholds modified by cosmological
PTs [78–80]. Furthermore, as in the case of the baryonic matter of the SM, its relic
density may be set via an asymmetry in the number of particles and anti-particles. In
non-thermal production, on the other hand, the DM abundance does not exhibit a ther-
mal distribution. It may for instance be generated from the decay of out-of-equilibrium
particles, or by coherently oscillating scalar fields [75]. Typical examples are axions and
sterile neutrinos, which we will briefly discuss in sections 2.3.2 and 2.3.3.
2.3.1. WIMP Dark Matter
An intriguing scenario for thermal particle DM is the so-called weakly interacting massive
particle (WIMP) paradigm. In this paradigm, the DM abundance is set via thermal
freeze-out. The interactions that change the number density of DM become inefficient
when the interaction rate drops below the Hubble rate (the rate at which the Universe
expands). The DM then chemically decouples, and its co-moving number density is
conserved. This process is called thermal freeze-out. The resulting DM relic abundance
can be calculated solving the corresponding Boltzmann equation [81]. We approximately
obtain [82]
2Ω 0.1 pb ch DM ' , (2.7)〈σv〉
where 〈σv〉 is the thermally averaged cross-section times velocity. This is the famous
WIMP miracle: the abundance of particle DM with weak-scale masses and cross-sections
set via thermal freeze-out coincides roughly with the experimentally observed DM density.
Over the past decades, WIMP DM has evolved into a standard DM paradigm. Nu-
merous models employing this scenario have been constructed. In models for SUSY for
instance, the lightest supersymmetric particle (LSP) may constitute a WIMP DM can-
didate if electrically neutral. The DM models considered in this dissertation also assume
thermal freeze-out. WIMP DM has the attractive feature that it provides promising
prospects for detecting DM, as the process setting the thermal abundance is generically
related to various processes used in astrophysical or laboratory probes. This is depicted
in fig. 2.1.
The same annihilation process that determines the DM relic density also leads to
the annihilation of DM into SM particles in regions of high local DM densities. Indi-
rect detection experiments aim at observing DM by detecting the annihilation products,
searching for γ-rays, cosmic rays of charged antiparticles, or neutrinos. Searches for pho-
tons produced either directly in the annihilation or radiated from charged annihilation
products are for instance performed at the Cherenkov telescopes Fermi-LAT , H.E.S.S.,
and MAGIC , observing nearby dwarf spheroidal galaxies in the Milky Way [83] or the
Galactic center [84–86].
Crossing symmetry further relates the annihilation process to DM scattering off SM
particles. This allows for direct detection of DM by measuring the corresponding recoil
of the scattering partner [87]. Experiments aiming at observing DM based on nuclear
recoil provide strong bounds on WIMP DM with masses in the range of a few GeV
12
2.3. Dark Matter
thermal freeze-out
indirect detection
DM SM Figure 2.1: Feynman diagram for WIMP DM produc-
tion and detection processes. For thermal production
via freeze-out and indirect detection via DM annihila-
tion into SM particles, the diagram has to be read from
left to right. Read from bottom to top, the diagram de-
DM SM scribes direct detection via DM-nucleus scattering, and
read from right to left it corresponds to production at
production at colliders colliders.
to a few TeV. These typically use Xenon or other nobel gases as targets. Currently,
the strongest limits are provided by XENON1T [88]. Light WIMP DM can further be
probed by cryogenic solid-state detectors such as CRESST or SuperCMDS, which may
probe masses as low as 1MeV using electron recoils [89]. Detectors based on charged
coupled devices (CCDs) such as DAMIC [90] and SENSEI [91] can explore sub-MeV DM
via scattering on electrons, and eV-scale dark-photon DM using absorption by electrons.
Finally, inverting the annihilation process, DM can be pair-produced in collisions of
SM particles, for example in proton collisions at the LHC . The DM then escapes the
detector, leading to missing energy signatures. Collider studies of DM therefore search
for missing energy recoiling against visible particles such as jets or photons [92–95]
Despite all these efforts to detect WIMP DM, no conclusive observation has been made
so far,7 challenging the standard WIMP scenario which generically predicts promising
detection prospects. As a result, alternatives to WIMP DM have become more and more
popular over the past years.
2.3.2. Axion-Like Particles
A common alternative to WIMP DM are axion-like particles (ALPs). These are very light,
neutral scalar or pseudo-scalar particles with weak couplings to matter and radiation,
often arising as (pseudo-)Goldstone bosons of a spontaneously broken U(1) symmetry [68].
If this breaking occurs after inflation, the corresponding PT may again be observable via
GWs. The term “axion” typically refers to the QCD axion arising from the U(1)PQ
Peccei-Quinn symmetry, whereas ALPs are more general variants.
7Note however that there are some debated hints. For instance, Fermi-LAT has observed an excess
of γ-rays from the Galactic Center in the few GeV range [96]. Whether this is to be attributed
to DM annihilation is an unsolved question (see e.g. ref. [97]). A similar excess can be found in
cosmic-ray anti-proton data [98, 99]. Furthermore, the DAMA/LIBRA collaboration has reported a
controversial detection of an annually-modulated DM annihilation signal [100], which conflicts with
the non-observation of this signal in other direct detection experiments [101].
13
direct detection
2. The Standard Model and Beyond
The QCD axion is a potential solution to the strong CP problem, which, similar to
the EW hierarchy problem, is a question of naturalness. It comes from the fact that the
symmetries of the SM do not forbid the existence of the so-called θ term,
g2L = − sθ 32 θ G̃
a Gaµν , (2.8)
π µν
where G (G̃) is the (dual) gluon field strength tensor. The θ term violates CP . Although
this term is in principle a total derivative, it cannot be discarded as it gives rise to
non-perturbative effects from instantons. The θ parameter can however be moved to a
complex phase in the quark mass matrix via a chiral rotation q → eiαγ5q due to the axial
anomaly of QCD. It therefore contributes to the electric dipole moment (EDM) of the
neutron [102]. Experimental limits on the neutron EDM [103] thus constrain |θ̄| < 10−10,
where θ̄ = θ + arg detM is the physical CP violating parameter originating from the θ
parameter and the phase in the quark mass matrix M . This is highly unnatural in the
technical sense.
In axion models, the strong CP problem is solved by adding a pseudo-scalar field φA,
the axion, that features a coupling of th(e form2 )
LA = −
gs θ̄ + φA G̃a Gaµν32 µν , (2.9)π fA
where fA the axion decay constant. Such a field can arise as a Goldstone boson from
the spontaneous breaking of a global symmetry, the so-called Peccei-Quinn symmetry
U(1)PQ [104, 105]. To understand how the axion solves the strong CP problem, let us
consider the vacuum energy of QCD, E(θ̄) = −m2 2πfπ cos(θ̄) [22], where mπ and fπ are
the pion mass and decay constant. In the presence of the axion field, the vacuum energy
is modified to E(θ̄) = −m2f2π π cos(θ̄ + φA/fA), i.e. the energy is now minimized if the
axion acquires a VEV that cancels the θ̄ parameter. The θ term therefore vanishes in the
vacuum.
The mass of the QCD axion is approximately given by mA ≈ mπfπ/fA [68], i.e. the
weaker it couples to the SM (the greater fA) the smaller the axion mass. Accelerator,
reactor, and cosmological constraints generally require fA > 107GeV, resulting in axion
masses below mA . 10meV [106]. Axions and ALPs therefore typically constitute very
light DM, mostly produced non-thermally [107].
Axions with such low coupling to the SM are referred to as invisible axions and are
inherently difficult to probe experimentally. Interestingly, axions or ALPs with extremely
large decay constants, fA ∼ 1017GeV, that are coupled to dark photons may again be
probed via GWs. If the axion was initially misaligned from its minimum in the early
Universe, it will roll down its potential and start to oscillate around the minimum once
the Hubble rate has dropped to the axion’s mass. One of the dark photon’s helicities then
experiences a tachionic instability, which exponentially amplifies vacuum fluctuations and
thereby generates a chiral SGWB [108, 109].
2.3.3. Sterile Neutrinos
Another possible DM candidate are sterile neutrinos, i.e. right-handed fermions that are
complete singlets under the SM. In contrast, the left-handed neutrinos in the SM lepton
14
2.3. Dark Matter
doublets are referred to as active neutrinos, as they interact via the weak force. Indeed,
the addition of right-handed singlet neutrinos is a well motivated extension of the SM.
Given the particle content of the SM, one would intuitively like to amend it by three
generations of right-handed neutrinos. Furthermore, and much more importantly, their
addition allows for the generation of masses for the SM neutrinos.
The origin and nature of neutrino masses is also an open problem of the SM. While
the left-handed neutrinos incorporated in the SM are exactly massless, the discovery of
neutrino flavor oscillations, for which Kajita and McDonald were awarded the Physics
Nobel Prize in 2015, necessarily requires that neutrinos have masses. Fits to oscillation
data allow to determine the differences ∑of the squared neutrino masses, and thereforeestab∑lish a lower bound on their sum of mν > 0.06 eV in the case of normal ordering,and mν > 0.1 eV for inverted ordering [110]. Observations of the CMB power spectrum∑and baryon acoustic oscillations by Planck, on the other hand, impose an upper limit of
mν < 0.12 eV [60].8 Measurements of the end point of the electron energy spectrum in
β decays with KATRIN [113] further yield an upper bound on the effective anti-electron-
neutrino mass meffν < 1.1 eV .
Normal and inverted ordering refer to the different scenarios for the neutrino mass hier-
archy allowed by the mass squared differences measured from oscillations. These indicate
one small mass splitting of ∆m2 ∼ 10−5 eV2, and a larger one of ∆m2 ∼ 10−3 eV2 [114].
In the normal ordering scenario, the small splitting is between the two lighter neutrino
mass eigenstates, whereas in the inverted hierarchy, it is between the two heavier ones.
The question which of the orderings is realized in nature is still on debate, however with
a preference for normal ordering in the fits [114].
The addition of Ns right-handed9 sterile neutrinos νiR, i = 1, . . . , Ns, provides a simple
way to generate mass terms for the active neutrinos. The SM Lagrangian can then be
extended by the terms
∆L = −(Y ) ¯̀α 1ν αi L H̃ νiR − 2(MM )ij ν̄
i
R ν
c j
R + h.c. , (2.10)
where α = e, µ, τ is a flavor index, and νcR denotes the charge conjugate sterile neutri-
nos. After SSB, th√e first term generates Dirac masses of the form (mD)
i α
iα ν̄RνL with
(mD)iα = (Y ∗ν )iα v/ 2, in the same way as the masses of the charged leptons and quarks
are generated. In the absence of the second term, the neutrino masses and mass eigen-
states are simply given by the eigenvalues and -vectors ofmD. For Ns = 3, we then obtain
three Dirac neutrinos. Why may however wonder why the neutrinos are so much lighter
than all other SM fermions, despite obtaining their masses via the same mechanism.
8This bound assumes three mass-degenerate neutrinos with no additional relativistic DOFs at low tem-
peratures. It may for instance be altered by the presence of additional neutrino species. Including
variations of Neff (see section 8.1.2 for a definition)∑still yields the same constraint [60], while also
fitting further parameters can relax the bound to mν < 0.515 eV [111]. Furthermore, as higher
neutrino masses lead to a lower Hubble rate today, including the data of [112], which is discrepant
with the Planck data at the 4.4σ level and conversely predicts a higher Hubble rate, leads to tighter
constraints [60].
9Note that the assumption that the sterile neutrinos are right-handed is not a restriction. Adding left-
handed sterile neutrinos leads to the same interactions terms with νR replaced by the charge conjugate
νcs , while adding both chiralities corresponds to adding twice as many single-handed fields.
15
2. The Standard Model and Beyond
If, on the other hand, the Majorana mass term for the sterile neutrinos (the second
term in eq. (2.10)) is included, the masses and eigenstates are obtained by diagonalizing
the combined mass matrix
( )
∆L ⊃ −1 c 
   0 mTDνL2 ν̄ ν̄R + h.c. , (2.11)L mD MM νcR
where we suppressed flavor indices. This induces a mixing between the active and charge-
conjugated sterile neutrinos. In general, we then obtain 3 + N Majorana neutrinos,10s
i.e. the mass eigenstates nk, k = 1, . . . , 3 +Ns, satisfy the Majorana condition nck = nk.11
Including the Majorana mass term has the attractive feature that it may explain the
smallness of the active neutrino masses via the (type-I) seesaw mechanism [116]. If we
take the Majorana masses much higher than the weak scale, i.e. MM  mD, we typically
obtain three light neutrinos with masses mν ' −mTDM−1M mD and NS heavy neutrinos
with masses ∼MM [110].
At least two sterile neutrinos are needed to generate the masses required to explain
the oscillation data. The lightest neutrino then remains massless. Adding a third right-
handed neutrino allows to provide masses for all three active flavors. We can thus arrange
a third or fourth right-handed neutrino to remain (mostly) sterile. As suggested in [117],
it may then constitute a non-thermal DM candidate with keV scale masses. This is for
instance incorporated in the neutrino minimal SM (νMSM) [118, 119], which extends the
SM by three right-handed neutrinos, of which two provide masses for the active neutrinos,
and the third one constitutes DM. The model can further generate the baryon asymmetry
of the Universe.
10Whether neutrinos are Dirac or Majorana states is another open question. An observation of neutri-
noless double-β-decay can shed light on this problem [115].
11 ∑ ∑Diagonalizing the mass matrix in eq. (2.11), we obtain ∆L = − mk c k k mk c
k 2 ν̄L νL + h.c. = − 2 n̄knk k,
where n k c kk = νL + νL . If MM has vanishing eigenvalues, some of these combine to Dirac fields.
16
Part I
New Physics at
Colliders

Prelude
In this first part of the dissertation we are going to consider various collider probes of
new physics. We will mostly focus on proton-proton collisions.
Currently, the most prominent and important collider facility is the LHC (Large Hadron
Collider) at CERN . It is the largest and most powerful accelerator in the world, colliding
protons or heavy ions at center-of-mass energies up to 13TeV. The LHC features the
four major experiments ATLAS, CMS, LHCb and ALICE, where the first two are the
ones most relevant in the following. During its second operational run that terminated
recently in the end of 2018, it has√delivered an integrated luminosity around ∼ 150 fb
−1
of proton-proton collision data at s = 13TeV to ATLAS and CMS each. In the first run
in 2√011 and 2012, roughly 6 fb
−1 and 23 fb−1 were collected per experiment at energies
of s = 7TeV and 8TeV, respectively. After the current maintenance shut-down, the
LHC is planned to resume operation in 2021 at a center-of-mass energy of 14TeV, with
the aim to record 300 fb−1 of pp collisions in run three until the end of 2024.
For the post-LHC era, various successor projects have been proposed. The LHC itself
will prospectively be upgraded to the HL-LHC (high-luminosity LHC) and subsequently
continue running at 14TeV around 2027, intending to reach an integrated luminosity of
3 ab−1 within a decade. More speculative future plans include for instance a HE-LHC
(high-energy LHC) with a center-of-mass energy of 27TeV, or an even more futuristic
100TeV proton collider denoted FCChh (Future Circular Collider).
In addition, future electron-positron collid√ers are also in debate. From 1989 to 2000,
electrons and positrons with energies up to s = 209GeV were collided at LEP (Large
Electron-Positron Collider) at CERN , providing data for precise determinations of the
properties of the Z and W boson. Although energy losses via Bremsstrahlung preclude
the acceleration of electrons to the energies reached in hadron machines, the cleaner
experimental conditions, in particular the information of the total momentum in the
events along the beam line, render electron colliders promising alternatives for future
experiments. As a consequence, proposals for new e+e− machines have been elaborated,
including both, linear accelera√tors such as the ILC (International Linear Collider) with
center-√of-mass energies up to s = 1TeV, as well as circular colliders like an FCCee with
up to s = 350GeV. Further ideas such as electron-hadron or muon colliders may also
be conceived. We will here however mostly focus on pp collisions.
Physics beyond the Standard Model (BSM) can be probed at colliders via two dif-
ferent paths. It can be searched for directly, trying to encounter new particles or their
corresponding missing-energy signature if they do not interact with the detector, or indi-
rectly via its effects on Standard Model (SM) processes. For the former case, a variety of
studies have been conducted by ATLAS and CMS, for historical reasons often presented
in the context of models of supersymmetry (SUSY). To obtain the corresponding limits
on a specific model, these searches then need to be adapted and reinterpreted, typically
19
Prelude: New Physics at Colliders
involving a recasting based on Monte Carlo (MC) simulations. In the latter case, on the
other hand, the limits obtained from data or projections are usually presented as bounds
on higher-dimensional operators contributing to the processes under consideration, em-
ploying the framework of effective field theories (EFTs).
In the following, we will present examples of both of these two approaches. Chapter 3
performs a direct search for Higgs-portal dark matter (DM). We reinterpret a CMS study
of invisible Higgs decays in vector-boson fusion (VBF) and provide a forecast for the HL-
LHC and HE-LHC sensitivity. In chapter 4 we entertain the top-pair associated Higgs
production channel to search for the h→ Zγ decay at future colliders and investigate the
corresponding indirect constraints on the new-physics contribution to the decay process.
We then conclude this part with a comprehensive study of an extension of the SM in
which lepton number is gauged, exploring the DM and collider phenomenology of the
model in chapter 5.
20
3. Probing Higgs-Portal Dark Matterwith Vector-Boson Fusion
This chapter is based on the publication [4] elaborated in collaboration with Jan Heisig,
Michael Krämer, and Alexander Mück. It mostly duplicates the structure and logic of the
publication.
To start our discussion of collider probes of new physics, we here present a search
for Higgs-portal dark matter (DM) at the LHC as well as its high-luminosity and high-
energy upgrades, entertaining the vector-boson fusion (VBF) Higgs production channel.
We reinterpret a study of invisible Higgs decays in VBF [120] and recast the corresponding
projections for the HL-LHC [121] by CMS. The latter is also used as a basis to obtain a
sensitivity forecast at the HE-LHC , including estimates of systematic uncertainties.
The discovery of the Higgs boson by ATLAS [16] and CMS [17] in 2012 marks one of
the greatest successes of the Standard Model (SM) of particle physics. Due to the relative
recency of its first observation, it is suggestive to assume that the SM Higgs sector may
reveal insights into new physics beyond the Standard Model (BSM). Indeed, the Higgs
bilinear H†H is the lowest-dimensional, Lorentz and gauge invariant operator in the SM,
allowing for a coupling to singlet extensions of the SM even at the renormalizable level.
This provides motivation for so-called Higgs-portal models [122, 123], which are DM
models in which the dark sector communicates with the SM primarily via its interaction
with the Higgs boson, i.e. the Higgs boson constitutes a portal between SM and dark
sector.
In this chapter we primarily focus on the scalar singlet Higgs-portal model [124–126],
in which the SM is augmented by a scalar DM field interacting exclusively1 via the Higgs
boson. The discussion of other DM spins is deferred to appendix 3.A. Regarding its
DM phenomenology, the parameter space of the model is rather strongly constrained by
direct detection bounds [88] on one hand and the DM abundance [60] on the other hand,
limiting the scalar mass to values close to half of the Higgs mass, mS ∼ mh/2, or values
above mS & 1TeV. Throughout this chapter we take the mass of the SM Higgs boson to
be mh = 125.09GeV [127].
While collider searches cannot reach the low portal couplings required to reproduce the
full DM relic density close to the Higgs resonance around mS ' mh/2 in the standard
thermal freeze-out scenario, the respective limits can still exclude a part of the viable
parameter space, for instance in the case that the Higgs-portal scalar only constitutes a
fraction of the total amount of DM. Such a scenario may be preferred when fitting the
model to the γ-ray Galactic center excess [128] or cosmic-ray anti-proton excess [129].
Furthermore, direct detection limits can be mitigated considering minimal extensions of
the model [130–132], potentially reopening larger regions of the parameter space above
the threshold, accessible to collider searches.
1Except for its quartic self-interaction.
21
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
While previous collider studies of Higgs-portal DM have either focused on the mass
region constrained by invisible Higgs decays [120, 133], or on the far off-shell (in terms of
the Higgs’ width) regime [123, 134, 135], we here also obtain limits for scalar masses close
to the resonance. For sizeable DM couplings and masses very close to mS ' mh/2, we
encounter an unphysical enhancement of the DM production cross-section caused by the
break-down of the fixed-width prescription of the propagator. This effect is fixed using a
running width in the propagator, giving consistent limits on the portal coupling.
This chapter proceeds as follows. Section 3.1 revisits the scalar singlet Higgs portal
model and the DM constraints on its parameter space. We then provide further details
on the failure of the fixed-width propagator arising in the vicinity of the Higgs resonance
in section 3.2. In section 3.3 we consider a 13TeV CMS search for invisible Higgs decays
in VBF [120]. We present the corresponding limits on the portal coupling in section 3.3.1
and validate our Monte Carlo (MC) setup for the following sections in section 3.3.2. Based
on ref. [121], section 3.4 then derives the prospective sensitivity of the HL-LHC as well as
HE-LHC . A careful estimate of the systematic uncertainties is obtained in section 3.4.1,
and the resulting constraints are presented in section 3.4.2. We conclude in section 3.5.
Appendix 3.A provides a reinterpretation for Higgs-portal models with other types of DM
fields.
3.1. The Scalar Singlet Higgs-Portal Model
In this chapter we are mainly going to focus on the scalar singlet Higgs-portal model [124–
126]. It is one of the simplest possible, ultraviolet (UV) complete extensions of the SM,
adding only a real scalar field S that transforms as a singet under the SM symmetries.
To render S a valid DM candidate we further have to impose a Z2 symmetry under which
S is odd and all other particles are even, assuring the stability of S. The Lagrangian of
this model then reads
L = LSM +
1 1 1 1
2 ∂µS ∂
µS − 2 2 4 2 †2 mS,0 S − 4 λS S − 2 λHP S H H , (3.1)
where the only possible interaction with SM fields at the renormalizable level is the
portal coupling ∼ λHP S2H†H to the Higgs bil√inear. When the Higgs field acquires its
vacuum expectation value (VEV), 〈H〉 = (0, v/ 2 )T with v ' 246GeV, the portal term
contributes to the physical scalar massm2S = m2S,0 + 12λ 2HPv , and induces the interactions
L ⊃ −1 12 λHPv hS
2 − λ h2 24 HP S , (3.2)
where h is the SM Higgs boson and we use unitary gauge. The first term in eq. (3.2)
then allows the Higgs boson to decay into a pair of DM particles if mS < mh/2. The
corresponding invisible decay width of the Higgs bos√on is given by
2 2 2
Γ λ v minv = Γ(h→ SS) = HP32 1− 4
S
2 . (3.3)πmh mh
The phenomenology of the model is primarily determined by two parameters: the DM
mass mS and the portal coupling λHP. The third parameter of the model, the scalar
22
3.1. The Scalar Singlet Higgs-Portal Model
self-coupling λS , only plays a minor role.2 As a result, the model is very simple and
highly predictive, but also rather strongly constrained. It has been studied extensively
in the literature (see e.g. ref. [123] and references therein).
The strongest constraints on the model arise from the combination of the DM relic
density measured by Planck [60] and direct detection limits on the scattering of DM
off heavy nuclei as for instance searched for by XENON1T [88]. In the thermal freeze-
out scenario, the abundance of DM is set by its annihilation into SM particles. Once
the annihilation rate drops below the expansion rate of the Universe, the DM cannot
maintain equilibrium with the SM. It then freezes out, i.e. decouples, and its number
density per co-moving volume is conserved. The lower the interaction cross-section the
earlier the freeze-out occurs, leading to a higher relic abundance as the DM experiences
less Boltzmann suppression (assuming that decoupling happens when the DM is non-
relativistic). The requirement that we do not produce more DM than observed therefore
places a lower bound on the annihilation cross-section as a function of the DM mass.
Direct detection experiments on the other hand put an upper bound on the DM-nucleus
scattering cross-section. In our simple model, both cross sections are controlled by the
portal coupling, so that λHP is constrained from above and below.
Figure 3.1 shows the relic density3 and direct detection constraints on the DM mass
mS and the portal coupling λHP in the scalar singlet Higgs-portal model, calculated
using MicrOMEGAs v5 [140, 141]. The solid black line depicts the parameters for which
the abundance of S coincides with the DM relic abundance h2ΩDM = 0.1200 ± 0.0012
measured by Planck [60]. For couplings above this line, S can only account for a fraction
of the DM density, whereas couplings below the line are excluded as they lead to an
overproduction of DM. This constrains the portal coupling to values above λHP & 0.04,
unless the scalar mass is around the Higgs resonance mS = mh/2 where the annihilation
of S through an s-channel Higgs boson is very efficient, allowing for portal couplings as
low as λ ' few× 10−4HP .
The current 90% confidence level (CL) upper limit from the XENON1T direct detec-
tion experiment is indicated by the solid blue line in fig. 3.1, excluding the blue shaded
region above the line. The direct detection bound severely limits the viable parameter
space of the model, leaving only two regions in which S is neither over-abundant nor
excluded by direct detection: the high-mass region with mS & 1TeV, and the resonance
region where mS ' mh/2.4 Future experiments will further narrow down these regions.
The projected sensitivities for LZ [142] and DARWIN [143] are indicated by the dashed
green and dotted purple lines, respectively. In addition to the direct detection constraints,
fig. 3.1 also depicts the current LHC limit from searches for invisible decays of the Higgs
boson, excluding Br(h → SS) > 19% at 95% CL [120]. Indirect detection experiments
may impose further limits in a very narrow region around mS ' mh/2 [146].
Note that the direct detection limits in fig. 3.1 assume that the scalar S accounts for the
full measured DM abundance. If it constitutes only a fraction of the total relic density,
2This parameter is however relevant in the context of DM self-interactions [136] or the stability of the
electroweak vacuum [137].
3See refs. [138, 139] for a more careful calculation, in particular regarding the region close to the Higgs
resonance.
4These are also the regions favored by global fits of the model, see e.g. refs. [144, 145].
23
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
100
2
h ΩD
M
10−1 2h Ω
=
S
1T
NO
N
10−2 Br (h→ inv) X
E
LZ
IN
− R
W
10 3 D
A
10−4
10 20 50 100 200 500 1000
mS [GeV]
Figure 3.1: DM constraints on the scalar singlet Higgs-portal model as a function of
the DM mass mS and portal coupling λHP. Along the solid black line, the scalar S
accounts for the full DM relic abundance measured by Planck [60], whereas the region
below the line is excluded as DM is overproduced. The current 90% CL exclusion
reach of XENON1T [88] is shown in blue. The dashed green and dotted purple lines
indicate the prospective reach of LZ [142] and DARWIN [143], respectively. The orange
line corresponds to the 95% upper bound on the invisible Higgs branching ratio by
CMS [120].
as it is the case in the region above the black line, the constraints are relaxed. Taking
this into account further opens up the parameter space of larger portal couplings in the
region of the resonance.
3.2. Threshold at Resonance
Previous studies of collider searches for Higgs-portal DM have focused either on the
mass region below the Higgs resonance, mS < mh/2, where the Higgs boson can decay
invisibly into a DM pair (see e.g. refs. [120, 133]) or on DM masses lying at least a few
GeV above mh/2, where the DM can only be produced from an off-shell Higgs boson (see
e.g. refs. [123, 134, 135]). In this chapter, we will also consider the transition between the
two regions with mS ' mh/2. In order to obtain consistent results in this region, special
care needs to be taken regarding the treatment of the Higgs-boson propagator. This is
due to the failure of the fixed-width prescription of the propagator caused by the invisible
decay channel opening just above the resonance. Before investigating current and future
collider limits on the model, let us therefore first review this problem in more detail and
explain how it is fixed by using a running width in the Higgs-boson propagator.
Neglecting electroweak (EW) corrections and higher-order corrections in the portal
coupling, the DM production cross-section factorizes into the production of an off-shell
Higgs boson and its sub∫sequent decay into a DM pair, i.e.dq2
σ 2inv = 2 σh(q ) |P (q
2)|2 2 q Γ 2inv(q ) Θ(q2 − 4m2
π S
) . (3.4)
24
λHP
3.2. Threshold at Resonance
Here, σh(q2) is the production√cross-section of an off-shell Higgs with invariant mass q
2,
P (q2) is the Higgs-boson propagator, and Γinv(q2) is the off-shell decay width given by
eq. (3.3) with m 2h replaced by q .
To obtain accurate predictions for s-channel resonances at momentum q2 around the
resonance, we need to resum one-particle irreducible (1PI) loop-corrections to the prop-
agator. The resulting dressed propagator can be written as
P (q2) = i
q2 −m2R + Σ( 2) +
, (3.5)
q iε
where mR is the renormalized mass and Σ(q2) is the 1PI self-energy of the propagating
particle. For momenta close to the resonance, we can usually approximate the self-
energy as constant, replacing Σ(q2) by Σ(m2P ), where mP is the pole mass defined by
m2P = m2R + Re Σ(m2P ), i.e. the real part of Σ enters the definition of the pole mass. The
imaginary part of Σ on the other hand is related to the total decay width of the (off-
shell) particle via the optical theorem, Im Σ(q2) = q Γtot(q2). We therefore obtain the
Breit-Wigner propagator
i
P 2f (q ) = 2 2 + Γ , (3.6)q −mP imP tot
where Γtot = Γtot(m2P ), which is the propagator commonly used for s-channel resonances.
This expression is typically valid if the width is sufficiently small, Γtot  mp. If the
propagating particle is kinematically allowed to go on-shell, i.e. if mS < mh/2 in our
case, we can further use the narrow-width approximation (NWA) and take
|Pf ( 2)|2 ≈
π
q 2Γ δ(q −m
2
P ) , (3.7)mP tot
so that the DM cross-section eq. (3.4) simply becomes the on-shell cross-section times
branching ratio, σ = σ (m2DM h h) × Brinv with Brinv = Γinv/Γtot. In particular, the total
Higgs production cross∫-section is then equal to the on-shell production cross-section,2
tot = dq ( 2) 2 q Γtot(q
2)
σ 2f 2 σh qπ (q2 −m2h)2 + 2Γ2 ( 2)
' σh(mh) . (3.8)mh tot mh
The fixed-width propagator eq. (3.6) presumes that the total decay rate is a smooth
function around the resonance and can be approximated as a constant. This is however
not the case if a large decay channel opens up close to the resonance, as it is the case in
the Higgs-portal model for mS ' mh/2 and λHP & 0.1. If this assumption is violated, the
fixed-width prescription breaks down and the total cross-section calculated using eq. (3.8)
can exceed the on-shell production cross-section. This effect is demonstrated in fig. 3.2,
where we show the fiducial DM production cross-section in VBF at the 13TeV LHC
(see section 3.3 for details) calculated using a fixed-width propagator (solid red line) for
λHP = 1 and DM masses close to mh/2. With such a large portal coupling, the Higgs
boson decays 100% invisibly if mS < mh/2, such that the DM cross-section is equal
to the on-shell Higgs cross-section below the resonance, whereas it falls off quickly for
25
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
16
−1 λHP = 1 1.0
mh
10 mS = 2 14
0.8 λHP = 1 12
10
on-shell 0.6 2
cross-section 2 q Γinv(q ) 8
10−2 0.4 |Pr(q
2)|2 6
|P (q2 2f )|
fixed width 40.2
running width 2
0.0 0
−200 −100 0 100 200 −4 −2 0 2 4 6 8 10
(mS −mh/2) [MeV] (q −mh) [MeV]
Figure 3.2: Fiducial cross-section for Figure 3.3: Squared fixed-width (dashed
DM production in VBF for λHP = 1 blue) and running-width (solid blue) prop-
in the fixed-width (solid red) and agator and decay width (red) as a function
running-width (dashed blue) prescrip- of the invariant mass of the Higgs boson.
tion.
DM masses above the resonance. However, for mS ' mh/2 the cross section displays an
unphysical feature, exceeding the on-shell cross-section by almost an order of magnitude.
This unphysical behavior can be fixed by using the running-width propagator which
keeps the momentum dependence in the imaginary part of the self-energy,
P (q2) = ir √
q2 −m2 + i q2 Γtot(q2
, (3.9)
P )
where we use Γtot(q2) = ΓSM 2 SMh + Γinv(q ) with Γh = 4.1MeV (i.e. we neglect the momen-
tum dependence of the visible width) as the dominant effect originates from the opening
of the invisible channel. As shown by the blue dashed line in fig. 3.2, the DM cross-section
is well-behaved if calculated using a running-width propagator, with σinv ≤ σh(m2h) for
all DM masses.
To further illustrate why the fixed-width prescription breaks down, let us consider the
squared propagators as well as the numerator 2 q Γinv(q2) from eq. (3.4) in the resonance
region q2 ' m2h for mS = mh/2 and λHP = 1 depicted in fig. 3.3. If the fixed-width
propagator (dashed blue line) is used, the suppression of off-shell momenta sightly above
the resonance is insufficient to overcome the rapidly-growing invisible width in the nu-
merator (red line). For momenta close to the resonance, q2 ≈ m2h, we therefore obtain
(cf. eq. (3.4))
2 Γ ( 2 q Γinv(q
2)
q inv q
2) |P 2 2f (q )| ' 2 2 2  1 , (3.10)mh Γtot(mh)
which can grow arbitrarily large and lead to an enhancement of the DM production
cross-section to values above the cross-section for on-shell Higgs production. If, on the
other hand, the running Higgs-width is used in the propagator (solid blue line), the
denominator also grows with q2. As a result, the opening invisible channel leads to an
26
σinv [pb]
|P (q2)|2 × (m Γ )2h tot
2 q Γ (q2 2inv ) [GeV ]
3.3. Current LHC Limits
q q
Z,W± S
h
Z,W± S
Figure 3.4: Feynman diagram for Higgs-portal
q̄ q̄ DM production in VBF at the LHC .
additional suppression of momenta above the threshold for DM pair production in the
denominator and we obtain
2 Γ 2 2 2 2 q Γinv(q
2) 2
q inv(q ) |Pr(q )| '
q2 Γ2 2 ∼ , (3.11)tot(q ) q Γinv(q2)
where we assumed that Γinv(q2) dominates the total width. This additional suppression
of momenta above the resonance prohibits the uncontrolled growth of the cross section
and restores σinv ≤ σh(m2h).
3.3. Current LHC Limits
Let us now investigate the constraints we can put on the Higgs-portal coupling λHP from
current LHC data. We will base our analysis on a search for invisible Higgs decays in the
VBF channel by CMS [120] using 35.9 fb−1 of data recorded at a center-of-mass energy
of 13TeV.
Vector-boson fusion (VBF) [147, 148] is one of the primary Higgs production-channels
at the LHC . In this channel, the Higgs boson is produced from the fusion of two Z or W
bosons radiated off quarks from the colliding protons in the process pp→ h+ 2 jets. The
corresponding Feynman diagram is depicted in fig. 3.4. It features a very characteristic
signature of two hard forward jets separated by a large gap in pseudo-rapidity and with
a large dijet invariant mass. While the cross section for VBF Higgs production is roughly
an order of magnitude below the cross section for Higgs production in gluon fusion, its
distinct topology allows for an efficient suppression of background processes via phase-
space cuts, which is of particular importance in searches for invisible decays at proton
colliders as the Higgs boson cannot be reconstructed from its decay products in this case.
VBF hence constitutes the most promising channel in searches for Higgs-portal DM and
other primarily Higgs-mediated DM models [123, 134, 135, 149–151]. In the following we
will therefore focus on the VBF channel only.
The CMS search [120] presents limits derived from a cut-and-count analysis, as well as
a shape analysis with respect to the jet-pair invariant mass and pseudo-rapidity difference
distributions, posing a 95% CL observed upper bound on the invisible branching ratio
of the SM Higgs-boson of Br(h → inv) < 58% and Br(h → inv) < 33%, respectively.5
CMS further interprets their results as limits on the signal strength of an additional Higgs
5Note that the cut-and-count limit is alleviated compared to the expected limit of Br(h→ inv) < 30%
due to a ∼ 2.5σ excess of events in the signal region with respect to the background-only prediction,
27
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
boson H with mass mH that is produced SM-like, decays invisibly, and does not mix with
the SM Higgs. We will use these limits to reinterpret the CMS search in the context of
the scalar singlet Higgs-portal model in section 3.3.1 based on eq. (3.4). This allows us
to impose limits on the Higgs-portal coupling λHP for DM masses below, around, and
above the resonance without the need of any MC simulation. In section 3.3.2 we then
perform a leading order (LO) MC recasting of the cut-and-count analysis, validating the
MC setup used for our HL- and HE-LHC projections.
3.3.1. Reinterpretation of Upper Limits
In this section, we will reinterpret the CMS search [120] in the context of the scalar singlet
Higgs-portal model to obtain upper limits on the portal coupling λHP. For DM masses
below the threshold for production from on-shell Higgs decays, mS < mh/2, these bounds
can be trivially obtained directly from the 95% CL upper limit on the invisible branching
ratio Br95%inv . For masses in the vicinity and beyond the threshold on the other hand, we use
eq. (3.4) to calculate the VBF DM-production cross-section σinv, where in this context
σh(q2) denotes the fiducial cross-section at detector-level (including acceptance t√imes
efficiency) for the VBF production of an off-shell Higgs boson with invariant mass q2.
In the case of a cut-and-count analysis, the upper bound on λHP can then be computed
by equating σinv to the limit on the signal cross-section σ95%inv = Br95%inv × S/L, where
S = 743 [120] is the predicted number of signal events for Brinv = 1, and L = 35.9 fb−1
is the integrated luminosity.
To obtain the off-shell Higgs production cross-section σh(q2) from the experimental
analysis, we here use the limits on the signal strength µH = σH/σSMH × Br(H → inv) of
an additional Higgs boson H with mass mH that does not mix with the 125GeV Higgs
boson and is produced as in the SM (cf. fig. 7 of ref. [120]). If next-to-leading order
(NLO) EW corrections are neglected, the SM prediction for the on-shell production of
H simply corresponds to the off-shell production of the 125GeV Higgs at q2 = m2H,
i.e. σSMH = σ 2h(q = m2H). We can therefore relate the 95% CL limit on µH from the
cut-and-count analysis to the off-shell Higgs production cross-section σh(q2),
95%
µ95%H (m2H) =
σinv
( 2 = 2 ) , (3.12)σh q mH
so that eq. (3.4) can be rewritten as
∫
σinv = dq
2 1
95% 2 95% |P (q
2)|2 2 q Γ 2 2 2
( 2) inv
(q ) Θ(q − 4mS) . (3.13)
σinv π µH q
As a given parameter point is excluded at 95% CL if σinv > σ95%inv , the corresponding
limit on λHP as a function of the DM mass can be obtained by equating eq. (3.13) to one
and solving the equation numerically.
attributed to a statistical fluctuation (see section 7.2 of ref. [120]). A similarly strong bound of
Br(h→ inv) < 53% can be obtained from the recent measurement of the total Higgs boson width,
Γ = 3.2+2.8tot −2.2MeV [152].
28
3.3. Current LHC Limits
(mS −mh/2) [GeV]
−0.01 0.00 0.01
obs., on-shell
obs., running width
1 1
obs., fixed width
0.1
obs., running width
obs., fixed width
exp., running width exp., running width
exp., fixed width 0.1 exp., fixed width
0.01
61 62 63 64 65 66 67 62.53 62.54 62.55 62.56
mS [GeV] mS [GeV]
Figure 3.5: Observed (black) and expected (green) 95% CL upper limits on the
Higgs-portal coupling λHP from the cut-and-count analysis for a wide range of DM
masses (left) and very close to the resonance (right). The solid lines use the running-
width prescription, whereas the dashed lines indicate the corresponding limits if a
fixed width is used. The gray band reflects the uncertainty on the signal cross-section
in ref. [120]. The dotted line in the right panel indicates the limits from the invisible
branching ratio.
The resulting 95% CL limits on the portal coupling λHP as a function of the DM mass
mS from the cut-and-count analysis are shown in fig. 3.5. The black (green) line indicates
the observed (expected) limit. The solid lines correspond to limits obtained employing
the running-width prescription of the propagator, whereas the dashed lines use a fixed
width. We also indicate the 17% uncertainty on the on-shell Higgs production cross-
section in ref. [120], reflected by the gray band obtained by solving σ /σ95%inv inv = 1± 0.17.
The left panel covers a mass range of several GeV around the resonance mS = mh/2,
whereas the right panel focuses on the very resonance.
For DM masses below the Higgs resonance, portal couplings as low as λHP ∼ 0.1 can be
excluded. In this region both descriptions of the width in the propagator give consistent
results that perfectly agree with the limits obtained from the invisible branching fraction
(i.e. using the NWA) shown as a dotted line in the right panel of fig. 3.5. However, as
we approach the resonance, for masses mS & m /2−ΓSMh h the fixed-width approximation
(as well as the NWA) breaks down. The limits obtained using a fixed-width propagator
(dashed lines) exhibit an unphysical feature at the resonance (cf. right panel), whereas the
running-width calculation yields the proper bounds, excluding λHP > 0.47 atmS = mh/2.
Above the threshold, the search rapidly becomes less sensitive and only λHP & 1 can be
probed, entering the non-perturbative regime. The loss of perturbative control is further
indicated by the large deviations in the observed limits between the two calculations at
DM masses well above the threshold, where a difference between the descriptions is not
expected. This difference is formally of higher-order in λHP and can be interpreted as a
lower bound on the theoretical uncertainty.
29
λHP
λHP
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
(mS −mh/2) [GeV]
−0.01 0.00 0.01
obs., on-shell
obs., running width
1 1
obs., fixed width
0.1
obs., running width
obs., fixed width
exp., running width exp., running width
exp., fixed width 0.1 exp., fixed width
0.01
61 62 63 64 65 66 67 62.53 62.54 62.55 62.56
mS [GeV] mS [GeV]
Figure 3.6: Same as fig. 3.5, but for the shape analysis.
We further apply eq. (3.13) to derive limits from the shape analysis in [120], assuming
that the dependence of the distributions of the dijet invariant-mass and pseudo-rapidity
difference on q2 can be neglected. The resulting bounds are shown in fig. 3.6. The analysis
excludes λHP > 0.30 for mS = mh/2 and yields limits in the perturbative regime up to
mS . 67GeV. As the couplings constrained in the shape analysis are lower than in the
cut-and-count case, the effects of the breakdown of the fixed-width description are less
pronounced and both calculations agree well within the uncertainties.
3.3.2. Recasting of the Cut-and-Count Analysis
To validate the Monte Carlo (MC) setup we use for the HL-LHC and HE-LHC projections
in section 3.4, let us now perform a MC recasting of the cut-and-count analysis of ref. [120].
We again calculate the DM cross-section using eq. (3.4), but we now obtain the cross
section for off-shell Higgs production from MC simulation. Sinc√e NLO EW correctionsare neglected, the off-shell cross-sections can be calculated from the on-shell cross-section
in the SM, setting the Higgs mass to the corresponding value of q2.
We use the following setup for our MC simulation. Events for VBF Higgs produc-
tion in the SM with different masses of the Higgs boson are generated at LO using
MadGraph5_aMC@NLO v2.6 [153] in the 5-flavor scheme. We employ the NNPDF3.0 [154]
LO parton distribution function (PDF) set with αs(mZ) = 0.118 provided by the
LHAPDF6 [155] library. The renormalization and factorization scales are set to the W
mass, µR = µF = mW [156]. The generated events are passed to Pythia v8.235 [157]
to model parton shower and hadronization. Detector effects are subsequently simulated
in Delphes v3.4.2 [158] with the CMS detector card. Jet clustering is performed by
FastJet v3.3.1 [159] using the anti-kT algorithm [160] with R = 0.4.
We adapt the cuts used in the CMS analysis [120], listed in the corresponding column in
table 3.1. At least two jets with |ηj | < 4.7 and a minimum pT of 80GeV (40GeV) for the
(second-)hardest jet are required. To single out VBF Higgs production events, the leading
jet pair is further required to be separated in pseudo-rapidity, |∆ηjj | = |ηj1 − ηj2 | > 4.0,
30
λHP
λHP
3.3. Current LHC Limits
√
s 13TeV 14TeV / 27TeV
p j1T > 80GeV > 80GeV
p j2T > 40GeV > 40GeV
|ηj | < 4.7 < 5.0
min (|ηj1 |, |ηj2 |) < 3.0 –
Mjj > 1.3TeV > 2.5TeV / > 6TeV
ηj1 · ηj2 < 0 < 0
|∆ηjj | > 4.0 > 4.0
|∆φjj | < 1.5 < 1.8
E/T > 250GeV > 190GeV
|∆φjE/ | > 0.5 (p
j
T > 30GeV) > 0.5 (p
j
T T
> 30GeV)
photon veto pγT > 15GeV, |ηγ | < 2.5 –
electron veto peT > 10GeV, |ηe| < 2.5 peT > 10GeV, |ηe| < 2.8
muon veto pµ µT > 10GeV, |ηµ| < 2.4 pT > 10GeV, |ηµ| < 2.8
τ -lepton veto pτ τT > 18GeV, |ητ | < 2.3 pT > 20GeV, |ητ | < 3.0
b-jet veto pbT > 20GeV, |ηb| < 2.4 pbT > 30GeV, |ηb| < 5.0
Table 3.1: Analysis cuts used in this paper. The cuts for 13TeV and 14TeV are taken
from refs. [120] and [121], respectively. The cuts for the 27TeV HE-LHC are identical
to the ones for the 14TeV HL-LHC except for the cut on Mjj .
with ηj1 · ηj2 < 0 (i.e. the jets are in different hemispheres of the detector), to have a large
invariant mass, Mjj > 1.3TeV, as well as to be close in azimuthal angle, |∆φjj | < 1.5.
As the DM particles are invisible to the detector, a lower cut on the missing transverse
energy (MET) of E/T > 250GeV is applied. Additional jets with pT > 30GeV are allowed
if they are well separated from the MET, |∆φjE/ | > 0.5. Photons, electrons, muons, asT
well as b- and τ -tagged jets are vetoed.
To account for the contribution of gluon-initiated Higgs production to the signal, as well
as to profit from the NLO corrections and more sophisticated detector effects included
in the CMS simulation, we rescale our results for the cross section to match the CMS
prediction for on-shell Higgs production. For Br(h → inv) = 1, CMS predicts a total of
743±129 signal events [120], corresponding to a fiducial cross-section of (20.7± 3.6) fb−1,
whereas our LO simulation yields 14.2 fb−1 with a scale uncertainty around 25%. We
therefore rescale our cross sections by a factor 1.46 for the 13TeV LHC .
31
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
13 TeV obs., CMS Figure 3.7: Observed 95% CL up-
10 13 TeV obs. per limit on the signal strength
14 TeV, 3 ab−1 µH = σ × Br(H → inv)/σh(mH)
27 TeV, 15 ab−1 for an invisibly decaying Higgs
1 boson with mass mH from
the CMS cut-and-count anal-
0.1 ysis in ref. [120] (solid black),
as well the predictions from
our simulation for the current
0.01
LHC (dashed black), HL-
100 200 500 1000 LHC (blue), and HE-LHC (red).
mH [GeV]
Our (rescaled) cross-section predictions are presented in terms of the corresponding
95% CL upper limits on the signal strength µH for additional Higgs bosons with mass
mH in fig. 3.7. The dashed black line depicts our result, whereas the CMS limit from
fig. 7 of ref. [120] is indicated by the solid black line. For mH below 150GeV, the
limits agree within the percent level, whereas the deviation between the two becomes
larger for higher mH. However, for the DM masses constrained by the CMS analy-
sis, the integral in eq. (3.4) is dominated by values of q2 well below (200GeV)2. For
mS ' 70GeV for instance, more than 75% of the cross section arises from contributions
with q2 < (200GeV)2. Therefore, the limits on λHP derived using our MC simulation
agree with those presented in fig. 3.5 within the statistical MC uncertainty (∼ 1%), both
below and above the Higgs threshold.
3.4. HL-LHC and HE-LHC Projections
We will now derive the projected upper limits on the Higgs-portal coupling λHP at
the 14TeV HL-LHC with an integrated luminosity of 3 ab−1 and the 27TeV HE-LHC
with 15 ab−1 of luminosity. Our projections are based on the sensitivity forecast for the
search for invisible Higgs decays in VBF at the HL-LHC by CMS [121], which predicts a
prospective 95% CL upper bound on the invisible branching ratio of the Higgs boson of
Br(h→ inv) < 3.8%. This CMS limit is obtained from a cut-and-count analysis apply-
ing the cuts listed in the right column of table 3.1. The cuts resemble those used in the
13TeV analysis [120], with a lower MET cut of E/T > 190GeV and an increased cut on
the invariant mass of the leading jet pair, Mjj > 2.5TeV.
We calculate the LO cross-section σh(q2) for off-shell Higgs production in VBF at
center-of-mass energies of 14TeV and 27TeV with the same MC setup as used for our
13TeV limits described in section 3.3.2, now using the HL-LHC detector card in Delphes.
For the HL-LHC predictions we adopt cuts from the corresponding CMS study [121].
The same cuts are also applied for our HE-LHC study, with the invariant jet mass cut
raised to Mjj > 6TeV to benefit from the higher center-of-mass energy and increased lu-
minosity (see section 3.4.1 for details). To incorporate gluon-inititiated production, NLO
corrections, and the more refined detector simulation of CMS, our fiducial detector-level
32
σ × Br(H → inv)/σSM
3.4. HL-LHC and HE-LHC Projections
cross-section for on-shell Higgs production is again rescaled to match the H√L-LHC CMS
prediction of 16.3 fb [121]. Our MC simulation yields σh(m2h) = 10.6 fb at s = 14TeV,
corresponding to a rescaling factor of 1.54, which differs from the 13TeV rescaling factor
by 5%. We take this as an indication that the rescaling factor does not vary substantially
with the center-of-mass energy and the applied cuts, and therefore use the same factor
to rescale our 27TeV results.
The resulting 95% CL projected HL-LHC upper limits on the signal strength µH of
additional Higgs bosons are depicted by the blue line in fig. 3.7. These bounds are derived
based on the corresponding limit on the invisible branching ratio of the 125GeV Higgs
boson, Br(h → inv) < 3.8% [121]. The red line denotes the projected sensitivity of the
HE-LHC , using Br(h → inv) < 2.1% obtained as explained in the following sections.
The respective limits on the portal coupling λHP derived using eq. (3.13) are discussed in
section 3.4.2 (cf. fig. 3.10).
3.4.1. Background Predictions and Systematic Uncertainties
While we could base our HL-LHC limits on the CMS predictions in ref. [121], there is
no corresponding experimental projection for a 27TeV machine. Therefore, to obtain the
HE-LHC limit on the invisible Higgs branching ratio and the signal strength µH, as well
as the corresponding bounds on the portal coupling λHP via eq. (3.13), further steps are
required. In particular, we need to predict the number of background events, estimate the
systematic uncertainty, and adapt the analysis to the detector setup (i.e. center-of-mass
energy and luminosity).
We generate events for the dominant backgrounds pp → V + jets at LO, where V
is either a Z or W boson decaying into two neutrinos (Z → νν) or a neutrino-lepton-
pair (W → `ν), respectively. The same MC tool-chain as for the signal simulation is
used. Samples with two and three jets in the final state are merged employing the MLM
matching procedure [161, 162]. For comparison with the CMS predictions at 14TeV, we
separately simulate processes in which the jets originate from EW interactions and quan-
tum chromodynamics (QCD) radiation, neglecting interference effects. To improve our
LO prediction, we also generate the background events at the HL-LHC and determine
rescaling factors to match the CMS predictions in ref. [121] for each of the four contri-
butions. Furthermore, the subleading background originating from top pair production
(below 4%) is obtained by rescaling theW + jets background, assuming similar kinematic
distributions. The same factors are then used to rescale the background cross-sections at
the HE-LHC .
Figure 3.8 shows the jet-pair invariant mass distribution of our simulated events for the
Z+jets (blue),W+jets (red) and top (green) backgrounds at the HL-LHC (left) and HE-
LHC (right). For the Z and W backgrounds, the EW (QCD) contributions are shown
in deep (light) colors. Our prediction for the distribution of events for on-shell Higgs
production in VBF is depicted by the solid thick line. In the HL-LHC case, the signal
(thick) and total background (thin) predictions of CMS [121] are indicated by dashed
lines. A good agreement between CMS and our (rescaled) simulation can be observed.
33
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
12000 50000
W → `ν (QCD) W → `ν (QCD)
10000 W → `ν (EW) 40000 W → `ν (EW)
Z → νν (QCD) Z → νν (QCD)
8000 Z → νν (EW) Z → νν (EW)
30000
top top
6000 h(125 GeV) h(125 GeV)
total bkg. CMS 20000
4000
h(125 GeV) CMS
2000 10000
0 0
2.5 3.0 3.5 4.0 4.5 5.0 5 6 7 8 9 10
Mjj [TeV] Mjj [TeV]
(a) 14TeV HL-LHC , 3 ab−1 (b) 27TeV HE-LHC , 15 ab−1
Figure 3.8: Jet-pair invariant mass distribution of the main backgrounds (histograms)
and on-shell Higgs production (solid line) at the HL-LHC (left) and HE-LHC (right).
The thick and thin dashed lines in the left panel indicate the CMS prediction for the
signal and total background, respectively.
Based on our predictions for the signal and background cross-sections, we derive the
95% CL upper limit on the number of signal events in the Gaussian limit from the
condition √ S = 1.96 , (3.14)
S +B + (∆sysB )2
where S and B are the number of signal and background events, respectively, and ∆sysB
is the systematic uncertainty on the b√ackground. For the latter we employ a simpleuncertainty model,
∆sysB = f1B + (f2B)2 , (3.15)
√
consisting of one part scaling with B, i.e. like a statistical Poisson uncertainty, and
one part scaling with B, i.e. a luminosity-independent relative uncertainty, added in
quadrature.
Equation (3.15) is motivated by data-driven background determination methods used
in experimental analyses, as it is case in the 13TeV analysis [120] and the 14TeV projec-
tions [121] by CMS. The background originating from a Z boson produced in association
with jets and decaying into neutrinos can for instance be obtained by measuring the same
process with the Z boson decaying into charged leptons in a control region. If statistics
dominated, the uncertainty on the number of events in the control region scales with the
square root of the number of events, giving rise to the first contribution in eq. (3.15). A
prefactor f1 > 1 then corresponds to control regions with smaller statistics than the signal
region. The second contribution in eq. (3.15) accounts for the systematic uncertainty on
the transfer factors relating the event numbers in the control and signal regions.
As the two contributions to the uncertainty scale differently with the integrated lumi-
nosity, the HL-LHC limits on the invisible Higgs branching ratio for 300 fb−1, 1 ab−1 and
34
Events
Events
3.4. HL-LHC and HE-LHC Projections
2.6 Figure 3.9: Projected 95% CL up-
channel-wise rescaling per limit on the invisible branch-
global rescaling
2.4 ing ratio of the 125GeV Higgs bo-
son at the HE-LHC as a function
of the cut on the invariant mass
2.2
M cutjj of the leading jet pair, using
a channel-wise (solid black) and
2.0 global (dashed blue) rescaling of
the background. The gray band
1.8 depicts the variation of the limit if
5.0 5.5 6.0 6.5 7.0 7.5 8.0 the number of background events
M cutjj [TeV] is changed by 10%.
3 ab−1 provided in fig. 5b of ref. [121] can be used to extract the coefficients f1 and f2.
This yields f1 = 1.5 and f2 = 1.3%. We use these results to estimate the systematic
uncertainty at the HE-LHC .
We now adapt the cut-and-count search from ref. [121] to the HE-LHC using eq. (3.14).
Due to the increased center-of-mass energy, we find the biggest potential for gaining
sensitivity in strengthening the cut on the invariant mass Mjj of the jet pair. Since the
higher Mjj cut is also the most notable difference between the cuts used at 13TeV and
14TeV shown in table 3.1, we optimize this cut only and keep all other cuts as in the
HL-LHC case. Although further improvement may be achieved applying a higher cut on
the pseudo-rapidity difference of the leading jets, |∆ηjj |, we here refrain from doing so as
we cannot estimate the detector performance at high ηj reliably.
Figure 3.9 shows the 95% CL upper limit on the invisible branching ratio of the Higgs
boson as a function of the jet-pair invariant-mass cutM cutjj at the HE-LHC , obtained from
eq. (3.14) and based on the distributions depicted in fig. 3.8b. The black line indicates
the limits obtained with the background event numbers rescaled for each contribution
individually, as described above. For comparison, the dashed blue line shows the bound
we get rescaling all background contributions with the same factor. This global rescaling
factor is again determined from the 14TeV simulations, requiring that the total number of
background events coincides with the CMS prediction [121]. The boundaries of the gray
band correspond to the limits (using channel-wise rescaling) if the systematic background
uncertainty eq. (3.15) is varied by ±10%.
The strongest limit on the invisible branching fraction is obtained for an invariant mass
cut around M cutjj ' 6TeV – 6.5TeV, with only marginal variation of the limit within
this range. We therefore adopt Mjj > 6TeV to derive our limits, yielding a total of
∼ 120 000 background and ∼ 150 000 signal events for on-shell Higgs production with
Br(h → inv) = 100% at an integrated luminosity of 15 ab−1. The relative systematic
uncertainty on the background from eq. (3.15) is 1.4%, dominated by the luminosity-
independent contribution f2. From eq. (3.14) we obtain the 95% CL upper limit on the
invisible branching ratio of the Higgs boson Br(h→ inv) < 2.1%.
35
Br(h→ inv) upper limit [%]
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
(mS −mh/2) [GeV]
−1 0 1 Figure 3.10: Projected 95% CL
upper limits on the Higgs-portal
coupling λHP at the HL-LHC
1 (blue curve) and HE-LHC (red
curve) with integrated luminosi-
ties of 3 ab−1 and 15 ab−1, respec-
0.1
tively. The shaded band indicates
the HL-LHC sensitivity if the sys-
0.01 − tematic uncertainty on the back-14 TeV, 3 ab 1
−1 ground is multiplied or divided by27 TeV, 15 ab
two. Note that the plot changes
61 62 63 64 70 80 90 100 from linear to logarithmic scaling
mS [GeV] of the abscissa at mS = 64GeV.
3.4.2. Sensitivity Projections
The HL-LHC and HE-LHC limits on the signal strength µH of an additional Higgs boson
with mass mH that does not mix with the 125GeV Higgs are indicated by the blue and
red line in fig. 3.7. Figure 3.10 shows the corresponding 95% CL upper limits on the
Higgs-portal coupling λHP obtained from eq. (3.13). The blue curve corresponds to the
prospective HL-LHC sensitivity assuming an integrated luminosity of 3 ab−1, whereas the
red line denotes the HE-LHC projection from 15 ab−1 of data. The shaded band shows
the shift of the HL-LHC bound if the systematic uncertainty on the background is varied
between half and twice the value obtained from eq. (3.15).
For DM masses less than mS . 61GeV, portal couplings around λHP ∼ 0.01 can be
excluded. At the resonance mS ' mh/2, the HL-LHC can probe λHP ' 0.09, whereas√the
HE-LHC may reach λHP ' 0.07. Above the resonance, perturbative couplings λHP < 4π
are accessible up to mS < 100GeV at the HL-LHC and mS < 120GeV at the HE-LHC ,
respectively.
As the uncertainty of our cut-and-count analysis is dominated by systematic uncer-
tainties, only little improvement on the limits can be seen comparing the HL-LHC and
HE-LHC results. Our projections can however be viewed as conservative. Stronger lim-
its may be achieved if more a sophisticated analysis e.g. using shape or multivariate
techniques is applied.
3.5. Conclusion
In this chapter we have presented a study of Higgs-portal DM at proton colliders, focusing
on the VBF channel. We have derived the 95% CL limits on the portal coupling λHP
in the scalar singlet Higgs-portal model from current LHC data and provide forecasts
for the sensitivity at the HL- and HE-LHC . Our observed and projected constraints are
based on a 13TeV search for invisible Higgs decays in VBF [120] and the corresponding
HL-LHC simulation [121] by CMS. The limits incorporate an estimate of the systematic
uncertainties achievable via data-driven background-determination methods. Particular
36
λHP
3.5. Conclusion
care was taken to consistently derive the bounds for DM close to the Higgs resonance,
requiring the use of the running Higgs-width in the propagator to avoid an unphysical
enhancement of the DM production cross-section. While this enhancement would lead to
an overestimation of the constraining power by 15% – 30% when the 13TeV LHC limits
are considered, the effect is negligible for the size of couplings constrained by the HL- or
HE-LHC .
The 95% CL upper limits on the invisible branching ratio of the Higgs boson pro-
vided by CMS are Brinv < 33% from current data [120] and Brinv < 3.8% for the
HL-LHC [121]. For the HE-LHC we obtain the corresponding limit of Brinv < 2.1%.
Note that the projected limits are based on cut-and-count analyses and may be improved
if more sophisticated methods are employed. Using 35.9 fb−1√ of pp collisions recorded
at s = 13TeV we can establish an upper limit on the portal coupling in the singlet
scalar model around λHP ' 0.04 below the resonance (at mS = 61GeV), while the limit
rapidly degrades when masses above the resonance are considered, excluding for instance
λHP & 2.5 at mS = 64GeV. At mS = mh/2, we obtain an upper bound of λHP ' 0.3.
The HL-LHC with 3 ab−1 improves these limits by a factor of ∼ 0.3, and we can gain
another 25% √in sensitivity at the HE-LHC with 15 ab
−1. Currently, perturbative cou-
plings λHP ≤ 4π can be probed for DM masses below mS . 67GeV, whereas the HL-
and HE-LHC may reach mS . 100GeV and mS . 120GeV, respectively.
We also present our limits as upper bounds on the signal strengh for the invisible decay
of an additional Higgs boson with mass mH that is produced SM-like and does not mix
with the 125GeV Higgs. The corresponding results allow for a simple reinterpretation
of the search for other models with invisible particles coupled to the Higgs boson only.
This is illustrated in appendix 3.A for the case of Higgs portal models with different
spin-choices for the DM field. Our numerical results are available in digital form as
supplementary material to the publication [4].
37
Appendix of Chapter 3
Appendix 3.A. Reinterpretation for Other Higgs-Portal
Models
Based on eq. (3.13) and the limits on the signal strength µH shown in fig. 3.7, our results
can be easily reinterpreted for any other DM model in which the DM is produced from an
s-channel Higgs boson exclusively, simply by replacing Γinv in eq. (3.13) by the respective
expression for the invisible decay width of an off-shell Higgs boson. To conclude this
chapter, we therefore here present limits on the portal coupling for other simple Higgs-
portal models with different types of DM fields. In particular, we consider DM described
by a Majorana fermion χ, a vector field Xµ, and an anti-symmetric rank-two tensor field6
Bµν , all singlets under the SM gauge groups. Further details on the models can be found
in refs. [134, 163, 165]. The respective portal interactions are given by
LχHP = −
λHP
Λ H
†H χ̄χ , (3.16a)
LX = −λHP †HP 2 H HX
µXµ , (3.16b)
LB = −λHP H†H BµνHP 2 Bµν . (3.16c)
We here neglect other possible interactions with SM fields such as a pseudo-scalar coupling
i χ̄γ5χH†H, or couplings to the hypercharge field strength tensor Fµν . The corresponding
decay rates of the Higgs boson into a DM pair are [134, 165]
(
2 2 2
) 3
2
Γ (h→ χχ̄) = λHPv
m
4 Λ2 mh 1− 4
χ
2 , √ (3.17a)π mh
Γ ( → ) = λ
2 v2HP m
4
h − 4m2hm2 + 12m4 m2h XX X X X128 4 √ 1− 4 2 , (3.17b)πmh mX mh
2 2
Γ ( → BB) = λHPv m
4
h − 4m2hm2B + 6m4 m2h B B16 4 1− 4 2 . (3.17c)πmh mB mh
In contrast to the scalar DM model discussed before, the models in eq. (3.16) are
not UV complete. They are non-renormalizable, as apparent in the fermion case from
the dimension of the portal-operator in eq. (3.16a), and violate perturbative unitarity
6We here consider the transverse mode of the anti-symmetric rank-two tensor, which can be used to
describe a massive spin-one resonance. Compared to the other Higgs-portal models studied here, this
model has the additional feature that no Z2 symmetry is required to stabilize the DM candidate. See
refs. [163, 164] and references therein for further details.
38
Appendix 3.A. Reinterpretation for Other Higgs-Portal Models
at high energies. To restore renormalizability and unitarity, additional fields need to
be introduced [166–170]. As a consequence, the invisible decay rates grow rapidly with
q2, and the integral in eq. (3.4) or eq. (3.13) receives large contributions from off-shell
Higgs momenta q2  (2mDM)2, in particular in the vector and tensor case. In UV
completions of the models, these unitarity violating contributions7 are expected to be
suppressed, e.g. via destructive interference with additional degrees of freedom (DOFs)
that unitarize the theory. We therefore impose an upper cut-off on the q2 integral in
eqs. (3.4) and (3.13) at the perturbative unitarity bound from Higgs-to-DM scattering,
hh → XX /BB. For vector DM we use q2 < 32πm2X/λHP [172], and for the tensor field
we obtain q2 < 16πm2X/λHP following the conventions of ref. [163]. In the fermion case,
no strong dependence of the portal-coupling limits on the upper bound of the integral is
observed.
The 95% CL upper limits on the portal couplings in the effective Higgs-portal models
are shown in fig. 3.11. The vector and tensor model are displayed in the left and right
lower panel, whereas the fermion case is shown in the upper right corner. For comparison,
we also show the singlet scalar model on the upper left. The black lines depict the current
limits from the CMS shape analysis [120], whereas the blue and red curves denote the
projected sensitivity at the HL-LHC and HE-LHC with 3 ab−1 and 15 ab−1 of integrated
luminosity, respectively. For the current bound, we also show the uncertainty on the
signal projections as a gray band. To illustrate the dependence of the limits on the
upper cut-off on the off-shell Higgs momentum, we also indicate the corresponding limits
applying a cut-off of q2 < 1TeV and without cut-off by the dashed and dotted light-red
curves for HE-LHC case.
Note that the lines indicating the 13TeV limits in the vector and tensor model end
at DM masses around ∼ 70GeV. This is caused by the running width dominating the
propagator for large portal couplings λHP. The cross section in eq. (3.4) then reaches a
maximum, and a further increase of λHP leads to a suppression of the cross section. To
obtain reliable limits in this parameter region, a more detailed study would be required,
taking into account the high-energy behavior of the models.
7 Note that the running width in the Higgs propagator also has a unitarizing effect as it suppresses
large-q2 contributions when the total Higgs-width dominates the propagator. This has for instance
been exploited in the context of the Higgsplosion mechanism in ref. [171].
39
3. Probing Higgs-Portal Dark Matter with Vector-Boson Fusion
(mS −mh/2) [GeV] (mS −mh/2) [GeV]
−1 0 1 −1 0 1
10−2
1
0.1 10−3
13 TeV, 35.9 fb−1 13 TeV, 35.9 fb−1
0.01 14 TeV, 3 ab−1 14 TeV, 3 ab−1
−4
27 TeV, 15 ab−1 10 27 TeV, 15 ab−1
61 62 63 64 80 100 150 200 61 62 63 64 80 100 150 200
mS [GeV] mS [GeV]
(a) scalar DM (b) Majorana fermion DM
(mS −mh/2) [GeV] (mS −mh/2) [GeV]
−1 0 1 −1 0 1
1
1
0.1
0.1
13 TeV, 35.9 fb−1 13 TeV, 35.9 fb−1
14 TeV, 3 ab−1 −10.01 14 TeV, 3 ab
27 TeV, 15 ab−1 27 TeV, 15 ab−1
0.01
q2 ≤ (1 TeV)2 q2 ≤ (1 TeV)2
no q2-cut no q2-cut
0.001
61 62 63 64 80 100 150 200 61 62 63 64 80 100 150 200
mS [GeV] mS [GeV]
(c) vector DM (d) tensor DM
Figure 3.11: Current 95% CL upper limits (black, with gray band indicating the signal
uncertainty) as well as HL-LHC (blue) and HE-LHC (red) projected sensitivity on the
portal coupling λHP in Higgs-portal models with different types of DM fields. For the
vector and tensor case in the lower panel, the dashed and dotted light-red curves depict
the HE-LHC limits if the upper bound of the integral in eq. (3.13) is set to 1TeV and
∞, respectively, whereas the solid lines cut the integral at the unitarity bound.
40
λHP λHP
λHP λHP/Λ [GeV
−1]
4. Discovering the h → Zγ Decay in tt̄Associated Production
The following chapter is based on the work [3] in collaboration with Florian Goertz, Pe-
dro Schwaller and Valentin Tenorth. The chapter resembles the publication in structure and
logic.
In continuation of our exploration of collider probes of physics beyond the Standard
Model (BSM), we now indirectly constrain new physics via its contribution to the decay
of the Higgs boson into a Z boson and a photon. Nowadays, almost a decade after the
first observation of the Higgs boson in 2012 [16, 17], the data collected in proton-proton
collisions at the CERN LHC has provided measurements of various Higgs properties, in
particular many of its production and decay modes. The h → Zγ decay considered in
this chapter has however not been measured so far. It furnishes a promising channel for
the determination of spin [173, 174] and CP [175, 176] properties of the Higgs, and can
potentially probe new physics that could be hidden in other observables [176–184].
Similar to the decay into a pair of photons, which, despite its low branching ratio
of Br(h→ γγ) = 2.27× 10−3 [156], was one of the primary channels in the Higgs dis-
covery, the h → Zγ decay is a loop-induced process, mainly mediated by top and W
loops. Although the corresponding branching fraction, Br(h→ Zγ) = 1.54× 10−3 [156],
amounts to roughly two thirds of the diphoton one, this decay is significantly harder to
observe as the Z boson decays predominantly to hadrons and therefore suffers from large
backgrounds at the LHC . An accurate reconstruction of the Z boson with a relatively
low background is possible if leptonic decays are considered, however at the price of the
additional factor Br(Z → `+`−) = 6.67% [68], where here and henceforth ` denotes an
electron or muon. As a consequence, even the most recent search for the h → `+`−γ
decay by ATLAS, using 139 fb−1 of data collected at a center-of-mass energy of 13TeV,
only provides an upper bound on the√cross section for h → Zγ of 3.6 times the Stan-
dard Model (SM) prediction [185]. At s = 14TeV and with an integrated luminosity of
100 fb−1, a significance of 2σ can be reached [173], while even at the HL-LHC with 3 ab−1
a 5σ observa√tion will be challenging [186]. A future electron-positron collider, such as the
FCCee with s = 240GeV and 10 ab−1 luminosity, may reach a significance of 3.6σ [187].
In this chapter we entertain the pp→ t t̄ h production mode to search for the h→ Zγ
decay, taking advantage of the presence of the top-pair in the final state to enhance the
signal-to-background ratio. The observation of top-pair associated Higgs-production has
been established recently [188–190], motivating its further use in collider studies, such
as the search for invisible Higgs decays [191] or the h → Zγ decay considered here.
Due to the large top-Yukawa-coupling, only a modest penalty is payed when radiating a
Higgs boson from a top-quark pair. Top-associated Higgs production therefore provides
a significantly larger signal-to-background ratio compared to the dominant production
41
4. Discovering the h→ Zγ Decay in tt̄ Associated Production
from gluon-fusion for instance, rendering it a promising channel in searches for rare Higgs
decays.
On the other hand, as the top quark is the heaviest particle in the SM, a large partonic
center-of-mass energy is required to produce the t t̄ h state. The corresponding inclusive
cross-section for top-associated Higgs production in proton collisions is therefore roughly
two orders of magnitude lower then the pp→ h channel from gluon-fusion. Hence, a high
integrated luminosity is required to observe the h→ Zγ decay in this channel, impeding
its potential in searches based on currently available data.
In the following, we thus investigate the potential to measure h → Zγ with a lepton-
ically decaying Z in top-pair associated production at the HL-LHC with an integrated
luminosity of 3 ab−1, as well as the 27TeV HE-LHC and a 100TeV hadron collider such
as the FCChh with luminosities of 15 ab−1 and 30 ab−1, respectively. We first provide a
rough estimate of the expected number of signal and background events at the HL-LHC
in section 4.1. Subsequently, we setup a simple cut-and-count analysis and calculate the
expected significance based on Monte Carlo (MC) event simulations in section 4.2, also
considering projections for the HE-LHC and FCChh. Finally, in section 4.3, we examine
constraints on new physics via the corresponding limits on the hZγ coupling. We provide
a summary of this chapter in section 4.4.
4.1. HL-LHC Sensitivity Estimate
√
The total cross-section for top-pair associated Higgs production at s = 14TeV including
quantum chromodynamics (QCD) and electroweak (EW) next-to-leading order (NLO)
corrections is [156]
σ(pp→ tt̄h) = 613 fb +6.0%−9.2% (scale) ± 3.5% (PDF + αs) , (4.1)
where the first uncertainty reflects the change of the cross section when varying the
factorization and renormalization scales between half and twice the central value, and
the second one combines the uncertainty on the parton distribution functions (PDFs)
and strong coupling strength. Assuming an integrated luminosity of 3 ab−1, and taking
the branching ratios [68, 156]
Br(h→ Zγ) = 1.54× 10−3 and Br(Z → `+`−) = 6.67% (4.2)
for the Higgs and Z decays, we obtain a total of S0 ≈ 190 signal events.
While the decay products of the Higgs boson, i.e. the photon and the electron or muon
pair from the Z decay, can be reconstructed efficiently, requirements imposed to tag top-
pair-associated production will degrade the signal count. To retain an observable number
of events after selection cuts have been applied, the analysis needs to be designed as
inclusive as possible. For our first estimate, let us here assume a selection efficiency of
10% – 15%, in the same ballpark as the corresponding efficiency in the diphoton case
(∼ 12% [192]). We therefore expect roughly S = (20 – 30) signal events in the analysis.
42
4.2. Analysis
The dominant irreducible background is t t̄ Z produ√ction with the radiation of an ad-
ditional photon. The corresponding cross section at s = 14TeV including NLO QCD
(but not EW) corrections is
σ(pp→ tt̄Zγ) = 9.3 fb +10.4%−11.5% (scale) ± 3.4% (PDF + αs) , (4.3)
where we used MadGraph5 [153] for the calculation and applied pγT > 10GeV and |ηγ | < 4
at generator-level. This amounts to B0 ≈ 1860 background events (with the Z boson
decaying leptonically) at the HL-LHC , i.e. roughly an order-of-magnitude more than the
number signal events.
In addition, further background events originate from reducible backgrounds, where
we expect the most important contribution to be pp → tt̄Zj with the jet misidentified
as a photon. This misidentification may for instance occur if most of the jet energy is
carried by a π0 or η meson decaying to a collimated pair of photons. In an experimental
analysis, this background will be determined using data-driven methods, e.g. applying
a two-dimensional side-band method based on the photon isolation criterion [193], or
floating the background normalization and fitting the invariant mass distribution of the
reconstructed Z γ pair below and above MZγ ∼ mh. Since a reliable simulation of this
background is challenging, we here simply enhance the number of background events by
50% to account for non-simulated backgrounds. Assuming the same selection efficiency
as for the signal, we obtain (280 – 420) background events. We here neglect further
background contributions that arise depending on the respective top-pair decay channel
considered, as we expect these to be sub-leading.
Due to the extremely narrow width of the Higgs boson, Γh = 4.1MeV [156], the in-
variant mass distribution of the Z γ pair in the signal events is sharply peaked at the
Higgs mass, whereas the background has a smooth distribution. To obtain a large signal-
to-background ratio we therefore restrict the invariant mass to lie within a 10GeV win-
dow around the Higgs mass, 120GeV < MZγ < 130GeV. While the signal is basically
unaffected by this cut, the background is reduced to roughly 7% of its original size,
based on the corresponding distribution in the MadGraph events. W√e therefore expect
B = (20 – 30) ≈ S background events, resulting in a significance of S/ B ' 4.5σ – 5.5σ.
Further improvement of the sensitivity compared to this simple cut-and-count analysis
may be achieved fitting the invariant mass distribution in the signal-plus-background and
background-only hypothesis. The t t̄ h channel therefore provides promising prospects for
an observation of the h→ Zγ decay.
4.2. Analysis
In this section we corroborate our estimated sensitivity, setting up a toy analysis for the
HL-LHC using MC simulations. The analysis in section 4.2.1 focuses on the semi-leptonic
top-pair decay channel, with t → bjj and t̄ → b̄`−ν̄` or vice versa. In section 4.2.2, our
results are then extrapolated to also include the all-leptonic and all-hadronic channel,
assuming the same efficiency as obtained for our analysis in the semi-leptonic case. We
extend the projections to the HE-LHC and FCChh in section 4.2.3.
43
4. Discovering the h→ Zγ Decay in tt̄ Associated Production
4.2.1. Semi-Leptonic Channel
√
We generate events at the HL-LHC with s = 14TeV and an integrated luminosity of
3 ab−1 for the signal process pp→ tt̄h and the irreducible background pp→ tt̄Zγ (without
the corresponding Higgs contribution) at NLO in QCD using MadGraph5_aMC@NLO [153,
194]. The renormalization and factorization scale are set to µR = µF = mt +mh/2 [156].
We use the PDF4LHC15 NLO PDF set [195] accessed via the LHAPDF6 interface [155].
Our inclusive cross-section reproduces the corresponding NLO QCD prediction of 603 fb
in ref. [156]. The subsequent h → Zγ and Z → `+`− decays, as well as parton-shower
simulation and hadronization are performed in Pythia8 [157], assuming the branching
ratios in eq. (4.2). The events are then passed to Delphes v3.4.2 [158] for detector sim-
ulation with the HL-LHC detector card. Jets are reconstructed in FastJet 3 [159] using
the anti-kt algorithm [160] with R = 0.4, imposing pT > 25GeV and |η| < 2.5.
Our analysis for the semi-leptonic channel is loosely based on ref. [192], which is a 8TeV
cut-and-count search for h→ γγ in top-pair associated production. Since we expect two
leptons (electrons or muons) from the Z boson decay and one lepton from the leptonically
decaying (anti-)top quark, we require exactly three leptons satisfying the reconstruction
criteria pT > 15GeV (10GeV) and |η| < 2.47 (2.7) for electrons (muons). We further
require at least three jets with pT > 30GeV and |η| < 2.5 and impose a minimum missing
transverse energy (MET) cut of E/T > 20GeV. Furthermore, we demand the presence of
at least one b-tagged jet, as well as at least one photon with pT > 15GeV and |η| < 2.37.
For the reconstruction of the Z boson, a pair of opposite-sign same-flavor (OSSF) leptons
with an invariant mass 76GeV < M`` < 106GeV is required. This lepton pair is then
used along with the hardest photon to reconstruct the Higgs boson. If more than one such
lepton pair is found, the pair with the invariant mass closest to the Z mass is used. To
suppress the irreducible background we finally restrict the invariant mass of the γ `+`−
system to a 10GeV window around the Higgs mass, 120GeV < Mγ`` < 130GeV.
Table 4.1 lists the cut-flow for the signal and background events. The spectrum of
the invariant mass Mγ`` of the reconstructed Higgs boson after applying the selection
cuts (except for the cut on Mγ``) are shown in fig. 4.1. The signal is broken down into
the respective contributions with all-leptonic (light orange), semi-leptonic (orange) and
all-hadronic (red) decays of the tt̄ pair, and stacked on top of the irreducible background
(blue). It can be seen that the analysis indeed picks out the semi-leptonic channel with
negligible contamination from the other top-pair decays. The signal is sharply peaked
around Mγ`` ≈ mh, so that we obtain a signal-to-background ratio of S/B ' 1 after
cutting on Mγ``. Note that the signal-to-background ratio remains constant during the
cut flow except for the last cut on the mass of the reconstructed Higgs boson, as these
cuts are meant to suppress the reducible backgrounds which we did not simulate.1 The
corresponding selection efficiencies for signal and background in the semi-leptonic channel
are N = Nfinal/(Br(semi-lept.)×Ninitial), where N = S,B are the number of signal and
1 We however confirmed in a leading order (LO) simulation that the reducible backgrounds tt̄W±γ,
W±bb̄jZγ, and tt̄tt̄γ are negligible, as well as that the background tZγjj (∼ 30% of the irreducible
background) is within the 50% enhancement used in our naive estimate and the extrapolation to all
top-channels.
44
4.2. Analysis
14TeV, 3 ab−1 27TeV, 15 ab−1 100TeV, 30 ab−1
Cut S B S B S B
Initial 186 1862 4.4k 47k 112k 1.3M
N(`) = 3 25 273 539 6.2k 16k 210k
N(j) ≥ 3, p jT > 30GeV 15 170 344 4.1k 12k 160k
E/T > 20GeV 14 160 322 3.9k 11k 150k
N(b) ≥ 1 12 137 276 3.3k 10k 140k
N(γ) ≥ 1, pγT > 15GeV 8.1 83 180 2.0k 6.7k 84k
Z-reconstruction 7.6 80 166 1.9k 6.3k 82k
Higgs-reconstruction 7.3 5.2 160 101 6.1k 3.2k
Table 4.1: Number of signal and background events after each selection cut for the
HL-LHC (14TeV, 3 ab−1), HE-LHC (27TeV, 15 ab−1) and FCC (100TeV, 30 ab−1hh ).
background events taken from table 4.1, and Br(semi-lept.) = 28.8% [68] is the branching
ratio into semi-leptonic decays of the top-pair. We obtain S = 14% and B = 0.97%.
4.2.2. All-Hadronic and All-Leptonic Channel
Let us now assume that the selection efficiencies are the same for all top channels. Taking
the initial number of signal and background events from table 4.1 we then arrive at the
total numbers of
S = 186× S ≈ 25 and B = 1.5× 1862× B ≈ 27 . (4.4)
7
14 TeV, 3 ab−16 Figure 4.1: Invariant mass
leptonic
5 distribution of the photon
semi-leptonic and OSSF lepton-pair system
4 hadronic (before Higgs-reconstruction
3 background cut) for the background (blue)
2 and signal events with fully-
hadronic (red, not visible),
1 semi-leptonic (orange) and
0 fully-leptonic (light orange)
100 150 200 250 decays of the top-quark pair at
Mγ`` [GeV] the HL-LHC .
45
# events
4. Discovering the h→ Zγ Decay in tt̄ Associated Production
In other words, we enhance the final numbers in table 4.1 by a factor 1/Br(semi-lept.) to
mimic an analysis that also considers the all-leptonic and all-hadronic decay channels. In
addition, we here emend the background by 50% to account for reducible backgrounds,
such as pp→ tt̄Zj. The result agrees well with ou√r estimate from section 4.1.
Considering the statistical uncert√ainty ∆B = B ≈ 5 only, our cut-and-count anal-
ysis establishes a significance of S/ B ≈ 4.8 for the observation of the h → Zγ decay
in t t̄ associated production alone. Top-pair associated Higgs production may therefore
contribute significantly to establish an experimental observation of this Higgs decay. The
sensitivity may be further improved employing top-reconstruction algorithms based on
boosted decision trees, as used in the observation of top-associated Higgs production
in other Higgs decay channels [188–190], with which hadronic top-decays can be recon-
structed at high efficiency. Our results may therefore be seen as a conservative estimate,
even in the case that the non-simulated background processes are underestimated.
4.2.3. Predictions for 27 and 100 TeV Colliders
Next we derive the corresponding prospective signficance at the 27TeV HE-LHC [196]
and the 100TeV FCChh [197] with the respective integrated luminosities of 15 ab−1 and
30 ab−1. Events are simulated with the same MC setup as in section 4.2.1, using the
HL-LHC (FCChh) Delphes detector card for the 27TeV (100TeV) collider. The signal
cross-sections for pp → tt̄h production are 2.9 pb at the HE-LHC [198] and 33 pb at the
FCChh [199], which we reproduce in our si√mulations. For the irreducib√le t t̄ Z γ back-
ground we obtain cross sections of 46 fb (at s = 27TeV) and 670 fb (at s = 100TeV),
respectively, again requiring pT > 10GeV and |η| < 4 for the photons in MadGraph.
For simplicity and comparability with the HL-LHC case, we apply the same selection
cuts as in section 4.2.1 for the analysis of the semi-leptonic top-pair channel. The resulting
event numbers are listed in the respective columns of table 4.1, and the Mγ`` invariant
mass spectra are depicted in fig. 4.2. In the semi-leptonic channel, we obtain signal and
background efficiencies of S = 13% (19%) and B = 0.75% (0.85%) at the HE-LHC
(FCChh). Enhancing the background by 50% and extrapolating to the other top decays
we therefore expect S ≈ 555 and B ≈ 526 at the HE-LHC over all channels. For the
FCChh we obtain S ≈ 21 000 and B ≈ 17 000 in total.
4.3. Constraints on New Physics
Measurements of (or limits on) the h → Zγ decay can be used to indirectly constrain
new physics via its effects on the decay rate. Since the hZ γ coupling is already loop-
suppressed in the SM, it provides particularly promising prospects for the observation of
BSM effects, e.g. from additional heavy states running in the loop.
Within the effective field theory (EFT) framework, the impact of new physics on SM
observables can be parameterized in terms of higher-dimensional operators. Let us here
46
4.3. Constraints on New Physics
150 7500
27 TeV, 15 ab−1 100 TeV, 30 ab−1
125 6250
leptonic leptonic
100 semi-leptonic 5000 semi-leptonic
hadronic hadronic
75 3750
background background
50 2500
25 1250
0 0
100 150 200 250 100 150 200 250
Mγ`` [GeV] Mγ`` [GeV]
(a) HE-LHC (b) FCChh
Figure 4.2: Same as fig. 4.1 for the HE-LHC (left) with a center-of-mass energy of 27TeV
and an integrated luminosity of 15 ab−1, as well as the FCChh (right) with 100TeV and
30 ab−1 of luminosity.
neglect possible new physics in the production of the Higgs boson and consider the leading
dimension-six operators relevant for the h→ Zγ decay rate,
O i g2 µ † ν aHW = 2 (D H) σa (D H) Wm µν ,W
O i g1 µHB = 2 (D H)
† (DνH) Bµν , (4.5)
mW
g2O = 1 H†H B Bµνγ
m2
µν ,
W
where H is the SM Higgs doublet, W aµν and Bµν are the weak and hypercharge field-
strength tensors, and g2 and g1 are the respective gauge couplings. We here neglect CP
odd operators. After electroweak symmetry breaking (EWSB) w√e can expand the Higgs
field around its vacuum expectation value (VEV), 〈H〉 = (0, v/ 2)T . The operators in
eq. (4.5) then contribute to the tree-level term
L ⊃ hc Z FµνZγ µν , (4.6)
v
where h is the physical Higgs mode, Fµν is the electromagnetic (EM) field strength tensor,
and Zµν the equivalent for the Z boson. In terms of the Wilson coefficients cHW , cHB
and cγ of the dimension-six operators,2[ the cZγ coupling is given b]y
c 2Zγ = − tan θW (cHW − cHB) + 8 sin θW cγ (4.7)
with θW denoting the weak mixing angle.
2We amend the SM Lagrangian by the terms ∆L = cHWOHW + cHBOHB + cγOγ .
47
# events
# events
4. Discovering the h→ Zγ Decay in tt̄ Associated Production
0.4 no systematic 14 TeV 0.4 5 % systematic 14 TeV
uncertainties 27 TeV uncertainties 27 TeV
0.3 100 TeV 0.3 100 TeV
1σ 1σ
0.2 0.2
0.1 0.1
2σ 2σ
0.0 0.0
0.6 0.8 1.0 1.2 1.4 0.6 0.8 1.0 1.2 1.4
κZγ κZγ
(a) without systematic uncertainty (b) 5% systematic uncertainty
Figure 4.3: Expected p-value for κZγ at the HL-LHC (green), HE-LHC (blue) and
FCChh (orange), assuming that the SM value is observed. In the left panel systematic
uncertainties are neglected, whereas the right panel includes a 5% uncertainty.
To constrain the new physics effects in the h → Zγ decay, we employ the so-called
coupling strength modifier or κ framework [200]. In this framework, the absolute value
of each coupling of the Higgs boson is modified by a factor κi, while the corresponding
tensor structure is assumed to be SM-like. The SM is therefore reproduced if all κi = 1.
Experimentally the coupling modifiers can be extracted from the ratios of production
cross-sections or decay rates to the respective SM prediction. In particular, the squared
modifier of the hZ γ coupling can be obtained from the h→ Zγ decay rate and expressed
in terms of the new-physics coupling cZγ in eq. (4.7) as [179]
κ2
Γ(h→ Zγ) 4π
Zγ = Γ( ) ' 1− 0.146 cos cZγ , (4.8)h→ Zγ SM α θW
where α is the EM fine-structure constant and the second equality only holds for small
cZγ . Again, we neglect modifications of the production cross-section.
We now obtain our limits on the coupling modifier κZγ from the h → Zγ decay in
tt̄ associated Higgs production as follows. The predicted total number of events for a
given value of κ 2Zγ is N(κZγ) = κZγ S + B, where S and B are the SM signal and
background event counts over all top decays, given by eq. (4.4) in the HL-LHC case or
by the respective numbers in section 4.2.3 for the HE-LHC and FCChh. We then assume
that the measured number of events corresponds to the SM prediction, N(κZγ =1), and
exclude values of κZγ at a significance of nσ if N(κZγ) deviates from the SM value by
more than n times the uncertainty.
Figure 4.3 shows the corresponding probability (times uncertainty) as a function of κZγ
in the Gaussian approximation, [ ]
1 2
p(κZγ) = √ exp −
∆N (κZγ)
2π 2
, (4.9)
σ2N
48
p(κZγ)
p(κZγ)
4.3. Constraints on New Physics
HL-LHC 1σ : 0.85 < κZγ < 1.15
2σ : 0.71 < κZγ < 1.30
14 TeV 1σ : 0.86 < κZγ < 1.14
3 ab−1 2σ : 0.71 < κZγ < 1.29
HE-LHC 1σ : 0.96 < κZγ < 1.04
2σ : 0.93 < κZγ < 1.08
27 TeV 1σ : 0.97 < κZγ < 1.03
15 ab−1 2σ : 0.94 < κZγ < 1.06
FCChh 1σ : 0.98 < κZγ < 1.03
2σ : 0.95 < κZγ < 1.05
100 TeV 1σ : 0.995 < κZγ < 1.005
30 ab−1 2σ : 0.991 < κZγ < 1.009
0.7 0.8 0.9 1.0 1.1 1.2 1.3
κZγ
Figure 4.4: Prospective 1σ and 2σ constraints on the coupling modifier κZγ from the pro-
cess pp→ tt̄h, h→ Zγ at the HL-LHC (top), HE-LHC (center) and FCChh (bottom),
assuming statistical uncertainties only (red) as well as a 5% systematic uncertainty on
the signal (blue).
where ∆N (κZγ) ≡ N(κZγ)−N(1) = (κ2Zγ − 1)S, and σN is the uncertainty. The colored
lines correspond to the HL-LHC (green), HE-LHC (blue) and FCChh (orange). The inter-
sections with the solid (dashed) black line indicate the values excluded with a significance
of 1σ (2σ).
If only statistical uncertainties are considered (fig. 4.3a), i.e. taking σ2N = N(κZγ), the
1σ (2σ) limits on κZγ are obtained as
HL-LHC : 0.86 ≤ κZγ ≤ 1.14 ( 0.71 ≤ κZγ ≤ 1.29 ) ,
HE-LHC : 0.97 ≤ κZγ ≤ 1.03 ( 0.94 ≤ κZγ ≤ 1.06 ) , (4.10)
FCChh: 0.995 ≤ κZγ ≤ 1.005 ( 0.991 ≤ κZγ ≤ 1.009 ) ,
which are displayed as red bars in fig. 4.4. With the low-background process considered
here, the signal can be established at the future HE-LHC and FCChh hadron colliders
with a significance far beyond 5σ, allowing for the measurement of the effective hZ γ
coupling at the percent-level and thus for a determination of spin and CP properties of
the Higgs boson.
At this level of precision, the assumption of considering statistical uncertainties only
is questionable and systematic errors need to be incorporated. We therefore take into
account an estimate of the theoretical uncertainties on the prediction of the signal cross-
section σ(pp→ tt̄h). Currently, this uncertainty is around 10% (cf. eq. (4.1)). Antic-
ipating some progress in the theoretical predictions, we assume a relative systematic
uncertainty of 5%, added to the statistical error in quadrature. The corresponding limits
49
4. Discovering the h→ Zγ Decay in tt̄ Associated Production
are depicted as blue bars in fig. 4.4, and the p-values are plotted in fig. 4.3b. For the 1σ
(2σ) constraints we then obtain
HL-LHC : 0.85 ≤ κZγ ≤ 1.15 ( 0.71 ≤ κZγ ≤ 1.30 ) ,
HE-LHC : 0.96 ≤ κZγ ≤ 1.04 ( 0.93 ≤ κZγ ≤ 1.08 ) , (4.11)
FCChh: 0.98 ≤ κZγ ≤ 1.03 ( 0.95 ≤ κZγ ≤ 1.05 ) .
Our projections are competitive to limits from other production modes [198], which are
around 10% at the HL-LHC and 3% – 4% at the HE-LHC (at 1σ), as well as to prospec-
tive constraints from the ILC of ∼ 5% [201].
The systematic uncertainties may be further reduced considering ratios of couplings,
such as κZγ/κγγ , where uncertainties on the production mode cancel to a large extend.
Since the h → Zγ rate in the SM is dominated by the W loop, so that heavy charged
fermions for instance have a larger effect on the h → γγ rate, this ratio is still highly
sensitive to BSM physics.
4.4. Conclusion
We have investigated the prospects for discovering the Higgs decay to a Z boson and a
photon in top-pair associated production at future proton colliders. Our projections are
based on a MC analysis of the semi-leptonic top channel. Assuming the same selection
efficiency for the fully-leptonic and fully-hadronic channel, we have demonstrated that tt̄
associated production can contribute significantly to establishing an observation of the
h → Zγ decay at the HL-LHC . Improved analysis techniques may even permit a ∼ 5σ
discovery in this production channel alone. The higher event rates at a potential HE-LHC
or FCChh should definitely lead to a 5σ observation in the considered channel.
We further evaluated the corresponding bounds on the modifier κZγ of the effective
hZ γ coupling, constraining κZγ at the level of 15%, 4% and 2% at the HL-LHC , HE-
LHC , and FCChh, respectively, if a systematic uncertainty of 5% is assumed. Our limits
are competitive to those obtained from other production channels, including electron-
positron colliders [187, 198, 201, 202]. It can be expected that a more sophisticated
analysis, for instance using advanced top-tagging techniques based on machine learning,
will strengthen these bounds.
50
5. Leptophilic Dark Matter fromGauged Lepton Number
This chapter is based on the publication [1]. Since the text in the paper (except for the
introduction) has been composed by the author, the sections 5.1 to 5.4 are copied word-for-
word. Minor modifications have been made to adjust to the structure, conventions and style
of this thesis.
While the previous chapters have described two rather generic searches for physics
beyond the Standard Model (BSM), let us now conduct a dedicated study of a specific
model of new physics with particular focus on dark matter (DM).
As discussed in section 2.3.1, DM models with a relic abundance set via thermal freeze-
out are in tension. Searches for DM at the LHC as well as direct detection experiments
based on DM scattering on nuclei strongly constrain its interactions with quarks, while
non-negligible interactions with at least a subset of the Standard Model (SM) particles
are required to reproduce the observed abundance. In this chapter we therefore consider
a model of leptophilic dark matter [203–205], i.e. DM that couples predominantly to
leptons. Further motivation for new physics primarily interacting with leptons is provided
by models of neutrino mass generation, the persistence of the muon g − 2 anomaly, as
well as by current hints for lepton-flavor-universality violation in B-physics observables.1
We here extend the SM by promoting lepton number to a U(1)` gauge group. Since
lepton number is anomalous if only the SM particle content is assumed, this requires the
introduction of additional fields to cancel the anomalies.2 In the model considered in
this chapter, anomaly cancellation is achieved adding two generations of fermions that
are vector-like under the SM gauge groups but have chiral interaction with respect to
lepton number [5]. A residual global symmetry surviving spontaneous symmetry breaking
(SSB) then ensures the stability of the lightest additional lepton, automatically providing
a candidate for leptophilic DM.
Since no massless gauge boson other than the photon is observed experimentally, a
mass term for the lepton number gauge boson needs to be generated. We therefore break
lepton number spontaneously. This may potentially lead to the generation of a stochastic
gravitational wave background (SGWB) in the early Universe if the corresponding phase
transition (PT) is of first-order, which might then be observable at LISA or other future
gravitational wave (GW) experiments. We will thus discuss the lepton number breaking
PT in this model as an example of how new physics can be probed via GWs in chap-
ter 7 of part II of this thesis. Particular focus will be put on the interplay between the
detectability of the SGWB and constraints from the collider and DM phenomenology of
the model.
1Recent work in this direction can for instance be found in refs. [206–218].
2Various ways of anomaly free gauging of lepton number can be found in the literature [5, 219–228].
51
5. Leptophilic Dark Matter from Gauged Lepton Number
This chapter is organized as follows. The model is introduced in section 5.1, discussing
constraints on the lepton number gauge coupling from renormalization group (RG) run-
ning. The corresponding DM phenomenology is investigated in section 5.2, and section 5.3
studies constraints from collider experiments. Intermediate conclusions are presented in
section 5.4. A discussion of the lepton number breaking PT, the resulting GW signal and
its detectability is deferred to chapter 7.
5.1. The Model
The model considered here has been introduced in [5]. In this model, the SM gauge
group is extended by an additional U(1)` lepton number gauge group under which all SM
leptons including three generations of right-handed neutrinos carry unit charge, whereas
the other SM fields are neutral. Lepton number is spontaneously broken by an SM singlet
scalar field, giving mass to the lepton number gauge boson. Additional fermionic fields
are added to cancel gauge anomalies. These additional fields are vector-like under the
SM gauge group.
5.1.1. Gauge Sector
The model is based on the gauge group3 SU(3)c ⊗ SU(2)W ⊗ U(1)Y ⊗ U(1)`. Omitting
quantum chromodynamics (QCD), the gauge sector of the Lagrangian is given by
L ⊃ −1W a aµν 1 µν 1 µν  µν4 µνW − 4B̂µνB̂ − 4 Ẑ` µνẐ` + 2B̂µνẐ` , (5.1)
where W a and B̂ are the gauge bosons of the SM weak and hypercharge gauge group,
respectively, and Ẑ is the lepton number gauge boson. The ` 2B̂
µν
µνẐ` term leads to
kinetic mixing between the hypercharge and lepton number U(1) gauge bosons. The
kinetic terms can be diagonalizedby aGL(2,R) transfor   
mation [229]
 B̂ 1 √ = 1−2B . (5.2)
Ẑ` 0 √ 11− Z2 `
The hats denote fields in the gauge basis and the unhatted fields are in the basis where
the kinetic terms are diagonal and canonically normalized.
The model further features two scalar fields: the SM Higgs doublet transforming under
SU(2)W ⊗ U(1)Y ⊗ U(1)` as H ∼ (2, 1/2, 0), and a complex scalar Φ ∼ (1, 0, LΦ) which
is an SM singlet with lepton number LΦ. W√e let both fields acq√uire a vacuum expec-
tation value (VEV) given by 〈H〉 = (0, vH/ 2) and 〈Φ〉 = vΦ/ 2, thus breaking the
electroweak (EW) and lepton number gauge group SU(2)W ⊗ U(1)Y ⊗ U(1)` to U(1)EM
electromagnetism (EM). The gauge bosons then obtain masses from the kinetic terms of
the scalar fields, with the covariant derivative given by
Dµ = ∂µ − ig2W a aµT − ig1Y B̂µ − ig`LẐ`. (5.3)
3To avoid confusion between the L denoting the weak gauge group of left-handed fields and lepton
number, we here use SU(2)W instead of the standard SU(2)L notation.
52
5.1. The Model
Here, g2, g1 and g` are the gauge couplings of the SU(2)W , U(1)Y and U(1)` gauge
groups, respectively. The W mass is the same as in the SM, m = 1W 2g2vH , whereas the
mass matrix for the remaining gauge fields in the kinetic eigenbasis (W 3, B, Z`) is given
by
  g
2 2 2
2vH −g1g2vH − g√1g2v
2
H

4 4 4 1−2 
M2 = −g1g2v2H g2 21vH g21v2 GB H 4 4 √ . (5.4)4 1−2 
g1g2v2 g2v2 g2L2 v2 2

− √ H √1 H ` Φ Φ + g1v
2
H
4 1−2 4 1−2 1−2 4(1−2)
The upper-left 2× 2 submatrix is diagonalized rotating by the SM weak mixing angle. If
 6= 0, the resulting ZSM boson is mixing with the Z` boson. The kinetic eigenstates are
related to the physical masseigenstates by   3
W  cW cξ sW −cW sξ Z
 B  
  
= −sW cξ cW s W sξ A , (5.5)
Z s 0 c Z ′` ξ ξ
√ √
where cW = cos θW = g2/ g2 21 + g2 and sW = sin θW = g1/ g2 21 + g2 are sine and cosine
of the weak mixing angle θW , whereas cξ = cos ξ and sξ = sin ξ are sine and cosine of the
Z − Z ′ mixing angle ξ. Defining M2 = (g2 + g2)v2 /4, M2 = g2L2 v2 , M2 = g2v2√ ZSM 1 2 H Z` ` Φ Φ B 1 H/4
and η = / 1− 2, the Z − Z ′ mixing angle and the neutral gauge boson masses are
2 √2MZ sin θW  1− 2 M2tan(2ξ) = SM ZSM
M2 −M2 (1− 2) + 2 sin2 2 ≈ 2 sin θW 2 , (5.6)Z` Z M θ  MSM ZSM W Z`
and ( √ )
2 = 1
( )
m M2 +M2 + η2M2 ± M2 2 2 2
2 2 2
Z(′) 2 Z(` )ZSM B Z +MZ + η M − 4M` SM B Z M` ZSM (5.7)
≈M2ZSM M
2
Z ,`
where the approximate expressions are expansions up to linear order in . Note that the
definition of the weak mixing angle θW and the EM coupling e in terms of the SM gauge
couplings g1 and g2 is not altered by the kinetic mixing.
5.1.2. Scalar Sector
The model has two scalar fields: the SM Higgs H ∼ (2, 1/2, 0) and the U(1)` breaking SM
singlet scalar Φ ∼ (1, 0, LΦ), where we choose LΦ = 3 as will be discussed in section 5.1.3.
The corresponding potential is given by
( ) ( )
V (H,Φ) = −µ2 † †
2 2 † † 2 † †
HH H + λH H H − µΦΦ Φ + λΦ Φ Φ + λpH H Φ Φ . (5.8)
53
5. Leptophilic Dark Matter from Gauged Lepton Number
Expanding the fields around their VEVs,
= 

H  ( G+ ) ( )and Φ = √1 v + φ̂+ ω̂0 , (5.9)
√1 v + ĥ+ Ĝ0 2
Φ
2 H
the would-be Nambu-Goldstone bosons G±, Ĝ0 and ω̂0 become the longitudinal degrees
of freedom of theW±,Z and Z
′ gauge bosons. The mass matrix for theremaining scalarsis −µ22 H + 3 λλ v2 pH H + 2 v2= Φ λpvHvΦMH  . (5.10)
− 2 + 3 2 + λλpvHvΦ µΦ λ v p 2Φ Φ 2 vH
The Higgs portal term λ H†p H Φ†Φ induces a mixing between the ĥ and φ̂ fields. The
mass eigenstates are definedby h = 
  cos θH − sin θHĥ (5.11)
φ sin θH cos θH φ̂
with the corresponding(masses ) √
m2 2
( )2
h,φ = λHvH + λΦv2 ± λ v2 − λΦv2Φ H H Φ + λ2pv2 2HvΦ , (5.12)
where we eliminated µ2 and µ2H Φ using the condition that the potential (5.8) has a min-
imum for ĥ = vH and φ̂ = vΦ. Here, h is the SM-like Higgs with mh = 125GeV, and φ
is the lepton number Higgs which will typically have a mass mφ > mh due to the VEV
hierarchy imposed by LEP constraints (see section 5.3.1).
5.1.3. Fermion Sector
With the SM fermion content only, lepton number is an anomalous symmetry. The
lepton-gravity U(1)` and pure lepton [U(1)`]3 anomalies are canceled by the presence of
three generations of right-handed neutrinos νR ∼ (1, 0, 1), whereas the cancellation of the
remaining anomalies requires additional fermions, to which we refer as exotic or dark4
leptons in the following. This can be realized in various ways (see e.g. refs. [219–228]).
Here, we add two sets of chiral fermions that combine to transform vector-like under the
SM gauge group, and thus do not spoil the cancellation of anomalies in the SM gauge
sector.     ′  ( ) ′′ ( )′ NL N`L = ∼ 2 1  R 1,− , L′ , `′′ =   ∼ 2,− ′′
E′ ( 2
R
E′′ 2
, L ,
L R (5.13)
ν ′ ∼ (1, 0, L′) ′′ ( ′′)R , νL ∼ 1, 0, L ,
e′
) ( )
R ∼ 1,−1, L′ , e′′L ∼ 1,−1, L′′ .
4Strictly speaking, the additional leptons are not “dark” in the typical sense since most of them carry EW
charge. However, as we will discuss in section 5.2, the lightest exotic lepton constitutes a candidate
for DM, so that, in a slight abuse of language, we also denote the other anomaly-canceling fermions
as dark.
54
5.1. The Model
L′ L′′ ∆L
-5 -2 ¯̀′′Φ∗` , ē′′Φ∗R L L e ′′ ∗R , ν̄LΦ νR
-4 -1 ¯̀′′ H̃νc , ν̄ ′′H†R R L `cL , ν̄ ′ ∗RΦ νcR
-3 0 ν̄ ′′ν ′′cL L
-2 1 ¯̀′′ ′′R`L , ēLe , ν̄ ′′R LνR
-3/2 3/2 ¯̀′′ H̃ν ′c , ν̄ ′′ † ′c ′′ ′′c ′ † ′′c ′ ∗ ′c ¯̀′ ′′cR R LH `L , ν̄LΦνL , ν̄RH `R , ν̄RΦ νR , LH̃νL
-1 2 ν̄ ′ νcR R
0 3 ν̄ ′ ν ′cR R
1 4 ¯̀′′Φ` , ē′′R L LΦe ′′R , ν̄LΦνR , ē′ †RH `L , ν̄ ′RH̃†` ¯̀′ ¯̀′L , LHeR , LH̃νR
2 5 ν̄ ′RΦνcR
Table 5.1: Lepton number charge assignments for L′ and L′′ that allow for the additional
terms ∆L in the Lagrangian.
The first set corresponds to a 4th generation of SM-like leptons but with lepton num-
ber L′, whereas the second set has opposite chirality and lepton number L′′. Impos-
ing the condition L′ − L′′ = 3, the remaining [SU(2) ]2W ⊗ U(1)`, [U(1)Y ]2 ⊗ U(1)` and
U(1)Y ⊗ [U(1) 2`] anomalies cancel.
In order to write Yukawa terms for the additional fermions involving the lepton number
breaking scalar Φ, LΦ = 3 must be chosen. The Yukawa sector is then given by
L ⊃− c ¯̀′′` RΦ`′ − c ′′ ′ ′′ ′L eēLΦeR − cν ν̄LΦνR
− y′ ¯̀′ He′ − y′′ ¯̀′′He′′ − y′ ¯̀′
(5.14)
e L R e R L ν LH̃ν
′
R − y′′ ¯̀′′ν RH̃ν ′′L + h.c. ,
Note that specific values of L′ allow for additional terms in the Lagrangian. For exam-
ple if (L′, L′′) = (1, 4) or (−2, 1), the dark fermions can mix with the SM ones, potentially
leading to flavor-changing neutral currents and threatening the stability of our DM can-
didate. Table 5.1 lists the lepton number charges that allow additional terms. We will
exclude these choices in the following. For any other pair of real numbers with L′′ = L′+3,
the Yukawa interactions of the exotic leptons are fully described by eq. (5.14).
After spontaneous symmetry breaking, mass terms for the additional fermions are
generated,
( )    N ′′ ( )R E′′L ⊃ − ν e RN̄ ′ ′′ M −L ν̄L LR  Ē′ ē′′ M + h.c. , (5.15)′ L L LR ν ′R eR
55
5. Leptophilic Dark Matter from Gauged Lepton Number
with the mass matrices given by   c∗ ′ν = √1  `vΦ yνvHMLR  e = √1  c∗ ′`vΦ yevH, MLR  . (5.16)2 y′′∗v c v 2 y′′∗ν H ν Φ e vH cevΦ
The matrices can be diagonalized via singular value decomposition, yielding the diagonal
matrices Mν = Uν†Mν Uν and M eD L LR R D = U
e† e e a
L MLRUR, where UC are unitary matrices.
For simplicity, and to avoid CP violating phases, let us assume that the Yukawa
couplings ci and ′(′)yi are real. The diagonalization matrices then become orthogonal.
The fermions combine to two charged (e4 and e5) and two neutral (ν4 and ν5) Dirac
fields, whichare given in terms of theoriginal fields byν4 
 
 cosαν sinα= νN ′L  cosβ ′′+ ν sin βνNR ,
ν5 − sinα ′′ν cosαννL − sin βν cosβν ν ′R        (5.17)e4  cosα sinα E′   cosβ sin β E′′= e e L + e e R .
e5 − sinαe cosαe e′′L − sin βe cosβe e′R
The right- and left-handed fields mix with different mixing angles unless we choose
y′ν = y′′ ′ν and ye = y′′e . Thus, the resulting fermions in general are chiral with respect to
both the SM and U(1)`.
In the absence of the Yukawa terms (5.14), the Lagrangian exhibits a global [U(1)]6
symmetry at the classical level, consisting of a U(1) symmetry for each additional lepton
field in eq. (5.13). The Φ Yukawa terms break this to three U(1) symmetries (one for the
doublets, one for the charged singlets, and one for the neutral singlets), whereas the H
Yukawa terms (in the absence of the Φ Yukawas) break the symmetry to U(1)L′⊗U(1)L′′ .
Hence, small values of ci  1 and ′(′)yi  1 are technically natural, rendering vector-
like masses civΦ  vΦ. Similarly, ySMν i  1 are technically natural. As we will see
later, vΦ & 2TeV, therefore we typically have civΦ  ′(′)yi vH . Consequently, the mix-
ing angles αe/ν √and βe/ν are usually sm√all, and the masses are approximately given by
me4 5 = c`/evΦ/ 2 and mν4 5 = c`/νvΦ/ 2./ /
To simplify the discussion of the model we will restrict to the case of symmetric mass
matrices, i.e. y ′ ′′e/ν ≡ ye/ν = ye/ν , in the following, so that αe/ν = βe/ν . Let us further
assume that ce = c`. The masse(s are then given by1 √ )
mν4 5 = √ (c` + cν)vΦ ± (c − c )2v2` ν Φ + 4y2 2/ 2 2 ν
vH , (5.18)
1
me4 5 = √ (c/ 2 `
vΦ ± yevH) . (5.19)
In particular, mν4 = (me4 +me5)/2 if yνvH  cl/νvΦ, and e4 and e5 are maximally mixed
with αe = βe = π/4.
Note that the SM neutrino masses in our model are pure Dirac masses generated from
small Yukawa couplings to the SM Higgs doublet. Majorana mass terms are forbidden
56
5.1. The Model
by the lepton number gauge symmetry, whereas mass terms arising from mixing with the
exotic leptons would spoil the DM stability and are therefore avoided by suitable choices
of the lepton number charges.
5.1.4. RG Running
Before exploring the phenomenology of the model, let us first consider the renormalization
group running of the lepton number gauge coupling g`.
The running of a gauge coupling g is governed by the beta function
= ∂ gβ log . (5.20)∂ µ
For a U(1) gauge group, the one-loopbeta function is 
g3 2 ∑ 1 ∑
β = 16 2 3 Q2f + 3 Q2s , (5.21)π
f s
where the sums run over all Weyl fermions and complex scalars charged under the gauge
group with charge Qf/s, respectively.
For the lepton number gauge group we get contributions from the SM leptons (with
unit charge), the two additional generations of vector-like leptons (with charge L′ and
L′′), and the lept[on number breaking scalar (with charge L ). Thus,
g3 8( ) ] 3 [
Φ ]
β = ` N + L′2 + 1L′′2 + L2 g` ′ 16 ′216 2 3 f 3 Φ = 16 2 35 + 16L + 3 L , (5.22)π π
where we used the lepton number charges L′′ = L′ + 3, LΦ = 3 and the number of SM
flavors Nf = 3.
The dependence of the gauge coupling on the scale µ is consequently given by
2 [ 2 ]
g2(µ) = g0 ' 2 1 + g0` 2 g0 8 2 b log
µ
, (5.23)
1− g0[ 8 2 b log
µ π µ0
π µ0 ]
where g0 = g ′ 16 ′2`(µ0) and b = 35 + 16L + 3 L . U(1)` has a Landau pole when the
second term in the bracket in eq. (5.23) is of o(rder )unity, i.e. at the scale µ = Λ with
2
Λ = µ0 exp
8π
2 . (5.24)bg0
We now choose µ0 = mZ′ = 3g0vΦ. Figure 5.1 shows the Landau pole Λ normalized
to the scalar VEV vΦ as a function of g0 = g`(mZ′) for different values of the charge
L′. Certainly, we want the Landau pole to occur significantly above the Z ′ mass and
above vΦ, otherwise the validity of our perturbative results would be questionable. This
requires choosing g`(mZ′) . 0.5 for most values of L′. The slowest running is obtained
for L′ = −3/2, which we however excluded since it allows for Majorana mass terms.
57
5. Leptophilic Dark Matter from Gauged Lepton Number
1010
L′ = −32
108 L′ = −12
L′ = 1
106 2
L′ = 3 Figure 5.1: Landau pole Λ2
L′ = 3 normalized to the scalar104
Λ = m VEV vΦ as a function ofZ ′
g0 = g`(mZ′) for differ-
102 ent values of the charge L′.
The gray, solid line indi-
100 cates the value of the Z ′
0.0 0.5 1.0 1.5 2.0 mass corresponding to g0.g`(mZ ′)
To prevent the gauge coupling from running into a Landau pole at low scales, we choose
L′ = −1/2 in the remainder of this paper.5 In this case g`(mZ′) ≈ 1 is acceptable, with
the Landau pole located almost two orders of magnitude above vΦ. For vΦ = 2TeV this
implies an upper bound of mZ′ . 6TeV for the mass of the Z ′ boson. For most of the
considerations that follow, the exact value of L′ merely matters anyway. An exception
are the DM constraints discussed in section 5.2, where we therefore also consider different
values for L′.
5.2. Leptophilic Dark Matter
Provided that we avoid the specific choices of L′ and L′′ discussed in section 5.1.3, the
model features a global U(1)L′+L′′ symmetry under which all SM fermions are neutral
whereas the exotic leptons have unit charge, and which is free of anomalies. This sym-
metry persists when the EW and U(1)` gauge symmetries are broken and ensures the
stability of the lightest dark lepton. If neutral, it is a candidate for dark matter.
For the remainder we identify ν5 as the DM candidate (which can always be achieved
by defining the mixing angles in eq. (5.17) accordingly) and relabel it as νDM ≡ ν5. Since
direct detection experiments exclude DM candidates with unsuppressed couplings to the
SM Z boson, νDM should be composed predominantly of the SM singlets ν ′′ ′L and νR.
Consequently αν and βν should be small. In particular, this requires cν < c`. Finally,
at least one of y′ν and y′′ν should be non-vanishing, otherwise an additional global U(1)
symmetry remains unbroken and the next-to-lightest dark fermion (either ν4 or e4/5)
would be stable as well.
The model is implemented in FeynRules [230], and subsequently mircOMEGAs [140]
was used to calculate the relic density as well as direct and indirect detection constraints.
5Note that picking a half-integer value is mostly for aesthetic reasons. We could have chosen any real
number not listed in table 5.1. Further requiring Λ . 100TeV for g` = 1 and vΦ = 2TeV restricts the
viable choices to L′ ∈ [−5/2,−1/2].
58
Λ/vΦ
5.2. Leptophilic Dark Matter
scalar sector gauge sector fermion sector
vΦ = 2TeV mZ′ = 1.5TeV mDM = 640GeV me4 = 2.0TeV
mφ = 2.5TeV  = 0 sin(θDM) = 0 me5 = 1.5TeV
sin(θH) = 0 L′ = −12
Table 5.2: Default values for the model parameters (assuming y′ = y′′ν/e ν/e and ce = cl)
used throughout this paper, unless specified otherwise. For negligible sin(θDM), the
mostly-doublet, heavy neutrino mass is given by mν4 ' (me4 +me5)/2 = 1.75TeV.
νDM HL
νDM `SM HL SM
Z ′ γ, Z,W ′
(Z ′
(Z ) Z,W
)
ν ′DM ¯̀SM HL SM
(a) (b) SM SM
(c)
Figure 5.2: Processes contributing to the depletion of the DM relic abundance.
Unless specified otherwise, we use the parameters listed in table 5.2. We here relabelled
θDM ≡ αν = βν .
5.2.1. Relic Abundance
Assuming that νDM is a thermal relic, its abundance is predominantly set by its anni-
hilation cross-section to two leptons through an s-channel Z ′ (fig. 5.2a). Other possible
channels are annihilation to gauge or scalar bosons through an intermediate h or φ, or
to fermions via a Z boson. The former is suppressed by the h − φ mixing, whereas the
latter can arise from Z − Z ′ mixing or by an admixture of the SM component in the
DM. The doublet-singlet mixing further allows for t-channel annihilation to two bosons,
and a small mass splitting between the DM and the other exotic leptons can lead to
co-annihilation. See ref. [5] for more details.
The parameter regions in the mDM−vΦ and mDM−mZ′ planes that reproduce the DM
relic abundance of h2ΩDM = 0.1200 ± 0.0012 measured by the Planck satellite [60] are
shown in fig. 5.3 for different values of L′. We assume a lepton number gauge coupling
of g` = 0.1 and a scalar self-coupling of λΦ = 0.5, as well as Yukawa couplings c` = 1.5
and ye = 0 when scanning over the VEV (left panel), and a scalar VEV of vΦ = 2TeV
when varying the Z ′ mass (right panel). The remaining parameters are set to the values
specified in table 5.2.
The colored regions yield a DM abundance that lies within two standard deviations
around the Planck measurement. For each value of mZ′ we typically obtain two viable
59
5. Leptophilic Dark Matter from Gauged Lepton Number
10 3.0
′ ′
g` = 0.1, λΦ = 0.5
ΓZ > 0.1mZ
9 ′ (L
′ = 3/2)
L =−12 2.5
8 L′= 12
7 2L′= 3 2.0 ′/2 mZ
=
6 DM
/2 1.5
m
5 ′
mZ
=
4 M 1
D mφ 1.0m = 2M
3 mD
0.5
2
250 500 750 1000 1250 1500 0 200 400 600 800 1000
mDM [GeV] mDM [GeV]
(a) g` = 0.1, λΦ = 0.5 (b) vΦ = 2TeV
Figure 5.3: Parameter regions reproducing the DM relic density h2ΩDM = 0.120± 0.001
measured by Planck [60] within two standard deviations for different values of L′, fixing
the gauge coupling (left) or scalar VEV (right). The dashed gray lines in the right plot
indicate the parameters for which the Z ′ width exceeds 10% of its mass.
values for the DM mass, one below and one above the mDM = mZ′/2 resonance. In
fig. 5.3a, we can in addition also see the scalar resonance at mDM = mφ/2. To guide the
eye, the resonances are indicated by dashed dark-gray lines.
Since the Z ′ predominantly decays into νDM and the other dark leptons, its width
increases with L′. For L′ = −1/2, the Z ′ is rather narrow and the DM mass is restricted
to values close to half of the Z ′ mass. For larger charges, the resonance becomes broader
and the DM mass can be lower. The light gray, dashed line in fig. 5.3b indicates the
regions in which the width of the Z ′ exceeds 10% of its mass for L′ = 3/2.
The dependence of the DM relic density on the masses of the non-DM exotic leptons
e4, e5, and ν4, denoted as heavy leptons (HLs) in the following, is shown in fig. 5.4,
assuming that they all have the same mass mHL. The colored regions again yield the
measured DM abundance, now assuming L′ = −1/2. The colors correspond to a relative
mass splitting ∆m ≡ (mHL − mDM)/mDM of 1% (blue), 2% (red), 5% (green), and
10% (purple) between the DM and the HLs.
For high DM masses, the HL masses affect the relic density only by changing the Z ′
width. However, for lower DM masses the abundance is no longer set by annihilation
of the DM to SM leptons as depicted in fig. 5.2a, but via co-annihilation. In this case,
the HL abundance is depleted by annihilation of e4, e5 and ν4 to SM particles through
electroweak processes as shown in fig. 5.2b. This depletion is transferred to the DM
abundance by the EW processes depicted in fig. 5.2c in which a DM particle scatters
off SM particles and changes into a HL. These processes require sin θDM 6= 0, but we
need this assumption anyways to ensure that there is only a single DM component. As
the diagram 5.2b is dominanted by EW processes, the relic density in this regime is
independent of the Z ′ mass. Figure 5.4 assumes sin θDM = 0. However, modifying the
60
vΦ [TeV]
mZ ′ [TeV]
L′= 3
2
L ′= 1
2
L ′=− 1
2
5.2. Leptophilic Dark Matter
3.0
2.5
Figure 5.4: Same as fig. 5.3b fix-
2.0 ′/2 ing L′mZ = −1/2, but including co-
=
DM annihilation with the heavy lep-
1.5 m tons (HLs) for a mass splitting of
∆m = 1% – 10%, assuming that e4,1.0
e5 and ν4 have equal mass mHL.
0.5 The colored regions reproduce the
measured relic abundance, the col-
0 200 400 600 800 1000 ors correspond to different values of
mDM [GeV] mHL/mDM = 1 + ∆m.
mixing within the range allowed by direct detection (see section 5.2.2) does not alter the
result.
Varying the remaining parameters of the model only has a minor effect on these results.
The scalar mass mφ and the h − φ mixing angle θH only have an effect in the region of
the φ resonance mDM = mφ/2 (or the h resonance).
5.2.2. Direct and Indirect Detection
Direct detection experiments strongly constrain DM couplings to the SM Z boson via
scattering off nuclei. For small values of the kinetic mixing parameter , the coupling is
given by
νDM
ie ig [ ]`
Z s2 ′ ′′ ′ ′′ 2 2DM 2 γµ+sξ 2 γµ (L + L ) + (L − L )(cDM − sDM)γ5 , (5.25)cW sW
νDM
where sW = sin θW , sξ = sin ξ, sDM = sin θDM and cDM = cos θDM. The first term
originates from the heavy doublet neutrino N = N ′L+N ′′R (which has vector-like couplings
to the SM Z) mixing into the DM, whereas the second part comes from the chiral DM−
Z ′ coupling. The axial part of the latter is also modified by the DM mixing via the
ν4ν4Z ′ vertex, while the vector part remains untouched by θDM since it here enters as
(c2 2DM + sDM).
Figure 5.5 shows the constraints on the Z − Z ′ and DM mixing as a function of the
Z ′ mass, obtained from direct detection limits on spin-independent DM-nucleus scatter-
ing. The solid lines correspond to current constraints from the XENON1T experiment
based on one tonne times year of data acquisition [88], the long dashed lines indicate the
prospective sensitivity of LZ [142], and the dash-dotted lines show the projected reach of
DARWIN [143]. At each parameter point, the DM mass is fixed to a value reproducing
h2ΩDM = 0.12 (chosing the value below the Z ′ resonance). The charges are taken to be
L′ = −1/2 (blue) or L′ = 3/2 (red), and the VEV is set to vΦ = 2TeV.
61
mZ ′ [TeV]
mHL/mDM = 1.1
mHL/mDM = 1.05
mHL/mDM = 1.02
mHL/mDM = 1.01
5. Leptophilic Dark Matter from Gauged Lepton Number
0.030
L′= 3 1T ′ 32 XENO
N L = 2
−1 L′=−1 0.025 L′=−110 2 T 2
ENON
1
X XENON1T
0.020
ARWI
N
LZ D
10−2
IN 0.015
LZ DARW
LZ
10−
0.010
3
0.005
DARWIN
10−4 0.000
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
mZ ′ [TeV] mZ ′ [TeV]
(a) kinetic mixing parameter  (θDM = 0) (b) DM mixing angle sin θDM ( = 0)
Figure 5.5: Direct detection limits on the Z − Z ′ and νDM − ν4 mixing for L′ = −1/2
(blue) and L′′ = 3/2 (red). Current constraints from the XENON1T experiment [88]
are shown as solid lines, the long-dashed lines indicate the projected sensitivity of
LZ [142], and the dash-dotted lines correspond to DARWIN [143]. The DM mass is
fixed by the requirement to reproduce the Planck relic density; all other parameters
are set according to table 5.2. The short-dashed lines in 5.5b indicate the prospective
reach of the CTA [231] indirect detection experiment. The light dotted lines show the
region where the Z ′ width grows above 10% of the mass.
Current direct detection experiments can probe kinetic mixing parameters in the per-
cent range and DM mixing angles of sin θDM & 0.015 – 0.025, depending on the Z ′ mass.
With LZ, DM mixing angles around sin θDM & 0.006 – 0.01 as well as kinetic mixing
in the sub-percent range can be reached, while DARWIN can prospectively exclude
sin θDM & 0.004 – 0.006, and sub-per-mill kinetic mixing for mZ′ . 1TeV. The con-
straints for L′ = 3/2 are stronger than for L′ = −1/2 since the latter case leads to higher
DM masses, whereas the former case gives DM masses below 500GeV.
Beside Z-mediated DM-quark interactions, nuclear scattering can also proceed via
Higgs (or φ) exchange, either through h − φ mixing or by direct DM-Higgs couplings.
However, since the Higgs only weakly couples to nuclei, the direct detection constraints on
the Higgs mixing angle are much weaker than on θDM or ξ. Formφ = 2.5TeV, XENON1T
can currently exclude sin θH & 0.1 – 0.4. LZ and DARWIN can prospectively probe
sin θH & 0.02 – 0.06 and sin θH & 0.007 – 0.02, respectively. Here, direct detection ex-
periments are more sensitive for L′ = −1/2 since the Yukawa couplings are proportional
to the DM mass, i.e. we benefit from the higher DM masses in the L′ = −1/2 case.
For lower φ masses on the other hand, the scattering cross section is reduced due to
φ − h interference effects, leading to weaker direct detection limits. The corresponding
constraints are shown in fig. 5.6.
A further, indirect way of probing DM is via its annihilation to SM particles. Since the
observation of charged particles suffers from large uncertainties associated with their prop-
agation through the Galactic halo, we here only consider indirect detection constraints
62

ΓZ′ > 0.1mZ′
sin(θDM)
CTA, τ+τ−
CTA, τ+τ−
ΓZ′ > 0.1mZ′
5.2. Leptophilic Dark Matter
L′= 3 mφ=200 GeV2
′ 1 m =2.5 TeV
XENON1T L =− φ2
XENON1T
0.1 0.1
LZ
LZ
DARWIN
0.01 0.01 DARWIN
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
mZ ′ [TeV] mZ ′ [TeV]
(a) mφ = 2.5TeV (b) L′ = −1/2
Figure 5.6: Direct detection limits on the Higgs mixing angle θH for lepton number
charges (left plot, mφ = 2.5TeV) of L′ = −1/2 (blue) and L′ = 3/2 (red), and scalar
masses (right plot, L′ = −1/2) of mφ = 200GeV (green) and mφ = 2.5TeV (blue). The
current constraints from the XENON1T experiment [88] are shown as solid lines, the
long-dashed and dash-dotted lines indicate the projected sensitivities of LZ [142] and
DARWIN [143], respectively. As before, the DM mass is set to the value reproducing
the measured relic abundance.
from γ-ray searches. As photons travel unperturbed by Galactic magnetic fields, they
can be traced back to their production site, allowing for constraints on DM annihilation
by observing photons from regions with a high DM density.
In our model, the DM typically annihilates through the lepton number gauge boson
Z ′ into SM leptons with equal branching ratios. Photons are thus predominantly pro-
duced as secondary products from annihilation to charged leptons. Direct production of
monochromatic photons is possible via annihilation through the scalar bosons h and φ,
however, this is suppressed unless resonant.
We tested the annihilation of thermally produced DM (i.e. satisfying the relic density
constraint) in our model against current limits from observations of dwarf spheroidal
galaxies by MAGIC and Fermi-LAT [83], and of the inner Galactic halo by H.E.S.S. [85],
as well as against γ line searches from Fermi-LAT [84] and H.E.S.S. [86]. The strongest
limits come from secondary produced photons from annihilation into τ leptons. However,
the current sensitivity reaches the level of the annihilation cross section required for
thermal production (which in addition is reduced by the branching ratio of 1/6 into
tauons) only for DM below 100GeV, which, even for light Z ′ masses, is below the DM
masses predicted by our model (cf. fig. 5.3). This also holds for the projected sensitivity
of Fermi-LAT , assuming a 15-year data set of 60 dwarf spheroidal galaxies [232]. On the
other hand, a next-generation γ-ray observatory such as the CTA [231] will be able to
exclude Z ′ masses between roughly 670GeV and 1.46TeV if L′ = 3/2. The corresponding
limit is indicated by the vertical dashed lines in fig. 5.5b.
63
sin(θH)
ΓZ′> 0.1mZ′
sin(θH)
5. Leptophilic Dark Matter from Gauged Lepton Number
In principle, our model can furthermore be probed through its neutrino sector. Modifi-
cations of the neutrino interactions with the SM leptons arise from Z ′ exchange or kinetic
mixing, and DM-neutrino interactions can be mediated by a Z or Z ′ boson. Neutrino
couplings to the lepton number breaking scalar and SM Higgs boson are suppressed by
the neutrino Yukawa couplings. For the range of Z ′ and DM masses considered here, no
constraints are obtained from current data [233, 234].
5.3. Collider Phenomenology
While new physics that couples directly to quarks and gluons is nowadays severly con-
strained by direct searches at the LHC , the situation is different for the leptophilic new
physics model we are considering here. In this model, constraints predominantly arise
from a combination of LEP limits as well as direct and indirect LHC measurements,
such as e.g. Higgs data. In the following we present an overview of the most important
constraints on the lepton number gauge boson, the extended Higgs sector and the new
leptons introduced in our model, and comment on the prospects for detection at the
HL-LHC and future colliders.
5.3.1. Z ′ Constraints
Since in the absence of kinetic mixing the lepton number gauge boson does not couple to
quarks, the strongest constraints on the Z ′ boson come from LEP II. These exclude Z ′
masses below the maximal LEP center-of-mass energy of 209GeV (except for tiny gauge
couplings g` < 10−2 [235, 236]) and severely restrict the lepton number breaking VEV vΦ
through 4-lepton contact interactions.
A heavy Z ′ induces effective contact interactions between electrons and charged (SM)
leptons. Since the Z ′ couplings to SM fermions are vector-like, the corresponding contact
interactions are given by (neglecting kinetic mixing)
g2 g2 2L ` µ geff ⊃ −2 2 ēγµe ēγ e−
`
2 ēγµe µ̄γ
µµ− `2 ēγµe τ̄γ
µτ , (5.26)
mZ′ mZ′ mZ′
which interfere destructively with the SM Z boson for center-of-mass energies above the
Z-pole. LEP puts a 95% lower bound of Λ > 20TeV [237] on the scale suppressing
this contact interaction, related to the model parameters by Λ2 = 4πm2Z′/g2` . Using
mZ′ = LΦg`vΦ with LΦ = 3, this gives a lower bound on the scalar VEV of
vΦ & 1880GeV . (5.27)
To be conservative, we set the VEV to vΦ = 2TeV in the following.
Future e+e− colliders have the potential to s√ubstantially tighten these bounds. For
instance, the ILC with a center-of-mass energy of s = 1TeV can exclude Z ′ masses below
1TeV (unless the gauge coupling is below g` . 7.6× 10−8), and constrain the VEV to be
above vΦ & 15TeV at 90% confidence level (CL) using muon contact interactions [208].
At the LHC , the Z ′ is rather hard to produce. In particular, in the absence of kinetic
mixing it is predominantly produced by pair-producing SM leptons that radiate off a Z ′.
The Z ′ can then be detected from its decays to charged SM leptons.
64
5.3. Collider Phenomenology
100 10−1
pp→ `` + (Z ′ → `+`−) pp→ Z ′ → `+`−
10−1 10 events @ 300 fb−1
10−2
10 events @ 3 ab−1
10−3 10−2
10−4 √ ATLAS, 1707.02424
s = 13 TeV
√ CMS, 1803.06292
10−5 s = 14 TeV√ LHC-14, 300 fb
−1
s = 100 TeV LHC-14, 3 ab−1
10−6 10−3
0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
mZ ′ [TeV] mZ ′ [TeV]
Figure 5.7: Z ′ production cross-section in Figure 5.8: Current LHC limits in the
the absence of kinetic mixing for the Z ′ mass vs. kinetic mixing parameter-
LHC-13 (blue), LHC-14 (red), as well as plane from ATLAS (solid red) and CMS
FCC-100 (green). (solid blue), as well as projections for
the 14TeV LHC with 300 fb−1 (dashed
green) and 3 ab−1 (dotted purple).
To obtain a rough estimate of the detection prospects, we calculate the parton-level
cross-section for pp→ ``Z ′ with CalcHEP 3.6 [238], where ` can be any SM lepton, includ-
ing neutrinos. For the decay we use the narrow-width approximation (NWA), assuming
that the Z ′ decays into S(M leptons o)nly. In this case, the corresponding branching ratio
to charged leptons is Br Z ′ → `+`− = 50%. The cross section as a function of the Z ′
mass is shown in fig. 5.7 for the LHC with center-of-mass energies of 13TeV (blue) and
14TeV (red), as well as for a 100TeV FCC (green). The gray lines indicate the cross
sections that would produce 10 Z ′s, assuming integrated luminosities of 300 fb−1 and
3 ab−1.
With the current LHC data, no constraints can be put on the Z ′ mass if kinetic mixing
is absent. With 300 fb−1, the LHC would not produce a sufficient amount of Z ′s. At
the HL-LHC , Z ′ masses below . 400GeV can be reached. The prospects for a 100TeV
collider are more promising. With 3 ab−1, more than 10 events are produced for masses
up to 2.5TeV, extending the reach into the multi-TeV region.
In the presence of kinetic mixing the situation is different. The lepton number gauge
boson then couples to quarks with couplings proportional to the kinetic mixing parameter
 and the quark hypercharge. It can thus be produced directly in proton-proton collisions
and be searched for as a dilepton resonance, giving constraints in the mZ′ vs.  plane.
Figure 5.8 shows constraints from current sear√ches for dilepton resonances using 36 fb
−1
of data collected at a center-of-mass energy of s = 13TeV by ATLAS [133] (solid red
line√) and CMS [239] (solid blue line). Projections for the LHC at a center-of-mass energy
of s = 14TeV with an integrated luminosity of 300 fb−1 (dashed green) and 3 ab−1
(dotted purple) are also shown. Currently, kinetic mixing can be probed in the percent
range, the HL-LHC can prospectively reach the sub-percent range for light Z ′. Again,
65
σ [fb]

5. Leptophilic Dark Matter from Gauged Lepton Number
the cross sections have been calculated with CalcHEP [238], assuming a narrow Z ′ width
with a branching ratio of 1/3 to light charged leptons (e or µ). The projections have been
obtained assuming that the limits on cross-section ratios provided by CMS [239] do not
change when increasing the center-of-mass energy from 13TeV to 14TeV, and that the
exclusion reach scales with the square root of the luminosity.
The kinetic mixing can also be probed via its effects on SM precision measurements
at electron-positron colliders [240]. However, as these effects are suppressed for high Z ′
masses, the LHC provides the strongest constraints in the mass range considered here.
5.3.2. Higgs Constraints
The scalar sector of our model is subject to constraints from measurements of the prop-
erties of the 125GeV Higgs boson at ATLAS and CMS, as well as from null-results of
searches for scalar bosons at different masses.
In our model, the SM Higgs properties can be modified by three effects: the h − φ
mixing which modifies all SM Higgs couplings, modifications of Higgs couplings to EW
gauge bosons (in particular to two photons) by loops of heavy charged leptons, and decays
to BSM states (if kinematically accessible). However, given the lower bound on the lepton
number breaking VEV (5.27), the new states are typically too heavy for the SM Higgs to
decay into, so that the last effect is absent in most of the parameter space.
The mixing between the lepton number breaking scalar and the SM Higgs boson given
by equation (5.11) reduces the Higgs couplings to SM fields by cos θH . ATLAS and CMS
provide limits on modifications of Higgs couplings compared to the SM values in terms
of signal strengths, defined by
= σ (pp→ h)× Br (h→ X)µX
σSM
. (5.28)
(pp→ h)× BrSM (h→ X)
Neglecting additional Higgs decay channels and further modifications of loop-induced
Higgs couplings discussed below, the production cross-sections are modified by a factor
cos2 θH , whereas the branching ratios remain unchanged as the cosine factors in the
partial and total widths cancel. Thus, the signal strengths are µ = cos2 θH . An estimate
of the limit on the mixing angle can be obtained from the global signal strength. The
current CMS measurement is µ = 1.17± 0.10 [241]. This gives a 95% exclusion of
| sin θH | < 0.16 . (5.29)
Loops of the dark electrons e4 and e5 can contribute sizeably to the h → γγ and
h→ Zγ rate. In the SM, t∣hese rates are given by [242]∣2
α2m3 ∣Γ(h→ γγ) = h ∣∣∑ ∣∣f 2 ∣256 3 2 ∣∣ Nc QfA1/2(τf ) +A1(τW )∣∣ , (5.30)π vH (f ) ∣ ∣2
2 3 3 ∣ ∣
Γ( → ) = αmWmh 1− m
2
h Zγ Z ∣∣∣∑ fQf v̂f ∣128 4 4 2 ∣ Nc A1/2(τf , λf ) +A1(τ , λ ∣W W )π v ∣H mh cf W ∣ , (5.31)
66
5.3. Collider Phenomenology
where α is the electromagnetic coupling constant, τi = 4m2i /m2h, and λi = 4m2i /m2Z . The
expressions for the form factors As for a spin s particle running in the loop can be found
in ref. [242]. The sums run over all charged fermions that couple to the Higgs. Nfc is
the color-representation of the fermion, Qf is its electric charge, and v̂f is the fermion’s
(reduced) vector-coupling to the Z boson. In the SM, the dominant contribution from
fermions comes from the top quark with N t 8 2c = 3, Qt = 2/3, and v̂t = 1− 3sW .
Equations (5.30) and (5.31) assume that the fermions couple to the Higgs with a vertex
factor proportional to their masses. The top quark in the SM for instance has a vertex
factor −i√yt2 = −i
mt
v . This is not true for the heavy charged leptons. The correspondingH
vertex factors are
e−4/5
h − √
i Yhe4 5 , Yhe4 5 = ±cHye − sHc` , (5.32)2 / /
e+4/5
for the SM-like Higgs.6 The correct result can then be obtained by rescaling the dark
electron contributions by a factor Y√sfvH2 , where s = h, φ and f = e4, e5. Due to the scalarmf
mixing, the SM contributions further get a factor of cH or sH for h and φ, respectively.
We thus obtain
2 ∣
Γ( → ) = α m
3 ∣
h ∣ 4h γγ 256 3 2 ∣ 3cHA1/2(τt) + cHA1(τW )π vH ∣ (5.33)
+ Y√he4v
2
H Yhe ∣5vH ∣
( 2me)A1/2(τe4) + √ A (τ ) ,4 ∣∣ 2
1/2 e5
m ∣e5
αm2 m3
3
Γ( → ) = W h m
2
Z ∣6 + 16s2h Zγ W128 1−π4v4 m2 ∣ 3 cHA1/2(τt, λt) + cHA1(τW , λW )H h cW (5.34)
+ 1− 4
2 (Y Y ) ∣s vH he he ∣2W √ 4A 5
c 2 m 1/2
(τ ∣e4 , λe4) + A1/2(τe5 , λe5) ∣ .
W e4 me5
The corresponding widths for the φ scalar can be obtained by replacing mh → mφ,
cH → sH , Y 2 2he → Yφe , and τi = 4mi /mφ. The leading QCD corrections can be includedi i
by multiplying the top contribution by (1− αs/π) [242].
We evaluated constraints from direct Higgs search√es using HiggsBou√nds 4.3.1 [243]. The
corresponding limits from LEP [244, 245] (purple), s = 7TeV and s = 8TeV searches
with ATLAS and CMS for a Higgs boson in the h→ ZZ/WW channel [246–249] (blue),
and a combination of CMS 7TeV and 8TeV searches in various final states [250] (green)
are shown as colored regions in fig. 5.9. The red line indicates limits from signal strength
measurements. These include measurements of Higgs boson properties in the h → 4`
and h → γγ channels by ATLAS [251, 252], and a CMS analysis combining different
6The corresponding interactions of φ are obtained by replacing cH → sH and sH → −cH .
67
5. Leptophilic Dark Matter from Gauged Lepton Number
1.0 0.5
0.4 c` = 0
c` = 0.10.8 0.3
c` = 0.5
0.2
ATLAS/CMS c` = 1
0.6 φ→ ZZ,WW 0.1 c` = 10
0.0
0.4 −0.1
LHC, 13 TeV, 36 fb−1 −0.2
0.2 global µ −0.3
−0.4
0.0 −0.5
0 200 400 600 800 1000 0.0 0.5 1.0 1.5 2.0 2.5 3.0
mφ [GeV] ye
Figure 5.9: Exclusion bounds on the mass Figure 5.10: 95% exclusion limits from
of the lepton number breaking scalar φ signal strength measurements by AT-
and the Higgs mixing angle θH from di- LAS and CMS on the Higgs mixing an-
rect searches (colored regions) and signal gle and the heavy electron Yukawa cou-
strength measurements (colored lines). plings.
channels [241], both at a center-of-mass energy of 13TeV, as well as the combination of
7TeV and 8TeV results from ATLAS and CMS [253]. The dashed orange line corresponds
to the naive estimate (5.29).
The constraints on the Higgs mixing angle θH are shown in fig. 5.9 for the parameter
values given in table 5.2. Signal strength measurements exclude sin θH & 0.27, i.e. the
limit in eq. (5.29) from the global signal strength overestimates the exclusion reach.
Direct searches for additional scalars provide somewhat weaker constraints of around
sin θH & 0.4 for a large range of mφ, but are stronger for scalar masses below the Higgs
mass. The Higgs signal strength fits are more involved for mφ near 125GeV and for
mφ < 62.5GeV where the Higgs may decay into φ-pairs. The signal strength constraint
shown in fig. 5.9 should be taken with a grain of salt in those regions.
If the new leptons have sizeable couplings to the Higgs boson, the Higgs signal strengths
in different channels can vary due to the loop contributions to h → γγ and h → Zγ
decays. Figure 5.10 shows the current LHC limits from refs. [241, 251–253] as a function
of the mixing angle and the heavy electron Yukawa couplings c` and ye. If the Φ Yukawa
coupling c` is small, the dark electrons gain their mass predominantly from EW symmetry
breaking and hence strongly contribute to the h→ γγ rate. Thus, the Yukawa coupling
to the Higgs doublet ye is also restricted to be small. For large c`, the heavy electron
contributions in eq. (5.33) are mass-suppressed, so that ye can take large values without
modifying Γ (h→ γγ) beyond the experimentally allowed limits.
Note that for c` = 0 the charged heavy lepton masses are given by m y√evHe4/5 = 2 .
The LEP limit on the mass (see section 5.3.3) then constrains the Yukawa coupling to
ye > 0.57, hence the entire region allowed by h → γγ and h → Zγ is excluded in this
case. Similarly, section 5.1.3 implies |ye| < 0.24 for c` = 0.1 and vΦ = 2TeV. Further
68
sin(θH)
LEP
sin(θH)
CMS combined
5.3. Collider Phenomenology
101 100
ν e+ mDM = 100 GeV4 4/5
100 e− e+4/5 4/5 80
− q̄qe ν̄
− 4/5 410 1 ν ν̄ 604 4 Z
10−2 40 h
10−3
ν̄ν
√ 20 `+`−
s = 13 TeV
10−4 0
200 400 600 800 1000 200 300 400 500
me /e /ν [GeV] mν [GeV]4 5 4 4
Figure 5.11: Production cross-sections Figure 5.12: Branching ratios of the neu-
for HL pairs at the LHC with a center-of- tral HL, assuming me4 = me5 ≈ mν4 ,
mass energy of 13TeV, assuming equal sin θDM = 0.01, and a DM mass of
masses for all HLs. 100GeV.
note that the exclusion for c` = 10 is shown despite severly challenging the bounds of
perturbativity to illustrate the constraints in the limit of large c`.
5.3.3. Constraints on Heavy Leptons
Whereas the DM candidate is mostly an SM singlet, the remaining heavy leptons (HLs)
carry EW charge and can hence be produced at colliders. Direct searches for heavy,
charged leptons at LEP set a lower limit of roughly 100GeV on the e4 and e5 mass [254].
LHC limits on the HLs can be obtained by recasting supersymmetry (SUSY) searches
for electroweakly produced charginos and neutralinos.
Figure 5.11 shows the cross section for the production of two charged heavy leptons
(dashed red), heavy positrons with a heavy neutrino (solid blue), heavy electrons with
a heavy anti-neutrino (dash-do√tted green), and pairs of heavy neutrinos (dotted purple)
in proton-proton collisions at s = 13TeV calculated using CalcHep [238]. We here take
the most optimistic scenario for HL production in which all HLs have the same mass. As
the DM mixing angle θDM is restricted to be small by the direct detection constraints in
section 5.2.2, and therefore only has a negligible effect on the production cross-section, it
can be set to zero.
Due to the U(1) symmetry that stabilizes the DM, the HLs can only decay amongst
themselves or to dark matter. Consequently, to allow the lighter dark electron to de-
cay, θDM 6= 0 is required. Otherwise the model would have a charged DM population
and therefore be excluded. However, even a (negligibly) small amount of DM mixing is
sufficient to let the exotic particles decay fast enough to avoid this problem.
The charged HLs typically decay into DM and a (potentially off-shell) W boson. De-
pending on the masses, decays to other exotic leptons can also be possible. These are
however suppressed by phase space. The heavy neutrino can decay to DM and a(n off-
shell) Z boson or, if mν4 > mDM+mh, to DM and an SM Higgs boson. In the latter case,
69
σ [pb]
Br(ν4 → νDM + X) [%]
mν4 = mDM +mZ
mν4 = mDM +mh
5. Leptophilic Dark Matter from Gauged Lepton Number
101 300
100
100 200
10−1
10−1 100 10−2
10−2 0 10−3
100 200 300 400 500 100 200 300 400 500
me [GeV] me [GeV]4/5 4/5
Figure 5.13: Total cross √section for Figure 5.14: CMS 95% exclusion limits
pp→ e±4/5ν4 production at s = 13TeV from [256], assuming equal masses for all
as a function of the HL mass. The col- HLs and that they decay within the de-
ors resemble the color-coding of the cross tector.
section in [256].
the branching rations to DM + Z and DM + h are roughly 50%. Figure 5.12 shows the
branching ratios of ν4 as a function of the mass for a DM mass of 100GeV and a mixing
angle of sin θDM = 0.01.
At the LHC , the HLs can be searched for by looking for missing transverse energy
(MET) in association with W , Z, or h bosons (or SM lepton pairs in the off-shell case).
These searches have been performed by ATLAS [255] and CMS [256] in t√he context of
simplified SUSY models, using 36 fb−1 of data recorded at the LHC with s = 13TeV.
They assume that the lightest neutralino χ̃01 is the lightest supersymmetric particle (LSP)
and consider the process pp→ χ̃± 01 χ̃2, where the lightest chargino χ̃±1 decays to the LSP
plus a W boson, and the next-to-lightest neutralino χ̃02 to the LSP plus Z or h. The
respective 95% CL exclusion bounds from CMS can be found in fig. 8 of ref. [256].
The production cross-section for the corresponding process in our model is depicted
in fig. 5.13, again assuming sin θDM = 0 and mHL ≡ me4 = m( ) e5
= mν4 . The respec-
tive exclusion bounds from CMS [256] are shown in fig. 5.14, taking the limits with
Br (χ̃02 → χ̃01Z) = 100(% for mDM < mHL < mDM +mh, and the limits assuming
Br χ̃0
)
2 → χ̃01Z = Br χ̃0 → χ̃02 1h = 50% for mHL > mDM +mh. The LHC can currently
exclude HL masses below mHL . 180GeV and DM masses below mDM . 140GeV. For
the co-annihilation region discussed in section 5.2.1, the mass splitting between the
charged states and the DM candidate becomes very small, so that the searches used
here become inefficient. Instead it has been shown that these regions can be probed
by mono-jet, mono-Z and disappearing track searches, with masses of up to 200GeV
reachable at the LHC and up to 1TeV at a future hadron collider [257–262].
70
σ [pb]
mDM [GeV]
m
e
4/5 =
m
m DM
e
4/5 =
m
DM +
m
h
95 % CL upper limit on cross-section [pb]
5.4. Intermediate Conclusion
5.4. Intermediate Conclusion
We have presented a comprehensive study of the DM and collider phenomenology of a
model in which lepton number is gauged, extending and updating the limits in ref. [5].
A mass for the lepton number Z ′ boson is generated via spontaneous breaking of the
corresponding gauge symmetry, induced by the VEV of an SM singlet scalar field Φ with
lepton number charge LΦ = 3. The model further features additional leptonic states, the
presence of which is forced upon us by the necessity to cancel gauge anomalies associated
with U(1)`. These exotic leptons naturally give rise to a candidate of leptophilic DM.
Assuming that the DM is a thermal relic, we identified the regions of the parameter
space in which the DM candidate can account for the full abundance measured by Planck.
We found that the correct relic density can be reproduced for a broad extent of DM masses
in the O(100GeV) to TeV range. This typically requires choosing mZ′ ∼ 2mDM.
Direct and indirect detection experiments put limits on the DM interactions with SM
fields. Direct detection constrains the various mixings that can give rise to DM-quark
interactions. These are the SM doublet admixture into the singlet DM characterized by
θDM, the kinetic mixing parameter  of the lepton number and hypercharge gauge groups,
and the mixing angle θH between the SM and the lepton-number Higgs. XENON1T can
exclude  and θDM in the percent range, and sin θH & O(0.1); LZ and DARWIN can
improve the limits by roughly an order of magnitude. Indirect detection on the other
hand mainly probes Z ′ mediated DM-SM interactions. However, even the next-generation
CTA is only sensitive for lepton number charges as large as L′ = 3/2.
We also investigated collider constraints. The most important ones are LEP limits.
These put a lower bound on the lepton number breaking scalar VEV vΦ & 1.88TeV,
and exclude Z ′ masses below mZ′ ' 200GeV, as well as charged exotic leptons be-
low 100GeV. The LHC can only put limits on the Z ′ mass if the kinetic mixing is
 & O(0.01− 0.1), the HL-LHC with 3 ab−1 can reach  ∼ 10−3 for low Z ′ masses and
even exclude mZ′ . 500GeV if  = 0. Current LHC measurements further exclude Higgs
mixing angles sin θH & 0.27 and constrain the exotic leptons’ Yukawa couplings.
Having mapped out the phenomenologically viable parameter space of the model, we
can now proceed and investigate the lepton number breaking PT as well as the detectabil-
ity of the corresponding SGWB in chapter 7. Let us therefore hereby conclude part I of
this thesis and move on to part II on constraining new physics using GWs generated in
cosmological first-order PTs.
71

Part II
Gravitational Waves
from Cosmological
Phase Transitions

Prelude
Gravitational waves (GWs) are perturbations in the metric of space-time propagating at
the speed of light, following as a consequence of Einstein’s theory of general relativity
(GR). Their existence was predicted by Albert Einstein in 1916 [263, 264].
First indirect evidence of GWs was provided in 1979 [265] through measurements of
the Hulse-Taylor binary [266], a binary system of a pulsar and a neutron star (NS), for
which Hulse and Taylor were awarded the 1993 Nobel prize in physics. Pulsars (see e.g.
ref. [267] for a review) are highly magnetized NS that rotate rapidly and emit electromag-
netic (EM) radiation along their magnetic axis. This beam of radiation then hits Earth
periodically, resulting in light-house-like pulse signals that provide very accurate clocks.
Timing of these pulses over a sufficiently large period of observation allows for a precise
determination of the masses and orbit of the binary system. As the system looses energy
due to emission of GWs, the orbit is expected to decay. The corresponding decrease of
the orbital period could be observed in the Hulse-Taylor binary at a rate consistent with
the predictions of GR [268] and is now established to agree with the GR prediction at a
level of a few per mill [269].
On September 14, 2015, almost a century after Einstein’s prediction, the first di-
rect observation of GWs was achieved by the Advanced LIGO (Laser Interferometer
Gravitational-Wave Observatory) interferometers [19]. The physics Nobel prize 2017 was
awarded to Rainer Weiss, Barry Barish, and Kip Thorne for this ground-breaking dis-
covery. The observed GW signal originated from the merger of two black holes (BHs)
into a single BH. Since then, the LIGO and Virgo collaborations have reported 9 further
observations of BH merger events as well as one NS merger during their first two obser-
vational runs, and 56 detection candidates from their third run. The direct detection of
GWs has inspired a drastic increase in the interest in GW physics and the various ways
it can be used to probe fundamental physics. It opens a new and unique window to the
early Universe, allowing us to see much further into the past than ever before.
We prevalently observe our Universe through light. The potential for direct observa-
tions is therefore limited to the eras during which the Universe was transparent to EM
radiation, to wit, the time after photon decoupling, a few hundred thousand years af-
ter the Big Bang at a temperature of roughly 1 eV [270].1 When electrons and protons
form neutral hydrogen atoms in the so-called recombination epoch, the number density
of free electrons in the Universe drops rapidly and the processes that keep the photons in
thermal equilibrium with the plasma of the early Universe (mainly Thompson scattering,
e− + γ → e− + γ) become inefficient. The photon interaction rate then drops below the
1Still, we have a reliable probe of an even earlier stage of the Universe — Big Bang Nucleosynthesis
(BBN). The concordance of the observed abundances of light elements with the corresponding predic-
tion of BBN indicates that it indeed proceeded as predicted in the Standard Models of particle physics
and cosmology at a temperature around 1MeV.
75
Prelude: Gravitational Waves from Cosmological Phase Transitions
Hubble rate, the rate at which the Universe expands. As a consequence the photons
decouple and are no longer in thermal equilibrium with the rest of the plasma; they now
stream freely through the Universe. The relic photons from the time of recombination
are today observed in the form of the cosmic microwave background (CMB). Although
direct observations via photons are limited to the time after photon decoupling, even
much earlier processes such as inflation may still be observable indirectly through their
imprints in the CMB.
Due to the very weak coupling strength of gravity, GWs on the other hand decouple
very early in the history of the Universe. The interaction rate Γ of particles in the
early Universe is Γ = nσv, where σ is the cross section, n is the number density (of the
interaction partner), and v the (relative) velocity. As all species are relativistic at high
temperatures, we can take n ∼ T 3 and v ∼ c, and the cross section for gravitons can be
estimated as σ ∼ T 2/M4P , where MP ∼ 2× 1018GeV is the Planck mass. The Hubble
rate during radiation domination on the other hand is roughly given by H ∼ T 2/MP .
Comparing the interaction rate of gravitons to the expansion rate of the Universe therefore
yields
Γ( ( )T ) 3
( ) ∼
T
,
H T MP
i.e. gravitons decouple around the Planck scale. After that, they can propagate freely
from the time of their production until today, therefore allowing us to directly observe
the very early Universe.
Whereas the GWs observed so far all originate from single events at specific positions
in the sky, namely mergers of BH or NS binaries, we are here interested in stochastic
backgrounds of GWs, i.e. the superposition of many statistically independent GW events.
In particular, GWs produced in the early Universe are typically of stochastic nature due
to causality, as the correlation length of the source is limited by the horizon size at
the time of production. A GW signal produced at a temperature of 100GeV (around
the time of the electroweak (EW) phase transition) for instance could today be observed
from ∼ 1024 independent regions on the sky [271]. Since the early Universe is (assumed to
be) homogeneous and isotropic on large scales, the initial conditions for generating GWs
are the same in all these patches, resulting in a stochastic gravitational wave background
(SGWB).
The following part of this thesis is focused on the SGWB from cosmological phase
transitions (PTs), and its potential for providing insights into new physics. The possibility
to probe dark sector physics using GW signals from PTs was proposed in [272] and
explored further in [273–281]. An introduction to SGWBs is provided in chapter 6,
describing its main characteristics and detection prospects. We explain how PTs generate
GWs and how the corresponding spectrum is calculated. As an example for the potential
of probing new physics via GWs, chapter 7 considers the lepton number breaking PT
in the gauged lepton number model introduced in chapter 5. Chapter 8 subsequently
investigates PTs in decoupled hidden sectors, with particular focus on sub-MeV sectors
and the interplay with constraints on the effective number of neutrino species.
76
6. Stochastic Gravitational WaveBackgrounds
A stochastic gravitational wave background (SGWB) is a gravitational wave (GW) signal
produced by a large number of independent, unresolved, and weak sources, characterized
only statistically (see e.g. refs. [271, 282–285]). It can be viewed as the GW equivalent
of the cosmic microwave background (CMB). SGWBs can be generated by astrophysical
sources, such as compact binaries of white dwarfs and super-massive black hole binaries
(SMBHBs), or be of cosmological origin, such as quantum fluctuations of the vacuum
generated during inflation, the decay of cosmic string loops, and cosmological first-order
phase transitions (PTs).
The main characteristics of SGWBs are summarized in section 6.1. We then give a brief
overview of current and future GW detectors in section 6.2, mostly focusing on those that
are sufficiently sensitive to detect SGWBs, and describe how the sensitivity is evaluated in
section 6.3. Section 6.4 subsequently focuses on SGWBs generated in cosmological first-
order PTs, explaining the generation mechanism and how the corresponding spectrum is
calculated. An integral ingredient for the calculation is the finite-temperature effective
potential, which is shortly reviewed in section 6.5.
6.1. Characterization
SGWBs generated in the early Universe are usually assumed to be
isotropic. Since the early Universe was homogeneous and isotropic, as reflected in the
isotropy of the CMB, a cosmological SGWB should also share this property.
unpolarized. As the Standard Model (SM) exhibits parity violation in the weak interac-
tions only, there is no reason why a generic SGWB should be polarized. There are
however mechanisms that can produce a polarized SGWB.
stationary. In other words, the statistical properties of the SGWB only depend on time
differences, but not on absolute time. Comparing the age of the Universe of roughly
14Gyr [60] to the maximal realistic observation period or around 10 yr, this assump-
tion is almost certain to be true.
Gaussian. According to the central limit theorem, the sum of a large number of statisti-
cally independent random variables follows a Gaussian distribution. Some produc-
tion mechanisms may however result in a non-Gaussian SGWB.
Based on these assumptions, SGWBs are typically described in terms of their fractional
energy density (or density parameter) power spectrum, i.e. the energy density ρGW per
77
6. Stochastic Gravitational Wave Backgrounds
logarithmic frequency interval normalized to the critical energy density ρc = 3M2PH2,
where MP is the reduced Planck mass and H is the Hubble rate,
Ω ( ) ≡ 1 d ρGW(f)GW f
ρc d log
. (6.1)
f
As eq. (6.1) depends on the value of the Hubble rate due to the normalization to ρc, one
typically rewrites the Hubble rate as H = h × 100 kmMpc−1 s−1 and reports limits on
the quantity h2ΩGW.
6.2. Gravitational Wave Experiments
Over the past century, several methods for detecting GWs have been proposed and de-
veloped, ranging from simple resonant mass detectors to sophisticated interferometers in
space.
The first GW detector was the Weber bar, proposed in 1960 [286] and constructed in
1966 [287] by Joseph Weber. It consisted of a 1.5 t aluminium cylinder with a length of
∼ 150 cm and a resonance frequency of 1660Hz. The basic idea was that an incident GW
with a frequency close to the resonance frequency would induce a detectable change in
the length of the cylinder. Using several of these cylinders, Weber claimed a detection of
GWs in 1969 [288]. However, his claim could eventually not be supported.
Nowadays, we have two important classes of GW observatories at our disposal: GW
interferometers and pulsar timing arrays (PTAs). The former can be subdivided into
ground- and space-based observatories. Each of these types of experiments covers a
different frequency range.
GW interferometers detect GWs by measuring the GW-induced motion of free-falling
test masses via laser interferometry. The best-known, currently operating,1 ground-based
observatories are LIGO [289, 290] and Virgo [291, 292], which were recently joined on
February 25, 2020, by the Japanese underground detector KAGRA [293, 294]. They have
arm lengths of a few kilometers and are sensitive in the high-frequency range around
10Hz – 104Hz. Whereas the sensitivity of the current experiments is typically insuffi-
cient to detect the SGWB from cosmological PTs, the next generation of ground-based
interferometers, such as ET [295] or CE [296], will have a much higher sensitivity.
Space-based interferometers have the advantage that they overcome the limitation of
ground-based interferometers to frequencies f & 1Hz due to seismic noise. They can
also have much longer arms and may therefore probe lower frequencies. The first space-
based GW observatory is LISA [20], which will prospectively be launched in the mid
2030s [271].2 It consists of three satellites arranged in a regular triangle with a side-
length of 2.5× 109m, orbiting the sun roughly 20◦ behind Earth. LISA will be sensitive
in the frequency range 0.1mHz – 0.1Hz. Various successor experiments targeting the
deci-Hertz region have already been proposed, including BBO [299], DECIGO [300, 301],
1Actually, the third observational run of LIGO and Virgo was suspended on March 27, 2020, (about a
month before the scheduled end of the run) due to the COVID-19 pandemic.
2 There are also plans for LISA-like Chinese projects, Taiji [297] and TianQin [298], which aim at a
similar launch date.
78
6.3. Detection and Sensitivity
and its scaled-down version B-DECIGO [302], the former two consisting of four copies of
LISA (with shorter arms).
PTAs are a network of millisecond pulsars that detects GWs by monitoring the pul-
sars’ times of arrival. An incident GW would emerge as a correlated change in the timing
residuals of the pulsars. PTAs are sensitive to SGWBs in the low frequency range around
10−9Hz – 10−7Hz, set by the total time of observation. Currently operating observa-
tories are EPTA [303, 304], NANOGrav [305, 306], and PPTA [307], as well as their
combination, IPTA [308, 309]. A planned, next-generation observatory, SKA [310], will
prospectively start taking data in 2028 [311].
6.3. Detection and Sensitivity
The detectability of a given SGWB power spectrum h2ΩGW is assessed based on the
signal-to-noise ratio (SNR) ρ. We consider an SGWB detectable if the corresponding
SNR exceeds a threshold value, ρ > ρthr, where ρthr depends on the experiment under
consideration.
For a network of detectors, such as PTAs or the LIGO-Virgo network, the optimal-filter
cross-correlated SNR is given by
f∫max [ ]2
2 = 2 d h
2ΩGW(f)
ρ Tobs f 2Ω ( ) , (6.2)h eff f
fmin
where Tobs is the observation period, (fmin, fmax) is the frequency bandwidth of the
detectors, and h2Ωeff is the effective noise of the network (see appendix 6.A for details) in
fractional energy density. The auto-correlated SNR for a single detector can be obtained
by omitting the factor 2 in eq. (6.2). The effective noise of various GW observatories is
shown in fig. 6.1, along with the expected background from SMBHBs [306, 312]. Analytic
expressions for the noise spectra as well as the corresponding values of the SNR threshold
used in this dissertation can be found in ref. [2].
While eq. (6.2) allows us to calculate whether a given GW spectrum is detectable
or not, the effective noise curves h2Ωeff are not suitable for a graphical evaluation of
the detectability. To provide a simple way to visualize if a SGWB spectrum can be
detected by a given experiment, one usually employs so-called power-law integrated (PLI)
sensitivity curves [313]. To construct the PLI curves, the signal is assumed to follow a
simple power-law, ( )
2Ω 2 f
β
h GW = h Ωβ , (6.3)
fref
where fref is an arbitrary reference frequency. For a fixed value of the exponent β and
a given experiment, one can then invert eq. (6.2) to calculate the minimal detectable
amplitude h2Ωthrβ for which ρ = ρthr. The PLI sensitivity h2ΩPLI is obtained by taking
the envelope of the minimal detectable powe[r-law sp(ectra)ov]er all values of β,β
h2ΩPLI( ) = max 2Ωthr
f
f h β . (6.4)
β fref
79
6. Stochastic Gravitational Wave Backgrounds
10−3 10−3
10−6 10−6
10−9 HBB −9SM 10
10− −
5 years
12 12
5 years 10
10 years
10 years
20 years
10−15 −1520 years 10
10−18 10−18
10−9 10−6 10−3 100 103 10−9 10−6 10−3 100 103
f [Hz] f [Hz]
Figure 6.1: Energy density noise h2Ωeff Figure 6.2: PLI sensitivity h2ΩPLI
The region enclosed by the PLI curve is then interpreted as the region to which the exper-
iment is sensitive, i.e. a spectrum that reaches into the region above the PLI sensitivity
curve is detectable. Although this is strictly speaking only true for simple power-law
spectra of the form in eq. (6.3), SGWBs can typically be approximated by power-laws
at least over a large fraction of the experiment’s frequency band, so that this method is
applicable.
Figure 6.2 shows the PLI spectra corresponding to the noise curves in fig. 6.1 along
with some example spectra (see section 6.4.3 for details). Note that we assume that the
SMBHB background can be resolved and subtracted. Throughout this work, we will use
the SNR, eq. (6.2), to assess the detectability of an SGWB numerically, and the PLI
curves in eq. (6.4) for graphical representation of the sensitivity.
6.4. Cosmological Phase Transitions
Throughout most of its evolution, our Universe is very well described as a hot plasma of
particles in local thermal equilibrium at a temperature T .3 As the Universe keeps expand-
ing adiabatically, its temperature decreases at a rate determined by the energy content.
During this process it most probably went through at least two PTs: the electroweak PT
(EWPT), and the confining PT of quantum chromodynamics (QCD).
A cosmological PT is a transition between different vacua, often associated with the
breaking of a global or local symmetry. More generally, PTs can be defined as “a line
in the (T, µ)-plane across which the grand canonical free energy density f(T, µ) is non-
analytic” [314], where µ is the chemical potential. This non-analyticity across the line
separating the phases is typically related to the change in an order parameter given by the
vacuum expectation value (VEV) of an elementary or composite field, such as the Higgs’
VEV in the case of the EWPT, spontaneously breaking SU(2)L × U(1)Y → U(1)EM, or
the quark condensate of QCD confinement breaking chiral symmetry.
3As we will discuss in chapter 8, the various components of the plasma do not need to be in thermal
contact with one-another and may therefore have different temperatures.
80
h2Ωeff
EPTA
NANOG
SK raA v
DECIG
B O
B-D
BO ECIGO
ET
h2ΩPLI
NA EPTANOGra
S vKA
DECIG
B OBO B-DECIG
E OT
ISAL
LIS
A
6.4. Cosmological Phase Transitions
φ φ
(a) cross-over (b) first-order
Figure 6.3: Illustration of a cross-over (left) and first-order PT (right).
In quantum field theory (QFT), vacua are given by the minima of the effective po-
tential Veff(φ, T ), which is the potential of the order parameter4 〈φ〉 incorporating quan-
tum and thermal corrections. Further details on the effective potential shall be deferred
to section 6.5. At high temperatures, it is typically dominated by terms of the form
φ2T 2, which then restore spontaneously broken symmetries. As a consequence, theo-
ries that experience spontaneous symmetry breaking (SSB) have generically undergone a
symmetry-breaking PT in the early Universe, as understood by Kirzhnits and Linde in
1972 [315].
We commonly distinguish between two types of PTs: first-order and higher-order tran-
sitions. Formally, a first-order PT is a PT in which the derivative of the free energy
density with respect to a thermodynamic parameter, e.g. temperature, is discontinuous.
Similarly, higher-order PTs have discontinuities in higher-order derivatives (and are con-
tinuous in the lower-order ones). Finally, in a cross-over all derivatives are continuous.5
Since only first-order PTs can generate GWs, we will henceforth only distinguish between
first-order and cross-over transitions, including all higher-order transitions in the latter
category. In the SM, both, the electroweak (EW) and QCD PT, are cross-overs. Their
nature may however change if new physics contributions are taken into account.
The important distinction between first-order and cross-over PTs lies in the way the
order parameter, i.e. the VEV of φ, changes. This is illustrated in fig. 6.3, where the
effective potential Veff is shown as a function of φ. The dashed green lines depict the
potential at high temperatures, which has a single minimum at the origin, whereas the
solid blue lines are the low-temperature potential.
In a cross-over (fig. 6.3a), as the Universe cools down, the high-temperature minimum
turns into a maximum at low temperatures and the potential develops a minimum at
non-vanishing field values. The field φ can then smoothly “roll down” the potential to
transition from the high- to the low-temperature vacuum. In the case of a first-order
4For simplicity, we here assume a single order parameter and vanishing chemical potential.
5Strictly speaking, a cross-over does therefore not correspond to a PT according to the definition above.
81
Veff
Veff
6. Stochastic Gravitational Wave Backgrounds
〈φ〉 = v 〈φ〉 = 0
〈φ〉 = v
〈φ〉 = v Figure 6.4: Illustration of a
first-order PT via the nucle-
ation of bubbles of the true vac-
uum inside the false-vacuum
phase.
PT (fig. 6.3b) on the other hand, the high-temperature minimum still persists at low
temperatures as a local minimum, but the global minimum again lies at non-vanishing
field values. However, the two minima are now separated by a potential barrier, such that
the field cannot smoothly evolve from the false (local) vacuum to the true (global) one.
Instead, in a first-order PT the field has to thermally fluctuate over or quantum tunnel
through the barrier.
It is this tunneling process through which first-order PTs generate a SGWB. We there-
fore provide more details on the process in section 6.4.1. Subsequently, section 6.4.2
introduces the parameters used to characterize the PTs. Finally, the corresponding gen-
eration mechanisms for the SGWB is explained in section 6.4.3.
6.4.1. Bubble Nucleation
A cosmological first-order PT proceeds through the nucleation of bubbles of the true
vacuum in the sea of the false vacuum. At high temperatures, the Universe, depicted as
a box in fig. 6.4, is in the false-vacuum phase, which we here assume to be characterized
by a vanishing VEV, 〈φ〉 = 0. As the Universe cools down, a second minimum, the
true vacuum, starts to form at 〈φ〉 = v. When the true vacuum becomes energetically
favorable, the field tunnels at random points of the Universe, forming spherical bubbles
inside of which the fields is in the true vacuum, shown in gray in fig. 6.4.
Driven by the energy release from the potential difference in the tunneling, the bubbles
subsequently expand, provided that the energy gain exceeds the surface energy of the
bubbles (otherwise they collapse). The nucleation of the expanding vacuum bubbles then
competes against the expansion of the Universe. If the bubbles are nucleated sufficiently
fast to overcome the Hubble expansion, the bubbles will collide and merge, and eventually
fill the whole Universe with the true vacuum.
82
6.4. Cosmological Phase Transitions
The bubble nucleation (rate pe)r unit volume is given by [316–320] −4 S 2E,4R0( 2 ) exp((−SE,4)) for quantum tunneling,Γ =  π 3 (6.5)4 SE,3 2 exp −SE,3T 2 for thermal fluctuations,πT T
where R0 is the radius of the nucleated bubble, and the d-dimensional Euclidean action
SE,d is given by ∫∞ [1 (d )2 ]φb
SE,d = Ωd dr 2 d + V (φb) , (6.6)r
0
evaluated at the O(d) symmetric bounce solution φb that satisfies the differential equation
d2 φb + d− 1 dφbd 2 d = V
′(φb) . (6.7)
r r r
Here, Ωd is the solid angle in d dimensions (Ω3 = 4π and Ω4 = 2π2), V is given by the
effective potential at finite or zero temperature, respectively, shifted such that V = 0 in
the false vacuum, and V ′ is its derivative with respect to φ.
Let us now define two characteristic temperatures of first-order PTs: the critical
temperature Tc and the nucleation temperature Tn. The critical temperature is sim-
ply the temperature at which the true and the false vacuum become degenerate, i.e.
Veff(φt, Tc) = Veff(φf , Tc), where φt and φf are the field values of the true and false vac-
uum, respectively. This is the temperature below which it is in principle possible to
nucleate bubbles of the true vacuum. However, as discussed above, the transition does
not occur unless bubbles are nucleated sufficiently fast to overcome the Hubble expansion.
This defines the nucleation temperature, at which the probability that on average one
bubble has been nucleated per Horizon volume is of order one.
The nucleation temperature can be roughly estimated by taking Γ(Tn)/H4(Tn) ∼ 1.
Assuming thermal tunneling with Γ ∼ T 4 exp (−SE,3/T ) and a radiation dominated Uni-
verse with Hubble rate and energy density given by
2( ) = ρ
2
rad(T ) π
H T 3 2 , and ρ
4
rad(T ) = 30 g?(T )T , (6.8)MP
where g? is the effective number of relati(vistic degr)ees of fre(edom (D)OFs), we obtain
SE,3(Tn) ∼ 146− 4 log Tn − 2 log g?(Tn)
Tn 100GeV 100
. (6.9)
Unless specified otherwise, we therefore use the condition SE,3(T )/T = 140 to determine
the nucleation temperature Tn.
We can further define the temperature at which the transition is completed as the
temperature at which an order one fraction of the Universe has transitioned to the true
vacuum. However, in this dissertation we are only going to deal with PTs with no signif-
icant super-cooling (i.e. the Universe is not going to be dominated by vacuum energy),
transitioning fast enough to assume an instant transition and ignore the change of tem-
perature during the PT.
83
6. Stochastic Gravitational Wave Backgrounds
6.4.2. Phase Transition Parameters
A cosmological first-order PT can be characterized by four parameters: the temperature
T∗ at which the transition occurs, the energy budget α, the inverse duration β, and
the wall velocity vw at which the bubble walls move. Here, we will only consider fast
transitions for which we can take T∗ = Tn. In the remainder of this section, all functions
of T should be understood to be evaluated at T = T∗.
The energy budget of a PT is defined by the energy released in the tunneling, given
by the latent heat E ,6 normalized to the energy density in radiation at the time of the
transition, ( )
α = E = 1 ∆ − ∂∆VeffVeff T , (6.10)
ρrad ρrad ∂ T
where ∆Veff ≡ Veff(φf (T ), T )− Veff(φt(T ), T ) is the potential difference between the false
and true vacuum at temperature T . This can be interpreted as the strength of the PT.
The inverse duration of the transition is approximately given by the relative change of
the nucleation rate, β = Γ̇/Γ. This parameter is usually normalized to the Hubble rate
at the time of the transition. Using H ≡ ȧ/a, and assuming an adiabatically expanding
Universe with s(T )a3 ∼ T 3a3 = const, we obtain
β = 1 d log Γd =
d log Γ
d log = −
d log Γ d SE,3
H H t a d log = T d , (6.11)T T T
where we neglected the time/temperature dependence of the prefactor of the thermal
tunneling rate in eq. (6.5).
While the quantities discussed above only depend on equilibrium properties and can
be directly calculated from the effective potential or the tunneling rate, the velocity vw
at which the bubble walls move is a more complicated beast. It depends on the friction
exerted on the bubble walls by the particles of the plasma that gain a mass in the
transition. The wall velocity therefore depends on microscopic properties of the plasma
and out-of-equilibrium dynamics [321–323]. However, the generation of an observable
SGWB requires the PT to be strongly first-order, which typically comes along with
large wall velocities. Unless specified otherwise, we will therefore simply assume vw ' 1
throughout this thesis.
Although we can be ignorant about the exact value of vw, the wall dynamics have an
important impact on the way GWs are generated in the transition, as it determines the
amount of latent heat that is transferred into bulk motion of the primordial plasma. One
therefore distinguishes between two regimes: runaway and non-runaway [324, 325]. In the
runaway regime, the friction exerted by the plasma is insufficient to prevent the bubble
walls from accelerating perpetually. As a consequence, only little energy is transferred to
the plasma and most of the energy release goes into the acceleration of the bubbles. If,
on the other hand, the friction is sufficiently strong, the bubbles reach a terminal boost
factor. The latent heat is then efficiently converted into plasma motion.
6Altern(atively, one often)defines α via the change in the trace of the energy-momentum tensor,
α ≡ ∆V − 1 ∂eff 4 ∆Veff /ρrad. For strong PTs, the ∆Veff part dominates and both definitions co-∂ T
incide.
84
6.4. Cosmological Phase Transitions
At leading order (LO), the latent heat required to enter a runaway regime (normalized
to the radiation energy densit(y) can be estimated as [321, 324] )
= 1
∑ T 2 2 ∑ T 2α n ∆m + n ∆m2∞ i 24 i i 48 i , (6.12)ρrad bosons fermions
where n and ∆m2i i are the number of DOFs and squared mass difference of species i
between the two phases, respectively. Transitions with α > α∞ run away, whereas the
ones with α < α∞ do not. However, it has been shown that in the case that the particles
gaining a mass are coupled to gauge bosons, next-to-leading order (NLO) corrections due
to transition radiation from the particles crossing the wall produce a friction proportional
to the wall boost factor γ [325]. Therefore, a runaway is prevented in this case. We can,
however, still treat vw ' 1 [325].
6.4.3. Generation of a Stochastic Gravitational Wave Background
The possibility of generating GWs in a cosmological first-order PT was realized in the
mid 1980’s by Witten [326] and Hogan [327]. The generation of the GWs occurs via
three mechanisms: collisions of the vacuum bubbles, collisions of sound waves in the
primordial plasma of the Universe, and turbulence. Each of these mechanisms results in
a contribution to the GW power spectrum,
h2ΩGW(f) = h2Ω 2 2φ(f) + h Ωsw(f) + h Ωturb(f) . (6.13)
Recall that a first-order PT proceeds via the nucleation of bubbles of the true vacuum
inside the false vacuum. The energy freed in the tunneling process can, however, not
be released into GWs due to the spherical symmetry of the bubbles. According to the
famous quadrupole formula, the generation of GWs requires a quadrupole moment that
varies in time. The latent heat can therefore only drive the expansion of the bubbles.
However, as soon as two of these bubbles collide, the spherical symmetry is broken and
GWs are generated. This yields the first contribution to the spectrum, the scalar field
contribution h2Ωφ. It is commonly calculated using the envelope approximation [328],
which assumes that most of the energy is stored in a thin shell around the bubble walls
and only considers the envelope of the collided bubbles as illustrated in black in fig. 6.5.
For the other two contributions we need to take into account that the transition happens
in the early Universe, where a thermal plasma of particles is present. If the scalar field (or
operator) that acquires a VEV couples to this plasma, the expanding bubbles will induce
acoustic waves in it. These sound waves also form bubbles, that expand at the speed
of sound, depicted in red in fig. 6.5. The sound bubbles then emit GWs upon collision,
producing the sound wave contribution h2Ωsw. As the collisions of acoustic waves provide
a much longer lasting source than the initial vacuum bubble collisions, this contribution
typically dominates the spectrum if present.
Finally, the expanding bubbles also induce vortical motions and eddies in the fluid,
depicted in blue in fig. 6.5. These give rise to GWs generated from magnetohydrody-
namic (MHD) turbulence. The corresponding GW spectrum is denoted as h2Ωturb. This
85
6. Stochastic Gravitational Wave Backgrounds
v 〈φ〉 = 0w
vs vw
vs
vw vs 〈 〉 vs vwφ = v
v Figure 6.5: Illustration of thes
v v
generation of GWs in a first-
s w
v order PT from vacuum bubblew collisions (black), sound waves
(red), and turbulences (blue).
contribution is typically negligible compared to the sound wave contribution, unless the
sound waves last less than a Hubble time.
The spectra for the respective contributions are obtained from analytic arguments and
numerical simulations. In terms of the parameters introduced in section 6.4.2, we will
use the following expressions throughout this work [329–332].
0.11v3 ( )2 ( )2Ω ( ) = R w H κφ α 2h φ f 0.42 + v2w ( β ) (1 + ) Sφ(f) , (6.14a)α 2
h2Ωsw(f) = R 0 159
H κsw α
. vw ( β ) (1 + ) Ssw(f) , (6.14b)α 3
2Ω ( ) = R 20 1 H κturb α
2
h turb f . vw 1 + Sturb(f) . (6.14c)β α
The spectral shapes S are given by
S ( ) = 3.8 (f/fφ)
2.8
φ f 1 + 2 8 ( , (6.15a)( ). [f/f )3.8φ3 7 ] 7f 2Ssw(f) = (fsw ) 4[+ 3 (f/fsw)2 ] , (6.15b)3 113
Sturb(f) =
f 1 1
. (6.15c)
fturb 1 + (f/fturb)2 1 + 8πf/h∗
The corresponding peak frequencies at production are roughly set by the transition
time scale β. After red-shifting to today they become
= 0.62
( ) ( ) ( )
h∗ β √2h∗ β 3.5h∗ βfφ 1.8 , f = , f = . (6.16)− 0 1 sw turb. vw + v2w H 3 vw H 2 vw H
86
6.4. Cosmological Phase Transitions
The Hubble rate at production red-shifted to today, h∗, and the red-shift factor R for
the amplitude (also accounting the change of the critical energy density ρc = 3M2 2PH )
are
( ) 1
a g0 3∗ −32 √ ( )( ) 1= = 3.2× 10 ?S ∗ = 16.5µHz T∗ g∗ 6h∗ H ?∗ ∗ g? T∗ 100GeV 100 , (6.17a)(a0 ) ( ) g?S ( ) 44 ( ) ( ) 1
R = a∗ H
2 0 3 ∗ ∗ −
∗ 3= 2.473× 10−5 g?S g? = 1.67× 10−5 g? , (6.17b)
a0 H100 g∗?S g
0
? 100
where quantities with index ‘0’ (‘∗’) are evaluated today (at emission), g? and g?S are
the relativistic and entropic effective DOFs, respectively, and H100 = 100 kmMpc−1 s−1.
We have inserted eq. (6.8) and assumed conservation of co-moving entropy,
a3(T )s(T ) ∝ a3(T )g (T )T 3?S = const, as well as T0 = 2.35× 10−13GeV [68, 333], g0? = 2
and g0?S = 3.909 [334] for the photon temperature and effective DOFs today. In the last
step we have assumed that the number of entropic and radiation DOFs at the time of the
transition do not differ, i.e. g∗ = g∗? ?S .
Finally, we need the efficiency factors κφ, κsw, and κturb for the conversion of latent heat
into acceleration of the bubble walls, bulk motion of the plasma, and MHD turbulence,
respectively. Whether the GW spectrum is dominated by the vacuum bubble collisions
or the plasma contributions depends on the behavior of the bubble walls. If the coupling
to the plasma is sufficiently strong to prevent runaway, the contribution h2Ωφ from the
collisions of vacuum bubbles can be neglected and only the plasma contributions are
relevant. In this case, we can set κφ = 0, and the efficiency factor for the sound wave
contribution is [321]
κsw = κ(α) =
α √
0.73 + 0.083 α+ , (6.18)α
provided that vw ∼ 1. If, on the other hand, the bubbles enter the runaway regime, only
a fraction α∞/α of the latent heat can be converted into bulk motion of the plasma,
with α∞ given by eq. (6.12). The surplus energy is then goes into the acceleration of
the bubble walls. The efficiency factors for the vacuum bubble and sound wave collisions
then become
= 1− α∞κφ , and
α∞
κsw = κ(α∞) . (6.19)
α α
In both cases, the amount of latent heat converted to turbulence is only a fraction of the
sound wave efficiency, κturb = εturbκsw, where εturb ' 5% – 10%.
The reader should be aware that the expressions for the GW spectrum quoted above
reflect the state of the art around the time the works [1] and [2] were published. As
the field is currently developing quickly, further progress has been achieved, including
more recent simulations of the scalar field and sound wave contributions [335, 336] and
improved perceptions regarding the energy budget of the transition and the life time of
sound waves in the plasma [320, 337–339] .
87
6. Stochastic Gravitational Wave Backgrounds
6.5. The Effective Potential
When being introduced to SSB, one typically studies the vacuum of a theory by minimiz-
ing the potential (i.e. the non-derivative part of the negative Lagrangian) with respect
to the fields, assuming that the vacuum states of the fields are constant in space-time.
While this is correct in classical field theory, the quantum nature of the fields in a QFT
gives rise to quantum corrections, which need to be taken into account. In the early
Universe, the presence of a thermal bath of particles further requires the incorporation
of thermal effects. This is done by the effective potential Veff, which is the potential of
the fields including quantum and thermal corrections. As should be apparent from the
previous section, the effective potential plays an integral role in the determination of the
parameters characterizing a cosmological PT and the corresponding GW spectrum.
In the following, the most important formulae for the calculation of the effective po-
tential will be summarized. Further details, including a formal definition of the effective
potential, are included in appendix 6.B.
At the one-loop level, the finite-temperature effective potential including daisy-resum-
mation7 is given by
Veff(φ, T ) = Vtree(φ) + VCW(φ) + ∆Vct(φ) + VT(φ, T ) + Vring(φ, T ) , (6.20)
where φ denotes the classical background fields (potentially more than one), Vtree is the
tree-level potential, VCW are the one-loop zero-temperature or Coleman-Weinberg [340]
corrections, ∆Vct includes counter-terms, VT are the one-loop thermal corrections, and
Vring resums the leading contributions from higher-loop diagrams (ring diagrams).
The dimensionally regularized and renormalized Coleman-Weinberg [340] corrections
in Landau gauge evaluate to
∑ [ ( ) ]
( ) = ηi n
2
i
V φ 4CW 64 2mi (
m
φ) log i (φ)2 − Ci , (6.21)π µ
i R
where the sum runs over all particle species that couple to the fields φ, m2i (φ) and
ni are the field-dependent squared masses and DOFs8 of species i, ηi = +1 (−1) for
bosons (fermions), and Ci = 3/2 (5/6) for scalars and fermions (gauge bosons). The
renormalization scale µR is typically chosen to be the magnitude of the zero-temperature
VEV of φ. In eq. (6.21), ultraviolet (UV) divergences are canceled using MS counter-
terms. If further renormalization conditions are imposed, we need to include the finite
part of the counter-terms (or the difference to the MS ones, to be more precise) represented
by ∆Vct(φ) in eq. (6.20).
The finite-temperature one-loop corrections are given by [341]∑ 4 ∫∞  √ η 2i ni T m (φ)
VT(φ, T ) = 22 2 dxx log
1− ηi exp− x2 + i 2 , (6.22)π T
i 0
7I.e. resumming the leading higher-loop corrections, see appendix 6.B.3.
8 Note that, as we are working in Landau gauge, both, massive gauge bosons and the corresponding
would-be Goldstone bosons, contribute with three and one polarization DOF, respectively.
88
6.5. The Effective Potential
which can be expanded for high temperatures as [341]∑ ( ) 3 [ ]2 2 2 ∑ 2
( ) = 1 m (φ) 1 m (φ) 1 m (φ)VT φ, T T 4 n  i − ii  4 i24 2 12 2 −T ni 2 +. . . , (6.23)bosons T π T fermions 48 T
where field-independent and higher-order terms in m2/T 2 have been dropped. As the
squared masses typically grow with the square of the fields, m2(φ) ∼ φ2, the dominant
field-dependent part at high temperatures then goes like T 2φ2 and therefore restores
broken symmetries in the early Universe.
In addition to the thermal one-loop corrections, we also take into account the so-called
ring or daisy contributions. These arise from the inclusion of the leading higher-loop cor-
rections by resumming the high-temperature thermal mass corrections to the Matsubara
zero-mode propagator in the one-loop po[tential. The corresponding corre]ctions are [342]∑ ( ) 3 ( ) 3
Vring(φ, T ) = −
T 2
12 ni m (φ) + Π(T )
2 − m2(φ) 2 , (6.24)
π i ibosons
where Π(T ) are the thermal Debye masses evaluated in the high-temperature limit, and
(m2(φ) + Π(T ))i is the i-th eigenvalue of the full (tree-level + thermal) mass matrix [343].
Note that for gauge bosons only the longitudinal mode receives thermal mass corrections,
whereas the Debye mass of the transverse modes vanishes.
89
Appendix of Chapter 6
Appendix 6.A. Signal-to-Noise Ratio
In this appendix we provide a brief outline of the derivation of the SNR eq. (6.2) based
on the reviews [282–284] as well as ref. [313]. To relate the SGWB power spectrum to
its response in a detector, we here start from the expression for the corresponding metric
perturbation hab, and then derive the pairwise-correlated optimal-filter SNR for a network
of detectors.
The plane wave expansion of a GW in transverse traceless (TT) gauge is given by
∫∞ ∫ ∑
hab(t, ~x) = df d2k̂ h (f, k̂) A (k̂) e2πif(t−k̂·~x/c)A ab , (6.25)
−∞ S2 A=+,×
where k̂ is the unit vector in the direction of propagation, Aab are the polarization tensors
for the + and × polarization, an〈d hA(f, k̂) are the corresponding Fourier modes. In thecase of a SGWB satisfying the assumption〉s from section 6.1, the latter are random fields
whose ensemble averages satisfy hA(f, k̂) = 0 and9
〈 〉 1
h∗A(f, k̂)hA′(f ′, k̂′) = 16 δ(f − f
′) δ 2AA′ δ (k̂ − k̂′)Sh(f) , (6.26)
π
where Sh(f) is the one-sided strain power-spectral density (PSD). Then, using that hab
is real and therefore h∗A(f, k̂) = hA(−f, k̂), as well as  abAab(k̂)A′(k̂) = 2δAA′ ,
〈 〉 ∫∞ ∫∞
h (t, ~x)habab (t, ~x) = df Sh(f) = 2 d(log f)h2c(f) , (6.27)
−∞ √ −∞
defining the characteristic strain amplitude hc(f) = fSh(f). Finally, the energy density
of a GW is [344]
M2 〈 〉 2 ∫∞
ρ PGW = 4 ḣab(
M
t, ~x)ḣab(t, ~x) = P d(log f) (2πf)2 h22 c(f) , (6.28)
−∞
and therefore, using eq. (6.1), we can relate the GW power spectrum to the characteristic
strain and the strain PSD by
2 2
ΩGW(f) =
2π 2 2
3 2 f hc(f) =
2π
f33 2 Sh(f) . (6.29)H H
9Regarding the definitions of Sh(f) and hc(f) we here follow ref. [313].
90
Appendix 6.A. Signal-to-Noise Ratio
Now consider the response h(t) of a detector at position ~x to an incoming GW. The
detector response may for instance be the GW-induced phase difference in an interfer-
ometer or the timing residuals in a PTA. For weak signals it is linear in the perturbation
and therefore given by ∫∞ ∫
h(t) = dt′ d3x′Rab(t′, ~x′)h ′ab(t− t , ~x− ~x′) , (6.30)
−∞
where Rab(t, ~x) is the impulse response of the detector (see ref. [284] for further details).
For a plane wave we correspondin∫gly obta∑in
h(f) = d2k̂ RA(f, k̂)hA(f, k̂) (6.31)
S2 A=+,×
in frequency space, with10 RA(f, k̂) = A (k̂)Rab(f, k̂) e−2πif k̂·~x/cab describing the detector
response to a sinusodial plane GW from direction k̂ with frequency f and polarization
A. The output d of the detector is then composed of the GW signal h and the respective
noise n, i.e. d(f) = h(f) + n(f).
Let us now suppose that we have a time series of measurements di(t) from a network
of detectors located at positions ~xi. Furthermore, assume that the noise in each detector
is stationary (i.e. its variance is time-independent) and Gaussian, as well as statistically
independent of the noise in the other detectors and the GW signal. In the frequency
domain, it is then charac〈terized by th〉e mean 〈ni(f)〉 = 0 and the (co-)variances
( ) 1ni f n∗ ′j (f ) = 2 δ(f − f
′) δij Pni(f) , (6.32)
where Pni(f) is the noise PSD in detector i. For the signal on the other hand, eqs. (6.26)
and (6.31) yield 〈 〉 1
h (f)h∗(f ′) = δ(f − f ′∫ ∑ i j 2 ) Γij Sh(f) , (6.33)where Γij = d2k̂ A A∗ARi (f, k̂)Rj (f, k̂) is the so-called overlap reduction function.
We now take advantage of the fact that the GW signal is correlated between the de-
tectors, whereas the noise is not. Therefore, the signal can be extracted by considering
correlations between∫the detector outputs. We thus define the pair-wise correlated de-
tector output D = ∞ij −∞df di(f)d∗j (f)Qij(f), where Qij(f) is a filter function which we
choose such that is maximizes the SNR. The corresponding SNR is given by ρij = µσ with
mean µ = 〈Sij〉 and variance σ2 = 〈S2ij〉− 〈S 2ij〉 . For a weak signal we can neglect the hi
in the variance a[nd we obtain (details of the ]calculation can be found in refs. [282–284])∫∞ 2
df Qij(f) Γij(|f |)Sh(|f |) ∫∞
2 = −∫∞ = 2 d Γij(f)S2h(f)ρij Tobs ∞ Tobs f ( ) ( , (6.34)d | ( P f P f)f Qij f)|2 P (|f |)P (|f |) ni njni nj 0
−∞
10We here adapt the definition of ref. [284] for RA(f, k̂), which includes the exponential factor. Rab(f, k̂)
denotes the Fourier transform of Rab(t, ~x).
91
6. Stochastic Gravitational Wave Backgrounds
where we maximized the SNR setting ( ) = Γij(|f |)Sh(|f |)Qij f P (|f |)P (|f |) in the second step. Theni nj
total s∑quared SNR of the network is simply obtained by summing over all detector pairs,
ρ2 = 2i>j ρij . Rewriting Sh(f) in terms of the fractional energy density h2ΩGW using
eq. (6.29), and defining the effective noise of the detector network
∑ − 12 2 Γ ( )  2Ωeff(f) = π 3 ij f3 2 f ( ) ( ) , (6.35)H P f P f
i>j ni nj
we arrive at the expression eq. (6.2).
In the case of a single detector such as LISA, a cross-correlated analysis is of course
not possible. For LISA one can however form combinations of the data from its six laser
links, in which the signal is highly suppressed and the noise can be measured, employing
a technique called time delay interferometry (TDI) [345, 346]. The noise can then be
subtracted, and the corresponding auto-correlated SNR is given by [313]
f∫max [ ]2
2 = d h ΩGW(
2
f)
ρ Tobs f
h2Ω ( ) . (6.36)n f
fmin
2π2f3Here, Ω (f) = Pn(f)n 3 2R( ) , where Pn(f) is again the detector noise PSD andR(f) = Γii(f)H f
is the polarization- and sky-averaged detector response.
Appendix 6.B. Further Details on the Effective Potential
This appendix provides further details on the effective potential. Appendix 6.B.1 gives
a formal definition, while appendices 6.B.2 and 6.B.3 sketch the derivation of the zero-
and finite-temperature one-loop corrections.
6.B.1. Formal Definition
The effective potential for QFTs was first introduced by Heisenberg and Euler [347], as
well as Schwinger [348], and applied to SSB by Jona-Lasinio [349]. It is the generating
functional for one-particle irreducible (1PI) Green’s functions at zero-momentum [350].
The generating functional W (J) for connected Green’s functions11 is defined through
the path integral via ∫
Z(J) = 〈0+|0−〉J = Dφ exp [iS(φ) + iJ ·φ] = exp [iW (J)] , (6.37)
where |0±〉 denote the va∫cuum state at t = ±∞, φ and J are fields and sources, and we
use the notation J ·φ = d4xφ(x)J(x). We here consider the case of a single field only,
following refs. [340, 350, 351], but the generalization is straight forward, see ref. [352].
11 I.e. when expanding W (J) in powers of J in the functional sense, the corresponding expansion co-
efficients are the sum of all connected Feynman diagrams with the respective number and types of
external legs.
92
Appendix 6.B. Further Details on the Effective Potential
Now, classical fields are given by
+
( ) = 〈0 |φ(x)|0
−〉J δW (J)
φc x 〈0+|0− = (6.38)〉J δJ(x)
and we can define the effective action Γ(φc) as the Legendre transform of W (J), i.e.
Γ(φc) = W (J)− J ·φc, which generates 1PI Green’s functions Γ(n)(x1, . . . , xn),
∑∞
Γ( ) = 1
∫
φc ! dx1 . . . dx Γ
(n)
n (x1, . . . , xn)φc(x1) . . . φc(xn)
n∑=0 n∞ ∫ ∏n [ ] (6.39)
= d pi(2 )4 φ̃c(−pi) (2π)
4δ(4)(p1 + · · ·+ p ) Γ(n)n (p1, . . . pn) ,
n=0 πi=1 ∫
where we changed to Fourier modes φ̃c(p) = d4x e−ip·x φc(x) in the second line. Let
us now consider a vacuum state that is constant in space-time, φc(x) = φ0. Then, the
effective potential can be defined via
∑∞ ∫
Γ(φ0) = (2π)4 δ(4)(0) Γ(n)(0, . . . , 0)φn0 = − d4xVeff(φ0) . (6.40)
∫ n=0
Using that d4x = (2π)4 δ(4)(0) is just the space-time volume factor, we can identify the
effective potential as the sum of all 1PI Green’s functions at zero-momentum.
Note that, when calculating the effective potential, we cannot simply expand in powers
of couplings, as diagrams with internal massless particles become more infrared (IR)
divergent when the number of external legs (and thereby the power of the couplings) is
increased [352]. One instead typically expands in the number of loops [340, 352], which
is equivalent to expanding in powers of ~ and provides the additional advantage that this
expansions is invariant under shifts of the fields φ → φ + φ̄. Using this expansion, the
effective potential can be represented diagrammatically as
+ + + +O(~3) . (6.41)
In finite temperature quantum field theory (FTQFT) the effective potential can be
defined in the same way, going from Minkowski to Euclidean time (where w∫e now de-∫fine Z(∫J) = exp [−W (J)], with ‘−’ instead of ‘i’) and replacing the integrals d4xE byβ 3
0 dτ d x with β = 1/T [314, 351].
Note that a simpler derivation of the effective potential was presented in ref. [353],
where the fields are shifted by a their zero-momentum component φ0 (i.e. a constant
background field), φ(x) = φ0 + φ′(∫x), and Veff is[ defi∫ned via the ]generating functional at
vanishing external source, Z(0) = ∞ dφ exp −i d4−∞ 0 xVeff(φ0) , such that[ ∫ ] ∫
exp −i d4
[ ]
xV ′eff(φ0) = Dφ exp iS(φ0 + φ′) . (6.42)
93
6. Stochastic Gravitational Wave Backgrounds
6.B.2. The One-Loop Effective Potential at Zero-Temperature
Let us now very briefly recapitulate the derivation of the zero-temperature one-loop effec-
tive potential eq. (6.21). It was first calculated diagrammatically by Coleman and Wein-
berg in 1972 [340, 352] and is hence often referred to as Coleman-Weinberg potential.
A computation based on functional methods was presented in by 1974 by Jackiw [350].
We here illustrate the calculation in λφ4 theory following the diagrammatic approach,
considering a real scalar field φ with the tree-level potential
2
V (φ) = m0 2 + λ2 φ 4!φ
4 . (6.43)
Further details of the calculation can be found in refs. [340, 351, 352].
As mentioned above, the effective potential is calculated expanding in the number of
loops, with an infinite series of diagrams contributing at each loop-level. For the potential
eq. (6.43), the Feynman diagrams contributing at one-loop are
+ + + + . . . , (6.44)
where there external legs correspond to classical background fields with zero momentum.
As the theory only features a quartic interaction, all diagrams have an even number of
external legs.
Each diagram in eq. (6.44) contributes to the one-loop potential with a term given
by the respective amplitudemultiplied by φ2n/(2n)!, where 2n is the number of external
legs. The corresponding diagram then has[n internal v]ertices and propagators, so that
the expression for the diagram is (−iλ)n× i/(p2 −m2 n0) , where we leave the +iε in the
propagator implicit. We further need to take into account a combinatoric factor from
the different ways of assigning momenta to the external lines. There are (2n)! ways to
assign the momenta. Interchanging the two momenta at any of the n vertices however
does not change the diagram, so that we need to multiply by a symmetry factor 1/2n.
Furthermore, momentum assignments related by rotations or reflections of the diagram
are equivalent, giving another symmetry factor 1/(2n). Summing over all diagrams and
using dimensional regularizatio∑n we the∫n obtain∞ [ ]dDp i λ/2φ2 n
V D−4CW(φ) = µR
=1 ∫ (2π)D 2n[ p2 −m2 + iεn1 D ] (6.45)= µD−4 d pE log p22 R (2 ) E +m2(φ)− iε ,π D
where µR is the renormalization scale and D is the number of dimensions. For the
second equality, we have performed a Wick rotation, identified the infinite sum as the
series representation of the logarithm, log(1 + x) = ∑∞n=1(−1)n−1xn/n, and dropped
φ-independent terms. Performing the in[tegra(tion we)then obtain] [351]
m4(φ) m2(φ) 3
VCW(φ) = 64 2 log 2 − 2 −∆ , (6.46)π µR
94
Appendix 6.B. Further Details on the Effective Potential
where ∆ = 2 − γ + log 4π and we defined m2(φ) = m24−D E 0 +λφ2/2. In the MS renormal-
ization scheme, the term proportional to ∆ is subtracted, giving the result in eq. (6.21)
for a single scalar DOF (ni = 1).
If N scalars with identical mass are considered, we further recover the factor ni = N
from the number of DOFs in eq. (6.21). A similar calculation can be carried out for
fermions propagating in the loop. The Feynman rule for fermion loops then gives an
additional over-all negative sign, so that the ηi factor in eq. (6.21) arises. The Dirac
trace further yields a factor corresponding to the number of spin DOFs, i.e. ni = 4 (2) for
Dirac (Weyl) fermions. The equivalent calculation for gauge bosons in the loop is typically
carried out in Landau gauge (ξ = 0) as the contributing diagrams are then simply given
by the ones in eq. (6.44) with internal lines replaced by gauge boson propagators, and
proceeds in analogy to the scalar case. In other gauges, additional diagrams with internal
ghost fields would contribute.12 The contraction of Lorentz indices then gives a factor
Tr( µν pµpνη − 2 ) = D − 1 from the numerator of the gauge boson propagator, which canp
be rewritten as D − 1 = 3(1 + (D − 4)/3). Taking D → 4, we see that each gauge boson
contributes with ni = 3 DOFs. The trace further combines with the 2/(4−D) divergence
in ∆, and we obtain that for gauge bosons Ci = 5/6 in eq. (6.21). Note that, since we are
working Landau gauge, the would-be Nambu-Goldstone bosons also contribute as scalar
DOFs.
While we cured UV divergences using dimensional regularization, eq. (6.21) may still
suffer from IR singularities in its second derivative generated by fields that become mass-
less for certain values of φ. These divergences are particularly problematic when imposing
renormalization conditions on the second derivative of the potential in the broken vac-
uum, where the Goldstone masses vanish. The Goldstone divergences can be treated in
different ways, e.g. including the one-loop zero-momentum self-energy of the Goldstone
bosons in their squared masses in eq. (6.21) [355], or including the self-energy of the
background field, which suffers from the same divergence, in the renormalization condi-
tion [356]. We follow the latter approach in chapter 7, whereas we simply numerically
regularize the singularity in chapter 8.
6.B.3. The One-Loop Effective Potential at Finite-Temperature
The thermal one-loop effective potential can be obtained in the same manner as the
zero-temperature one, evaluating the Feynman diagrams in FTQFT. An introduction
to FTQFT can for instance be found in refs. [314, 351, 357] and is clearly beyond the
scope of this work. For the following it shall be sufficient to state that the path integral
representation of the partition function in the imaginary time formalism of FTQFT can
be obtained from its zero-temperature QFT equivalent via the following steps [314].
1. Perform a wick rotation t→ −iτ .
2. Introduce the Euclidean Lagrangian LE = −L(t=−iτ).
3. Restrict τ to the interval (0, β), where β = 1/T .
12See e.g. chapter 21 of ref. [21] for gauge-fixing in spontaneously broken gauge theories. The effective
potential of scalar electrodynamics in arbitrary Rξ gauges is for instance discussed in ref. [354].
95
6. Stochastic Gravitational Wave Backgrounds
4. Require periodicity (anti-periodicity) in τ for bosonic (fermionic) fields,
i.e. φ(β) = φ(0) (ψ(β) = −ψ(0)).
We then obtain a path integral representation similar to eq. (6.37), but with iS replaced by
−SE and imposing the r∫espective boundary conditions on the fields. Here, the Euclidean
action is given by S = β
∫
E 0 dτ dx3 LE . The corresponding Feynman rules can be derived
from LE in analogy to the zero-temperature case.
This recipe can be motivated comparing the quantum mechanics (QM) partition func-
tion Z(T ) to the va[cuum]matrix element 〈0, tf |0, ti〉 betwee[n times ti and]tf ,
Z(T ) = Tr exp −βĤ and 〈0, tf |0, ti〉 = 〈0| exp −i (tf − ti) Ĥ |0〉 . (6.47)
Defining ∆τ = i(tf − ti) the exponential in the matrix element becomes exp(−∆τĤ).
We can then identify β with ∆τ , motivating the Wick rotation. The negative sign in
the definition of LE is just convention.13 Furthermore, we usually send tf/i → ±∞ in
the matrix element, such that the time variable t is not restricted. In the thermal case
however, β is fixed and we need to restrict τ ∈ (0, β). Finally, since the trace evaluates
the exponential at equal final and initial states, we have to impose periodic boundary
conditions in τ on bosonic fields, whereas fermionic fields are anti-periodic due to their
anti-commutativity.
Due to the restriction of the imaginary time variable τ to the finite interval (0, β),
the zero-component of the Euclidean four-momentum can only take discrete values given
by ωn = 2πnT for bosons and ωn = (2n+ 1)πT for fermions, respectively, where n is an
integer. These are called M∫atsubara freque∑ncies. The p0 integrals then turn into sumsover the Matsubara modes, p02πf(p0)→ T n f(p0 = iωn).
Using theses rules, the one-loop potential at finite-temperature summing the diagrams
in eq. (6.44) becomes
∞ ∫ D−1 [ ]
V ( ) = 1φ, T µD−4
∑ d p 2 2
1-loop 2 R T =−∞ (2 ) −1
log ωn + ω , (6.48)π D
n
where ω2 = p~ 2 + m2(φ). The Matsubara sum can be evaluated analytically, giving a
temperature-independent part that reproduces the Coleman-Weinberg potential, and a
part that contains the thermal corrections. The latter was first calculated in 1974 by
Jackiw and Dolan [341] in a functional approach and by Weinberg [358] (the other one)14
with diagrammatic methods. The finite-temperature part is UV-finite, so that we can
execute the limitD → 4. On∫e obtains, after integration over the solid(angle [34)1, 351, 358],∑ ∞ [ ] ∑ 4 2
VT(φ, T ) =
T
ηi ni 2 2 dp p
2 log 1− η e−βωii =
T m
n J i
(φ)
i 2 2 η̄π π i T 2 , (6.49)
i 0 i √
where we now sum over all contributing species i with ni DOFs, and ω = p~ 2 +m2i (φ).
We recover eq. (6.22). Again, ηi = +1 (−1) for bosons (fermions)[, and η̄i = −ηi.13 ]With this convention, the Euclidean Lagrangian for a scalar field is L = 12 (∂ φ)2 + (∇φ)2τ + V (φ) .
14The finite-temperature corrections [358] were calculated by Steven Weinberg, whereas the Coleman-
Weinberg potential [340] is due to Erick Weinberg.
96
Appendix 6.B. Further Details on the Effective Potential
(a) . .
.. .
.. .
.
(c) (d)
.
.
.
(b)
Figure 6.6: One- (a), two- (b+c) and dominant
multi-loop (d) contributions to the φ2 vertex. Figure 6.7: Daisy or ring diagram.
We have defined the thermal loop functions for bosons (J−) and fermions (J )∫∞ [ ( +√ )]
J (x2∓ ) = ± dy y2 log 1∓ exp − y2 + x2 . (6.50)
0
These admit the high-temperature expansions [341, 351, 357]
π4 π2 π ( ) ( )3 x4 x2 ( )
J (x2) = − + x2 − x2 2 6− 45 12 6 − 32 log( )+O x , (6.51a)a−
7π4 π2 x4 x2 ( )
J (x2) = − + 2 6+ 360 48x + 32 log +O x , (6.51b)a+
where a− = 16 a+ = 16π2 exp(3/2− 2γE).
Note that the x4 log(x2) terms in eq. (6.51) combine with the m4 log(m2) terms in the
zero-temperature part eq. (6.21), so that the only non-analytic dependence on φ is in
the (x2)3/2 term in eq. (6.51a). The latter term becomes imaginary for negative squared
masses, indicating a breakdown of the perturbative expansion due to IR singularities in
the zero Matsubara modes of the bosonic contributions [351, 359]. Indeed, the breakdown
of fixed-order perturbation theory in the restoration of spontaneously broken theories can
be expected, since the thermal loop-corrections overpower the tree-level potential, and is
related to the fact that FTQFT has two scales, the mass scale m and temperature T , so
that large ratios T/m need to be resummed [360]. This can be achieved resumming the
most-IR-divergent higher-loop corrections [360–362].
Let us inspect the higher-loop corrections to the first term in eq. (6.44) quadratic in
φ, cf. fig. 6.6, in λφ4 theory. The high-temperature behavior can be derived from the
superficial degree of divergence d of the diagrams [351, 360]. Diagrams with d > 0 scale
with T d, whereas diagrams with d ≤ 0 scale linearly in T due to the T prefactor of the
Matsubara sum. The one-loop diagram fig. 6.6a has d = 2 and therefore behaves like
λT 2. The two-loop correction from the sunrise diagram fig. 6.6b has two logarithmically
divergent loops (each contributing a factor T ) and scales like λ2T 2, whereas the diagram
in fig. 6.6c has a quadratically divergent loop stacked on a logarithmically divergent loop,
and hence goes like λ2T 3/m, where factors ofm are added on dimensional grounds. Thus,
in the high-temperature limit, the dominant two-loop correction to the two-point function
97
6. Stochastic Gravitational Wave Backgrounds
comes from adding a quadratically divergent loop to the one-loop diagram, i.e. fig. 6.6c.
Similarly, at the N+1 loop level, the dominant contribution originates from the diagram
in which N bubbles are attached to the main loop, see fig. 6.6d, which behaves like
λT (λT 2)N/m2N−1 = λTm(λT 2/m2)N .
The N -loop corrections to the higher-point contributions can be obtained by attaching
additional external legs to the loop diagrams of the two-point function. One again finds
that the dominant contributions are the ones adding bubbles to the main loop, where
each additional quadratically divergent loop contributes a factor f = λT 2/m2 at high
temperatures. At the critical temperature, the thermal corrections to the potential are
on the order of the tree-level contributions, so that we expect f ∼ 1. Hence, powers
of f need to be resummed to all orders. This corresponds to resumming multi-loop
contributions to the effective potential of the form depicted in fig. 6.7, called daisy or
ring diagrams, where the small loops are evaluated in the high-temperature limit.
Daisy resummation is achieved by resumming the one-loop thermal self-energy cor-
rections Πi(T ), called Debye mass, to the propagator at high-temperature and vanishing
external momentum (i.e. in the IR limit). This amounts to replacingm2i (φ) in the effective
potential by m2i (φ) + Πi(T ).15 Performing this replacement in the full one-loop poten-
tial however requires the introduction of temperature-dependent counter-terms [356]. To
avoid this problem, the shift is only carried out in the zero-modes. To this end, we rewrite
the logarithm in eq.[ (6.48) as [342]] [ ] [ ]
log ω2n + ω2 + Πi = log ω2n + ω2 + log 1 +
Πi
2 + 2 . (6.52)ωn ω
The first term then gives the usual one-loop potential, whereas the second one is evaluated
only for the bosonic n = 0 M[atsubara mod]e, yielding∫∞ [ ]
d 2 log 1 + Πi π
3
p p 2 + 2 = − (m
2
i + Πi) 2 −m3p m i , (6.53)
0 i
3
where divergent but field-independent terms have been dropped. We thus obtain the ring
correction eq. (6.24). The second term in eq. (6.53) then cancels with the cubic term in
the high-temperature expansion eq. (6.51a).
A few comments are in order. First, note that thermal corrections to fermion masses
are not resummed as there is no fermionic Matsubara zero-mode. However, the fermionic
one-loop self-energy diagrams are at worst logarithmically divergent, and can therefore
be neglected as they only scale linearly in T . Further note that for gauge bosons only
the longitudinal modes receive a thermal mass correction Π ∼ T 2, so that the Daisy
correction vanishes for the transverse modes. Finally, it shall be emphasized that the
(m2i +Π )3/2i term should actually read (
3/2
m2 +Π)i , i.e. we need to add the Debye masses
first and then diagonalize the corrected mass matrix.
15This is done similar to the resummation of 1PI corrections to the propagator in zero-temperature QFT,
which leads to the replacement m20 → m2R − imRΓ.
98
7. Gravitational Wave Signatures fromLepton Number Breaking
This chapter is based on the sections 5 to 7 as well as appendices A and B of the paper [1].
It contains text composed by the author taken verbatim from the publication. Minor modifi-
cations have been made to fit the structure, conventions and style of this dissertation.
Let us now return to the model of gauged lepton number introduced in chapter 5 and
investigate whether the lepton number breaking phase transition (PT) can be arranged to
be sufficiently strongly first-order to be detectable by LISA or other future gravitational
wave (GW) observatories. Since we set the vacuum expectation value (VEV) of the
lepton number Higgs, which roughly determines the overall scale of the transition, to
vΦ = 2TeV to satisfy LEP constraints, we can expect transition temperatures in the
TeV range. A potential stochastic gravitational wave background (SGWB) generated
in the transition will therefore end up in the frequency range accessible to space-based
experiments. This first-order PT could further provide the out-of-equilibrium condition
necessary for successful baryogenesis, as was demonstrated recently in a model of non-
abelian gauged lepton number [363].
In the following we aim to identify the regions of parameter space of the model in
which the lepton number PT generates a detectable SGWB while at the same time be-
ing consistent with the collider and dark matter (DM) constraints discussed previously
in chapter 5. The assumption of producing DM as a thermal relic in particular implies
thermal equilibrium between the dark lepton sector and the Standard Model (SM). The
scenario of a decoupled dark sector is studied in a general context in chapter 8. Nonethe-
less, the breaking of lepton number occurs separated from the electroweak PT (EWPT)
for a large part of the viable parameter space, with a GW spectrum independent of the
nature of the latter. Indeed, the EWPT mostly proceeds as a weak cross-over, like it is
also the case in the pure SM.
In section 7.1 we discuss the nature of the lepton number breaking PT. We calculate
the effective potential and determine the regions of parameter space in which the PT is of
first order. The corresponding SGWB and its detectability are evaluated in section 7.2.
Conclusions are presented in section 7.3.
7.1. The Lepton Number Breaking Phase Transition
In the early Universe, the spontaneously broken symmetries, viz. the electroweak (EW)
and lepton number gauge symmetries, are typically restored due to thermal effects induced
by finite-temperature corrections to the effective potential of the scalar fields whose VEVs
break the symmetries. These are the scalar φ̂ breaking the U(1)` lepton number gauge
symmetry, and the Higgs field ĥ which breaks SU(2)W⊗U(1)Y . As a consequence, during
99
7. Gravitational Wave Signatures from Lepton Number Breaking
its history the Universe must have undergone the corresponding PTs associated with the
breaking of these symmetries.
At high temperatures, the global minimum of the finite-temperature effective potential
is at the origin, i.e. SU(2)W ⊗U(1)Y ⊗U(1)` is unbroken. When the temperature drops,
the potential changes and at some point develops a minimum at non-vanishing field
values. Whether both fields develop non-zero VEVs at the same time or independently
at different temperatures depends on the parameters of the model. Since the portal
interaction between the two scalars is restricted to be small (see sections 5.2.2 and 5.3.2),
and due to the hierarchy of the VEVs (vH = 246GeV, vΦ & 1.9TeV), the lepton number
breaking PT however typically occurs first at temperatures in the TeV range, leaving the
EW symmetry unbroken. EW symmetry breaking subsequently proceeds like in the SM
at a temperature of T ' 160GeV [34].
While the EWPT then mostly proceeds as a cross-over, the lepton number PT can be
first-order and may therefore generate GWs. In the following, we will thus briefly discuss
the full finite-temperature effective potential of the lepton number and EW Higgs fields,
and then focus on the lepton number breaking scalar only, neglecting interactions with
the SM fields.
7.1.1. Finite-Temperature Effective Potential
The daisy-resummed one-loop finite-temperature effective potential takes the form (cf.
eq. (6.20))
Veff(h, φ, T ) = Vtree(h, φ) + VCW(h, φ) + ∆Vct(h, φ) + VT(h, φ, T ) + Vring(h, φ, T ) , (7.1)
where h and φ are the classical background fields. Note that these in general do not
coincide with the mass eigenstates but with the gauge interaction eigenstates ĥ and φ̂.
We here however drop the hats for convenience.
The corresponding tree-level potential is
V (h, φ) = −1µ2 1 1h2 − µ2 φ2 + λ h4 + 1 4 1tree 2 H 2 Φ 4 H 4λΦφ + 4λph
2φ2 . (7.2)
The non-MS parts of the counter-terms are
∆V (h, φ) = −1 1 1 1 1δµ2 h2 − δµ2 φ2 + δλ h4 + δλ φ4 + δλ h2 2ct 2 H 2 Φ 4 H 4 Φ 4 p φ , (7.3)
which we fix by imposing that the VEV as well as the mass-squared matrix of h and φ
in the broken vacuum remain at the tree-level values.
The general expressions for the Coleman-Weinberg, thermal one-loop and daisy-resum-
mation potentials are given in section 6.5. The field-dependent masses are obtained by
diagonalizing the mass matrices eqs. (5.4), (5.10) and (5.16), as well as the corresponding
mass matrices for the SM fermions, replacing vH and vΦ by h and φ, respectively. The
Debye masses for the scalars and the lepton number gauge boson are given further below
in eq. (7.6), while the corresponding corrections for the SM gauge bosons can be found
in [342]. To consider effects of kinetic mixing, the thermal masses must be corrected for
the mixing in eq. (5.2).
100
7.1. The Lepton Number Breaking Phase Transition
In fig. 7.1 we show the effective potential at different temperatures, to illustrate the
individual steps of the symmetry breaking process. The model parameters are given in
the figure caption.
At high temperatures, the global minimum of the effective potential is in the symmetric
(unbroken) vacuum (h, φ) = (0, 0). As the Universe cools down, a second minimum starts
to form at non-vanishing values of φ. At Tc ' 835GeV, the two minima are degenerate.
At lower temperatures, the second minimum (h, φ) ∼ (0, 1.1TeV) is the global minimum
and breaks lepton number, whereas the EW symmetry remains unbroken. This minimum
is separated from the symmetric minimum by a potential barrier. Thus, to transition to
the global minimum, the field has to tunnel (or remain in the symmetric vacuum until
the barrier disappears).
As the Universe cools further, the minimum at the origin disappears at some point,
and the global minimum moves towards the zero-temperature lepton-number-breaking
VEV (h, φ) = (0, 2TeV). Subsequently, at T . 160GeV, the minimum starts to shift to
non-vanishing Higgs field values, breaking the EW symmetry in a cross-over transition.
Eventually, the Universe ends up in today’s vacuum (h, φ) ' (246GeV, 2TeV).
7.1.2. A First-Order Lepton-Number-Breaking Phase Transition
In this section, we examine the lepton number breaking PT in the limit of negligible
portal coupling λp between the SM Higgs and the scalar Φ. Further assuming that the
kinetic mixing of the gauge bosons as well as the exotic Yukawa couplings ci and yi of
the dark leptons are small, we can study a simplified version of the effective potential in
which only the lepton number breaking scalar and the lepton number gauge boson are
considered.
In this case, the tree-level potential simplifies to
V (φ) = −1µ2 2 + 1tree 2 Φφ 4λΦφ
4 . (7.4)
Setting the lepton number breaking VEV to vΦ = 2TeV, in agreement with the LEP
constraint, the model is therefore fully specified by mZ′ and mφ.
The field dependent masses of the scalar, the gauge boson, and the Goldstone boson
are given by
m2 = −µ2 + 3λ φ2φ Φ Φ , m2ω0 = −µ2Φ + λ 2 2 2 2Φφ , and mZ′ = 9g`φ . (7.5)
The thermal(mass correc)tions are (Πφ = Πω0 = ΠΦ)
Π = 1 9 2
( )
λ 2 2 2 2 2 2 ′2 ′′2Φ 3 Φ +
′
4g` T and ΠZ = 3g`T + 3g`T 3 + L + L , (7.6)L
where the first part of ΠZ′ comes from the scalar, and the second part from the SM andL
exotic leptons. The subscript L of ΠZ′ indicates that only the longitudinal part of the
Z ′
L
boson receives a thermal correction.
We further use an on-she∣ll scheme, imposing the conditions
∂ (VCW + ∆Vct) ∣∣∣ ∣= 0 and ∂2 (VCW + ∆Vct) ∣∂ φ ∂ φ2 ∣∣ = −∆Σ . (7.7)φ=vΦ φ=vΦ
101
7. Gravitational Wave Signatures from Lepton Number Breaking
2.5 2.5
T = 1000 GeV T = 835 GeV
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
0 100 200 300 400 0 100 200 300 400
h [GeV] h [GeV]
2.5 2.5
T = 825 GeV T = 160 GeV
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
0 100 200 300 400 0 100 200 300 400
h [GeV] h [GeV]
2.5 2.5
T = 150 GeV T = 0 GeV
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
0 100 200 300 400 0 100 200 300 400
h [GeV] h [GeV]
Figure 7.1: Effective potential as a function of temperature. The colored equipoten-
tial lines correspond to Veff = (30GeV)4, (60GeV)4, (90GeV)4, . . . , (600GeV)4 (from
dark-purple to yellow). The red dot denotes the global minimum of the potential at
Veff = 0GeV4. The model parameters are vΦ = 2TeV, mφ = 500GeV, sin θH = 0.05,
mZ′ = 1.5TeV,  = 0, mDM = 590GeV, sin θDM = 0, me4 = me5 = 650GeV, and
L′ = −1/2.
102
φ [TeV] φ [TeV] φ [TeV]
φ [TeV] φ [TeV] φ [TeV]
7.1. The Lepton Number Breaking Phase Transition
This ensures that the VEV and the scalar mass at zero temperature remain at their tree-
level values. Here, ∆Σ ≡ Σ(m2φ)−Σ(0) is the difference of the scalar self-energy evaluated
at the tree-level mass and at zero-momentum, see appendix 7.A. The second derivative
of the Coleman-Weinberg potential in the vacuum suffers from logarithmic divergences
originating from the vanishing Goldstone masses. These are infrared (IR) divergencies
and are due to the fact that the effective potential is evaluated at vanishing external
momentum. However, the scalar self-energy at zero-momentum suffers from the same
divergences, hence its presence in the second condition above [356]. The divergences in
∆Σ and ∂2V /∂φ2CW then cancel, ensuring that we obtain (IR-)finite counter-terms.
We use the numerical package CosmoTransitions [364] to evaluate the effective potential
and to analyze the PT. Fixing the VEV to vΦ = 2TeV (and setting the renormalization
scale to µR = vΦ), we identify the region in the mφ −mZ′ parameter space at which a
first-order PT occurs.
In this model, the potential barrier between the vacua is generated by thermal cor-
rections from gauge boson loops (note the cubic term in the high-T expansion of the
bosonic thermal contribution in eq. (6.23)), i.e. the larger the gauge coupling (and hence
also the Z ′ mass) the higher and wider the barrier. Increasing the scalar mass on the
other hand increases the quartic coupling, which in turn reduces (the relative size of)
the barrier. Thus, first-order PTs can be obtained for mZ′ & mφ; strong transitions
occur for mZ′ & 2mφ. The term “strong” here refers to transitions in which the VEV
(or more precisely the distance between the two degenerate minima in field space) at the
critical temperature is larger than the critical temperature itself, i.e. 〈φ〉c /Tc & 1, where
〈φ〉c = 〈φ(Tc)〉. This measure is often employed in the context of baryogenesis [351].
Figure 7.2a shows the regions in the mφ −mZ′ plane in which the effective potential
develops degenerate minima at a critical temperature T for L′c = −1/2. The colors
indicate the corresponding Tc. In the colored region above the black line, the measure
〈φ〉c /Tc implies strong transitions. The parameter points which actually lead to a first-
order PT through bubble nucleation are shown in fig. 7.2b along with the corresponding
nucleation temperature, again for L′ = −1/2. Here, the black line indicates 〈φ〉n /Tn & 1
evaluated at the nucleation temperature.
Although the renormalization conditions eq. (7.7) ensure that the zero-temperature
potential has a minimum at φ = vΦ, this minimum is not necessarily the global minimum.
In particular, if the gauge boson mass mZ′ is much bigger than the scalar mass mφ,
the potential develops a global zero-temperature minimum at φ = 0, i.e. the Coleman-
Weinberg corrections restore the symmetry already at T = 0.1 This is the case in the
white area labeled by “〈φ〉0 = 0” above the colored region in fig. 7.2a (and above the
dotted line in fig. 7.2b), which is of course excluded since it would imply the existence
of a second massless gauge boson with significant couplings to leptons. Furthermore,
even a global minimum at φ = vΦ does not automatically ensure that today’s Universe
has transitioned to the true vacuum. If the barrier is very large with a small potential
difference between the two vacua, which is the case close to the region in which the
1Of course, the physical scalar and Z′ m√asses become mφ = 0 and mZ′ = 0 in this region. Hence, the x
and y axes should be interpreted as 2λΦv2Φ and 3g`vΦ respectively, where vΦ = 2TeV then has no
physical meaning.
103
7. Gravitational Wave Signatures from Lepton Number Breaking
3.0 T [GeV] 3.0c T [GeV]
0 0 lin
g n
3000 ne 30002.5 〉 = 2.5 〉 = n〈φ 0 〈φ 0 t
u
no
2.0 2.0
1
> 2000 1 2000
T c >
1.5 〈φ〉
/
c 1.5 /T
n
〈φ〉n
1.0 1000 1.0 1000
0.5 no barrier 0.5 cross-over
0 0
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(a) Critical temperature Tc (b) Nucleation temperature Tn
Figure 7.2: Parameter points with two phases separated by a potential barrier at a
critical temperature Tc (left), and points that give rise to a cosmological first-order PT
with a nucleation temperature Tn (right). The colored regions above the solid black
line feature strong transitions with 〈φ(T )〉/T > 1 at Tc or Tn, respectively. The dotted
line in the right plot denotes the border at which φ = 0 becomes a global minimum.
potential has a global zero-temperature minimum at φ = 0, the tunneling probability
is too low. Therefore the field is stuck in the false vacuum and does not tunnel. This
corresponds to the parameter region labeled “no tunneling” in fig. 7.2b.2
On the other hand, for mZ′ . mφ no significant barrier is induced and there is no
temperature at which the potential has degenerate minima. Also, if the potential barrier
separating the phases is very shallow, it might disappear before bubbles are nucleated. In
both cases the transition occurs without tunneling as a cross-over3 and no gravitational
waves are generated. This happens in the areas labeled “no barrier” or “cross-over” in
fig. 7.2.
So far, to simplify the parameter space to two dimensions, we neglected the contribu-
tions from the dark Yukawa couplings of the fourth and fifth generation leptons to the
effective potential. However, if the leptons are heavy, the Yukawa couplings are large
and the potential can be modified significantly. This is in particular the case for large Z ′
masses, where the exotic leptons are required to be heavy in order to obtain the correct
DM relic abundance.
2Note that CosmoTransitions only evaluates the thermal tunneling probability. Quantum tunneling is
not taken into account.
3We here rely on the ability of CosmoTransitions to identify cross-over transitions. A proper determina-
tion of whether a transition is cross-over may involve non-perturbative calculations and is beyond the
scope of this work. Here, we are mainly interested in the region where strong first-order transitions
occur.
104
mZ ′ [TeV]
mZ ′ [TeV]
7.1. The Lepton Number Breaking Phase Transition
3.0 Tn [GeV]
3.0 Tn [GeV]
0 3000 0 3000
2.5 〉 = 2.50 〉 =0 red〈φ 〈φ oest 1
2.0 2.0 rΦ >v n
1 2000 〉 =
/T 2000
> 〈φ 0 〈φ
〉n
1.5 /Tn〉 1.5〈φ n
1.0 1000 1.0 un 1000stabl
0.5 ecross-over 0.5 cross-over
0 0
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(a) mDM = 200GeV, mHL = 210GeV (b) mDM = 500GeV, mHL = 1TeV
Figure 7.3: Parameter points giving rise to a cosmological first-order PT with a nu-
cleation temperature Tn, including the contribution from DM and the exotic leptons.
The dashed lines indicate the corresponding region neglecting the fermion contribu-
tions (cf. fig. 7.2b). In the gray shaded region, the potential becomes unstable below
φ = 100GeV; above the gray dotted line it is stable up to φ = 106TeV.
For simplicity, we here again assume that the exotic electrons and the exotic neutrino
have equal masses, mHL ≡ me4 = me5 = mν4 , i.e. that the SM Higgs Yukawa couplings
yν and ye in eq. (5.14) vanish. The field dependent masses are then given by
cν c`
mDM = √ φ , mHL = √ φ . (7.8)2 2
The Yukawa couplings further c(ontribute to the scalar the)rmal mass correction eq. (7.6),which becomes
Π = 1 9λ + g2 + 1 2 1 2Φ 3 Φ 4 ` 12cν + 4c` T
2 . (7.9)
Figure 7.3 shows the values of the scalar and Z ′ masses that lead to a first-order PT
with the corresponding nucleation temperature, for two different choices of the DM and
heavy lepton (HL) masses. The region that gives rise to a first-order PT for vanishing
fermion couplings (cf. fig. 7.2b) is indicated by the dashed lines.
As expected, for light HLs (i.e. low Yukawa couplings) the situation changes only
marginally with respect to the case assuming vanishing Yukawas. However, for higher
fermion masses, the region that yields a first-order PT changes, and the nucleation tem-
perature decreases.
In the parameter region labeled “〈φ〉0 = vΦ restored” in fig. 7.3b, the bosonic loop
corrections to the zero-temperature potential induce a global minimum at φ = 0 in the
absence of fermions. If the dark sector leptons are included, their contributions have the
opposite sign and partially cancel the bosonic ones, and the global minimum at φ = vΦ is
restored. Hence, the region allowing for a first-order PT is extended. On the other hand,
if the fermionic corrections overcome the bosonic ones at high field-values, the potential
105
mZ ′ [TeV]
mZ ′ [TeV]
7. Gravitational Wave Signatures from Lepton Number Breaking
3.0 Tn [GeV]
0 3000
2.5 〉 = d〈φ 0 or
e
es
t
2.0 rvΦ
= 1 2000
1.5 〈φ
〉 0 >
〉 /T
n
〈φ n
1.0 1000
Figure 7.4: Same as fig. 7.3, but at each
0.5 cross-over parameter point the DM mass is set to a
0 value that yields the correct relic abun-
0 200 400 600 800 1000 dance. The masses of e4, e5, and ν4 are
mφ [GeV] set to mHL = 1.5×mDM.
is destabilized as it is not bounded from below. This occurs for low Z ′ and φ masses.
The gray shaded regions are excluded since the potential becomes unstable at field values
below φ = 100TeV, i.e. Veff(100TeV) < Veff(〈φ〉0) at T = 0. Above the gray dotted curve
the potential is stable even up to φ = 106TeV. Note however that a reliable evaluation
of the potential at such high field values requires the inclusion of renormalization group
(RG) effects.
At high temperatures, the loop corrections from the fermions give a positive contri-
bution ∼ φ2T 2, whereas they do not contribute to the cubic terms (note that there is
no (m2/T 2)3/2 term in the fermionic sum of eq. (6.23)). As a consequence, the finite-
temperature corrections restore the symmetric minimum at lower temperatures, reducing
the nucleation temperature.
Finally, to properly connect to the DM picture, let us require that the DM candidate
has the correct thermal abundance.4 Figure 7.4 shows the nucleation temperature for the
corresponding PT, assuming that mHL = 1.5 × mDM. At each parameter point in the
mφ −mZ′ plane we use micrOMEGAs to find the value of the DM mass that yields the
measured abundance, picking the value below the Z ′ resonance (i.e. we are sitting on the
upper branch of the blue line in fig. 5.3). Again, the dashed lines indicate the parameter
region that provides a first-order PT if the fermions are neglected.
As the DM mass required to obtain the correct abundance increases with the Z ′ mass
and is mostly independent of the scalar mass, the effects of including the dark leptons
are stronger for larger Z ′ masses. Hence, the fermionic corrections restore the T = 0
minimum at φ = vΦ for high mZ′ , whereas this effect is absent in the low mZ′ range.
Furthermore, since mDM < mZ′/2 (and mHL = 1.5 × mDM), the bosonic contributions
are sufficiently large to circumvent the destabilizing effects of the fermionic corrections
in the full parameter space shown in fig. 7.4.
4Note that the DM in fig. 7.3a already has the measured abundance by co-annihilation for most values
of mZ′ , cf. the green line in fig. 5.4.
106
mZ ′ [TeV]
7.2. Gravitational Waves Signature
10−9
mφ 200GeV
LISA mZ′ 1.4TeV
10−12 vΦ 2TeV
L′ −12
10−15 Tc 487GeV
rb
Ω tu2 Tn 198GeV
h
10−18 α 0.18
10−4 10−2 100 102
f [Hz] β/H∗ 570
Figure 7.5: GW spectrum (black solid line) from the U(1)` breaking phase transition
for mφ = 200GeV and mZ′ = 1.4TeV. The contributions from different production
mechanisms are indicated by the dashed green and gray lines. The colored regions
indicate the power-law integrated (PLI) sensitvity of LISA (blue), B-DECIGO (red),
DECIGO (dark orange), and BBO (orange).
7.2. Gravitational Waves Signature
Having explored the parameter regions that give rise to a first-order PT, we now calcu-
late the remaining transition parameters, to wit, the energy budget α and the relative
transition scale β/H∗, and determine the corresponding GW spectrum as well as its de-
tectability at future GW experiments. As the scalar field φ is coupled to the lepton
number gauge boson, the corresponding friction prevents the bubbles from entering the
runaway regime [325]. Therefore, only the plasma contributions to the spectrum are con-
sidered throughout this chapter, assuming a turbulent fraction of εturb = 5%. We further
take vw = 1 in this section, discussing the impact of the wall velocity in appendix 7.B.
An example of the GW spectrum generated by the lepton number breaking PT for a
scalar mass of mφ = 200GeV and a gauge boson mass of mZ′ = 1.4TeV (with vΦ = 2TeV
and L′ = −1/2, neglecting the dark lepton contributions) is shown in fig. 7.5 (black
curve). The contributions from acoustic production (green) and magnetohydrodynamic
(MHD) turbulence (gray) are indicated by dashed lines. For this choice of parameters,
the transition occurs at a nucleation temperature of Tn ∼ 200GeV with a peak frequency
of the spectrum of f ∼ 22mHz.
Figure 7.5 also shows the sensitivity curves of the future space-based GW interferometer
LISA and its potential successors (B-)DECIGO and BBO as colored regions. Note that
these curves are not the noise curves, but the PLI curves defined in eq. (6.4), which
indicate that a GW background should be detectable by the experiment if the spectrum
touches or reaches into the region above the respective sensitivity curve. Thus, the
spectrum shown in the figure can be probed by all four experiments.
107
h2ΩGW
h 2
Ω
h 2 G
Ω W
sw
B
D -DEC EI CBB G
I
O GO O
7. Gravitational Wave Signatures from Lepton Number Breaking
3.0 α 3.0 β/H∗
0 100 0 106
2.5 = 2.5 =
〈φ〉 0 φ〉 010−1 〈
2.0 2.0 10
5
1 1
> 10−2 >
1.5 /Tn〉 1.5 〉 /
Tn 104
〈φ n − 〈φ n10 3
1.0 1.0
3
10−4
10
0.5 cross-over 0.5 cross-over
10−5 102
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(a) energy budget α (b) relative transition scale β/H∗
Figure 7.6: Energy budget α and inverse relative time scale β/H∗ as a function of the
U(1)` breaking scalar mass mφ and the U(1)` gauge boson mass mZ′ in the case of
negligible portal coupling λp.
While the specific parameter point depicted in fig. 7.5 exhibits good prospects for ob-
serving the generated SGWB at GW experiments, the majority of the parameter regions
with a first-order PT gives rise to transitions which are too weak for a detection. Nonethe-
less, a significant fraction of the parameter space may be probed at least at the far-future
observatories DECIGO and BBO, as we will demonstrate in the following.
Figure 7.6 shows the parameters α and β/H∗ for the lepton number breaking PT
as a function of the φ and Z ′ mass in the simplified case of negligible portal coupling
λp considered in section 7.1.2, calculated using CosmoTransitions [364]. Most choices of
masses give rise to a rather short first-order PT (high β/H∗) with few energy released
(low α). However, large values of α and small values of β/H∗ can be obtained in the
mZ′ & 2mφ region, which is the region we identified to give rise to strong first-order PTs in
section 7.1.2. As the amplitude of the sound-wave contribution to the GW spectrum 6.14b
is proportional to α2/(1 +α)2 and H∗/β, this is indeed the region in which the stochastic
background can be expected to be detectable.
The corresponding parameter points for which the SGWB generated by the lepton
number breaking PT is accessible to space-based GW interferometers are depicted in
fig. 7.7a. The blue, red, dark orange, and orange regions can be detected by LISA, B-
DECIGO, DECIGO, and BBO, respectively, whereas in the gray region the generated
GW background is not detectable. If a parameter point is detectable by more than one
experiment, the color corresponds to the experiment named first in the list above.
If the φ Yukawa couplings (and the Higgs portal coupling) are neglected, LISA and
B-DECIGO are only sensitive in the mZ′  mφ margin of the parameter space that has
a first-order PT. DECIGO can probe a small portion of the parameter space, also only
close to the mZ′  mφ edge, BBO is slightly more sensitive. Still, the majority of the
parameter space is inaccessible to GW experiments.
108
mZ ′ [TeV]
mZ ′ [TeV]
7.2. Gravitational Waves Signature
3.0 3.0
0 0
2.5 〉 = 2.5 =〈φ 0 〈φ〉 0
2.0 2.0
1.5 1.5
1.0 1.0
0.5 cross-over 0.5 cross-over
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(a) neglecting Yukawa terms (b) mDM = 150GeV, mHL = 200GeV
3.0 3.0
0 sin θDM = 0.024
2.5 = 2.5
〈φ〉 0 0
〈φ〉
=
0
2.0 2.0
sin θDM = 0.022
1.5 1.5
sin θDM = 0.02
1.0 un 1.0stable cross-over
0.5 cross-over 0.5 sin θDM = 0.015
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(c) mDM = 500GeV, mHL = 1TeV (d) h2ΩDM = 0.1198, mHL = 1.5×mDM
LISA B-DECIGO DECIGO BBO
Figure 7.7: Sensitivity of space-based GW observatories to the stochastic background
from the lepton number breaking PT (for negligible portal couplings) in the mφ−mZ′
plane. In the colored regions, the SGWB can be detected by LISA (blue), B-DECIGO
(red), DECIGO (dark orange), and BBO (orange), respectively. In the gray region,
the stochastic background is not detectable. The gray shaded regions (below the dark-
gray solid line) in figs. 7.7b and 7.7c are excluded as the potential is unstable. In
fig. 7.7d, the DM mass is set to a value reproducing the measured DM abundance at
each parameter point. The dashed lines indicate the exclusion reach of XENON1T for
different values of the DM mixing angle θDM.
109
mZ ′ [TeV] mZ ′ [TeV]
mZ ′ [TeV] mZ ′ [TeV]
7. Gravitational Wave Signatures from Lepton Number Breaking
So far we neglected the contributions of the dark leptons to the effective potential. In-
cluding them can significantly improve the detection prospects. The detectability of the
SGWB for two different choices of exotic lepton masses are shown in figs. 7.7b and 7.7c.
For light dark leptons (small Yukawas) with mDM = 200GeV and mHL = 210GeV
(mHL ≡ me4 = me5 = mν4), the detectable part of the parameter space barely changes.
For heavier leptons with mDM = 0.5TeV and mHL = 1TeV however, a significant fraction
of the first-order transitions can be probed. Still, the sensitivity of LISA is restricted to a
band near the mZ′  mφ edge of the PT region, the size of the detectable region however
increases notably compared to the case with vanishing Yukawa couplings. B-DECIGO
can probe additional parameter points, mostly for low scalar masses. DECIGO and BBO
can reach mZ′ & 1TeV for mφ ∼ 50GeV and mZ′ & 2.8TeV for mφ ∼ 1TeV.
Last but not least, fig. 7.7d shows the detectability of the SGWB from the lepton
number breaking PT requiring that the DM accounts for the full thermal abundance
measured by Planck, assuming mHL = 1.5×mDM as in fig. 7.4. Again, the effects of the
dark leptons significantly enhance the parameter space to which future space-based GW
observatories are sensitive. The dashed lines indicate the exclusion reach of XENON1T
(cf. fig. 5.5b) for DM mixing angles of sin θDM = 0.015, 0.02, 0.22 and 0.024. The white
region in the upper left part of the plot is excluded as the lepton number gauge group
remains unbroken. Although not specifically mentioned, this applies to all sub-figures of
fig. 7.7.
7.3. Conclusion
In this chapter we have continued our study of the gauged lepton number model intro-
duced in chapter 5, investigating the lepton number breaking PT in the early Universe.
If the portal coupling between the SM and dark Higgs is sufficiently small, the lepton
number and EW PTs happen independently from one another. Due to the VEV hierar-
chy imposed by the LEP constraints, the former typically occurs first. We found that
in a large fraction of the parameter space the lepton number transition is first order. It
can thus generate a stochastic background of GWs. We calculated the corresponding GW
spectrum and evaluated the detection prospects for future space-based GW observatories.
While LISA can only probe a rather small fraction of the parameter space, its possible
successors BBO and DECIGO are able to explore a significant fraction of the parameter
points that give rise to a first-order PT. Notably, the exotic leptons significantly enhance
the detection prospects, particularly when requiring that the measured relic abundance
is reproduced. This is due to two effects. First, the presence of additional particles lowers
the nucleation temperature, and second, the fermionic contributions restore the broken
minimum in a part of the parameter space in which the bosonic corrections alone would
shift the vacuum back to the origin.
Further interesting effects may arise if one considers non-vanishing portal couplings
between the dark and SM scalar sectors. The transition can then proceed diagonally in
field space, breaking the EW and lepton number gauge groups simultaneously. We leave
this subject for future work. Another possible direction would be the investigation of the
phase transition in the context of Baryogenesis.
110
Appendix of Chapter 7
Appendix 7.A. Goldstone Divergences
In this appendix we address the cancellation of the IR divergence in the second derivative
of the effective potential, originating from the vanishing Goldstone mass in Landau gauge.
We follow the treatment in ref. [356]. This leads to the renormalization condition eq. (7.7).
We calculate the self-energy Σ(p2) of the lepton number breaking scalar φ in Landau
gauge, using dimensional regularization in D = 4−2 dimensions. More precisely, we are
interested in the difference of the self-energy evaluated at the scalar mass p2 = m2φ and
at p2 = 0, ∆Σ ≡ Σ(m2φ)− Σ(0), where p2 is the external momentum squared.
7.A.1. Scalar Self-Energy
We consider the Lagrangian ( )
L = D Φ†Dµ Φ + µ2 Φ† †
2
µ Φ Φ− λΦ Φ Φ , (7.10)
where Dµ Φ = ∂µΦ− ig ′`LΦZ Φ. We rewrite t(he complex sc)alar Φ in terms of its real and
imaginary parts and its VEV vΦ as Φ = √12 vΦ + φ+ iω
0 .
The (bare) self-energy of φ receives contributions from loops of φ itself, the Goldstone
boson ω0, and the gauge boson Z ′. As the tadpole diagrams depicted in fig. 7.8 are
independent of the external momentum, we only need to evaluate the remaining bubble
diagrams. These are
−iΣS0 (p2) = φ φ, ω0 φ , (7.11)
Z ′
−iΣZ′(p20 ) = φ Z ′ φ + φ φ . (7.12)
ω0
We perform the Passarino-Veltman reduction [365] of the corresponding integrals using
FeynCalc 9.3.0 [366, 367], y(ielding2 2 )
ΣS0 (p2) = −
λSvΦ
32 2 B0 {p
2 2 2
π [,mS ,mS , ] ( ) (7.13)2 2
ΣZ′(p2) = − LΦg` 4 2 2 4 2 2 20 32π2m2 p − 4mZ′p +(12mZ′ )B0 p ,mZ(′ ,mZ′Z′ ) } (7.14)
− p4B p2, 0, 0 − 2p2A p2 − 8m40 0 Z′ ,
111
7. Gravitational Wave Signatures from Lepton Number Breaking
φ, ω0 Z ′
φ φ φ φ
Figure 7.8: Tadpole diagrams contributing to the self-energy of φ.
where λS = 6λΦ for S = φ and λS = 2λΦ for S = ω0, respectively. The scalar one- and
two-point functions A0 and B0 are defined by
( )
2 = (2πµ )
4−D ∫
R dD 1A m q , (7.15)
( 0 ) iπ2 q2 −m2 + iε
2 2 2 = (2πµ )
4−D ∫
R 1 1
B0 p ,m1,m2 dDq , (7.16)iπ2 q2 −m2 + iε (q + p)21 −m22 + iε
where µR i(s th)e renorm( al)ization s(cale). The fu(ll s)elf-energy is then given by
Σ 20 p = Σφ 2 + Σω
0 ′
p p2 + ΣZ p20 0 0 + tadpole contributions . (7.17)
The renormalized self-energy( is)related(to )the bare one by
Σ p2R = Σ0 p2 + δm2 − p2δZ , (7.18)
where δm2 and δZ are the mass and field renormalization counter-terms for φ,
L ⊃ 1 µ2δZ∂µφ∂ φ−
1
δm2φ22 . (7.19)
When calculating ∆Σ, δm2 cancels in( the)difference but δZ remains, i.e.
∆Σ = Σ0 m2φ − Σ0 (0)−m2φδZ . (7.20)
In particular, this means that ∆Σ is independent of the renormalization conditions we
impose on the counter-terms δm2 and δλ when calculating the effective potential. We
can now fix δZ by requiring canonical normalization of the field φ, i.e.
∂ Σ ( ) ( )R
2 m
2
φ = 0 =⇒ =
∂ Σ0
δZ m2 . (7.21)
∂ p ∂ p2 φ
Again, the tadpole diagrams in fig. 7.8 do not contribute as they are independent of p2.
Finally, the difference in the self-energy used in the renormalization condition eq. (7.7)
is obtained by plugging eqs. (7.17) and (7.21) into eq. (7.20). We use LoopTools 2.13 [368,
369] to evaluate the finite part of the scalar integrals and their derivatives.
112
Appendix 7.A. Goldstone Divergences
7.A.2. On-Shell Renormalization of the Effective Potential
The momentum-dependent mass o(f φ)is given by ( )
m2 p2 2φ = mφ,R + Σ 2R p , (7.22)
where mφ,R is the renormalized mass parameter in the Lagrangian, which is related to
the physical (pole) mass m2φ ≡ m2φ(m2φ) by ( )
m2 2 2φ,R = mφ − ΣR mφ . (7.23)
Since the effective potential is defined at vanishing external momentum, we now impose
the conditions
∂ Veff(φ, T = 0) ∣∣∣∣∣ = 0 , (7.24)∂ φ φ=vΦ
∂2 Veff(φ, T = 0) ∣∣∣ = m2 (0) = m2 −∆Σ . (7.25)
∂ φ2 ∣ φ φ
φ=vΦ
We further want the VEV vΦ and the scalar mass mφ to be identical to the values inferred
from the tree-level potential, i∣.e. ∣
∂ V ∣tree(φ) ∣∣∣ ∂2= 0 Vtree(φ) ∣,∂ φ φ=v ∂ φ2 ∣∣ = m2φ , (7.26)Φ φ=vΦ
hence, using Veff(φ, T = 0) = Vtree(φ) +VCW(φ) + ∆Vct(φ), we obtain the renormalization
conditions in eq. (7.7).
Note that ∆Σ has an IR divergence coming from the Goldstone contribution eq. (7.13)
to the self-energy at zero-momentum,
2 2 ( )
Σω0(0) = −λΦvΦ0 8 2 B0 0,m
2 2
ω0 ,mω0 . (7.27)π
In Landau gauge mω0 = 0, but we keep it as a regulator. Taking the analytic expression
for the scalar two-point function from refs. [370, 371],
( ) m2 − iε
B0 0,m2ω0 ,m2ω0 = ∆− log ω
0
2 , (7.28)µR
where ∆ = 1 − γE + log 4π, we obtain the IR divergent part
λ2 2 m2−∆Σ = ΦvΦ8 2 log
ω0
2 + finite terms . (7.29)π µR
On the other hand, the Goldstone contribution to the Coleman-Weinberg potential
eq. (6.21) is given by
m4
[ 2 ]
( ) ⊃ ω0
(φ) m
V φ log ω0
(φ)
− 3CW 64 2 2 2 , (7.30)π µR
113
7. Gravitational Wave Signatures from Lepton Number Breaking
where m2ω0(φ) = λΦφ2 − µ2Φ with µ2 2Φ =[λΦvΦ, and its d]erivatives are
∂ VCW λΦφ m
2
ω0(φ) m2ω0(φ)⊃ 16 2 [ log 2 −]1 , (7.31)∂ φ π µR
∂2 V λ m2 (φ) m2 (φ) λ2 φ2 m2CW ⊃ Φ ω0
(φ)
2 16 2 log
ω0 Φ
2 − 1 + 8 2 log
ω0
2 . (7.32)∂ φ π µR π µR
Whereas the parts with the square brackets go to zero when taking the limit φ −→ vΦ,
the second part in the second derivative gives the same IR divergence we encountered in
eq. (7.29). Hence, the IR divergences on both sides of the second condition in eq. (7.7)
cancel and we obtain IR-finite counter-terms.
Appendix 7.B. Bubble Wall Velocity
The wall velocity is rather difficult to compute and can in general not be determined from
the finite-T effective potential alone, as it involves out-of-equilibrium dynamics. It can
be obtained by a microscopic treatment of the fluid solving Boltzmann equations or at
the macroscopic level adding an effective friction term to the scalar equation of motion,
see refs. [321, 322] and references therein for more details, or ref. [323] for recent results.
Assuming Chapman-Jouget detonations [372], vw can be calculated as a function of α,
yielding values ranging from the speed of sound c 1s = √3 in the plasma to the speed of
light. However, this assumption is not justified and typically incorrect [373].
Since we expect that the bubbles do not run away in our model, the GW spectrum
is given by h2Ω (f) = h2GW Ωsw(f) + h2Ωturb(f). For both, sound wave and turbulence
contribution, cf. eqs. (6.14b) and (6.14c), the amplitudes of the spectra are proportional to
vw and the peak frequencies shift as 1/vw, i.e. order one changes in the wall velocity only
have an order one effect on the spectrum and peak frequencies. Hence, the detectability
of the generated stochastic background eventually only has a mild dependence on vw. As
a detectable signal further typically requires strong transitions with large wall velocities,
we simply take the most optimistic estimate vw = 1.
Figure 7.9 shows the dependence of the detectability on the bubble wall velocity in the
case of negligible portal and Yukawa couplings. Compared to fig. 7.7a above, where vw = 1
was assumed, we here show the sensitivity for a slightly lower wall velocity (vw = 0.9), a
wall velocity close to the speed of sound (vw = 0.6), and subsonic bubbles (vw = 0.3 and
vw = 0.1). Again, the colored regions are detectable by LISA (blue), B-DECIGO (red),
DECIGO (dark orange) and BBO (orange). The GW spectra generated by the first-order
PTs in the gray region are not detectable. For supersonic bubbles, the detectable regions
barely change when varying vw. Taking vw to subsonic values decreases the sensitivity
visibly.
114
Appendix 7.B. Bubble Wall Velocity
3.0 3.0
0 0
2.5 〉 = 2.5 =〈φ 0 〈φ〉 0
2.0 2.0
1.5 1.5
1.0 1.0
0.5 cross-over 0.5 cross-over
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(a) vw = 0.9 (b) vw = 0.6
3.0 3.0
0 0
2.5 〉 = 2.5 =〈φ 0 〈φ〉 0
2.0 2.0
1.5 1.5
1.0 1.0
0.5 cross-over 0.5 cross-over
0 200 400 600 800 1000 0 200 400 600 800 1000
mφ [GeV] mφ [GeV]
(c) vw = 0.3 (d) vw = 0.1
LISA B-DECIGO DECIGO BBO
Figure 7.9: Same as fig. 7.7a, but with different wall velocities. Figure 7.7a assumes a
wall velocity of vw = 1.
115
mZ ′ [TeV] mZ ′ [TeV]
mZ ′ [TeV] mZ ′ [TeV]
8. Constraining Secluded HiddenSectors with Gravitational Waves
This chapter is based on work conducted in collaboration with Moritz Breitbach,
Joachim Kopp, Toby Opferkuch, and Pedro Schwaller [2]. It closely resembles the publi-
cation.
So far we have only considered phase transitions (PTs) occurring within the Standard
Model (SM) itself or in sectors beyond the Standard Model (BSM) that are in thermal
equilibrium with the SM. This assumption is however not necessarily true. In this chapter,
we will discuss the case of PTs in hidden sectors that are decoupled from the SM.
Despite the strong motivation for new physics, the extensive search for beyond the
Standard Model (BSM) phenomena at the LHC did not provide any clear hints for the
existence of new particles so far. We can therefore conclude that the BSM states are either
too heavy to be produced at the energy scales currently accessible, or too weakly coupled
to be generated at a detectable rate. This provides motivation for so-called hidden or
dark sectors, i.e. a group of particles that interact only very weakly, maybe even only
gravitationally, with the SM. While such a sector is typically very challenging to detect
directly, gravitational waves (GWs) from a PT within the sector may provide a possibility
to assess these models. In particular, if the sector interacts with the SM via gravity only,
GW probes may be the only way to study such a case.
In this chapter, we will mainly focus on sub-MeV hidden sectors. While PTs at such
low temperatures have the advantage that there are less degrees of freedom (DOFs)
contributing to the radiation energy density in eq. (6.8), so that the relative amount of
energy released into GWs is typically larger than at high temperatures, hidden sectors
featuring additional particles with sub-MeV masses are subject to strong constraints on
the effective number of neutrino species Neff. These constraints require light hidden
sectors to be decoupled from the SM and to be colder than the photon bath at low
temperatures. We discuss how this affects the stochastic gravitational wave background
(SGWB) generated by a first-order PT in such a sector and its detectability. Due to the
low temperatures considered here, the GW spectrum is peaked at frequencies accessible
through pulsar timing arrays (PTAs).
To assess whether it is possible to construct sub-MeV hidden sector models that gen-
erate an observable SGWB while at the same time consistent with Neff constraints, two
simple benchmark models are considered. The first model consists of two SM singlet
scalars, the other one is a Higgsed dark photon model. As the number of additional
DOFs at low temperatures is strongly constrained, these two models should cover a large
fraction of the model space for the relevant DOFs in sub-MeV sectors.
This chapter is structured as outlined below. We first provide an introduction to de-
coupled hidden sectors in section 8.1, reviewing the decoupling of neutrinos in the SM
in section 8.1.1, as well as constraints from the effective number of neutrino species in
section 8.1.2. In section 8.1.3 we describe the hidden sector scenarios considered in this
116
8.1. Decoupled Hidden Sectors
chapter. Section 8.2 then discusses how the the SGWB of a first-order PT is altered
if it occurs in a secluded sector. The dependence of the parameters characterizing the
transition on the temperature ratio between the two sectors is elaborated in section 8.2.1,
and section 8.2.2 presents the corresponding effect on the detectability of the spectrum.
In section 8.3 we finally demonstrate that decoupled hidden sectors that satisfy the cos-
mological constraints on Neff may still feature a first-order PT observable via GWs by
considering two toy models. We then conclude in section 8.4.
8.1. Decoupled Hidden Sectors
As already stated above, the radiation energy density of BSM sectors at temperatures
below an MeV is rather strongly constrained. These constraints are typically phrased in
terms of the effective number of neutrino species, Neff. It parametrizes the new physics’
contributions to the energy density of the early Universe as if these were originating
from additional neutrino generations. In other words, the energy density is rewritten by
splitting off the photon contribution and tr[eating a(ll ot)her sp]ecies as neutrinos,
π2 ∑ 43
ρrad(T ) = 30 g
4
iTi = 1 +
7 4
8 11 Neff ργ(T ) . (8.1)
i
Here, the sum runs over all relativistic species, which can in principle all have different
temperatures Ti, and gi are the respective effective number of DOFs. As the energy,
entropy and number density of relativistic fermions is by a factor of 78 lower than the one
of a boson, the effective DOFs for fermions include this 78 factor.
As we will argue in the following, the constraints onNeff require sub-MeV hidden sectors
to be decoupled from the photon bath, so that they can have a different temperature. The
constraints can then be evaded by assigning the hidden sector a lower temperature. Since
the radiation energy density goes with the fourth power of temperature, this suppresses
the new physics contribution to Neff quite efficiently. We therefore define the temperature
ratio ξh as the ratio of the hidden sector to the photon temperature,
= Thξh (8.2)
Tγ
and consider ξh < 1 in the following.
To understand the process of particle decoupling as well as how the temperature ratio
ξ3ν = 4/11 in eq. (8.1) arises, we will first briefly recapitulate the decoupling of neutrinos
in the SM in section 8.1.1. Subsequently, we will review Neff and the various constraints
on this parameter in section 8.1.2. Finally, section 8.1.3 is devoted to the different hidden
sector scenarios we are going to consider.
8.1.1. Neutrino Decoupling and Electron-Positron Annihilation
In order to maintain equilibrium, the interactions between a given particle species and
the other particles in the plasma of the early Universe have to occur sufficiently fast to
compete against the Hubble expansion. A species therefore decouples from the plasma
117
8. Constraining Secluded Hidden Sectors with Gravitational Waves
when its interaction rate Γ drops below the Hubble rate H. Recall from our discussion of
the decoupling of gravity in the beginning of chapter 6 that the interaction rate is given
by Γ = nσv. For neutrinos, these interactions are annihilation to and scattering off SM
leptons with σ ∝ G2F , where GF is Fermi’s constant. Due to their extremely low mass,
neutrinos are still relativistic upon decoupling, such that n ∼ T 3, v = c, and σ ∼ G2 2FT
on dimensional grounds. Further taking H ∼ T 2/MP we obtain that neutrinos decouple
at a temperature around ( )− 1
T 2 3ν−dec. ∼ GFMP ∼ 1MeV . (8.3)
After decoupling, the decoupled species and the particles in the thermal bath can in
principle have different temperatures. These are determined by the conservation of co-
moving entropy density, which is conserved separately in the two sectors. In the case of
neutrino decoupling
2π23 = 3 3 2π
2
a sν a 45 gν Tν = const and a
3s 3 3th = a 45 gth Tth = const , (8.4)
where Tν (Tth) and gν (gth) are the temperature and effective number of DOFs of the
neutrinos (thermal bath), respectively. Then g = 7 21ν 8 × Nν × 2 = 4 , where Nν = 3 is
the number of SM neutrino generations, and the remaining thermal bath is composed of
photons and electrons, i.e. g = 2 + 7th 8 × 4 =
11
2 .
As the neutrinos and the bath have the same temperature at decoupling, they will also
have the same temperature directly after decoupling. This however changes as soon as
some particle species becomes non-relativistic and annihilates, altering the effective num-
ber of DOFs. The entropy of the annihilating species is then transferred to the remaining
particles, causing the temperature of the sector to drop slower than corresponding to
the usual 1/a dependence. Indeed, electrons and positrons become non-relativistic at
T ' me = 511 keV, just after neutrino decoupling, causing gth to drop to gγ = 2. As the
photons are the last particles remaining in the thermal bath, we will always characterize
it in terms of the photon temperature Tth = Tγ from now on.
Let us now compare the photon bath just before e± annihilation (with quantities in-
dexed by ‘(1)’) and at the time when the annihilation has completed (with quantities
indexed by ‘(2)’). If we assume that the heating of the photon bath due to the entropy
transfer from the annihilating electrons and positrons is quasi-instantaneous, we can ne-
glect the change of the scale factor a of the Universe during this process. Conservation
of co-moving entropy in the bath then implies that the photon temperature has changed
by
(2) ( (1)) 1 ( ) 1
Tγ = g
3
th 3
(1) (2) =
11
. (8.5)
T g 4γ th
In the neutrino sector on the other hand, the number of effective DOFs remains constant,
and so does the temperature, i.e. (2) = (1) = (1)Tν Tν Tγ . Therefore, they now have a
temperature different from the one of the photons(, wit)h a temperature ratio of(2) (1) 1
≡ Tν = Tγ = 4
3
ξν (2) (2)
T 11
. (8.6)
γ Tγ
118
8.1. Decoupled Hidden Sectors
After e± annihilation, the temperatures of photons and neutrinos again evolve as T ∼ 1/a,
and the temperature ratio of (4/11)1/3 is preserved.
Note that we have assumed that neutrino decoupling happens (quasi-)instantaneously,
and that the full electron entropy is dumped into photons. However, as the time at which
electrons become non-relativistic is very close to the time of neutrino decoupling, this is
not entirely true. When electrons and positrons annihilate, the neutrinos are not com-
pletely decoupled yet, and a small fraction of the electron entropy is also transferred to the
neutrinos. As a result, the neutrino spectra feature small non-thermal distortions [374].
8.1.2. Effective Number of Neutrino Species
Let us now consider the energy den[sity of the U]nivers[e after e
± annihila]tion,
π2 ( ) 4 ( ) 4
ρ = g T 4 + g T 4 = 1 + gν Tν 7 4
3
rad 30 γ γ ν ν 4 ργ = 1 + 8 Nν 11 ργ . (8.7)gγ Tγ
We can now interpret additional relativistic DOFs in terms of additional neutrino gener-
ations, defining the effective number of neutrino species as
8 − ( ) 4= ρrad ργ 11 3Neff 7 4 . (8.8)ργ
In the SM we have NSMeff = 3.046 [374], where the deviation from Nν = 3 originates from
entropy leakage due to non-instantaneous neutrino decoupling.
As the effective number of neutrino species parameterizes new physics’ contributions
to the radiation energy density, it directly effects the expansion rate of the Universe. It
is therefore cosmologically constrained from two types of observations: measurements of
the relative abundance of light elements at the time of Big Bang Nucleosynthesis (BBN),
and the power spectrum of the cosmic microwave background (CMB) [60, 375].
The production of light elements, in particular of 4He, from free neutrons and protons
in the early Universe occurred at temperatures around TBBN ∼ 1MeV. This process of
BBN [376] is well understood [377–379] and provides the earliest stage of our Universe that
we have probed reliably [68, 380]. As eventually all free neutrons end up bound in 4He
nuclei to a good approximation, the final helium abundance (relative to the total nucleon
abundance) is essentially determined by the neutron-to-proton ratio at BBN. This ratio
depends on the time or temperature at which the interactions interconverting neutrons
and protons freeze out, which in turn is sensitive to the Hubble rate around T ∼ 1MeV.
Additional relativistic species lead to a higher freeze-out temperature and thereby a larger
neutron-to-proton ratio, increasing the helium abundance. Measurements of the relative
abundance of 4He can therefore be used to put limits on the effective number of neutrino
species [381, 382], imposing a 95% confidence level (CL) constraint of [60]
Neff = 2.95+0.56−0.52 (BBN) , (8.9)
assuming that Neff is constant during BBN [236, 378, 379, 381] (see e.g. ref. [383] for the
impact of relaxing this assumption).
119
8. Constraining Secluded Hidden Sectors with Gravitational Waves
Complementary constraints on Neff around the time of recombination at Trec ∼ 0.3 eV
can be extracted from the temperature and polarization power spectrum of the CMB.
Deviations from NSMeff lead to changes in the heights and positions of the acoustic peaks,
as well as in the damping tail of the CMB spectrum. These modifications are caused by an
increase of the photon diffusion scale (Silk damping scale) at recombination as well as the
temperatures of photon decoupling and matter-radiation-equality due to the presence of
additional relativistic DOFs [68, 270]. The current 95% CL constraint from 2018 Planck
data including polarization and baryon acoustic oscillation (BAO) measurements is [60]
Neff = 2.99+0.34−0.33 (CMB) . (8.10)
Further taking into account data from local measurements of the Hubble rate H0 to-
day [112], which provide a value for H0 that is conflict with the one determined by
Planck, the CMB 95% CL limit on Neff relaxes to [60]
Neff = 3.27± 0.30 (CMB+H0) . (8.11)
8.1.3. Hidden Sector Cosmology
The Neff bounds discussed above strongly constrain the relativistic particle content of
hidden sector models at sub-MeV temperatures. In the following, we will review various
generic scenarios for such models and the corresponding constraints. The effective number
of relativistic DOFs in the hidden sector shall be denoted by gh. For the case of sectors
that are decoupled from the photon bath, let ξh further be the ratio of the hidden-sector
to photon temperature, as defined in eq. (8.2).
Hidden sectors in thermal contact with the SM
Any additional relativistic DOF that is in thermal equilibrium with the photon bath
throughout the BBN (and e± annihilation) epoch is inconsistent with the bounds on
Neff. Even a single real scalar DOF (gh = 1) would produce a deviation of Neff from
the SM value of ∆Neff = 2.2 and thereby be in conflict with the limits from eqs. (8.9)
to (8.11). We therefore discard this scenario in the following.
If, however, around the time of neutrino decoupling thermal contact between the hidden
sector and the SM is predominantly established via its interactions with the neutrinos, the
hidden sector will decouple from the photon bath along with the neutrinos, remaining in
thermal equilibrium with the latter. In this ca(se ξh = ξ)ν = (4/11)
1/3 after e± annihilation,
and Neff is modified to1
Neff = NSMeff 1 +
gh
. (8.12)
gν
Once the hidden sector particles become non-relativistic, they annihilate and transfer
their entropy into neutrinos. This heats the neutrino sector, modifying the neutrinos’
temperature ratio by a factor [(gh + gν)/gν ]1(/3, i.e. [)384]4
= SM 1 + gh
3
Neff Neff . (8.13)gν
1Throughout this chapter, we will use NSMeff = 3.046 instead of Nν = 3 despite neglecting the entropy
leakage in our calculations, such that we reproduce the SM value in the limit gh → 0.
120
8.1. Decoupled Hidden Sectors
7 7
BBN 95% CL BBN 95% CL
6 CMB+H0 6
5 5
Higgsed Dark Photon Higgsed Dark Photon
4 4
CMB+H0
3 3
Singlet Scalars Singlet Scalars
2 2
CMB
1 1
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
ξ ξinith h
(a) decoupled hidden sector (b) ν-quilibration
Figure 8.1: 95% CL limits on Neff from BBN (eq. (8.9), orange shaded region),
CMB (eq. (8.10), blue dashed line), and CMB+H0 (eq. (8.10), blue shaded region)
in the decoupled hidden sector (left) and ν-quilibration (right) scenario, as a function
of the hidden-sector effective number of DOFs gh and temperature ratio ξh. The dashed
and dash-dotted line indicate the value of gh in the singlet scalars (section 8.3.1) and
Higgsed dark photon (section 8.3.2) toy model, respectively.
Even the least stringent 95% CL constraint considered here, eq. (8.11), limits the number
of hidden sector DOFs to gh . 0.90 (0.66) if they are (non-)relativistic at recombination.
As a consequence, no additional light DOF can remain in thermal equilibrium with the
SM (with neither photons nor neutrinos) after neutrino decoupling.
Completely decoupled hidden sectors
Let us therefore consider the case that the hidden sector is completely decoupled from
the SM with an arbitrary temperature ratio ξh. This scenario can arise if the the hidden
sector and the SM never were in equilibrium at all, in which case the temperature ratio
is determined by the initial conditions after inflation, or if it decoupled from the SM
early on (well before BBN), so that ξh is set via subsequent annihilation processes of
non-relativistic species in the two sectors. We then obtain
( ) 4
3
Neff =
4 11
NSM 4eff + 7 4 gh ξh . (8.14)
Note that, when the particles of such a sector become non-relativistic, they typically need
to annihilate into a form of dark radiation to avoid over-closure of the Universe.
The BBN and CMB constraints on the temperature ratio ξh and number of effective rel-
ativistic DOFs in a fully decoupled hidden sector are shown in fig. 8.1a. The solid orange
and blue lines depict the 95% CL bound from BBN (eq. (8.9)) and CMB+H0 (eq. (8.11)),
respectively, the CMB only limit (eq. (8.10)) is shown as the dashed blue line. As can be
seen in the plot, even with only a single additional relativistic DOF (gh = 1), the hidden
121
gh
gh
B
CM
8. Constraining Secluded Hidden Sectors with Gravitational Waves
( )
T h-dec T ν-dec h-eq h-annγ γ T TBBN γ γ
high T low T
T e
±-ann
γ
Figure 8.2: Sequence of events in the ν-quilibration scenario, labeled by the corre-
sponding temperature Tγ of the photon bath. In chronological order: hidden sector
decoupling (T h−decγ , if ever in thermal contact with the photons), neutrino decou-
pling (T ν−decγ ) and electron-positron annihilation ( e
±
T −annγ ), hidden sector-neutrino
(re-)equilibration (T h−eqγ ), and hidden sector annihilation (T h−annγ ).
sector is required to be at least a factor of ξh ' 0.6 – 0.7 colder than the visible sector in
order to satisfy the constraints.
Hidden sectors equilibrating with neutrinos (ν-quilibration)
While we established above that the hidden sector cannot be in thermal equilibrium
with the SM neutrinos around the time of BBN, it may still equilibrate with the latter
after they decoupled from the photon bath. This scenario, dubbed ν-quilibration in the
following, has been considered in refs. [385, 386] and has the advantage that the hidden
sector can annihilate into neutrinos when becoming non-relativistic, as we will discuss
later. Let us consider the sequence of events depicted in fig. 8.2.2
The hidden sector either never was in thermal contact with the photon bath, or de-
coupled from it well before BBN at the (photon) temperature T h−decγ , and may therefore
have a temperature different from the visible sector. We denote the hidden sector tem-
perature ratio at the time of neutrino decoupling (at the photon temperature T ν−decγ ) by
ξinith . Shortly after that, electrons and positrons become non-relativistic and annihilate
at e±T −annγ , heating up the photon bath. Neglecting the leakage of entropy from the
electrons into the neutrinos, this process changes the temperature ratio of the neutrinos
from ξν = 1 to the SM value ξν = ξSMν = (4/11)1/3 determined in eq. (8.6). The hidden
sector temperature ratio is modified in the same way to ξ = ξSM ξinith ν h .
Subsequently, towards the end of the BBN era or later, the hidden sector (re-)enters
equilibrium with the neutrinos at the photon temperature T h−eqγ . For simplicity we as-
sume that no hidden sector DOFs have become non-relativistic after neutrino decoupling,
so that ξh is only modified by e± annihilation. Further assuming that equilibration oc-
curs quasi-instantaneously, the process is governed by conservation of energy. This implies
that g 4 4 4νTν + ghTh = (gν + gh)Tν+h, where Tν and Th are the neutrino and hidden sector
temperature at the beginning of equilibration, and Tν+h is the temperature of the com-
2Since we neglect the leakage of entropy from e± annihilation into neutrinos, it actually does not matter
whether the equilibration occurs between neutrino decoupling and e± annihilation or shortly after
the latter event: the thermalization process does not care about the photon bath, and the photon
heating due to annihilation is independent of the decoupled sectors. The resulting Neff constraints are
therefore identical.
122
8.2. Gravitational Waves from Decoupled Hidden Sectors
bined sector after the process has completed. The corresponding temperature ratio ξν+h
is then given by
 ( )4 ( )  1g ξSM + g ξSMξinit
4 4 [ ]− 1 [ ( ) ] 1ν ν h ν h gh 4 4 4
ξν+h = +  = ξSMν 1 + 1 + gh ξinit . (8.15)g g g g hν h ν ν
With respect to the SM, the effective number of neutrino species is then modified by
a factor (ξ /ξSMν+h ν )4 accounting for the different temperature ratio, and by a factor
(gh + gν)/gν from the additional DOFs. [Therefore, after] the hidden sector thermalizedwith the neutrinos,
g ( )h 4
N SM initeff = Neff 1 + ξh . (8.16)gν
Subsequently, the hidden sector particle content will become non-relativistic at some
temperature T h−annγ . If the hidden sector had lost thermal contact to the neutrino bath by
then, it would have frozen-out while still relativistic, and therefore most likely overclose
the Universe. We will hence assume that the hidden sector is still in equilibrium with the
neutrinos when becoming non-relativistic. Its entropy is then transferred to the neutrino
bath, heating it by a factor [(gν + gh)/g ]1/3ν . The temperature ratio of the neutrinos
after hidden sector a[nnihilat]ion then becom[es+ 1 ] 1 [ ( ) ] 1gν gh 3
ξ SM
gh 12 gh init 4 4
ν = ξν+h = ξν 1 + 1 + ξh . (8.17)gν gν gν
Unless the hidden sector contains sub-eV particles, annihilation will occur before the time
of recombination, and the value of Neff constrained by the CMB is[ ] 1 [ ( ) ]
N = NSM gh
3
1 + 1 + gh ξinit
4
eff eff h . (8.18)gν gν
The BBN (orange shaded region) and CMB (blue shaded region and blue dashed line)
95% CL constraints on Neff in this scenario are shown in fig. 8.1b. The abscissa shows
the hidden sector temperature ratio ξinith before the e± annihilation, and the ordinate is
the effective number of hidden sector DOFs gh. We employ eqs. (8.16) and (8.18) for
the BBN and CMB limits, respectively, assuming that the hidden sector annihilates after
BBN3 and before recombination.
8.2. Gravitational Waves from Decoupled Hidden Sectors
We will now explore how first-order PTs as well as their respective GW signal and its
detectability are modified if they occur in a decoupled hidden sector with a temperature
different from the one of the photon bath. While all properties of the hidden sector are
naturally described in terms of its temperature Th, we assume that our Universe remains
dominated by the visible sector, so that it is more intuitive to characterize the latter
3 More precisely: after the initial formation of light elements (D and 4He) at Tγ ∼ 0.1MeV [387].
123
8. Constraining Secluded Hidden Sectors with Gravitational Waves
in terms of the photon temperature Tγ . In particular, we usually express the radiation
energy density of the Universe, which eventually determines the Hubble rate, in terms of
Tγ , i.e.
2
ρ ( ) = πT g (T )T 4 , with g (T ) = gSM 4rad γ 30 ? γ γ ? γ ? (Tγ) + gh ξh . (8.19)
We will mostly consider PTs occurring between the epoch of BBN and before recombina-
tion, so the the effective number of relativistic DOFs is then given by g? = gγ+g ξ4ν ν+gh ξ4h.
Similarly, we express the total entropy density of the Universe in terms of the photon
temperature, s(T ) = 2π2γ 45 g?S(Tγ)T 3γ with g 3?S = gγ + gν ξν + gh ξ3h at low temperatures.
A phenomenologically important consequence of a decoupled hidden sector is that it
cannot interact efficiently with the SM plasma. Whereas this could in principle mean that
the vacuum bubbles run away, and that the corresponding SGWB is generated from the
vacuum-bubble collisions only, we will assume in the following that the models considered
here feature a hidden plasma which exerts friction on the vacuum bubbles and in which
sound waves can be induced in the same way as in the SM plasma. We therefore work on
the premise that the formulae for the plasma contributions to the GW spectrum (at the
time of production) presented in section 6.4.3 still apply. With respect to hidden sectors
in equilibrium with the SM but otherwise identical internal properties, the spectrum is
then only altered through the change of the parameters entering the calculation of the
spectrum and modifications of the red-shifting, derived in the following.
8.2.1. Temperature Ratio Dependence
To facilitate comparison with the case of a hidden sector in thermal contact with the
photon bath (ξh = 1), we here express all PT parameters in terms of the hidden sector
temperature Th and the temperature ratio ξh. All parameters internal to the dark sector
(i.e. masses, couplings, etc.) are kept fixed.
The effective potential Veff(φ, Th), the critical bounce action SE,3(Th)/Th, and the ther-
mal tunneling rate per unit volume Γ(T 4h) ∼ Th exp (−SE,3/Th) only depend on the hidden
sector properties. They therefore do not require any changes in their definition (apart
from making clear that they should be considered as functions of the hidden sector tem-
perature Th). The first modification required for decoupled hidden sectors is therefore
in the nucleation condition eq. (6.9). Recall that the nucleation temperature is defined
as the temperature at which Γ(Tn,h) ∼ H4(Tn,γ), i.e. by comparing the nucleation rate
to the Hubble rate. Here, Tn,h and Tn,γ are the temperatures of the hidden and visible
sector, respectively, at the time of nucleation. As H ∼ T 2γ we therefore pick up a factor ξ8h
when expressing everything in terms of the hidden sector temperature, so that eq. (6.9)
is modified to ( ) ( )
SE,3(Tn,h) ∼ 146− 4 log Tn,h g?,n100GeV − 2 log 100 − 8 log ξh , (8.20)Tn,h
where g?,n = g?(T −1n,γ) = g?(ξh Tn,h) is the effective number of radiative DOFs at nucle-
ation. Due to the weak logarithmic dependence on ξh, the corresponding change in the
nucleation temperature is typically negligible.
124
8.2. Gravitational Waves from Decoupled Hidden Sectors
The most prominent effect on the SGWB stems from the energy budget given by
eq. (6.10). Recall that the latent heat E is defined in terms of the effective potential,
and therefore natively depends on the hidden sector temperature. The radiation energy
density on the other hand is given by eq. (8.19). Thus, compared to the case of a hidden
sector in equilibrium with the SM, the energy budget is suppressed by the fourth power
of the temperature ratio if we keep E and Tn,h fixed, i.e.
≡ E(Tn,h) ≈ E(Tn,h)α 4( ) −4 = ξh αh , (8.21)ρrad Tn,γ ρrad(Tn,h) ξh
where αh ≡ [α]ξ =1 is the energy budget if both sectors have the same temperature, andh
we neglected potential changes in g?.
The inverse time scale of the PT normalized to the Hubble rate, β/H∗, on the other
hand, is independent of the temperature ratio. This can be easily seen from eq. (6.11).
Once normalized to the Hubble rate, the time derivative in the definition of β = Γ̇/Γ can
be traded for a derivative with respect to the scale factor a. As co-moving entropy is
conserved in the hidden and visible sector separately, the scale-factor derivative can be
expressed as a derivative with respec[t to either ted ]
mperature, so that we can take
β = SE,3Thd . (8.22)H∗ Th Th Th=Tn,h
Beside the input parameters which characterize the PT discussed above, the tempera-
ture ratio also modifies the amount of red-shifting the GW spectrum experiences. While
the temperature factors in the red-shifting of the amplitude in eq. (6.17b) cancel, the
frequency red-shift times the Hubble rate at the time of the transition is proportional
to the nucleation temperature in the photon bath, and therefore picks up a factor of
1/ξh compared to the ξh = 1 case. Adapting the normalization to the temperature range
considered here, and keeping the differentiation between g? and g?S as these differ due to
decoupling, eq. (6.17) becomes
( )( )10 3 ( ∗)12
h∗ = 68.8 pHz
Tn,h g?S g? −1
1MeV ∗ 2 ξh , (8.23a)( ) g(?S40 3 ∗)
R = 2.473× 10−5 g?S g?
g∗?S 2
. (8.23b)
Note that in the effective number of entropic DOFs today (or, to be more precise, at
matter-radiation equality), g0 = 2 + 7N (ξ0?S 4 ν ν)3 + g 0 3h(ξh) , the values of the temperature
ratios ξ0ν and ξ0h may differ from the values at the time of the PT if a species becomes
non-relativistic and transfers its entropy to the remaining particles. Further note that, if
the hidden sector decouples from the photon bath before the PT and remains decoupled
until today, it is sufficient to consider the SM entropic DOFs only, as co-moving entropy
is then conserved separately in each sector.
Finally, in the definition of the run-away criterion we need to take into account that the
explicit temperature dependence displayed in eq. (6.12) is on Tn,h, whereas ρrad implicitly
depends on Tn,γ . Therefore, the ratio α/α∞ is independent of the temperature ratio.
125
8. Constraining Secluded Hidden Sectors with Gravitational Waves
10−9
10−12 ξ−1h
10−15
10−18 Figure 8.3: Illustration of the
10−21 dependence of the SGWB from
a hidden sector PT on the tem-
10−24 − − − − − − perature ratio ξh = Th/Tγ ,10 6 10 5 10 4 10 3 10 2 10 1 100 keeping Tn,h, β/H∗ and αh
f [Hz] fixed.
The impact of increasing the photon temperature with respect to the hidden sector
temperature is illustrated in fig. 8.3. When the nucleation temperature Tn,h in the hidden
sector, the latent heat release (i.e. the energy budget in the case that ξh = 1, αh), and
the inverse time scale normalized to the Hubble rate, β/H∗, are kept constant, and the
temperature ratio ξh is decreased, the GW spectrum is affected in two ways. First, the
amplitude recedes due to the ξ4h suppression of the energy budget α in eq. (8.21). Second,
the spectrum is shifted to slightly higher frequencies by the ξ−1h dependence of the peak
frequency from red-shifting through eq. (8.23a).
8.2.2. Sensitivity
As discussed in section 8.1, constraints on the effective number of neutrino species require
sub-MeV hidden sectors to be colder than the photon bath. We will therefore now discuss
how the detectability of first-order PTs in hidden sectors via GWs is affected if the hidden
sector is decoupled from the SM with a temperature ratio ξh < 1.
Figure 8.4 depicts the sensitivity of various GW observatories to the SGWB produced
in a PT occurring at a nucleation temperature Tn,h in a decoupled hidden sector with
temperature ratio ξh. The colored regions can be probed by the respective experiments.
We present the projected regions of sensitivity for SKA, LISA, (B-)DECIGO, BBO,
and ET . For SKA we assume observation periods of 5, 10 and 20 years. The hidden
sector temperature around which BBN occurs, i.e. corresponding to a photon temperature
of TBBNγ = 1MeV, and below which Neff constraints apply, is indicated by the black
line.4 Throughout this chapter we assume that the SGWB from super-massive black
hole binaries (SMBHBs) will be resolved and subtracted from our signal, otherwise the
sensitivity of SKA is diminished significantly.
The left panel of fig. 8.4 shows the case of a runaway transition, whereas the right plot
considers a non-runaway GW spectrum dominated by the sound wave and turbulence
contributions in a hidden plasma. We fix the latent heat and the transition rate divided
by the Hubble rate such that αh = 0.1 and β/H∗ = 10, where αh is the corresponding
value of α assuming equal temperatures in the hidden and visible sector, and assume
4Note that the jagged features in the sensitivity curves, such as the particularly distinct spike close to
the BBN line, are due to approximating the SM number of relativistic DOFs by a step function.
126
h2ΩGW
8.2. Gravitational Waves from Decoupled Hidden Sectors
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 BBN 0.2 BBN
10−6 10−3 100 103 106 109 10−6 10−3 100 103 106 109
Tn,h [GeV] Tn,h [GeV]
(a) runaway bubbles with α α∞ (b) non-runaway bubbles (α < α∞)
Figure 8.4: Projected sensitivity reach (colored regions) of PTAs (SKA), space-based
(LISA, (B-)DECIGO, BBO), and ground-based (ET) GW observatories as a function
of the nucleation temperature Tn,h in the hidden sector and the temperature ratio
ξh. The SKA limits are shown for 5, 10, and 20 years of observation. The black lines
indicate the hidden sector temperature below which BBN occurred and Neff constraints
apply. The left (right) panel assumes a (non-)runaway transition. We fix αh = 0.1 and
β/H∗ = 10 (see text for details).
g  gSMh ? . Note that fixing these parameters corresponds to fixing the properties of
the hidden sector, whereas α ∼ ξ4αh also depends on the visible sector temperature. We
further optimistically take the fraction of bulk motion converted into turbulence to be
εturb = 10%, and assume a bubble wall velocity of vw = 1 in both cases. The latter two
assumptions will be retained throughout this chapter.
If the hidden sector has the same temperature as the photon bath, an ample range
of nucleation temperatures can be probed. In particular, in the runaway case the entire
range 100 eV . Tn,h . 106TeV is accessible for the parameter values considered in fig. 8.4.
However, once the hidden sector temperature is reduced with respect to the photon bath,
the sensitivity recedes, mostly due to the ξ4h suppression of the energy budget. For very
low temperature ratios ξh . 0.15, the SGWB produced in the PT becomes undetectable
even in the most optimistic scenario.
As we are interested in the interplay between GW and Neff constraints, let us now focus
on PTs occurring after the onset of BBN, with a nucleation temperature in the photon
bath below Tn,γ . 1MeV. Since this is the region in which PTAs are sensitive, we only
consider projections for SKA in the following.
We have shown that decreasing the temperature ratio ξh reduces the prospects for
observing the SGWB from hidden sector PTs. Let us therefore adopt the maximal value
127
ξh
SKA
ξh
SKA
DECIGO
ET
O
DEC
IG
B- OBB
LISA
5 years
0 yea
rs
1
years20
ET
O O
B-DE
CIG IG
DE
C
LISA
O
BB
5 yea
rs
ears
10 y s
yea
r
20
8. Constraining Secluded Hidden Sectors with Gravitational Waves
103 102
2 0.2 0.310
SKA 0.3 10
1
0.4
SKA
101 0.4 s
ea
r 0.5
0.5 y
20
0.6 0.6
100 − 10
0
10 4 10−3 10−2 10−1 100 10−2 10−1 100
Tn,h [GeV] Tn,h [GeV]
(a) runaway bubbles with α α∞ (b) non-runaway bubbles (α < α∞)
Figure 8.5: Range of nucleation temperatures Tn,h and numbers of relativistic DOFs gh
in completely decoupled hidden sectors featuring PTs observable through GWs with
SKA (purple regions) after an observation period of 5, 10, and 20 years, respectively,
assuming αh = 0.1 and β/H∗ = 10. The temperature ratio ξh is set to the maximal
value complying with the BBN constraint on Neff in eq. (8.9), see eq. (8.24). The
constraint does not pertain to the gray colored region where the PT occurs before
BBN.
of ξh consistent with the BBN constraints on Neff, eq. (8.9). Assuming a completely
decoupled hidden sector, we can solv(e eq). (8(.14) for ξ)h, yielding1 14 3= 7 ∆Neff 4ξh 11 4 , (8.24)gh
where the 95% CL upper limit from BBN is ∆Neff < 0.46.
The values of the nucleation temperature Tn,h and number of relativistic DOFs gh
in the hidden sector that give rise to a GW signal observable in SKA, saturating the
Neff limit eq. (8.9) from BBN5 are shown in fig. 8.5. We again assume αh = 0.1 and
β/H∗ = 10. The left (right) plot considers a transition with (non-)runaway bubble
walls. The regions shaded in purple can be probed by SKA after 5, 10, and 20 years of
observation, respectively. In the gray colored region, the PT occurs before the onset of
BBN and is therefore unconstrained by Neff.
For transitions in the runaway regime, hidden sectors with only few relativistic DOFs
and nucleation temperatures Tn,h ∼ 1MeV may be probed by SKA after only 5 years of
observation, whereas in the non-runaway case, even hidden sectors with a single relativis-
tic DOF require observation periods of at least 10 years. The observational prospects
are most promising for early transitions close to the beginning of BBN. PTs at lower
temperatures require longer observation periods.
Whereas it appears odd at first sight that, according to fig. 8.5, for PTs close to BBN,
SKA is able to probe models with arbitrarily many additional DOFs, this is just an
5I.e. sitting on top of the orange line in fig. 8.1a.
128
gh
5 years
gh
10 years
ξh
> TBB
N
Tn,γ
V
= 10
0 ke
Tn,γ
ξh
TBBN
T >n,γ
10 yea
rs
100 ke
V
T ,γ =n
10 keV
T ,γ =n
20 yea
rs
1 keV
Tn,γ =
8.3. Toy Models
artifact of saturating the N 4eff bound. Equation (8.24) keeps gh ξh (and thereby Neff and
the total energy density of the Universe) fixed. Furthermore, keeping αh (the energy
budget for ξh = 1) constant does no longer correspond to fixing the latent heat E because
we vary gh. As ρ SM 4rad ∝ (g? + gh)Tn,h for ξh = 1, the actual value of α is not suppressed
by ξ4h, but approaches a constant for large gh, i.e.
( E ) (gSM= = ? + gh) ξ4h α SM 4h ghα 2 4 SM + 4 −−−−g−?→ gh ξhπ gSM + g ξ T 4 g g ξ gSM + 4 αh . (8.25)30 ? h g ξh n,γ ? h h ? h h
With eq. (8.9) we obtain gh ξ4h = 0.21 and α = 0.059 (for αh = 0.1). Since we further
fix β/H∗, the spectrum depends on ξh only via the red-shifting in eq. (8.23). Again,
saturating the Neff bound keeps g? constant, so that the only dependence on gh is in the
frequency red-shift.6 As a result, for g  gSMh ? the GW spectrum does not change along
lines of constant Tn,γ , where Tn,γ = 1/4Tn,h/ξh ∼ Tn,h gh .
8.3. Toy Models
To assess the question whether it is possible to construct cosmologically viable (i.e. satis-
fying the constraints on the effective number of neutrino species) sub-MeV hidden sector
models that feature a first-order PT observable through the corresponding SGWB, let
us explore the situation on the example of concrete benchmark models. We will consider
two simple toy models in the following. The first hidden sector consists of two singlet
scalars with two hidden DOFs and a potential barrier at tree-level, whereas the second
model is a Higgsed dark photon model with four DOFs and a loop-induced barrier. To
let the hidden sectors thermally decouple from the photon bath as required by the Neff
constraints, we neglect all portals to the visible sector. A discussion of the potential size
of the portal couplings can be found in ref. [388]. As Neff severely limits the number of
relativistic DOFs at the MeV scale, we expect that the models considered here provide
a low-temperature effective description of most viable, perturbative, ultraviolet (UV)
complete models with non-trivial sub-MeV dynamics.
8.3.1. Singlet Scalars
Our first toy model consists of two scalar particles that are singlets under the SM. In this
case, a barrier between two phases can be generated a tree-level from a cubic coupling
in the potential. As we will see in the following, the second scalar is required to let the
Universe first evolve into the false vacuum, so that it can subsequently tunnel into the
true one, producing a first-order PT. The model therefore introduces two hidden sector
DOFs.
The simplest possible hidden sector model would consist of only a single real scalar
particle S with tree-level potential
µ2 2
V (S) = S 2tree 2 S +
κS λS
3 S
3 + S44 = −
κSvS + λSvS 2 κS 3
2 S + 3 S +
λS
S44 , (8.26)
6 Recall that for decoupled sectors co-moving entropy is conserved in each sector. Therefore, in the g?S
factors we only need to consider the SM entropic DOFs.
129
8. Constraining Secluded Hidden Sectors with Gravitational Waves
T = 0
T1 > 0
T2 > T1
0 vS 0 vS
S S
(a) single real scalar field (b) two real scalars
Figure 8.6: Sketch of the effective potential and its evolution with temperature for a
single real scalar field (left, eq. (8.26)) and two real scalars (right, eq. (8.27)) as a
function of the field S.
where we imposed that the potential has a minimum at S = vS > 0 and used the minimum
condition V ′tree(vS) = 0 to eliminate µ2S . We need to require λS > 0 for the potential to be
bounded from below. The field dependent mass is m2S(S) = κS (2S−vS) +λ 2 2S (3S −vS),
so that κS > −2λSvS in order for S = vS to be a minimum. The potential further also
has extrema at S = 0 and S = −(κS + λSvS)/λS . At high temperatures, the potential is
approximately given by V 2 2eff ' VT ∼ Th mS , cf. eq. (6.23). It therefore only has a single
minimum at S = − κS3λ .S
We can now distinguish two cases. In the first case, S = 0 also corresponds to a mini-
mum. This requires κS < −λSvS . The global minimum is at S = vS for κS > −32λSvS ,
and at S = 0 otherwise. In the other case, the origin is a maximum, i.e. κS > −λSvS .
Then, the global minimum is at S = vS for κS < 0 and at S < 0 otherwise. Now, com-
paring the position of the high-temperature minimum to the position of the maximum,
we see that in both cases the high-temperature minimum is always at the same side of
the barrier as the global minimum of the tree-level potential. Therefore, as the Universe
cools down, the high-temperature minimum will always evolve into the true vacuum and
no first-order PT occurs.
This behavior is illustrated in fig. 8.6a. At high temperatures (red line), the effective
potential Veff(S, T ) only has a single minimum. Due to the cubic term in eq. (8.26),
this minimum is displaced from the origin. As the temperature drops (dark red line),
the minimum shifts towards S = vS and the potential develops a barrier between the
minimum and the origin. Finally, the temperature dependent vacuum evolves into the
global minimum of the zero-temperature potential (black line) at S = vS . The Universe
always resides in the true vacuum and no PT occurs.
To obtain a first-order PT we therefore need to add additional fields. Their field-
dependent masses can then be arranged such that the high-temperature minimum evolves
into the false vacuum, so that the field subsequently has tunnel to reach the true vacuum.
130
Veff(S, T )
Veff(S, T )
T2 > T1
T
1 > 0
T = 0
8.3. Toy Models
Thus, let us consider a hidden sector with two real scalar fields, S and A, both singlets
under the SM. For simplicity we impose a Z2 symmetry under which S is even and A is
odd. The corresponding potential reads
µ2S 2 κS 3 λS 4 µ
2
A 2 λA 4 2 λSAV 2 2tree(S,A) = 2 S + 3 S + 4 S + 2 A + 4 A + κSA S A + 2 S A . (8.27)
To simplify the analysis of the potential, we now only let S acquire a vacuum expectation
value (VEV), 〈S〉 = vS , while 〈A〉 = 0 so that the Z2 symmetry remains unbroken. Hence,
we impose µ2A, κSA ≥ 0, as well as λS , λA > 0 to ensure stability of the potential. We
fix A = 0 at all temperatures for the remainder of this chapter and treat the potential
as function of S alone. We checked that this is indeed valid for the parameter values
considered here.
The field dependent masses are given by
m2S(S) = µ2S + 2κSS + 3λ S2S and m2 2 2A(S) = µA + 2κSAS + λSAS , (8.28)
and from the minimum condition ∂ Vtree∂ S (vS) = 0 we can eliminate µS = −(κS +λS vS) vS .
At high temperatures, the potential now behaves as Veff ' VT ∼ T 2 2 2h (mS + mA). The
high-temperature minimum is at S = −(κS + κSA)/(3λS + λSA), so we can adjust κSA
and λSA to shift it towards the origin and obtain a first-order PT, as depicted in fig. 8.6b.
At high temperatures, the minimum of the potential (blue line) is close to the origin.
When the temperature decreases (dark blue line), a second minimum develops. At zero
temperature (black line), this new minimum has become the global one, whereas the
original high-temperature minimum still persists. We therefore now obtain a first-order
PT.
For a quantitative investigation of the PT in this model we use the numerical code
CosmoTransitions [364], implementing the daisy-resummed one-loop thermal effective po-
tential eq. (6.20). The counter-term potential is given by
∆ ( ) = δµ
2
S 2 + δκS 3 + δλSVct S 2 S 3 S 4 S
4 , (8.29)
on which we impose the renorm∣ alization conditions∣∣ ∣∂ (V + ∆Vct) ∣ 2 ∣CW = 0 and ∂ (VCW + ∆Vct)2 ∣∣∣ = 0 , (8.30)∂ S S=v ∂ SS S=vS
to ensure that the scalar VEV vS and mass mS remain at their tree-level values. We
additionally require
VCW(vS)− VCW(0) + ∆Vct(vS)−∆Vct(0) = 0 (8.31)
to fix the vacuum structure. However, as the one-loop quantum corrections shift the
local minimum at S = 0 slightly away from the origin, the latter condition may not be
sufficient if the minima are almost degenerate. Finally, the thermal masses of S and A
entering the daisy[corrections]Vring are [ ]
Π ( ) = λS + λSAT T 2S h 4 12 h and Π
λA λSA 2
A(Th) = 4 + 12 Th . (8.32)
131
8. Constraining Secluded Hidden Sectors with Gravitational Waves
1.4 α 1.4 β/H∗
〈 −1S〉0 = 0 10 〈S〉0 = 0
1.3 1.3
104
10 years 10 years
1.2 SKA 20 years 1.2 SKA 20 years
1.1 1.1 3
10−2 10
1.0 1.0
0.9 0.9 102
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
λSA λSA
(a) energy budget α (b) inverse time scale β/H∗
Figure 8.7: Energy budget α (left) and inverse time scale β/H∗ (right) for the singlet
scalars model in the κ̄ vs. λSA plane, where κ̄ ≡ −κS/(λSvS). We assume vS = 50 keV
and ξh = 0.66, complying with the Neff constraints in the ν-quilibration scenario
(cf. figs. 8.1b and 8.8). The remaining model parameters are set to λS = λA = 0.1 and
µA = κSA = 0.1 vS . The hatched regions enclosed by the solid black contours can be
probed by SKA after 10 and 20 years of observation, respectively. The dotted lines are
contours of constant α/α∞, with a runaway transition occurring for α/α∞ > 1.
The parameter space regions giving rise to a first-order PT as a function of the cubic
coupling κ̄ and the mixed quartic coupling λSA are shown in fig. 8.7, where we defined
κ̄ ≡ − κSλ v . Recall that, according to our tree-level analysis of the single-scalar model, theS S
potential has a local minimum at the origin and a global one at S = vS for 1 < κ̄ < 3/2.
We assume a scalar VEV of vS = 50 keV, as well as λS = λA = 0.1 and µA = κSA = 0.1 vS .
To avoid tension with the constraints on the effective number of neutrino species, we
further assume a temperature ratio of ξh = 0.66, which saturates the CMB+H0 bound
eq. (8.11) in the ν-quilibration scenario discussed in section 8.1.3.
A first-order PT occurs in the colored regions, where the color-coding in the two panels
indicates the respective value of the energy budget α (left panel) and inverse time scale
β/H∗ (right panel). In the white space above the colored region, the Universe remains in
the vacuum at S = 0, either because it is trapped in the false vacuum since the tunneling
probability is too low, or because quantum corrections render the minimum at S = 0 the
global minimum. The hatched region enclosed by the solid black lines is accessible by
SKA with an observation period of 10 and 20 years, respectively.
As κ̄ corresponds to the cubic self-coupling of S, it controls the height and width of the
barrier separating the false- and true-vacuum phase. It therefore provides the primary
handle to control the relative transition time scale β/H∗. The higher κ̄ the wider the
barrier, and thus, the slower the PT. Increasing κ̄ leads to a decrease in β/H∗. For κ̄ & 1.3,
the Universe is trapped in the S = 0 vacuum. The mixed quartic λSA (as well as the
mixed cubic κSA) on the other hand critically influences the high-temperature behavior of
the potential, in particular the location of the high-temperature minimum. Higher values
132
κ̄
κ̄
nawa
y
non-
ru
1
α/α∞
=
1.2
α/α∞=
α/α∞= 1.4
y
-run
awa
non
1
α/α∞
=
α/α∞=
1.2
α/α∞= 1.4
8.3. Toy Models
1.0 1.0
0.9 5 SKA
SKA
1 y
0.9
0 ea
0.8 20 y
rs
y ee a
0.8
a rr s
0.7 CMB+H s0 0.7 CMB+H0
0.6 CMB+H
ν-quilibration
0 0.6 CMB+H
ν-quilibration
0
BBN BBN
0.5 CMB 0.5 CMB
allowed for
0.4 fully decoupled hidden sector 0.4 allowed for
fully decoupled hidden sector
0.3 0.3
1.10 1.15 1.20 1.25 1.30 10−3 10−2 10−1 100
κ̄ vS [MeV]
(a) vS = 300 keV (b) κ̄ = 1.275
Figure 8.8: Impact of the temperature ratio ξh at the time of the PT on the sensitivity
of SKA (purple shaded regions) to first-order PTs in the singlet scalars model as a
function of κ̄ (left, for vS = 300 keV) and the VEV of S (right, for κ̄ = 1.275). We
set λSA = 2, λS = λA = 0.1, and µA = κSA = 0.1 vS . The Neff constraints exclude
the regions above the respective dashed lines. The regions excluded by the BBN and
CMB+H0 limit are shaded in gray.
of λSA lower the temperature at which the thermal corrections dominate.7 It therefore
directly influences the critical and nucleation temperature. Increasing λSA decreases Tn,h,
and thereby rises the energy budget α. Note that the latent heat release required to enter
a runaway regime, cf. eq. (6.12), grows as T 2n,h, so that runaway transitions occur for
lower values of λSA. Contours of constant α/α∞ are shown as dotted lines in fig. 8.7.
Figure 8.8 illustrates the impact of the temperature ratio ξh at the time of the PT on
the detectability of the corresponding SGWB by SKA as a function of the cubic coupling
κ̄ (left panel) and the VEV vS of S (right panel). We fix the mixed quartic coupling to
λSA = 2, as well as vS = 300 keV and κ̄ = 1.275 in the left and right plot, respectively.
The remaining parameters of the model are set as in fig. 8.7. SKA is sensitive to the
purple colored regions, where the different shades of purple correspond to 5, 10 and
20 years of observation time, respectively. Temperature ratios above the dashed lines are
excluded by the respective Neff constraints, cf. eqs. (8.9) to (8.11), if the hidden sector
is completely decoupled. The regions excluded by the BBN and CMB+H0 constraints
are further shaded in gray. The white space between the dotted lines is in agreement
with Neff limits if the ν-quilibration scenario is assumed. In the gray colored area at the
right-hand side of fig. 8.8a, the Universe remains in the vacuum located at the origin and
no PT occurs. In fig. 8.8b on the other hand, the gray region in the right corresponds to
PTs before BBN, alleviating the constraints on Neff.
While SKA may probe a large fraction of the parameter space displayed in fig. 8.8 if the
hidden sector has the same temperature as the photon bath, the sensitivity significantly
7In the high-temperature limit it contributes as ∼ λ 2SA Th S2 to the potential.
133
ξh
〈S〉0 = 0
ξh
e±-annih
T .n,γ > TBBN
years5
0 ye
ars
1
year
s
20
8. Constraining Secluded Hidden Sectors with Gravitational Waves
decreases when the temperature ratio is reduced. If we assume that the hidden sector is
completely decoupled from the SM throughout the post-BBN evolution of the Universe,
the Neff limits, eqs. (8.9) to (8.11), constrain the temperature ratio to values below 0.57
(BBN), 0.59 (CMB+H0), and 0.50 (CMB), respectively, cf. fig. 8.1a. If, on the other
hand, the hidden sector (re-)equilibrates with the neutrino sector between the on-set of
BBN and the PT, temperature ratios around ξh ∼ 0.66 can be realized. Note that this
value corresponds to the temperature ratio at the time of the transition, which relates
to the value before BBN constrained in fig. 8.1b via eq. (8.15). If the PT further occurs
between neutrino decoupling and e± annihilation, the values of ξh constrained by Neff
and at the time of the transition also differ in the completely decoupled scenario, with
the former related to the latter by a factor ξSMν . The time of e± annihilation is indicated
by a dark gray line in the right side of fig. 8.8b.
8.3.2. Dark Photon
As a second toy model, let us consider a complex scalar field charged under a dark U(1)
gauge group. In contrast to the singlet scalars model, the barrier between the phases here
originates from the loop corrections to the potential, in particular from the transverse
modes of the dark photon, as can be seen from eq. (6.23) and the fact that barriers from
the scalar and the longitudinal modes are cancelled by the corresponding ring corrections
in eq. (6.24). This model is very similar to the gauged lepton number model considered
in chapter 7. The SGWB from the gauge-symmetry breaking PT in this model has also
been studied in refs. [277, 389] for transitions at super-MeV scales, whereas we here focus
on sub-MeV PTs. The model features four physical DOFs in the hidden sector.
The most general Lagrangian of the hidden sector including its interactions with the
SM is given by
L ⊃ |D S|2 − 1 µ 4F
′ F ′µν ′ µνµν − 2FµνB − V (S, H) (8.33)
where S and H are the dark and SM Higgs fields, respectively. The covariant derivative of
the complex scalar S is Dµ S = (∂µ + ig ′ ′DAµ)S, where A denotes the dark photon with
the corresponding gauge coupling gD, and F ′µν = ∂µA′µ − ∂ ′νAµ is the dark field strength
tensor, whereas Bµν is the SM hypercharge field strength. The tree-level potential reads
V (S,H) = −µ2S |S|2 − 2 | |2 +
λS |S|4 + λHµ H 4H 2 2 |H| + λ |S|
2
HS |H|2 . (8.34)
√
We decompose the dark Higgs into its real and imaginary part, S = (S + iσ)/ 2, and
choose the phase of S such that it develops a VEV along its real part only, with 〈S〉 = vS .
Since we want the hidden sector to decouple from the SM, the kinetic mixing parameter
 and the Higgs portal coupling λHS need to be negligibly small. We therefore assume
λHS =  = 0 in the remainder of this chapter.
The field dependent masses of the dark Higgs boson S, the Goldstone boson σ and the
dark photon A′ are
m2S(S) = −µ2 +
3
λ S2 , m2S 2 S σ(S) = −µ
2 + 1λ S2S 2 S , and m
2
A′(S) = g2 S2D , (8.35)
134
8.3. Toy Models
where we can eliminate µ2 = λS 2S 2 vS from the minimum condition ∂V/∂S = 0.
We again use CosmoTransitions [364] to investigate the PT in this model. We impose
the renormalization conditions in eq. (8.30) to fix the paramters of the counter-term
potential,
2
∆Vct(S) = −
δµS 2 + δλS2 S 8 S
4 . (8.36)
The Debye masses of the sca[lars and th]e longitudinal mode A
′
L of the dark photon are
2 2
ΠS(Th) = Π
λS gD 2 gD 2
σ(Th) = 6 + 4 Th and ΠA
′ (Th) = 3 Th . (8.37)L
Since the bubble wall friction induced by gauge bosons acquiring a mass in a PT, such
as the dark photon in the case under consideration, hinder the bubbles from entering the
runaway regime [325], we only consider non-runaway bubbles in this model.
In fig. 8.9 the energy budget α (left panel) and inverse relative time scale β/H∗ (right
panel) are shown as a function of the quartic coupling λS and the gauge coupling gD.
The VEV of S is set to vS = 40 keV. Since the CMB+H0 constraints exclude this model
if we assume the ν-quilibration scenario, cf. fig. 8.1b, we now consider the case of a
completely decoupled hidden sector. The Neff constraints then require a temperature
ratio of ξh = 0.48 at the time of the PT. A first-order PT occurs in the colored regions,
where the color indicates the respective value of α and β/H∗. In the white space above
the colored region, the dark gauge symmetry remains unbroken, i.e. 〈S〉 = 0 at T = 0,
whereas in the white area in the lower right corner, the transition is a cross-over. The
solid black lines indicate the prospective reach of SKA after an observation period of 10
and 20 years, respectively, where the accessible regions are hatched.
As already discussed in the context of the gauged lepton number model in section 7.1.2,
the potential barrier is generated by the thermal corrections (cf. eq. (6.23)) from the
transverse modes of the dark photon field A′. Therefore, increasing the gauge coupling
gD increases the barrier, such that the transition becomes slower and more energetic.
For very large values of gD, the Universe is stuck in the symmetric phase, whereas for
low gD the barrier is shallow and disappears before tunneling, leading to a smooth cross-
over. Increasing the quartic λS on the other hand enhances the tree-level potential,8 and
thereby decreases the relative size of the barrier. As a result, a compensating increase of
gD is required to sustain the PT dynamics.
The effect of the temperature ratio ξh at the time of the transition is shown in fig. 8.10,
varying the dark gauge coupling gD (left, for vS = 300 keV) and VEV vS (right, for
gD = 0.7). The quartic couping is set to λS = 0.01 in both cases. The parameter regions
accessible to SKA are colored in purple, indicating the required period of observation
(5, 10 or 20 years) via the respective shading. The Neff limits on ξh, assuming that
the hidden sector is decoupled at the time of BBN and thereafter, are indicated by the
horizontal dashed back lines. Values of ξh above these lines are excluded at 95% CL
by the respective constraint. In the case of the BBN and CMB+H0 limit, the excluded
regions are further shaded in gray.
8Since we fix the VEV we can rewrite the tree-level potential as Vtree(S) = λS8 (S
2 − 2 v2 )S2S .
135
8. Constraining Secluded Hidden Sectors with Gravitational Waves
1.0 α 1.0 β/H∗
100
0.8 0.8 105
0 −1 0
〉 =
10 =
0.6 0〈S 0.6 〉
0
〈S
10−2 104
0.4 0.4
A A
SK 10−3 SK
0.2 ss-ov
er 0.2 ss-ov
er 3
cro ro
10
− c4
10−4 10−3 10− −
10
2 10 1 10−4 10−3 10−2 10−1
λS λS
(a) energy budget α (b) inverse time scale β/H∗
Figure 8.9: Energy budget α (left) and inverse time scale β/H∗ (right) for the dark
photon model in the gD vs. λS plane. We assume vS = 40 keV and ξh = 0.48, com-
plying with the Neff constraints for a completely decoupled hidden sector (cf. figs. 8.1a
and 8.10). The hatched regions enclosed by the solid black contours can be probed by
SKA after 10 and 20 years of observation, respectively.
1.0 1.0
20
y
0.8 ea SKArs 0.8 SKA
0.6 0.6
CMB+H0 CMB+H0
BBN BBN
0.4 CMB 0.4 CMB
allowed for
0.2 fully decoupled hidden sector 0.2 allowed for
fully decoupled hidden sector
0.50 0.55 0.60 0.65 0.70 0.75 10−3 10−2 10−1 100 101
gD vS [MeV]
(a) vS = 300 keV (b) gD = 0.7
Figure 8.10: Impact of the temperature ratio ξh at the time of the PT on the sensitivity
of SKA (purple shaded regions) to first-order PTs in the dark photon model as a
function of gD (left, for vS = 300 keV) and the VEV of S (right, for gD = 0.7), setting
λS = 0.01. The Neff constraints exclude the regions above the respective dashed lines.
The regions excluded by the BBN and CMB+H0 limit are shaded in gray.
136
ξh gD
10
2 y0 ea
y re sars
〈S〉0 = 0
ξh gD
10
2 y0 ea
y re sars
T ±n, eγ > -annT ih.BBN
years5
10 ye
ars
ars
20 ye
ar
s
ye rs5
ye
a
10
8.3. Toy Models
Similar to the situation in the singlet scalars model shown in fig. 8.8, fig. 8.10 indicates
that SKA can access a large fraction of the parameter space displayed in the figure if
both sector have the same temperature (ξh = 1), whereas the sensitivity is reduced as ξh
decreases. In order to comply with the constraints on Neff, eqs. (8.9) to (8.11), the hidden
sector has to be colder than the photon bath by a factor of 0.48 (BBN), 0.49 (CMB+H0),
and 0.42 (CMB), respectively, cf. fig. 8.1a. For vS = 300 keV (and λS = 0.01), detectabil-
ity by SKA then requires gD & 0.65, even with 20 years of observation. For gD = 0.7, the
PT can be observed for vS & 100 keV after 5 years, and for vS & 5 keV after 20 years. In
the gray area in the right of fig. 8.10b, the PT occurs before BBN, whereas for parameters
between the gray region and the solid dark-gray line, the PT occurs between neutrino
decoupling and e± annihilation, so that the value of ξh constrained by Neff is reduced by
ξSM = (4/11)1/3ν compared to the value at the time of the transition.
8.3.3. Parameter Scans
In the discussion of the numerical results for our toy models, we so far only focused
on specific slices through the parameter space. To achieve a consideration of the full
spectrum of parameters giving rise to first-order PT, let us now present results obtained
from random scans over the parameter spaces of the two models.
We calculate the nucleation temperature Tn,h, the energy budget αh for ξh = 1, and
the inverse relative time scale β/H∗, scanning 4000 random points for each model. In
the singlet scalars model described by eq. (8.27) we scan 0 < λSA < 3 and 0.7 < κ̄ < 1.5
linearly. The remaining parameters, µA/vS , κSA/vS , λS and λA, are scanned logarith-
mically in the range 10−3 – 1. In the dark photon model, cf. eqs. (8.33) and (8.34), we
scan 10−4 < λS < 0.1 logarithmically and 0 < gD < 1 linearly, setting λHS =  = 0. The
VEV is kept fixed in the scans. The results are then subsequently rescaled to obtain the
desired value of the nucleation temperature.
The corresponding values of the energy budget α and in inverse time scale β/H∗ for
the scanned parameter points in the toy models are shown in fig. 8.11. Green and blue
dots correspond to the singlets scalars and Higgsed dark photon model, respectively. The
parameter regions accessible to different GW experiments are indicated by the shaded
regions. In the top panel, a temperature ratio of ξh = 1 is assumed, and the nucleation
temperature in the hidden sector is set to Tn,h = 200GeV (left) and Tn,h = 50 keV
(right), respectively, whereas the bottom panel takes Tn,h = 50 keV with temperature
ratios ξh = 0.66 (left) and ξh = 0.48 (right).
For a nucleation temperature of Tn,h = 200GeV, cf. fig. 8.11a, the hidden sectors are
not affected by the constraints on the effective number of neutrino species, and we can
savely assume that they have the same temperature as the photon bath, provided that
the hidden sector entropy can be transferred to the SM or dark radiation when becoming
non-relativistic. For the chosen transition temperature, the SGWB may be probed by
space-based experiments. While a first-order PT in the singlet scalars model remains
undetectable for most parameter points, a large fraction of the parameter points in the
dark photon model may be probed by DECIGO and BBO. LISA and B-DECIGO are
mostly insensitive to the toy model PTs at the chosen temperature.
137
8. Constraining Secluded Hidden Sectors with Gravitational Waves
100 100
ars EPTA NANO-
5 ye Grav
SKA
10−1 LISA 10−1
10 yea
rs
0 years
10−2 10−2 2
singlet scalars 3 singlet scalars 7
Higgsed dark photon 3
10−3 10−
Higgsed dark photon 7
3
100 102 104 100 102 104
β/H∗ β/H∗
(a) Tn,h = 200GeV, ξh = 1 (b) Tn,h = 50 keV, ξh = 1
100 0
year
s EPTA 10NANO-
5 ears
EPTA NANO-
Grav 5 y Grav
SKA SKA
10−1 10−1
0 years1 10 yea
rs
20 yea
rs
− − 20 year
s
10 2 10 2
singlet scalars 3 singlet scalars 3
Higgsed dark photon 7 Higgsed dark photon 3
10−3 10−3
100 102 104 100 102 104
β/H∗ β/H∗
(c) Tn,h = 50 keV, ξh = 0.66 (d) Tn,h = 50 keV, ξh = 0.48
Figure 8.11: Spectrum of GW parameters α and β/H∗ covered by the singlet scalars
(green dots) and Higgsed dark photon (blue dots) model for the nucleation temperatures
Tn,h in the hidden sector and temperature ratios ξh stated in the respective subcaptions.
The SGWB in the shaded regions can be probed by the corresponding future GW
observatory. A black tick mark (3) indicates that the model complies with the Neff
constraints (assuming the ν-quilibration scenario in the singlet scalars model and a
completely decoupled sector in the dark photon case), whereas a red cross (7) and a
lower opaqueness of the dots denote tension with these limits.
138
α α
DE
B CB IO GO
α α
ET
GO
EC
I
B-
D
8.4. Conclusion
When considering a nucleation temperature of Tn,h = 50 keV on the other hand, the
generated SGWB falls into the region accessible by PTAs. For a temperature ratio of
ξh = 1, cf. fig. 8.11b, both models feature PTs lying within the potential reach of SKA
after only five years of observation. A small fraction of the parameter space is even
excluded by the currently available EPTA and NANOGrav data. However, for such a
low nucleation temperature, our toy models are excluded by Neff.
For a temperature ratio of ξh = 0.66, as depicted in fig. 8.11c, the singlet scalars
model with only two additional DOFs is viable in the ν-quilibration scenario. Comparing
figs. 8.11b and 8.11c, we can see how the parameter points are shifted to lower α due
to the ξ4h suppression in eq. (8.21). As a result, the parameter space giving rise to an
observable PT is reduced, and the detection of the corresponding GW signal in the singlet
scalars model now typically requires at least ten years of observation with SKA.
In order for the dark photon model, which features four hidden DOFs, to be cosmolog-
ically viable, we need to further reduce the temperature ratio to ξh = 0.48, see fig. 8.11d.
The model then complies with the constraints on Neff if the hidden sector is completely
decoupled. However, the parameter points are further shifted to lower α, and only very
a small portion remains detectable in the dark photon model, whereas the singlet scalars
model is now almost undetectable. Still, there are a few parameter points that remain
detectable by SKA even after only five years of observation.
In conclusion, our toy models indicate that it is possible to have a first-order PT
observable via GWs in a sub-MeV scale hidden sector, while at the same time satisfying
the BBN and CMB(+H0) constraints on the effective number of neutrino species.
8.4. Conclusion
In this chapter we have studied the detectability of SGWBs generated from cosmological
first-order PTs occurring in decoupled hidden sectors. Particular focus was put on sub-
MeV scale sectors. These are subject to strong constraints from the effective number
of neutrino species. We have discussed the corresponding bounds on the number of
relativistic DOFs and temperature of the hidden sector. These require the hidden sector
to be colder than the photon bath by an O(1) factor.
We have then investigated the effect of the temperature ratio between the two sectors
on the PT in such a hidden sector, finding that a lower hidden sector temperature with
respect to the SM mitigates the SGWB generated in the transition, primarily by suppress-
ing the energy budget. The detection prospects at current and future GW observatories
are therefore diminished, rendering the transition unobservable if the dark sector if too
cold. Nonetheless, it is possible to construct sub-MeV hidden sector models that satisfy
the Neff constraints but feature a first-order PT observable in GWs using PTAs.
To corroborate these statements, we have considered two concrete toy realizations of
decoupled sub-MeV hidden sectors, to wit, a model with two singlet scalars, and a gauged
dark photon model. We find that, even after reducing the hidden sector temperature to
a level compatible with Neff, a detectable SGWB can still be obtained in parts of the
parameter space in both of these models.
139

Epilogue

9. Conclusion and Summary
In this thesis, we have studied the phenomenology of various models of physics beyond
the Standard Model (BSM). We have predominantly explored two paths to constrain
new physics. Part I of this work considered searches at particle (in particular proton)
colliders. In part II on the other hand, BSM physics was probed via the generation of
gravitational waves (GWs) in cosmological first-order phase transitions (PTs). These two
paths provide complementary ways to probe new physics, potentially with an interesting
interplay.
We started our discussion of collider studies of new physics in chapter 3 with an in-
vestigation of the scalar singlet Higgs-portal dark matter (DM) model at the LHC . In
contrast to previous works, we here did not only consider the low-mass region in which
the Higgs boson can decay invisibly into DM, and the high-mass region where the DM is
produced via an off-shell Higgs boson, but also the transition between these two regimes,
i.e. mS ' mh/2. We found that, if the kinematic threshold for the decay of an off-shell
Higgs boson to a DM pair is very close to the on-shell Higgs mass, the fixed-width approx-
imation in the Breit-Wigner propagator fails. This leads to an unphysical enhancement
of the DM production cross-section, potentially exceeding the on-shell Higgs production
rate. Therefore, the momentum-dependence of the width in the propagator needs to
be retained to obtain consistent results. We then derived the current 95% confidence
level limits on the Higgs-portal coupling as a function of the DM mass, reinterpreting
the CMS search for invisible decays of the Higgs boson produced in vector-boson fu-
sion [120]. Furthermore, based on the corresponding HL-LHC forecast by CMS [121],
projections for the sensitivity of the high-luminosity and high-energy LHC upgrades were
presented. Our projections include an estimate of the systematic uncertainties on the
background, assuming that the latter is determined by measurements in control regions.
Finally, we also presented our bounds as limits on the signal strength of additional Higgs
bosons that decay invisibly, which allows for a simple reinterpretation in other dominantly
Higgs-mediated DM models, as we illustrated for various effective Higgs-portal models
with DM candidates of different spin in the appendix.
In chapter 4, this dissertation then proceeded with an investigation of the prospects
to observe the Higgs decay into a Z boson and a photon in top-pair associated pro-
duction. Due to the low branching ratio of the decay, in particular when leptonic Z
decays are considered to allow for an accurate reconstruction of the decay products at
hadron colliders, an observation in the dominant Higgs production channels is difficult,
even at high luminosities. In top-pair associated production, we however expect that
top-tagging techniques can significantly suppress the reducible backgrounds, such that
the large Yukawa coupling of the top quark promises a sizable signal-to-background ratio
in this channel. Still, an inclusive analysis and high luminosity are required to sustain
an observable number of signal events. We have therefore set up a toy analysis searching
143
9. Conclusion and Summary
for the pp → t̄th, h → Zγ process, focusing on the semi-leptonic decay channel of the
top-quark pair. Based on Monte Carlo simulations, and an extrapolation to also include
the fully-hadronic and fully-leptonic top channels, we found that the process under con-
sideration can contribute significantly to establishing an observation of the h→ Zγ decay
at the HL-LHC , and may allow for precise measurements at the HE-LHC and a future
100TeV FCChh. We further assessed the indirect constraints that can be put on the
contribution of new physics to the h → Zγ decay rate, constraining the corresponding
coupling modifier at the level of 15%, 4% and 2% at the HL-LHC , HE-LHC , and FCChh,
respectively.
Chapter 5 then concluded the part on colliders by presenting a comprehensive study of
an extension of the Standard Model (SM) in which lepton number is gauged, exploring
the DM and collider phenomenology of the model. The cancellation of lepton number
gauge anomalies requires the existence of additional leptons. The lightest of these exotic
leptons is stable and establishes a candidate for particle DM. We identified the regions of
parameter space in which the model can account for the full DM relic abundance measured
by Planck, finding that a large range of DM masses from O (100GeV) to O (fewTeV) is
possible. We also evaluated direct and indirect detection limits, constraining the kinetic
and DM mixing parameters. Furthermore, we assessed the bounds from the LHC and
LEP on the Z ′ lepton number gauge boson, the lepton number breaking scalar field φ,
as well as the exotic leptons, providing limits on the masses of these particles and their
mixing with SM fields. LEP data further puts a lower bound on the vacuum expectation
value (VEV) of the lepton number Higgs of vΦ ≥ 1.88TeV.
We then started part II of this thesis after a short introduction to stochastic gravita-
tional wave backgrounds (SGWBs) from cosmological first-order PTs in chapter 6, investi-
gating the lepton number breaking PT of the gauged lepton number model in chapter 7.
As the VEV that breaks lepton number is roughly an order of magnitude larger than
the SM Higgs VEV, the breaking of lepton number and the electroweak (EW) gauge
symmetry typically proceeds via two separate transitions. While the latter remains a
cross-over, the former can be of first-order and may be observed with GW observatories.
We investigated the parameter regions in which a first-order PT occurs and evaluated
the detectability of the corresponding SGWB at LISA and other future GW experiments.
While the PT is mostly too weak to produce observable GWs if the contributions from
the exotic leptons are neglected, the latter significantly enhance the detectability. We
assessed the detection prospects in the parameter range where the model accounts for
the full DM abundance, finding that a SGWB detectable by LISA or possible successor
experiments is produced in a large fraction of the viable parameter space.
Finally, we conducted a study of PTs in decoupled dark sectors in chapter 8 with partic-
ular focus on sub-MeV hidden sectors. BSM particles with masses at the MeV-scale and
below are constrained by limits on the number of relativistic degrees of freedom (DOFs)
at the times of Big Bang Nucleosynthesis and photon decoupling. These constraints are
typically phrased in terms of the effective number of neutrino species, Neff, and exclude
additional relativistic DOFs that are in thermal equilibrium with the SM at tempera-
tures below . 1MeV. Sub-MeV hidden sectors therefore need to be decoupled from the
photon bath. We discussed how this affects potential PTs in such a hidden sector and
the detectability of the corresponding SGWB, deriving the dependence of the parame-
144
9. Conclusion and Summary
ters characterizing the transition on the temperature ratio ξh = Th/Tγ between the two
sectors. We found that the most prominent effect is a suppression of the energy budget
α of the transition by a factor ξ4h if the hidden sector is colder than the photons. As a
result, PTs are harder to detect if they occur in cool decoupled sectors. It is however still
possible to construct models that comply with the constraints on Neff and still feature a
PT observable in GWs using pulsar timing arrays, as we demonstrated on the example
of two toy models.
In conclusion, we have presented various searches exploring the phenomenology of new
physics at particle colliders and via GWs. Both provide powerful tools for constraining or
maybe even discovering BSM physics in the future. With the increase of luminosity and
energy at coming colliders, heavier and more weakly coupled particles may be produced
directly or observed indirectly via their effects on SM observables. Even in the case
that the interactions of new physics with SM particles are too low to be detectable at
colliders, GWs may still provide a path for probing such models, for instance via the
SGWB generated in a cosmological first-order PT. Given the amount of planned and
proposed experiments in both of these directions, the future bears bright prospects for
unveiling the nature of physics beyond the Standard Model.
145
Acknowledgements
All names and personal references were removed from the acknowledgements in the electronic
publication of this dissertation.
First and foremost, I would like to express my gratitude towards my thesis advisor for
the opportunity to conduct my PhD studies under his supervision, and for his support
and advise. I am very happy to say that I do not only appreciate him as a supervisor,
but also as a friend, and that I could not imagine a better ‘Doktorvater’ for myself.
I am further deeply indebted to all my collaborators, from whom I learned a lot over that
past years. Moreover, I would like to thank all the great members of the THEP working
group, who made me feel welcome since the first day in Mainz and who make this group
such a kind an enjoyable place. I am truly grateful to all of you, and I could not wish
for better colleagues. Furthermore, I am thankful to our invaluable administration who
helped me navigating through the jungle of travel request, reimbursement and insurance,
as well as any other administrative question I could think of.
I also gratefully acknowledge support from the graduate school “Symmetry Breaking”
(DFG/GRK 1581), financially as well as through interesting lectures and a nice summer
school and retreat. In addition, I thank the ZDV for the computing time granted at
MOGON, and their technical support.
Finally, I would like to thank my non-physicist friends, who helped to keep me en-
tertained and mentally sane over the past months of social distancing, and, last but
not least, my parents and my sister, without whose continuous support, motivation and
understanding none of this would be possible.
146
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172
List of Abbreviations
1PI one-particle irreducible MET missing transverse energy
ALP axion-like particle MHD magnetohydrodynamic
BAO baryon acoustic oscillation MS minimal subtraction
BBN Big Bang Nucleosynthesis NLO next-to-leading order
BH black hole νMSM neutrino minimal SM
BSM beyond the Standard Model NS neutron star
CB compact binary NWA narrow-width approximation
CCD charged coupled device OSSF opposite-sign same-flavor
CDM cold dark matter PBH primordial black hole
CKM Cabibbo-Kobayashi-Maskawa PDF parton distribution function
CL confidence level PLI power-law integrated
CMB cosmic microwave background PMNS Pontecorvo-Maki-Nakagawa-
COVID-19 Coronavirus disease 2019 Sakata
DM dark matter PSD power-spectral density
DOF degree of freedom PT phase transition
EDM electric dipole moment PTA pulsar timing array
EFT effective field theory QCD quantum chromodynamics
EM electromagnetic, electromagnetism QFT quantum field theory
EW electroweak QM quantum mechanics
EWBG electroweak baryogenesis RG renormalization group
EWPT electroweak phase transition SGWB stochastic GW background
EWSB electroweak symmetry breaking SM Standard Model
FTQFT finite temperature QFT SMBHB super-massive black hole binary
GR general relativity SNR signal-to-noise ratio
GW gravitational wave SSB spontaneous symmetry breaking
HL heavy lepton SUSY supersymmetry
IR infrared TDI time delay interferometry
LO leading order TT transverse traceless
LSP lightest supersymmetric particle UV ultraviolet
MACHO massive astrophysical compact VBF vector-boson fusion
halo object VEV vacuum expectation value
MC Monte Carlo WIMP weakly interacting massive particle
173
List of Experiments
Colliders and Detectors
FCC Future Circular Collider
HE-LHC high-energy LHC
HL-LHC high-luminosity LHC
ILC International Linear Collider
LEP Large Electron-Positron Collider
LHC Large Hadron Collider
ALICE A Large Ion Collider Experiment
ATLAS A Toroidal LHC ApparatuS
CMS Compact Muon Solenoid
LHCb LHC-beauty
RHIC relativistic heavy ion collider
Dark Matter Experiments
direct detection
CMDS Cryogenic Dark Matter Search
CRESST Cryogenic Rare Event Search with Superconducting Thermometers
DAMA/LIBRA DArk MAtter / Large sodium Iodide Bulk for RAre processes
DAMIC Dark Matter in CCDs
DARWIN DARk matter WImp search with liquid xenoN
LUX Large Underground Xenon dark matter experiment
LZ LUX-ZEPLIN
SENSEI Sub-Electron-Noise Skipper-CCD Experimental Instrument
XENON1T
ZEPLIN ZonEd Proportional scintillation in LIquid Noble gases
gamma-ray telescopes (indirect detection)
CTA Cherenkov Telescope Array
Fermi-LAT Fermi Large Area Telescope
H.E.S.S. High Energy Stereoscopic System
MAGIC Major Atmospheric Gamma Imaging Cherenkov Telescopes
Gravitational Wave Experiments
ground-based observatories
CE Cosmic Explorer
ET Einstein Telescope
KAGRA Kamioka Gravitational Wave Detector
LIGO Laser Interferometer Gravitational-Wave Observatory
Virgo
174
List of Experiments
pulsar timing arrays
EPTA European Pulsar Timing Array
IPTA International Pulsar Timing Array
NANOGrav North American Nanohertz Observatory for Gravitational Waves
PPTA Parkes Pulsar Timing Array
SKA Square Kilometre Array
space-based observatories
B-DECIGO Scaled-down version of DECIGO, “B” stands for “Basic” or “Base”
BBO Big Bang Observer
DECIGO DECi-hertz Interferometer Gravitational Wave Observatory
LISA Laser Interferometer Space Antenna
Other
CERN European Council for Nuclear Research
KATRIN KArlsruhe TRItium Neutrino experiment
Planck Planck satellite
175
This dissertation is typeset with LATEX2ε in the KOMA-Script
book class. All Feynman diagrams are drawn using the TikZ-
FeynHand [390, 391] package. Figures 6.4, 6.5 and 8.2 are created
with TikZ. The remaining figures are generated in Python3 using
the Matplotlib [392] library.