Exotic Phenomenology
of DFSZ Axion Models
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
Julien Laux
born in Boppard, Germany
Mainz, August 23, 2023
Date of submission: 23.08.2023
Date of examination: 18.01.2024
For my family
“Sapere aude! - Dare to know!”
Immanuel Kant
“A physicist is just an atom’s way of looking at itself.”
Niels Bohr
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Abstract
In this thesis we introduce exotic properties of the highly motivated axion particle which
lead to a better accessibility in experimental searches. The axion is an essential ingredient
of theories beyond the Standard Model of particle physics (SM) which aim to solve the
strong CP problem manifesting in the absence of an electric dipole moment of the neutron
(nEDM).
The axion is mainly characterized by its mass ma and its coupling to SM particles with
a special emphasis on the axion-diphoton coupling Gaγγ . Our focus lies on axion models
which fall into the class of the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) construction
which is specified by direct axion-fermion couplings. This is in contrast to the Kim-
Shifman-Vainshtein-Zakharov model which only accounts for axion couplings to gauge
bosons.
While the first part of this work is dedicated to new effects which arise in the axion
potential with an influence on ma, the second part deals with new interactions described
within an effective field theory framework. These interactions comprise enhanced axion
fermion couplings induced by a Dark Matter scalar field as well as axion couplings to the
Higgs and Z boson which are generated through kinetic mixing effects in gauged U(1)′
extensions.
On one hand, the axion potential is sensitive to corrections by instanton and QCD effects,
leading to a constant shift in the axion mass. On the other hand, additional operators
which explicitly break the global Peccei-Quinn symmetry associated to the axion have a
more severe impact. We show that these operators can lead to a much lighter axion mass
than expected from traditional models. Taking into account current nEDM bounds, we
estimate the maximal extent of the DFSZ-line in the {ma, Gaγγ}-plane. Under the use of
a UV completion we provide an example which naturally induces these effects.
Heavy new particles induce effective axion interactions at energies below their mass scale
which we calculate in a general framework accounting for possible flavor-changing effects
in the fermion couplings. We show that these effective operators can be generated from
a heavy Dark Matter field and lead to a chiral enhancement in the fermion couplings.
Constraining our parameter space under the use of Dark Matter effects, we find that the
axion interactions to charged leptons and heavy up-type quarks can naturally be enhanced
by a factor of order O(10− 100) while couplings to light quarks and neutrinos are barely
enhanced.
Finally, we present non-trivial mixing effects between axions and additional U(1)′ gauge
bosons, leading to a reduced parameter space as the mass of the additional gauge boson
mZ′ depends on the properties of the axion field. We demonstrate the phenomenological
consequences in a gauged baryon number extension and constrain the parameter space in
the mass vs. coupling planes on the basis of collider data from ATLAS and CMS.
v

List of Publications
This thesis is based on the following list of projects, of which three papers [1–3] are
already published and two in preparation [4,5]. We provide a brief overview of the topics
and highlight the authors’s contributions.
[1] A. Kivel, J. Laux and F. Yu, Supersizing axions with small size instantons, JHEP
11 (2022) 088 [2207.08740].
At collider scales, the QCD axion in its canonical form is already excluded. Never-
theless, effects from additional SU(N) gauge groups which confine at a high energy
scale can lead to small size instanton (SSI) effects. These SSI effects affect the re-
lation between the axion mass ma and the decay constant fa allowing for a shift of
the QCD axion band to higher masses for fixed fa. This includes a region which is
accessible by collider experiments.
In our work we calculate these effects quantitatively with a new approach by using
the t’Hooft determinantal operator. The calculation involves the diagonalization
of the mass matrix of the pseudoscalar particles such as the axion, the pion, the
η′, and an additional axieta. We present our results in the mass vs. diphoton
coupling-plane with two bands for the axion and the axieta covering the parameter
space for collider searches. The author constructed an order by order analysis
for the eigenvalues and derived several limiting cases to compare with established
results and to point out the main dependences on our parameters.
[2] A. Kivel, J. Laux and F. Yu, Axion couplings in gauged U(1)’ extensions of the
Standard Model, JHEP 03 (2023) 078 [2211.12155].
The Standard Model Lagrangian is invariant under two anomalous global U(1)
symmetries related to baryon number conservation and lepton number conservation.
These symmetries can be protected by being gauged, involving a new gauge boson
which we call Z ′. Since anomalous symmetries cannot be gauged, we need to intro-
duce new heavy fermions which cancel the gauge anomalies and are therefore called
anomalons.
The author constructed a UV completion which gives both, a Z ′ from gauged
baryon number, as well as a DFSZ-type axion-like particle (ALP) which is charac-
terized by its direct coupling to SM fermions through a two Higgs doublet model.
Integrating out the heavy anomalons leads to an effective theory of the ALP which
contains new and enhanced operators such as an ALP coupling to photon and
Z ′ or to Higgs and Z boson. The author was also responsible for performing the
calculations in a general way accounting for possible flavor-violating ALP couplings
and defining a basis which is invariant under chiral fermion transformations. In the
end, we constrain our model by comparing simulated cross sections with recent LHC
measurements. Here, the author provided the calculation for the branching ratios
of the ALP and Z ′ decays as well as the FeynRules model which was implemented
in MadGraph5 aMC@NLO for our simulation.
vii
[3] F. Elahi, G. Elor, A. Kivel, J. Laux, S. Najjari and F. Yu, Lighter QCD axion from
anarchy, Phys. Rev. D 108 (2023) L031701 [2301.08760].
A problem which many axion models must face is the axion quality problem: In
order to solve the strong CP problem exactly, we cannot have additional terms which
break the associated anomalous, global Peccei-Quinn (PQ) symmetry. However,
for example, gravity effects would break global symmetries and lead therefore to a
misalignment in the strong CP phase, spoiling the axion solution to the strong CP
problem.
In this work, we explore the effects of a term which breaks the PQ symmetry
softly but is still consistent with current nEDM measurements. We find in the
case of the DFSZ model that this leads to changes in the low ma region of the
QCD axion band, allowing us to find the axion at a lower mass for constant
fa, and to set an upper bound on fa. In this project, the author constructed a
general potential for a toy model as well as for the DFSZ model. In addition, the
author calculated the mass basis of the Goldstone bosons and pointed out the new
fa dependence. Finally, we implemented the result into the axion diphoton coupling.
[4] J. Laux, S. Najjari and F. Yu, in preparation.
Besides the axion-diphoton coupling, experimental data also provide constraints
for other axion couplings, especially axion couplings to fermions. Analogously to
the theoretical approaches to solve the muon g-2 tension, we can consider chirally
enhanced axion-fermion couplings by introducing an additional heavy fermion and
heavy scalar.
We work out the manifestations of chiral enhancement in KSVZ and DFSZ models
and use constraints from colliders and Dark Matter (DM) observables for the heavy
scalar to identify new parameter space. The author’s part was to define the corre-
sponding model and to describe the chiral enhancement as well as flavor changing
effects. In addition, the author estimated the chiral enhancement in the DFSZ model
accounting for DM effects and tested an alternative model involving neutrino masses.
[5] A. Kivel, J. Laux and F. Yu, in preparation.
We discovered in our project [3] that PQ-breaking operators can lead to a much
lighter axion than in traditional approaches. However, our solution was still subject
to the axion quality problem, since we only achieved a sizable deviation from the
canonical QCD axion band for extreme values of the CP -violating parameters in the
UV.
In this project, we construct a UV completion for the anarchic axion model which
avoids generating a new fine-tuning problem under the help of a Nelson-Barr model
extension. In addition, we present possible origins of the additional PQ-breaking
operator. The author helped identifying the required field content and constructed
a mechanism which provided the additional PQ-breaking operator.
If not stated otherwise, the figures presented in this thesis were created by the author
himself. Figures which were produced by collaborators refer to the respective paper.
Some figures were recreated to achieve a uniform color scheme and plot layout. Feynman
viii
diagrams in this thesis were produced using JaxoDraw [6]. We performed our calculations
primarily with the help of Mathematica [7] where we used the packages Package X [8] and
FeynRules [9]. We simulated collider and DM data using MadGraph5 aMC@NLO [10] and
micrOMEGAS-5.3 [11].
ix

Contents
Abstract v
List of Publication vii
Contents xi
Prologue 1
I Introduction 3
II Theoretical Background 5
II.1 The Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . 6
II.1.1 Open Problems and New Physics . . . . . . . . . . . . . . . . . . . 8
II.2 Formalism of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . 9
II.2.1 Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . . 10
II.2.2 Electroweak Symmetry Breaking . . . . . . . . . . . . . . . . . . . 12
II.2.3 Chiral Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . 14
II.3 Axions and Axion-like Particles . . . . . . . . . . . . . . . . . . . . . . . . 15
II.3.1 KSVZ Axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II.3.2 DFSZ Axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
II.3.3 Axion Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
II.3.4 Axion Diphoton Coupling and Experimental Searches . . . . . . . 19
Part I: Exotic Axion Potential 21
III Axion Potential and Goldstone Basis 23
III.1 Generalized Scalar Lagrangian and Goldstone Basis . . . . . . . . . . . . . 23
III.1.1 Application to DFSZ Model . . . . . . . . . . . . . . . . . . . . . . 25
III.2 Scalar Potential from Fermion Condensates . . . . . . . . . . . . . . . . . 27
III.2.1 Scalar Potential from SU(N) Instantons . . . . . . . . . . . . . . . 28
III.2.2 Application to DFSZ Model . . . . . . . . . . . . . . . . . . . . . . 29
IV Anarchic Axion 33
xi
CONTENTS
IV.1 Light Axion from additional PQ breaking Operator . . . . . . . . . . . . . 33
IV.2 Operators with Anarchic Axion Solution . . . . . . . . . . . . . . . . . . . 36
IV.2.1 Explicit soft breaking Term . . . . . . . . . . . . . . . . . . . . . . 37
IV.2.2 PQ breaking from Yukawa Interactions . . . . . . . . . . . . . . . . 37
IV.2.3 PQ breaking from SU(N) Instantons . . . . . . . . . . . . . . . . . 38
IV.3 UV Completion with natural light Axion Solution . . . . . . . . . . . . . . 39
Part II: Exotic Axion Couplings 45
V Axion Couplings and ALP-EFT Basis 47
V.1 Generalized ALP-EFT Basis and Fermion Transformations . . . . . . . . . 48
V.1.1 ALP-EFT Terms from Scalar Potential . . . . . . . . . . . . . . . . 49
V.1.2 Application to DFSZ Model . . . . . . . . . . . . . . . . . . . . . . 50
V.2 Calculation of ALP-EFT Wilson Coefficients . . . . . . . . . . . . . . . . 51
V.2.1 Loop-induced Axion Coupling to Gauge Bosons . . . . . . . . . . . 51
V.2.2 Loop-induced Axion Coupling to Scalar and Gauge Boson . . . . . 53
V.2.3 Loop-induced Axion Coupling to Fermions . . . . . . . . . . . . . . 55
VI Chiral Enhancement of Axion Couplings 57
VI.1 Chiral Enhancement in Axion Models . . . . . . . . . . . . . . . . . . . . 57
VI.2 Effective Axion Interactions and Flavor Effects . . . . . . . . . . . . . . . 58
VI.2.1 Axion-Gauge Boson Interactions . . . . . . . . . . . . . . . . . . . 59
VI.2.2 Axion-Fermion Interactions and Flavor Effects . . . . . . . . . . . 59
VI.3 Chiral Enhancement and Dark Matter . . . . . . . . . . . . . . . . . . . . 61
VI.3.1 Relic Abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
VI.3.2 Direct Detection Constraints . . . . . . . . . . . . . . . . . . . . . 64
VI.3.3 Chiral Enhancement of Axion Quark Couplings . . . . . . . . . . . 66
VI.3.4 Chiral Enhancement of Axion Lepton Couplings . . . . . . . . . . 69
VI.4 Chiral Enhancement and Neutrino Masses . . . . . . . . . . . . . . . . . . 69
VI.4.1 Effective Couplings from Neutrino Interactions . . . . . . . . . . . 71
VII Axion Couplings in gauged U(1)’ Extensions 73
VII.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
VII.1.1 Scalar Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
VII.1.2 Fermion Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
VII.1.3 Gauge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
VII.2 Effective Theory and Wilson Coefficients . . . . . . . . . . . . . . . . . . . 80
VII.3 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
VII.3.1 Collider Constraints for the Z ′ Boson . . . . . . . . . . . . . . . . 83
VII.3.2 Collider Constraints for the Axion . . . . . . . . . . . . . . . . . . 85
xii
CONTENTS
Epilogue 89
VIII Conclusion and Outlook 91
Appendix 95
A Decay Widths and Cross Sections . . . . . . . . . . . . . . . . . . . . . . . 95
A.1 Production Cross Section for Axions/ALPs . . . . . . . . . . . . . 97
List of Figures 100
List of Tables 102
List of Abbreviations 103
List of Experiments 104
Bibliography 106
xiii

Prologue

CHAPTER I
Introduction
Over the last century, our understanding of the universe and its tiniest constituents has
undergone remarkable progress. We learned that atoms were composed out of elementary
particles which interact via fundamental force carriers and that radiation from the depths
of our universe provides information about the origins of our universe. Furthermore, we
gained the knowledge to reproduce the conditions of the early universe in experiments like
the Large Hadron Collider (LHC).
All these achievements would not have been possible without an underlying theory as
a mathematical description that allows us to formulate our expectations as measurable
observables. The most successful theory which explains the processes in the atom with
incredible precision was constructed more than fifty years ago and is called the Standard
Model (SM) of particles physics. One of the most important predictions of the SM was
the existence of the Higgs boson, which was finally observed in 2012 at the LHC [12,13].
Despite its tremendous success, there is various evidence of an incompleteness of the SM.
Phenomena that are observed but not described by the SM include, for example, gravity,
Dark Matter (DM), Dark Energy, neutrino masses, or the asymmetry between matter and
anti-matter. On the other hand, there are phenomena that are generally expected in the
SM but are not observed in nature. One important example is the absence of an electric
dipole moment of the neutron (nEDM).
The nEDM is an observable associated with the behavior of the strong nuclear force
under charge conjugation C and parity inversion P . The SM incorporates two mechanisms,
instantons and chiral fermion transformations, which lead to an asymmetry under a CP
transformation, quantified by a CP violating angle θ̄. This angle induces an nEDM and
is constrained to θ̄ < 10−10 [14]. Hence, the two contributions from instantons and chiral
fermion transformations cancel out, hinting toward a new symmetry beyond the SM.
The most extensively studied solution to this strong CP problem was proposed by
Roberto Peccei and Helen Quinn in 1977 [15,16] and is therefore called the Peccei-Quinn
(PQ) mechanism. It is based on a spontaneously broken global symmetry at high ener-
gies that renders θ̄ dynamically to zero. Of particular interest in this mechanism is the
prediction of a new particle, the axion [17,18].
The axion is characterized by its mass ma and its decay constant fa which follow the
constant relation m2 2afa ≡ Λ4QCD determined by effects from quantum chromodynamics
(QCD), the theory of the strong nuclear interactions. The couplings of the axion to
the fundamental particles of the SM are described by two classes of models, which are
called Kim-Shifman-Vainshtein-Zakharov (KSVZ) and Dine-Fischler-Srednicki-Zhitnitsky
(DFSZ). In KSVZ constructions [19, 20] the axion only couples to the force carriers of
3
CHAPTER I. INTRODUCTION
the SM with special attention on the axion-diphoton coupling Gaγγ , which is inversely
proportional to fa. On the other hand, DFSZ models [21,22] include additional interactions
with the fermionic matter fields in the SM.
Since the axion was proposed, there has been an extensive effort to prove its existence
in the laboratory and with astronomical observations. Various experiments search for
the axion in the sun (Helioscopes), in the local DM density (Haloscopes), behind walls
(“Light-shining-through-wall”), or at colliders. The challenges of all these experiments
are, on one hand that the axion can acquire masses within the whole mass range from
almost massless up to collider scale masses. On the other hand, the couplings to SM
particles become much weaker for small axion masses due to their inverse fa scaling.
This brings us to ask the following question, whether these challenges are generic for
the axion or whether other known effects beyond the SM modify our expectations, leading
to better accessibility of the axion in experiments. In this thesis, we pursue this question
within the DFSZ axion model, where we present different methods to either change the
axion mass or its couplings.
In order to find possible modifications of the axion mass, we want to investigate the
impact of additional operators which break the PQ-symmetry explicitly. These kinds of
operators are in general expected from gravity effects and change the potential in such a
way, that the minimum is moved out of the CP conserving phase, leading to a quality
problem of the axion [23]. In our work, we pursue the idea that these operators could also
change the canonical axion mass relation as long as the residual CP violating phase is still
consistent with the nEDM bound.
In the second project, we intend to learn about enhancement effects in axion-fermion
couplings. We address the question of whether axions are sensitive to chiral enhancement
effects, which were analogously motivated to solve the tension in the muon magnetic dipole
moment. In particular, we want to know whether a heavy DM scalar or heavy right-handed
neutrinos can induce such effects and how we can estimate the enhancement.
Lastly, we search for new effects arising in axion models that also involve an additional
U(1)′ gauge symmetry. We aim to include kinetic mixing effects and incorporate effective
interactions induced by heavy fermions, which are needed to cancel gauge anomalies and
are therefore called anomalons. In order to constrain the associated observables, we apply
our results to collider data from LHC experiments.
We organize this thesis in the following way: First, we present in Chapter II the theoret-
ical background of our work, where we introduce the SM and the original axion extensions.
We then move on with part I about modifications of the axion potential. In this part, we
first provide in Chapter III a general discussion about contributions to the axion potential
from complex scalar fields, Yukawa interaction to fermions, and instanton effects. After-
wards, we use this description in Chapter IV to define a class of axions with exceptionally
light masses that solve the strong CP problem with in current nEDM bounds.
The second part of this thesis dedicates exotic axion couplings primarily induced by
new heavy particles. Hence, we describe the resulting interactions in an effective field
theory (EFT) framework and calculate the respective Wilson coefficients in Chapter V.
In Chapter VI we derive one-loop effects which chirally enhance axion-fermion couplings
and constrain the parameter space by using DM bounds on the heavy fields. Another
application of the EFT framework is presented in Chapter VII where we introduce a new
Z ′ gauge boson associated with a gauged version of baryon number along with heavy
anomalons generating new axion couplings to various bosons. Finally, we conclude our
work in Chapter VIII.
4
CHAPTER II
Theoretical Background
In this chapter we present the context and motivation of this thesis within the field of
high-energy physics. The research which is pursued in the field is driven by the questions
of how the world works in its smallest components and how the universe evolved to provide
an environment in which stars and planets can form to pave the way for a habitable earth.
There are two main concepts which allow us to find the answers to these questions at
high energies: special relativity and quantum mechanics. Special relativity postulates that
light travels with a constant velocity c ≡ 299792458m/s, such that cosmic radiation opens
a window to the past as the light we see now was emitted at an earlier stage of the universe.
Observations like the cosmic microwave background (CMB) lead to the conclusion, that
the universe used to be at a much denser and hotter stage, a stage of high energies.
Quantum mechanics on the other hand tells us that there is a minimal uncertainty
between the measurements of distances and momenta, ∆x × ∆p ≥ ℏ/2, set by Planck’s
constant 2πℏ ≡ 6.62607015 × 10−34Js. Together with the relation between energy, mass,
and momentum in special relativity, E2 = p2c2+m2c4, it signifies that phenomena at very
small distances can only be described by theories at very high energies. Since relativistic
effects play an important role at these energies, we use the framework of a relativistic
quantum field theory (QFT) to formulate a theory of high-energy physics, describing the
behavior of physical particles as quantum fields.
The model which describes the behavior of the known fundamental particles best is
called the Standard Model of particle physics (SM). In fact, it is so successful that precision
experiments have not been able to rule out the SM so far, but observations showed an
incompleteness in multiple aspects. The research area which aims to find completions of
the SM is called physics beyond the Standard Model (BSM).
In order to test the SM and search for new building blocks of a more complete theory
of high energy physics, there are two ways of building up experiments. The first option
is to use signals which nature provides from the depths of the universe. For this purpose
we can use telescopes which capture the light from distant stars and galaxies as well as
primordial radiation like the CMB. More recently we can also use earth based detectors
to detect gravitational waves and cosmic particles like neutrinos as messengers from the
early universe.
On the other hand, we can invent technical machines which produce and accelerate
particles at very high energies, therefore simulating the conditions which were present in
the beginnings of the universe. The facility which reaches the highest particle energies up
to now was built in Geneva by the European Organization for Nuclear Research (CERN)
and is called the Large Hadron Collider (LHC).
5
CHAPTER II. THEORETICAL BACKGROUND
In the following we present in section II.1 the historical evolution of the SM and the
state of the art in high-energy physics introducing open problems which can not be solved
within the framework of the SM. In section II.2 we then build up the formal description of
the SM as a QFT on the Lagrangian level. Finally, in section II.3 we introduce the BSM
theory of an axion which was invented originally to solve the strong CP problem of the
SM.
II.1 The Standard Model of Particle Physics
The field of particle physics began to evolve when Ernest Rutherford discovered in 1911 [24]
that atoms are composed out of a heavy nucleus and a light shell consisting of subatomic
particles. The force which binds the two components together is given by the electric
force between the positively charged protons (p) in the nucleus and the negatively charged
electrons (e) in the shell. This force is mediated by photons (γ) which also transmit energy
between different atoms through their role as electromagnetic radiation observed as visible
light in the optical frequency spectrum.
The atomic nucleus is characterized by two numbers, the nuclear charge number Z and
the atomic mass number A. These two numbers provide information of the content of
subatomic particles in the nucleus: The number of positively charged protons (p+) is
given by Z, while the difference A−Z specifies the number of electrically neutral neutrons
(n0). Since neutrons are slightly heavier than protons, they can decay via the β-decay
into a proton and an electron, hence increasing Z by one unit.
The behavior of the nucleus required an explanation involving two new forces. The
“strong” nuclear forces binds protons and neutrons together. It is mediated by pions (π)
and was first described by Hideki Yukawa in 1935 [25]. The “weak” nuclear force mediates
the β-decay and was theorized by Enrico Fermi in 1933 [26, 27] as a contact interaction
involving an additional charge- and mass-less particle, the neutrino (ν). Both new particles
could later be experimentally observed, the pion in 1947 [28] and the neutrino in 1956 [29].
The next step was to formulate the nuclear forces within the framework of a QFT.
The underlying theory of the strong force is called quantum chromodynamics (QCD). It
postulates that the nucleons p+, n0 as well as the pions consist of fundamental particles,
the quarks (q) [30, 31]. Quarks are characterized by two properties, color and flavor.
Color is responsible for binding quarks together and is mediated by mass-less gluons (g),
whereas flavor categorizes masses and electrical charges of quarks. Up-type quarks have
an electrical charge of 2/3 and are called up, charme and top (u, c, t) ordered by increasing
mass, while down, strange and bottom (d, s, b) represent down-type quarks with a charge
of −1/3.
The weak force was unified with the electric force to an electroweak force coming along
with three new mediator particles, the electrically charged W± bosons and the neutral Z0
boson [32]. These weak mediators have high masses, and therefore energies which are much
heavier than the momentum transfer in the β-decay. An external observer is therefore not
able to resolve their impact, leading to the aforementioned description of Fermi’s theory as
a contact interaction. The mathematical procedure of removing high-energy modes from a
low-energy theory is called “integrating out” the heavy fields and the the resulting theory
is described as an effective field theory (EFT).
The weak interaction is able to change the flavor of quarks through the charged W
interaction. It also couples to three generations of leptons, which comprise the electrically
charged electron, muon and tau (e, µ, τ) as well as three respective neutrinos (νe, νµ, ντ ).
6
II.1. THE STANDARD MODEL OF PARTICLE PHYSICS
However, in 1956 Chien-Shiung Wu demonstrated that the weak interaction violates parity
by only coupling to neutrinos with a spin-projection opposite to the momentum direction
[33]. Hence, the weak interaction effects only particles with left-handed chirality and
possible right-handed neutrinos remained unobservable.
The chiral behavior and the short range of the weak interaction were difficult to explain
within the framework of a QFT, since matter fields (fermions) with chiral interactions
and mediator fields (gauge bosons) were considered to be massless to preserve a gauge
symmetry in the QFT. In order to solve this problem, three independent groups proposed
a new particle which gives a mass to both, fermions and gauge bosons, while breaking the
gauge symmetry of the QFT [34–36]. This particle was later named Higgs particle (h)
after one of the inventors.
With this new ingredient, Steven Weinberg and Abdus Salam were able to finalize the
electroweak theory [37, 38] which then together with QCD formed the Standard Model
of Particle Physics (SM). The discovery of the Higgs particle by the ATLAS and CMS
collaborations at the Large Hadron Collider (LHC) in 2012 [12,13] completed the particle
spectrum of the SM, leaving the SM as the most successful theory in high-energy physics,
which still is consistent with the overwhelming majority of precision measurements.
leptons νe νµ ντ
Q = 0 mνe < 1.0 eV/c
2 mνµ < 0.19 MeV/c
2 mντ < 18.2 MeV/c
2
leptons e µ τ
fermions Q = −1 me = 0.511 MeV/c2 mµ = 105.658 MeV/c2 mτ = 1.776 GeV/c2
s = 12 quarks u c t
Q = 23 mu = 2.16 MeV/c
2 mc = 1.27 GeV/c
2 mt = 172.69 GeV/c
2
quarks d s b
Q = −13 md = 4.67 MeV/c
2 ms = 93.4 MeV/c
2 mb = 4.18 GeV/c
2
electroweak γ Z0 W±
bosons s = 1 mγ = 0 mZ = 91.188 GeV/c
2 mW = 80.377 GeV/c
2
s = 0, 1 gluons g Higgs h
s = 1 mg = 0 s = 0 mh = 124.97 GeV/c
2
baryons n0 p+ ...
bound s = 12 mn = 0.939 GeV/c
2 mp = 0.938 GeV/c
2
states mesons π0 π± ...
s = 0 mπ = 134.976 MeV/c
2 mπ± = 139.570 MeV/c
2
Table II.1: Overview of SM particles and light QCD bound states. The table contains
mass m, electric charge Q and spin s of all SM fermions and gauge bosons as well as the
lightest baryons and mesons which are composed out of up- and down-quarks. The masses
are given up to three decimal places and can be found in reference [39].
Table II.1 gives an overview of the particles of the SM and some noteworthy low-energy
bound states. It contains the masses of the particles as well as their electrical charge and
spin. We express masses of particles in units of eV, referring to the kinetic energy of a
single electron with elementary charge 1e ≡ 1.602176634 × 10−19C which is accelerated
from rest in an electric potential of 1V. Hence, the conversion factor into the International
System of Units (SI) is given by 1eV/c2 = 1.78266192 × 10−36kg. For the remainder of
the thesis we use natural units, for which we set the fundamental constants to c = 1 = ℏ.
7
CHAPTER II. THEORETICAL BACKGROUND
II.1.1 Open Problems and New Physics
Although precision measurements strongly support the SM, there are several observations
which hint towards an incompleteness of the SM. First of all, there is the simple observation
that there is gravity [40]. The gravitational force has not yet been described by the SM and
has its own fundamental theory called general relativity (GR) [41]. It is widely believed
that in agreement with the weakness of the gravitational force, quantum effects would
play an important role at energies above the Planck mass MPl ≈ 1.2209× 1019 GeV [39],
where a theory of quantum gravity is needed.
Important applications of GR comprise the description of gravitational waves as well as
the expansion of the universe through a cosmological constant. It also gives a prediction
for the rotation curves of galaxies which is in tension with the observations. This leads to
the conclusion that there are matter particles which do not have interactions described by
the SM and are therefore called Dark Matter (DM).
Another area which reveals open problems of the SM is the cosmological evolution of the
universe. One important source of information in this field is the CMB, which was emitted
380000 years after the Big Bang. It shows temperature anisotropies in the angular power
spectrum which can be fitted under the use of the ΛCDM model [42], where Λ represents
the cosmological constant and CDM stands for cold Dark Matter. It not only supports
the existence of DM and a cosmological constant but also gives an estimate of the energy
budget of the universe. Within this energy budget, ordinary (mainly baryonic) matter
makes up 4.9%, Dark Matter provides 26.8% and dark energy given by the cosmological
constant contributes with 68.3% [39].
It is also evident that there is an asymmetry between matter and anti-matter, since
we predominantly observe matter particles in nature but not their corresponding anti-
particles. To account for this asymmetry, the cosmological evolution of the universe has
to fulfill the Sakharov conditions [43] which include the violation of baryon number B,
CP violation and interactions out of thermal equilibrium. The process which leads to
the asymmetry between matter and anti-matter is called baryogenesis and relies on BSM
extensions, since there is not enough B and CP violation within the SM.
Lastly, there are problems concerning parameters which are predicted by the SM but
are observed differently. Most relevant for this thesis is the strong CP problem concerning
the absence of CP violation in QCD manifesting in an apparently vanishing electric dipole
moment (EDM) of the neutron [14]. This is surprising as the SM contains two mechanisms
which lead to a CP violation in the strong sector as we will see in subsection II.2.2.
Another contradiction was found in neutrino physics. Since right-handed neutrinos are
unobserved, the neutrinos remain as the only purely left-handed particles and are described
mass-less within the SM. However, the observed phenomenon of neutrino oscillations [44]
can only be explained by having massive neutrinos, which require BSM extensions with
additional particle content.
A discrepancy is also present for the anomalous magnetic dipole moment of the muon.
Recent studies show a deviation between theoretical expectation and experimental obser-
vation of 4.2σ [45]. This gap provides room for BSM theories involving heavy fields which
are integrated out. Of course there are many more open problems, but to mention all of
them would clearly lead far beyond the scope of this thesis.
8
II.2. FORMALISM OF THE STANDARD MODEL
II.2 Formalism of the Standard Model
In order to calculate measurable observables we formulate our models as field theories,
characterized by their Lagrangian. The fundamental Lagrangian of a theory contains all
possible operators Oi of interactions between particles respecting the gauge symmetry
which are renormalizable for four spa∑ce-∑time dimensions,
L ⊃ Ci Odim=d
Λd−4 i
, (II.1)
d≤4 i
with dimensionless couplings Ci and mass scale Λ. Higher dimensional operators (d ≥ 5)
will be regarded as effective interactions, which are suppressed by a UV mass scale Λ ≫ mt.
A renormalizable theory at the corresponding UV mass scale is called a UV completion of
the effective theory.
We distinguish three types of particles, varying by their spin properties. Fermions are
spin-1/2 particles which are formally represented by spinors ψ. In the case of the SM
these are quarks and leptons. Gauge bosons are spin-1 particles, described by 4-vectors
in space-time Aaµ, where the group index a refers to the generators of the corresponding
gauge group. These vector bosons respect a gauge symmetry, where the physics is invariant
under a transformation Aµ → Aµ+∂µα in the case of a U(1) gauge group. Massless gauge
bosons as photons and gluons have two transversal polarization modes, while massive gauge
bosons as W and Z bosons have a third, longitudinal mode. Finally, spin-0 particles like
the Higgs boson are expressed by (in general complex) scalar fields Φ.
For example, for a theory containing a fermion ψ with mass m respecting an SU(N)
gauge symmetry with gauge coupling g, the (classical) Lagrangian can be written as
L ⊃ ψ̄(iD/ −m)ψ − 1F a F aµν , (II.2)
4 µν
where Dµ = ∂µ + igt
aAaµ is the covariant derivative and F
a
µν = ∂
a
µAν − ∂ Aa + gfabcν µ AbµAcν
the field strength tensor. The structure constants fabc depend on the generators ta of the
gauge group and vanish in the abelian U(1) case. We implicitly sum over Lorentz indices
and group indices by using the Einstein sum convention.
Physical observables can then be calculated with a path integral, which gives the vacuum
expectation value of the time-ordered∫produ∫ct of a∫n operator O via
⟨Ω| 1T{O(x1, ..., xn)} |Ω⟩ = Dψ Dψ̄ DAaµO(x1, ..., xn) exp(iS) , (II.3)Z[0] ∫
where Ω denotes the physical vacuum and S = d4xL describes the action. The generating
functional Z is d∫efined∫by ∫ ( ∫ )
Z[η, η̄, Jaµ ] = Dψ Dψ̄ DAaµ exp iS[ψ, ψ̄, Aaµ] + i d4x(ψ̄η + η̄ψ + JaµAaµ) ,
where η, η̄ and Jaµ represent classical source fields.
We notice that symmetries of the classical Lagrangian are not necessarily symmetries of
the quantized theory. Especially, the total derivative of the Chern-Simons current [46]
Kµ = 2ϵµναβ(Aa∂ Aa
g
+ fabcν α β A
aAb c
3 ν α
Aβ) (II.4)
9
CHAPTER II. THEORETICAL BACKGROUND
can change the a∫ction by a topological winding number
g2 g24 µ ⇒ L 1
2
δS = d x∂ K δ = ϵµναβF a
g
F a = F a F̃ aµν . (II.5)
32π2
µ
32π2 2 µν αβ 32π2 µν
A gauge field configuration which changes the winding number is called an instanton. The
true vacuum is given by a superposition of all possible classes of instanton configurations
and is parametrized by an angle θ ∈ {0, 2π}, giving an effective term θ × δL in the
Lagrangian.
The same term can be generated through anomalous fermion transformations of the
form
ψ → e−i
X αγ X
2 5ψ, ψ̄ → ψ̄e−i αγ2 5 . (II.6)
This transformation changes the fermionic path integral measure in the partition function.
The resulting Jacobian factor has to be regularized following Fujikawas method [47], which
leads to a shift proportional to δL in the Lagrangian.
In particular, the fermionic part of the Lagrangian changes after such an anomalous
transformation to
ψ̄(i∂/−m)ψ → Xψ̄iγµ∂µψ + (∂µα)ψ̄γµγ5ψ − ψ̄me−iXαγ5ψ − 2αAXQQ × δL , (II.7)
2
where AXQQ denotes the anomaly coefficient {
2
AXQQ = T (R)(X 2LQL − 2
XQ U(1)
XRQR) = . (II.8)X/2 SU(N)
Here, we considered the fermion to transform vector-like under the gauge sy[mmetr]ies
(QL = QR) and T (R) represents the Dynkin index [48] of a representation R, tr T
a
RT
b
R ≡
T (R)δab. In section V.1 we will generalize the anomalous transformations to multiple
flavors and different chiralities.
II.2.1 Standard Model Lagrangian
In order to analyze the interactions of the different particles in the SM, we split the
Lagrangian of the SM into four parts,
LSM ⊃ LSM + LSM + LSM + LSMgauge scalar fermion Yukawa , (II.9)
where LSMgauge describes the kinetic part of the gauge bosons, LSMscalar the kinetic part and
potential of the scalar Higgs boson, LSMfermion the kinetic part of the fermions and LSMYukawa
the Yukawa interactions between the fermions and the Higgs.
The underlying gauge symmetry of the SM is given by
GSMgauge = SU(3)C × SU(2)L × U(1)Y , (II.10)
where SU(3)C represents the color gauge group of QCD and SU(2)L × U(1)Y the elec-
troweak gauge group containing the left-handed gauge group and hypercharge Y .
For each subgroup of the SM gauge group we have a set of gauge bosons, one for each
generator. The kinetic terms including possible self-interactions can then be written under
the use of the field strength tensors
2
LSMgauge ⊃ −
1
B Bµνµν −
1 1 g
W a aµν a aµνµνW − GµνG + θ s Ga aµνQCD4 4 4 32π2 µν
G̃ , (II.11)
10
II.2. FORMALISM OF THE STANDARD MODEL
with G̃aµν = 1/2ϵµναβGaαβ. We neglect possible θ terms for SU(2)L and U(1)Y since they
appear to be unphysical as we will see in the next subsection.
Next, we have the scalar part of the SM Lagrangian, consisting of the kinetic part which
is described by a covariant derivative DH,µ containing gauge interactions, and a quartic
potential with coupling λ, ( )
2 2
LSMscalar ⊃ |
v
DH,µH|2 − λ |H|2 − . (II.12)
2
We account for gauge invariance by only having absolute value squares in the scalar La-
grangian. The quartic potential has√its minima away from |H| = 0 leading to a vacuum
expectation value (vev) ⟨|H|⟩ = v/ 2. This vev breaks the gauge symmetry as we will
see in the next subsection.
We describe the fermions as chiral fields in the Weyl representation, such that we can
treat the gauge interactions for left- and right-handed fields separately. The kinetic terms
including the gauge interactions then read
LSM ⊃ i / ij j i / ij j i / ij j ifermion iQ̄LDQQ + iL̄LDLL + iūRDu u + id̄RD/
ij j ij
L L R d dR + iē
i
RD/ e e
j
R , (II.13)
where L T TL = (νL, eL) and QL = (uL, dL) represent the left-handed doublets which
transform under the left-handed SU(2)L gauge group.
Each fermion has a flavor index i ∈ 1, 2, 3 which runs over the number of fermion
generations Ng. Each generation has the same quantum numbers and transformation
properties, shown in Table II.2.
SU(3)C SU(2)L U(1)Y
QiL 3 2 1/6
uiR 3 1 2/3
diR 3 1 -1/3
LiL 1 2 -1/2
eiR 1 1 -1
H 1 2 1/2
Table II.2: Fermionic and scalar field content of the SM. Shown are the transformation
properties under the SM gauge groups for three generations of Weyl fermions with flavor
index i. Quarks transform fundamentally under the color gauge group SU(3)C , denoted
by a “3”, while left handed fields transform fundamentally (“2”) under the weak gauge
group SU(2)L. The last column contains the hypercharges Y of the fields.
This means the Lagrangian possesses at this stage a global symmetry of the form
GSMglobal = U(3)Q × U(3)u × U(3)d × U(3)L × U(3)e , (II.14)
meaning that we can transform each Weyl fermion individually under a global U(3) group.
From Table II.2 we can now deduce the covariant derivatives for the fermions and the
Higgs, giving ( )
Dij
gY
Q,µ = (∂µ − i Bµ − ig τaL W a − ig a a ijsλ G δ ,6 µ µ
ij g
)
Y
D = ∂ a a ijL,µ µ + i Bµ − igLτ Wµ δ ,2
11
CHAPTER II. THEORETICAL BACKGROUND
Dij
( )
f,µ = ∂µ − iYfgY Bµ − i(1− δfe)gsλ
aGa δijµ , f ∈ {e, u, d} ,
gY
DH,µ = ∂µ − i Bµ − igLτaW a , (II.15)
2 µ
with τa and λa being the generators of the respective gauge groups. Due to the global
symmetry the gauge interactions are flavor-conserving, which will change in the broken
phase of the SM.
Finally, there are the Yukawa interactions of the fermions to the Higgs boson,
LSM ⊃ −yij i j ijYukawa u Q̄LH̃uR − yd Q̄
i
LHd
j
R − y
ij
e L̄
i
LHe
j
R + h.c. , (II.16)
with H̃a = ϵabH
∗
b describing the Higgs doublet contracted with the totally anti-symmetric
tensor of SU(2)L. The Higgs vev will introduce masses to the fermions which also breaks
the global symmetry explicitly.
II.2.2 Electroweak Symmetry Breaking
Having introduced the ingredients of the SM, we can now discuss the effects of the vev v
of the Higgs field, defined by its scalar potential. In particular, the symmetry under which
H transforms is broken by the vev to its diagonal subgroup, SU(2)L × U(1)Y → U(1)Q,
where U(1)Q denotes the gauge group of QED with the electric charge Q.
In order to study the changes in th(e Lagran)g(ian we )express the Higgs doublet in itsnon-linear form
φaσa 0
H = exp i v√+h , (II.17)v 2
where h denotes the radial mode of the Higgs field and φa the angular modes contracted
with the SU(2) Pauli matrices σa.
The scalar potential can then be expanded using the radial and angular modes by
LSM →1(∂ h)2 − λv2(h2∣− λvh3
λ
scalar µ − h42 4 ∣ ( ) )2 2
g2L(v + h)
2 ∣∣∣ + − D + 0µφ ∣ 1 ∂µφ+ W ∣µ ∣ + Zµ − , (II.18)4 mW 2 cos2 θW mZ
where we used the weak mixing angle tan θW = g
±
Y /gL. The fields Wµ and Zµ are the mass
eigenstates of the electroweak gauge bosons with masses mW = gLv/2 and mZ = mW /cW .
We can express(the)se fie(lds in terms of the)or(igina)l fields via 1 2
Aµ cos θW sin θW Bµ ± Wµ ∓ iW√ µ= − 3 , Wµ = . (II.19)Zµ sin θW cos θW Wµ 2
The angular modes φ± are constructed analogously. We see, that the field Aµ does not
appear quadratically in the scalar Lagrangian, such that Aµ does not get a mass, describing
the massless photon of QED.
In order to factor out the fermion mass matrices we remove the angular modes from the
Yukawa potential by performing chiral rotations of the left-handed fermions and succes-
sively a vect(or-like r)otation to account for the mixing(of Bµ and W)3µ ,
φaσa φ0
F iL → exp i F iL , F ∈ {L,Q} , f i → exp −is2WQf f i , f ∈ {e, u, d} ,v v
(II.20)
12
II.2. FORMALISM OF THE STANDARD MODEL
which leads to the desired form of the Yukawa potential,
LSM vYukawa → − √
+ h v + h v + h
ūLYuuR − √ d̄LYddR − √ ēLYeeR + h.c. , (II.21)
2 2 2
where the Yukawa matrices are still in general non-diagonal and complex and fermions are
represented as vectors in flavor-space.
The transformation in equation (II.20) changes the kinetic part of the fermionic La-
grangian as well. Expressing the gauge interactions in terms of their conserved fermionic
currents Jµ we get
LSM i i i i i i ifermion →iū ∂/u + id̄(∂/d + iē ∂/e)+ iν̄L∂/νi + e(A(Jµ + g GaJ)a,µL µ QED s µ QCD )
e ∂µφ
0 e D φ+µ
+ Z µµ − JZ + √ W
+ −µ
µ − JW + h.c. , (II.22)sW cW mZ 2sW mW
where e = gL/sW = gY /cW is now the electromagnetic coupling constant and f
i ≡ f iL+f iR
denote Dirac spinors with f ∈ {e, u, d}.
We notice that in both, the scalar and fermionic Lagrangian the massive gauge bosons
W±µ and Zµ appear in the same way in respect to the angular modes φ
± and φ0. Hence,
we can use the gauge freedom of the original symmetry to shift the fields by
0 + −
→ ∂µφZ Z + , W+ → W+ Dµφ+ , W− → W− Dµφµ µ
m µ µ µ µ
+ . (II.23)
Z mW mW
This fixes the gauge to the so-called unitary gauge, which means, that the gauge symmetry
is broken. The angular fields of the Higgs are therefore called the Goldstone bosons of the
broken gauge symmetry and are absorbed (“eaten”) as longitudinal modes of the massive
gauge bosons. Note, that LSMgauge stays invariant under this transformation.
Finally, we want to diagonalize the Yukawa terms in order to obtain mass eigenstates
of the fermions. Hence we use the U(3) transformations of the original global symmetry
to bring the Yukawa matrices into a real and diagonal form YD,f . We express the trans-
formations in terms of the unitary matrices Uf for the left-handed fields and Vf for the
right-handed fields,
fL → U † † †f fL , fR → Vf fR , UfUf = 1 = VfVf , YD,f = UfYfVf , (II.24)
√
such that we get diagonal fermion mass matrices of the form Mf = YD,fv/ 2.
These chiral transformations have two effects on the remaining SM parameters. On one
hand, the weak gauge current becomes flavor-changing,
J−µ = ūi µ ij j i µ ij jW Lγ δ dL → ūLγ VCKMdL , (II.25)
where the flavor-changing CKM matrix VCKM ≡ U†uUd also contains a CP -violating
phase.
On the other hand, the θ term changes due to the anomalous nature of the chiral
transformation, ( )
det(UuUd)
θQCD → θ̄SM = θQCD − arg = θQCD − arg (det(YuYd)) . (II.26)
det(VuVd)
Since the other gauge groups, SU(2)L and U(1)Y are chiral, a corresponding transforma-
tion would not contain a matching number of phases of the matrices Uf and Vf . Hence,
13
CHAPTER II. THEORETICAL BACKGROUND
we can always use one matrix to cancel the corresponding θ terms, while keeping the total
phase differences, making the θ terms for SU(2)L × U(1)Y unphysical.
The CP -violating parameter θ̄SM feeds linearly into the nEDM, which constrains θ̄SM <
10−10 [14]. Hence, θQCD and arg(det(YuYd)) would need to cancel almost exactly, hinting
towards a new symmetry. This is called the strong CP problem and will be discussed in
section II.3.
II.2.3 Chiral Symmetry Breaking
The gauge coupling gs of the remaining gauge symmetry SU(3)C grows for larger distances,
leading to confinement of quarks, such that we can not detect free quarks at measurable
distances. Hence, the quarks form a so-called quark condensate for energies below the
QCD confinement scale, which we can write as a vacuum expectation value ⟨q̄q⟩ = v3χ.
This vev breaks the chiral symmetry of QCD under which the light quarks u, d (and s)
transform independently for left- and right-handed fields.
In particular, the up- and down-quarks form left- and right-handed doublets (uL, dL)
T
and (uR, d
T
R) transforming under a global SU(2)L×SU(2)R symmetry. The vev vχ breaks
this symmetry to its diagonal subgroup SU(2)I , called isospin.
The Goldstone bosons of this spontaneous symmetry breaking can be combined as a
field Σ containing the pions πa, ( )
a a
Σ = √vπ π σexp i , (II.27)
2 fπ
√
which transforms as a bifundamental under SU(2)L×SU(2)R. The parameter fπ = 2vπ
represents the pion decay constant.
Since the up- and down-quarks are not exactly massless, the chiral symmetry is broken
also explicitly by the mass matrixM = diag(mu,md). Hence, the pions acquire a mass and
are therefore called pseudo-Nambu-Goldstone bosons. Treating the Mass matrix as a spu-
rion which transforms as a bifundamental as well, we can construct the chiral Lagrangian
as
v3
( )
3
L 2 χ πchiral ⊃Tr(|DΣ,µΣ)| ) + Tr(MΣ+ h.c.) +O
fπ f3π
1
= (∂ π0µ )
2 + |D +µπ |2 + v3χ(mu +md)2
v3
( )
χ(mu +md) (π
0)2 + (π+)2− + (π
−)2 π3
+O , (II.28)
2 f2 3π fπ
where we neglect higher order interactions.
We see, that the pions indeed get a mass, leading to the Gell-Mann-Oakes-Renner rela-
tion [49] between pion mass and pion decay constant
m2πf
2
π = v
3
χ(mu +md) . (II.29)
We remark, that including the strange quark as well as corrections from higher order QCD
effects add further mesons and correct the pion masses, leading to slightly different masses
between neutral and charged pions.
14
II.3. AXIONS AND AXION-LIKE PARTICLES
II.3 Axions and Axion-like Particles
In order to solve the strong CP -Problem there are multiple attempts to introduce new
symmetries, which either forbid such a CP -violating term (i.e. Nelson-Barr mechanism
[50, 51]) or render it dynamically to zero (i.e. Peccei-Quinn mechanism [15, 16]). In this
thesis we mainly focus on the latter. The Peccei-Quinn mechanism introduces a global
U(1)PQ symmetry which is broken at a high mass scale va and dynamically cancels the θ̄SM
parameter. The corresponding Goldstone boson was identified by Weinberg and Wilczek
in [17,18] and is called the axion a.
The U(1)PQ symmetry can be realized as the internal continuous shift symmetry of the
angular mode of a complex scalar field ( )
vΦ√+ hΦ aΦ aΦ aΦ = exp i , ∼ , (II.30)
2 vΦ vΦ va
where hΦ denotes the radial mode and aΦ the angular mode which is mainly aligned with
the axion. The vev vΦ breaks the PQ symmetry, leaving a discrete shift symmetry for the
axion. For a sufficiently high mass scale, vΦ dominates the PQ breaking scale va which
is related to the axion decay constant fa via fa = Xva, where X is the PQ charge of the
field Φ. ∑
WritingA f f fgg = f Ng (XL−XR)/(2X) as the canonically normalized anomaly coefficient
for the U(1) × SU(3)2 anomaly with NfPQ C g being the number of generations of fermion
type f , we acquire a CP -conserving t(erm at dimension)5 of the form
L ⊃ − g
2
s A a a aµνagg 2X gg − θ̄SM GµνG̃ , (II.31)32π2 fa
which also respects a discrete shift symmetry of the axion, since the path integral does
not change for δS → δS + 2π. We see, that the axion cancels the θ̄SM term exactly if it
acquires a vev
⟨ θ̄SMfaa⟩ = . (II.32)
2XAgg
In subsection II.3.4 we will show that this is indeed the true minimum of the axion potential
as long as the GG̃ term is the only term which breaks the discrete shift symmetry of the
axion. In the following we mainly consider the canonical form of Lagg, where we use the
canonical PQ charge normalization X = (2A )−1gg .
There are two types of models which generate Lagg from the complex scalar Φ. A KSVZ
model contains an additional heavy quark, which can be anomalously transformed to
generate Lagg, while a DFSZ model contains a second Higgs doublet, such that anomalous
transformations of the SM fermions generate Lagg due to a mixing of the angular modes
of the scalar fields. In the following we will present both types of models.
II.3.1 KSVZ Axion
In the KSVZ Model [19,20] the PQ Anomaly is mediated by an additional heavy quark q,
with its mass generated through a Yukawa coupling to the complex scalar Φ. An additional
gloabal Zn symmetry with n > 4 forbids polynomials of Φ in the scalar potential which
would break the PQ symmetry explicitly. The corresponding field content is shown in
Table II.3.
Since the neutral angular mode of the Higgs doublet is entirely absorbed by the Z
boson, the axion aligns exactly with the angular mode aΦ of the complex scalar. Hence,
15
CHAPTER II. THEORETICAL BACKGROUND
SU(3)C SU(2)L U(1)Y Zn U(1)PQ
QiL 3 2 1/6 0 XQ
uiR 3 1 2/3 0 XQ
diR 3 1 -1/3 0 XQ
LiL 1 2 -1/2 0 XL
eiR 1 1 -1 0 XL
H 1 2 1/2 0 0
qL 3 1 Yq 0 -X/2
qR 3 1 Yq n-1 X/2
Φ 1 1 0 1 -X
Table II.3: Fermionic and scalar field content of the KSVZ model. In addition to the
SM we have a heavy quark field q and a gauge singlet complex scalar Φ. Both fields are
charged under a Zn symmetry which protects a global PQ symmetry.
the Higgs doublet is not charged under the global PQ symmetry and the SM fermions
are only charged vector-like under U(1)PQ such that they do not contribute to the PQ
Anomaly.
The Yukawa and scalar Lagrangia(ns extend t2 )
o
2 ( )( )2 2
LKSVZ ⊃LSM
v v
scalar scalar + |∂µΦ|2 − λΦ |Φ|2 − Φ − λHΦ |H|2 −
v |Φ|2 − Φ ,
2 2 2
LKSVZ SMYukawa ⊃LYukawa − yq q̄LΦqR + h.c. , (II.33)
with the scalar Potential only effecting the radial modes of the complex fields.
Performing an axial transformation on(the heavy q)uarks,
q → X aexp −i γ5 q , (II.34)
2 fa
generates the PQ anomaly in the CP conserving Lagg interaction. The corresponding
anomaly coefficient and PQ charge normalization evaluate to
AKSVZ 1= ⇒ XKSVZgg = 1 . (II.35)2
We show the electromagnetic PQ anomaly in subsection II.3.4.
II.3.2 DFSZ Axion
In contrast to the KSVZ model, the DFSZ model [21, 22] relies on a two Higgs doublet
description, where down-like quarks acquire their mass through the Yukawa interaction
with a Higgs doublet Hd and up-like quarks through the interaction with a second Higgs
doublet Hu. While Hd has the same gauge properties as the SM Higgs doublet H, Hu
transforms like H̃.
A Zn symmetry with n > 4 ensures that the quarks only have Yukawa couplings to their
respective Higgs fields and that no polynomials in Φ break the PQ symmetry explicitly.
The corresponding field content is shown in Table II.4.
We choose the Zn charges for a Type II two Higgs doublet model [52] in which the
charged leptons couple to Hd, such that the Yukawa Lagrangian becomes
LDFSZ ⊃ −yijYukawa u Q̄i
j ij i j
LHuuR − yd Q̄LHddR − y
ij i j
e L̄LHdeR + h.c. . (II.36)
16
II.3. AXIONS AND AXION-LIKE PARTICLES
SU(3)C SU(2)L U(1)Y Zn U(1)PQ
QiL 3 2 1/6 0 XQ
uiR 3 1 2/3 1 XQ-Xu
diR 3 1 -1/3 0 XQ-Xd
LiL 1 2 -1/2 0 XL
eiR 1 1 -1 0 XL-Xd
Hu 1 2 -1/2 n-1 Xu
Hd 1 2 1/2 0 Xd
Φ 1 1 0 1 -X
Table II.4: Fermionic and scalar field content of the DFSZ model. The model contains two
Higgs fields Hu and Hd which break the electroweak symmetry as well as an additional
gauge singlet Φ. A Zn defines the Higgs couplings to fermions and protects a global PQ
symmetry.
Contracting SU(2)L indices implicitly, the scalar Lagrangian of this model reads
LDFSZ 2scalar ⊃|DH̃,µHu| + |DH,µH |
2 + |∂ Φ|2 − µ2 |H |2d µ u u − µ2 2 2 2d|Hd| − µΦ|Φ|
− λ |H |4 − λ |H |4 − λ |Φ|4 − λ |H |2|H |2 − λ′ |H H |2u u d d Φ ud u d ud u d
− λ 2 2uΦ|Hu| |Φ| − λdΦ|H |2|Φ|2d − µAHuHdΦ+ h.c. , (II.37)
where the term proportional to µA leads to a mixing between the neutral angular modes
of Hu, Hd and Φ, providing a mass for one linear combination A and leaving the axion as
the massless linear combination which is not absorbed by the Z boson.
The mixing of angular modes leads to a direct coupling of the SM fermions to the axion
which also manifests in their non-vanishing axial PQ charges. Hence, in the DFSZ model
the PQ ano(maly Lagg is)generated via an(axial transf)ormation of the S(M fermion, )
i → − Xu au exp i γ ui Xd a Xd a5 , di → exp −i γ i i5 d , e → exp −i γ5 ei ,
2 fa 2 fa 2 fa
(II.38)
with Xu+Xd = X, such that the DFSZ model does not require an additional heavy quark.
The resulting anomaly coefficient and PQ charge normalization are
ADFSZ Nggg = ⇒ XDFSZ
1
= . (II.39)
2 Ng
The electromagnetic anomaly is shown in subsection II.3.4.
Besides the mixing of the angular modes the DFSZ model also introduces a mixing
between radial modes, since it relies on a two Higgs doublet models. The three additional
scalar degrees of freedom are H0 and H±. The neutral boson is defined as the orthogonal
combination to the SM Higgs field h in the Higgs basis. It is aligned with the SM vev v
such that the mixing matrix re(ads ) ( )( )
h ≡ sβ cβ hu0 − , (II.40)H cβ sβ hd
where β describes the mixing angle between vu and vd. We assume that H
0 and H± are
much heavier than the electroweak scale, such that further mass mixing between H0 and
h is suppressed.
17
CHAPTER II. THEORETICAL BACKGROUND
II.3.3 Axion Potential
In both, the KSVZ- and DFSZ-type models we can write the axion Lagrangian after
anomalous transformation of the fermions as ( )
L 1 ∂µa g
2 a
axion ⊃ (∂ 2µa) + Jµ − sPQ 2XA − θ̄ G
a G̃aµνgg SM
2 fa 32π2 f
µν
a
− A e
2 a ( )
X F µν ±γγ 2 µνF̃ +O h, Zµ,W , (II.41)(4π) f µa
where we neglect for now all couplings to the heavy fields h, Zµ and W
±
µ . We see, that the
axion coupling to fermions in this basis is given by the global PQ current JµPQ. This has
the advantage, that integrating out the fermions does not give additional contributions to
the anomalous axion-gauge boson couplings. On the other hand, taking the opposite limit
of light fermion masses in respect to the axion mass will change the anomalous axion-gauge
boson couplings.
Of special interest for experimental searches is the axion-diphoton coupling, which under
canonical charge normalization X = (2A −1gg) is given by
e2 2L a Gaγγ e 1 Aγγaγγ ⊃ −XA F µν µνγγ 2 µνF̃ ≡ − aFµνF̃ , Gaγγ = . (II.42)(4π) f 2a 4∑ 8π fa Agg
The anomaly coefficients Agg and A f f f f 2 f f 2γγ = f Ng Nc (XL(QL) − XR(QR) )/X with N
f
c
being the number of colors of fermion f are model dependent and differ therefore in KSVZ
and DFSZ models.
In order to determine the axion mass, we can use the chiral Lagrangian of the pions for
a leading order result. Therefore, we remove the axion from the GG̃ operator and shift it
into the mass m(atrix(of up- and)dow)n quark via the(trans(formation ) )
→ X
′
u a X
′ a
u exp i − θ̄SM γ d5 u , d → exp i − θ̄SM γ5 d , (II.43)
2 fa 2 fa
where the global(charges (fulfil X(′ ′u +Xd =)1.)The resul(ting q(uark mass)m)a)trix becomes
a a
M → diag mu exp iX ′u − θ̄SM ,md exp iX ′d − θ̄SM , (II.44)fa fa
where the axion field couples exponentially to the masses.
Choosing X ′u = md/(mu+md) and X
′
d = mu/(mu+md), the chiral Lagrangian changes
to (
3 ( ) )v (m +m ) (π0)2 + (π+ 2L →− χ u d ) + (π−)2 m m a 2d uchiral + − θ̄SM ,
2 f2π (md +mu)
2 fa
(II.45)
where the choice of the global charges X ′ and X ′u d prevented any mixing of the axion with
the neutral pion. Here, we neglected the kinetic part of the fields as well as higher order
interactions.
For the axion mass we now get a relation
2 2 mdmum 2 2afa = m f ≡ Λ4 , (II.46)(md +mu)2 π π QCD
18
II.3. AXIONS AND AXION-LIKE PARTICLES
where we introduced Λ4QCD as the mass contribution from QCD. We note, that higher
order QCD corrections as well as instanton contributions will correct this result. For the
vanilla KSVZ and DFSZ model, these corrections can be taken to be small. We refer to
the case that the mafa relationship is a free parameter as an axion-like particle (ALP)
which does not have its origin in the strong CP problem.
Since we neglect higher order interactions of the axion, we can regard the axion potential
as a leading order approximation of (a cosine po)tential(( ) )
4 a a
4
Vaxion = −ΛQCD cos − θ̄SM +O − θ̄SM , (II.47)fa fa
which is the canonical form for an instanton induced potential. If this term is the only
contribution to the axion potential, the minimum for the field a is exactly at ⟨a⟩ = θ̄fa as
we required to solve the strong CP problem exactly.
II.3.4 Axion Diphoton Coupling and Experimental Searches
In order to get a prediction for the axion-diphoton coupling, which we can use to constrain
the axion mass, we need to express the diphoton coupling in the same basis as the potential.
Thus, we transform up- and down-quark anomalously with PQ charges X ′ and X ′u d and
obtain ( )
e2 1 A AKSVZ ADFSZγγ
Gaγγ → −
2 4md +mu γγ 2 γγ 8, = 6Yq , = . (II.48)8π2 f A 3 m +m AKSVZ ADFSZa gg d u gg gg 3
The change in Gaγγ is only a leading order approximation and is sensitive to higher order
QCD corrections as well as instanton calculations.
Figure II.1 shows the canonical DFSZ axion line in the {ma, Gaγγ} plane, where we used
the results from reference [54], ( )
e2 1 Aγγ
ΛQCD = 75.5(5) MeV , Gaγγ = − 1.92(4) , (II.49)
8π2 fa Agg
which also include higher order QCD corrections.
There are multiple ways to search for the anomalous axion-diphoton coupling in experi-
ments and through astrophysical observations. Experiments which are primarily dedicated
to the search for axions make use of the Primakoff effect [55], in which the application
of a strong magnetic field can convert axions into photons and vice versa. In the lab-
oratory this can be achieved by Light-shining-through-walls experiments [56–61], where
one magnetic field converts a beam of light into axions which then can pass through a
light-absorbing barrier (“wall”) and get reconverted by a second magnetic field and hence
could be measured.
On the other hand, there are experiments which aim for detecting axions from natural
sources, and therefore only need to reconvert the axions into photons by applying a mag-
netic field. There are two main examples for this kind of experiments, Helioscopes and
Haloscopes [62]. The goal of Helioscopes [63–65] is to find axions which are produced in
the sun. Hence, the respective experiments track the apparent movement of the sun while
being shielded from sunlight. Haloscopes instead search for axions which are present in
the Dark Matter Halo [66–101]. Thus, Haloscopes rely on axions contributing to the local
Dark Matter density, i.e. through the axion misalignment mechanism [102–104].
19
CHAPTER II. THEORETICAL BACKGROUND
10-5
10-9
10-13
10-17
10-20 10-15 10-10 10-5 1 105
Figure II.1: Expected QCD axion band for ΛQCD = 75.5(5) MeV in the axion mass
vs. diphoton coupling parameter space. The band corresponds to the possible region
which is motivated by both KSVZ and DFSZ models. The constraints are extracted from
reference [53] and are explained in more detail in the main text.
Heavier axions can be probed in collider and beam dump experiments [105–121]. In
collider experiments axions can be produced as intermediate particles, leaving an analysis
of the decay products. Beam dump experiments on the other hand, search for axions
as long-lived particles which decays within a dedicated decay volume. Shielding devices
separate the long-lived particles from SM particles in order to obtain a clean signal.
Finally, there is a large variety of constraints which are based on astrophysical [122–186]
and cosmological [187–189] observations. These mainly indirect measurements involve
diverse radiation and neutrinos emitting sources as, for example, globular clusters or
supernovae. The cosmological bounds rely on our understanding of the cosmic history
including information about big bang nucleosynthesis (BBN) as well as CMB data.
Taking all these constraints into account, Figure II.1 gives an impression of how much
parameter space is still open for experimental searches and theoretical motivation. While
there are many motivations for finding axion-like particles over the full {ma, Gaγγ} range, it
is more difficult to motivate QCD axions apart from the canonical coupling band. Multiple
proposals in this direction have been shown references [190–197].
In principle we can deviate from the canonical band by either modifying the axion
potential to move to different axion masses, or changing the axion couplings, moving along
the Gaγγ-axis. In this thesis we are going to present novel approaches in both directions.
The first part will be about exotic axion potentials based on the work in references [1,3,5],
while the second part deals with exotic axion couplings as presented in references [2, 4].
20
QCD
Axion
Part I
Exotic Axion Potential

CHAPTER III
Axion Potential and Goldstone Basis
The first part of this thesis is dedicated to the axion potential Vaxion. As we discussed in
section II.3, the QCD(axion pote)ntial i(s g(iven at lea)din)g order by
− 4 a − O a
4
Vaxion = ΛQCD cos θ̄SM + − 4
mdmu
θ̄SM , Λ
2 2
QCD = mπfπ .fa f 2a (md +mu)
(III.1)
Nevertheless, corrections from QCD, instantons and additional PQ breaking operators
can change this picture drastically. While QCD and instantons give mainly a correction
to ΛQCD, the implications from additional PQ breaking operators are more severe. In
particular, these operators move the minimum out of the CP conserving phase of QCD,
spoiling the axion solution of the strong CP problem. This is known as the quality problem
of the axion [23].
Both types of corrections lead to additional mixing effects between neutral Goldstone
bosons. The QCD and instanton corrections mainly couple global Goldstone bosons as
the axion a to the pions π0 of chiral symmetry breaking. Additional PQ breaking terms
lead to a mixing between different global Goldstones as well as Goldstone bosons of gauge
symmetries, like the φ0.
In this chapter we provide with a general description of a theory with multiple complex
scalar fields, focusing on the angular Lagrangian Lang for which we disregard radial degrees
of freedom. With this description we can find a basis in which we can distinguish between
neutral Goldstone bosons of spontaneously broken U(1) gauge symmetries, spontaneously
broken global U(1) symmetries and pseudo Goldstone bosons of explicitly broken global
U(1) symmetries.
In section III.1 we present a generalized scalar Lagrangian and point out linear trans-
formations to obtain a useful basis. We demonstrate the procedure for the vanilla DFSZ
model. Afterwards, in section III.2, we derive the parametric dependence from additional
effects coming from fermion interactions and SU(N) instantons.
III.1 Generalized Scalar Lagrangian and Goldstone Basis
We start our discussion of the angular Lagrangian by constructing a generalized version of
a scalar Lagrangian containing NΦ complex scalars Φi. The kinetic and potential terms
of this theory then read
∑ ∑ (∏( )ni ( )∗)n̄iΦ a aL 2 4 i Φiscalar ⊃ |Di,µΦi| − λaΛ + h.c. , (III.2)
Λ Λ
i a i
23
CHAPTER III. AXION POTENTIAL AND GOLDSTONE BASIS
where a runs over all terms in the potential with corresponding dimensionless coupling λa
and mass scale Λ.
We impose a U(1)NA gauge symmetry on our theory with NA being the number of
corresponding gauge bosons AI,µ. Hence, th∑e covariant derivative of a scalar Φi is givenby
D ii,µ = ∂µ − i gIQIAI,µ , (III.3)
I
with gI being the gauge coupling of gauge field AI,µ and Q
i
I being the charge of the scalar
field under the respective gauge symmetry. In general, the U(1) gauge symmetries can be
subgroups of SU(N) symmetries, but since we are only interested in neutral Goldstone
bosons we disregard further SU(N) gauge bosons.
The terms in the scala∑r p(otential )are characterized∑by( their di)mension and total U(1)charges,
d ≡ nia a + n̄ia ≤ 4 , QI,a ≡ ni i ia − n̄a QI . (III.4)
i i
In order to preserve gauge invariance we require QI,a = 0 for each gauge symmetry.
Since the U(1) gauge symmetries can be in general subgroups of SU(N) symmetries, we
forbid kinetic mixing of the field strength tensors, such that there is only a mass mixing
between the gauge bosons and the mass basis can be found by performing an orthogonal
rotation,
G ⊃∑ N O−(N→A) m 0 m 0Agauge U(1) U(∑1)NA × U(1)N IA −∑→ U(1)NA , (III.5)
g QiA = g QiI I I,µ I IA
i
I,µ + gJQJAJ,µ , (III.6)
I mI ̸=0 mJ=0
where NmA corresponds to the number of massive gauge bosons and N
0
A to the number
of massless gauge bosons. In the SM this orthogonal transformation corresponds to the
mixing between the Z boson and the photon through the weak mixing angle. We discuss
possible deviations through kinetic mixing in section VII.1.3.
In order to match pseudoscalar degrees of freedom of the complex scalars with longitu-
dinal modes of the massive gauge bosons, we parametrize the complex fields non-linearly
in their radial and angular modes, ( )
vi√+ ϕi aiΦi = exp i , (III.7)
2 vi
√
where ϕi represent the radial modes, ai the angular modes and vi = 2⟨|Φi|⟩ the vevs
set by the scalar potential.
Disregarding for now the explicit symmetry breaking by the scalar potential, we can
perform a corresponding orthogonal transformation on the angular modes,
G ⊃ N O−(NΦ) N 0φ v 0Φglobal U(1) → U(1) × U(1)Na × U(1)N −→iΦ U(1)NΦ , (III.8)
where N0Φ denotes the number of complex scalars with vanishing vev vi = 0, Nφ the
number of Goldstone bosons which get absorbed as longitudinal modes of the gauge fields
and Na the number of global Goldstone bosons with decay constants fK.
Without loss of generality we assume that all spontaneously broken gauge symmetries
follow a Higgs mechanism (NmA = Nφ) and that all vevs are non-vanishing (N
0
Φ = 0), such
24
III.1. GENERALIZED SCALAR LAGRANGIAN AND GOLDSTONE BASIS
that the orthogonal transformation can be written as
a ∑ φ0 ∑i aK
= gIQ
i I i
I + Q . (III.9)vi m
K
I fK
mI≠ 0 K
The corresponding rotation matrix and orthogonality conditions are given by
g i iT IQIviR T
QKvi
iI = ∑ , RiK = , ∑ ∑ (III.10)mI fKgIQi i i i i iIvi gJQJvi QKvi QLvi gIQ vi Q viδIJ = , δKL = , 0 = I K . (III.11)
∑mI mJ fK f m fi i L i I K
From m2I = g
2 (QiI i I)
2v2i we can deduce that the gauge charges to massless gauge bosons
vanish for vi ̸= 0, such that the angular(part of th(e kinetic term)s becomes∑ ∑ )2
| |2 (vi + ϕi)
2 ∑ ∂ φ0 ∑i µD I iaKi,µΦi = gIQI A − − Q
∑ 2 ( )
I,µ
m KI fi i I K K
2
ϕ−i→→0 m
2
I − ∂µφ
0 ∑ 1
A II,µ + (∂
2
µaK) . (III.12)
2 m 2
I I K
We notice, that indeed the fields φ0I get absorbed as longitudinal modes for the massive
gauge bosons AI,µ while the fields aK represent scalar Goldstone bosons for the sponta-
neously broken U(1)Na symmetry.
We call the new basis the Goldstone basis in order to emphasize that this is neither
the so-called Higgs basis, nor the mass basis of the global Goldstones aK. The Higgs
basis describes the basis in which each radial mode is aligned to a gauge boson mass.
It coincides with the Goldstone basis, if the charges of the complex scalars are of equal
j
magnitude (|QiI| = |QI|).
In order to switch to the physical mass basis we consider the angular potential where
we param∣ etrize the coupling(s in t(heir no)n-linea)r form( λa = |λa| exp(−iθa),∣ )∣∣ ∑ ∏ v nia+n̄ia ∑| | 4 √ i ( ) aiV = 2 λa Λ ( co)s ( n
i i
a − n̄a − θa
ϕ →0 ∑ ( ) 2Λ∏ vi a i i i )|λ | √a 4−da i i ∑ aK
= 2Λ (v )na+n̄ai cos QK,a − θa . (III.13)
2 f
a i K K
We notice, that the Goldstone bosons φ0I of the gauge symmetries drop out due to the
vanishing total gauge charge and that terms which are only sensitive to the absolute
values of the complex scalars |Φi| do not contribute to the angular potential, since they
have nia = n̄
i
a for each scalar.
In the special case where the terms in the angular potential contain orthogonal linear
combinations of global Goldstone bosons, we can write QK,a ≡ QaδKa and the Goldstone
basis aligns with the mass basis.
III.1.1 Application to DFSZ Model
In the case of the DFSZ Model we have three complex scalar fields Hu, Hd and Φ, one
massive neutral gauge boson Zµ and one term in the scalar potential with coupling µA
25
CHAPTER III. AXION POTENTIAL AND GOLDSTONE BASIS
which breaks the global symmetry explicitly. Hence, we have three pseudoscalar degrees of
freedom, the longitudinal mode φ0 of the Zµ, a heavy pseudo Goldstone boson A from the
explicitly broken global symmetry and a massless Goldstone boson of the remaining global
PQ symmetry, the axion a. We discuss the QCD induced potential which furthermore
breaks the PQ symmetry in section III.2.
In order to identify the linear combinations of angular modes which represent the heavy
pseudoscalar A and the axion a we first perform an orthogonal transformation on the
initial global U(1)3 in the unbroken phase,
O(3) µ
U(1) AHu × U(1)H × U(1)d Φ −→ U(1)Y × U(1)PQ × U(1)Z −→ U(1)Y × U(1)PQ , (III.14)
where we align one linear combination with the U(1)Y gauge symmetry of hypercharge and
one with a global U(1)Z symmetry which is explicitly broken by the term proportional to
µA. The alignment with U(1)Y instead of the spontaneously broken subgroup of U(1)Y ×
SU(2)L is sufficient since the charged Goldstone bosons φ
± do not mix with the chargeless
fields and the T3 charges of the Higgs components in the broken phase match the respective
hypercharges.
We parametrize the orthogonal rotation matrix by three angles β1, β2 and β3. Abbre-
viatingsineandcosine by sβ and cβ we can writei i  φ0  sβ1cβ3 −cβ1cβ3 −sβ3 aua = cβ1cβ2 − s  β1sβ2sβ3 sβ1cβ2 + cβ1sβ2sβ3 −sβ2cβ3 ad , (III.15)
A cβ1sβ2 + sβ1cβ2sβ3 sβ1sβ2 − cβ1cβ2sβ3 cβ2cβ3 aΦ
where we can now evaluate the mixing angles under the use of equation (III.10),
sβ3 = −
gY YΦvΦ
= 0 ⇒ cβ
m 3
= 1 (III.16)
Z
−gY Yuvu mZ vutβ1 = = ⇔ v
−1
usβ = v = v c
−1
d β (III.17)m 1 1Z gY Ydvd vd
XΦvΦ fA X fA
t −1 −1 −1 −1β2 = − = ⇔ faX s = vΦ = fAZ c , (III.18)fa ZΦvΦ Z f β2 Φ β2Φ a
with ZΦ being the charge of Φ under U(1)Z . We can choose ZΦ freely, since it normalizes
the decay constant fA of the heavy pseudoscalar.
Under the requirement that the axion has to be mainly composed out of the angular
mode aΦ in order to ensure a high PQ breaking scale fa ≫ fA, v, we choose for the charge
normalization
f2A faZΦ ≡ X ⇒ tanβ2 = ≫ 1 . (III.19)
f2a fA
Since the opposite case, in which aΦ mainly mixes into A is already ruled out [198], we
will treat f2 2A/fa ≪ 1 as a small parameter for the remainder of this thesis.
Finally, we can use Xu+Xd = X and the orthogonality conditions from equation (III.11)
to find a relation between fa and the SM vev v.
gY Yuvu Xuvu gY Ydvd Xdvd
0 = + ⇔ X 2u = cβ X , Xd = s2β X , (III.20)m f m f 1 1Z a Z a
c2Xuvu β Xsβ1v
cβ1cβ2 = =
1 ⇔ facβ
f f 2
= Xvsβ1cβ1 , (III.21)
a a
where we get a large scale separation for β2 → π/2 and β1 → π/4.
26
III.2. SCALAR POTENTIAL FROM FERMION CONDENSATES
III.2 Scalar Potential from Fermion Condensates
Another contribution to the angular potential comes from Yukawa interactions to fermions
which form a condensate of the form ⟨ψ̄jψj⟩ = v3j . This condensate breaks the chiral
symmetry of the ferm(ions ψ
j .)The corresponding(pions)form a (matr)ix valued field
f a a 0 0
2
ψ ψ π t fψ πj πjΣ = exp 2i , Σψjj ≡ exp i +O  , (III.22)2 fψ 2 fψ fψ
where ta represent the generators of the chiral symmetry, fψ the decay constant of the
pions and π0j the neutral pions from the diagonal entries of the pion matrix.
Using this notation and neglecting flavor-changing effects we can express the Yukawa
potential by
∑ (∏( ) ( ) )ni ∗ n̄i ∏( ) j j ñjbb b ψ̄ ψL − Φi ΦYukawa = y 4 i L R bΛ + h.c.
Λ Λ Λ3
b
∑ (
i
∏( ) ( ) )
j
∏( )ñj ( ) jni ∗ n̄i Σψ b v3 ñbb b
→ − y Λ4 Φi Φi  jj j b + h.c. ,
Λ Λ f Λ3
b i j ψ
(III.23)
where b runs over all Yukawa interactions and vector-like masses and yb represent the
corresponding Yukawa couplings.
The ex∑pre(ssions fo)r the∑dimension and total∑char(ges of th)e operat∑ors extend to
d ≡ ni + n̄i + 3 ñjb b b b ≤ 4 , QI,b ≡ nib − n̄i i
j
b QI + 2 ñbQ
j
I,A , (III.24)
i j i j
where we require QI,b = 0 for gauge symmetries.
We can remove the Goldstone bosons of the gauge fields by performing an axial trans-
formation of the fermions ( ∑ )
j → φ
0
ψ exp i g Qj II I,A γ5 ψ
j . (III.25)
m
I I
Expressing the Yukawa couplings non-linearly by yb = |yb| exp(−iθb) the angular potential
extends∣ to∣∣ (∣ ∑ ∏( )
)
v n
i
b+n̄
i
b ∏( )ñjv3 bV = 2|y | i jΛ4 √ 

b
ϕ →0  2Λ 2Λ3i b ∑i j ( ) ( ) × i − i a ∑ π0 ∑ φ0i jcos n j j I b n̄b + ñb + 2 gIQv I,A − θb
i ( i j ) fψ mI  I∑ |y | (√ )4−d ∏ ∏(√ )ñjb b
= 2Λ (v )n
i i
b+n̄b  2v3 b
 i2 jb i j∑ ∑ ( ∑ ) a 0× K πjcos Q j j aKK,b + ñ − 2 Q − θ b . (III.26)
f b f K,A f
K K j ψ K K
27
CHAPTER III. AXION POTENTIAL AND GOLDSTONE BASIS
We see, that the dependence on the gauge Goldstone bosons φ0I again drops out due to
QI,b = 0 and that we have an additional term in the cosine which leads to a mass mixing
between the global Goldstone bosons aK and the neutral pions π
0
j .
From the Y∣ukawa interaction we ca(n now deduce the f)ermion mass mj , defined by∣∣ ∑ ∑L j aKYukawa∣ ⊃ − mjψ̄ exp −2i Qj jK,A γ5 ψ +O (θb, QK,b) , (III.27)
ϕi→0 ∑j ( ) (
f
K
|y | √ 4−db ∏
K )
b i i⇒ mj = ñjb √ 2Λ (vi)
nb+n̄b , ñjb ∈ {0, 1} . (III.28)
2
b i
We notice, that only the global Goldstone bosons have a coupling in the mass sector of
the fermions, which we will make use of in the ALP-EFT description in section V.1.
Hence, the mass mixing potential for π0j and aK is given by
∑ π0j aK
Vmix = 2Q
j 3
K,Avjmj +O (θb, QK,b) . (III.29)fψ fj, KK
We neglect θb, which will be absorbed by the minima of the fields aK, as well as QK,b,
since in this thesis we do not consider Yukawa interactions which break global symmetries
explicitly.
III.2.1 Scalar Potential from SU(N) Instantons
In the case that the fermion condensate is induced by the confinement of an SU(N)
symmetry, we obtain corrections to the angular potential induced by instanton effects.
We recall the gauge sector of the SU(N) Lagrangian from section II.2,
L 1 g
2
gauge ⊃ − F a F aµνµν + θF a F̃ aµνµν , (III.30)4 32π2
where g is the corresponding gauge coupling and θ a CP violating phase. We remove
the couplings to global Goldstone bosons aK from the Yukawa potential via a fermion
transformation ( ∑ ) ∑
ψj → aKexp i Qj j jK,A γ5 ψ , AKQQ ≡ − Q , (III.31)f K,A
K K j
where AKQQ represent the corresponding anomaly coefficients as defined in equation (II.8).
The SU(N) Lag(rangian proportiona)l to θ then changes to
2 ∑
Lθgauge → −
g A aK a aµν 2 KQQ − θ FµνF̃ (III.32)32π2 fK ∏
K ( )
→ K4 K−3 det ψ̄j ψj  ( ( ∑ ))aKL R exp −i 2 AKQQ − θ + h.c. , (III.33)fK
mj≲vj K
where we replaced the FF̃ interaction by the t’Hooft determinantal operator from instan-
tons [199,200], with K being the corresponding effective instanton amplitude [199–201].
28
III.2. SCALAR POTENTIAL FROM FERMION CONDENSATES
This induces an angular potential from instantons of the form∑  ∏ ( ) ñj ( )ñj ( )n̂jc c cΣψ v3 m Λ2 ( )4 jj j j j Vinst = K  δñj j − 1n̂ 
c m( ( fψ K3 K3 c cj≲vj ∑ ))
× aKexp −i 2 AKQQ − θ + h.c.∑ ∏K ( )
fK
ñj3 c
4−3N0 vj ( ) ( )j= 2K ψ m Λ2 n̂cj j δ − 1  (III.34) 2 ñ
j
cn̂
j
c
c
 ∑
mj≲vj ∑ aK π0× jcos 2 A jKQQ − ñc − θ , (III.35)f
K K
fψ
mj≲vj
where λj defines a cut-off scale for the instanton size integration, c runs over all contribut-
ing instanton diagrams and N0ψ denotes ∑the n(umber o)f light fermions, defined via
N0 ≡ ñj + n̂jψ c c . (III.36)
mj≲vj
The instanton potential now also introduces a mass mixing of the global Goldstone bosons
to the pions of an approximate global SU(N0ψ) symmetry. This mixing can be suppressed,
if the instanton contribution to the pion mass is much smaller than the contribution from
the fermion masses.
III.2.2 Application to DFSZ Model
In the DFSZ model, the PQ symmetry is broken by the quark condensate vχ,
vχ
U(1)Y × U(1)PQ −→ U(1)Y . (III.37)
This explicit breaking of the global PQ symmetry can be evaluated in two different ways,
via the Yukawa induced quark masses of the chiral Lagrangian Lchiral or through instanton
effects.
In the vanilla QCD axion calculation from section II.3 we used chiral transformations
to shift the axion interaction into the first generation quark mass matrix. In this case,
where we have
mj
θb = 0 , QK,b = 0 , 2Q
j
K,A = 1− , π
0
j = (δju − δjd)π0 , (III.38)mu +md
the mass mixing between the axion a and the neutral pion π0 in equation (III.29) cancels
and we recover the leading order axion potential with m2f2a a = Λ
4
QCD from equation (III.1).
Higher order QCD corrections include further neutral mesons and therefore change the
potential and mass mixing.
On the other hand, if we calculate the axion potential through instanton effects we
obtain additional corrections, which can have a strong impact on the {ma, fa} parameter
space. The starting point for the instanton calculation for the QCD axion is the anomalous
SU(3)C Lagrangian, ( )
g2L ⊃ − s a − θ̄ Ga G̃aµνagg , (III.39)
32π2
SM
f µνa
29
CHAPTER III. AXION POTENTIAL AND GOLDSTONE BASIS
where we can make use of the axion shift symmetry by shifting a → a + θ̄SMfa to define
⟨a⟩ = 0 as the CP conserving phase of QCD.
At the QCD confinement scale there are three light quarks, the up-, down- and strange-
quark (u, d, s). These three quarks respect an approximate global U(3)L×U(3)R symmetry
which gets broken by the quark condensate v3χ. The associated Goldstone bosons which
are neutral in all quantum numbers are the pion π0 ∼ ūu − d̄d as well as the mesons
η ∼ ūu+ d̄d− 2s̄s and η′ ∼ ūu+ d̄d+ s̄s with their respective decay constants fπ, fη and
fη′ .
We now construct the potential of the neutral Goldstone bosons following the procedure
from subsection III.2.1. We neglect the mixing with the η meson as well as mixing effects
induced by the strange quark which are suppressed by 1/ms compared to 1/mu and 1/md.
Including the mass contributions to mπ and mη′ from the chiral Lagrangian leads to an
instanton induced(potenti(al of ) ( ))
v9 a η′ v6 ∑ a η′ 0
V DFSZinst =−
χ χ π
2K−5 cos − 2 + mjΛ2 cos − − (δju − δjd)
6( f ja ) fη′ 4 ( ) fa fη′ fπj=u,d′ 0 ′ 0
− η π η πmuv3χ cos + −m v3d χ cos − . (III.40)fη′ fπ fη′ fπ
From this potential we can now derive a mass mixing matrix between the fields a, η′ and
π0.
In the following we present our calculation from reference [1], where we reproduced and
extended the discussion in [201]. Under the use of the abbreviations
mumd v
9
χ
m± = md ±mu , µ = , Λ4
m +m η
′ = ,
u d 4K5
v6χ
µΛ3 = (m Λ2 +m Λ2) , 0 ≈ m Λ2 −m Λ2inst u u d d u u d d , (III.41)4K5
we obtain the mass contribution1 Λ4( ) ′+2µΛ
3 4
inst 2Λ ′+2µΛ
3
η η inst
 −   f
2
a faf
0
η′  a
1 ′  2Λ4DFSZ ⊃ 0 +2µΛ3 m v3+4Λ4 +2µΛ3 3 Vinst a η π − η′ inst + χ η′ inst m−vχ η′ ,2 faf −η′ fη′ fη′fπ  0
m−v3 m 3

− χ +vχ+2µΛ
3 π
0 instfη′fπ f2π
(III.42)
which is hierarchical in the decay constants fa, fη′ and fπ.
The diagonalization of the mass matrix at leading order in 1/fa leads to the pseudoscalar
mass relations
m v3 + 4Λ4 3 3 3
2 + χ η′
+ 2µΛinst m+vχ + 2µΛ
m instπ,η′ = √ +√( 2f2 2f2√ η
′ π
√ )m v3 + 4Λ4 + 2µΛ3 2+ ′ 3χ η inst m+vχ + 2µΛ3 m2 v6∓ − inst − χ2 + , (III.43)2f 2f2 f2 f2η′ π η′ π
1We refer in reference [1] to this calculation as the KSVZ case, since we assume a more general axion
Lagrangian, in which we have not necessarily shifted the axion couplings into Lagg.
30
III.2. SCALAR POTENTIAL FROM FERMION CONDENSATES
( )2
(m v3 3 4+ χ + 2µΛinst) 2Λη′ + 2µΛ
3
2 2 4 3 instmafa =Λη′ + 2µΛinst − , (III.44)(m 3 4 3 3 3 3 2+vχ + 4Λη′ + 2µΛinst)(m+vχ + 2µΛinst)− (m−vχ)
which we now further expand in different limiting cases.
For this purpose we define a parameter
m v3+ χ + 4Λ
4 3
η′ + 2µΛinst m v
3 + 2µΛ3+ χ
∆2 ≡ − instm 2 , (III.45)f 2η′ fπ
which for small K ≪ vχ becomes large as it scales as ∆m → 4Λ4 2η′/fη′ .
Using a series representation of the square root in equation (III.43) allows us to express
the masses via ( )( )
m v3 + 4Λ4 + 2µΛ3 ∆2 ∑∞ 1/2 4m2 6 k2 + χ η′ inst m −vχmη′ = + , (III.46)f2 4 2 2η′ (2 k ∆k=)1( )mfπfη′3 km+vχ + 2µΛ3 2 ∑∞∆ 1/2 4m2inst m −v6χm2π = − , (III.47)f2 2 k ∆4 2 2π k=1 mfπfη′
such that the limit Λη′ ≫ mu,md is given by dropping the last term containing the series.
Similarly, we can rewrite the relation for the axion from equation (III.44) using a series
of the form
(m+v
3
χ + 2µΛ
3 2 2 6
m2f2 = inst
) −m−vχ
a a 4(m 3 3+vχ + 2µΛinst) ( )k
(4µ2Λ6inst +m
2 6 2 6 2 ∑∞ 2 6 3 3 2
− +
vχ −m−vχ) m−vχ − (m+vχ + 2µΛinst)
16Λ4
. (III.48)
η′(m v
3
+ χ + 2µΛ
3 2 4 3 3
inst) 4Λk=0 η′
(m+vχ + 2µΛinst)
Again, we obtain the limit Λη′ ≫ mu,md by dropping the term which contains the series.
Under the use of the leading order expressions from equations (III.47) and (III.48), we
can express the axion mass relation by ( )
m4 4πfπ −m2−v6 m2 Λ3 (m 3 32 2 χ 4 − inst +vχ + µΛinst)mafa = = Λ4m2f2 QCD
1 + 4 . (III.49)m m f4π π + π0 π
We see, that the leading expression gets corrected by a small factor proportional to the
quark mass difference. Under the use of the experimental values [202]
fπ0 ≈ 131 MeV, fη′ ≈ 121 MeV,
mπ0 = 134.9768(5) MeV, mη′ = 957.78(6) MeV,
mu = 2.16
+0.5
−0.26 MeV , md = 4.67
+0.48
−0.17 MeV , (III.50)
we can derive the mass scales vχ = 336.3MeV, Λη′ = 239.3 MeV and Λinst = 261.7 MeV.
This provides an estimate of the small correction factor of m2f2a a = Λ
4
QCD(1 + 0.037).
31

CHAPTER IV
Anarchic Axion
Every QCD axion model suffers from having a quality problem [23], meaning that addi-
tional PQ-breaking operators move the axion minimum out of the CP conserving phase of
QCD. This is especially problematic since Planck mass suppressed operators are expected
break global symmetries in general [203–205].
In this chapter, we show that such additional operators also drastically change the be-
havior in the {ma, Gaγγ} plane, allowing for heavier and much lighter axions than usual.
Therefore, we first show the general concept in section IV.1 and possible operators which
generate this behavior in section IV.2. Finally, we present in section IV.3 a UV com-
pletion which relaxes the quality problem. While the general concept was published in
reference [3], the UV completion is treated in a follow-up work [5].
IV.1 Light Axion from additional PQ breaking Operator
In this section we demonstrate the modified {ma, Gaγγ} behavior in the DFSZ model from
an additional PQ breaking operator that has a linear dependence on fa. We start with
the angular potential of the D(FSZ mo(del, 2 ) ) ( )
V DFSZ
|µA| 1 A f
= − √ v v v cos 1 + A − θ − Λ4 a Aang u d Φ A QCD cos + − θ̄SM ,
2 N 2g fA fa fa fA
(IV.1)
where we expressed µA = |µA| exp(−iθA) in its non-linear form.
To this angular potential we add a PQ brea(king o(perator of)the for)m
break fa 1 a AVang = −Λ4QCDN2g cos + x − θx , (IV.2)fmax Ng fa fA
where the factor fmax depends on the mechanism which generates this term and marks the
maximal fa for which V
break
ang has a smaller axion mass contribution than the PQ anomaly.
The parameters x and θx depend on the mechanism as well and symbolize the dependence
on the pseudoscalar A and the phase of the additional operator, respectively. The factors
of Ng stem from the normalization of fa and fmax.
We take advantage of the initial shift symmetry in order to move the constant phase
offsets into V breakang . The corresponding transformations read
→ NgθAfA NgθAfaA A+ 2 , a → a+ θ̄SMfa − 2 , (IV.3)f f
1 + A 1 + A
f2 f2a a
such that the CP conserving phase is at ⟨A⟩ = 0 = ⟨a⟩.
33
CHAPTER IV. ANARCHIC AXION
The resulting angular potential, where we also replaced vu, vd and vΦ by the expressions
obtained in subsection III.(1.1 is the)n given by
| | 2 −1
( (
µ f 2
)) ( )
V →− √A vf f N2 1+ A
1 A f a A
ang a A g cos 1 +
A − Λ4
f2 ( N )f f2 QCD
cos +
2 a g A a fa fA
− Λ4 N2 fa cos

1 a A θ̄SM (1− x)θA
QCD g + x − θx + − 2 . (IV.4)fmax Ng fa fA Ng f1 + A
f2a
We see that the dependence on θA drops out for x = 1, hence any deviation from ⟨A⟩ = 0
would be absorbed by a corresponding shift of θA, leaving only ⟨a⟩ ≠ 0 in this case.
In general, we can combine the different phases to a parameter θ̄ and can express the
residual PQ anomaly in the minimum of a and A by an effective phase θ̄eff. These two
phases are defined by
≡ − − (1− x)θA ≡ −⟨a⟩ − ⟨A⟩Ng θ̄ θ̄SM Ngθx Ng , N θ̄ , (IV.5)f2 g eff
1 + A fa fA
f2a
where we defined both parameters with an additional factor of Ng to account for the full
2π period of θx and θA.
For a sufficiently large |µA| ≫ ΛQCD, the mass of the heavy pseudoscalar is dominantly
given by the µA term, ( 2 ( ))
2 |√µA| fa f ΛQCDmA = v 1 + A +O . (IV.6)
2 f 2A fa |µA|
Consequently, the mixing between A and a in the mass basis is negligibly small and the
minimum of A is only set by the µA term to ⟨A⟩ = 0, independently of the choice of x.
Hence, we can treat the axion potential separately from the heavy pseudoscalar by
setting A → 0, ∣∣∣ ( ) ( )4 a 4 2 fa 1 aVaxion = Vang∣ = −ΛQCD cos − Λf QCDNg cos + θ̄ . (IV.7)A→0 a fmax Ng fa
This potential equals the potential in equation (III.1) of the vanilla axion model for fmax →
∞. Since the phases of the two terms differ by θ̄ the minimum of a gets moved out of the
CP conserving phase by ⟨a⟩ = −Ng θ̄efffa.
In order to find the resulting shift in ⟨a⟩ we determine the minimum of the potential
with respect to t∣∣he axion a,∂Vaxion ∣∣ ( )⇒ − fa ( )∂a ∣ = 0 sin Ng θ̄eff +Ng sin θ̄ − θ̄eff = 0 , (IV.8)∣ a→⟨a⟩ ( ) fmax∂2Vaxion ∣∣ ( ) 2 2= m2a ⇒ fa m fcos −Ng θ̄eff + cos θ̄ − θ̄ a aeff = 4 . (IV.9)∂2a a→⟨a⟩ fmax ΛQCD
We can make use of these relations to constrain the physically allowed parameter space.
Subs(tituting θ̄)leaves us with
1 m2f2 ( ) 4 4 2
1− cos2(N θ̄ )− 2 a a m f f 1g eff 4 cos Ng θ̄ +
a a − aeff 8 + = 0 , (IV.10)N2g Λ 2 2QCD ΛQCD fmax Ng
which depends on the deviation of m2f2a a from Λ
4
QCD and of fa from fmax.
34
IV.1. LIGHT AXION FROM ADDITIONAL PQ BREAKING OPERATOR
We display the allowed parameter space in m2af
2
a/Λ
4
QCD and fmax/fa in Figure IV.1 for
different values of θ̄eff. It shows that we can either deviate from the canonical DFSZ line
to heavier axion masses for small θ̄ and fa > fmax, or to much lighter axion masses for
θ̄ → π and f → f . On the other hand, for f ≪ f we recover m2f2 4a max a max a a = ΛQCD. This
also implies, that in the absence of V breakang (fmax → ∞) we restore the canonical axion
solution as expected.
1000 1000
500
100
100
50
10
10
1 5
0.1 1
0.1 0.5 1 5 10 0.8 0.9 1.0 1.1 1.2
Figure IV.1: Deviation from the canonical axion mass relation in the anarchic axion model.
The allowed parameter space is shown in white for different residual strong CP phases θ̄eff
compared to the axion mass relation in the θ̄ = π case. The left panel shows higher values
of θ̄eff while the right panel represents a zoomed-in version with lower values of θ̄eff.
The maximal extend of a line of constant θ̄ is sensitive to the effective CP violating
phase, which is constrained by Ng θ̄eff < 10
−10 [14]. This residual phase can be determined
from equation (IV.8) and reads for θ̄ close to π
2√(π − θ̄) ( )θ̄ = +O (π − θ̄)2eff . (IV.11)
2 2
−1 + 1 + 4mafmax
Λ4QCD
Since the deviation of θ̄ from π is now of the same order of magnitude as the effective
CP violating parameter θ̄eff we only get lighter axion masses for an accidentally small
difference of θ̄ and π. We see this behavior in Figure IV.1 as the allowed parameter space
approaches the θ̄ = π line for smaller θ̄eff. In section IV.3, we show a model extension that
naturally leads to such a small difference and therefore prevents the reintroduction of a
fine-tuning problem.
In order to see the corresponding behavior in the {ma, Gaγγ} plane, we solve equa-
tion (IV.9) for 1/fa,
( ) √√√√( ) ( ) ( ( ) )21 cos θ̄ − θ̄eff m2a cos θ̄ −( θ̄− eff= + + ) . (IV.12)
f 2f 4a max cos Ng θ̄eff ΛQCD cos Ng θ̄eff 2fmax cos Ng θ̄eff
From this equation we see, that for θ̄ → π and ma → 0 the 1/fa dependence reaches a
constant. For a vanishing θ̄eff this constant is exactly 1/fmax.
35
CHAPTER IV. ANARCHIC AXION
Neglecting the mixing between A and a of order O(ΛQCD/|µA|), the axion diphoton
coupling stays in its canonical form, such that we get new model lines for different fmax,
shown in Figure IV.2. We see that the model lines branch off close to fa ≈ fmax and reach
ma → 0 for fa → fmax. Lines of light axions are already excluded for fmax ≲ 107 GeV by
astrophysical bounds and helioscope experiments such as CAST [63,64].
= -810k
10-10
f 7max-=3*10 GeV= 410k
10-12
f 9max=3*10 GeV =1k
10-14
fmax=3*1011 GeV
FS
Z
D
10- k = (π-θ -1016
Ng|θeff| ≤)/1010-10
fmax=3*1013 GeV
10-18
10-9 10-7 10-5 0.001 0.100 10 1000
Figure IV.2: Deviations from the canonical DFSZ axion line in the {ma, Gaγγ} plane. We
show the axion mass relations for θ̄ close to π for different values of fmax in comparison
to the canonical DFSZ line. The dashed lines show the maximal deviations from π to not
spoil the nEDM bound of N |θ̄ | ≤ 10−10g eff . The limits are extracted from reference [53].
For 107 GeV ≲ f 11max ≲ 10 GeV the model lines for light axions reach a region that
is relevant for Haloscope experiments and telescope-based searches. Our approach not
only shows that additional PQ breaking operators change the QCD axion line drastically
but also constrains the maximal extend of the canonical DFSZ line for an additional PQ
breaking term in the Lagrangian to fa ≲ fmax. In the next section, we discuss how we can
generate such an additional potential.
IV.2 Operators with Anarchic Axion Solution
In order to construct a PQ breaking term of the form of V breakang in equation (IV.2), we
can make use of the various options which we presented in Chapter III. This includes a
term in the scalar potential which explicitly breaks the PQ symmetry. We show such an
example in subsection IV.2.1. Additionally, we can generate V breakang from a fermion con-
densate through either the Yukawa interaction in subsection IV.2.2 or SU(N) instantons
in subsection IV.2.3.
36
IV.2. OPERATORS WITH ANARCHIC AXION SOLUTION
IV.2.1 Explicit soft breaking Term
In the DFSZ model the axion as the Goldstone boson of the spontaneously broken PQ
symmetry is composed of the angular modes of the complex scalars Hu, Hd, and Φ. Hence,
we can define an additional operator which breaks the PQ symmetry explicitly by adding
a new term in the scalar potential(, )( ) ( )
V break − 2 − hu hdsoft = µBHuHd + h.c. = 1 + 1 + |µB|2
au ad
vuvd cos + − 2θB ,
vu vd vu vd
(IV.13)
which describes a soft PQ breaking since it is proportional to a complex mass scale µB =
|µB| exp(−iθB).
We reexpress the vevs and angular modes in terms of the physical scales fa, fA, v and
the fields a and A under the use of the mixing angles tanβ1 = vu/vd and tanβ2 = fa/fA
(see subsection III.1.1). The angular potential is(then(given by ) )
V break 2
1 a A
ang → −|µB| Ngfav cosβ2 cos + − 2θB (IV.14)Ng fa fA
where the amplitude of the term is naturally suppressed by cosβ2 ≪ 1.
The corresponding model dependent parameters read
N 4gΛQCD
x = 1 , θx = 2θB , fmax = . (IV.15)|µ |2B v cosβ2
We notice, that since x = 1 the dependence of θA drops out and the minimum of the heavy
pseudoscalar is exactly at ⟨A⟩ = 0.
From the plateau region, where we have fa ≲ fmax, we can find a constraint on the
magnitude of µB,
N2gΛ
4
| 2 QCDµB| sin(2β1) ≲ 2 ≈ 10−8 GeV2 . (IV.16)
v2
Especially in the case of a large scale separation between v and fa (β1 → π/4) we get
|µB| ≲ 10−4 GeV.
IV.2.2 PQ breaking from Yukawa Interactions
The appearance of a fermion condensate can enter in two different ways into the scalar
potential. The first option is through a Yukawa interaction with one of the complex scalars.
The second option is through instanton effects which we discuss in the next subsection.
We add a fermion ψ which does not transform under the SM gauge group but has chiral
Zn charges with a difference of 1. We then can write down(a Yukaw)a interaction of theform
break |√yψ|→ 3 aΦVYukawa = yψψ̄LΦψR + h.c. vψvΦ cos − θψ , (IV.17)
2 vΦ
with coupling yψ = |yψ| exp(−iθψ). This term leads to an angular potential for the axion
via the angular mode aΦ of the complex(scala(r field Φ with) )
break |√yψ|− 3 √Ngfa 1 a f2 AVang = vψ cos − A + θψ − π , (IV.18)
2 f21 + A N
2
g fa fa fA
f2a
where we replaced aΦ and vΦ according to subsection III.1.1. We see, that V
break
ang has a
linear dependence on fa which is crucial for the deviation in the {ma, Gaγγ} plane.
37
CHAPTER IV. ANARCHIC AXION
The related model dependent parameter are then given by √
f2 √ N Λ4 f2
− A − g QCDx = , θx = π θψ , f Amax = 2 1 + , (IV.19)
f2a |y 3ψ|vψ f2a
where now only the large parameter µA ensures the decoupling of A from the axion poten-
tial. The experimental limits from Figure IV.2 now set a constraint on fmax ≳ 107 GeV
which can be converted for a O(1) Yukawa coupling into a limit on the fermion condensate
v ≲ 10−7ψ GeV.
IV.2.3 PQ breaking from SU(N) Instantons
We now consider the case that the complex scalar field couples to Nψ fermions ψ
j , which
are singlets under the SM gauge group but transform under an additional SU(N) gauge
group. We then can calculate the operator which breaks the PQ symmetry explicitly
through instanton effects.
We start by e∑xtending the Yukawa La∑grangian for multiple fer(mio(ns ψj , ) )
Lbreak ⊃ − j j − |yj | j aΦ jYukawa yjψ̄LΦψR + h.c. = √ (vΦ + hΦ)ψ̄ exp i, − θj γ5 ψ ,2 v
j j Φ
(IV.20)
with Yukawa couplings√yj ≡ |yj | exp(−iθj). The Yukawa interactions induce fermions
masses of mj = |yj |vΦ/ 2.
We then use a chiral transformatio(n of t(he form ) )
ψj → i aΦ iexp − − θ jj γ5 ψ (IV.21)
2 vΦ 2
in order to shift the axion coupling into the anomalous θ termof SU(N),
Lθ g
2 g2 a ∑
= θF a
Φ
gauge µνF̃
aµν → − Nψ − θj − θF a aµν
32π2 32π2 v µν
F̃ . (IV.22)
Φ j
The factor Nψ appears since all fermions ψ
j which transform under SU(N) contribute.
The total CP violating θ parameter also shifts by the sum of complex phases θj .
We assume that only one fermion ψ0 has a mass below the confinement scale mψ ≤ vψ.
Hence, we can use equation (III.35) with N0ψ = 1 to obtain the corresponding instanton
potential ( )  ∑ 
V break
aΦ
inst = −K v3ψ + 2mψΛ2ψ cosNψ − θj − θ , (IV.23)vΦ j
with instanton amplitude K and Λψ being a cut-off in the instanton size integration. Since
ψ0 is the only light fermion, we do not have an approximate global SU(N0ψ) symmetry
and therefore no associated pions.
Reexpressing the fermion mass mψ as well as the angular mode aΦ and vev vΦ of the
complex terms in terms of the axion a, the pseudoscalar A and the associated decay
constants, the resulting angular potential reads  
√
V break = −Kv3ang ψ + 2|y 2ψ|Λψ√ ( 2 )N ∑gfa  Nψ a fcos − A A + θ + θ .
f2 2
j
1 + A Ng fa fa fA
f2 ja
(IV.24)
38
IV.3. UV COMPLETION WITH NATURAL LIGHT AXION SOLUTION
We notice, that we not only generate a PQ breaking operator linear in fa but also get a
constant correction to ma of order Kv
3
ψ/Λ
4
QCD. This dependence opens up the parameter
space allowing for higher and lower axion masses, as shown in Figure IV.3.
1000
100
10
1
0.1
0.1 0.5 1 5 10
Figure IV.3: Deviation from the canonical axion mass relation for an instanton induced
additional PQ breaking. We show the allowed parameter space in white for a fixed
Ng θ̄eff = 0.8. The different regions are obtained for a variation of the constant instanton
contribution Kv3ψ from the PQ anomaly Λ
4
QCD.
The figure is based on a modified version of equation (IV.10), taking into account the
c(orrection)proportional to Kv
3
ψ/Λ
4
QCD as well as the m(ultiplicity Nψ of the f)ermions,
N2 m2ψ f
2 ( ) 2 3 2 2
− 2 − a a m
4f4 fa Nψ Kvψ Nψ
1 cos (Ng θ̄eff) 2 cos Ng θ̄eff +
a a − N2 + + = 0 .
N2 4g ΛQCD Λ
8 ψ f N2 Λ4 N2QCD max g QCD g
(IV.25)
It becomes evident, that not only the parameter space opens up, but also that the fmax/fa
ratio gets shifted to smaller values for increasing Kv3 4ψ/ΛQCD.
Thus, the maximal decay constant fmax is defin√ed in the limit where the constant piecevanishes and is given by
N Λ4g QCD f
2
f = √ 1 + Amax . (IV.26)
2|y |KΛ2 f2ψ ψ a
For f ≳ 107max GeV this sets an upper limit on the parameters 2|y 2 −7ψ|KΛψ ≲ 10 GeV.
This is consistent with the assumption, that the Yukawa coupling |yψ| is associated with
the only light fermion which transforms under SU(N), keeping fmax large.
IV.3 UV Completion with natural light Axion Solution
In Figure IV.2 we saw that in order to achieve a lighter axion than usual, the phase θ̄
has to be closely aligned with π, such that light axion solutions would still have a quality
problem. In our work [5] we aim to find a UV completion that avoids this problem.
For this purpose we can embed our model into a Nelson-Barr solution [50, 51] of the
strong CP problem. Nelson-Barr models regard CP as a conserved symmetry in the UV,
39
CHAPTER IV. ANARCHIC AXION
such that the strong CP phase is parametrically zero and can only emerge radiatively or
through higher dimensional operators. Including an axion in our case allows for a second
solution of Nelson-Barr with a strong CP phase of π in the UV. This second case coincides
with our light axion construction.
Since the SM also includes a CP violating phase in the weak sector, a viable Nelson-Barr
extension must incorporate a mechanism that violates CP spontaneously. This sponta-
neous CP violation (SCPV) are minimally realizable by adding a vector-like quark q and
an additional complex scalar ΦC , as was shown in [206]. The corresponding field content
is depicted in Table IV.1.
SU(3)C SU(2)L U(1)Y Zn U(1)PQ
QiL 3 2 1/6 0 XQ
uiR 3 1 2/3 1 XQ-Xu
diR 3 1 -1/3 0 XQ-Xd
LiL 1 2 -1/2 0 XL
eiR 1 1 -1 0 XL-Xd
Hu 1 2 -1/2 n-1 Xu
Hd 1 2 1/2 0 Xd
Φ 1 1 0 1 -X
qL 3 1 2/3 - δqd n/2 + δqu Xq
qR 3 1 2/3 - δqd n/2 + δqu Xq
ΦC 1 1 0 n/2 0
Table IV.1: Field content of the Nelson-Barr extension in the anarchic axion model. We
add a vector-like quark q and a complex scalar ΦC to the field content of the DFSZ model.
The quark can either have up-type charges (δqu = 1, δqd = 0) or down-type charges
(δqu = 0, δqd = 1).
In addition to our requirement of n > 4 to secure the DFSZ model, we also assume that
n represents an even number which is necessary to generate SCPV. This becomes clear by
investigating the scalar Lagrangian of the NB extension,
LNB DFSZ 2 2 2 4scalar ⊃Lscalar + |∂µΦC | − µC |ΦC | − λC |ΦC |
− (λ |H |2|Φ |2 − λ |H |2|Φ |2uC u C dC d C − λΦC |Φ|2|Φ 2C | )
− µ̃2C + λ̃C |ΦC |2 + λ̃uC |H |2u + λ̃ |H 2 2 2dC d| + λ̃ΦC |Φ| ΦC − λ′CΦ4C + h.c. ,
(IV.27)
where the las√t line only appears for even n and induces a complex vev ⟨ΦC⟩ ≡
vC exp(−iθC)/ 2 for the scalar field ΦC including a CP violating phase θC .
In order to move the CP violating phase into the weak sector of the SM, we choose the
vector-like quark either being up-type (δqu = 1, δqd = 0) or down-type (δqu = 0, δqd = 1),
such that the Yukawa Lagra(ngian becomes) ( )
LNB DFSZ i iYukawa ⊃ LYukawa − q̄L yCΦC + ỹCΦ∗ i iC δquuR + δqddR − µq q̄LqR + h.c. . (IV.28)
This choice however, not only opens up a decay channel for the additional quark but it
also introduces an off-diagonal mass mixing term proportional to the Yukawa couplings
yi iC and ỹC in addition to a vector-like mass µq of the quark.
40
IV.3. UV COMPLETION WITH NATURAL LIGHT AXION SOLUTION
Since the N(elson-Barr construction assures CP to be a conserved symmetry in( the UV),the couplings can only)depend on phases which are multiples of π: yi ≡ |yiC C | exp −ikiCπ ,
ỹi ≡ |ỹiC C | exp −ik̃iCπ and µq ≡ |µq| exp(−ikqπ), with {kiC , k̃iC , kq} ∈ Z. The correspond-
ing quark mass matrix thenreads( ) ( − ) v iθ iθ v
LNB ⊃ − ūL d̄L q̄L Y
u
u√ 0 δqu (yCe C + ỹCe C) √C2 2
0 Y √vd δ y e−iθC + ỹ eiθ √vC CC C 
 
uR
Yukawa d 2 qd
d
2 R
+ h.c. .
0 0 |µ |e−ikqπq qR
(IV.29)
From this Lagrangian we can extract the four dimensional Yukawa matrices Yqf for both
up- and down-type quarks, f ∈ {u, d}. The Yukawa matrices and their hermitian products
Y†qfYqf read( √ )  
v2
Y 2f δqf v m
f † vf †
C
Y ≡ √ f , Y† 2
Y Yf δqf √ Y mC
Y =  2 f 2 f qf 2 −ikqπ qf qf0 v2 ,√vf †v |µq|e f δqf mCYf (|µq|2 + δ |m |2)f 2 qf C
√ (IV.30)
where we used m −iθC iθCC ≡ (yCe + ỹCe )vC/ 2 as a short-hand.
Assuming |µq|2 + |mC |2 ≫ v2, we can now find the three-dimensional Yukawa matrices
in the DFSZ model YDFSZf by diagonalizing equation (IV.30), which leads to
( ) Y†m m†
DFSZ † DFSZ † − f
C CYf
Yf Yf = YfYf δqf . (IV.31)|µ |2 2q + |mC |
We notice that we only get a correction for those quarks which couple directly to q (δqf =
1).
Since the Nelson-Barr construction forbids CP violation in the UV, the Yukawa matrices
Yf are purely real, such that a complex phase only enters through the SCPV phase θC in
mC . We transform the SM quarks according to subsection II.2.2 using U(3) matrices to
obtain real and diagonal Yukawa matrices YDFSZD,f ,
fL → Uf fL , fR → Vf fR , U†fU
†
f = 1 = VfV
DFSZ † DFSZ
f , YD,f = UfYf Vf . (IV.32)
This then leads to the CKM matrix VCKM ≡ U†uUd, which contains a CP violating phase
as expected, since either the up-type or down-type quarks have a complex Yukawa matrix
before the transformation.
In order to determine the CP violating phase of QCD we only need to take into account
the quark sector which couples to the heavy quark q, since the Yukawa matrix of the other
sector is purely real. Since the parameter θQCD is also forbidden by CP conservation in
the Nelson-Barr model, the parame∑ter θ̄SM changes to
θ̄SM → − δqfarg (det(Yqf )) = kqπ , (IV.33)
f=u,d
such that the θ̄ parameter from equation (IV.5) is given by Ng θ̄NB ≡ kqπ. Strictly speaking
are all CP phases discretized as kπ with k ∈ Z, but we can absorb all integers k into kq
without loss of generality.
In addition to the kqπ phase we need to take corrections from higher dimensional opera-
tors as well as radiative corrections into account. As it was pointed out by reference [207]
41
CHAPTER IV. ANARCHIC AXION
dimension-5 operators contributing to a strong CP phase are given by
y ∑ yi
Ld=5 ⊃ − qC 2 − fCYukawa ΦC q̄LqR δqf ΦCQ̄iLHfqR + h.c. . (IV.34)Λ Λ
f=u,d
Hence, we get a correction to θ̄NB of Ng∆θ̄NB ∼ θCvC/Λ. One obtains a comparable
contribution from radiative corrections that involves multi-scalar interactions from equa-
tion (IV.27).
We can now compute the residual CP violating phase under the use of equation (IV.11).
For small axion masses it reads
Λ4 m2f2| QCD θCvC faθ̄eff| = ∆θ̄NB ≤ 10−10 ⇔ a a
m2 2 4
≳ . (IV.35)
afmax ΛQCD |θ̄eff|Λ fmax
Hence, for fa close to fmax, v < vC = 1 TeV and a Planck suppressed higher dimension-5
operator (Λ = MPl) we can deviate from the canonical axion mass relation by six orders
of magnitude, m2f2 −6 4a a ≳ 10 ΛQCD.
Since θ̄NB now only takes discrete values, we need to specify the generation of the
V breakang in more detail to obtain the light axion solution. We show this necessity based on
an instanton induced PQ breaking operator as was introduced in subsection IV.2.3.
In particular, one cannot choose the number of fermions Nψ freely anymore. This
becomes evi∣dent as the respective version of equation (I(V.9) now contains)factors of Nψ,
∂2Vaxion ∣∣∣ ( ) fa N m2 2ψ= m2 ⇒ cos −N θ̄ + cos k π −N θ̄ = afa2 a g eff q ψ eff ,∂ a a→⟨a⟩ fmax Ng Λ4QCD
(IV.36)
where the θ̄ dependence was replaced by the discretized version containing kq.
In Figure IV.4 we show the allowed parameter space for Nψ ∈ {1, 2, 3, 4} accounting for
the deviations of m2f2a a from Λ
4
QCD and fa from fmax. We also depicted the discretized
model lines achievable for different kq. We notice that we can only reach the light axion
solution for an odd number of SU(N) fermions.
In conclusion, we showed that one indeed can obtain light axion solutions without rein-
troducing a fine-tuning problem. However, we only reach extreme deviations for sup-
pressed higher-order corrections like Planck suppressed dimension-5 operators. We leave
the construction of mechanisms which relax these bounds further for future work.
42
IV.3. UV COMPLETION WITH NATURAL LIGHT AXION SOLUTION
104 104
1000 1000
100 100
kq=0
10 kq=3 kq=0 10 kq=1
k
kq=2 q=1
1 1
0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2
104 104
1000 1000
=1 k = kq=0100 kq q 0 100
kq=1
10 10
1 1
0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2
Figure IV.4: Deviation from the canonical axion mass relation for an instanton induced
additional PQ breaking with Nelson-Barr extension. We show the allowed parameter space
in white for a fixed N θ̄ = 0.1 and Kv3 ≪ Λ4g eff ψ QCD and vary Nψ between the panels. The
red lines show the expected mass relations for different classes of kq denoted by their lowest
possible value.
43

Part II
Exotic Axion Couplings

CHAPTER V
Axion Couplings and ALP-EFT Basis
In the second part of this thesis we investigate various modifications of axion couplings.
For this discussion we first establish in section V.1 a general ALP-EFT basis which includes
all possible linear axion couplings to fermions, scalars and gauge bosons up to O(1/fa).
Afterwards we calculate in section V.2 the respective Wilson coefficients from integrating
out heavy fermion. The results of this chapter are mainly published in [2].
In Chapter III we saw already that in the flavor-conserving case the global Goldstone
bosons couple to the mass sector of the fermions,
∣∣ ∑ ( ∑ )
L ∣ aKYukawa∣ ⊃ − mjψ̄j exp −2i QjK,A γ5 ψj , (V.1)
ϕ →0 fi j K K
where we set the focus for this chapter on the axion aK ≡ a with the corresponding PQ
charges QK,A ≡ XA. Further global Goldstone bosons as the A from the DFSZ model will
be regarded as much heavier and therefore decouple from the particle spectrum.
In order to take effects into account which can change the flavor of the fermions, we
express the parameters as vectors and matrices in flavor space. The vector Ψ then contains
all fermion flavors and Mψ represents the fermion mass matrix. Expressing also the PQ
charges as vector-like and axial charge matrices, Xψ and XψV A, the axion couplings in the
fermion mass sector be(come ) ( )
LψM ⊃ −
a a
Ψ̄ exp i(Xψ −XψV Aγ ) M
ψ
5 exp −i(XψV +X
ψ
Aγ5) Ψ . (V.2)fa fa
We notice that in the flavor-conserving case the dependence on the vector-like charges
cancels and we recover equation (V.1).
Expanding LψM in leading order in 1/fa leads to the canonical expression
a ( ) ( )Lψ ⊃ 1M i ∑Ψ̄[(M
ψ,XψV ]Ψ + Ψ̄{M
ψ,Xψ
f A
}γ5Ψ +O
a f2a ) ( )a 1
= i (mi −mj)ψ̄ Xiji V ψj + (m
ij
i +mj)ψ̄iXA γ5ψj +O . (V.3)fa f2ij a
In the following section we use the axion interaction in the mass sector of the fermions as
a starting point to obtain the full ALP-EFT basis.
47
CHAPTER V. AXION COUPLINGS AND ALP-EFT BASIS
V.1 Generalized ALP-EFT Basis and Fermion Transforma-
tions
In this section we construct a general ALP-EFT basis for the axion field up to O(1/fa).
We start by identifying interactions which are induced by chiral transformations of fermion
fields and add terms from the scalar Lagrangian in subsection V.1.1. Finally we apply the
general description in subsection V.1.2 to the DFSZ model.
First we need the most general fermionic Lagrangian at dimension 4 including the expo-
nential axion coupling in the fermion mass sector from equation (V.2). It involves gauge
interaction in unitary gau∑ge as well as Yukawa interactions o∑f all fermions and reads
Lψ ⊃ µ ϕiiΨ̄γ ∂µΨ(+ gIΨ̄AI,µγ
µ(Qψ +Qψ √ ψ)I,V I,(Aγ5)Ψ− Ψ̄Yi2I i )Ψ
− Ψ̄ exp i(Xψ a aV −X
ψ
Aγ5) M
ψ exp −i(XψV +X
ψ
Aγ5) Ψ , (V.4)fa fa
where the matrices Yψi represent the Yukawa interactions to the real scalar fields ϕi. We
neglect angular scalar fields besides the axion as they are either absorbed by the gauge
fields in unitary gauge or are assumed to be much heavier.
The charge matrices Qψ ψI,V and QI,A represent the couplings of the fermions to the
respective gauge groups labeled by index I. The gauge interactions are further specified
by their gauge couplings gI and gauge bosons AI,µ = A
a a
I,µtI in unitary gauge, with t
a
I
being the corresponding generators. In case of an SU(N) symmetry the charge matrices
comprise unitary matrices as the CKM matrix in case of the SM SU(2)L gauge group.
From Lψ we can deduce general ALP-EFT terms by performing a chiral transformation
on the fermions(of the form ) ( )
Ψ → exp i(Xψ ψ a aV +XAγ5) Ψ, Ψ̄ → Ψ̄ exp −i(X
ψ ψ
V −XAγ5) . (V.5)fa fa
The resulting Lagrangian can then be used to write down a basis for which the physical
observables are invariant under the given chiral transformation.
Among other terms, the transformation in equation (V.5) leads to an anomaly in the
gauge sector due to the regularization in the Jacobian factor of the path integral follow-
ing Fujikawa’s method [47]. The anomaly appears as a flavor changing generalization of
equation (II.∑8) with anomaly coefficients given by
APQIJ = T (R )(XijQikIk R IRQ
kj ij ik kj
∑ JR
−XL QILQJL)
i,j,k
= 2 T (RIk)(X
ij ik kj
V (QIV QJA +Q
ik QkjIA JV ) +X
ij(Qik kj ikA IV QJV +QIAQ
kj
JA)) , (V.6)
i,j,k [ ]
where T (RIk) again is defined as the Dynkin index [48] via tr T
aT bR R ≡ T (R)δab and RIk
denotes the representation of fermion k under gauge group I.
The transformed Lagrangian then contains
Lψ → ∂ a a
∑
µ gIgJ
iΨ̄γµ∂ Ψ− Ψ̄γµ(Xψ +Xψµ ( V Aγ5)Ψ)− Ψ̄M
ψΨ+ A a a,µν(PQIJ Ff f (4π)2 IµνF̃J∑ a a I,J )a a
+ gIΨ̄A γ
µ exp −i(Xψ +Xψγ ) (Qψ +QψI,µ V A 5 IV IAγ5) exp i(X
ψ
V +X
ψ
Aγ5) Ψfa fI a
48
V.1. GENERALIZED ALP-EFT BASIS AND FERMION TRANSFORMATIONS
∑ ( ) ( )
− √ϕi Ψ̄ exp − a ai(Xψ ψ ψV −XAγ5) Yi exp i(X
ψ
V +X
ψ
Aγ5) Ψ , (V.7)2 f f
i a a
involving the derivative coupling of the axion to the PQ current as well as the the anoma-
lous axion coupling to gauge bosons.
Expanding the gaug(e and Yukawa interaction at leading order in 1/fa leads to∑ )
Lψ µ ψ ψ a µ ψ ψ ψ ψgauge ⊃ gIAI,µ (Ψ̄γ (QIV +QIAγ5)Ψ + i Ψ̄γ [QIV +QIAγ5,XV )+X∑ f A
γ5]Ψ ,
I a
Lψ ⊃ vi√+ ϕi − a aYukawa Ψ̄Y
ψΨ− i Ψ̄[Yψ ψ ψi i ,XV ]Ψ− i Ψ̄{Yi ,X
ψ
A}γ5Ψ . (V.8)2 f f
i a a
We notice that the gauge bosons acquire a new coupling proportional to the commutator
of gauge charges and PQ charges, which vanishes in the flavor-conserving limit.
After pointing out the axion couplings which are generated through fermion transfor-
mations, we can write down the full fermion-related axion Lagrangian with generalized
coefficients at order 1/fa, ∑
Lψ ∂µa µ a gIgJaxion ⊃− Ψ̄γ (C
ψ +Cψ γ )Ψ + CIJ F a F̃ a,µν5
2f 1V 1A f 3a a (4π)2
Iµν J
I,J
i a i a
+ Ψ̄∑[M
ψ,Cψ ]Ψ + Ψ̄{Mψ,Cψ
2 f 2V 2 f ∑2A}γ5Ψa ai a √ϕi ψ ψ i a √ϕi+ Ψ̄[Yi ,C ]Ψ + Ψ̄{Yψi ,Cψ }γ5Ψ2 fa ∑ 2 iV 2 f iAi a 2i ( )
− i a g Ψ̄AIγµ 1I µ [Q
ψ
IV +Q
ψ
IAγ5,C
ψ
IV +C
ψ
IAγ5]Ψ +O . (V.9)2 f f2a I a
This set of operators describes a closed basis under chiral fermion transformations and is
needed to fully calculate the Wilson coefficients in an effective theory where we integrate
out heavy fermions. The operators in the EFT then follow the same pattern, but also
include terms from the scalar Lagrangian, which we show in the next subsection.
Applying a chiral fermion transformation of the form of equation (V.5) changes the
coefficients in equation (V.9) according to
Cψ ψ ψ ψ ψ ψ1V/A → C1V/A + 2XV/A , C2V/A → C2V/A − 2XV/A ,
Cψ → Cψ − 2Xψ , Cψ → Cψ − 2Xψ ,
iV/A iV/A V/A IV/A IV/A V/A
CIJ → CIJ3 3 +APQIJ . (V.10)
We will see in section V.2 that these transformation properties assure that the effective
theory is invariant under chiral fermion transformations.
V.1.1 ALP-EFT Terms from Scalar Potential
The Lagrangian in equation (V.9) does not comprise all axion couplings in the EFT
at order 1/fa, since integrating out heavy fermions also mediates axion interactions to
scalars and gauge bosons which originate at tree-level from the kinetic part of the scalar
Lagrangian.
49
CHAPTER V. AXION COUPLINGS AND ALP-EFT BASIS
We recall from equation (III.12) the expression for the kinetic terms in the Goldstone
basis in unitary gauge,
∑ ∑  2
Lϕ ⊃ | |2 (vi + ϕi)
2  ∑ ∑i aK
kin Di,µΦi = gIQIA
i
I,µ − Q 
2 K f
i i KmI ̸=0 K ( )
⊃ −∂µa
∑ ( ) ∑ 1
Xi 2v
2
iϕi + ϕi gIQ
i
IAI,µ +O , (V.11)fa f2i m aI≠ 0
where we again set the focus on the axion aK ≡ a with PQ charges QiK ≡ Xi. The
orthogonality conditions assure that there is no tree-level mixing between the axion and
the gauge bosons.
The resulting axion interaction can then be generalized by introducing a set of coeffi-
cients CIi4 , defined by ∑ ( ) ( )2
Lϕaxion ⊃ −
∂µa
CIi
ϕ 1
4 g A
i
I I,µ viϕi + +O , (V.12)
fa 2 f2I,i a
which is by construction invariant under chiral fermion transformations. Finally, the full
axion coupling basis at 1/fa is given by L ψ ϕaxion = Laxion + Laxion.
V.1.2 Application to DFSZ Model
After we introduced a full basis for axion couplings we can apply our derivation to the
DFSZ axion model at tree-level. In the Higgs basis we only generate axion couplings to
fermions via the mixin∑g of an(gular m)odes in(the Yukawa in)teraction(s, )
L 1 2 h aaxion ⊃ (∂µa) − mi 1 + ψ̄i exp −2iXiA γ5 ψi +O A,H0, H± , (V.13)2 v f
i∈ aSM
where we neglect terms proportional to A, H0 and H±, assuming that these fields are
much heavier then the electroweak scale and therefore decouple from the particle spectrum.
We note that we do not have a CZh4 coupling in the Higgs basis at tree-level, since the
alignment of h with the SM vev v allows us to apply the orthogonality conditions of the
angular mixing to the Higgs couplings.
We then can generate the other terms of the axion interaction basis by performing the
chiral trans(formation fro)m equation (II.3(8), ) ( )
i → Xu au exp −i γ ui , di5 → −
Xd a
exp i γ di , ei5 → exp −
Xd a
i γ i5 e .
2 fa 2 fa 2 fa
This transformation results in the canonic(al DFSZ axion)Lagrangian at order 1/fa,
L →1 ∂µa g
2 a
(∂ a)2 + Jµ − s 2XA − θ̄ Ga G̃aµνaxion µ
2 f PQ
gg
32π2 f µνa a
e2− A a
2
X F µν
e 1 a µν
γγ 2 µν
F̃ −XAγZ Z F̃
(4π) f s c (4π)2
µν
a W W fa
2 g2
− e 1 a aXA Z Z̃µνZZ 2 2 µν −XA
L µν
2 WW
WµνW̃
sW cW (4π) fa (4π)
2 fa
a e ( )
+ i √ (W−(XdJ+µ −X +µ 0 ±uJ ) + h.c.) +O A,H ,H , (V.14)
f µa 2s
W W,/l
W
50
V.2. CALCULATION OF ALP-EFT WILSON COEFFICIENTS
where we added with respect to equation (II.41) the couplings to the heavy SM fields Zµ
and W±µ proportional to the respective normalized anomaly coefficients AIJ ≡ APQIJ/X.
We see that since the charged current interactions are flavor-violating, we generate a
commutator interaction as defined in equation (V.8), which does not depend on the leptons
in the J+µ/ current, since neutrinos are not charged under PQ.W,l
In Chapter VII we show that we can generate exotic interactions like an effective CeffZh
coupling by integrating out heavy fermions. For this purpose we present in the next section
a calculation of the ALP-EFT Wilson coefficients.
V.2 Calculation of ALP-EFT Wilson Coefficients
In this section we provide with general calculations of Wilson coefficients in an ALP-EFT
where we integrate out heavy fermions. The interactions of the heavy fermions can be
flavor-changing and are given by the terms defined in equations (V.4) and (V.9).
The resulting effective operators are then represented by the full ALP-EFT basis con-
sisting of the terms in equations (V.9) and (V.12). Especially the terms with coefficients
CIJ3 and C
Ii
4 are newly generated from only having fermion couplings in the UV. Hence, we
show the calculation for an effective axion coupling to two gauge bosons in subsection V.2.1
and to one gauge boson and one scalar in subsection V.2.2.
Finally, we discuss in subsection V.2.3 additional contributions to axion interactions
with SM fermions from integrating out heavy fermions and scalars, leading to effective
C1 and C2 terms. We do not discuss effective contributions to the vertices in Lψgauge
and LψYukawa, which describe interactions of four particles and do not play a role in our
phenomenological studies.
V.2.1 Loop-induced Axion Coupling to Gauge Bosons
We start our discussion on fermion loop induced axion couplings by considering the anoma-
lous axion couplings to gauge bosons. It is given for U(1) gauge bosons by
Leff ⊃ − gIgJ aCeff F F̃µνaxion IJ Iµν J , (V.15)(4π)2 fa
where we define the Wilson coefficient CeffIJ excluding the loop factor of (4π)
−2, in order
to match the parametrics with the tree-level CIJ3 term. This is motivated by the fact that
an anomalous fermion transformation can shift the UV coupling entirely from the fermion
sector in the gauge sector, which therefore has to match with the one-loop contribution.
A AIµ AIµ Iµψi ψj
CeffIJ
a a ψk a ψk
ψj ψi
AJν AJν AJν
Figure V.1: Effective vertex and one-loop diagrams for anomalous coupling of an axion to
two gauge bosons. The loop consists of three fermions ψi, ψj and ψk connecting an axion
a with two gauge bosons AI,µ and AJ,ν .
In order to determine the Wilson coefficient we calculate the triangle diagrams shown in
Figure V.1. We consider three flavors of fermions ψi, ψj and ψk with masses mi, mj and
51
CHAPTER V. AXION COUPLINGS AND ALP-EFT BASIS
mk, which have in general flavor violating couplings to the axion and the gauge bosons
defined by the corresponding coupling coefficients and charge matrices.
We consider three limiting cases. In the first limit we consider the fermion masses being
much larger than the boson masses, mi,mj ,mk ≫ ma,mI,mJ. The resulting Wilson
coefficient read∑s
eff IJ − C0(0, 0, 0,mi,mj ,mk)CIJ =C3 2 N( k
×
λ(m2a,m
2 2
I(,mJ)i,j,k
× (mi +m )(Cijj 1A + C
ij
2A) (Q
ik
IV Q
kj ik
JV −QIAQ
kj
JA)mkm
2
a(m
2
a −m2 −m2I J) )
+ (Qik Qkj ik kj 2 2 2 2 2 2 2 2IV JV +QIAQJA)(m(imJ(ma +mI −mJ) +mjmI(ma −mI +mJ))
+ (m −m )(Cij + Cij ) (Qik Qkj −Qik kji j 1V 2V IA JV IV QJA)mkm
2
a(m
2 2 2
a −mI −mJ) ))
+ (Qik QkjIV( JA +Q
ik kj 2 2 2 2 2
IAQJV )(mimJ(ma +mI −mJ)−mjmI(m
2 2 2
∑ a
−)mI +m(J))2 )
− N Cij (Qik Qkj +Qik Qkj ) + Cij (Qik Qkj ik kj
ma,I,J
k 1A IV JV IA JA 1V IV JA +QIAQJV ) +O 2 ,m
i,j,k i,k,j
(V.16)
with Nk being the multiplicity of the fermions accounting for the number of colors and
flavors. The function λ represents the Källén function, while the function C0 denotes the
three-point Passarino-Veltman function [208], given in the heavy fermi(on limit by2 )m
C (m2 ,m2 2
1 b ,b ,b
0 b  b ,mb ,m(f1 ,m)f2 ,mf3) = C0(0, 0, 0,(mf1 ,)mf2 ,m
1 2 3
1 2 3 f3) + 2 Omf m22 f1,f2,f3 m
2
f m
2
1
( ln )(2 ) ( ln )(
f3 ( )
1 mf m
2
f m
2 
2 2 b
= + ) +O 1,b2,b3 2 , (V.17)m m2 m2f f m2 m2 2 f2 1− 3 1− m1 1− f1 1− f3 f1,f2,f3
m2 2 2 2f mf mf m1 2 3 f2
with boson masses mb and fermion masses mi f .i
From equation (V.16) we see, that the result is indeed invariant under a chiral fermion
transformation as defined in equation (V.5). In order to obtain the result for SU(N) gauge
bosons, which necessarily requires I = J to conserve the respective color charge, we just
replace the number of colors by the Dynkin index T (RIk).
The second limit is characterized by the flavor conserving limit with i = j = k ≡ f . The
W(ilson)coefficient in∑this lim((it is given by ( )
Ceff = CIJIJ 3 − 2 N C
f
f 1A(Q
f
IV Q
f
JV +Q
f
IAQ
f f f
JA) + 2 C1A + C2A ×
f ( f f f
f f Q Q m
2
× (Q Q m2C (m2 2 2
IA JA f
IV JV f 0 a,mI,mJ,mf ,mf ,mf )− λ(m2,m2,m2
×
a I J)
× (m4 − (m2 −m2)2a I J )C0(m2a,m2I,m2J,mf ,mf ,mf ) + 4m2aB 20(ma,mf ,m)f )))
− 2(m2a +m2I −m2)B (m2J 0 I,mf ,mf )− 2(m2 2a −mI +m2 2J)B0(mJ,mf ,mf ) .
(V.18)
52
V.2. CALCULATION OF ALP-EFT WILSON COEFFICIENTS
The result now depends on the two-point Passarino-Veltman function B0 [208]. The 1/ϵ
divergence in B0 from dimensional regularization cancels in this case due to a vanishing
prefactor for the constant pieces of the B0 functions.
Finally, we have the limit for the SM W bosons, in which the axion coupling is flavor-
conserving (i = j ≡ f) and the gauge couplings are left-handed to gauge bosons with
equal masses (I = J ≡ W ).(In this case, the W(ilson coefficient reads∑ m2 ffk kf f (C1A + Cfeff 2A)C = N Q Q 4 B (m2WW f WL WL 2 0 a,mf ,mf )−B (m20 W ,mf ,m2 − ) ) k
)
m 4m
f,k a W
+ (m2 −m2 +m2W f k)C0(m2 2 2a,mW ,mW ,mf ,mf ,mk) − C
f WW
1A + C3 , (V.19)
where the charge matrices QWL represent the CKM matrix VCKM. The divergence in B0
again cancels and the result is finite.
V.2.2 Loop-induced Axion Coupling to Scalar and Gauge Boson
In this subsection we discuss the interaction of an axion to a gauge boson AIµ and a real
scalar field ϕi, which is induced by a fermion loop. Although this interaction is not part
of the fermion-related axion interactions at tree-level, it generically is generated through
the scalar Lagrangian from equation (V.11).
We write down the effective Lagrangian analogously to the CIi4 interaction from equa-
tion (V.12),
µ
Leff eff ∂ aaxion ⊃ −CIigIviϕiAI,µ , (V.20)fa
where we neglect the ϕ2i interactions, since it is a four-point vertex which does not play a
role in our phenomenological studies.
A A AIµ Iµ Iµψi ψj
CeffIi
a a ψk a ψk
ψj ψi
φi φi φi
(a) (b) (c)
AIµ A AIµ Iµ
ψi ψi AJµ
a ψk a ψk a
ψj ψj
φi φi φi
(d) (e) (f)
Figure V.2: Effective vertex and one-loop diagrams for coupling of an axion to a gauge
boson and a scalar. The loop consists of three fermions ψi, ψj and ψk connecting an axion
a with one gauge boson AI,µ and a scalar ϕi.
We find five one-loop diagrams involving up to three flavors of fermions ψi, ψj and ψk
with masses mi, mj and mk. These diagrams are shown in Figure V.2. The first two loop-
diagrams in subfigures V.2b and V.2c show the usual triangle contribution from three-point
53
CHAPTER V. AXION COUPLINGS AND ALP-EFT BASIS
fermion interactions. Diagrams V.2d and V.2e include the four-point contact interactions
in the gauge and Yukawa sector including flavor-violating commutator interactions.
The last diagram in subfigure V.2f shows an additional contribution from an off-shell
internal gauge boson line. The diagram cancels as soon as the internal gauge boson is
on-shell due to the orthogonality relations from the mixing of angular fields. We remark
that the effective coupling has also been studied in reference [209] to analyze the CP
properties of a new scalar, but since we additionally account for flavor-violating fermion
interactions we include Figure V.2e as an additional diagram.
In the limit where we have fermions which are much heavier then the external bosons,
mi,mj ,mk ≫ ma,mI, i, the Wilson coefficient for the acion-scalar-gauge boson coupling
is given by (
eff Ii 1
∑
√1 C0(0, 0, 0,mi,mj ,mk)CIi =C4 + Nk 2 ×(4π)2 2v λ(m2i a,m2,m2)i,j,k i I
× ((mi +mj)((Cji + Cji1A 2A)Qik Y kjIA i + (Cij + Cij )Y jkQki1A 2A i IA)× )
× m2(m2 2 2 2I I −ma −mi)(mi −mj)(mj +mk)− λ(ma,m2,m2)(m m +m2i I i k j )
− 2(mi +mj)((Cji + Cji ik kj ij1A 2A)QIAYi + (C1A + C
ij
2A)Y
jk
i Q
ki
IA)B0(0,mk,mi)
+ (∑m ji ji ik kjj −mk)((C1A + CIA)QIAYi + (Cij + Cij )Y jk1A IA i QkiIA)B0(0,mj ,mk)
− (m +m ji jii j)((C1A + C2A)Q
ik
JAY
kj ij
j + (C1A + C
ij )Y jk2A j Q
ki
JA)B0(0,mi,mj)
J,j
g2
) ( )
QiQiJ I Jvivj m
2
× O a,I,i+ , (V.21)
m2J m
2
i,k,j
where we used the scalar coupling to two gauge bosons from equation (V.11). The result
is again invariant under a chiral transformation as defined in equation (V.5).
The two-point Passarino-Veltman functio(n B0 in the)heavy fermion limit is given by
m2
B0(m
2 b
b ,mf1 ,mf2) = B0(0,mf1 ,mf2) +O( ) m2 2f ,m1 f2 ( ) ( )
2 2 2 21 − µ − 1
m 2f +mf mf m
= γE + ln(4π) + ln + 1
1 2
2 2 ln
1 b
2 +O 2 2 ,ϵ mf1mf2 2mf −m m m ,m1 f2 f2 f1 f2
(V.22)
where the divergent piece does not cancel trivially in this case. In order to find the
cancellation we extrac(t the(divergent piece from equ)ation (V(.21), )
Ceff
1 1 1
Ii = √ T− Yi(,M,QIA,C2A −CIA +)T+) Yi,M,Q ,C∑2 IA 1A
+C2A
(4π) 2vi ϵ
g2JQ
i i
− I
QJvivj
2 T+ Yj,M,Q
0
JA,C1A +C2A +O(ϵ ) , (V.23)
m
J,j J
where T− and T+ represent the traces of commutator and anti-commutator relations, given
by
T+(A,B,C,D) = Tr[{A,C}{B,D}] + Tr[{B,C}{A,D}] , (V.24)
T−(A,B,C,D) = Tr[{A,C}{B,D}]− Tr[{B,C}{A,D}] = Tr[[B,A][C,D]] . (V.25)
54
V.2. CALCULATION OF ALP-EFT WILSON COEFFICIENTS
We notice, that we need the contribution from diagram V.2f to be able to cancel the T+
term, while the T− term cancels for vanishing commutators.
Applying the orthogonality conditions from angular mixing, we can replace
g QiJ Ivj = δIJδij , (V.26)
mJ
such that the T+ terms cancel exactly(and we are left with )
eff 1 1 1CIi = √ T− Yi,M,QIA,C2A −CIA +O(ϵ0) . (V.27)(4π)2 2vi ϵ
This contribution cancels in the case where the Yukawa matrices commute with the mass
matrix, [Yi,M] = 0, and the divergence vanishes.
V.2.3 Loop-induced Axion Coupling to Fermions
Lastly, we discuss the loop-induced axion interaction to fermions. This coupling provides
the leading contribution for fermions which are not charged under the PQ symmetry and
therefore do not have a tree-level coupling (to axions. Th)e effective Lagrangian for thiscase is given by
Leff ∂µa i eff eff jaxion ⊃ − f̄ CV,ij + CA,ij f , (V.28)2fa
where f i and f j now represent light fermions with masses mi and mj . Alternatively, we
can replace this interaction by an axion coupling in the mass sector of the fermions.
f i
Ceff f
i
ij ψk
a
a φS
j ψlf f j
(a) (b)
f i f i
ψk
a φS a φS
ψl
f j f j
(c) (d)
Figure V.3: Effective vertex and one-loop diagrams for anomalous coupling of an axion to
two fermions. The loop consists of up to two fermions ψk, ψl and a scalar ϕS connecting
an axion a with two fermions f i and f j .
In order to calculate the Wilson coefficients Ceff and CeffV A we close a loop containing two
heavy fermion flavors ψk and ψl and one heavy real scalar field ϕS . The corresponding
diagrams are shown in Figure V.3. In addition to the triangle diagram in subfigure V.3b,
we get two diagrams with loop corrections on each outgoing fermion-line due to the four-
point contact interaction defined in equation (V.9).
55
CHAPTER V. AXION COUPLINGS AND ALP-EFT BASIS
Since we are interested in effective interactions to the SM fields and do not want to
generate any SU(2)L anomalies, we couple the heavy scalar to right-handed light fermions
and left-handed heavy fermions,
L ⊃ −ykiψ̄k iYukawa S LϕSfR + h.c. , (V.29)
where we assume the Yukawa matrix YS to be purely real. The corresponding masses for
the heavy fields are denoted by mk, ml and mS , respectively.
Since the effective interaction is only mediated to right-handed light fermions, we get
matching expressions for both, vector-like coefficient CeffV and the axial coefficient C
eff
A .
In the limit where we integrate out the heavy fermions and scalar the Wilson coefficients
read ∑ ( ( )
Ceff
1 Nk
V/A,ij =C
ij − yik1V/A S y
lj 8 Ckl + CklS 1A 2A mkmlC0(0, 0, 0,mk,m2 ( l
,mS)
( (4π) 2k,l )
+ Ckl + Ckl − Ckl − Ckl1V 2V 1A 2A (m2k − 4m 2kml +ml )C0(0,)0, 0,mk,ml,mS)
(+m
2
kC0(0, 0, 0,mk,mk,mS))(+m
2
lC0(0, 0, 0,ml,ml,mS)
+ Ckl kl kl kl1V + CSV − C1A − CSA 1−B0(0,mk,mS)−B0(0),m)l,mS()
m2
)
+m2 2
a,i,j
kC0(0, 0, 0,mk,mk,mS) +mlC0(0, 0, 0,ml,ml,mS) +O 2 .mS,k,l
(V.30)
From the transformation properties in equation (V.10) we can deduce that the coefficients
are invariant under the chiral transformation defined in equation (V.5). We further notice,
that in the flavor-conserving axion limit (ψk = ψl) the dependence on the vector-like
couplings drops out. This matches our expectations since the tree-level vector-like axion
couplings cancel in this case.
The divergence in the two-point Passarino-Veltman functions B0 only vanishes if the
contributions from diagrams V.3c and V.3d cancel the contribution from the derivative
interaction. This holds true in the case that the heavy fermions ψk and ψl acquire their
axion couplings originally in the mass sector.
56
CHAPTER VI
Chiral Enhancement of Axion
Couplings
In the previous chapter we saw that we can modify axion couplings in the framework of
an EFT, where we integrate out fields which are much heavier than the energy scale of
interest. The focus of this chapter is on the axion couplings to SM fermions which get
an additional contribution from heavy fermions and scalars. Since the SM fermions are
chiral, the heavy fields couple differently to left- and right-handed states. An increase in
the fermion coupling is therefore called chiral enhancement.
In general, Chiral enhancement describes a loop correction to a fermionic coupling which
is enhanced by the ratio of a heavy scale and the fermion mass. Such types of corrections
are usually used to explain discrepancies between theoretical expectations and experimen-
tal measurements such as in the muon anomalous magnetic dipole moment [210]. In the
case of axion models, chiral enhancement mainly interferes with flavor effects as well as
the properties of the additional field content.
To demonstrate this, we first show in the concept of chiral enhancement within axion
models section VI.1. Afterwards in section VI.2, we present the corresponding effec-
tive couplings and flavor effects. In section VI.3 we embed the chiral enhancement into
the DFSZ model with a DM scalar and constrain our parameter space according to DM
bounds. Finally, we discuss in section VI.4 additional effects accounting for neutrino
masses.
VI.1 Chiral Enhancement in Axion Models
In this section we discuss the possibility and implications of having corrections through
chiral enhancement in an axion model. Even though these corrections are generally consid-
ered to be sub-dominant effects, chiral enhancement gets interesting in axion physics as it
can compete with the tree-level coupling in DFSZ models and even provide the dominant
contribution in KSVZ models. We demonstrate this fact in a model with a heavy fermion
F and a gauge singlet scalar S.
The tree-level coupling to SM fermions f and the heavy fermion F can be parameterized
by ∑ C0 C0
L ⊃ − aff µaxion (∂µa)f̄γ γ aFF µ5f − (∂µa)F̄ γ γ5F , (VI.1)
2fa 2f
f∈ aSM
where a represents the axion field with decay constant fa and dimensionless couplings
C0 0aff and CaFF .
57
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
We consider the heavy fermion F getting its mass from the complex scalar Φ in a
KSVZ-like manner, where the angular mode of Φ is dominantly represented by the axion.
Assuming the fermion F ∈ {E,U,D} having the same quantum numbers as one of the
right-handed SM fermions (f ∈ {e, u, d}), the Yukawa couplings of F and S read
LFSYukawa ⊃ −yF F̄LΦFR − yiSF̄ Sf iL R − y
ij i j
efff̄LSfR + h.c. , (VI.2)
where yijeff needs to be generated through a dimension-5 interaction of the form Q̄
i H SdjL d R,
Q̄iLH Su
j
u R or L̄
i
LHdSe
j
R, respectively.
Figure VI.1: Feynman diagrams for axion couplings to SM fermions.
The tree-level coupling of the axion to the SM fermions as well as the one-loop corrections
induced by S are shown in Figure VI.1. We split the total coupling at one-loop into three
parts,
Ctotaff ≡ C0
f F
aff + Caff + Caff , (VI.3)
where Cfaff stands for the one-loop correction with the SM fermion in the loop and C
F
aff
with the heavy fermion in the loop. The(parametric)depen(dence)is given by
|y |2 m2 m2 4f eff f f mfCaff = C
0
aff 2 1 + ln 2 +O ( 4 , ) (VI.4)4π2 mS ( mS ) mS4
F |y |2 m2S 0 F m2 mF f m4C Faff = C 1 + ln +O4π2 aFF m2 m2 m4
, 4 . (VI.5)
S S S mS
Here, we can see that in the DFSZ-case (C0aff ̸= 0) the correction from the heavy fermion
is enhanced compared to the correction with only having f by a factor m2 /m2F f . For large
Yukawa couplings |yS | ∼ 4π/3 it even competes with the tree-level coupling. In the KSVZ-
case (C0aff = 0) the effective one-loop contribution from the heavy F gives the leading
contribution, since the tree-level contribution vanishes.
In the next section we set up CFaff in a more general way and include flavor-changing
interactions. In section VI.3, we also account for the case mF ≫ mS which leads to even
larger contributions.
VI.2 Effective Axion Interactions and Flavor Effects
In this section we calculate the effective interactions of the axion a to gauge bosons and
SM fermions. In addition to the canonical DFSZ contribution, the couplings to gauge
bosons are determined by integrating out the heavy fermion F in a KSVZ-like manner
as presented in subsection V.2.1. The couplings to SM fermions are enhancend by their
58
VI.2. EFFECTIVE AXION INTERACTIONS AND FLAVOR EFFECTS
Yukawa interactions with the heavy fermion F and the heavy scalar S which are both
integrated out according to our calculation in subsection V.2.3.
We present the resulting EFT interactions in subsections VI.2.1 and VI.2.2. In subsec-
tion VI.2.2 we also show that the effective axion fermion interactions involve in general
flavor changing effects. These effects are strongly constrained [106,211,212] and therefore
narrow down the available parameter space for chiral enhancement.
VI.2.1 Axion-Gauge Boson Interactions
We start with the axion gauge boson couplings which are induced at one-loop order by
integrating out the heavy fermion F . The corresponding Feynman diagrams are shown
in Figure V.1, replacing the fields ψi,j,k with F . Since F does not couple to the SU(2)L
gauge group we only generate effective couplings to the neutral bosons γ, Z and the gluon
g, given by
LF ⊃− F a µν − F a a aC F F̃ C Z F̃µν − CF Z Z̃µν − CF Ga G̃aµνaxion aγγ µνf aZγ µνf aZZ µνf agg f µν
.
a a a a
(VI.6)
The superscript F denotes that we integrated out the heavy fermion.
In order to calculate the effective coefficients we express the axion couplings to F in
terms of the Yukawa interaction to the c(omplex s)calar Φ,( ( ) )
LF hΦ a AYukawa ⊃ −yF F̄LΦFR + h.c. = −mF 1 + F̄ exp iX + γ5 F, (VI.7)vΦ fa fA
where we re-expressed the angular mode aΦ according to the procedure from subsec-
tion III.1.1. In this basis we do not generate anomalous axion couplings to gauge bosons
at tree-level.
Taking the mass of F to be proportional to the axion decay constant fa which is in usual
axion models well above the weak scale, we neglect all terms of order O(1/m2F ). Using
our calculation from subsection V.2.1 the coefficients then read
Y 2 2F F 2 F Y e
2 Y 2 e2
Caγγ = X e , CaZγ = X
F , CF F
(4π)2 (4π)2 s c aZZ
= X ,
W W (4π)2 s2W c
2
W
1− δ 2F FE gC = X sagg . (VI.8)2 (4π)2
We see, that the induced coupling to gluons vanishes in the case of heavy electron (F = E).
The loop-induced couplings lead to a small shift of the DFSZ axion line in the {ma, Gaγγ}-
plane, since we effectively add a KSVZ model on top of the DFSZ construction.
VI.2.2 Axion-Fermion Interactions and Flavor Effects
In addition to the canonical axion-fermion couplings, we generate also one-loop contribu-
tions for the SM fermions which couple to the heavy fermion F and the real scalar S via a
right-handed Yukawa interaction. Since the mass of the scalar mS is a free parameter, we
can take it to be at a scale well above the electroweak scale. This allows us to integrate out
the scalar, leading to a chirally enhanced effective axion-fermion coupling. The respective
Feynman diagrams for this process are shown in Figure V.3, after replacing the fields ψk,l
with F and ϕS with S.
59
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
This can be added to the effective Lagrangian analogously to our discussion of subsec-
tion V.2.3 as a derivative coupling to the axion via an effective PQ current
LS ⊃ −∂µa f̄γµ(CS,V +CS,Aaxion 2f aff aff
γ5)f . (VI.9)
a
Here, the superscript S for the effective coupling matrices CSaff symbolizes that we inte-
grated out the scalar S. We split the coupling matrices into a vector-like part CS,Vaff and
an axial part CS,Aaff .
Since S only couples to right-handed SM fermions, the vector-like and axial parts of the
PQ current have the same charge. Again, we neglect all terms of order O(1/m2F ) as well
as O(1/m2S). For now, we also neglect terms which are sensitive to the difference between
mF and mS . The resulting effective charge matrix in the m(ass basis)of the SM fermionsreads i j 2
CS,V
( ) y y m
aff = C
S,A
aff ≡ −X
m , Xm
ij S S F
f f = X +O 1− 2 . (VI.10)8π2 mS
It depends dominantly on the couplings yiS of the SM fermions to the real scalar. We see
that for each flavor-diagonal axion-fermion coupling we also generate the corresponding
mixed couplings between different flavors.
In order to further study the flavor structure we include the PQ current in the covariant
derivative of the respective right-handed fermion f ∈ {e, u, d}. In matrix form regarding
flavor space and mass basis of the fermions this reads
Dmf,µ = ∂µ1−
(∂µa)
iYfgY Bµ1− i(1− δfe)gsλaGaµ1− i Xmf , (VI.11)fa
where only the PQ current is flavor-changing.
Going back to gauge basis via the transformation from subsection II.2.2
f → V†f , X = V Xm †R f R f f f Vf , (VI.12)
we recover flavor-diagonal gauge interactions, but have flavor-changing Higgs-Yukawa in-
teractions. The PQ current is also still flavor-changing due to the dependence on the
fermion-scalar interactions. Hence, there is no basis where Yukawa interactions and PQ
current are generically flavor-diagonal at the same time.
The combined flavor effects of the Higgs Yukawa interactions and the PQ interactions can
be seen by performing a chiral transformations on the fermions following the description
in section V.1 w(ith ) ( )
→ i a i af exp (Xf +Xfγ5) f, f̄ → f̄ exp − (Xf −Xfγ5) . (VI.13)
2 fa 2 fa
This is equivalent to a trans(formatio)n of only the right-handed fields, yielding
L ⊃ −v√+ h af̄(LYf exp i Xf fR + h.c.2 fa ) ( )
= −v√+ h a a 1f̄Yff + i f̄ [Yf ,Xf ]f + i f̄{Yf ,Xf}f +O . (VI.14)
2 f 2a fa fa
We see again, that we can not diagonalize the Yukawa coupling to the Higgs and to the
axion at the same time.
60
VI.3. CHIRAL ENHANCEMENT AND DARK MATTER
The resulting flavor changing effects in equation (VI.10) are strongly constrained as
shown in references [106, 211, 212]. Hence, for the flavor changing couplings between two
SM fermions with flavor indices i ̸= j we take a conservative limit
√
yiSy
j
f 2 m yia F y
j
1
X S ≲ 10−8 = 10−8X sinβ2 ⇔ S S ≲ 10−9 GeV−1 , (VI.15)
8π2 TeV yF TeV mF yF
where we rewrite fa in terms of vΦ following subsection III.1.1 with β2 → π/2 for a large
scale separation fa ≫ v.
VI.3 Chiral Enhancement and Dark Matter
Having introduced the concept of chiral enhancement in axion models, we now want to
estimate the impact of such an enhancement within the DFSZ model. For this purpose we
add a heavy fermion F and a heavy gauge singlet scalar S to the field content of the DFSZ
model. Again we choose the quantum numbers of the heavy fermion F ∈ {E,U,D} to be
aligned with the quantum numbers of one of the right-handed SM fields, f ∈ {e, u, d}.
The corresponding field content is presented in Table VI.1, where the Zn charges with
4 < n ∈ 2N are chosen in such a way that for mF > mS the fermion F can decay into SM
fermions while the decay of the scalar S is forbidden. Thus, the heavy scalar serves as a
DM candidate. The full Lagrangian of this model is given by
LDFSZ ⊃LDFSZ 1+ (∂ S)2DM ( µ + iF̄D/FF − yF F̄LΦF − y
i
R SF̄LSf
i
R + h.c.2
1 ( ))− S2 m2S + λ 2 2 2 0 ±SS + λHS(h + 2vh) + λΦS(hΦ + 2vΦhΦ) +O H ,H ,2
(VI.16)
where hΦ was defined in equation (II.30) as the radial mode of Φ and h describes the
radial mode of the two Higgs doublets from equation (II.40) which is aligned with the SM
Higgs field. We neglect the couplings to the heavy DFSZ fields H0 and H±. Note, that a
tree-level coupling of S to the pseudoscalars a and A is forbidden due to the Zn symmetry.
SU(3)C SU(2)L U(1)Y Zn U(1)PQ
QiL 3 2 1/6 0 XQ
uiR 3 1 2/3 1 XQ-Xu
diR 3 1 -1/3 0 XQ-Xd
LiL 1 2 -1/2 0 XL
eiR 1 1 -1 0 XL-Xd
Hu 1 2 -1/2 n-1 Xu
Hd 1 2 1/2 0 Xd
ΦA 1 1 0 1 -X
FL RF 1 YF n/2 + δFU XF
FR RF 1 YF (n− 2)/2 + δFU XF+X
S 1 1 0 n/2 0
Table VI.1: Field content of the DFSZ model with an additional heavy fermion F and
a DM scalar S. The SU(3)C representations RF and hypercharges YF of the additional
fermion correspond to one of the right-handed SM fermions.
Since the heavy scalar contributes to the DM relic density, we obtain further limits on
the Yukawa couplings yiS . On one hand, the Yukawa coupling controls the rate for pair
61
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
annihilation of the heavy scalar into SM particles in the early universe. If this process is
not efficient enough, the relic abundance of S exceeds the measured value for DM in the
energy budget of the universe. Thus, this provides a lower limit on yiS as we present in
subsection VI.3.1.
On the other hand, experiments dedicated for direct DM detection provide an upper
bound on interactions between DM and SM particles. We discuss the corresponding
constraints on yiS from the LUX-ZEPLIN (LZ) and XENONnT experiments [213–215]
in subsection VI.3.2. Afterwards, we show our estimates on chiral enhancement for quarks
in subsection VI.3.3 and for charged leptons in subsection VI.3.4.
VI.3.1 Relic Abundance
From cosmological observations like rotation curves of galaxies and the cosmic microwave
background (CMB) we know that 26.8% of the energy density of the universe is given by
cold Dark Matter [39]. The term “cold” refers to a non-relativistic particle species which
is mainly characterized by its mass, whereas “dark” signifies that the particles do not
transform under the SM gauge group. Furthermore, DM has to be stable on cosmological
time scales.
In our model, the scalar field S fulfills the requirements on a DM candidate as it is a
heavy gauge singlet and cannot decay due to the Zn symmetry. We can estimate its fraction
of the energy density of the universe by defining the corresponding density parameter
≡ ρS mSnS 8πmSnSΩS = = 2 . (VI.17)ρ 2c ρc 3H MPl
Here, ρS describes the energy density of S which is given by the mass mS times the number
density nS since we assume the scalar to be non-relativistic. The critical energy density
ρc describes the energy density of an entirely flat universe. We express ρc in terms of
the Planck Mass MPl and the Hubble parameter H which measures the expansion of the
universe [216,217].
The behavior of the number density nS is sensitive to the pair annihilation rate of S into
SM particles as it moves the particles into a state of thermal equilibrium. This behavior
can be described using a Boltzmann equation of the form [218]
d ( )
nS + 3HnS = −⟨σv⟩ n2 2S − n̄S , (VI.18)dt
where n̄S denotes the number density in thermal equilibrium and ⟨σv⟩ describes the ther-
mal average of the pair annihilation cross section σ times the velocity v.
As soon as the pair annihilation rate becomes smaller than the expansion rate of the
universe, the scalar S decouples from thermal equilibrium. This is called thermal freeze-
out and occurs when
n̄S⟨σv⟩ = H , (VI.19)
where the left-hand side describes the pair annihilation rate and the right-hand side the
expansion rate of the universe given by H.
Following the standard freeze-out calculation as, for example, shown in reference [218]
we obtain for the DM relic abundance ΩDM and the freeze√-out condition the expressions
2 xS 1.07× 109 GeV−1 √
√
8 g∗
h ΩDM ≳ √ , x e−xSS = 2π3 , (VI.20)
g∗ MPl⟨σv⟩ 45mSMPl⟨σv⟩
62
VI.3. CHIRAL ENHANCEMENT AND DARK MATTER
with h being the scaling factor for the Hubble parameter (H = 100hkms−1Mpc−1), g∗
describing the number of relativistic degrees of freedom and xS = mS/TS expressing the
dependence on the freeze-out temperature TS . The total DM relic abundance is determined
from CMB data using the ΛCDM model and is given by h2ΩDM = 0.12± 0.012 [42].
In order to find the dependence of the freeze-out temperature TS on the DM mass mS
we express ⟨σv⟩ in terms of the relic density and rewrite the freeze-out condition as
≡ 3/2 − x 2 2.4436× 10
−8 GeV
δ SS xS e g∗h ΩDM = 0 . (VI.21)mS
This equation does not have an analytical solution for xS . Hence, we approximate the
solution in our mass range of interest using a graphical approach. In Figure VI.2 we show
δS for DM masses between mS = 1 GeV and m = 10
6
S GeV. We assume that freeze-out
occurs at temperatures above the electroweak symmetry breaking scale, such that all SM
particles are relativistic with g∗ = 106.75.
200
150
100
50
0
15 20 25 30 35
Figure VI.2: Graphical determination of the freeze-out temperature TS for different DM
masses mS . The zeroes of the lines represent values of xS = mS/TS for which the freeze-
out condition δS = 0 is fulfilled.
The zeroes of δS represent values of xS = mS/TS for which the freeze-out condition
δS = 0 is fulfilled. Since the zeroes in Figure VI.2 are relatively even distributed we
approximate xS with a logarith(mic dependence on m(S ,m ))≈ SxS 19.4645 + 1.0606 ln . (VI.22)
GeV
With this expression we can find a(lower limit on the thermally averaged cross section
⟨ ⟩ ≥ xS 1.07× 10
9 GeV−1 (≈ m ))√ Sσv 1.3759 + 0.0750 ln × 10−9 GeV−2 , (VI.23)
g∗ MPlh2ΩDM GeV
for which the pair annihilation is efficient enough to not overproduce DM particles.
In our model the annihilation cross section is mainly mediated through the interaction
of S with the Higgs and the SM fermions. We show the contributing Feynman diagrams in
Figure VI.3. The channels which are mediated by the Higgs are sensitive to the coupling
λHS and are given by [219–221] √
Nfλ2 m2 2 2 2c HS f (mS −mf ) m⟨ fσv⟩Hff ≃ [ ] 1− 2 , (VI.24)4πm2S (4m2S −m2)2h +m2Γ2h h mS
63
mS=106GeV
m 5S=10 GeV
=104m GeVS
mS=103GeV
= 02GeVmS 1
01GeV
mS=1
0GeV10mS=
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
H δV λ
2
HS([4m4 − 4m2m2 4⟨σv⟩ ≃ S S V + 3mVV V ( ]
√
) m2
1− V , (VI.25)
8πm2S (4m
2 −m2)2 +m2 2 2S h hΓh m)S2 √
λ2 4 4HS 4mS −mh − λ v2HS (4m2 2S −mh) m2
⟨σv⟩H hhh ≃ [ ] 1− , (VI.26)8πm2S(m2h − 2m2S)2 (4m2S −m2)2h +m2 2 2hΓh mS
where Γh symbolizes the total decay width of the Higgs, f ∈ {e, u, d} denotes SM fermions
with color multiplicity NfC and V ∈ {Z,W} represents SM gauge bosons with δW = 1 and
δZ = 1/2.
S yf fS h S SM S
h
F
λ λ gHS HS SM
S h S SM f
S yS f̄
Figure VI.3: Feynman diagrams for DM matter pair annihilation into SM particles.
Analogously we can find the annihilation cross section for the last diagram which is
mediated through the heavy fermion F , l(ead)ing to4
Nf yf ( 2 )c m 3/2S
⟨σv⟩Fff ≃ 2 1−
f
2 . (VI.27)4πmF mS
Summing over all contributions, th∑e total thermally averaged annihilation cross sectionreads ∑( )
⟨σv⟩ = ⟨σv⟩H + ⟨σv⟩H + ⟨σv⟩Hhh V V ff + ⟨σv⟩Fff . (VI.28)
V f
Assuming λ to be sufficiently small (λ ≲ 10−3HS HS ), we can neglect the Higgs contribu-
tions and find an estimate of the Yukawa couplings yfS .
After applying the limit from equation (VI.23) to one fermion flavor f , we can use the
flavor limit from equation (VI.15) to find a bound on the Yukawa couplings of the other
flavo(rs i)≠ f ,4 ( ) ( )4
yf 4 yi yfS 4π i f−9 −2 yS S S m
2 N
≳ c2 10 GeV ⇒ = ( F2 4 )4 ≲ 2 10−27 GeV−2 .mF Nfc mF mF 4πyyf FS
(VI.29)
We see that the flavor limit leads to a strong suppression of the other Yukawa couplings,
such that we only can have one sizable Yukawa coupling at a time.
VI.3.2 Direct Detection Constraints
Dark Matter direct detection experiments like XENONnT and LUX-ZEPLIN [213–215]
provide constrains on the scattering cross section σnS between DM particles and nucleons.
The tree-level scattering is again mediated via the Higgs and Yukawa interactions as
64
VI.3. CHIRAL ENHANCEMENT AND DARK MATTER
depicted in the last two diagrams in Figure VI.3. The corresponding cross sections read
λ2 4 2
σH = HS
mnfn
nS ( , (VI.30)4πm4 2h)(mS +mn)
yq
4
m4 (fn)2Fq S n qσnS = 2 , (VI.31)4πmFm
2
q (mS +mn)
2
where m nn ≈ 0.939 GeV refers to the avera∑ge nucleon mass, fq represents the individual
fraction of quark q in a nucleon and f = nn q fq the total quark fraction in a nucleon.
We approximate the quark fractions following references [222–224] by
fnu ≈ 0.02 , fnd ≈ 0.045 , fn n n ns ≈ 0.043 , fc = fb = ft ≈ 0.06 , (VI.32)
such that the total quark fraction is given by fn ≈ 0.3.
10-40 10-40
10-42 10-42
10-44 10-44
10-46 10-46
10-48 10-48
10-50 10-50
100 200 500 1000 100 200 500 1000
Figure VI.4: DM direct detection limits on DM-nucleon cross sections with chirally en-
hanced quark couplings. The light red region is excluded by the LUX-ZEPLIN experiment,
while the red dashed line shows the 20 ton-year projection for XENONnT [213–215]. The
superscript in the Yukawa coupling denotes which DM Yukawa couplings are non-zero.
The left panel uses non-zero Yukawa couplings of up-type quarks and the right panel of
down-type quarks. The calculation was performed under the use of micrOMEGAS-5.3 [11].
The figure is taken from reference [4].
Figure VI.4 shows the limits from the LZ experiment in the light red region and the 20
ton-year projection from the XENONnT experiment for different quarks coupling to the
DM scalar with Yukawa couplings yqS . We depi∑ct the total DM-nucleon cross section
σnS = σ
H
nS + σ
Fq
nS , (VI.33)
q
where we again assume that the contribution of the Higgs boson is negligible due to a
small coupling λ −3HS = 10 .
Under the assumption that we can only switch on one Yukawa coupling at a time, we
get a bound on the Yukawa c(oup)lings ofq 4yS 4πm(2q(m)S +mn)2≲ exp
m2 n 2
σnS , (VI.34)
F f 4q mn
where σexpnS symbolizes the limits obtained by the experiments.
65
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
We can also find a limit on the DM-cross section in the case of chirally enhanced axion
lepton couplings. But since leptons do not contribute to the nucleonic cross section, we
only get a contribution from σHnS , such that the bound depends entirely on λHS . In
Figure VI.5 we show the case for λHS = 10
−3 as an example.
10-40
10-42
10-44
10-46
10-48
10-50
100 200 500 1000
Figure VI.5: DM direct detection limits on DM-nucleon cross sections with chirally en-
hanced lepton couplings. The light red region is excluded by the LUX-ZEPLIN experiment,
while the red dashed line shows the 20 ton-year projection for XENONnT [213–215]. The
superscript in the Yukawa coupling denotes which DM Yukawa couplings are non-zero.
The calculation was performed using micrOMEGAS-5.3 [11]. The figure is taken from ref-
erence [4].
VI.3.3 Chiral Enhancement of Axion Quark Couplings
We now make use of the DM bounds from the previous sections to establish an estimate on
the chiral enhancement of quark couplings. Neglecting all terms of order O(m2f/m2S) and
O(m2f/m2F ) but keeping now further dependences on the heavy masses, the total coefficient
for the pseudoscalar axion quark couplings reads( ) ( )2 
yq
2 m
m2 m
2
Fm
2 S
S ln m2
Ctot = C0 S,Aaqq aqq + Caqq = −Xu/d −X S  F + F 4π2 2 − 2 ( )m m 2 − 2 2 , (VI.35)F S mF mS
where the tree-level coupling is given for up-type quarks as Xu = cos
2 β1X and for down-
type quarks as Xd = sin
2 β1X with tanβ1 = vd/vu as shown in subsection III.1.1. Hence,
the chiral enhancement for up-type quarks is larger at β1 → π/2 and for down-type
quarks at β1 → 0. The vector-like axion quark coupling does not play a role here as we
only consider flavor conserving couplings due to the smallness of flavor-changing couplings.
We extract the limits on the DM Yukawa couplings from relic abundance and direct
detection as presented in equations (VI.23) and (VI.34), leading to
( )
− 2 −3/2 ( )× 9 1 q 44π xS 1.07 10 GeV m 2q yS 4πm (mS +m )2√ 1− n2 ≲ 2 ≲ q2 ( ) σexp . (VI.36)3 g∗ MPlh ΩDM mS m fn 2 nSF q m4n
66
VI.3. CHIRAL ENHANCEMENT AND DARK MATTER
We see that small quark mass are stronger constrained by the direct detection experiments
than large quark masses. The heavy fermion mass on the other hand relaxes the direct
detection bound for higher values, but increases the lower bound from relic abundance.
We show the resulting parameter space for different quarks in Figure VI.6, where we
also account for an upper limit at which the Yukawa couplings become non-perturbative.
The heavy fermion mass is taken to be mF = 50 TeV for all quarks besides the up-quark,
where we show the case for mF = 100 TeV. The high values are motivated by the fact,
that F has to be heavier than S to forbid direct decays of the heavy scalar. As expected,
we see that the parameter space for light quarks is much more constrained than for heavy
quarks due to the direct detection bounds which are sensitive too the quark mass.
The mixing angle β1 is taken to be β1 = 0.45π for up-type quarks and β1 = 0.05π for
down-type quarks to get a sizable chiral enhancement of order O(10). In fact, for up-type
quarks this represents a natural choice as this angle corresponds to the case where the
SM Higgs Yukawa couplings to top- and bottom-quark are of the same size. On the other
hand, the limit for the down-type quarks is more difficult to construct, since the top-quark
would need a larger SM Yukawa coupling. Hence, we conclude that chiral enhancement is
more preferable for up-type quarks than down-type quarks.
Finally, we notice that for any non-zero Yukawa coupling yiS the limits from relic abun-
dance and direct DM searches overlap, leading to a lower bound on the DM mass mS .
For example, assuming an enhanced axion-charm quark interaction Ctot 0acc > Cacc with
mF = 50 TeV and β1 = 0.45π, we obtain a lower limit of mS > 1 TeV.
67
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
50 50
yu dS>4π/3 yS>4π/3
10 10
5 Z 5 LZL
NnT T
h2ΩDM>0.12 XENO 2Ω nh DM>0.12 ENONX
1 1
0.5
100 1000 104
0.5
105 100 1000 104 105
50 50
yc sS>4π/3 yS>4π/3
nT
10 10 NO
N
E
5 5 X
h2ΩDM>0.12
1 1 h
2ΩDM>0.12
0.5 0.5
100 1000 104 105 100 1000 104 105
50 > 50ytS 4π/3 ybS>4π/3
10 10
5 5
h2Ω 2DM>0.12 h ΩDM>0.12
1 1
0.5 0.5
100 1000 104 105 100 1000 104 105
Figure VI.6: Chiral enhancement of axion couplings to quarks. The red region is ex-
cluded by the LUX-ZEPLIN experiment, while the red dashed line shows the 20 ton-year
projection for XENONnT [213–215]. In the blue region the scalar S would provide an over-
abundance of DM and in the gray region the Yukawa couplings become non-perturbative.
The left panels use non-zero Yukawa couplings of up-type quarks and the right panels of
down-type quarks.
68
L
X ZENONnT
LZ
XENONnT
LZ
XENONnT
LZ
VI.4. CHIRAL ENHANCEMENT AND NEUTRINO MASSES
VI.3.4 Chiral Enhancement of Axion Lepton Couplings
In the case of leptons the tree-level axion lepton coupling is given by C0all = −Xd as it
was the case for down-like quarks. However, in contrast to down-type quarks the chiral
enhancement is less constrained. This has multiple reasons. On one hand, the direct
detection constraints are weaker, since they are only sensitive to the Higgs coupling λHS .
On the other hand, we could have also chosen a Zn charge of 1 for the right-handed
charged leptons, such that they would receive their mass contribution from Hu. This case
is called the flipped two Higgs-doublet model [52] in which the tree-level coupling to leptons
becomes C0all = −Xu, leaving the chiral enhancement preferable to all charged leptons.
Figure VI.7 shows the corresponding parameter space for mF = 50 TeV, β1 = π/8 and
λHS = 10
−2.
50
ye,μ,τS >4π
10
5
1 h2ΩDM>0.12
0.5
100 1000 104 105
Figure VI.7: Chiral enhancement of axion couplings to leptons. The red region is ex-
cluded by the LUX-ZEPLIN experiment, while the red dashed line shows the 20 ton-year
projection for XENONnT [213–215]. In the blue region the scalar S would provide an over-
abundance of DM and in the gray region the Yukawa couplings become non-perturbative.
VI.4 Chiral Enhancement and Neutrino Masses
In the previous sections we showed how the axion couplings to quarks and charged leptons
are enhanced in presence of new heavy fields. In this section we briefly investigate axion
couplings to neutrinos and possible chiral enhancement factors from heavy right-handed
neutrinos.
In order to be able to explain the phenomenon of neutrino oscillation [44] we need to
include a mass term for the neutrinos in our theory. Such a mass term can be written down
using the SM (or DFSZ) field content at dimension five via the Weinberg operator [225]
( )
L ⊃ Cν cν L̄ H ∗L u HuLL + h.c. . (VI.37)Λν
We can find a UV completion for this operator by introducing right-handed fermion fields
νiR which are gauge singlet under the SM gauge group. We show the extended field content
in Table VI.2.
69
LZ
XENONnT
CHAPTER VI. CHIRAL ENHANCEMENT OF AXION COUPLINGS
SU(3)C SU(2)L U(1)Y Zn U(1)PQ
QiL 3 2 1/6 (n− 1)/2 XQ
uiR 3 1 2/3 (n+ 1)/2 XQ-Xu
diR 3 1 -1/3 (n− 1)/2 XQ-Xd
LiL 1 2 -1/2 (n− 1)/2 XL
eiR 1 1 -1 (n− 1)/2 XL-Xd
Hu 1 2 -1/2 n-1 Xu
Hd 1 2 1/2 0 Xd
ΦA 1 1 0 1 -X
νiR 1 1 0 (n+ 1)/2 XL-Xu
Table VI.2: Field content of the DFSZ model with three additional right-handed neutrinos
νiR.
The Zn charges allow for two Yukawa couplings for the right-handed neutrinos, one
being the usual Higgs coupling which is the same as for the other SM fermions and the
other one being a coupling to the complex scalar Φ involving the charge conjugated field
ν̄c = (ν )c = (νcR R )L. The full Lagrangian is given by
LDFSZ ⊃ LDFSZν + ν̄iRi∂/νiR − yijν L̄iLH ν
j
u R − y
ij ν̄c,iΦ∗N R ν
j
R + h.c. , (VI.38)
where the term proportional to the Yukawa coupling yijν leads to a Dirac mass while the
term proportional to yijN provides a Majorana mass for the neutrinos. Since the last term
includes the field Φ with vev vΦ ≫ v, we expect the right handed-field to be much heavier
than the left-handed fields.
In order to find the mass basis of the neutrinos, we first diagonalize the matrix Yν
analogously to the procedure in the SM,
νL → Uνν † † †L , νR → VννR , UνUν = 1 = VνVν , YD,ν = UνYνVν . (VI.39)
Together with the same transformations for the charged leptons we now get a mixing
matrix in the weak sector V †PMNS ≡ UνUe which is responsible for neutrino oscillation.
Assuming yijN = y
i
Nδ
ij for simplicity, we find two mass eigensta(tes f)or the neutrinos,i
c,i 1 yD,νXs v v
2
νi − i β1= (ν ν c,i iL L) + (νR − νR) +O( ) , (VI.40)2 yi f f2N a ai 2
N i = (νc,i
1 yD,νXsβ1v
+ νi ) + (νc,i
v
R R L + ν
i
L) +O , (VI.41)2 yiNf f2a a
with sβ1 being the sine of the mixing angle between vu and vd. The corresponding masses
are up to O(v2/f2a )
(yi )2 Xs2 2fa 1 D,ν β v 1 (y
i 2
D,ν) Xs
2 2
i √ √ 1 √β
v
mN i = yN + , m
1
i = . (VI.42)
X 2 4 yi
ν
N 2fa 4 y
i
N 2fa
The fields νi correspond to the observed neutrino states with direct couplings to the
SU(2)L gauge bosons, while the coupl√ings of the heavy fields N i to the gauge bosons are
suppressed by an additional factor of miν/(m
i i
N −mν) ∼ v/fa.
70
VI.4. CHIRAL ENHANCEMENT AND NEUTRINO MASSES
Now, we can write down the couplings of the pseudoscalar a in the mass basis of the
fermions, where we expand in large fa, dropping all terms of order O(1/f2a ). The corre-
sponding Lagrangian reads
Lν a aaxion ⊃− iXmi N̄ i iN γ5N √+ iXmi ν̄iγ iν 5νfa fa
i
− a miX(mi −mi ν i i i iN ν) (ν̄ N − N̄ ν ) . (VI.43)f mi ia N −mν
We already see that loop-effects from the neutrino couplings are either suppressed by miν
or their corresponding couplings to gauge bosons. It also shows that although the heavy
neutrinos N i have a strong coupling to the axion, the axion couplings to the light neutrinos
νi are not enhanced.
VI.4.1 Effective Couplings from Neutrino Interactions
Having introduced the tree-level axion couplings to neutrinos, we discuss the effective
couplings induced by neutrino effects. Since the neutrinos only couple to the weak bosons
Z, W+ and W− and charged leptons l the effective couplings are
L ⊃CN a Z Z̃µν N a µν − ∂µa+ C W W̃ l̄γµ(CN,V +CN,AEFT aZZ µν aWW µν all all γ5)l . (VI.44)fa fa 2fa
The superscript N denotes that we integrated out the heavy neutrinos. The corresponding
diagrams are shown in Figure VI.8.
Z/W ℓ
N N
a a
N/ℓ W
N N
Z/W ℓ̄
Figure VI.8: Feynman diagrams for effective couplings induced by heavy right-handed
neutrinos.
Although the heavy neutrinos N i have a mass proportional to the axion decay constant
and have therefore a large coupling to the axion at tree-level, the corresponding couplings
to the gauge bosons are each suppressed by at least one power of v/fa. On the other hand
the light neutrinos have a mass which is already suppressed by a factor v2/f2a . Hence the
effective couplings are all suppressed by
2 2 2 2 2 2
CN
1 e v N 1 e v N,V N,A 1 e v
aZZ ∼ (4π)2 s2 c2
, C ∼ , C = −C ∼ .
f2 aWWW W a (4π)
2 s2W f
2 all all
a (4π)
2 s2 f2W a
(VI.45)
This means that although we have predestined candidates for a loop induced axion-W
boson coupling as well as chirally enhanced charged lepton couplings of the same size,
these couplings are much smaller than the couplings induced by the heavy fermion F
in the previous model. In addition, the flavor-violating couplings induced by the heavy
neutrinos are suppressed by the unitarity of the PMNS matrix.
71

CHAPTER VII
Axion Couplings in gauged U(1)’
Extensions
In this chapter we study the interplay between the axion and additional U(1)′ gauge
bosons. It mainly follows our discussion in reference [2]. In the SM there are two residual
global symmetries, baryon number U(1)B and lepton number U(1)L. Since both of these
symmetries are anomalous, they are expected to be violated through non-perturbative
effects. Only their combination U(1)B−L does not generate an anomaly.
In order to protect either U(1)B or U(1)L, we can associate it to a U(1)
′ gauge symmetry,
introducing a new electrically neutral gauge bosons, called Z ′. This additional gauge field
receives a contribution from the hypercharge gauge boson through kinetic mixing which
is consistent with gauge invariance. The kinetic mixing is induced by additional heavy
fermions which are needed to cancel the U(1)′×SU(2)2L and U(1)′×U(1)2Y gauge anomalies
and are therefore called anomalons.
Here, we present the implications on axion physics in the case of a DFSZ model in
combination with a gauged U(1)B symmetry with gauge coupling gB. While the SM sector
consists of two Higgs fields to enable axion couplings to SM fermions, the additional sector
also includes two complex scalars which spontaneously break the PQ symmetry as well as
baryon number. This leads to a non-trivial mixing between the axion and the longitudinal
mode of the Z ′.
This connection between the angular fields reduces the parameter spaces in the mass
coupling planes as the {m , f−1a a } and {mZ′ , gB} spaces are no longer independent. We
constrain our parameters by applying collider limits from ATLAS and CMS and present
our results in the {ma, Gaγγ} parameter space.
In the following we start in section VII.1 by setting up the model and discussing the
mixing of the angular fields. Afterwards, in section VII.2 we describe our theory in an
ALP-EFT framework, where we integrate out the heavy fermions. Finally, in section VII.3
we present our phenomenological studies which are based on collider experiments.
VII.1 Model
We start with the construction of the model in general, followed by a more detailed
description of the different particle sectors. To construct a DFSZ model with a gauged
U(1)B extension, we first take the canonical two-Higgs doublet model with Higgs fields
Hu, Hd and add two complex scalar fields Φ1 and Φ2. These two complex scalars are
singlets under the SM gauge group but are charged under U(1)B with baryon numbers
B1 = −3 and B2 = 3.
73
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
We assume that both additional scalars acquire a non-vanishing vev, such that they not
only fulfill the role of the complex scalar Φ in the DFSZ model, but also break U(1)B
spontaneously. Hence, the Z ′ becomes massive and the angular modes of Φ1 and Φ2
provide both the axion degree of freedom as well as the longitudinal mode of the Z ′.
Of the SM fields only the quarks have a baryon number Bq = 1/3, such that we need
fermionic anomalon fields which cancel the U(1)B × SU(2)2L and U(1)B × U(1)2Y gauge
anomalies. We adapt the respective field content from references [226,227] which introduce
six chiral anomalon fields L′L, L
′
R, E
′
L, E
′ ′
R, NL and N
′
R.
Table VII.1 shows the complete fermionic and scalar field content. We introduce a
discrete Z4 symmetry, which is mainly needed to define the Yukawa couplings in the
two-Higgs doublet model and to protect the global PQ symmetry, as we discussed in
subsection II.3.2.
SU(3)C SU(2)L U(1)Y U(1)B Z4 U(1)PQ
QiL 3 2 1/6 1/3 0 XQ
uiR 3 1 2/3 1/3 1 XQ-Xu
diR 3 1 -1/3 1/3 0 XQ-Xd
LiL 1 2 -1/2 0 0 XL
eiR 1 1 -1 0 0 XL-Xd
Hu 1 2 -1/2 0 3 Xu
Hd 1 2 1/2 0 0 Xd
L′L 1 2 -1/2 -1 0 X
′
L′R 1 2 -1/2 2 3 X
′-X2
E′L 1 1 -1 2 3 X
′-Xd-X2
E′R 1 1 -1 -1 0 X
′-Xd
N ′L 1 1 0 2 0 X
′-Xu-X2
N ′R 1 1 0 -1 1 X
′-Xu
Φ1 1 1 0 -3 2 −X1
Φ2 1 1 0 3 3 −X2
Table VII.1: Field content for a DFSZ-like model with gauged baryon number. In addi-
tion to the SM fermions we have two Higgs doublets Hu and Hd for electroweak symmetry
breaking (EWSB), two Higgs fields Φ1 and Φ2 for breaking baryon number and heavy
anomalon fields which cancel gauge anomalies. Besides the SM charges, we assign baryon
number and a Z4 charge. The PQ charge results from the global symmetry of the La-
grangian.
Hence, the scalar Lagrangian of our model involving(all complex )sc(alars is given by∑ ∑ ∑ 2)
L 2 v
2 vj
scalar ⊃ |Di,µHi| + |D Φ |2 − λ |H |2 − ii,µ i ij i |Hj|2 −
2 2
i=u,∑d ( i=1,2 )( i,j=)u,d2 ∑ ( )( )
− 2 v
2 v 2 v2
λ i 2
j 2 vi 2 j
( ij
|Φi| − |Φj| − − λij |Hi| − |Φj| −
2 ) 2 2 2i,j=1,2 i,j
− λ TA Hu HdΦ1Φ2 + h.c. , (VII.1)
where we contract SU(2)L indices implicitly. The last term proportional to λA fulfills
the role of the µA term in the vanilla axion model, separating the axion from the heavy
pseudoscalar A. We discuss this in more detail in subsection VII.1.1.
74
VII.1. MODEL
The Z4 charges of the anomalons are chosen in such a way, that they get mass contribu-
tions through Yukawa couplings with the Higgs doublets and the complex scalar Φ2. The
corresponding Yukawa Lagrangian reads
L ij iYukawa ⊃− yu Q̄LHuu
j − yijR d Q̄
i
LH d
j
d R − y
ij i j
e L̄LHdeR
− y ′ ′ ′ ′ ′ ′LL̄RΦ2LL − yEĒLΦ2ER − yN N̄LΦ2NR
− y1L̄′LH E′d R − y L̄′2 RH E′d L − y ′ ′ ′3L̄LHuNR − y4L̄RHuN ′L + h.c. . (VII.2)
The Yukawa interactions not only generate fermion masses but also mediate axion cou-
plings to the anomalon sector, such that we induce corrections to axion couplings by
integrating out the heavy fermions. Since the Yukawa interactions also induce a mass
mixing between the anomalons, the axion couplings will change the flavor of the heavy
fields. We demonstrate this in subsection VII.1.2.
Finally, in subsection VII.1.3 we show that the anomalons also fulfill a trace-condition
which allows for a finite loop-induced kinetic mixing term
L ⊃ ϵeff µνkin-mix BµνK , (VII.3)
2
where Kµ describes the U(1)B gauge boson before diagonalization and canonical normal-
ization of the gauge boson mass basis. In principle, we could also write down a kinetic
mixing at tree-level, but since the U(1) gauge groups generally can be subgroups of SU(N)
symmetries in the UV we forbid a tree-level kinetic mixing.
In the following we include a more detailed discussion of the model setup from refer-
ence [2] which was written by the author of this thesis. The discussion consists of three
parts: In subsection VII.1.1 we present the scalar sector, where we define the Goldstone
basis taking into account the mixing of angular fields. Subsequently, we discuss the mixing
between anomalon fields in subsection VII.1.2 leading to flavor-violating axion couplings.
In the last subsection VII.1.3 we then show the impact of kinetic mixing on the interactions
involving the gauge fields.
VII.1.1 Scalar Sector
To define the Golds(tone ba)sis(of the angu)lar fields, we param(eterize t)he(complex fi)elds via
1 φaσa v + h 1 φaσa 0
Hu = √ exp i u u u , Hd = √ exp i d ,
2 vu (0 ) 2 (vd ) vd + hd
v1
Φ1 = √
+ h1 a1 v2√+ h2 a2exp i , Φ2 = exp i . (VII.4)
2 v1 2 v2
Here, vi denote the vevs of the scalar fields, hi are the radial modes, and ai and φ
a
i are
angular modes, while σa denote the Pauli matrices as generators of SU(2)L. We define
au ≡ φ3u, ad ≡ −φ3d as neutral angular mod√es ofHu andHd. Th√e vevs which spontaneously
break SU(2)L×U(1)Y and U(1)B are v ≡ v2u + v2d and v
′ ≡ v2 21 + v2, respectively, with
tanβ ≡ v ′u/vd and tanβ ≡ v1/v2.
Since the PQ∑symmetry is ortho∑gonal to the gauge U(1) s∑ymmetries, we have the rela-tions
0 = YiXiv
2 , 0 = B X v2 2 2 2 2i i i i , X va = Xivi . (VII.5)
{Hi} {Φi} {Hi,Φi}
75
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
Together with the requirement of X ≡ Xu + Xd = X1 + X2 to keep the λA term PQ
invariant, the PQ charges of the scalar fields evaluate to
Xu = X cos
2 β , Xd = X sin
2 β , X 21 = X cos β
′ , X2 = X sin
2 β′ . (VII.6)
Consequently√, the PQ symmetry is spontaneously broken by the effective scale va, where
2 ′ 2 ′ v sinβ cosβv 2 2 2a ≡ v sin β cos β + v sin β cos2 β′ , tan γ ≡ ′ ′ ′ . (VII.7)v sinβ cosβ
The PQ charge normalization X is then fixed by the axion decay constant fa ≡ Xva.
To identify the axion a, the heavy SU(2) pseudoscalar A, and the two Goldstones φ0, φB
for the longitudinalZ and Z ′ bosons, we perform the followingorthogonal transformation,
 a  cβsγ sβsγ −cβ′cγ −sβ′cγ au A  cβcγ sβcγ c  = β′sγ sβ′sγ ad0 −  , (VII.8)φ sβ cβ 0 0 a1
φB 0 0 −sβ′ cβ′ a2
where the φ0 and φB Goldstones are easily identified as aligning with the Higgs basis of
each sector. The shorthand notation of s and c represents cosine and sine. We remark that
for v ≪ v′, γ ≈ 0, we reproduce the invisible axion of the DFSZ model which is dominantly
composed of a1 and a2 SM gauge singlets. We can also reproduce the Weinberg-Wilczek
model [17,18] by considering the other limit, v ≫ v′, γ ≈ π/2.
The heavy pseudoscalar A gets a mass from the λA term given by
2
m2
λA va
A = . (VII.9)2 sβcβsβ′cβ′
A mass for the axion is only induced by instanton effects, which are quantified by the
topological susceptibility χ [54]
mumd
χ = m2af
2
a , χ
4 2 2
QCD = ΛQCD = m(m +m )2 π
fπ . (VII.10)
u d
For an ALP, χ remains a free parameter, while a vanilla QCD axion demands χ = χQCD,
although recent studies have demonstrated that χ can be enhanced by non-QCD sources
and still preserve the axion solution to the strong CP problem [1, 193, 194, 196]. As long
as λA is sufficiently large, the basis rotation in equation (VII.8) coincides with the mass
basis of a and A.
For the CP even Higgs bosons, we perform the orthogonal transformation to the Higgs
basis in the alignment limit, giving     h  sβ cβ 0 0 huH0 ≡ cβ −s 0 0 β ′ h d . (VII.11)h 0 0 sβ′ cβ′ h1
H ′0 0 0 cβ′ −sβ′ h2
We will assume that the Higgs basis is aligned with the mass basis and neglect further
scalar mixing, since our focus is the phenomenology of the light axion and Z ′ boson. Large
deviations from the alignment limit are also strongly constrained by Higgs observables [228,
229].
76
VII.1. MODEL
VII.1.2 Fermion Sector
In this section, we calculate the anomalon masses and couplings to the axion and other
scalars. From equation (VII.2), the masses of the anomalons arise from the vevs v2, vu and
vd, where we parameterize the Yukawa couplings by yk = |yk| exp iδk. After accounting
for rephasing freedom, we have two complex phases signifying CP violation which we shift
into the Yukawa terms with Hu and Hd,
L ′ ′anom ⊃− |yL|L̄RΦ2LL − |y |Ē′E LΦ ′2ER − |y1|eiδ12L̄′ ′ −iδ12 ′ ′LHdER − |y2|e L̄RHdEL
− |y ′N |N̄LΦ N ′ − |y |eiδ342 R 3 L̄′ ′LHuNR − |y |e−iδ34L̄′4 RHuN ′L + h.c. , (VII.12)
where the physical complex phases δ12 and δ34 are given by
δ1 − δ2 + δL − δE δ3 − δ4 + δL − δN
δ12 = , δ34 = . (VII.13)
2 2
The induced mass parameters are
|√yL|m = c ′v′ |√yE | |yN |L β , mE = cβ′v′ , mN = √ c ′β′v ,
2 2 2
|√y1| |y2| |y3| |y4|m1 = c12cβv , m2 = √ c12cβv , m3 = √ c34sβv , m4 = √ c34sβv , (VII.14)
2 2 2 2
where c12 ≡ cos δ12 and c34 ≡ cos δ34 reflect the impact of the CP violating phases. We
introduce the shorthand notation
≡ mi +mj ≡ mi −mjmij , ∆ij , tij ≡ tan δij , (VII.15)
2 mi +mj
after which t(he mass m(ixing matrices become )( ′ )
L ⊃− ē′ ′
) mL m12(1 + ∆12)(1 + it12) e
anom Ē
( L L )(m12(1−∆12)(1− it12) mE )(
R
′ (VII.16)ER
m m (1 + ∆ ′
)
− ν̄ ′ N̄ ′ L 34 34)(1 + it34) νRL L m34(1−
+ h.c. .
∆ ′34)(1− it34) mN NR
For v′ ≫ v, the off-diagonal terms are at least suppressed by v/va. The ∆12 and ∆34
terms are also suppressed by the difference in the Yukawa couplings, which is generally
negligible unless the couplings are hierarchical, and so we will assume ∆12 = ∆34 = 0 for
the remainder of this work.
The CP violation is encoded via the tangent of the CP violating phases and will cause
mixing between the axion a and the SM Higgs h. Since we are aligned in the Higgs basis,
we will set t12 = t34 = 0 and leave a study of small deviations inducing mixing between a
and h to future work.
After these simplifying assumptions, we can now rotate the symmetric mass matrices of
th(e an)omal(ons in equation (V)II(.16))using(αE a)nd α(N mixing angles d)efi(ned)via
E1 cosα sinα e
′ N cosα sinα ν ′
= E E , 1− =
N N . (VII.17)
E2 sinα cosα E
′
E E N2 − sinαN cosαN N ′
The masses of E(1, E2√, N1 and N2 a)re given by ( √ )
2 2
m = m 1∓ ∆2 m12 2 m34E1,2 LE LE + , mN = mLN 1∓ ∆ + , (VII.18)m2 1,2 LN 2LE mLN
77
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
using the shorthand notations in equation (VII.15).
We now evaluate the couplings of the axion field a and the SM Higgs field h to the
anomalons. The couplings to the axion are at order 1/fa
Lanom, a ⊃
a
iX2 cos(2αE)(mE1Ē1γ5E1 −mE2Ē2γ5E2)fa
a mE +mE
+ iX 1 22 sin(2αE) (Ē1γ5E2 + Ē2γ5E1)
fa 2
a
+ iX2 cos(2αN )(mN1N̄1γ5N1 −mN2N̄2γ5N2)fa
a mN +mN
+ iX sin(2α ) 1 22 N (N̄1γ5N2 + N̄2γ5N1)
fa 2
a mE1 −mE+ iXd sin(2α ) 2E (Ē1E2 − Ē2E1)
fa 2
a mN −mN
+ iX sin(2α ) 1 2u N (N̄1N2 − N̄2N1) . (VII.19)
fa 2
Here, we see that the terms proportional to X2 are the canonical axial couplings propor-
tional to fermion masses, while the remaining terms proportional to Xd or Xu scale as the
difference of fermion masses and arise generically in flavor violating axion models, as we
discussed in section V.1.
Separately, the interactions of the SM Higgs h to the anomalons are
Lanom, h ⊃
mE −mE h
sin(2α ) 1 2E (cos(2αE)(Ē1E2 + Ē2E1)− sin(2αE)(Ē1E1 − Ē2E2))
2 v
mN1 −mN h+ sin(2α ) 2N (cos(2αN )(N̄1N2 + N̄2N1)− sin(2αN )(N̄1N1 − N̄2N2)) .
2 v
(VII.20)
At dimension 5 we also get mixed operators
L mE −mE h aanom, a, h ⊃ iXd sin(2α ) 1 2E (Ē1E2 − Ē2E1)
2 v fa
mN −mN h a
+ iX 1 2u sin(2αN ) (N̄1N2 − N̄2N1) . (VII.21)
2 v fa
The last two terms are due to the fact that the interactions of the axion proportional to Xu
and Xd are induced by the Higgs doublets and are needed for a complete set of operators
at order 1/fa. In the case with CP violation, the linear Higgs interactions would mix with
the linear axion interactions proportional to Xu and Xd.
VII.1.3 Gauge Sector
Finally, we discuss the Z and Z ′ interactions, which necessarily includes the kinetic mixing
effect in equation (VII.3). The effective kinetic mixing parameter ϵeff is determined by
calculating the one-loop contribution to the two point interaction between the hypercharge
gauge field Bµ and baryon number gauge field Kµ, giving
2
L ⊃ ϵeff µν
m
B eff µµνK + BµK , (VII.22)
2 2
wheremeff corresponds to a possible mass mixing. The mass mixing vanishes if all fermions
in the loop are vector-like under one of the U(1) gauge symmetries [227]. The divergence in
78
VII.1. MODEL
the two-point loop diagram is c
fermions, ∑ancelled after imposing the trace condition on the mediator
N (Y fBf f ff V V + YABA) = 0 , (VII.23)
f
where Nf denotes the multiplicity factor of fermion f . In the unbroken phase of elec-
troweak symmetry, we can consider the SM fermions to be massless, such that the only
remaining contribution comes from the anomalons. Form2 2 2L , mE ≫ p the effective kinetic
mixing parameter reads
∑ ( ( ) ( )2 )gY gB 4 5 m 2f p
ϵ feff = (YV B
f + Y f fV ABA) + ln +O(4π)2 3 ′ (′ ′ ( ) ( 3 ) p2 mf∈{L ,E ,N } ( ))2f
−eg
2 2 2
B 1 2 10 mL m p= + ln + ln E +O . (VII.24)
cW (4π)2 3 3 p2 p2 m2 2L,mE
The large logarithms in ϵeff roughly cancel the loop factor such that the dominant para-
metric dependence is given by ϵ −1eff ≈ egBcW . In the following we will see that we get new
interactions proportional to ϵeff.
We recall from Ref. [230] that kinetic mixing is removed by shifting the gauge fields into
a diagonal and canonically normalized basis, using the replacement rules
m2 ′ ′ ( )ZSM =Z − ϵ s Z Z +O ϵ2µ µ eff W 2 2 µ eff , (VII.25)mZ′ −mZ
m2 ( )
Kµ =Z
′ Z 2
µ − ϵeffsW 2 2 Zµ +O ϵeff (VII.26)mZ −mZ′
to shift to the mass basis. Assum(ing mK > mZ,SM, the correspon)ding masses are
ϵ2 s2 m2
m = m eff
W Z,SM
Z Z,S(M 1 + 2 2 +O(ϵ
4
eff) ), (VII.27)2 mZ,SM −mK
ϵ2 (m2K − c2Wm2eff Z,SM)mZ′ = mK 1 + 2 2 +O(ϵ
4
eff) , (VII.28)2 mK −mZ,SM
with sW , cW being sine and cosine of the weak angle θW . We see that the mass correction
only appears at order ϵ2eff and is hence typically negligible.
We apply the shifts to the gauge bosons in equations (VII.25) and (VII.26) and obtain
for the scalar Lagrangian
Ld≤4 1 µ 1 µ 1 ′ µ ′ 1 ′ µ ′ ′ ′scalar ⊃ ∂µh∂ h+ ∂µH0∂ H0 + ∂µh ∂ h + ∂µH0∂ H0 − V (h,H0, h ,H2 2 2 2 0
)
1 1 m2 1 m2 A4
+ ∂µa∂
µa+ ∂ A∂µA− A 2µ ( A +
A
2 ( 2 ) 2 s2c2γ γ v2a 4! )( )1 e2 2 2
+ (h+ v)2 +H2
mZ′ ′ µ mZ′ ′µ
8 s2 2 0W(cW )(
Zµ − ϵeffsW 2 2 Zm −m )µ( Z − ϵeffsW m2 ZZ′ Z Z′ −m2Z )
9 m2 m2
+ g2B (h
′ +(v
′)2 +H ′2 Z ′ − ϵ s Z ′µ Z µ0 µ eff W )2 Z− 2 µ Z − ϵeffsW Z2 mZ m 2 2Z′ mZ −mZ′
− e
m2 ′
H0 Zµ − ϵ Z ′ µ µeffsW 2 2 Zµ (sγ∂ a+ cγ∂ A)sW cW mZ′ −mZ
79
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
( 2 )
+ 6g H ′ ′
m ( )
Z − ϵ Z µ µ 2B 0 µ effsW 2 2 Zµ (cγ∂ a− sγ∂ A) +O ϵm −m eff
, (VII.29)
Z Z′
where φ0 is absorbed by ZSMµ , φB by Kµ, and A is the only angular mode which gets a
mass from the term proportional to λA as defined in (VII.9).
Finally, we discuss the gauge interactions of the anomalons. Following Ref. [230], the
interactions of the neutral gauge bosons are given(at O(ϵeff) by
1. )
L ⊃ µ √ e
2
eA J + (W− +µ
e µ mZ µ
gauge µ( Q µ JW + h.c.) + Zµ ) JZ − ϵeffsW gB2s s c m2 JW W W Z −m2 BZ′2
+ Z ′ g Jµ
e m ′
µ B B + ϵeffeJ
µ − ϵ Z µQ eff c 2 2
JZ , (VII.30)
W mZ′ −mZ
with the gauge currents given by
JµQ ⊃− (Ē1γ
µE1 + Ē
µ
2γ E2) , (VII.31)
J+µ ⊃(c c Ē γµN + c s Ē γµN + s c Ē γµN + s s Ē γµW E N 1 1 E N 1 2 E N 2 1 E N 2 N2) , (VII.32)
1
Jµ 2 2 µ 2 2 µZ ⊃ ((2sW − cE)Ē1γ E1 + (2sW − sE)Ē2γ E2 − s
µ µ
EcE(Ē1γ E2 + Ē2γ E1))
2
1
+ (c2 N̄ γµN + s2N 1 1 N N̄ γ
µ
2 N2 + s c (N̄ γ
µ
N N 1 N2 + N̄2γ
µN1)) , (VII.33)
2
µ ⊃1JB (Ē
µ
1γ E1 + Ē
µ
2γ E2 + N̄1γ
µN1 + N̄ γ
µ
2 N2)
2
3
+ (cos(2αE)(Ē
µ µ µ µ
1γ γ5E1 − Ē2γ γ5E2) + sin(2αE)(Ē1γ γ5E2 + Ē2γ γ5E1))
2
3
+ (cos(2α µN )(N̄1γ γ5N − N̄ γµ1 2 γ5N2) + sin(2αN )(N̄ γµ1 γ5N2 + N̄2γµγ5N1)) .
2
(VII.34)
There are two extreme cases: αi → 0 corresponds to minimal mixing, while αi → π/4
describes maximal mixing. In the minimal mixing case, we recover flavor-conserving axion
and Z ′ couplings, while in the maximal mixing case, the axion, Z and Z ′ bosons all change
the flavor of the anomalons. Another feature is given by the fact that the anomalons
give rise to new contributions to the Higgs decay to two gauge bosons which are not
excluded [232].
VII.2 Effective Theory and Wilson Coefficients
In order to study the phenomenology of our model at collider scales we describe the cou-
plings of the axion in an ALP-EFT framework where we integrate out the heavy anoma-
lons, while keeping the SM fermions dynamically. In addition, we need to deviate from
the DFSZ axion band to higher masses, m2f2 ≫ Λ4a a QCD. This can be either achieved by
enhancing χ for the QCD axion through non-QCD sources [1, 193, 194, 196] or by a free
topological susceptibility in case of an ALP.
In both cases we assume that the QCD contribution is negligible, such that we can work
in the ALP-EFT basis in which we do not use the axion digluon coupling for the axion
mass generation and neglect QCD corrections to the axion diphoton coupling. The axion
1In contrast to Refs. [227, 231], our convention for gB in this work uses L = 1g Z′ (q̄γµq), and thus3 B µ
our gB is half the value used in Refs. [227,231].
80
VII.2. EFFECTIVE THEORY AND WILSON COEFFICIENTS
Lagrangian consists of an effective contribution from integrating out the anomalons as well
as an SM contribution and reads
L 1
2
µ ma 2 ∂µa µ eff v µ eff v ′ µ
axion ⊃+ ( (∂µa)(∂ a))− a + JPQ,S(M − CZh h)Zµ∂ a− CZ′h hZ2 2 f µ∂ aa fa fa2 2
+ CSM + Ceff
e a e 1 a
( γγ γγ ) FµνF̃
µν + CSM + Ceff µν
(4π)2 f Zγ Zγ
ZµνF̃
a ( sW cW)(4π)2 fa
SM eff e
2 1 a
+ C + C Z Z̃µν + CSM + Ceff
gBe a ′ µν
( ZZ ZZ s)2 2 Z F̃c (4π)2 µν ′ ′W W f Z γ Z γ 2 µνag2 ( ) (4π) fa
+ CSM + Ceff B
a ′ ′µν gBe 1 a
( Z′Z′ Z′Z′) ZµνZ̃ + C
SM + Ceff ′ µν
(4π)2
′
f Z Z Z
′Z Zs c (4π)2 f µν
Z̃
a W W a
2 2
+ CSM + Ceff
gL a g a
WW WW WµνW̃
µν + CSM s Ga G̃aµν
(4π)2 f gg (4π)2 f µνa a
a √ e+ i (W−(X J+µ −X J+µ
( )
µ d W u /) + h.c.) +O h
′, H ,H ′ , A,H±0 0 , (VII.35)f 2s W,la W
where we neglect contributions from other scalars, which we assume to be much heavier
than the collider scale.
Under the use of our calculations of the Wilson coefficients in the heavy fermion limit
in equations (V.16) and (V.21) we obtain ( )
2
Ceff
8 ∆E m 1
γγ =− Xs2 cos(2α ) aβ′ 3 E +O , (VII.36)3 Σ f2E a f3a
Xs2 2 2
eff − β
′ Xs Xs
C = , Ceff
β′
= −( (1− 2s2 eff β
′
γZ ZZ W ) , CWW = )− , (VII.37)4 4 2
Xs2 2 2 2 2
eff − β
′
C = (Σ2
m ′(m +m −m ′)
hZ′ M +∆
2
M ) 1− 6 Z
a h Z
2 2 , (VII.38)2 λ(mZ′ ,m
2
a,mh)
eff ϵeffe m
2
Z′ eff eff g ϵ
2 2
− − B eff
s cW m
CγZ′ = 2 2 CγZ , C
W Z eff
hZ = 2 C ′ , (VII.39)gBcW mZ′ −mZ ( e mZ −m2 hZZ′2 2 )2
eff ϵeffe m ′ ϵeffe m ′ ( )CZZ′ =− Z Ceff , Ceff Z eff2 2 ZZ Z′Z′ = − 2 2 CZZ +O ϵ3 ,gBc effW mZ′ −mZ gBcW mZ′ −mZ
(VII.40)
where we drop all terms of order 1/fa, ϵ
2 , ∆2eff E , ∆
2 2
N and ∆EN , if not mentioned otherwise.
The parameters Σ and ∆(decribe mass sums and difference)s and are given by
m12 +m34 1 |y1|√+ |y2| |y3|√+ |y4| cβ + sβΣM = = ( c12cβ + c34sβ) ≈ √ , (VII.41)v 2 2 2 2
m12 −m34 1 |y1|√+ |y2| − |y3|√+ |y4|
c
≈ β
− sβ
∆M = = c12cβ c34sβ √ , (VII.42)
v 2 2 2 √ 2
mE1 +mE2 2mLE |yL|√+ |yE | cγ ≈ 4π 2ΣE = = = , (VII.43)
fa fa 2 Xsβ′ 3 Xsβ′√
mN +mN 2mLN |yL|+ |yN | cγ 4π 2
ΣN =
1 2 = = √ ≈ , (VII.44)
fa fa 2 Xsβ′ 3 Xsβ′ √
mE1 −mE 2c∆ = 2 − 1 2m12 v ΣM +∆M v βE = = − ≈ − , (VII.45)
fa sin(2αE) fa fa sin(2αE) fa sin(2αE)√
mN1 −mN2 − 1 2m34 − v ΣM −∆M v
2sβ
∆N = = = ≈ − , (VII.46)
fa sin(2αN ) fa fa sin(2αN ) fa sin(2αN )
81
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
− 2mLE − 2mLN |yE |√− |yN | cγ∆EN = ΣE ΣN = = . (VII.47)
fa 2 Xsβ′
Here, we further assumed that the Yukawa couplings |yL|, |yE | and |yN | are 4π/3, while
the other Yukawa couplings are of order 1. As it was motivated previously, we used the
small angle approximation for the angles γ, δ12 and δ34.
We highlight that the axion diphoton coupling is significantly suppressed due to the
opposite sign in the flavor-conserving interactions of the charged anomalons. This is
remarkable as it is in contrast to usual QCD axion models where the diphoton and digluon
couplings are dominating. In fact, in our model the coefficients Ceff , Ceff and CeffγZ ZZ WW
become the most important axion gauge boson couplings for energies above 2mt where
the SM contributions to the axion couplings to photons and gluons decrease. We discuss
this in more detail in section VII.3.
We also notice that the coupling of the axion to the Higgs and the SM Z boson is
proportional to the effective kinetic mixing parameter ϵeff. This meets our expectations as
in absence of kinetic mixing the orthogonality conditions from angular mixing forbid the
respective term in the Lagrangian.
Lastly, we point out that the Z ′ mass as well as the anomalon masses implicitly depend
on the axion decay constant fa, since in the invisible axion limit, as described in subsec-
tion VII.1.1, fa is mainly composed out of v
′ with f = Xc−1 ′a γ v sβ′cβ′ . The mass relations
of the Z ′ and the anomalons are then given by
′ 3cγgBfa ΣE,N ±∆≡ E,N 4π famZ′ = 3gBv = , manom fa ≈ √ . (VII.48)
Xsβ′cβ′ 2 3 Xsβ′ 2
We find that in order to obtain a Z ′ mass which is lighter than the anomalons but allows
for a sizable gauge coupling g we need to consider the limit β′B → π/4. Substituting the
PQ charge normalization of the DFSZ model and taking the small angle approximation
for γ then leads to mZ′ ≈ 36gBfa and manom ≈ 8πfa.
Hence, we are left with two independent parameters for the QCD axion, gB and fa, and
a third parameter χ for an ALP. We can trade χ or fa to have a free axion mass ma or
analogously gB or fa to obtain a free Z
′ mass mZ′ . Having established the parameters in
the EFT, we proceed in the next section discussing the phenomenological implications of
our model.
VII.3 Phenomenology
We have seen in equation (VII.35) that the effective Lagrangian contains new exotic op-
erators involving the axion, the Z ′ boson as well as heavy SM particles. These operators
are of particular interest for collider experiments which cover energy scales in the GeV to
TeV range.
In order to find constraints on our model parameter from collider experiments we investi-
gate different production and decay channels and compare the results with existing narrow
resonance searches at the LHC. The analysis includes the calculation of the branching ra-
tios of the possible two-body decays as well as a simulation of the collider events using
MadGraph5 aMC@NLO [10]. We provide an exemplary calculation of a narrow resonance
cross section in section A in the appendix.
First, in subsection VII.3.1, we constrain the {mZ′ , gB} parameter space by looking at
different Z ′ decays. Afterwards, we discuss in subsection VII.3.2 possible axion decays
and present our results in the {ma, Gaγγ}-plane.
82
VII.3. PHENOMENOLOGY
VII.3.1 Collider Constraints for the Z ′ Boson
The phenomenology of the Z ′ gauge boson is mainly determined by its mass mZ′ and
gauge coupling gB. We can find constraints on these parameters from collider experiments
by varying the mass in the accessible energy range while comparing the expected cross
sections for a given gauge coupling with the actual experimental limits. We focus on
narrow resonance searches, since we expect the total decay width of the Z ′ to be much
smaller than its mass. In this narrow width approximation, the cross section is dictated
by the branching ratios of the different Z ′ decays.
In order to find the most relevant production processes as well as interesting decay
channels, we first determine partial two-body decay widths and set up the corresponding
branching ratios. Since the quarks are the only particles in the SM which transform under
U(1)B, we expect the corresponding decay√widths to be(dominatin)g,
N2 2q gBB
2
qmZ′ m
2 2
q mq
ΓZ′→q̄q = 1− 2
12π m2
1 + 2 , (VII.49)
Z′ m
2
Z′
where Bq denotes the baryon number of quark q, mq its mass and Nq its multiplicity.
The coupling to other SM particles is only induced via kinetic mixing or integrating
out the heavy anomalons. In case of SM leptons the tree-level coupling is mediated by
the mixing of the Z ′ with the photon and the Z boson. Hence, the decay width to two
leptons with mass ml is controlled by the electric charge Ql and the SU(2)L isospin charge
T l3 = ±1/2 associated to(the third)gene(rator, √−2 )
e2ϵ2 2 ( ) ( )2effmZ′ m 2 1 m2ΓZ′→l̄l =( 2 1−
Z 2 l 2 l l
12πc m2
QlsW − T3Q)lsW + T3 1− 22 m2W Z′ Z′
m2 (Q s2 2l l W ) − T l3Q s2 − (T l 2× l1 + 2 w 3
) /4
2 . (VII.50)mZ′ (Qls
2 2 l
W ) − T3Q s2l w + (T l)23 /2
We notice, that ΓZ′→l̄l is suppressed by the square of the effective kinetic mixing parameter
ϵeff and has a resonance at mZ′ ≈ mZ .
The Z ′ decay widths involving the axion can be derived from the effective operators
in equation (VII.35), while for the Z ′ decays into gauge bosons we use the procedure
presented in reference [232].
Especially the decay of the Z ′ into a photon and a Z boson can be calculated analogously
to [232] by interchanging the role(of the Z ′ a)nd the Z,
3 e4g2B m
2
Z m
4
ΓZ′→γZ = 1− Z
2∣∣04∑8π5 s2 c2 m(′ m4∣ W W Z Z′∣∣ m2 ′ ( )− 2T q3Q ZqBq 2 2 B0(m2 2− Z′ ,mq,mq)−B0(mZ ,mq,mq)m m
q Z′ Z )
m2 ′
( +2m
2 Z
q 2 C0(0,m
2
Z ,m
2
Z′ ,mq,mq,mq)mZ
m2 ′ ( )
+ Z2 2 B0(m
2 2
− Z′
,manom,manom)−B0(mZ ,manom,manom)mZ′ mZ )∣∣2
+ 2m2anomC (0,m
2 ,m20 Z Z′ ,manom,manom,manom) ∣∣ ,
(VII.51)
where we neglect corrections of order O(∆2 , ∆2 , ∆2E N EN ).
83
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
Figure VII.1: Branching ratios of two-body decays of the Z ′ gauge boson for mZ′ between
100 GeV and 5 TeV. The ALP mass is fixed at ma = 1 GeV and the gauge coupling at
gB = 0.5. The figure is taken from reference [2] and was created by the author.
In Figure VII.1 we show the branching ratios of two-body decays of the Z ′ for a varying
mZ′ between 100 GeV and 5 TeV, while keeping the gauge coupling fixed at gB = 0.5.
The axion mass is taken to be ma = 1 GeV. We display the decay into two top quarks
separately from the other quarks. We see that the most dominant decay is into two quarks
as we expected, while the decay into leptons is heavily suppressed.
The exotic decays Z ′ → ah, Z ′ → WW , Z ′ → γZ and Z ′ → γh represent the most
important sub-dominant contributions. The decay into axion and Higgs can only appear
in a model construction like ours, with non-trivial mixing effects between the axion and
the Z ′. The decays into SM particles are tested by ATLAS and CMS and can therefore
be used to constrain our parameter space.
For this purpose we simulate detector events in the respective decay channels using
MadGraph5 aMC@NLO [10] and compare the results with the published LHC limits [233–
235]. We then exclude regions in the parameter space which are not consistent with the
experimental results.
We show the excluded regions in Figure VII.2, where we display for comparison the
limits from dijet resonance searches [227]. We also include a limit on the charged anomalon
masses, manom < 90 GeV from the ALEPH and L3 collaborations [236,237], which is linked
to the Z ′ mass via their dependence on fa as shown in equations (VII.18) and (VII.48).
While the Z ′ → hγ and Z ′ → WW channels can only probe unrealistically large gauge
couplings of order O(10), the Z ′ → Zγ constraint competes with the dijet limit at low
masses. The enhanced sensitivity in the Z ′ → Zγ channel reflects a better efficiency and a
smaller background for signal photon and leptons. The limits get weaker with increasing
Z ′ mass since the corresponding branching ratios in Figure VII.1 fall off for higher masses
as the Z ′ → t̄t channel opens up.
84
VII.3. PHENOMENOLOGY
10
5
1
0.5
Z ′ Z Z ′ h Di-Jet limits
Z ′ WW ALEPH/L3 limit
150 350 600 1000 2000 3000
mZ ′[GeV]
Figure VII.2: Excluded regions in the {mZ′ , gB} parameter space from narrow resonance
searches in the Zγ [233], hγ [234] and WW [235] channels. Shown are also a limit on the
anomalon masses, manom < 90 GeV excluded from searches by ALEPH and L3 collabo-
rations [236, 237], as well as constraints from dijet resonance searches [227]. The figure is
taken from reference [2].
VII.3.2 Collider Constraints for the Axion
In case of the axion phenomenology we are mainly interested in constraining the {m −1a, fa }
parameter space, which provides information about the topological susceptibility χ and
therefore the underlying axion or ALP model. In order to exclude parameter space based
on existing collider searches we first compute the branching ratios of the axion two-body
decays and simulate the cross section for different decay channels. We compare the cross
sections with the experimental limits and present our results in the {ma, Gaγγ}-plane.
The branching ratios are dictated by the operators of the EFT basis defined in equa-
tion (VII.35) and are shown in Figure VII.3 for a varying axion mass. We set the axion
decay constant to fa = 500 GeV and the Z
′ mass to mZ′ = 1 TeV. Consequently, the
gauge coupling gB and the effective kinetic mixing parameter ϵeff are fixed as well. Again,
we display the decay into two top quarks separately from the other quarks.
We see that for small axion masses the decays into SM fermions, gluons and photons are
most dominant as it is true for most QCD axion models. In addition, the exotic a → hZ
decay channel becomes enhanced near the top threshold providing approximately 10% of
the total decay width. Since this decay is only induced through kinetic mixing effects, this
channel serves as an exciting opportunity for discovery.
Above the top threshold the decays a → WW , a → ZZ and a → γZ become more
important, since they are induced by integrating out the anomalons, while the axion
diphoton and the axion digluon couplings cancel out at high energies. The a → hZ ′
decay mode is suppressed by the Z ′ mass and therefore does not pl√ay a big role. The
corresponding branching ratio has a minimum at m2a = m
2
h + 4m
2
Z′ + 16m
2 2
hmZ′ + 9m
4
Z′
as the Wilson coefficient in equation (VII.38) changes sign at this point.
We make use of the branching ratios by simulating collider events in different decay
85
gB
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
Figure VII.3: Branching ratios of two-body decays of the axion for ma between 100 GeV
and 5 TeV. The Z ′ mass is fixed at mZ′ = 1 TeV and the ALP decay constant at fa =
500 GeV. The figure is taken from reference [2] and was created by the author.
channels using MadGraph5 aMC@NLO [10]. We then compare the result with the LHC limits
published in references [233,235,238,239]. Figure VII.4 shows the excluded regions in the
{ma, f−1a } parameter space for which the simulated cross section is not consistent with
the experimental limits.
In addition to the collider constraints we display the limit on charged anomalon masses,
manom < 90 GeV from the ALEPH and L3 collaborations [236,237] for the mixing angles
β = 0.05 and β′ = π/4. The anomalon masses directly depend on fa as shown in equa-
tions (VII.18) and (VII.48). We also show two lines of constant mafa with χ = (10
2 GeV)4
and χ = (103 GeV)4.
We see that the diphoton channel gives the strongest constraints. This is reasonable,
since the γγ resonance searches are more sensitive than the other channels and at the
top threshold there is still a significant contribution to the axion diphoton coupling from
SM particles. In contrast, the γγ constraints get weaker for higher axion masses as the
effective contribution from anomalons cancels out.
Finally, we study the impact of our results on the {ma, Gaγγ} parameter space. For this
purpose we first express the axion diphoton coupling thro√ugh∑  22e2 m2 2m2f −m2a + m4 2 2f f a − 4mamfGaγγ = N 2 fQfXA ln 2 , (VII.52)π2f 2
∈ a
m
f SM,E a
2mf
1,2
E
where the axial PQ charges X 1,2A of the charged anomalons are extracted from equa-
tion (VII.19). We disregard possible corrections from mass mixing with pions since we are
interested in axions and ALPs with enhanced masses m2f2 ≫ Λ4a a QCD.
Figure VII.5 shows the {ma, Gaγγ}-plane for axion masses between m = 10−2a GeV
and ma = 10
6 GeV. We convert our results from Figure VII.4 to the Gaγγ coupling and
display for comparison the canonical limits extracted from reference [53]. In addition we
show four lines of constant topological susceptibility. The dashed lines represent the model
lines without threshold corrections through the quark masses.
86
VII.3. PHENOMENOLOGY
101
100
10 1
a ALEPH/L3 limit
10 2 a Z m f = (102a a  GeV)2
a hZ m f = (103 GeV)2a a
a ZZ
10 3
150 350 600 1000 2000 3000
ma[GeV]
Figure VII.4: Excluded regions in the {m , f−1a a } parameter space from narrow resonance
searches in the γγ [238], Zγ [233], ZZ [235] and hZ [239] channels. Shown are also a limit
on the anomalon masses, manom < 90 GeV excluded from searches by ALEPH and L3
collaborations [236, 237], as well as lines of different mafa relations. The figure is taken
from reference [2].
The behavior of the model lines reflects the mass thresholds of the quarks and shows a
smaller diphoton coupling at higher energies due to the cancellation effects in the anomalon
induced couplings. We notice, that our limits from the diphoton resonance are stronger
than the canonical bounds in this channel around the top threshold but weaker for higher
axion masses. In fact, for our model the decay channels a → Zγ, a → hZ and a → ZZ
compete with the diphoton limits and become even more important for axion masses larger
than 1 TeV.
87
f 1a [GeV 1]
CHAPTER VII. AXION COUPLINGS IN GAUGED U(1)’ EXTENSIONS
0.01
10-4 )4eV
2 =(10 G m2 f 4a ) an
- ma =( GeV o2 m10 <96 2 010 G2 fa e
ma V
)4
- =( V03 Ge10 8 f 2 12 a
ma
4
10-10 2 =(
)
04 G
eV
1
2 fa
ma
10-12
0.1 10 1000 105
Figure VII.5: Implications of an additional U(1)B gauge symmetry in the
{ma, Gaγγ} parameter space for axion masses between m = 10−2a GeV and ma =
106 GeV. Displayed are four lines of constant topological susceptibility with χ ∈
{(10 GeV)4, (102 GeV)4, (103 GeV)4, (104 GeV)4}, where the dashed lines represent the
model without threshold corrections through quark masses. We extract the canonical con-
straints from reference [53] and overlay our limits from Figure VII.4.
88
Epilogue

CHAPTER VIII
Conclusion and Outlook
In this thesis we worked out various phenomenological implications in DFSZ axion models
taking into account consequences from several observations and theoretical expectations
which can not be described within the SM. Our goal was to identify effects that lead to a
better accessibility of the DFSZ axion in experimental searches, including dedicated axion
experiments like Haloscopes and Helioscopes as well as collider and DM experiments.
In the prologue we pointed out that the challenges in axion physics comprise an extensive
possible mass range with a decrease in the coupling strength of axion-SM interactions for
small axion masses. This decrease in the coupling strength is due to an inverse proportion-
ality to the axion decay constant fa in combination with a constant axion mass relation
m2af
2 4
a = ΛQCD. We concluded that we can find deviations from the standard procedure
either in the axion potential which fixes ma or in the axion interactions themselves.
We started in part I by discussing possible modifications of the axion potential. For this
purpose we introduced in Chapter III a general description of the angular scalar potential
involving the pseudoscalar degrees of freedom of complex scalar fields. We pointed out
three contributions to the angular potential. The first contribution originated from com-
plex scalar fields with non-vanishing vevs in the scalar Lagrangian and led to a mixing
between neutral Goldstone bosons of gauge symmetries and global Goldstone bosons like
the axion.
Secondly, the angular potential consisted of terms that were induced by a fermionic
condensate. In general, the fermion condensate entered through the Yukawa interactions
with the complex scalar fields. The third contribution was induced by fermion condensates
that are generated by the confinement of an SU(N) gauge symmetry. Here, we used
instanton calculus to find corrections in the potential. It turned out that in the standard
axion case the correction was of order O(1%) and therefore negligible.
We then used the results from Chapter III in Chapter IV to find a procedure in which
additional PQ-breaking operators led to a branch-off point in the {ma, Gaγγ}-plane at
which the axion decay constant saturated at a model dependent value fmax. This value
not only provided an estimate of the maximal extent of the expected Gaγγ line in presence
of an additional PQ-breaking operator, but also allowed for much lighter axion masses
with m2f2a a ≪ Λ4QCD. This corresponds to an enhancement in the axion-diphoton coupling
for small axion masses, reaching for 107 GeV ≲ fmax ≲ 1011 GeV the preferred region of
Haloscope and Helioscope searches.
Following the procedure from the previous chapter, we were able to identify several
constructions which naturally provide fmax in our range of interest. On the other hand,
in order to achieve a sizable deviation from the canonical DFSZ line, we needed a CP
91
CHAPTER VIII. CONCLUSION AND OUTLOOK
violating parameter θ̄ ≈ π in the UV to not generate a residual CP phase θ̄eff that was in
conflict with current nEDM measurements. We solved this problem with a Nelson-Barr-
like UV completion, leading to a deviation of m2f2 ≳ 10−6Λ4a a QCD in presence of Planck
mass suppressed operators.
In part II we discussed modifications of the axion couplings using an EFT approach.
For this purpose we constructed in Chapter V a general ALP-EFT basis accounting for
anomalous fermion transformations and flavor changing effects. We identified operators
which do not appear in the flavor-conserving limit, including a five-dimensional commu-
tator interaction between the axion, two fermions, and a gauge boson. Subsequently, we
calculated the Wilson coefficients of effective three-point interactions, where we integrate
out heavy fermions and scalars. We presented our results for axion couplings to two gauge
boson, one gauge boson and one scalar, and two fermions.
The effective axion-fermion vertices played a role in Chapter VI in the context of chiral
enhancement. Here, we compared the tree-level axion-fermion coupling to the one-loop
contribution from integrating out a heavy DM scalar S and a heavy fermion F . We
observed that the effective one-loop contribution led to an enhancement in the total axion-
fermion interaction proportional to the Yukawa couplings between S, F and SM fermions.
We constrained these couplings using collider limits on flavor-violating axion interactions
and DM bounds on the scalar S.
We found out that the flavor limits led to a large hierarchy between the individual
Yukawa couplings, allowing for only one sizable yiS at a time. Furthermore, after applying
limits on mS from DM relic abundance and DM direct detection, we concluded that chiral
enhancement of order O(10− 100) appears preferably for heavy up-type quarks as well as
for charged leptons. In addition, we noticed that any non-zero chiral enhancement implied
a lower limit of mS as a result of an overlap between the two DM limits.
Afterwards, we briefly discussed possible enhancement factors in models with neutrino
masses. In general, heavy right-handed neutrinos are predestined candidates for generating
chiral enhancement, since they have a large coupling to the axion and a chiral coupling
to leptons through a W interaction. However, this W interaction is suppressed by the
light-neutrino mass, since W bosons only couple to right-handed neutrinos through their
mixing with their left-handed counterparts. This leads to a large suppression in the chiral
enhancement of order v/fa.
Finally, we studied in Chapter VII new effects in axion models with an additional U(1)B
gauge symmetry. The model contained a new gauge boson Z ′ and heavy fermionic anoma-
lon fields to cancel gauge anomalies. After integrating out the anomalons and accounting
for kinetic mixing effects between the gauge bosons, we obtained new exotic operators in-
volving the axion, the Z ′ boson and heavy SM fields. In particular, kinetic mixing effects
induced a three-point interaction between axion, Higgs and Z boson which provides up to
O(10%) of the total axion decay width but vanishes in canonical axion models.
Another peculiarity of our model were cancellation effects in the effective axion-diphoton
coupling from integrating out the anomalons. We simulated collider events in different
axion and Z ′ channels and compared our results with narrow resonance searches at the
LHC. We found out that due to the suppression in the axion-diphoton coupling, the decay
channels a → Zγ, a → hZ and a → ZZ provided the strongest bounds in the {ma, f−1a }
parameter space for axion masses above 1 TeV.
In summary, we discovered new ways to search for DFSZ axions that are accessible
for experiments in the near future but lie outside of the canonical expectation range.
On one hand, we obtained enhancement effects in the axion-diphoton and axion-fermion
92
couplings which generically arose from the interplay of the DFSZ axion model with Planck-
mass suppressed operators and heavy DM particles. On the other hand, we found new
operators describing axion interactions with heavy SM particles and additional fields from
other BSM theories such as right-handed neutrinos, DM scalars or Z ′ gauge bosons.
In this thesis, we were able to answer the question of whether generic BSM features
modify the DFSZ axion solution to the strong CP problem towards a better accessibility
in experiments. Nevertheless, there are still open problems that are inspirational for future
research. One remaining question is whether we can reach arbitrarily small axion masses
from PQ-breaking effects which do not interfere with radiative corrections and Planck-
suppressed operators. Secondly, chiral enhancement effects could only affect individual
axion-fermion couplings with a preference for charged leptons and heavy up-type quarks.
Here, we can ask the question of finding a different description which also allows for
multiple enhanced interactions at the same time including also light quarks and neutrinos.
Finally, we found new interactions between the axion and the Higgs boson, which can be
even better constrained at future colliders and therefore motivate more extensive studies
at collider experiments.
93

Appendix
A Decay Widths and Cross Sections
In this section we want to recap the procedure of calculating decay widths and cross section
to estimate observables for collider searches. This summary was prepared by the author
for a previous work and is mainly based on the textbooks [240, 241]. We start with the
decay width of the axion/ALP or analogously∫for a gauge boson which can be calculatedby
Γ(A → F 1) = dΠ |M |2F A→F , (IX.1)
2mA
where A stands for the decaying particle and F for the final state. Here, the Lorentz-
invariant phase space element is g∏iven byd3pj 1 ∑
dΠ 4F = δ (pA − pj). (IX.2)
(2π)3 2E
j∈F j j∈F
For a two-particle final state th∫is Lorentz∫-invariant phase space becomes
dΩCM 1 2|p⃗|
dΠ2 = , (IX.3)
4π 8π ECM
where |p⃗| is the absolute value of the three-momenta of the outgoing particles B and C in
the center-of-mass frame (CM) of the initial particles. Since there is only one decaying
particle, the center-of-mass energy is given by ECM = mA. For all external particles being
on-shell (p2A = m
2
A, p
2 2
B = mB, p
2
C = m
2
C), |p⃗| ca√n be calculate√d by
mA = E√CM = EB + EC = |p⃗|2 +m2B + |p⃗|2 +m2C (IX.4)
λ(m2 ,m2 ,m2)
⇔ | | A B Cp⃗ = . (IX.5)
2mA
Hence, the Lorentz-invari√ant phase space becomes∫ √
1 λ(m
2
A,m
2 2 ∫
B,mC) dΩ 1 λ(m
2 ,m2 ,m2
CM A B C)
dΠ2 =
8π m2
=
4π 8π m2
. (IX.6)
A A
Next, we calculate the squared matrix elem∑ent which is defined by
|M 2 1A→F | = M ∗A→FM
d A→F
, (IX.7)
A
pol.
95
APPENDIX
with dA being the number of degrees of freedom for the incoming particle. Furthermore,
the sum going over all polarizations of the external gauge bosons. Inserting the effective
couplings c effeff ≡ C12g1g2(4π)2( from subsection V.)2.1(gives with da = 1∑ )
|M |2 ceff ceff= 2 ϵµναβp p ϵ∗ ϵ∗ 2 ϵρσγδa→A1A2 1α 2β 1µ 2ν p 1γ
p2δϵ1ρϵ2σ
fa fa
pol. 
c2 ∑ ∑
= 4 eff ϵµναβϵρσγδp ∗ ∗
2 1α
p2βp1γp 2δ ϵ1µϵ1ρ ϵ2νϵ2σ . (IX.8)fa pol. pol.
The polarization sum yields for mass∑less gauge bosons
ϵ∗µϵν = −gµν (IX.9)
pol.
and for massive gauge bosons ∑
∗ pµpνϵµϵν = −gµν + . (IX.10)m2
pol.
Since there are ϵ-tensors in the matrix element, the additional term in equation (IX.10)
vanishes and equation (IX.9) holds true for all gauge bosons. Therefore, the squared
matrix element becomes under the use of Package X [8]
c2 (p2, p2, p2)
|Ma→A1A2 |2 = 4 eff 1 2 ϵµναβϵρσγδp1αp2βp1γp2δ(−g2 µρ)(−gνσ)fa
c2eff(p
2, p2 21, p2)= 2 λ(p2, p2, p21 2). (IX.11)f2a
Having gauge bosons in the initial state changes the result only by factors of d1 and
d2, since C0(p
2, p2, p2,m21 2 f ,m
2 2
f ,mf ) is symmetric under interchanging p
2, p2 21 and p2 and,
therefore, ceff(p
2, p2 2 2 21, p2) is also symmetric. This is true as well for λ(p , p1, p
2
2). Hence, the
general expression for the triangle diagram with one external axion/ALP and two external
gauge bosons is
c2 (p2 , p2 , p2)
|M 2 2A→BC | = eff A B C λ(p2 , p2 , p2A B C). (IX.12)dA µ2
Inserting the previous expressions into the decay width (IX.1) and using the on-shell
condition (p2A = m
2
A, p
2
B =∫m
2 , p2B C = m
2
C) leads to ∫
1 1
Γ(A → BC) = dΠ2|MA→BC |2 = |M 2A→BC | dΠ2
2mA 2mA √
2 2 2 2 λ(m2 ,m2 21 2 c (m ,m ,m ) 1 A B,mC)
= eff A B C λ(m2 2 2
2m d µ2 A
,mB,mC)
A A 8π m2A
c2eff(m
2
A,m
2
B,m
2
= C
)
(λ(m2 23 A,mB,m
2))3/2C (IX.13)dA8πµ2mA
for the triangle coupling.
96
A. DECAY WIDTHS AND CROSS SECTIONS
A.1 Production Cross Section for Axions/ALPs
In order to get predictions for hadron colliders like the LHC, the production cross section
for an axion/ALP has to be evaluated. Since for high energies the parton distribution
function (PDF) of the gluon gives the largest contribution in proton-proton collision, the
partonic cross section for two gluons in the initial state has to be calculated. The full
process would be gg → a → A1A2. This is displayed in Figure IX.1.
g1µ A1ρ
pg1 p1
p
a
pg2 p2
g2ν A2σ
Figure IX.1: Feynman diagram for the s-channel cross section of two incoming gluons
g1 and g2 with momenta pg1 and pg2 and two outgoing gauge bosons A1 and A2 with
momenta p1 and p2. Inbetween an axion/ALP a with momentum p is produced. The
vertices are the effective couplings from section V.2.
The axion/ALP can be treated as an unstable particle with the total decay width being
much smaller than its real pole mass Γtot ≪ mP . These assumptions lead to an imaginary
part in the propagator
i
iG(p2) = 2 . (IX.14)p2 −mP + imPΓtot
In the squared matrix element this gives a Breit-Wigner distribution, which can be ap-
proxi∣mated for Γtot ≪ mP∣ by the so-called narrow-width approximation∣∣∣ i ∣∣∣2 1 π= 22 2 ≈ δ(p −m2P ). (IX.15)p2 −m 2 2 2P + imPΓtot (p −mP ) + (mPΓtot) mPΓtot
The matrix element for the process in Figure IX.1 is
iM igg→A1A2 = iMgg→a iM2 − 2 a→A1A2
(IX.16)
p ma + imaΓtot
and, therefore, the squared matrix element gives
|Mgg→A1A2 |2 ≈ |M 2
π 2 2 2
gg→a| δ(p −m )|Ma→A A | . (IX.17)
m Γ a 1 2a tot
Similar to the decay width, the cross section of a process AB → F with A,B being the
initial particles and F being the final state is defined∫ by
σ(AB → F 1) = dΠ |M |2, (IX.18)
2EA2EB|v⃗A − v⃗B|
F AB→F
97
APPENDIX
with v⃗ ≡ p⃗/E. Hence, the partonic cross section for gg → a → A1A2 is given by
1
σ̂(gg → A1A2) ≈
2∫Eg12Eg2 |v⃗g1 − v⃗g2 |
π
dΠ2|M 2gg→a| δ(p2 −m2a)|M 2a→A1A2 | . (IX.19)maΓtot
The external gluons/gauge bosons are on-shell, while the internal axion/ALP can be taken
to be on-shell as well due to the δ-function. With this,
p2g = p
2 2
g = 0, p = m
2 2
a, p1 = m
2 2 2
1 2 1, p2 = m2, (IX.20)
the squared matrix elements do not depend on the momenta anymore and the partonic
cross section can be re-expressed using equation (IX.13) by
σ̂(gg → A1A2) ≈
1 |M |2 Γ(a → A1A2)gg→a 2πδ(p2 −m2)
2Eg12E
a
g2 |v⃗g1 − v⃗g2 | Γtot
:= σ̂(gg → a)BR(a → A1A2), (IX.21)
where the branching ratio of the decay a → A1A2 is defined by
Γ(a → A1A2)
BR(a → A1A2) = . (IX.22)
Γtot
The squared matrix elements of gg → a and a → gg are related in the following way
2
|M |2 1= |M |2 1 m= 2m 8π√ a πmagg→a a→gg a Γ(a → gg) = Γ(a → gg)
d2 162 λ(m2g a, 0, 0) 16
(IX.23)
with dg = 2(N
2
C − 1) = 2(32 − 1) = 16 for NC = 3 colors. Thus, the partonic production
cross section for an axion/ALP is given by
π2maΓ(a → gg)
σ̂(gg → a) = δ(ŝ−m2) (IX.24)
32Eg1Eg2 |v⃗g1 − v⃗
a
g2 |
≈ πBR(a → gg) (m Γ
2
a tot)
(IX.25)
32Eg1Eg2 |v⃗g1 − v⃗g2 | (ŝ−m2a)2 + (m 2aΓtot)
and, therefore, the total product∫ion cros∫s section for a proton-proton collision is1 1
σ(pp → a) = dx1 dx2fg(x1)fg(x2)σ̂(gg → a), (IX.26)
0 0
with fg(x) being the parton distribution function for the gluons. x denotes the Bjorken
variable and gives the fraction of the proton momentum P , which contributes in the cross
section via th√e gluons pg = xP . For the center-of-mass energy being much larger than the
proton mass s ≫ mp, the partonic squared center-of-mass energy is given by
ŝ = (x P + x P )2 = x2m2 + x2m2 + x x (s− 2m21 1 2 2 1 p 2 p 1 2 p) ≈ x1x2s. (IX.27)
In the center-of-mass frame of the protons (P1 = (E, p⃗) and P2 = (E,−p⃗)), the denomi-
nator in equation (IX.24) simplifies to ∣∣ ∣2p⃗ ∣ s
Eg1Eg2 |v⃗g1 − v⃗ | = x x 2 ∣ ∣ 2g2 1 2E ∣ ∣ ≈ x1x22E = x1x2 . (IX.28)E 2
Hence, the partonic cross section becomes
→ πBR(a → gg) (maΓ
2
tot)
σ̂(gg a) = . (IX.29)
16x1x2s (x 2 2 21x2s−ma) + (maΓtot)
Finally, we remark that the calculation for the Z ′ works analogously by interchanging the
scalar propagator by a vector propagator.
98

List of Figures
Chapter II
II.1 Expected QCD axion band in the axion mass vs. diphoton coupling param-
eter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Chapter IV
IV.1 Deviation from the canonical axion mass relation in the anarchic axion model 35
IV.2 Deviations from the canonical DFSZ axion line in the {ma, Gaγγ} plane . . . 36
IV.3 Deviation from the canonical axion mass relation for an instanton induced
additional PQ breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
IV.4 Deviation from the canonical axion mass relation for an instanton induced
additional PQ breaking with Nelson-Barr extension . . . . . . . . . . . . . . . 43
Chapter V
V.1 One-loop diagrams for anomalous coupling of an axion to two gauge bosons . 51
V.2 One-loop diagrams for coupling of an axion to a gauge boson and a scalar . . 53
V.3 One-loop diagrams for anomalous coupling of an axion to two fermions . . . . 55
Chapter VI
VI.1 Feynman diagrams for axion couplings to SM fermions. . . . . . . . . . . . . 58
VI.2 Graphical determination of the freeze-out temperature TS for different DM
masses mS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
VI.3 Feynman diagrams for DM matter pair annihilation into SM particles . . . . 64
VI.4 DM direct detection limits on DM-nucleon cross sections with chirally en-
hanced quark couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
VI.5 DM direct detection limits on DM-nucleon cross sections with chirally en-
hanced lepton couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
VI.6 Chiral enhancement of axion couplings to quarks . . . . . . . . . . . . . . . . 68
VI.7 Chiral enhancement of axion couplings to leptons . . . . . . . . . . . . . . . . 69
VI.8 Feynman diagrams for effective couplings induced by heavy right-handed neu-
trinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter VII
VII.1 Branching ratios of two-body decays of the Z ′ gauge boson . . . . . . . . . . 84
100
LIST OF FIGURES
VII.2 Excluded regions in the {mZ′ , gB} parameter space from narrow resonance
searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
VII.3 Branching ratios of two-body decays of the axion . . . . . . . . . . . . . . . . 86
VII.4 Excluded regions in the {m , f−1a a } parameter space from narrow resonance
searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
VII.5 Implications of an additional U(1)B gauge symmetry in the {ma, Gaγγ} pa-
rameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Appendix
IX.1 Feynman diagram for the s-channel cross section of gg → A1A2 . . . . . . . . 97
101
List of Tables
Chapter II
II.1 Overview of SM particles and light QCD bound states . . . . . . . . . . . . . 7
II.2 Fermionic and scalar field content of the SM. . . . . . . . . . . . . . . . . . . 11
II.3 Fermionic and scalar field content of the KSVZ model . . . . . . . . . . . . . 16
II.4 Fermionic and scalar field content of the DFSZ model . . . . . . . . . . . . . 17
Chapter IV
IV.1 Field content of the Nelson-Barr extension in the anarchic axion model . . . . 40
Chapter VI
VI.1 Field content of the DFSZ model with an additional heavy fermion F and a
DM scalar S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
VI.2 Field content of the DFSZ model with three additional right-handed neutrinos
νiR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Chapter VII
VII.1 Field content for a DFSZ-like model with gauged baryon number . . . . . . . 74
102
List of Abbreviations
ΛCDM Standard Model of cosmology
ALP axion-like particle
BBN BigBang Nucleosynthesis
BR branching ratio
BSM beyond the Standard Model
C charge conjugation
CDM cold dark matter
CKM Cabibbo-Kobayashi-Maskawa matrix
CMB cosmic microwave background
DFSZ Dine-Fischler-Srednicki-Zhitnitsky
DM dark matter
EDM electric dipole moment
EFT effective field theory
EWSB electroweak symmetry breaking
GR general relativity
KSVZ Kim-Shifman-Vainshtein-Zakharov
P parity inversion
PDF parton distribution function
PMNS Pontecorvo–Maki–Nakagawa–Sakata matrix
PQ Peccei-Quinn
QCD quantum chromodynamics
QED quantum electrodynamics
QFT quantum field theory
SCPV spontaneous CP violation
SM Standard Model of particle physics
SSI small size instantons
UV ultraviolet
vev vacuum expectation value
WIMP weakly interacting massive particles
103
List of Experiments
CERN - Conseil europeén pour la recherche nucléaire
LHC - Large Hadron Collider since 2010
– ATLAS - A Toroidal LHC Apparatus
– CMS - Compact Muon Solenoid
LEP - Large Electron-Positron Collider 1989 - 2000
– ALEPH - Apparatus for LEP Physics
– L3 - Third LEP Experiment
Axion Experiments
Haloscopes
– ADMX - Axion Dark Matter Experiment
– ABRACADABRA - A Broadband/Resonant Approach to Cosmic Axion
Detection with an Amplifying B-field Ring Apparatus
– SHAFT - Search for Halo Axions with Ferromagnetic Toroids
Helioscopes
– CAST - CERN Axion Solar Telescope
– IAXO - International Axion Observatory
LSW - Light-shining-through-wall
– ALPS - Any Light Particle Search
– CROWS - CERN Resonant WISP Search
104
LIST OF EXPERIMENTS
Further Experiments
Dark Matter Direct Detection
– LUX-ZEPLIN - Collaboration of Large Underground Xenon Experiment and
Zoned Proportional Scintillation in Liquid Noble Gases
– XENONnT
Neutrino Experiments
– Super-Kamiokande - Super-Kamioka Neutrino Detection Experiment
– MiniBooNE - Mini Booster Neutrino Experiment
105

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