MSc PHYSICS AND ASTRONOMY GRAVITATION, ASTRO-, AND PARTICLE PHYSICS MASTER THESIS

Linking the B-physics Anomalies and Muon g − 2 A Phenomenological Study Beyond the Standard Model

By

Anders Rehult 12623881

February - October 2020 60 EC

Supervisor/Examiner: Second Examiner: prof. dr. Robert Fleischer prof. dr. Piet Mulders

i

Abstract

The search for physics beyond the Standard Model is guided by anomalies: discrepancies between the theoretical predictions and experimental measurements of physical quantities. Hints of new physics are found in the recently observed B-physics anomalies and the long-standing anomalous magnetic dipole moment of the muon, muon g − 2. The former include a group of anomalous measurements related to the quark-level transition b → s. We investigate what features are required of a theory to explain these b → s anomalies simultaneously with muon g − 2. We then consider three kinds of models that might explain the anomalies: models with leptoquarks, hypothetical particles that couple leptons to quarks; Z0 models, which contain an additional fundamental force and a corresponding force carrier; and supersymmetric scenarios that postulate a symmetry that gives rise to a partner for each SM particle. We identify a leptoquark model that carries the necessary features to explain both kinds of anomalies. Within this model, we study the behaviour of 0 muon g − 2 and the anomalous branching ratio of Bs mesons, bound states of b and s quarks, into muons. 0 ¯0 0 ¯0 The Bs meson can spontaneously oscillate into its antiparticle Bs , a phenomenon known as Bs −Bs mixing. We calculate the effects of this mixing on the anomalous branching ratio. We then identify parameter space 0 ¯0 that explains the anomalous branching ratio and muon g − 2 within Bs − Bs constraints. Furthermore, we study the potential for CP violation, the breaking of a discrete symmetry of our universe, subject to the same constraints and find that sizeable mixing-induced CP violation is possible within the model.

Acknowledgements

First of all, I would like to thank Prof. Dr. Robert Fleischer, whose corrections and nudges have repeatedly pulled me out of the mud and refocused my aim. Your guidance and expertise have been invaluable in sharpening my understanding of particle physics phenomenology and all that comes with it, and I greatly look forward to continuing to learn from your mentorship during the years ahead.

I would like to thank all the people at Nikhef and elsewhere who have been generous with their time and help. Thank you to Dr. Melissa van Beekveld, Dr. Anastasiia Filimonova, Eleftheria Malami, and Ruben Jaarsma for your guidance and technical support. Thank you to Dr. Darren Scott for never turning down an opportunity for discussion. It is funny how you can learn more from thirty minutes spent in front of a blackboard than from three hours behind a computer screen.

I would like to thank Prof. Dr. Bob van Eijk, who ignited my interest in particle physics and inspired me to take the leap from engineering into physics. Without you, I would not be anywhere near Nikhef, I would not have gone to CERN, and I would not be studying particle physics. Thank you for all these things.

The past half year has been a peculiar time, not least for carrying out a research project. I cannot help but think how difficult working remotely would have been thirty years ago, and I suppose I should thank the staff of Zoom for providing the means to do so. I could not have written this thesis without the support of the people around me. I would like to thank my family for keeping me sane back home during the first three months after COVID-19 hit. Without you, I would have been climbing my walls within weeks. Finally, I would like to thank Jonathan Ågren. Our frequent conversations routinely remind me of why I study physics. Particularly during the final stretch of this work, your drive has fuelled my drive. When a candle falters, another candle burning bright is all that is needed to light its flame again. ii Contents

1 Introduction 1

1.1 Conventions ...... 3

2 Quantum field theory and the Standard Model 5

2.1 Quantum electrodynamics ...... 5

2.2 Electroweak theory ...... 7

2.3 Quantum chromodynamics ...... 8

2.4 Generating masses—the Higgs mechanism ...... 9

2.5 Field content of the Standard Model ...... 13

3 Flavour physics 15

3.1 The CKM matrix ...... 15

3.2 CP violation ...... 18

3.3 B physics ...... 20

4 Effective field theory 21

4.1 Operator product expansion ...... 21

4.2 Calculating Wilson coefficients ...... 22

4.3 Probing new physics through effective field theory ...... 23

0 ¯0 5 Bs − Bs mixing 25

5.1 Mixing amplitude and phase ...... 25

5.2 Theoretical vs. time-integrated branching ratios ...... 26

6 The B-physics anomalies 29

iii iv CONTENTS

6.1 Lepton flavour universality violation ...... 29

+ − 6.2 The rare decay Bs → µ µ ...... 31

6.3 CP asymmetries ...... 34

7 Muon g − 2 35

7.1 Classical to quantum ...... 35

7.2 Radiative corrections ...... 37

8 Model-dependent analysis 39

8.1 Leptoquarks ...... 39

8.2 The scalar leptoquark S1 ...... 40

8.3 Calculation of Wilson coefficients ...... 41

+ − 8.4 Calculation of the theoretical branching ratio of Bs → µ µ and muon g − 2 ...... 42

8.5 Calculation of Aµµ and the CP asymmetries S and C ...... 44 ∆Γs µµ µµ

0 ¯0 8.6 Bs − Bs mixing constraints ...... 44

8.7 Results for Aµµ and S ...... 46 ∆Γs µµ

+ − 8.8 Results for the time-integrated branching ratio of Bs → µ µ ...... 47

8.9 Z0 models...... 51

8.10 Supersymmetry ...... 53

9 Conclusions 55

Appendices 57

A Standard Model Lagrangian 59

B SARAH and SPheno 61

C SARAH model file for the S1 scalar leptoquark model 67

References ...... 69 1 | Introduction

One does not need to go far back in time to find an understanding of fundamental physics radically different from ours. When Dalton hypothesized and subsequently discovered the building blocks of molecules in the early 1800s [1], he named them atoms, reintroducing a term meaning indivisible and coined by the Greek philosopher Democritus some two thousand years beforehand. After the discoveries of electrons, protons, and neutrons by Thomson1 [2], Rutherford [3], and Chadwick2 [4] respectively, it was clear that Dalton had jumped the gun by assigning this name to his newly-discovered particles. Following the discovery of quarks at SLAC3 [5], we know that a rich phenomenology is contained even within protons and neutrons. The contemporary understanding of the fundamental constituents of matter is as excitations of quantum fields, structures that permeate space and time. Our current knowledge of these fields, and of particles that arise from them, is encapsulated in the Standard Model (SM) of particle physics. With the 2012 discovery of the last missing piece of the SM, the Higgs boson4, by the ATLAS [6] and CMS collaborations [7], all particles of the SM are verified to exist.

The SM describes many phenomena with remarkable accuracy. However, it does not offer a complete description of experimentally observed phenomena. It does not explain dark matter, the substance that outweighs the “normal” matter described by the SM five times to one, making up 27% of the energy content of the observable universe as opposed to the 5% that normal matter accounts for [8]. Nor does it explain dark energy, the source of the acceleration of our universe’s expansion that makes up the remaining 68% [9]. It does not account for the observation of massive neutrinos [10], electrically neutral particles that interact very weakly with normal matter and are modelled as massless in the SM. In the SM, matter can only be created and annihilated in equal amounts with corresponding antimatter that carries opposite charges—e.g. electric charge—to its matter counterpart. Because of this, the SM does not explain the remarkable absence of antimatter in our observed universe [11]. Finally, the SM does not include a description of gravity, which becomes important for particle physics at very high energy scales, nor is it consistent with general relativity, the theoretical framework currently used to describe gravity at such scales [12]. Physics aimed at explaining these phenomena by extension of the SM is referred to as beyond the SM (BSM) physics, and any physics that is not part of the SM is called new physics (NP).

As the precisions of our theoretical predictions and experimental measurements have increased over the years, a new avenue of particle physics research has opened. In addition to trying to produce NP particles directly in experiments at the high-energy frontier, we can probe the potential effects of NP indirectly by studying low-energy processes with very high precision. In such processes, evanescent NP particles may pop in and out of existence as virtual particles, quantum fluctuations that cause small but measurable effects. This high-precision frontier poses a challenge for theorists and experimentalists alike, and its exploration is the motivation behind experimental endeavours such as the upcoming high-luminosity upgrade of the Large Hadron Collider (HL-LHC) at CERN planned for this decade [13]. To search for NP at the high-precision frontier, physicists look for where the experimental values of measurable quantities, observables, differ from

1Nobel prize 1906: Thomson 2Nobel prize 1935: Chadwick 3Nobel prize 1980: Friedman, Kendall, Taylor 4Nobel prize 2013: Englert, Higgs

1 2 CHAPTER 1. INTRODUCTION the values predicted by the SM. Such inconsistencies are known as anomalies and are smoking-gun signals of NP.

A promising field of study at the high-precision frontier is that of B physics. This field concerns the study of particles called B mesons, bound states of two quarks whereof one is a bottom quark. Certain decays of such particles into other, lighter particles are very sensitive to NP effects, and several anomalies commonly referred to as the B-physics anomalies have recently been observed in the decays of B mesons [14]. One of these anomalies, a recent addition stemming from measurements by the LHCb, ATLAS, and CMS collaborations 0 ¯0 published in 2020 [15], is related to the decays of electrically neutral B mesons called Bs and Bs into two muons, the “heavier brother of the electron”. The fact that muons play a part in these decays raises questions about possible links to another, long-standing anomaly: the anomalous magnetic dipole moment of the muon denoted by muon g − 2, pronounced “muon G minus two” [16]. The theoretical precision of the observable in question has only in 2020 reached the level of experiment [17], and physicists eagerly await results from the ongoing “Muon g − 2” experiment at Fermilab [18] that may smooth out the anomaly as a statistical fluctuation or cement it as an unambiguous signal of NP.

For an anomaly to be considered unambiguously discovered, the statistical significance of the anomalous measurement needs by convention to exceed 5σ, “five-sigma”. This corresponds, assuming a normal distribu- tion of observable values, to a p-value of 3 × 10−7. Equivalently, if one assumes the SM to perfectly describe some physical process and a large series of measurements of an observable within that process converge on an anomalous value, the chance of those measurements returning the observed value or a more extreme one should be lower than about 1 in 3.5 million. The B-physics and muon g − 2 anomalies currently exist at the levels of 2.5 − 4σ. Confirming or disproving these anomalies by increasing theoretical and experimental precision is the aim of much current effort in particle physics. If any anomaly is confirmed beyond doubt, the SM needs to be reworked to accommodate it. The BSM theory that would extend the SM would be heavily constrained; it would need to not only explain the anomaly but also be consistent with experimental data for all observables that are not in conflict with SM predictions. Considering multiple anomalies at once allows one to further constrain BSM theories; if several anomalies are unambiguously confirmed, a candidate BSM theory should ultimately explain all of them.

In this thesis, we examine what features of BSM theories are needed to explain some of the B-physics anomalies simultaneously with muon g − 2, focusing on the class of B-physics anomalies containing the 0 ¯0 decays of Bs and Bs into muons. We then study three categories of models that carry the necessary properties to explain both kinds of anomalies: models with leptoquarks, particles that connect two specific species of SM particles; models with an additional fundamental force and a corresponding force carrier called the Z0 particle; and supersymmetric models, models that postulate a symmetry that gives rise to a partner for each SM particle. For a promising leptoquark model, we calculate observables related to the decays of 0 ¯0 neutral B mesons, taking into account effects of the spontaneous oscillation of Bs mesons into Bs mesons, a 0 ¯0 phenomenon known as Bs −Bs mixing. We also study potential effects of the leptoquark on CP violation, the breaking of a discrete symmetry of our universe. Such breaking is necessary to explain the matter-antimatter asymmetry [19], and the level to which the symmetry is observed to be broken puts constraints on BSM theories.

This thesis is structured as follows. We introduce quantum field theory and outline the SM in Chapter 2, covering the key features relevant to this thesis. In Chapter 3 we discuss CP violation and flavour physics, 0 ¯0 the study of the weak interaction that enables e.g. the decays of B mesons and Bs − Bs mixing. In Chapter 4, we introduce effective field theory, a mathematical framework used to parameterize the effects of NP at low 0 ¯0 energies and an important tool for the exploration of the high-precision frontier. Chapter 5 covers Bs − Bs mixing and introduces several relevant observables. In Chapters 6 and 7, we present the B-physics anomalies and muon g − 2. In chapter 8, we discuss how leptoquarks, Z0 particles, and supersymmetry might explain the considered anomalies and study a specific leptoquark model in depth. Finally, we conclude in Chapter 9. 1.1. CONVENTIONS 3

1.1 Conventions

Throughout this thesis,

• We use natural units where ~ = c = 1. • Spacetime coordinates are denoted by x ≡ (t, ~x), where t and ~x denote temporal and spatial coordinates. • We use the following signature of the Minkowski spacetime metric:

gµν = diag(+1, −1, −1, −1).

• We use the Einstein summation convention for repeated spacetime indices, i.e.

µ 0 1 2 3 a bµ = a b0 − a b1 − a b2 − a b3

∂ • We use a shorthand notation for partial derivatives, ∂µ ≡ ∂xµ . • The Dirac gamma matrices are defined according to the Dirac-Pauli representation as

I 0   0 τ i 0 I γ0 ≡ , γi ≡ , i = 1, 2, 3, γ5 ≡ iγ0γ1γ2γ3 = , 0 −I −τ i 0 I 0

with I being the 2x2 identity matrix and with the Pauli matrices

0 1 0 −i 1 0  τ 1 = , τ 2 = , τ 3 = . 1 0 i 0 0 −1

• We define the adjoint Dirac spinor as ψ¯ ≡ ψ†γ0, where ψ† denotes the hermitian conjugate—the transposed complex conjugate—of ψ. • Dirac spinors are decomposed into chiral components by

ψ = PLψ + PRψ = ψL + ψR,

where 1 1 P ≡ (1 − γ ),P ≡ (1 + γ ) L 2 5 R 2 5 are the chiral projection operators.

µ • We employ Feynman slash notation for four-vectors, a/ ≡ γµa . 4 CHAPTER 1. INTRODUCTION 2 | Quantum field theory and the Stan- dard Model

The Standard Model (SM) is the flagship theory of particle physics. It is a quantum field theory (QFT), merging the short-distance effects of quantum mechanics with the high-velocity effects of special relativity. A QFT treats fundamental particles as excitations of fields that permeate spacetime. These fields are functions of spacetime coordinates x. The dynamics, masses, and interactions of fields are encoded in Lagrangian densities L, hereafter referred to as Lagrangians. From a Lagrangian, one can construct mathematical expressions for observables. Particles in the SM can be grouped in many ways, but the only division needed for now is the separation of bosons, particles of integer spin, from fermions, particles of half-integer spin. The SM models the electromagnetic, weak, and strong forces. In this chapter, we present the field and particle content of the SM before reviewing how the fundamental forces are incorporated in the SM and how particle masses are generated through the Higgs mechanism.

In QFT, forces are mediated by the exchange of spin-1 particles known as gauge bosons. The electrostatic repulsion between two electrons as illustrated in Fig. 2.1, for example, is in the SM mediated by the emission and absorption of photons by the electrons. Particles exchanged in this way are virtual particles, meaning that they for do not necessarily fulfill the energy-momentum relation E2 −|~p|2 = m2. Such particles can only 1 1 exist for the very short times allowed by Heisenberg’s uncertainty principle [20] ∆t ≥ 2∆E , where ∆ refers to the uncertainty with which a quantity is known. Fig. 2.1 is an example of a Feynman diagram. Such diagrams are not only helpful visualizations but also serve as computational aids; they can be translated into mathematical expressions for transition amplitudes that give the probability of the interaction. Feynman diagrams are drawn in momentum space, and by convention time runs from left to right. In Feynman diagrams, fermions are denoted by straight lines adorned with arrows. Arrows that face to the right denote particles, and arrows facing to the left denote antiparticles of opposite charge quantum numbers, more loosely known as charges, to their matter counterparts. The meaning of the charge quantum numbers will be made clear in this chapter.

2.1 Quantum electrodynamics

A central concept in QFT is that of gauge invariance: the invariance of Lagrangians under symmetry group transformations. To illustrate this concept, we construct the Lagrangian of quantum electrodynamics2 (QED), the chronologically first quantum field theory and one that models electromagnetism. We start from the Dirac Lagrangian for a free fermion3,

1Nobel prize 1932: Heisenberg 2Nobel prize 1965: Tomonaga, Schwinger, Feynman 3Nobel prize 1933: Schrödinger, Dirac

5 6 CHAPTER 2. QUANTUM FIELD THEORY AND THE STANDARD MODEL

e e

γ

e e

Figure 2.1: Feynman diagram illustrating the electromagnetic repulsion between two electrons. The electromagnetic force is mediated by the exchange of a virtual photon.

L(x) = ψ¯(x)(i∂/ − m)ψ(x), (2.1) where ψ(x) is a fermion field and m the mass of the corresponding particle. If a Lagrangian is to describe QED, it should be invariant under U(1) gauge transformations, where U(1) denotes all unitary matrices of dimension 1 [21]. Equivalently, the Lagrangian should be invariant under multiplication of the field ψ with any complex number of absolute value 1. Such gauge transformations are either global or local, depending on whether the transformation is independent of spacetime location. Global U(1) gauge transformations take the form

ψ(x) → eiQαψ(x), ψ¯(x) → e−iQαψ¯(x), (2.2) where α and Q are real constants, Q being the generator of U(1). The Lagrangian of Eq. (2.1) is already invariant under this transformation. If we impose a local gauge transformation, however, of the form

ψ(x) → eiQα(x)ψ(x), ψ¯(x) → e−iQα(x)ψ¯(x), (2.3) where α(x) depends on spacetime location, the Lagrangian of Eq. (2.1) acquires an extra term under the transformation,

L(x) → L0(x) = L(x) − ψ¯(x)∂α/ (x)ψ(x). (2.4)

To ensure the vanishing of this extra term, we replace the derivative ∂µ by the so-called covariant derivative Dµ, defined as

∂µ → Dµ ≡ ∂µ + iQAµ(x), (2.5)

where Aµ is a new field that transforms as

1 A (x) → A0 (x) = A (x) − ∂ α(x). (2.6) µ µ µ Q µ

Substituting the covariant derivative Dµ for ∂µ in the Lagrangian of Eq. (2.1), we obtain the gauge-invariant Lagrangian 2.2. ELECTROWEAK THEORY 7

L(x) = ψ¯(x)(iD/ − m)ψ(x) (2.7) ¯ ¯ µ = ψ(x)(i∂ψ/ (x) − m) − QAµ(x)ψ(x)γ ψ(x). (2.8)

We have in the process of constructing a gauge invariant Lagrangian added a massless gauge field to our theory. This field is the electromagnetic vector potential, i.e. the photon field that mediates the electromag- netic interaction.4 We can identify Q as the electric charge of the field ψ. To complete the QED Lagrangian, we need to add one final term: the kinetic term for the photon field that leads to the Maxwell equations in vacuum,

kinetic 1 µν L (x) = − Fµν (x)F (x), (2.9) Aµ 4 with the electromagnetic field strength tensor Fµν ≡ ∂ν Aµ(x)−∂µAν (x). Adding this term to the Lagrangian of Eq. (2.8) and making spacetime dependencies of fields implicit by denoting ψ ≡ ψ(x) etc., we finally end up with

¯ 1 µν LQED = ψ(iD/ − m)ψ − FµµF 4 (2.10) 1 = ψ¯(i∂/ − m)ψ − QA ψγ¯ µψ − F F µν . µ 4 µµ

This is the QED Lagrangian. It contains contains one massive fermion and one vector (spin-1) boson and is invariant under global and local U(1) gauge transformations.

2.2 Electroweak theory

The SM models not only the electromagnetic force but also the weak and strong forces. To discuss the weak force properly, we must first introduce the concepts of charge conjugation, parity, and helicity. The charge conjugation (C) operation reverses the sign of the charges of a field, turning the corresponding particle into its antiparticle. The parity (P) transformation is an operation that multiplies all spatial coordinates of a field by -1, turning ψ(t, ~x) into ψ(t, −~x) and vice versa. It is geometrically equivalent to the combination of a reflection in a plane through the origin and a 180-degree rotation around an axis through the origin, perpendicular to that plane. Helicity is defined as the projection of a particle’s spin onto its momentum. This definition separates massless particles, which travel at the speed of light, into two different kinds: those of negative helicity (left-handed particles) and those of positive helicity (right-handed particles). Left-and right-handed particles transform into one another through parity transformations. For a massive particle, one can always choose a reference frame traveling faster than the particle, reversing its helicity. In such cases, an innate quality of particles called their chirality determines their “preferred” helicity, meaning the helicity the particle would have if it were massless.

The electromagnetic and strong forces conserve C- and P-symmetry. As the weak force mediates interactions between fields of different charges, it violates C-symmetry. In 1956, Lee and Yang hypothesized that the weak force could also violate P-symmetry [22]. Half a year afterwards, Wu et al. confirmed this hypothesis in experiment5 [23]. The weak force violates parity “maximally” in that the gauge boson W couples only to left-handed particles and right-handed antiparticles. For this reason, as can be seen in Table 2.1 left-handed

4 The photon field of the SM is slightly different than this; it is a mixture of two fields owing to the breaking of SU(2)L ⊗U(1)Y symmetry by the Higgs boson as discussed later in this chapter. 5Nobel prize 1957: Lee, Yang 8 CHAPTER 2. QUANTUM FIELD THEORY AND THE STANDARD MODEL fermions are in the SM grouped in SU(2) doublets, meaning states that transform into one another through SU(2) transformations, while right-handed fermions are SU(2) singlets. For the same reason, the gauge group of the weak force is termed SU(2)L, where L denotes the “left-handedness” of the force. The S of SU(2) means “special” and denotes matrices of determinant 1. The invariance of a Lagrangian under SU(2) transformations means that the Lagrangian stays the same when the fields in it are multiplied by any unitary 2 × 2 matrix of determinant 1.

It has been shown [24, 25] that the electromagnetic and weak forces are two aspects of the same underlying, 6 fundamental electroweak force . This force is associated with the combined symmetry SU(2)L ⊗U(1)Y . The index Y denotes hypercharge, the charge quantum number and generator of the new gauge group U(1)Y . We introduce hypercharge because in the unification of the electromagnetic and weak forces, the electric 1 charge Q is no longer identical to the generator of a gauge group; instead, it is found to be Q = T3 + 2 Y , with T3 being the third component of weak isospin, a quantum number of SU(2)L. This unification also means that the gauge field of U(1)Y is not identical to the photon. Instead, we call the gauge field Bµ, and looking ahead, this field will mix with the gauge fields of SU(2)L to form the photon and the intermediate vector bosons W and Z7. The electroweak Lagrangian before EWSB can be constructed analogously to the QED Lagrangian by starting from the Dirac equation and imposing global and local gauge invariance under SU(2)L ⊗ U(1)Y . It is given by [26]

1 1 L = ψ¯ iD/ ψ + ψ¯ iD/ ψ − W a W a,µν − B Bµν , (2.11) EW L L L R R R 4 µν 4 µν with the electroweak field strength tensors

a a a b c W = ∂µW − ∂ν W − gabcW W , µν ν µ µ ν (2.12) Bµν = ∂µBν − ∂ν Bµ.

The chiral covariant derivatives Dµ,L,Dµ,R are given by

i i D = ∂ + g~τ · W~ + g0Y B , µ,L µ 2 µ 2 ψL µ (2.13) 0 Dµ,R = ∂µ + ig YψR Bµ,

a where YψH (H = L, R) is the hypercharge of the fermion field ψH . The fields Wµ (a = 1, 2, 3) are the three 0 gauge fields of SU(2)L, and the vector W~ contains all of them. g and g are electroweak coupling constants; ~τ is a vector containing the three Pauli matrices, the generators of SU(2)L; and abc is the Levi-Civita symbol.

2.3 Quantum chromodynamics

The final force described in the SM is the strong force. It is mediated by gluons, massless gauge bosons that couple to particles carrying colour charge. There are six different quantum numbers of colour charge known as red, antired, green, antigreen, blue, and antiblue. Under charge conjugation, the colour(s) of a particle will turn into the anticolour(s) of the corresponding antiparticle. In the SM, the only particles that carry colour charge are quarks and the gluons themselves. The strong force binds quarks together through the exchange of virtual gluons into composite, colourless states known as hadrons. Hadrons consisting of two quarks—a quark and an antiquark of matching colour-anticolour charges—are called mesons, and hadrons comprising three (anti-)quarks of different (anti-)colour charges are called (anti-)baryons. At all energy scales other than

6Nobel prize 1979: Glashow, Weinberg, Salam 7Nobel prize 1984: Rubbia, van der Meer 2.4. GENERATING MASSES—THE HIGGS MECHANISM 9 extremely high ones, quarks cannot escape being bound in hadrons; as the distance between quarks increases, the strong force that binds them together grows stronger, a phenomenon known as confinement. However, at small distance scales such as within hadrons, quarks behave almost as free particles, a phenomenon known as asymptotic freedom8 [27, 28]. The framework that describes the strong force is called quantum chromodynamics (QCD), and the QCD Lagrangian can be written as [29]

1 L = ψi¯ D/ ψ − F a F a,µν , (2.14) QCD QCD 4 µν,QCD QCD with the covariant derivative

λa D = ∂ + ig Ga (2.15) µ,QCD µ s 2 µ and the strong field strength tensor

a a a abc b c Fµν,QCD = ∂µGν − ∂ν Gµ − gsf GµGν . (2.16)

a a Here, λ are the eight Gell-Mann matrices, the generators of SU(3)c [29]. Gµ are eight vector fields, the gluons of different colour-anticolour combinations. f abc are real numbers called the structure constants of QCD that appear in the commutators of Gell-Mann matrices.9

The incorporation of the electroweak and strong forces requires the SM Lagrangian to be invariant under the SM gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y . The charge quantum numbers of this gauge group are colour charge c, the third component of weak isospin T3, and hypercharge Y . The full SM Lagrangian includes aspects of QED, electroweak theory, and QCD and can be found in Appendix A.

2.4 Generating masses—the Higgs mechanism

The QED Lagrangian of Eq. (2.10) describes a massive fermion field and a massless “photon” field. It contains a mass term for the fermion field as well as kinetic and interaction terms for both fields. For a Lagrangian that needs only be invariant under a U(1) gauge group, this is all well and good. However, the SM Lagrangian needs to be invariant under the full SM gauge group. The kinetic and interaction terms of the QED Lagrangian can easily be made so by slightly modifying the covariant derivative to decompose into those of Eq. (2.13). However, mass terms of vector bosons and fermions constitute a problem as they are not invariant under SU(2)L ⊗ U(1)Y . For vector bosons, this can be seen by trying to add a mass term for the “photon” field Aµ of the QED Lagrangian. Such a mass term takes the form

1 m2 A Aµ (2.17) 2 γ µ and transforms under U(1)Y as

1 1 1 1 1 m2 A Aµ → m2 (A + ∂ α)(Aµ + ∂µα) 6= m2 A Aµ. (2.18) 2 γ µ 2 γ µ Q µ Q 2 γ µ

8Nobel prize 2004: Gross, Politzer, Wilczek 9 The fact that the Gell-Mann matrices, the generators of SU(3)c, do not commute identifies QCD as a non-abelian theory and leads to self-interaction between the gluons. The same holds for the Pauli matrices τi of the electroweak theory, causing self-interactions between the intermediate vector bosons W and Z. 10 CHAPTER 2. QUANTUM FIELD THEORY AND THE STANDARD MODEL

¯ Fermionic mass terms are of the form −mf ψψ. Such terms also break gauge invariance, something that can be seen by decomposing the fermion field ψ into helicity states:

¯ ¯ ¯ −mf ψψ = −mf (ψR + ψL)(ψL + ψR) ¯ ¯ (2.19) = −mf (ψRψL + ψLψR)

¯ † 0 † 0 1 ¯ 5 2 ¯ as ψRψR = ψ PRγ PRψ = ψ γ PLPRψ = 4 ψ(1−(γ ) )ψ = 0 and similarly for ψLψL. Since the left-handed SU(2) doublet ψL transforms under SU(2) rotations while the SU(2) singlet ψR does not, the term is not gauge invariant.

The solution comes in the form of the Higgs mechanism [30], less commonly the Englert-Brout-Higgs- Guralnik-Hagen-Kibble mechanism, named after the physicists of three teams who concurrently and in- dependently formalized the mechanism. The introduction of a complex scalar SU(2)L doublet φ, the Higgs doublet, with a carefully chosen potential can be used to construct gauge invariant bosonic and fermionic mass terms. The mechanism also causes gauge bosons to mix between one another after electroweak sym- metry breaking (EWSB). We cover EWSB here only briefly but describe the fermionic mass part in detail as the construction of quark mass terms is relevant to the discussion of flavour physics, the topic of the next chapter. See e.g. Ref. [31] for a pedagogical discussion of EWSB.

The Higgs doublet is a complex scalar SU(2)L doublet, SU(3)c singlet of hypercharge 1. It initially takes the form

 +   φ 1 φ1 + iφ2 φ = 0 = √ . (2.20) φ 2 φ3 + iφ4

The Lagrangian describing it is

† µ LHiggs = (Dµ,Lφ) (DLφ) − V (φ), (2.21) with the chiral covariant derivative of Eq. (2.13).

The potential term V (φ) is given by

V (φ) = µ2(φ†φ) + λ(φ†φ)2, µ2 < 0, λ > 0. (2.22)

This potential contains a four-point self-interaction term and a mass term with a seemingly imaginary mass. q −µ2 2 v2 Defining v ≡ λ , it has a local maximum at |φ| = 0 and local minima in the “valley” |φ| = 2 . At |φ| = 0, SU(2)L ⊗ U(1)Y symmetry is unbroken. For the local minima, however, the Higgs doublet acquires a vacuum expectation value (VEV) and the Higgs Lagrangian is no longer invariant under SU(2)L ⊗ U(1)Y ; electroweak symmetry is broken.10 We choose the VEV to be

1 0 h0|φ|0i = √ , (2.23) 2 v and gauge fixing to the unitary gauge to remove one degree of freedom, we end up with

10The physical significance of this is the prediction that above a critical temperature, the electromagnetic and weak forces are indistinguishable. Furthermore, as EWSB is needed to generate masses in the SM, above this temperature all particles are massless! 2.4. GENERATING MASSES—THE HIGGS MECHANISM 11

1  0  φ = √ , (2.24) 2 v + h where h is the neutral Higgs field.

Inserting the Higgs doublet of Eq. (2.24) into the potential term of Eq. (2.21), we obtain a real, positive mass for the Higgs boson. Inserting the same doublet into the kinetic term, we end up with mass terms for a the gauge bosons Bµ and Wµ as well as interaction terms between them and the Higgs boson. However, the mass terms contain linear combinations of the different gauge fields. To obtain mass terms that correspond + − to physical particles, we define new gauge fields Aµ, Zµ, Wµ , and Wµ as the linear combinations of Bµ and a Wµ needed to obtain well-defined mass terms. The new gauge fields are mass eigenstates identified as the photon, the Z-boson, and the two W -bosons. They take the form

3 Aµ = cos θW Bµ + sin θW Wµ , Z = − sin θ B + cos θ W 3, µ W µ W µ (2.25) 1 W ± = √ (W 1 ∓ iW 2), µ 2 µ µ where θW is the weak mixing angle, also called Weinberg angle, related to the weak coupling constants as cos θ = g . W pg2+g02

For fermions, we can construct gauge invariant Lagrangian terms called Yukawa terms by contracting the Higgs doublet with the left-handed fermion doublets. The Higgs doublet by construction carries exactly the right quantum numbers to achieve this. We add interaction terms between fermion fields and the Higgs doublet of the form

¯ ¯ ¯ ¯ − Yf ψφψ = −Yf [ψLφψR + ψRφψL]. (2.26)

The quantity Yf is called a Yukawa coupling and parameterizes the strength of the fermion-Higgs interaction. Terms of this form, in addition to describing fermion-Higgs interactions, generate masses for the fermions. To see how, consider the electron, ψL = LL, ψR = eR:

¯ Le = −Ye[L¯LφeR +e ¯RφLL]       1 0 νL = −Ye √ (¯νL, e¯L) eR +e ¯R(0, v + h) 2 v + h eL Ye(v + h) = − √ [¯eLeR +e ¯ReL] 2 (2.27) Y (v + h) = − e √ ee¯ 2 Y v Y = − √e ee¯ − √e hee,¯ 2 2 where the first term is an electron mass term and the second a three-point interaction term between electrons and the Higgs boson.

In the quark sector, the story is a bit more involved. Due to the shape of the QL doublet, terms of the type of Eq. (2.26) will only give mass to the down-type quarks. However, another type of gauge-invariant Yukawa term can be constructed with the help of the Pauli spin matrix 12 CHAPTER 2. QUANTUM FIELD THEORY AND THE STANDARD MODEL

0 −i τ = . (2.28) 2 i 0

The term has the shape

¯ ∗ − Yf ψiτ2φ ψ. (2.29)

We can now write out the full Yukawa Lagrangian consisting of all possible terms of the forms of Eqs. (2.26) and (2.29). Before EWSB, it takes the form

ij ¯I ij ¯I ∗ ij ¯I I LY ukawa = −Yd QLiφdRj − Yu QLiiτ2φ uRj − Ye LijφeRj, (2.30) where the indices i, j sum over all three quark and lepton generations and the index I indicates that we are working with the interaction eigenstates, also called flavour eigenstates, of the fields. We see that the I second term, of the form of Eq. (2.29), enables the up-type quark field of the QLi doublet to interact with the Higgs doublet, the latter conjugated to ensure gauge invariance. The third term will generate masses 11 and Higgs interactions for charged leptons. As all three quark and lepton generations are included, Yu, Yd, and Ye are 3 × 3 matrices. This leads, as will be shown below, to a mixing between quark and lepton flavours loosely similar to the mixing in the gauge boson sector.

After EWSB, the quark mass terms generated by the Yukawa Lagrangian are

quark masses ij ¯I v I ij I v I LY ukawa = −Yd dLi √ dRj − Yu u¯Li √ uRj + h.c. 2 2 (2.31) ij ¯I I ij I I = −Md dLidRj − Mu u¯LiuRj + h.c., where we have defined the mass matrices M ≡ √v Y and the notation + h.c. indicates the addition of f 2 f the hermitian conjugates of all terms in the equation. Generally, the mass matrices Md and Mu are not diagonal and will give rise to ill-defined mass terms involving quarks from different generations. To obtain well-defined mass terms corresponding to physical particles, we diagonalize the mass matrices through the introduction of the quark rotation matrices Uu, Ud, Vu, and Vd. These are unitary 3 × 3 matrices that by design diagonalize the mass matrices as

diag † diag † Md = UdMdVd ,Mu = UuMuVu . (2.32)

We now finally obtain well-defined quark mass terms by writing

quark masses ¯I ij I I ij I LY ukawa = −dLiMd dRj − u¯LiMu uRj + h.c. ¯I † ij † I I † ij † I = −dLiUd UdMd Vd VddRj − u¯LiUuUuMu Vu VuuRj + h.c. (2.33) ¯ ij ij = −dLi(Md )diagdRj − u¯Li(Mu )diaguRj + h.c., where in the last step we have absorbed the rotation matrices into the quark fields, defining the quark mass eigenstates

11Note that as no right-handed neutrino fields are part of the SM, no mass terms for neutrinos can arise! This is a known flaw of the SM as neutrinos have been observed to be massive. 2.5. FIELD CONTENT OF THE STANDARD MODEL 13

I I dLi = (Ud)ijd , dRi = (Vd)ijd , Lj Rj (2.34) I I uLi = (Uu)ijuLj, uRi = (Vu)ijuRj, which are linear combinations of the interaction eigenstates.

2.5 Field content of the Standard Model

The SM contains 12 fermions. For each fermion, an antiparticle exists of opposite charge quantum numbers. The fermions comprise quarks, which carry colour charge and therefore interact through the strong force, and leptons, which do not. The quarks fragment into six flavours: up (u), down (d), charm (c), strange (s), top (t), and bottom (b). u, c, and t are up-type quarks that share a common electric charge and are separated only by their masses. Similarly, d, s, and b are down-type quarks. The leptons also come in six flavours: electron (e), electron neutrino (νe), muon (µ), muon neutrino (νµ), tau (τ), and tau neutrino (ντ ). They are grouped into charged leptons and neutrinos. The SM also contains five gauge bosons, spin-1 particles that mediate the three fundamental forces described by the SM. These gauge bosons are the photon (γ) for the electromagnetic interaction, the intermediate vector bosons W ± and Z for the electroweak interaction, and the gluon (g) for the strong interaction. Finally, the scalar (spin-0) Higgs field is added to give masses to the other particles. The fields and particles of the SM are listed in Table 2.1.

Particles Fields Content Spin Electric charge SU(3)c ⊗ SU(2)L ⊗ U(1)Y

Quarks QL (u, d)L 1/2 (2/3, −1/3) 3 ⊗ 2 ⊗ 1/6 (3 generations) uR uR 1/2 2/3 3 ⊗ 1 ⊗ 2/3 dR dR 1/2 −1/3 3 ⊗ 1 ⊗ −1/3

Leptons LL (ν, e)L 1/2 (0, −1) 1 ⊗ 2 ⊗ −1/2 (3 generations) eR eR 1/2 −1 1 ⊗ 1 ⊗ −1 0 B boson Bµ Z , γ 1 0, 0 1 ⊗ 1 ⊗ 0 a ± 0 W bosons Wµ W ,Z , γ 1 ±1, 0, 0 1 ⊗ 3 ⊗ 0 b Gluons Gµ g 1 0 8 ⊗ 1 ⊗ 0 Higgs boson (φ+, φ0) h 0 0 1 ⊗ 2 ⊗ 1/2

Table 2.1: Field and particle content of the SM, including spin, electric charge, and representa- tion under the SM gauge group. The first two, bold numbers under the gauge groups refer to the multiplicity of states under that gauge group; e.g, the left-handed quarks QL are triplets under SU(3)c and doublets under SU(2)L. The final number denotes the U(1)Y hypercharge. 3 The fields Wµ and Bµ mix through EWSB into the mass eigenstates of the Z boson and the photon γ. 14 CHAPTER 2. QUANTUM FIELD THEORY AND THE STANDARD MODEL 3 | Flavour physics

Seemingly serving only to separate fermions by their masses, the existence of different fermion flavours puzzled physicists upon discovery. This perplexion is well summarized by Nobel laureate Isidor Rabi’s quip at the discovery of the muon, the “heavier brother of the electron”: “Who ordered that?” Now, we know that processes that change the flavours of fermions are promising grounds in the search for NP.

3.1 The CKM matrix

Out of the fundamental forces, the weak interaction, specifically the charged-current interaction mediated by the W bosons, is the only one observed to change the flavours of fermions. To see how the charged-current interaction changes quark flavours, we return to the electroweak Lagrangian of Eq. (2.11), focusing on the kinetic term for left-handed fermions

i i L = ψ¯I iD/ ψI = ψ¯I iγ (∂µ + g~τ · W~ µ + g0Y Bµ)ψI , (3.1) kinetic,L L L L L µ 2 2 ψL L

I Focusing on the interaction term containing the W bosons and inserting the SU(2)L quark doublet QiL summed over flavour indices i, we get

g Lquarks,W = − Q¯I γ ~τ · W~ µQiI int.,L 2 iL µ L iI g I u = − u d
γ ~τ · W~ µ iL µ d 2 L (3.2) g g g g = − u¯I γ W µdiI − d¯I γ W µuiI + i u¯I γ W µdiI − i d¯I γ W µuiI 2 iL µ 1 L 2 iL µ 1 L 2 iL µ 2 L 2 iL µ 2 L g g − u¯I γ W µuiI + d¯I γ W µdiI . 2 iL µ 3 L 2 iL µ 3 L

Ignoring the terms containing W 3 (these will go into neutral-current interaction terms) and switching to the W ± boson and quark mass eigenstates of Eqs. (2.25) and (2.34), we obtain the charged-current Lagrangian

15 16 CHAPTER 3. FLAVOUR PHYSICS

g I −µ iI g ¯I +µ iI LCC = −√ u¯ γµW d − √ d γµW u 2 iL L 2 iL L g I −µ iI = −√ u¯ γµW d + h.c. 2 iL L (3.3) g u d† −µ j = −√ u¯iL(U U )ijγµW d + h.c. 2 L L L g −µ j = −√ u¯iL(VˆCKM )ijγµW d + h.c. 2 L

ˆ u d† 1 The matrix VCKM ≡ (ULUL )ij is called the Cabibbo-Kobayashi-Maskawa (CKM) matrix [32, 33]. It is a unitary matrix that parameterizes the mixing between quark flavour and mass eigenstates caused by the charged-current interaction. By convention, the up-type flavour and mass eigenstates are chosen to be equal, while the down-type flavour eigenstates are “rotated” into mass eigenstates:

I u = uj, i (3.4) I ˆ di = VCKM dj, or, writing out the down-type quark flavours:

 I      d Vud Vus Vub d I s  = Vcd Vcs Vcb  s . (3.5) I b Vtd Vts Vtb b

The transition amplitude from an u to a d quark is proportional to Vud, while that of the reverse process is ∗ proportional to Vud. The absolute values of CKM matrix elements have been extracted from studies of many different decays as [34]

 +0.00011  0.97401 ± 0.00011 0.22650 ± 0.00048 0.00361−0.00009 ˆ +0.00083 VCKM = 0.22636 ± 0.00048 0.97320 ± 0.00011 0.04053−0.00061  . (3.6) +0.00023 +0.00082 +0.000024 0.00854−0.00016 0.03978−0.00060 0.999172−0.000035

We see that the diagonal elements are close to unity and that elements increasingly far off the diagonal are increasingly small. Nothing in the SM predicts the magnitudes of the CKM matrix elements. For this reason, this hierarchy of absolute values is unexpected, and theorists look for underlying reasons for the observed pattern. A way of parameterizing the CKM matrix elements that nicely illustrates the puzzling hierarchy of values is the Wolfenstein parameterization [35], expressed in terms of the parameters A ≈ 0.82, λ ≈ 0.23, ρ ≈ 0.14, and η ≈ 0.37 to leading order in λ as

 1 2 3  1 − 2 λ λ Aλ (ρ − iη) ˆ 1 2 2 VCKM =  −λ 1 − 2 λ Aλ  . (3.7) Aλ3(1 − ρ − iη) −Aλ2 1

The unitarity of the CKM matrix,

ˆ † ˆ ˆ ˆ † VCKM VCKM = VCKM VCKM = I, (3.8) 1Nobel prize 2008: Kobayashi, Maskawa, Nambu 3.1. THE CKM MATRIX 17

Figure 3.1: The “non-squashed” unitarity triangle of the CKM matrix at leading order. This triangle corresponds to either of the orthogonality relations (3.11) and (3.14). Adapted from Ref. [36] and modified. where I is the 3 × 3 identity matrix, gives rise to 12 equations whereof six are normalization relations and six are orthogonality relations. The orthogonality relations are [36]

∗ ∗ ∗ VudVus + VcdVcs + VtdVts = 0, (3.9) | {z } | {z } | {z } O(λ) O(λ) O(λ5) ∗ ∗ ∗ VusVub + VcsVcb + VtsVtb = 0, (3.10) | {z } | {z } | {z } O(λ4) O(λ2) O(λ2) ∗ ∗ ∗ VudVub + VcdVcb + VtdVtb = 0, (3.11) | {z } | {z } | {z } (ρ+iη)Aλ3 −Aλ3 (1−ρ−iη)Aλ3 ∗ ∗ ∗ VudVcd + VusVcs + VubVcb = 0, (3.12) | {z } | {z } | {z } O(λ) O(λ) O(λ5) ∗ ∗ ∗ VcdVtd + VcsVts + VcbVtb = 0, (3.13) | {z } | {z } | {z } O(λ4) O(λ2) O(λ2) ∗ ∗ ∗ VudVtd + VusVts + VubVtb = 0. (3.14) | {z } | {z } | {z } (1−ρ−iη)Aλ3 −Aλ3 (ρ+iη)Aλ3

Here, the leading-order terms of the CKM matrix element products in the Wolfenstein parameterization are given. The orthogonality relations can be expressed as six triangles of equal areas in the complex plane, the unitarity triangles of the CKM matrix. Of the orthogonality relations, only Eqs. (3.11) and (3.14) contain terms that are all of equal order in λ. As a result, only the unitarity triangles of these two relations have sides of comparable lengths. At leading order, these two relations are identical and yield the same unitarity triangle, often referred to as the unitarity triangle of the CKM matrix. This triangle, obtained after dividing the relation by Aλ3 to make one of the side lengths unity, is shown in Fig. 3.1.

In the current era of high precision, it is necessary to expand the CKM matrix elements to higher orders of λ. Expanding the matrix elements of Eq. (3.11) to next-to-leading order and defining

 1   1  ρ¯ ≡ ρ 1 − λ2 , η¯ ≡ η 1 − λ2 , (3.15) 2 2 18 CHAPTER 3. FLAVOUR PHYSICS

Figure 3.2: The two non-squashed unitarity triangles of the CKM matrix at next-to-leading order in λ. The two triangles correspond to the orthogonality relations of Eq. (3.11) and (??), respectively. The angle γ0 satisfies γ = γ0 + δγ. Adapted from Ref. [36]. one obtains

[(¯ρ + iη¯) + (−1) + (1 − ρ¯) − iη¯]Aλ3 + O(λ7) = 0. (3.16)

Doing the same for Eq. (3.14), one finds

 λ2   1     1 − − (1 − λ2)ρ − i(1 − λ2)η + − 1 + − ρ λ2 − iηλ2 + ρ + iη Aλ3 + O(λ7) = 0. (3.17) 2 2

After dividing these expressions by the normalization factor Aλ3 and introducing

p 2 2 p 2 2 Rb ≡ ρ¯ +η ¯ ,Rt ≡ (1 − ρ¯) +η ¯ , (3.18) one obtains the two distinct unitarity triangles of Fig. 3.2. The angle δγ ≡ λ2η = O(1◦) at which the second 0 ¯0 triangle is raised from the real axis plays a role in our discussion of Bs − Bs mixing in Chapter 5. Precise determination of the sides and angles of the unitarity triangles is important to constrain the complex phases of CKM matrix elements and therefore the amount of CP violation in the SM as will be discussed in the next section. The collaborations CKMfitter [37] and UTfit [38] have used experimental data for various B meson decays to create the contours of Fig. 3.3.

3.2 CP violation

A central theme in flavour physics is CP-violation, the breaking of invariance of Lagrangians under the combined transformation CP . After C and P violation was discovered in the weak interaction, it was long thought that physics was symmetric under CP . In 1964, however, the first conclusive evidence of CP violation in the weak interaction was found in neutral K-mesons2 [39], hadrons comprising a strange (s) and a down (d) quark, one of which being an antiquark. In 2001, CP violation was observed in neutral Bd

2Nobel prize 1980: Cronin, Fitch 3.2. CP VIOLATION 19

Figure 3.3: Contours in the ρ¯−η¯ plane by the CKMfitter [37] and UTfit [38] collaborations. The tiny circles where the contours overlap shows the allowed region for the apex of the unitarity triangle arising from Eq. (3.11).

(b, d) mesons by the Belle and BaBar collaborations [40, 41], in 2013 in neutral, strange Bs (b, s) mesons by the LHCb collaboration [42], and in 2019 in neutral D (c, u) mesons, again by the LHCb collaboration [43].3 CP violation is parameterized in the SM as complex phases in the CKM matrix; in the Wolfenstein parameterization, CP-violating effects are modelled by the imaginary iη. In BSM theories, couplings between NP and SM particles can in general be complex, leading to new sources of CP violation unrelated to the CKM matrix that need to be taken into account to gauge the consistency of the theories with experimental measurements.

CP violation is also a necessary criterion to explain the matter-antimatter asymmetry. In 1967, Sakharov postulated three conditions necessary for this asymmetry to exist [19]. As most non-dark matter is made up of baryons, the Sakharov conditions focus on these. The Sakharov conditions, and their motivations, are

• Baryon number violation. This is necessary to produce an excess of baryons over antibaryons. • C violation and CP violation. Without C violation, any process that produces baryons in excess over antibaryons would be accompanied by a process that produces excess antibaryons over baryons. Without CP violation, there would for each process producing left-handed baryons be one producing right-handed antibaryons. • Interactions out of thermal equilibrium. For processes in thermal equilibrium, CPT symmetry conserves baryon number.

CP violation can be split up into three categories: direct CP violation, mixing-induced CP violation, and CP violation in interference between a decay with and without mixing [44]. Direct CP violation is a difference between the decay rate Γ of a particle P into a final state f and the rate of the CP-conjugate decay of the

3It is now generally believed that physics is symmetric under the transformation CPT , where T denotes the time reversal operation T (ψ(t, ~x)) = ψ(−t, ~x). In fact, the CPT theorem states that any Lorentz-invariant QFT, i.e. any QFT in which the laws of physics are identical in different reference frames, necessarily obeys CPT symmetry [44]. Lorentz invariance is required for a QFT to be consistent with special relativity, being one of Einstein’s two postulates of the theory. 20 CHAPTER 3. FLAVOUR PHYSICS antiparticle P¯ into the antiparticle final state f¯: Γ(P → f) 6= Γ(P¯ → f¯). The decay rate Γ(P → f) is the probability per unit time that the particle P will decay into the final state f. Mixing-induced CP violation refers to when the oscillation of a neutral meson into its antiparticle is different from the reverse process: Prob(P 0 → P¯0) 6= Prob(P¯0 → P 0). Observables related to mixing-induced CP violation are discussed in Chapter 5. Finally, CP violation can occur in the decay of a neutral meson P 0 that shares a final state f with its antiparticle P¯0. The decay rate will then get contributions from two amplitudes, A(P 0 → f) and A(P 0 → P¯0 → f).

3.3 B physics

B mesons are mesons containing at least one bottom quark. A bottom quark can form a meson together with any antiquark other than the top quark, which has a very short lifetime and decays into other, lighter 0 ¯ ¯0 ¯ 0 ¯ ¯0 quarks before it can hadronize. The neutral B mesons are B (db), B (bd), Bs (sb), Bs (bs¯), and bottomonium states of different masses, which are excited states of bottom-antibottom (b¯b) quark pairs. The charged B + ¯ − + ¯ − mesons are B (ub) and B (bu¯), and the charmed B mesons are Bc (cb) and Bc (bc¯).

Rare decays of B mesons play a central role at the high-precision frontier. A rare decay is one that proceeds only at loop-level in the SM, meaning through Feynman diagrams involving at least one closed loop of particles. Such diagrams are suppressed relative to tree-level diagrams, diagrams containing no loops, because each interaction point between particles in a diagram, called a vertex, contributes a tiny factor to the amplitude of the diagram. Due to their suppression in the SM, rare decays of B mesons are very sensitive to any NP effects that lift their suppression. The suppression of a decay manifests as a very small branching ratios for that decay. A branching ratio can be thought of as the probability for a particle P to decay into a particular final state f and is defined as

Γ(P → f) BR(P → f) ≡ , (3.19) Γtot where Γtot is the sum of the decay rates for all possible final states that the particle P can decay into. The 0 + − ¯0 + − rare decays Bs → µ µ and Bs → µ µ are treated in detail in Section 6.2.

Neutral B mesons can spontaneously turn into their antiparticles as illustrated in Fig. 5.1. This phenomenon provides important constraints on any NP that contributes to decays of neutral B mesons and is discussed in detail in Chapter 5. 4 | Effective field theory

Effective theories are a pragmatic concept. The idea behind an effective theory is to focus on effects rather than causes—to truncate our speculation about what the ultimate reason behind a phenomenon is, instead assuming a theory that describes the phenomenon accurately enough to be useful. Such a theory is useful in two ways. First, it allows us to make meaningful predictions in areas described well enough by the effective theory. Second, any mismatch between the predictions of the effective theory and observations indicates where our understanding is lacking. A good example of an effective theory is classical mechanics. Though we know that classical mechanics is a subset of relativistic mechanics obtained by setting the speed of light to infinity, it is useful to neglect relativistic effects when e.g. designing bridges. In the same way, it is useful to be agnostic as to what happens at very high energy scales when considering particle physics interactions. In fact, as the SM does not incorporate gravitational effects, the SM is an effective theory valid at low energies.

The theoretical framework for the study of weak decays of hadrons such as B mesons is that of an effective quantum field theory (EFT) [45]. In an EFT, one does not assume knowledge of the full theory behind a decay. Instead of calculating the contributions to a decay from all virtual particles that mediate it, one looks at the external, visible particles and “integrates out” the mediators, parameterizing the interaction in effective operators and what are known as Wilson coefficients. The set of particles that are integrated out can contain NP particles. As with any effective theory, the utility of the EFT approach is twofold. First, it allows one to make reliable predictions at low energy scales. Second, one can compare the SM predictions of Wilson coefficients to experimental data to find out in which processes NP contributions are needed to explain the data and which processes are strongly constrained to not contain NP. Then, one can hypothesize full theories that provide the correct contributions.

4.1 Operator product expansion

The basis of the EFT approach is to write the transition amplitude of a weak decay as the sum shown in Eq. (4.1). From a transition amplitude, one can obtain e.g. the branching ratio of a decay. Writing the amplitude in this way is called operator product expansion, and Heff is an effective Hamiltonian that i encodes the physics of the decay. GF is the Fermi constant and VCKM contain the relevant CKM matrix elements. hf|Oi(µ)|ii is a hadronic matrix element encoding long-distance, low-energy effects, and Oi are local, non-perturbative operators describing the structure of the current mediating the decay. Ci are Wilson coefficients: perturbative, generally complex scalars that describe the short-distance, high-energy physics of virtual particles mediating the decay, be they SM or BSM ones. µ is the energy scale at which the operators and Wilson coefficients are evaluated; its significance will be made clear soon.

GF X i hf|Heff |ii = √ VCKM Ci(µ) hf|Oi(µ)|ii (4.1) 2 i

As an example of operator product expansion, consider the quark-level charged-current transition b →

21 22 CHAPTER 4. EFFECTIVE FIELD THEORY

c

c b W ±

d b d

u u

Figure 4.1: Tree-level diagrams of the charged-current transition b → cdu¯ in the SM (left) and an EFT wherein the heavy W boson is integrated out (right). cdu¯ shown in Fig. 4.1. Ignoring QCD corrections—gluon exchanges between the quarks—one obtains by computing the left Feynman diagram of Fig. 4.1

G M 2 √F ∗ W ¯ VcbVud 2 2 (¯cb)V −A(du)V −A, (4.2) 2 k − MW with the “vector minus axial” currents

µ (¯q1q2)V −A ≡ q¯1γ (1 − γ5)q2. (4.3)

The propagator term can be expanded as

2 2 MW k 2 2 = −1 + O( 2 ), (4.4) k − MW MW and because in a typical interaction the momentum transfer k is small relative to the mass of the W - 2 2 boson MW , terms of order O(k /MW ) can safely be neglected. This series truncation is what is meant by “integrating out” the W -boson. It illustrates an important requirement for the EFT approach to be valid: the integrated-out particles need to be heavy. Eq. (4.2) becomes

GF ∗ ¯ GF ∗ Heff = √ VcbV (¯cb)V −A(du)V −A = √ VcbV O2, (4.5) 2 ud 2 ud an effective Hamiltonian of the same form as Eq. (4.1), with the current-current operator ¯ O2 ≡ (¯cb)V −A(du)V −A. (4.6)

4.2 Calculating Wilson coefficients

Wilson coefficients can be calculated through perturbation theory. They depend on the assumed nature of NP; different models give different expressions for Wilson coefficients. The calculation involves matching: equating the EFT-computed transition amplitude of a process with the same amplitude computed assuming a full theory. The process of matching is illustrated in Fig. 4.2. Comparing Eq. (4.5) with the corresponding term of Eq. (4.1), we see that the operator in this case is O2 and that the Wilson coefficient is simply unity. If we were to include QCD corrections, an additional operator and corresponding Wilson coefficient would 4.3. PROBING NEW PHYSICS THROUGH EFFECTIVE FIELD THEORY 23

Figure 4.2: A low-energy EFT containing a light field φ of mass m is matched onto a high- energy theory. The high-energy theory contains in addition to φ a heavy field Φ with mass M. Adapted from Ref. [46]. be added and the value of the preexisting Wilson coefficient would change [45]. It should be noted that by integrating out the W -boson, we have neglected higher-dimensional operators of Heff that could contribute to the decay. However, because of the large mass of the W -boson, as seen above these contributions are small.

The operators Oi and Wilson coefficients Ci can be evaluated at different energy scales µ. The choice of scale serves to separate the high-energy contributions of particles of mass M > µ from the low-energy contributions of those with M < µ. At high energy scales, the momentum transfer k is not small enough compared to the masses of virtual particles in a process for them to be integrated out. At progressively lower energy scales, more and more virtual particles can be integrated out and their effects parameterized in Wilson coefficients. The matching of an effective theory onto a full theory is first done at high energy scales. Wilson coefficients are then evolved down to lower energy scales for the EFT to be used for low-energy processes such as B meson decays. This evolution of Wilson coefficients is done by performing matching at the energy scale of each particle that is integrated out. Of course, the physical results predicted by an EFT should be independent on the choice of scale at which the Wilson coefficients and operators are evaluated. For leading-order perturbation theory like that used for Eq. (4.5), this is not the case; rather, the results will depend on some degree on the energy scales chosen in the calculation. This represents a theoretical uncertainty which is reduced by expanding the propagator terms to higher order and involving higher-order Feynman diagrams in the calculation [45].

4.3 Probing new physics through effective field theory

NP can enter the effective Hamiltonian of Eq. (4.1) in several ways [36]. First, NP can contribute to the Wilson coefficients Ci of operators that exist in the SM by adding channels through which a process 0 ¯0 can occur. An example of this is a leptoquark contributing to Bs − Bs mixing as investigated in Section 0 ¯0 8.1; while Bs − Bs mixing occurs in the SM, a leptoquark carrying the right couplings can enhance or suppress the relevant Wilson coefficients relative to the SM through interference between SM diagrams and diagrams involving the leptoquark. In such cases, the relevant Wilson coefficients can be split up before SM NP renormalization group evolution as Ci = Ci + Ci . Second, NP can give rise to operators that are not part of the SM, mediating entirely new processes. An example of this is a Z0 boson mediating the NP lepton flavour-violating process µ → eγ. This would require the addition of a non-SM operator Oi and NP NP a corresponding Wilson coefficient Ci to the effective Hamiltonian. Generally, Ci can carry complex CP-violating phases unrelated to those of the CKM matrix. 24 CHAPTER 4. EFFECTIVE FIELD THEORY 0 ¯0 5 | Bs − Bs mixing

± W b s b s u, c, t 0 ± ± 0 0 0 Bs W W B¯ Bs u, c, t u, c, t B¯ s s u, c, t W ± s s b b

0 ¯0 Figure 5.1: Leading-order diagrams for Bs − Bs mixing in the SM.

Through the exchange of virtual particles as in Fig. 5.1, a neutral B meson can spontaneously turn into its antiparticle. This leads to time-dependent oscillations between the two states, a phenomenon called B0 −B¯0 (“B zero—B zero bar”) mixing. Several observables can be extracted from B0 − B¯0 mixing that constrain any NP that contributes to the quark-level transition b → d or b → s, depending on whether the neutral 0 0 0 ¯0 B meson under consideration is a B type meson or a Bs type meson. B − B mixing also complicates the discussion of neutral B meson decays, necessitating the definition of and conversion between two kinds 0 ¯0 of branching ratios. In this chapter, we introduce these two kinds of branching ratios along with Bs − Bs observables used in our analysis in Chapter 8.1.

5.1 Mixing amplitude and phase

0 ¯0 Oscillations between Bs and Bs mesons can be modelled by an effective Hamiltonian that when sandwiched ¯0 0 between the initial and final states of a Bs → Bs transition yields [47]

0 ∆B=2 ¯0 s Bs Heff Bs = 2MBs M12, (5.1)

s where M12 is related to the mass difference ∆Ms between the heavy and light neutral Bs-meson mass eigenstates,

s s 1 0 ∆B=2 ¯0 s ∆Ms ≡ MH − ML = Bs Heff Bs = 2|M12| (5.2) |MBs |

Similarly to how quark mass eigenstates are obtained by the diagonalization of the Yukawa couplings de- scribed in Section 2.4, the neutral Bs meson mass eigenstates arise through the diagonalization of the effective ∆B=2 0 ¯0 Hamiltonian Heff . Any NP that opens additional channels for Bs − Bs mixing to proceed through will contribute to the mass difference ∆Ms. The current experimental results for ∆Ms agree with SM predic- tions, making this observable an important constraint on BSM theories containing such NP. A recent SM prediction is given in Ref. [48] as

25 0 ¯0 26 CHAPTER 5. BS − BS MIXING

SM −1 −11 (∆Ms) = (18.3 ± 2.7) ps = (1.20 ± 0.18) × 10 GeV. (5.3)

The current world-average experimental result is [49]

exp −1 −11 (∆Ms) = (17.749 ± 0.020) ps = (1.1683 ± 0.0013) × 10 GeV, (5.4) a value clearly in agreement with the SM prediction considering the relatively large theoretical uncertainty.

0 ¯0 In any process involving the decay of a Bs or Bs -meson, the meson might oscillate into its antiparticle before 0 ¯0 decaying. For this reason, Bs − Bs mixing is a source of mixing-induced CP violation. The CP violation is parameterized by the phase φs, defined as [47]

s 0 ∆B=2 ¯0 φs ≡ arg M12 = arg Bs Heff Bs . (5.5)

SM NP The phase splits up into φs = φs + φs , where [50]

SM 2 φs − 2δγ = −2λ η (5.6) parameterizes the SM contribution to the phase. Here, δγ ≡ λ2η is the angle at which the second unitarity triangle of Fig. 3.2 is raised from the real axis and λ and η are Wolfenstein parameters of the CKM matrix. The current SM-predicted value of φs is [49]

SM +0.0007 +0.04 ◦ φs = −0.0369−0.0010 = (−2.114−0.057) , (5.7)

0 0 + − The current world-average experimental value is, extracted from studies of Bs → J/ψφ, Bs → J/ψK K , 0 + − and Bs → J/ψπ π decays [49],

◦ φs = −0.021 ± 0.031 = (−1.2 ± 1.8) (5.8) which yields the phase coming from NP

NP SM ◦ φs = φs − φs = 0.016 ± 0.031 = (0.92 ± 1.8) , (5.9) where the uncertainty denotes the root mean square of the theoretical and experimental uncertainties.

5.2 Theoretical vs. time-integrated branching ratios

0 ¯0 Because Bs − Bs mixing can occur in neutral Bs meson decays, the branching ratios of such decays need to be carefully handled. Ignoring the mixing leads to what are known as theoretical branching ratios for the 0 ¯0 processes Bs → f and Bs → f, where f denotes an arbitrary final state. Taking the mixing into account involves integrating over time and leads to time-integrated, also called experimental, branching ratios. As 0 ¯0 one cannot “turn off” Bs −Bs mixing in experiment, the time-integrated branching ratio is the only one that can be directly measured. Furthermore, determining the initial state of a B meson requires analysis of by- products of its decay and is difficult to do in experiment. In some cases, knowledge of the initial state of the 5.2. THEORETICAL VS. TIME-INTEGRATED BRANCHING RATIOS 27 meson is important. In others, this information is not needed. Then, the untagged decay width, an average 0 ¯0 over Bs and Bs mesons, is used for the calculation of branching ratios. We denote the untagged, theoretical branching ratio by BR(Bs → f) and the untagged, time-integrated branching ratio by B¯(Bs → f). One can convert between the theoretical and time-integrated branching ratios through [51]

 2  1 − ys BR(Bs → f) = B¯(Bs → f), (5.10) 1 + Af y ∆Γs s

∆Γs where ys ≡ = 0.0645 ± 0.003 [49] depends on the decay width difference ∆Γs between the Bs mass 2Γs eigenstates and the mean decay rate of a Bs meson, Γs = 1/τBs , which is the inverse of the Bs meson lifetime. The quantity Af is final-state dependent. In the next chapter, this quantity is given for the specific case ∆Γs + − of Bs → µ µ . 0 ¯0 28 CHAPTER 5. BS − BS MIXING 6 | The B-physics anomalies

A class of anomalies that has seen much attention in recent years is the B-physics anomalies, also known as the B-decay anomalies or flavour anomalies. These anomalies are related through involving the same quark-level transitions. There are two quark-level transitions relevant to the B-physics anomalies: the flavour-changing neutral current (FCNC) b → s`` and the flavour-changing charged current (FCCC) b → c`ν¯`. Figure 6.1 shows the leading-order SM contributions to these transitions1. One can group the B-physics anomalies into four categories [52]:

1. Suppression of exclusive branching ratios based on the b → sµ+µ− FCNC, measured by LHCb [15, 53, 54].

2. Deviations from SM values of angular observables in the decay B → K∗µ+µ−, based on the same FCNC b → sµ+µ−, measured by LHCb, ATLAS, and CMS [55–57].

3. Apparent lepton flavour universality violation, specifically deviations from µ−e universality, in b → s`` transitions, measured by LHCb [58, 59].

4. Apparent LFUV, specifically deviations from τ − µ and τ − e universality, in b → c`ν¯` transitions, measured by BaBar, Belle, and LHCb [60–62].

In this thesis, we focus on BSM theories that contribute to the FCNC process b → s`+`−. These theories could be viable for explaining anomalies within the first three categories listed above. In particular, we treat 0 + − ¯0 + − the dimuon decays of neutral Bs mesons, Bs → µ µ and Bs → µ µ . The branching ratios of these decays have very recently been found to be in conflict with the SM prediction with a statistical significance of about 2.5σ as explained below. In this section, we discuss these branching ratios and lepton flavour universality violation while introducing observables relevant to our analysis in Chapter 8.1.

6.1 Lepton flavour universality violation

Lepton flavour universality violation refers to differences between branching ratios of processes that have different leptons in their final states but are otherwise identical. This is measured in the ratios

B → K(∗)µ+µ− R (∗) = (6.1) K B → K(∗)e+e− and

1The two upper diagrams are called “penguin” diagrams—can you see the resemblance?

29 30 CHAPTER 6. THE B-PHYSICS ANOMALIES

W u, c, t s s u, c, t W u, c, t W b b Z, γ, h Z, γ, h

` `

` ` c

b W

`

ν`

Figure 6.1: Leading-order SM diagrams for the FCNC b → s`` (top) and the FCCC b → c`ν¯` (bottom). The FCNC b → s`` occurs only at loop-level in the SM due to the GIM mechanism forbidding FCNCs at tree-level [63].

(∗) B¯ → D τν¯τ R (∗) = , ` = µ, e, (6.2) D (∗) B¯ → D `ν¯` where K∗ and D∗ are excited states of charged K± and D± mesons. In the SM Lagrangian, leptons appear in the gauge and Yukawa sectors. The gauge sector is lepton flavour universal; the three lepton generations carry identical quantum numbers under the SM gauge group, meaning that all terms in the gauge part of the SM Lagrangian are invariant under exchanges of lepton flavours and that the photon, W boson, and Z boson couple with the same strength to all three lepton generations. The Yukawa sector contains interaction terms between fermions and the Higgs doublet which after EWSB give rise to fermion mass terms through the Higgs mechanism. These terms contain lepton masses and therefore differentiate between lepton flavours. However, the differences do not propagate to noticeable differences between the branching ratios in RK(∗) , and the SM predicts RK = RK∗ = 1. In contrast, the LHCb collaboration has measured [58, 59]

+0.060 +0.016 RK = 0.846−0.054 −0.014, +0.11 +0.03 2 2 RK(∗) = 0.66−0.07 −0.03, 0.045 < q < 1.1 GeV , (6.3) +0.11 +0.05 2 2 RK(∗) = 0.69−0.07 −0.05, 1.1 < q < 6.0 GeV , where q denotes the dilepton invariant mass and the uncertainties are statistical and systematic, respectively. The ratios RK and RK(∗) are theoretically clean as their dependencies on hadronic form factors are suppressed relative to stand-alone branching ratios and because the CKM matrix elements in the branching ratios cancel. To explain these anomalies, NP that suppresses the muonic decay channels B → K(∗)µ+µ− relative to the electronic ones is needed. + − 6.2. THE RARE DECAY BS → µ µ 31

u, c, t b µ Z, h 0 ± Bs W s µ u, c, t W ± b µ

0 Bs u, c, t ν W ± s µ

0 + − Figure 6.2: Feynman diagram of leading-order contributions to Bs → µ µ in the SM. Top: Penguin diagram mediated by the exchange of a virtual W boson and a virtual Z boson. Naively, the Z or Higgs boson could be replaced by a photon, but the photon contribution vanishes when computing the amplitude due to a Ward identity [26]. Bottom: Box diagram with two virtual W bosons.

+ − 6.2 The rare decay Bs → µ µ

Any NP that contributes to the transition b → sµ+µ− will enhance or suppress the leptonic B meson decays 0 + − ¯0 + − 0 Bs → µ µ and Bs → µ µ . In the SM, these decays are heavily suppressed, with only around three Bs mesons out of every billion decaying into a µ+µ− pair [64]. There are three reasons for this. First, the process requires the FCNC b → s. In the SM, due to the GIM mechanism [63] FCNCs do not occur through tree- level Feynman diagrams but only in loop-level ones like those of Fig. 6.2. As the amplitudes of electroweak loop-level diagrams contain more vertex factors than tree-level diagrams do, and the vertex factors in turn contain the small electroweak coupling constant g = e/ sin θW , loop-level diagrams are suppressed relative to tree-level ones. Furthermore, as the process in the SM requires at least one transition between quark generations, its amplitude contains at least one off-diagonal CKM matrix element, suppressing the process 0 + − further. Finally, the Bs meson is a spin-0 particle. When it decays into a µ µ pair, the spin projections of 0 the muons along any arbitrary axis need to add up to 0. In the rest frame of the Bs , the muons will travel in opposite directions. For their spin projections to add up to 0, one of the muons needs a helicity opposite to its chirality. This is impossible for massless particles and becomes increasingly probable for more massive particles. Because of the small mass of the muons, the process is suppressed also by this helicity suppression.

+ − The heavy suppression of BR(Bs → µ µ ) makes it very sensitive to NP, which might contribute to the decay at tree-or loop-level through the involvement of some new particle, change the CKM matrix elements in the decay, or lift its helicity suppression. The process is also theoretically clean; as gluons do not couple to the purely leptonic final state, the only hadronic parameter entering its branching ratio is the decay constant fBs , a parameter that can be calculated through numerical methods such as lattice QCD [65]. The 0 + − 0 + − 0 + − similar processes Bs → e e , Bs → τ τ , and Bd → ` ` are also objects of study [64], although the branching ratios of the processes with τ +τ − final states have not yet been measured in experiment due to the propensity of τ leptons to decay before reaching detectors.

¯0 + − For the decay Bs → µ µ , the effective Hamiltonian is [64]

GF ∗ 0 0 0 0 0 0 Heff = −√ V Vtbα[C10O10 + CSOS + CP OP + C O + C O + C O ], (6.4) 2π ts 10 10 S S P P

where GF is the Fermi constant, the Vq1q2 are CKM matrix elements, α is the fine structure constant, and ∗ ∗ −2 terms proportional to VusVub/VtsVtb ∼ O(10 ) have been omitted. The operators are four-fermion operators given by 32 CHAPTER 6. THE B-PHYSICS ANOMALIES

µ 0 µ O10 = (¯sγµPLb)(¯µγ γ5µ),O10 = (¯sγµPRb)(¯µγ γ5µ), 0 OS = mb(¯sPRb)(¯µµ),OS = mb(¯sPLb)(¯µµ), (6.5) 0 OP = mb(¯sPRb)(¯µγ5µ),OP = mb(¯sPLb)(¯µγ5µ), where mb is the b-quark mass and with O10, OS and OP denoting axial vector, scalar, and pseudoscalar operators, respectively. The primed operators are chirality-flipped versions of unprimed ones; if Oi describes 0 a right-handed current, Oi describes a left-handed current. PL, PR are the chiral projection operators defined in Section 1.1.

¯0 + − Starting from the effective Hamiltonian for Bs → µ µ decays given in Eq. (6.4) and following the calcu- lation in [26], one obtains the untagged, theoretical branching ratio

s |M|2τ 4m2 BR(B → µ+µ−) = Bs 1 − µ , (6.6) s 16πM M 2 Bs Bs

¯0 2 with τBs = 1/Γtot being the lifetime of the Bs meson, equal to the inverse of its total decay width. |M| is given by, after computing the relevant Feynman diagrams and summing over lepton spins,

G2 α2f 2 1 4m2  M 2 2 |M|2 = F Bs |V V ∗ |2 1 − µ Bs |C m − C0 m |2 π2 tb ts 4 M 2 m + m S b S s Bs b s M 2 2 1 Bs 0 0 + (CP mb − CP ms) + mµ(C10 − C10) . (6.7) 2 mb + ms

In the SM, the only non-negligible operator out of Eq. (6.5) is O10. In the SM, then, the branching ratio of Eq. (6.6) becomes

2 2 2 2 s 2 G α MB f τB m 4m BR(B → µ+µ−) = F s Bs s µ |V V ∗ |2 1 − µ |C |2. (6.8) s SM 16π3 tb ts M 2 10 Bs

The value of the untagged, time-integrated branching ratio has been measured by ATLAS, LHCb, and CMS [15]. The average value of these measurements is as of 2020

¯ + − +0.37 −9 B(Bs → µ µ ) = (2.69−0.35) × 10 . (6.9)

The most recent SM prediction of the time-integrated branching ratio is given in Ref. [66] as

+ − SM −9 B¯(Bs → µ µ ) = (3.66 ± 0.14) × 10 , (6.10) which is 2.5σ away from the experimental value, using the root sum of the squared uncertainties. This intriguing strain makes it important to calculate Aµµ in BSM theories to be able to properly account for ∆Γs 0 ¯0 Bs − Bs mixing effects by converting between theoretical and time-integrated branching ratios. Analyses of the full LHC Run I and II datasets from LHCb, data from Belle II, and eventually data from the LHC Run III and the HL-LHC are expected to cement or smooth out both this strain and those of the LFUV observables. + − 6.2. THE RARE DECAY BS → µ µ 33

The quantity Af is given for a dimuon final state f = µ+µ− as [67] ∆Γs

|P |2 cos (2ϕ − φNP ) − |S|2 cos (2ϕ − φNP ) Aµµ = P s S s . (6.11) ∆Γs |P |2 + |S|2

P and S are defined as the following combinations of Wilson coefficients:

C − C0 M 2  m C − C0  10 10 Bs b P P iϕP P ≡ SM + SM ≡ |P |e , (6.12) C10 2mµ mb + ms C10

s 2 2   0  4mµ M mb CS − C S ≡ 1 − Bs S ≡ |S|eiϕS , (6.13) M 2 2m m + m CSM Bs µ b s 10 and ϕP and ϕS are CP-violating phases of P and S, respectively. In the SM, [67]

SM −2 C10 = −ηY sin θW Y0(xt) = −4.134, (6.14) where ηY represents QCD corrections, θW is the Weinberg angle, and

  xt xt − 4 3xt Y0(xt) = + 2 ln xt (6.15) 8 xt − 1 (xt − 1)

2 2 is an Inami-Lim function, with xt = mt /MW .

0 0 0 Also in the SM, C10 = CP = CP = CS = CS = 0, leaving us with P |SM = 1 and S|SM = 0, yielding Aµµ = 1. Because of this, Aµµ is, in addition to being necessary for converting between theoretical ∆Γs ∆Γs and time-integrated branching ratios through Eq. (5.10), an interesting observable in its own right. In BSM theories, the observable can assume values between −1 and 1, and any certain deviation from the SM-predicted Aµµ = 1 would signal the existence of NP. It is also a theoretically clean observable as the ∆Γs 0 + − ¯0 + − hadronic decay constant fBs that appears in the branching ratios of Bs → µ µ and Bs → µ µ cancels in Aµµ . Experimental constraints on the observable can be obtained through measurements of the effective ∆Γs s + − lifetime τµµ of Bs → µ µ decays. This parameter is defined as [67]

R ∞ + − thΓ(Bs(t) → µ µ )idt 0 (6.16) R ∞ + − 0 hΓ(Bs(t) → µ µ )idt and relates to Aµµ as [67] ∆Γs

2 2 2 1 (1 − y )τ − (1 + y )τB  Aµµ = s µµ s s . (6.17) ∆Γs 2 s ys 2τBs − (1 − ys )τµµ

The effective lifetime has been measured in 2017 [68] as

s τµµ = [2.04 ± 0.44(stat) ± 0.05(syst)]ps, (6.18) yielding the value 34 CHAPTER 6. THE B-PHYSICS ANOMALIES

Aµµ = 8.24 ± 10.72, (6.19) ∆Γs

s with an uncertainty fully dominated by that of the τµµ measurement. As the experimental precision of effective lifetime measurements increases, Aµµ will be a useful probe of NP. ∆Γs

6.3 CP asymmetries

Two observables for the study of CP violation are related to Aµµ : the CP asymmetries S and Cλ , given ∆Γs µµ µµ as [67]

2 NP 2 NP |P | sin (2ϕP − φs ) − |S| sin (2ϕS − φs ) Sµµ = (6.20) |P |2 + |S|2 and

  λ 2|PS| cos (ϕP − ϕS) Cµµ = −ηλ ≡ −ηλCµµ, (6.21) |P |2 + |S|2 where ηλ parameterizes the muon helicity: nL = +1, nR = −1. No measurements exist to date of either of these observables. In the SM, Sµµ = Cµµ = 0. A nonzero Sµµ would signal CP-violating phases not originating from the CKM matrix. A nonzero Cµµ would signal scalar NP. The CP asymmetries, just like Aµµ , are theoretically clean observables as they do not depend on the decay constant f . The three ∆Γs B observables Aµµ , S , and C are spherically related as ∆Γs µµ µµ

(Aµµ )2 + (S )2 + (C )2 = 1. (6.22) ∆Γs µµ µµ 7 | Muon g − 2

Muon g − 2 or aµ is shorthand for the anomalous magnetic dipole moment of the muon. All charged leptons carry magnetic dipole moments. Magnetic dipole moments are remarkable physical quantities as they are well understood classically, but when one looks deeper, all sorts of quantum mechanical effects come into play through radiative corrections [16].

7.1 Classical to quantum

Figure 7.1: A lepton of charge e and mass m orbiting an atom at a radius r and at velocity v.

In classical electrodynamics, the magnetic dipole moment of an charged lepton can be calculated by con- sidering the lepton as a charged point-like particle orbiting a point-like atom as in Fig. 7.1. Denoting the magnetic dipole moment by µ and considering the current loop constituted by the orbiting particle, we have

µ = IA, (7.1)

eω where I = ef = 2π (e elementary charge, f and ω frequency and angular frequency of orbit, respectively) is the current through the loop and A = πr2 the loop area. Noting that for circular motion ω = v/r, we can calculate the scalar angular momentum

L = pr = mvr = mωr2. (7.2)

Going back to Eq. (7.1), we get

eω (7.2) e µ = IA = = L. (7.3) 2 2m

35 36 CHAPTER 7. MUON G − 2

In vector form and taking into account that the charged leptons have different masses, this becomes [16]

e ~µ` = L.~ (7.4) 2m`

This classical result does not tell the whole story. First, the classical result only considers orbital angular momentum. Since the Stern-Gerlach and Uhlenbeck-Goudsmit experiments of the 1920s [69], we know that charged leptons carry the additional angular momentum component of spin. Replacing L~ in the right-hand side of Eq. (7.4) by the total angular momentum J~, we get a spin-correction to ~µ` denoted by g`, a Landé g-factor.

e ~µ` = g` J.~ (7.5) 2m`

Second, we know that charged leptons are not point-like particles interacting only with atoms; rather, like any elementary particle, they are in the modern framework of quantum field theory treated as excitations of fields constantly interacting with virtual particles. Due to the nonzero electric charge of charged leptons and the infinite range of the electromagnetic force, we can always consider any given charged lepton to interact with a photon as in Fig. 7.2. This diagram is the basis of the contemporary view of magnetic dipole moments of charged leptons.

γ

µ µ

Figure 7.2: Tree-level interaction between a photon and a muon. This diagram corresponds to the Dirac equation-predicted gµ = 2.

The diagram of Fig. 7.2 shows the tree-level interaction of a photon and a charged lepton. There are also further, loop-level interactions known as radiative corrections. The Dirac equation takes into account only the tree-level interaction and predicts g` = 2 for charged fermions [70]. Any radiative corrections will cause the value of g` to deviate from 2. Such deviations are parameterized in the parameter a`, defined as

g − 2 a = ` . (7.6) ` 2

The anomalous magnetic dipole moment of the muon, aµ, is then

g − 2 a = µ , (7.7) µ 2 and any deviation from aµ = 0 is due to effects beyond the tree-level photon contribution.

Among the charged leptons, the muon is particularly interesting in this context due to a long-standing discrepancy between the SM prediction of and experimental results for aµ. Currently, the most precise 7.2. RADIATIVE CORRECTIONS 37 experimental value comes from the Brookhaven National Laboratory E821 experiment from 2004 [71]. This experiment reported the value

exp −11 aµ = 116 592 089(63) × 10 , (7.8) wherein the uncertainty combines the statistical and systematic ones.

The precision of the SM prediction has only recently caught up to the experimental one. An international theoretical effort involving more than 100 physicists resulted in June 2020 in the current SM prediction of [17]

SM −11 aµ = 116 591 810(43) × 10 , (7.9) yielding the strain between experimental and theoretical results

−11 ∆aµ = 279(76) × 10 , (7.10) a discrepancy of 3.7σ.

7.2 Radiative corrections

The radiative corrections to aµ are split up into three major parts containing the five topologies shown in Fig. 7.3. The quantum-electrodynamical (QED) corrections are purely electromagnetic, involving only photons and charged leptons. These have been calculated to the five-loop order in the fine-structure constant [72] and are very well understood, with a theoretical uncertainty much smaller than the experimental one. The electroweak (EW) corrections, involving the SM vector bosons W and Z and all leptons, has been computed to the two-loop level [73]. Here, too, the uncertainty is smaller than that of experiment. The hadronic, or quantum-electrodynamical (QCD) part, involves quarks and is where the largest theoretical uncertainty lies. Because of this, the QCD part, particularly the determination of non-perturbative quantities through numerical methods such as lattice QCD, is where most current theory work in this area is focused [17]. The QCD part is further split up into its two dominant constituents: hadronic vacuum polarization (HVP) and hadronic light-by-light scattering (HLbL). The current theoretical values of all these corrections are [17]

QED EW HVP HLbL aµ = aµ + aµ + aµ + aµ , QED −11 aµ = 116 584 718.931(104) × 10 , EW −11 aµ = 153.6(1.0) × 10 , (7.11) HVP −11 aµ = 6845(40) × 10 , HLbL −11 aµ = 92(18) × 10 .

In order to either cement the anomaly as an unambiguous signal of NP or to smooth it out, ruling out the current anomaly as a fluctuation, experimental and theoretical precisions both need to increase. The experimental value needs to be precise enough to clearly diverge from or converge with the SM prediction, and the theoretical SM prediction needs to evolve to match this improved precision. The theory initiative is in full swing as reviewed in Ref. [17]. On the experimental front are the ongoing Fermilab Muon g − 2, [18], the upcoming J-PARC E34 [74], and the proposed CERN MUonE [75] experiments. 38 CHAPTER 7. MUON G − 2

Figure 7.3: Radiative corrections to aµ. Diagrams in order from top-left to bottom-right: quan- tum electrodynamical, electroweak Z, electroweak W, hadronic vacuum polarization, hadronic light-by-light scattering.

For NP to explain the muon g − 2 anomaly, a NP particle needs to couple lepton-flavour-diagonally, meaning that it allows for a three-point vertex involving the NP particle and two muons. Furthermore, in order for the contributions to muon g − 2 not to vanish, the NP particles need to couple non-chirally to the muon, coupling to both left-and right-handed muons [16]. The contribution of NP to the anomalous magnetic dipole moment of a lepton is proportional to the mass of the lepton [16]. This implies that heavier leptons are more sensitive to NP effects than lighter ones; muons are more sensitive than electrons, and taus are more sensitive than muons. The anomalous magnetic dipole moment of tau has not seen much attention for 0 + − the same reason that the decay Bs → τ τ has not; tau leptons have a very short mean lifetime and decay before they can be measured in detectors. 8 | Model-dependent analysis

We have in the previous chapters identified the features required of a BSM theory to explain the B-physics anomalies related to b → s and to explain the muon g − 2 anomaly. To explain both kinds of anomalies, a BSM theory needs to include a particle (or particles) that carries couplings to b and s as well as to both left- and right-handed muons. In this chapter, we investigate three different kinds of models that contain such particles. We first consider leptoquarks, bosons that couple leptons to quarks. We consider a specific scalar leptoquark in detail. We then briefly consider models with a Z-like boson called a Z0 and supersymmetric models that postulate a symmetry between fermions and bosons.

8.1 Leptoquarks

Leptoquarks (LQs) are hypothetical bosons that couple quarks to leptons. They carry both baryon number and lepton number. LQs that couple exclusively to left-or right-handed particles are called chiral LQs. LQs appear in the context of larger models such as the Georgi-Glashow model [76], a grand unified theory that seeks to unify the electroweak and strong forces by treating the SU(3)c and SU(2)L ⊗ U(1)Y subgroups of the SM gauge group as components of a larger SU(5) group. One can constrain the possible quantum numbers of LQs by making the simplifying assumption that interaction terms involving the LQs and the SM fermions are of zero mass dimension and invariant under the SM gauge group. In this case, the remaining, possible LQs are given in Table 8.1.

Symbol Interacts with SU(3)c ⊗ SU(2)L ⊗ U(1)Y ¯c c ¯ S1 QLLL, u¯ReR 3 ⊗ 1 ⊗ 1/3 ˜ ¯c ¯ S1 dReR 3 ⊗ 1 ⊗ 4/3 R2 u¯RLL, Q¯LeR 3 ⊗ 2 ⊗ 7/6 ¯ R˜2 dRLL 3 ⊗ 2 ⊗ 1/6 ¯c ¯ S3 QLLL 3 ⊗ 3 ⊗ 1/3

Table 8.1: Possible scalar leptoquarks, their interactions with SM fermions, and their quantum numbers. The hermitian conjugate of each listed interaction is also allowed. ψc = Cψ¯T is the charge conjugate of the spinor ψ, with the eigenvalue C = ±1 depending on the spinor. The numbers under the SU(3)c and SU(2)L SM gauge groups refer to the multiplicity of states under that gauge group; e.g, the first row in the table denotes a leptoquark that is an antitriplet under SU(3)c and a singlet under SU(2)L. Under the U(1)Y gauge group, the hypercharge of the LQ is given. Information sourced from Ref. [77].

Explaining the muon g − 2 anomaly through a LQ requires the LQ to couple non-chirally to the muon, i.e. couple to both helicity states of the muon. Such couplings can be obtained either through scalar LQs that mediate these couplings directly or through mixing between multiple scalar LQ states which are individually chiral [77]. In the simplest case of one LQ, this leaves the candidates S1 and R2 of Table 8.1. To explain the

39 40 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

+ − B-physics anomaly of BR(Bs → µ µ ), a LQ needs to mediate a flavour-changing coupling between b and s to be able to suppress the branching ratio through interference with the SM leading-order contributions (∗) B→K(∗)µ+µ− to the decay. To explain RK = B→K(∗)e+e− , a LQ that suppresses the muonic process relative to the (∗) + − electronic one is needed. Explaining muon g − 2, RK , and BR(Bs → µ µ ) simultaneously through one LQ then requires a scalar LQ that couples non-chirally to muons, mediates a coupling between b and s, and (∗) + − (∗) + − suppresses B → K µ µ relative to B → K e e . The LQ S1 has been shown by Bauer and Neubert [78] to carry these properties and is the main object of study in this chapter.

8.2 The scalar leptoquark S1

In Ref. [78], the LQ S1 has been shown to be able to explain the anomalous RD∗ and RK . The authors 0 ¯0 also show that the model can be consistent with experimental results for the Bs − Bs mixing amplitude and phase and can also explain the muon g − 2 anomaly, in addition to other results not covered in this thesis. + − They also claim that the model can explain the tension in the branching ratio BR(Bs → µ µ ). Here, we study the effects of B0 − B¯0 mixing on BR(B → µ+µ−) in the model by calculating Aµµ as well as s s s ∆Γs investigate mixing-induced CP-violating effects of the LQ by calculating its impact on Sµµ, two things that have not been done in the literature. To determine what values of complex couplings are allowed, we turn 0 ¯0 to Bs − Bs mixing constraints. For this analysis, we set the mass of the LQ and the couplings that do not enter Aµµ and S to values that explain muon g − 2, using results from Ref. [78]. ∆Γs µµ

We use the notation of Ref. [78], referring to the LQ S1 as φ. The LQ φ is described before EWSB by the Lagrangian

† 2 2 2 2 Lφ =(Dµφ) Dµφ − M |φ| − ghφ|Φ| |φ| φ (8.1) ¯c L ∗ c R ∗ + QLλ iτ2LLφ +u ¯Rλ eRφ + h.c..

Here, Φ is the Higgs doublet, λH (H = L, R) are 3 × 3 weak eigenstate coupling matrices, and ψc = Cψ¯T is the charge conjugate of the spinor ψ. In the fermion mass basis, the fermionic couplings of the LQ φ are described by

c L ∗ ¯c L ∗ c R ∗ Lφ ⊃ u¯LλueeLφ − dLλdν νLφ +u ¯RλueeRφ + h.c., (8.2)

H where λq` are fermion mass eigenstate coupling matrices of the form

 L L L  λue λuµ λuτ L L L L λue = λce λcµ λcτ  , L L L λte λtµ λtτ λL λL λL  dνe dνµ dντ L λL λL λL λdν =  sνe sνµ sντ  , (8.3) λL λL λL bνe bνµ bντ  R R R  λue λuµ λuτ R R R R λue = λce λcµ λcτ  . R R R λte λtµ λtτ

These relate to their weak eigenstate counterparts through 8.3. CALCULATION OF WILSON COEFFICIENTS 41

L T L L T L R T R λue = U u λ U e, λdν = U d λ , λueV u λueV e, (8.4)

where U f and V f are unitary rotation matrices of left-and right-handed fermion fields, respectively.

8.3 Calculation of Wilson coefficients

We first seek expressions for the relevant Wilson coefficients. The LQ φ contributes only to C10. The reason that the scalar LQ contributes only to this Wilson coefficient, corresponding to the axial operator, rather than CS of the scalar operator can be understood as follows. In the Lagrangians of Eqs. (8.1) and (8.2), the quark and lepton fields appear in the LQ interaction terms in the order q`. In a diquark-dilepton operator of an effective Hamiltonian, the fields are written in the order qq``. This enables the separation of QCD corrections, parameterized in a hadronic matrix element, from the theoretically cleaner, purely leptonic part. When constructing a diquark-dilepton effective operator by combining interaction terms of a Lagrangian, one needs to reorder the fermion fields from q`q` to qq``. This reordering is called a Fierz transformation, and in its process gamma matrices appear which end up in the operator. In the present case of the LQ φ, the resulting operator is the axial C10.

To obtain an expression for C10, we first convert between the operator bases used by the authors of Ref. [78] and that used in Eqs. (6.12) and (6.13). For this, we use the expressions of Ref. [79], the authors of which studied this same LQ:

µ µ µ µ µ µ CLL = C9 − C10 ,CLR = C9 + C10, (8.5) or, equivalently,

C + C C − C C = LR LL ,C = LR LL . (8.6) 9 2 10 2

2 2 Working in the limit Mφ  mt,W , the authors of Ref. [78] obtain the contributions of the leptoquark φ to CLL and CLR as

2 (φ) mt L 2 CLL = 2 λtµ 8παMφ √ L L† 1 2 (λ λ )bs L,† L − 2 ∗ (λ λ )µµ , 64πα GF Mφ VtbVts (8.7) 2  M 2  (φ) mt R 2 φ CLR = 2 λtµ ln 2 − f(xt) 16παMφ mt √ L L† 1 2 (λ λ )bs R† R − 2 ∗ (λ λ )µµ. 64πα GF Mφ VtbVts

−5 Here, α = 1/137 is the fine-structure constant evaluated at low energy scales, GF = 1.166 × 10 is the 3 ln xt
2 2 Fermi constant, and f(xt) = 1 + − 1 , where xt = m /M using the running top quark mass xt−1 xt−1 t W m (m ) = 162.3 GeV and M 2 = 80.38, evaluates to f(x ) = 0.47. (λLλL†) ≡ P λL λL∗ . Working out t t W t bs i bνi sνi (φ) the expression for C10 using Eqs. (8.6) and (8.7), we obtain 42 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

   M 2   (φ) 1 2 1 R 2 φ L 2 C10 = 2 mt λtµ ln 2 − f(xt) − λtµ 16παMφ 2 mt √ (8.8) L L†   2 (λ λ )bs R,† R L,† L − ∗ (λ λ )µµ − (λ λ )µµ . 8GF VtbVts

+ − 8.4 Calculation of the theoretical branching ratio of Bs → µ µ and muon g − 2

φ b µ

0 Bs ν u, c, t φ s µ φ b µ

ν u, c, t W ± µ s

Figure 8.1: Leading-order contributions of the leptoquark φ to the transition b → sµ+µ−.

FCNCs contributing to b → s are induced by φ first at the one-loop-level through the diagrams of Fig. 8.1. + − Using Eqs. (6.6) and (6.7), we can compute the value of BR(Bs → µ µ ), using recent parameter values from Refs. [34, 65], as

2 2 2 2 s 2 G α MB f τB m 4m 2 BR(B → µ+µ−) = F s Bs s µ |V V ∗ |2 1 − µ CSM + Cφ s 16π3 tb ts M 2 10 10 Bs (8.9) 2 SM φ ≈ 0.0537 C10 + C10 .

For the following analysis, however, we instead used the expression of Ref. [79]:

BR(B → µ+µ−) s φ + − SM = 1 − 0.24C10 . (8.10) BR(Bs → µ µ )

+ − When solving for BR(Bs → µ µ ), these expressions yield very similar but not identical results. For a more up-to-date analysis, the expression of Eq. (8.9) should be used instead. Using the expression of Eq. (8.8) for the contribution of φ to C10, we obtain + − 8.4. CALCULATION OF THE THEORETICAL BRANCHING RATIO OF BS → µ µ AND MUON G − 2 43

Figure 8.2: Leading NP contributions to aµ from the leptoquark φ.

 + − (φ) BR(Bs → µ µ ) + − SM = BR(Bs → µ µ )    M 2   0.24 2 1 R 2 φ L 2 1 − 2 mt λtµ ln 2 − f(xt) − λtµ 16παMφ 2 mt √ L L†   2 2 (λ λ )bs R,† R L,† L − ∗ (λ λ )µµ − (λ λ )µµ . (8.11) 8GF VtbVts

Regarding muon g − 2, φ opens up the contributions of Fig. 8.2. Computing these diagrams, the authors of Ref. [78] obtain by working in the limit Mφ  mt:

 M 2  (φ) X mµmq φ 7  R L∗ aµ = 2 2 ln 2 − Re λqµλqµ 4π Mφ mq 4 q=t,c (8.12) 2   mµ L† L R† R − 2 2 (λ λ )µµ + (λ λ )µµ , 32π Mφ

2 where (λH†λH ) ≡ P λH∗ λH = P λH µµ i uiµ uiµ i uiµ An example parameter point stated in Ref. [78] to be able to explain the muon g − 2 anomaly, here referred to as “parameter point 1”, is

Mφ = 1 TeV, (Parameter point 1) 0 0 0 0 0 0 L R (8.13) λue = 0 2.4 0 , λue = 0 0.01 0 . 0 0.5 0 0 0.005 0

Using these couplings and the values mµ = 0.106 GeV, mc = 1.275 GeV taken from Ref. [34], we find the contribution of the leptoquark φ to aµ to be

(φ) −11 aµ = 299.8 × 10 , (8.14) which indeed explains the anomaly by falling within the 1σ uncertainty range of Eq. (7.10). 44 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

8.5 Calculation of Aµµ and the CP asymmetries S and C ∆Γs µµ µµ

Having access to the expressions for the model’s contribution to the Wilson coefficient C10, we can compute its impact on Aµµ and the CP asymmetries S and C . Using Eqs. (6.11) through (6.21) and noting ∆Γs µµ µµ that the leptoquark contributes only to C10 with no impact on either CP or CS, we find the expressions

(φ) C10 1 1 P = 1 + SM = 1 + SM 2 C10 C10 16παMφ    M 2   2 1 R 2 φ L 2 m λ ln − f(xt) − λ t 2 tµ m2 tµ t √ L L†   2 (λ λ )bs R,† R L,† L − ∗ (λ λ )µµ − (λ λ )µµ , (8.15) 8GF VtbVts

S = 0, (8.16)

Aµµ = cos (2ϕ − φNP ), (8.17) ∆Γs P s

NP Sµµ = sin (2ϕP − φs ), (8.18) and

Cµµ = 0. (8.19)

As the LQ does not contribute to the scalar Wilson coefficient CS, |S| and ϕS will always be zero in this model, leading to a vanishing Cµµ. Despite not providing a contribution to the pseudoscalar CP , the LQ will generally contribute to |P | through affecting the axial C10 and to ϕP by making C10 complex.

0 ¯0 8.6 Bs − Bs mixing constraints

To determine what values of complex couplings are consistent with experimental data, we turn to constraints 0 ¯0 on the Bs − Bs mixing parameters ∆Ms and φs defined in Eqs. (5.2) and (5.5). The authors of Ref. [78] have obtained an expression for the mixing amplitude and phase expressed in the parameters CBs and φBs . These parameters are defined through [38]

B0 Hfull B¯0 2iφ s eff s C e Bs ≡ . (8.20) Bs 0 SM ¯0 Bs Heff Bs

Converting between these parameters and those of Eqs. (5.2) and (5.5) we find the following relationships, ∆B=2 full 0 identifying Heff = Heff and Bq = Bq :

∆Ms CBs = SM (8.21) ∆Ms 0 ¯0 8.6. BS − BS MIXING CONSTRAINTS 45

(a) ∆Ms (b) φs

L L,† L (λ λ )bs 0 ¯0 Figure 8.3: Parameter space for the ratio R = ∗ consistent with the Bs − Bs mixing bs VtbVts parameters ∆Ms (left) and φs (right) for the scalar leptoquark φ of mass 1 TeV. The coloured regions show the parameter space excluded by experimental measurements at 1σ. For ∆Ms, the white region represents values that keep ∆Ms within 1σ of the SM prediction away from the experimental value of Eq. (5.4). For φs, the white region represents values within the bounds of Eq. (5.9). and

1 φ = φNP . (8.22) Bs 2 s

The expression obtained by the authors of Ref. [78] is

2  L L† 2 (φ) 2iφ(φ) 1 MW (λ λ )bs C e Bs = 1 + , (8.23) Bs 4 2 ∗ g S0(xt) Mφ VtbVts

2 3 √ 4xt−11xt +xt where S0(xt) = 2 ≈ 2.30 is an Inami-Lim function and g = 4πα/ sin θW is one of the SU(2)L 4(1−xt) gauge couplings. Converting to ∆Ms and φs, we obtain

1 M 2 (λLλL†) 2 ∆M = ∆M SM 1 + W bs (8.24) s s 4 2 ∗ g S0(xt) Mφ VtbVts and

 2  L L† 2 NP 1 MW (λ λ )bs φs = arg 1 + 4 2 ∗ . (8.25) g S0(xt) Mφ VtbVts

L L,† L (λ λ )bs Fig. 8.3 shows what real and imaginary values of the ratio R ≡ ∗ are consistent with recent bs VtbVts NP L L experimental measurements of ∆Ms and φs . Here, the couplings λcµ = 2.4, λtµ = 0.5 are taken from parameter point 1, the example point to explain muon g − 2 given in Eq. (8.13). 46 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

8.7 Results for Aµµ and S ∆Γs µµ

Computing Aµµ and S through Eqs. (8.17) and (8.18), where ϕ is the complex phase of the P ∆Γs µµ P 0 ¯0 given in Eq. (8.15), we obtain the results illustrated in Fig. 8.4. The allowed values, within Bs − Bs mixing constraints, are illustrated as white regions; the areas where the white regions overlap contain values consistent with constraints on both ∆Ms and φs. Disregarding the tiny slivers of overlap near ReRLbs = 10, ImRL = 0, we find that Aµµ is bounded by 0.93 < Aµµ ≤ 1 and S by |S | < 0.37. bs ∆Γs ∆Γs µµ µµ

(a) Aµµ (b) S ∆Γs µµ

(c) Aµµ zoomed in (d) S zoomed in ∆Γs µµ Figure 8.4: Aµµ (left) and the CP asymmetry S (right) for a span of complex values of ∆Γs µµ L L,† L (λ λ )bs the ratio R = ∗ . The white “infinity symbol” marks the values consistent with bs VtbVts experimental measurements of ∆Ms, and the white “cross” marks those consistent with φs. 0 ¯0 The overlap of the white regions marks values consistent with both Bs −Bs mixing parameters. Here, the only nonzero up-type LQ couplings are taken from parameter point 1 of Eq. (8.13) L L as λcµ = 2.4 and λtµ = 0.5.

L L To investigate the robustness of these results for different values of the couplings λcµ and λtµ, we repeat our analysis for four additional parameter points that push aµ within 1σ of the experimental value, keeping Mφ constant as 1 TeV. We pick the parameter points to satisfy Eq. (24) of Ref. [78], which for Mφ = 1 TeV states that to explain muon g − 2, the LQ couplings must satisfy

 R L∗  R L∗ Re λcµλcµ + 20.7 Re λtµλtµ ≈ 0.08. (8.26)

The authors further state that parameter values satisfying this equation ensure that radiative corrections to the muon mass stays within the perturbative regime. The new parameter points are given in Eqs. (8.27-8.30) + − 8.8. RESULTS FOR THE TIME-INTEGRATED BRANCHING RATIO OF BS → µ µ 47

and the results in Fig. 8.5 and Eq. (8.31). The bounds coming from ∆Ms and φs are constant as none of the couplings that vary between the parameter points appear in the expressions for these observables. The bounds on Aµµ and S for the full set of parameter points are given in Eq. (8.31). Using the smallest ∆Γs µµ allowed Aµµ , the theoretical and time-integrated branching ratios of B → µ+µ− differ by 5.2% as obtained ∆Γs s from Eq. (5.10). From the bounds on Sµµ, it is clear that sizeable mixing-induced CP violation is possible within this model. To more precisely determine the bounds on Sµµ, investigating more thoroughly the L L parameter space that explains the muon g − 2 anomaly by scanning over the couplings λcµ and λtµ would be in order.

Mφ = 1 TeV, (Parameter point 2) 0 0 0 0 0 0 L R (8.27) λue = 0 2.0 0 , λue = 0 0.005 0 . 0 0.7 0 0 0.005 0

Mφ = 1 TeV, (Parameter point 3) 0 0 0 0 0 0 L R (8.28) λue = 0 2.8 0 , λue = 0 0.02 0 . 0 0.2 0 0 0.004 0

Mφ = 1 TeV, (Parameter point 4) 0 0 0 0 0 0 L R (8.29) λue = 0 3.0 0 , λue = 0 0.02 0 . 0 0.5 0 0 0.002 0

Mφ = 1 TeV, (Parameter point 5) 0 0 0 0 0 0 L R (8.30) λue = 0 3.2 0 , λue = 0 0.015 0 . 0 0.2 0 0 0.002 0

Point 1: 0.93 < Aµµ ≤ 1, |S | < 0.37 ∆Γs µµ Point 2: 0.96 < Aµµ ≤ 1, |S | < 0.28 ∆Γs µµ Point 3: 0.88 < Aµµ ≤ 1, |S | < 0.48 (8.31) ∆Γs µµ Point 4: 0.83 < Aµµ ≤ 1, |S | < 0.55 ∆Γs µµ Point 5: 0.79 < Aµµ ≤ 1, |S | < 0.61 ∆Γs µµ

µµ µµ 2 2 As described by Eq. (6.22), A∆Γ and Sµµ satisfy the relation (A∆Γ ) +(Sµµ) = 1 when Cµµ = 0, describing s µµ s a unit circle in the plane of Sµµ and A∆Γ. Fig. 8.4 shows what parts of this circle can be populated in the 0 ¯0 present model consistently with Bs − Bs mixing constraints while assuming the parameter point that yields the most extreme values, point 5.

+ − 8.8 Results for the time-integrated branching ratio of Bs → µ µ

Using Eqs. (5.10) and (8.17), we can convert the theoretical branching ratio of Eq. (8.11) into a time- ¯ + − L integrated one. Fig. 8.7 shows aµ and the time-integrated B(Bs → µ µ ) for values of the ratio Rbs 0 ¯0 L that are consistent with Bs − Bs mixing constraints. The couplings appearing in Rbs are independent of those contributing to aµ, leading to aµ being static when varying this ratio. We see that there exists 48 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

¯ + − 0 ¯0 parameter space consistent with muon g − 2, experimental constraints on B(Bs → µ µ ), and Bs − Bs mixing parameters. For three of the five parameter points that explain muon g − 2, namely points 3, 4, L and 5, there are values of Rbs that yield the correct contributions to the branching ratio to perfectly resolve 0 ¯0 L the tension while being consistent with Bs − Bs mixing parameters. These values are Rbs = 2.2 + 0.4i, L L Rbs = 1.9 + 0i, and Rbs = 1.7 + 0.6i for points 3, 4, and 5, respectively.

(a) Aµµ , parameter point 2 (b) S , parameter point 2 ∆Γs µµ

(c) Aµµ , parameter point 3 (d) S , parameter point 3 ∆Γs µµ + − 8.8. RESULTS FOR THE TIME-INTEGRATED BRANCHING RATIO OF BS → µ µ 49

(e) Aµµ , parameter point 4 (f) S , parameter point 4 ∆Γs µµ

(g) Aµµ , parameter point 5 (h) S , parameter point 5 ∆Γs µµ Figure 8.5: Aµµ (left) and the CP asymmetry S (right) for a span of complex values of the ∆Γs µµ L L,† L (λ λ )bs ratio R = ∗ . Here, the nonzero up-type LQ couplings are given by parameter points bs VtbVts 0 ¯0 2, 3, 4, and 5 of Eqs. (8.27) to (8.30). The Bs − Bs mixing bounds are constant as the LQ couplings that vary between the parameter points do not appear in the expressions for ∆Ms or φs. 50 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

Figure 8.6: Aµµ vs. S . The dotted line shows the circle of values satisfying (Aµµ )2 + ∆Γs µµ ∆Γs 2 (Sµµ) = 1. The blue, shaded region shows the region of values that can be attained consistently 0 ¯0 with Bs −Bs mixing constraints for parameter point 5 given in Eq. (8.30). These values match those that fall within the overlap of the white regions of Figs. 8.5g and 8.5h.

Figure 8.7: The anomalous magnetic dipole moment of the muon aµ vs. the time-integrated + − branching ratio B¯(Bs → µ µ ). ∆aµ denotes the difference between the experimental and theoretical values of aµ, equal to the hypothetical NP contribution. The vertical lines represent 0 ¯0 parameter space consistent with Bs − Bs mixing parameters for the parameter points of Eqs. (8.13), (8.27), (8.28), (8.29), and (8.30). The cyan and green dotted lines mark the experimental values, and the shaded regions are the 1σ error bands. The SM prediction of the values of the observables is given, with error bars representing the theoretical uncertainties. 8.9. Z0 MODELS 51

8.9 Z0 models

Another way to potentially explain both muon g −2 and the b → s class of B-decay anomalies is to introduce a new spin-1 gauge boson that couples in the necessary ways. Such a boson needs to mediate the FCNC coupling b → s and couple non-chirally to muons. This new boson could mediate the FCNC transition b → s at the loop-level, like the SM Z, or at the tree-level. The simplest choice is the tree-level. In this section, the properties required of a Z0 model to explain both classes of anomalies are explored.

As discussed in Chapter 2, gauge bosons are associated with gauge groups. To add a new gauge boson, an extra gauge group needs to be added to the SM or an existing one modified. The simplest gauge groups are of dimension U(1), and the extension of the SM by adding an additional U(1) gauge group gives rise to a Z-like particle often called a Z0. The additional gauge group is often called U(1)0, and models of this kind are called Z0 models or U(1)0 models. Due to their simplicity, they make up one of the most popular classes of SM extensions and have been heavily studied [80]. Z0 bosons have also been the targets of many experimental searches, yielding high limits on their masses; most kinds of Z0 bosons are constrained to have masses of at least several TeV [34].

Because a Z0 needs to couple non-chirally to muons to explain the muon g − 2 anomaly, a natural way to give rise to the needed couplings in a Z0 model is to also make the Z0 couple non-chirally to b and s quarks. Including only these minimal interactions at tree level by making only the b, s, and muon fields charged under U(1)0, we can construct the Lagrangian terms

Z0 0 L ¯ µ R ¯ µ L µ R µ Lint = Zµ(λbsbLγ sL + λbsbRγ sR + λµµµ¯Lγ µL + λµµµ¯Rγ µR) + h.c., (8.32) where λH are dimensionless coupling strengths. Naively, these interaction terms would be enough to f1f2 explain the considered anomalies. However, additional couplings are needed to ensure gauge invariance of the Lagrangian. As follows from Noether’s theorem, the charge under the new U(1)0 gauge group must be conserved in all interaction vertices present in the theory in order for the Lagrangian to be gauge invariant 0 under the extended SU(3)c ⊗ SU(2)L ⊗ U(1)Y ⊗ U(1) gauge group. Generally, this is not possible when a Z0 is the only NP particle in a theory [80].

A solution to the problem of charge conservation is to add extra fermion and/or scalar fields to the theory, known as exotic particles. This is done in all fully-fledged Z0 models, c.f. refs. [81–84]. Extra scalar fields that acquire VEVs are also required to push the masses of the Z0 and the exotic fermions to TeV scales [83]. It is not uncommon for such exotic fermions to be hypothesized to be dark matter as done in Ref. [83].

To mediate the FCNC b → s, a Z0 needs to carry the difference in U(1)0 charge between the b and s quarks. However, for the Z0 to couple non-chirally to muons, the Z0 needs to carry zero U(1)0 charge; the Z0 boson needs to be neutral under U(1)0. The only possibility of contributing directly to muon g − 2 and b → s then is in the case when the b and s quarks carry equal U(1)0 charge. Another possibility comes from a Z0 that only mediates the FCNC b → s, not coupling directly to muons. Such a Z0 could contribute to muon g − 2 indirectly through kinetic mixing, a feature of any model involving multiple U(1) gauge groups. The gauge boson of one group can mix into that of the other through loop diagrams involving fermions charged under both gauge groups [85]. In this way, a Z0 could contribute to muon g −2 at the multiple-loop-level by mixing with the SM Z boson through a b or s quark charged under the U(1)0 gauge group.

This is where the present analysis stops. It would be interesting to take the analysis of the two kinds of Z0 models mentioned in the above paragraph further. In the first case, this would entail ensuring U(1)0 charge conservation in all SM interaction vertices, adding the minimum required amount of exotic particles in the process, one of which being a new scalar particle that acquires a VEV. In the second case, it would mean investigating whether kinetic mixing effects can be made large enough to explain muon g − 2 without violating constraints from nonanomalous observables. Doing this, one would find out how complicated of a Z0 model is needed to explain the muon g − 2 and B-physics anomalies related to the b → s transition, or 52 CHAPTER 8. MODEL-DEPENDENT ANALYSIS indeed, if the construction of such a model is even possible. 8.10. SUPERSYMMETRY 53

8.10 Supersymmetry

A class of models that have been studied for their ability to explain the muon g − 2 anomaly is that of supersymmetric models. Supersymmetry (SUSY) is a hypothetical symmetry between fermions and bosons. In SUSY, each SM field is contained in a supermultiplet along with a supersymmetric partner field giving rise to sparticles of opposite spin type to the SM particles; each SM fermion has a bosonic partner called a squark or a slepton, and each SM gauge boson field has a fermionic superpartner with the suffix “-ino”, i.e. the bino, wino, and gluino. To prevent interactions that violate SU(2)L ⊗ U(1)Y gauge invariance, a second scalar Higgs SU(2)L doublet is added along with the superpartners, higgsinos, of both Higgs doublets [86].

An argument for the proposal of supersymmetric models is that they neatly solve the fine-tuning problem of the Higgs mass [86]. In short: the observable mass of the Higgs boson will receive radiative corrections proportional to the masses of any NP particles that couple directly or indirectly to the Higgs boson—indeed, 0 the scalar leptoquark S1 and a massive Z boson would give rise to such corrections. As NP particles have not been directly observed in collider experiments, NP particles either are very heavy or interact very weakly with SM particles. If they are heavy, their contributions to the Higgs boson mass would be large. While the contributions of different particles can carry opposite signs and might cancel out to yield the relatively small Higgs mass of 125 GeV observed at the LHC, it seems strange that the large radiative corrections arising from heavy NP particles should not drive the Higgs mass up to a very high scale, that the Higgs mass instead is fine-tuned to the observed value. To be clear: the fine-tuning problem only arises when we posit the existence of heavy NP particles. This means that the SM does not contain the fine-tuning problem. The problem is one of naturalness, but it is a remarkable one as it is the only case we know of where a low-energy observable—the Higgs mass—is sensitive to high-energy phenomena. Returning to the discussion about effective field theory at the start of Chapter 4, as a rule we do not usually concern ourselves with high-energy phenomena when considering low-energy processes; we do not consider gravitational effects when considering particle physics interactions, nor do we consider relativistic effects when designing bridges. The fact that the Higgs mass is sensitive to the high masses of hypothetical NP particles that would appear only at high energies is a remarkable exception to this rule. In SUSY, sparticles precisely cancel the contributions of particles to the Higgs mass, solving the fine-tuning problem.

If SUSY were an unbroken symmetry, sparticles would have identical masses to their particle counterparts. As no sparticles have been observed in experiment to date [34], we know that if SUSY exists it must be broken. Many different ways to break SUSY have been proposed that proceed analogously to EWSB in that a new particle or particles is added that couples to sparticles and acquires a VEV at some high energy scale, causing the interaction terms between the new particle and the sparticles to give rise to additional mass terms for the sparticles that push their masses up to high scales [86]. In the process, the bino and neutral 3 wino interaction eigenstate fields mix analogously to the SM Bµ and Wµ fields into the photino and zino mass eigenstates. The Higgsinos mix with the winos to form electrically neutral neutralinos and charged charginos.A goldstino fermion also emerges in the breaking. Generally, supersymmetric models are agnostic to the exact mechanism behind SUSY breaking and instead directly assume the existence of these mass eigenstates.

The Minimally Supersymmetric SM (MSSM) adds only the sparticles mentioned above to the SM. The interactions of the sparticles are assumed to conserve R-parity, a quantity defined as

3(B−L)+2s PR ≡ (−1) , (8.33) where B, L, and s denote the baryon number, lepton number, and spin of a particle. R-parity serves to separate particles from sparticles; particles carry PR = 1 while sparticles carry PR = −1. The conservation of R-parity implies the conservation of baryon and lepton numbers in interactions involving sparticles.

Returning to the anomalies under consideration in this thesis, the MSSM has been used to explain muon g − 2, c.f. Ref. [87]. However, it cannot explain the B-decay anomalies related to the FCNC b → s. The 54 CHAPTER 8. MODEL-DEPENDENT ANALYSIS

MSSM, not containing any source of quark flavour violation beyond that of the charged-current interaction parameterized by the CKM matrix, does not contribute enough to the relevant Wilson coefficients C9 and C10 to explain these anomalies [88]. Instead, extensions of the MSSM with R-parity violation (RPV) are considered [89]. Models with RPV contain extra interaction terms that link quarks and leptons through sparticles that behave like leptoquarks. For instance, the authors of Ref. [89] consider muon sneutrinos together with right-handed sbottom quarks. Both of these sparticles contribute individually to the FCNC b → s through RPV interactions. Finally, the authors of Ref. [90] consider a RPV SUSY scenario that can explain the B-physics anomalies of RK(∗) and RD(∗) together with the muon g − 2 anomaly. 9 | Conclusions

Anomalies have been observed at the 2.5 to 4σ level in the B-physics anomalies and muon g − 2. These could be signals of NP and will, with increasing theoretical and experimental precision, be confirmed or denied as such. On the front of the B-physics anomalies, improved precision is needed in the determination of hadronic decay constants and form factors as well as in experiment. Further progress in the hadronic form factor determination is expected from numerical methods such as lattice QCD [65]. In experiment, analyses of the full LHC Run I and II datasets from LHCb, data from Belle II, and eventually data from the LHC Run III and the HL-LHC will clarify the picture. Regarding muon g − 2, as the precision of the theoretical SM prediction of muon g − 2 has in June 2020 reached and surpassed that of experiment, the results from Fermilab’s Muon g − 2 experiment [18] are highly anticipated. Either with these results or after the muon g − 2 experiments at J-PARC [74] and CERN [75] get underway, the fate of this long-standing anomaly will be determined.

In this thesis, we have investigated what properties of BSM theories are necessary to explain the B-physics anomalies related to the flavour-changing neutral current b → s simultaneously with the muon g−2 anomaly. + − Of the B-physics anomalies, we have focused on the anomalous, untagged branching ratio BR(Bs → µ µ ), 0 + − an average over the branching ratios of the rare leptonic decay Bs → µ µ and its CP-conjugate process. For a theory to explain both kinds of anomalies, a NP particle (or particles) that carries couplings to b and s as well as couplings to both left-and right-handed muons is needed. To explain in particular the anomalous + − BR(Bs → µ µ ) and the lepton flavour universality violation observed in the ratios RK and RK∗ , NP needs + − (∗) + − to suppress the muonic decay channels Bs → µ µ and B → K µ µ .

We have studied the scalar leptoquark S1 of representation 3¯ ⊗ 1 ⊗ 1/3 under the SM gauge group SU(3)c ⊗ SU(2)L ⊗ U(1)Y . This LQ carries the correct couplings to explain the B-physics anomalies related to the FCNC b → s and the muon g − 2 anomaly. Going beyond what has been done in the literature, we have calculated the contribution of this LQ to the observable Aµµ that parameterizes the effects of ∆Γs 0 ¯0 + − Bs − Bs mixing on the branching ratio of Bs → µ µ . We have constrained the allowed values of complex 0 ¯0 couplings by requiring consistency with experimental data on the Bs − Bs mixing observables ∆Ms and φs. Furthermore, we have calculated the CP asymmetries Sµµ and Cµµ within the LQ model and found that sizeable mixing-induced CP violation as measured by a nonzero Sµµ is allowed. Notably, Cµµ will always be zero within the model. As experimental precision on Aµµ increases and S is measured, these ∆Γs µµ theoretically clean observables can be used to further test the S1 model. Finally, we have used our expression for Aµµ to convert from the theoretical branching ratio of the decay B → µ+µ− to the time-integrated ∆Γs s one measured in experiment. We have computed this time-integrated branching ratio for different parameter points that explain the muon g − 2 anomaly, finding a difference of about 5% between the theoretical and time-integrated branching ratios for the most extreme parameter point. After taking this conversion into + − account, we have identified parameter space that explains the BR(Bs → µ µ ) anomaly and the muon g −2 0 ¯0 anomaly consistently with Bs − Bs mixing constraints.

Finally, we have briefly considered models with Z0 bosons and supersymmetry. A Z0 model might be able to explain the considered anomalies, either by both mediating the FCNC b → s and coupling non-chirally to muons or by only mediating the FCNC and contributing to muon g − 2 through kinetic mixing with

55 56 CHAPTER 9. CONCLUSIONS the SM Z boson. Further investigation is needed to find out whether either of these scenarios can explain the anomalies. Regarding SUSY, models beyond the MSSM that include R-parity violation are needed to explain the B-physics anomalies related to b → s. Such models are also viable for explaining muon g − 2.

In our analysis of the scalar LQ S1, we have not considered the LFUV observables RK and RK∗ . As these + − observables stem from the same FCNC as Bs → µ µ , it would be natural to investigate how the LQ impacts + − them. This is particularly true as the suppression of the dimuon decay channel Bs → µ µ will also suppress (∗) + − BR(B→K µµ) B → Kµ µ , contributing in the right way to RK(∗) ≡ BR(B→K(∗)ee) to explain the anomalies. An analysis of RK has been done for the LQ S1 in Ref. [78], but the resulting bounds on couplings have not been included 0 (∗) + − in this thesis. We also have not considered the anomalous branching ratios of Bs → K µ µ [53] and 0 + − Bs → φµ µ [54] related to this same FCNC, nor have we studied the anomalous angular observables in the 0 (∗) + − decay Bs → K µ µ [55–57]. To extend the range of the analysis to cover all of these observables, a more efficient approach than to consider them one by one would be to match the computed Wilson coefficient to global fits that include all the observables, making full use of the EFT machinery.

An interesting direction for further study of the scalar LQ S1 is to investigate its effects on the anomalous magnetic dipole moment of the electron, electron g − 2. A recent measurement of the fine-structure constant [91] has led to a SM prediction in tension at the level of 2.4σ with current experimental data [92]. It would be straightforward to extend the analysis of this work to include electron g − 2 and constraints coming from + − Bs → e e . The experimental value of electron g − 2 pulls in the opposite direction from its SM prediction relative to muon g − 2, and it would be interesting to explore whether the LQ S1 can accommodate for this while staying consistent with bounds on non-anomalous observables. In the same vein, although no experimental data for the process exist yet, it would be straightforward to investigate the contributions of the LQ to the magnetic moment of tau.

An original intent of the work described in this thesis was to use the computational programs SARAH and SPheno to investigate links between the B-physics anomalies and muon g − 2. Much of the work done during the master’s project described in this thesis has been in implementing BSM theories in SARAH and analyzing them in SPheno. As the results obtained using the programs have been lackluster, they are not included in the body of this thesis. However, an introduction to SARAH and SPheno is included in Appendix B, and our implementation of a model extending the SM by including the LQ S1 can be found in Appendix C. Appendices

57

A | Standard Model Lagrangian

Figure A.1: Write-out of the full SM Lagrangian after EWSB, compiled by T.D. Gutierrez at http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html using appendices of Martinus Veltman’s book Diagrammatica: The Path to Feynman Diagrams [93].

59 60 APPENDIX A. STANDARD MODEL LAGRANGIAN B | SARAH and SPheno

B.1 SARAH

SARAH is a Mathematica package designed for the study of BSM models [94]. Like many other computa- tional tools in the field [95–99], its roots are in the study of supersymmetric (SUSY) models. What separates SARAH from the other tools cited is that it is very general. An original purpose of SARAH was to enable “easy, fast and exhaustive” [94] study of non-minimal SUSY models, meaning SUSY models beyond the MSSM. Since its advent, SARAH has evolved to also support non-SUSY BSM models, coming pre-loaded not only with the MSSM and extensions of it but also e.g. the leptoquark model studied in Section 8.1.

B.1.1 Implementing a model in SARAH

To implement a model in SARAH, a user must specify the following aspects of the model:

1. Symmetries, global and local

2. Field content, including the charges of the fields under the gauge groups of the model

3. Potential part of the Lagrangian

Given this information, SARAH can derive the Lagrangian of the model. Given additional information about which scalar fields acquire vacuum expectation values (VEVs) and which fields mix in the resulting gauge symmetry breaking, SARAH can analytically calculate the renormalization group equations to the two-loop level, mass matrices for fermions and scalars, self-energy and tadpole corrections to the two-loop level, and the vertex factors for all interactions in the theory [94].

A model implementation in SARAH is split up into three files. The first is the model file wherein the bulk of the information about the model is specified. Second is a parameters file containing supplementary information such as definitions of mixing matrices and relations between parameters. Third is a particles file with more detailed information about the particles in the model. For a more illustrative look at how a model is implemented in SARAH, we here go through the model file of the SM.

(* Gauge Groups *)

Gauge[[1]]={B, U[1], hypercharge, g1,False}; Gauge[[2]]={WB, SU[2], left, g2,True}; Gauge[[3]]={G, SU[3], color, g3,False};

61 62 APPENDIX B. SARAH AND SPHENO

The model file is split up into blocks. The first block, Gauge Groups, introduces the symmetries of the model. A gauge symmetry is defined by specifying the name or the gauge boson of the group, the dimension of the gauge group (U(1), SU(2), etc.), the name appended to the dimension of the group (e.g. U(1)-hypercharge), the name of the gauge coupling, and whether or not the indices of the gauge group should be expanded. The expansion of indices is used to make some components of a gauge boson behave differently under EWSB than others. In the second line of the block above, the “True” indicates that the components of the SU(2) gauge boson W are to be treated separately in EWSB. Breaking gauge symmetries can be done either by assigning scalar fields nonzero VEVs in the EWSB block below.

(* Matter Fields *)

FermionFields[[1]] = {q, 3, {uL, dL}, 1/6, 2, 3}; FermionFields[[2]] = {l, 3, {vL, eL}, -1/2, 2, 1}; FermionFields[[3]] = {d, 3, conj[dR], 1/3, 1, -3}; FermionFields[[4]] = {u, 3, conj[uR], -2/3, 1, -3}; FermionFields[[5]] = {e, 3, conj[eR], 1, 1, 1}; ScalarFields[[1]] = {H, 1, {Hp, H0}, 1/2, 2, 1};

The matter fields, both fermionic and scalar, are defined in the block Matter Fields. For each field is first specified the name of the field, the number of generations, and the components of the field; e.g. the Q field of the SM contains both the left-handed up quark generations and the left-handed down-quark generations. Then, the behaviour of the fields under gauge transformations is specified, in order of definition in the Gauge Groups block. For U(1) groups, the negative charge of the field under the group is specified. For all other groups, the number denotes the number of eigenstates of the field under the group; the left-handed Q field is a doublet under SU(2) and a triplet under SU(3), while the right-handed d and u fields are singlets under SU(2) and antitriplets under SU(3).

(* ----- Before EWSB ----- *)

DEFINITION[GaugeES][LagrangianInput]= { {LagHC, {AddHC->True}}, {LagNoHC,{AddHC->False}} };

LagNoHC = -mu2 conj[H].H - 1/2 \[Lambda] conj[H].H.conj[H].H; LagHC = -(Yd conj[H].d.q + Ye conj[H].e.l + Yu u.q.H);

In the block Before EWSB, the Lagrangian part involving interactions of scalar fields of the model is written down in a short-hand notation. Indices are automatically added to fields and parameters, index contraction is performed, and hermitian conjugates of terms in the “LagHC” part are added to the full Lagrangian. In the case of interactions between the SU(2) doublets Q and `, SARAH automatically determines the correct combinations of doublet components by requiring the expanded interaction terms to be gauge invariant. The SU(2) doublet nature of the Q and ` fields means that any coupling between uL and νL also induces a coupling between dL and dL, and any coupling between uL and eL also induces one between dL and νL. All other parts of the Lagrangian are derived from the symmetries, the field content, and the charges of the fields under the gauge groups.

(* EWSB *)

NameOfStates={GaugeES, EWSB}; B.2. SPHENO 63

DEFINITION[EWSB][VEVs]= {{H0, {v, 1/Sqrt[2]}, {Ah, \[ImaginaryI]/Sqrt[2]},{hh, 1/Sqrt[2]}}};

DEFINITION[EWSB][GaugeSector] = {{{VB,VWB[3]},{VP,VZ},ZZ}, {{VWB[1],VWB[2]},{VWp,conj[VWp]},ZW}};

DEFINITION[EWSB][MatterSector]= {{{{dL}, {conj[dR]}}, {{DL,Vd}, {DR,Ud}}}, {{{uL}, {conj[uR]}}, {{UL,Vu}, {UR,Uu}}}, {{{eL}, {conj[eR]}}, {{EL,Ve}, {ER,Ue}}}};

After some blocks specifying the VEVs of scalar fields and which eigenstates we want to study, normally gauge eigenstates and mass eigenstates after EWSB, the EWSB rotation of gauge fields into mass eigenstates is treated. This is done both for gauge bosons and matter fields. In the SM, the U(1) B boson mixes with the third, longitudinal component of the SU(2) W boson into the mass eigenstates of the photon and the Z boson through a rotation matrix here called ZZ, defined in the parameters file. Similarly, left-and-right- handed down-type quark gauge eigenstates mix into left-and-right-handed mass eigenstates through the quark rotation matrices Vd and Vu. The same process occurs for up-type quarks and for leptons. The quark and lepton rotation matrices are computed by SARAH to ensure diagonal mass eigenstates and therefore well-defined mass terms.

(* Dirac-Spinors *)

DEFINITION[EWSB][DiracSpinors]={ Fd ->{ DL, conj[DR]}, Fe ->{ EL, conj[ER]}, Fu ->{ UL, conj[UR]}, Fv ->{ vL, 0}};

DEFINITION[EWSB][GaugeES]={ Fd1 ->{ FdL, 0}, Fd2 ->{ 0, FdR}, Fu1 ->{ Fu1, 0}, Fu2 ->{ 0, Fu2}, Fe1 ->{ Fe1, 0}, Fe2 ->{ 0, Fe2}};

Finally comes a block specifying how the Weyl spinors defined in the Matter Fields block fit together into Dirac spinors.

B.2 SPheno

SPheno, an abbreviation of “Supersymmetric Phenomenology”, is a computational tool acting as a spectrum- generator, meaning that it can compute the mass spectra of elementary particles for different models [100]. It can also compute many other observables and includes some flavour physics observables through the built-in module FlavorKit [101]. SARAH can write input files for SPheno and has for this reason been described by 64 APPENDIX B. SARAH AND SPHENO

Figure B.1: All tree-and one-loop-level diagrams used by FlavorKit to calculate Wilson coef- ficients for four-fermion operators. All possible permutations of external states are taken into account, and all possible internal fields in the model are included. Adapted from Ref. [94]. its authors as a “spectrum-generator-generator” [94]. This connection between SARAH and SPheno is useful because SPheno contains computation routines that SARAH does not [102]. These allow for calculations of additional observables, such as those included in FlavorKit. SPheno is also numerically fast due to being built in Fortran. The output of SPheno is stored in a spectrum file containing numerical values of all observables calculated by SPheno.

B.2.1 Flavour physics in SPheno: the FlavorKit module

The FlavorKit module comes with a collection of EFT operators and generic expressions for the corresponding Wilson coefficients, as well as generic expressions for many flavour observables listed in Appendices B and C of Ref. [101]. When a model is implemented in SARAH and ran through SPheno, FlavorKit will compute values of flavour observables. This is done through the following steps [101]:

1. An effective Lagrangian is set up that includes the operators relevant for a considered observable: X Leff = CiOi (B.1) i Working with an effective Lagrangian is equivalent to working with the effective Hamiltonian of Eq. (4.1). 2. Wilson coefficients are computed by performing matching of EFT amplitudes for relevant processes to amplitudes computed assuming the implemented model. For four-fermion processes, all tree-level and one-loop diagrams are taken into account. These are listed in Fig. B.1. The Wilson coefficients are if needed evolved down to a low energy scale. 3. The Wilson coefficients are plugged into generic expressions for the considered flavour observable.

The spectrum file generated by running the default SARAH implementation of the SM through SPheno lists + − the following result for the untagged, theoretical BR(Bs → µ µ ):

−9 BR(Bs → µµ)SM,SP heno = 2.99779624 × 10 , (B.2) B.2. SPHENO 65 a value significantly lower than the current SM prediction of Eq. (6.10).

SPheno computes the NP contribution to aµ at the one-loop level. For this, SPheno uses rather dated QCD results; the QCD input parameter values are taken from papers from 1999 and 2000 [97]. As mentioned in Chapter 7, much theoretical progress has been made in the past 20 years. Nevertheless, without matching present-day precision, the SPheno computation of aµ can be used to study the behaviour of the observable in different models of NP.

NP,SP heno To explain the muon g − 2 anomaly, the SPheno-computed NP contribution (aµ) should equal about the strain between the SM and experiment given in Eq. (7.10) and repeated here for convenience:

−11 ∆aµ = 279(76) × 10 , (B.3)

The SM spectrum file lists the following value:

NP,SP heno −14 (∆aµ) |SM = 2.07281881 × 10 , (B.4) a value that should be exactly zero but gets a negligible contribution from an unknown, possibly numerical, source. 66 APPENDIX B. SARAH AND SPHENO C | SARAH model file for the S1 scalar leptoquark model

Model‘Name = "ScalarLeptoquark"; Model‘NameLaTeX ="Scalar leptoquark S_1"; Model‘Authors = "A. Rehult"; Model‘Date = "2020-08-31";

SetOptions[MakeAll,IncludeCalcHep->False,IncludeWHIZARD->False];

(* Implementing the scalar leptoquark S1 of 1511.01900. Based on the implementation of 1512.06828 by T. Opferkuch with the scalar singlet S and DM candidate Chi stripped away.*)

(*------*) (* Particle Content *) (*------*)

(* Gauge Groups *)

Gauge[[1]]={B, U[1], hypercharge, g1,False}; Gauge[[2]]={WB, SU[2], left, g2,True}; Gauge[[3]]={G, SU[3], color, g3,False};

(* SM Fields *)

FermionFields[[1]] = {q, 3, {uL, dL}, 1/6, 2, 3}; FermionFields[[2]] = {l, 3, {vL, eL}, -1/2, 2, 1}; FermionFields[[3]] = {d, 3, conj[dR], 1/3, 1, -3}; FermionFields[[4]] = {u, 3, conj[uR], -2/3, 1, -3}; FermionFields[[5]] = {e, 3, conj[eR], 1, 1, 1};

ScalarFields[[1]] = {H, 1, {Hp, H0}, 1/2, 2, 1};

(* New Field *)

(* Scalar Leptoquark *) ScalarFields[[2]] = {Phi, 1, Phi0, -1/3, 1, 3};

(*------*) (* DEFINITION *) (*------*)

NameOfStates={GaugeES, EWSB};

67 68 APPENDIX C. SARAH MODEL FILE FOR THE S1 SCALAR LEPTOQUARK MODEL

(* ----- Before EWSB ----- *)

DEFINITION[GaugeES][LagrangianInput]= { {LagSMYuk, {AddHC->True}}, {LagSMHiggs, {AddHC->False}}, {LagLQNoHC, {AddHC->False}}, {LagLQ, {AddHC->True}} };

LagSMHiggs = mu2 conj[H].H - 1/2 \[Lambda] conj[H].H.conj[H].H ;

LagSMYuk = -(Yd conj[H].d.q + Ye conj[H].e.l + Yu H.u.q );

LagLQNoHC = -(MPhi2 conj[Phi].Phi + gHP conj[H].H.conj[Phi].Phi + 1/2 LPhi Delta[col1,col2] Delta[col3,col4] conj[Phi].Phi.conj[Phi].Phi );

LagLQ = -(lamL q.l.conj[Phi] + lamR conj[u].conj[e].conj[Phi]);

(* Gauge Sector *)

DEFINITION[EWSB][GaugeSector] = { {{VB,VWB[3]},{VP,VZ},ZZ}, {{VWB[1],VWB[2]},{VWp,conj[VWp]},ZW} };

(* ----- VEVs ---- *)

DEFINITION[EWSB][VEVs]= { {H0, {v, 1/Sqrt[2]}, {Ah, \[ImaginaryI]/Sqrt[2]}, {hh, 1/Sqrt[2]}}};

DEFINITION[EWSB][MatterSector]= { {{{dL}, {conj[dR]}}, {{DL,Vd}, {DR,Ud}}}, {{{uL}, {conj[uR]}}, {{UL,Vu}, {UR,Uu}}}, {{{eL}, {conj[eR]}}, {{EL,Ve}, {ER,Ue}}}};

(*------*) (* Dirac Spinors *) (*------*)

DEFINITION[EWSB][DiracSpinors]={ Fd ->{ DL, conj[DR]}, Fe ->{ EL, conj[ER]}, Fu ->{ UL, conj[UR]}, Fv ->{ vL, 0}};

DEFINITION[EWSB][GaugeES]={ Fd1 ->{ FdL, 0}, Fd2 ->{ 0, FdR}, Fu1 ->{ FuL, 0}, Fu2 ->{ 0, FuR}, Fe1 ->{ FeL, 0}, Fe2 ->{ 0, FeR}}; Bibliography

[1] H. Hartley. “John Dalton, F.R.S. (1766-1844) and the Atomic Theory-A Lecture to Commemorate his Bicentenary”. In: Proceedings of the Royal Society of London. Series B, Biological Sciences 168.1013 (1967), pp. 335–359. url: http://www.jstor.org/stable/75700. [2] J. Thomson. “XL. Cathode Rays”. In: Philosophical Magazine Series 5 44 (1897), pp. 293–316. doi: 10.1080/14786449708621070. [3] E. Rutherford. “The scattering of alpha and beta particles by matter and the structure of the atom”. In: Phil. Mag. Ser. 6 21 (1911), pp. 669–688. doi: 10.1080/14786440508637080. [4] J. Chadwick. “Possible Existence of a Neutron”. In: Nature 129 (1932), p. 312. doi: 10 . 1038 / 129312a0. [5] E. Riordan. “The Discovery of quarks”. In: Science 256 (1992), pp. 1287–1293. doi: 10.1126/science. 256.5061.1287. [6] ATLAS Collaboration. “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC”. In: Physics Letters B 716.1 (2012), pp. 1–29. doi: 10. 1016/j.physletb.2012.08.020. url: http://dx.doi.org/10.1016/j.physletb.2012.08.020. [7] CMS Collaboration. “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC”. In: Physics Letters B 716.1 (2012), pp. 30–61. doi: 10.1016/j.physletb.2012.08.021. url: http://dx.doi.org/10.1016/j.physletb.2012.08.021. [8] J. L. Feng. “Dark Matter Candidates from Particle Physics and Methods of Detection”. In: Ann. Rev. Astron. Astrophys. 48 (2010), pp. 495–545. doi: 10.1146/annurev-astro-082708-101659. [9] E. J. Copeland, M. Sami, and S. Tsujikawa. “Dynamics of dark energy”. In: Int. J. Mod. Phys. D 15 (2006), pp. 1753–1936. doi: 10.1142/S021827180600942X. [10] Y. Fukuda et al. “Evidence for oscillation of atmospheric neutrinos”. In: Phys. Rev. Lett. 81 (1998), pp. 1562–1567. doi: 10.1103/PhysRevLett.81.1562. [11] L. Canetti, M. Drewes, and M. Shaposhnikov. “Matter and Antimatter in the Universe”. In: New J. Phys. 14 (2012), p. 095012. doi: 10.1088/1367-2630/14/9/095012. [12] S. Capozziello and M. De Laurentis. “Extended Theories of Gravity”. In: Phys. Rept. 509 (2011), pp. 167–321. doi: 10.1016/j.physrep.2011.09.003. [13] G. Apollinari, I. B. Alonso, P. F. O. Brüning, M. Lamont, L. Rossi, and L. Tavian. “High-Luminosity Large Hadron Collider (HL-LHC): Technical Design Report V. 0.1”. In: 4/2017 (2017). doi: 10 . 23731/CYRM-2017-004. [14] D. London. Anomalies in B Decays: A Sign of New Physics? 2019. url: https://arxiv.org/abs/ 1911.06238. 0 + − [15] LHCb Collaboration. “Combination of the ATLAS, CMS and LHCb results on the B(s) → µ µ decays”. In: LHCb-CONF-2020-002. CERN-LHCb-CONF-2020-002 (2020). url: https://cds.cern. ch/record/2727207.

69 70 BIBLIOGRAPHY

[16] F. Jegerlehner. The Anomalous Magnetic Moment of the Muon. Vol. 274. Springer, 2017. doi: 10. 1007/978-3-319-63577-4. [17] T. Aoyama et al. The anomalous magnetic moment of the muon in the Standard Model. 2020. url: https://arxiv.org/abs/2006.04822. [18] J. Grange et al. Muon (g-2) Technical Design Report. 2015. url: https://arxiv.org/abs/1501. 06858. [19] A. Sakharov. “Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe”. In: Sov. Phys. Usp. 34.5 (1991), pp. 392–393. doi: 10.1070/PU1991v034n05ABEH002497. [20] W. a Heisenberg. “Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”. In: Z. Phys. 43 (1927), pp. 172–198. doi: 10.1007/BF01397280. [21] M. Merk, I. van Vulpen, and W. Hulsbergen. Lecture notes for the course Particle Physics 1. 2019. url: https://www.nikhef.nl/~wouterh/teaching/PP1/LectureNotes2019.pdf. [22] T. D. Lee and C. N. Yang. “Question of Parity Conservation in Weak Interactions”. In: Phys. Rev. 104 (1 1956), pp. 254–258. doi: 10.1103/PhysRev.104.254. url: https://link.aps.org/doi/10. 1103/PhysRev.104.254. [23] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson. “Experimental Test of Parity Conservation in Beta Decay”. In: Phys. Rev. 105 (4 1957), pp. 1413–1415. doi: 10.1103/PhysRev. 105.1413. url: https://link.aps.org/doi/10.1103/PhysRev.105.1413. [24] S. Weinberg. “A Model of Leptons”. In: Phys. Rev. Lett. 19 (21 1967), pp. 1264–1266. doi: 10.1103/ PhysRevLett.19.1264. url: https://link.aps.org/doi/10.1103/PhysRevLett.19.1264. [25] A. Salam. “Weak and Electromagnetic Interactions”. In: Conf. Proc. C 680519 (1968), pp. 367–377. doi: 10.1142/9789812795915_0034. 0 + − [26] L. van Dijk. “Branching ratio calculations for Bs → µ µ ”. MA thesis. Radboud University Nijmegen, 2020. [27] D. J. Gross and F. Wilczek. “Ultraviolet Behavior of Nonabelian Gauge Theories”. In: Phys. Rev. Lett. 30 (1973), pp. 1343–1346. doi: 10.1103/PhysRevLett.30.1343. [28] H. Politzer. “Asymptotic Freedom: An Approach to Strong Interactions”. In: Phys. Rept. 14 (1974), pp. 129–180. doi: 10.1016/0370-1573(74)90014-3. [29] P. Christakoglou. Lectures on Quantum Chromodynamics. Part of the course Particle Physics 2. 2019. [30] P. W. Higgs. “Broken Symmetries and the Masses of Gauge Bosons”. In: Phys. Rev. Lett. 13 (16 1964), pp. 508–509. doi: 10.1103/PhysRevLett.13.508. url: https://link.aps.org/doi/10.1103/ PhysRevLett.13.508. [31] M. E. Peskin and D. V. Schroeder. An Introduction to Quantum Field Theory. Addison-Wesley, 1995. [32] N. Cabibbo. “Unitary Symmetry and Leptonic Decays”. In: Phys. Rev. Lett. 10 (12 1963), pp. 531–533. doi: 10.1103/PhysRevLett.10.531. url: https://link.aps.org/doi/10.1103/PhysRevLett.10. 531. [33] M. Kobayashi and T. Maskawa. “CP-Violation in the Renormalizable Theory of Weak Interaction”. In: Progress of Theoretical Physics 49.2 (1973), pp. 652–657. doi: 10 . 1143 / PTP . 49 . 652. url: https://doi.org/10.1143/PTP.49.652. [34] Particle Data Group. “Review of Particle Physics”. In: Progress of Theoretical and Experimental Physics 2020.8 (2020). 083C01. doi: 10.1093/ptep/ptaa104. url: https://doi.org/10.1093/ ptep/ptaa104. [35] L. Wolfenstein. “Parametrization of the Kobayashi-Maskawa Matrix”. In: Phys. Rev. Lett. 51 (21 1983), pp. 1945–1947. doi: 10.1103/PhysRevLett.51.1945. url: https://link.aps.org/doi/10. 1103/PhysRevLett.51.1945. [36] R. Fleischer. “Flavour Physics and CP Violation”. In: 2005 European School of High-Energy Physics. 2006. url: https://arxiv.org/abs/hep-ph/0608010. BIBLIOGRAPHY 71

[37] CKMfitter Group. “CP violation and the CKM matrix: Assessing the impact of the asymmetric B factories”. In: Eur. Phys. J. C 41.1 (2005), pp. 1–131. doi: 10.1140/epjc/s2005-02169-1. and online update at http://www.utfit.org/. [38] UTfit Collaboration. “Model-independent constraints on F= 2 operators and the scale of new physics”. In: Journal of High Energy Physics 2008.03 (2008), pp. 049–049. doi: 10.1088/1126-6708/2008/ 03/049. url: http://dx.doi.org/10.1088/1126-6708/2008/03/049. and online update at http://www.utfit.org/. 0 [39] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay. “Evidence for the 2π Decay of the K2 Meson”. In: Phys. Rev. Lett. 13 (4 1964), pp. 138–140. doi: 10.1103/PhysRevLett.13.138. url: https://link.aps.org/doi/10.1103/PhysRevLett.13.138. [40] Belle Collaboration. “Observation of Large CP Violation in the Neutral B Meson System”. In: Phys. Rev. Lett. 87 (9 2001), p. 091802. doi: 10.1103/PhysRevLett.87.091802. url: https://link.aps. org/doi/10.1103/PhysRevLett.87.091802. [41] BaBar Collaboration. “Observation of CP Violation in the B 0 Meson System”. In: Phys. Rev. Lett. 87 (9 2001), p. 091801. doi: 10.1103/PhysRevLett.87.091801. url: https://link.aps.org/doi/ 10.1103/PhysRevLett.87.091801.

[42] LHCb Collaboration. “First observation of CP violation in the decays of Bs mesons”. In: Physical Review Letters 110.22 (2013). doi: 10.1103/physrevlett.110.221601. url: http://dx.doi.org/ 10.1103/PhysRevLett.110.221601. [43] LHCb Collaboration. “Observation of CP Violation in Charm Decays”. In: Physical Review Letters 122.21 (2019). doi: 10.1103/physrevlett.122.211803. url: http://dx.doi.org/10.1103/ PhysRevLett.122.211803. [44] N. Tuning. Particle Physics 2: Lecture notes on CP violation. 2020. url: https://www.nikhef.nl/ ~h71/Lectures/2020/ppII-cpviolation-14022020.pdf. [45] A. J. Buras. Weak Hamiltonian, CP Violation and Rare Decays. 1998. url: https://arxiv.org/ abs/hep-ph/9806471. [46] W. Beenakker. Reader for the course Quantum Field Theory. url: https://www.hef.ru.nl/~wimb/ dictaat_QFT_old.pdf. [47] P. Ball and R. Fleischer. “Probing new physics through B mixing: Status, benchmarks and prospects”. In: The European Physical Journal C 48.2 (2006), pp. 413–426. doi: 10.1140/epjc/s10052-006- 0034-4. url: http://dx.doi.org/10.1140/epjc/s10052-006-0034-4. 0 [48] M. Artuso, G. Borissov, and A. Lenz. CP Violation in the Bs system. 2019. url: https://arxiv. org/abs/1511.09466. [49] Heavy Flavor Averaging Group. Averages of b-hadron, c-hadron, and τ-lepton properties as of 2018. 2019. url: https://arxiv.org/abs/1909.12524. and online update at http://www.slac.stan- ford.edu/xorg/hfag. 0 0 0 [50] K. De Bruyn and R. Fleischer. “A Roadmap to Control Penguin Effects in Bd → J/KS and Bs → J/Ψφ”. In: Journal of High Energy Physics 2015.3 (2015). doi: 10.1007/jhep03(2015)145. url: http://dx.doi.org/10.1007/JHEP03(2015)145. [51] K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk, and N. Tuning. “Branching ratio measurements of Bs decays”. In: Physical Review D 86.1 (2012). doi: 10.1103/physrevd.86.014027. url: http://dx.doi.org/10.1103/PhysRevD.86.014027. [52] J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl, and D. M. Straub. “B- decay discrepancies after Moriond 2019”. In: The European Physical Journal C 80.3 (2020). doi: 10.1140/epjc/s10052-020-7817-x. url: http://dx.doi.org/10.1140/epjc/s10052-020-7817- x. [53] LHCb Collaboration. “Differential branching fractions and isospin asymmetries of B → K(∗)µ+µ− decays”. In: Journal of High Energy Physics 2014.6 (2014). doi: 10.1007/jhep06(2014)133. url: http://dx.doi.org/10.1007/JHEP06(2014)133. 72 BIBLIOGRAPHY

0 + − [54] LHCb Collaboration. “Angular analysis and differential branching fraction of the decay Bs → φµ µ ”. In: Journal of High Energy Physics 2015.9 (2015). doi: 10.1007/jhep09(2015)179. url: http: //dx.doi.org/10.1007/JHEP09(2015)179. [55] CMS Collaboration. “Angular analysis of the decay B0 K0+ from pp collisions at sqrt(s)=8 TeV”. In: Physics Letters B 753 (2016), pp. 424–448. doi: 10.1016/j.physletb.2015.12.020. url: http://dx.doi.org/10.1016/j.physletb.2015.12.020. 0 ∗ + − √ [56] ATLAS Collaboration. “Angular analysis of Bd → K µ µ decays in pp collisions at s = 8 TeV with the ATLAS detector”. In: ATLAS-CONF-2017-023 (2017). url: http://cds.cern.ch/record/ 2258146. 0 0 ∗0 + − [57] CMS Collaboration. “Measurement√ of the P1 and P5 angular parameters of the decay B → K µ µ in proton-proton collisions at s = 8 TeV”. In: CMS-PAS-BPH-15-008 (2017). url: http://cds. cern.ch/record/2256738. [58] LHCb Collaboration. “Search for lepton-universality violation in B+ → K+`+`− decays”. In: Physical Review Letters 122.19 (2019). doi: 10.1103/physrevlett.122.191801. url: http://dx.doi.org/ 10.1103/PhysRevLett.122.191801. [59] LHCb Collaboration. “Test of lepton universality with B0 → K∗0`+`− decays”. In: Journal of High Energy Physics 2017.8 (2017). doi: 10.1007/jhep08(2017)055. url: http://dx.doi.org/10. 1007/JHEP08(2017)055. − (∗) − [60] BaBar Collaboration. “Measurement of an excess of B → D τ ντ decays and implications for charged Higgs bosons”. In: Physical Review D 88.7 (2013). doi: 10.1103/physrevd.88.072012. url: http://dx.doi.org/10.1103/PhysRevD.88.072012. − [61] Belle Collaboration. “Measurement of the τlepton polarization and R(D∗) in the decay B → ∗ − D τ ντ ”. In: Physical Review Letters 118.21 (2017). doi: 10 . 1103 / physrevlett . 118 . 211801. url: http://dx.doi.org/10.1103/PhysRevLett.118.211801. 0 ∗− + 0 ∗− + [62] LHCb Collaboration. “Measurement of the ratio of the B → D τ ντ and B → D µ νµ branch- ing fractions using three-prong τ-lepton decays”. In: Physical Review Letters 120.17 (2018). doi: 10. 1103/physrevlett.120.171802. url: http://dx.doi.org/10.1103/PhysRevLett.120.171802. [63] S. L. Glashow, J. Iliopoulos, and L. Maiani. “Weak Interactions with Lepton-Hadron Symmetry”. In: Phys. Rev. D 2 (7 1970), pp. 1285–1292. doi: 10.1103/PhysRevD.2.1285. url: https://link.aps. org/doi/10.1103/PhysRevD.2.1285. 0 + − [64] R. Fleischer, R. Jaarsma, and G. Tetlalmatzi-Xolocotzi. “In pursuit of new physics with Bs,d → ` ` ”. In: Journal of High Energy Physics 2017.5 (2017). doi: 10.1007/jhep05(2017)156. url: http: //dx.doi.org/10.1007/JHEP05(2017)156. [65] Flavour Lattice Averaging Group. “FLAG Review 2019”. In: The European Physical Journal C 80.2 (2020). doi: 10.1140/epjc/s10052-019-7354-7. url: http://dx.doi.org/10.1140/epjc/s10052- 019-7354-7. [66] M. Beneke, C. Bobeth, and R. Szafron. “Power-enhanced leading-logarithmic QED corrections to +− Bq → ”. In: Journal of High Energy Physics 2019.10 (2019). doi: 10.1007/jhep10(2019)232. url: http://dx.doi.org/10.1007/JHEP10(2019)232. [67] R. Fleischer, D. G. Espinosa, R. Jaarsma, and G. Tetlalmatzi-Xolocotzi. “CP violation in leptonic 0 rare Bs decays as a probe of new physics”. In: The European Physical Journal C 78.1 (2017). doi: 10.1140/epjc/s10052-017-5488-z. url: http://dx.doi.org/10.1140/epjc/s10052-017-5488- z. 0 + − [68] LHCb Collaboration. “Measurement of the Bs → µ µ branching fraction and effective lifetime and search for B0 → µ+µ− decays”. In: Physical Review Letters 118.19 (2017). doi: 10 . 1103 / physrevlett.118.191801. url: http://dx.doi.org/10.1103/PhysRevLett.118.191801. [69] R. G. Milner. A Short History of Spin. 2013. url: https://arxiv.org/abs/1311.5016. [70] S. Weinberg. The Quantum theory of fields. Vol. 1: Foundations. Cambridge University Press, 2005, pp. 454–457. BIBLIOGRAPHY 73

[71] G. W. Bennett et al. “Final report of the E821 muon anomalous magnetic moment measurement at BNL”. In: Physical Review D 73.7 (2006). doi: 10 . 1103 / physrevd . 73 . 072003. url: http : //dx.doi.org/10.1103/PhysRevD.73.072003. [72] T. Aoyama, T. Kinoshita, and M. Nio. “Theory of the Anomalous Magnetic Moment of the Electron”. In: Atoms 7.1 (2019), p. 28. doi: 10.3390/atoms7010028.

[73] C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim. “The electroweak contributions to (g −2)µ after the Higgs boson mass measurement”. In: Phys. Rev. D88 (2013), p. 053005. doi: 10.1103/PhysRevD. 88.053005. [74] M. Abe et al. “A new approach for measuring the muon anomalous magnetic moment and electric dipole moment”. In: Progress of Theoretical and Experimental Physics 2019.5 (2019). 053C02. doi: 10.1093/ptep/ptz030. url: https://doi.org/10.1093/ptep/ptz030. [75] G. Abbiendi. “Letter of Intent: the MUonE project”. In: CERN-SPSC-2019-026. SPSC-I-252 (2019). url: https://cds.cern.ch/record/2677471. [76] H. Georgi and S. Glashow. “Unity of All Elementary Particle Forces”. In: Phys. Rev. Lett. 32 (1974), pp. 438–441. doi: 10.1103/PhysRevLett.32.438. [77] I. Dorner, S. Fajfer, and O. Sumensari. “Muon g − 2 and scalar leptoquark mixing”. In: JHEP 06 (2020), p. 089. doi: 10.1007/JHEP06(2020)089.

[78] M. Bauer and M. Neubert. “Minimal Leptoquark Explanation for the RD(∗) , RK , and (g − 2)µ Anomalies”. In: Physical Review Letters 116.14 (2016). doi: 10.1103/physrevlett.116.141802. url: http://dx.doi.org/10.1103/PhysRevLett.116.141802.

[79] G. Hiller and M. Schmaltz. “RK and future b → s`` BSM opportunities”. In: Physical Review D 90.5 (2014). doi: 10.1103/physrevd.90.054014. url: http://dx.doi.org/10.1103/PhysRevD.90. 054014. [80] P. Langacker. “The physics of heavy Z0 gauge bosons”. In: Reviews of Modern Physics 81.3 (2009), pp. 1199–1228. doi: 10 . 1103 / revmodphys . 81 . 1199. url: http : / / dx . doi . org / 10 . 1103 / RevModPhys.81.1199. [81] M. Chala, M. Duerr, F. Kahlhoefer, and K. Schmidt-Hoberg. “Tricking Landau-Yang: How to obtain the diphoton excess from a vector resonance”. In: Physics Letters B 755 (2016), pp. 145–149. doi: 10. 1016/j.physletb.2016.02.006. url: http://dx.doi.org/10.1016/j.physletb.2016.02.006. [82] T. Appelquist, B. A. Dobrescu, and A. R. Hopper. “Nonexotic neutral gauge bosons”. In: Physical Review D 68.3 (2003). doi: 10.1103/physrevd.68.035012. url: http://dx.doi.org/10.1103/ PhysRevD.68.035012. [83] R. Martínez, J. Nisperuza, F. Ochoa, and J. P. Rubio. “Some phenomenological aspects of a new U(1)0 model”. In: Physical Review D 89.5 (2014). doi: 10.1103/physrevd.89.056008. url: http: //dx.doi.org/10.1103/PhysRevD.89.056008.

[84] D. Borah, L. Mukherjee, and S. Nandi. Low Scale U(1)X Gauge Symmetry as an Origin of Dark Matter, Neutrino Mass and Flavour Anomalies. 2020. url: https://arxiv.org/abs/2007.13778. [85] K. Schmitz. Kinetic Mixing in Field Theory. DESY Workshop Seminar, Winter Semester 2009/2010. 2009. url: https://bib-pubdb1.desy.de/record/88159/files/Kinetic_Mixing_in_Field_ Theory.pdf. [86] S. P. Martin. “A Supersymmetry primer”. In: Perspectives on supersymmetry. Vol.2. Vol. 21. 2010, pp. 1–153. doi: 10.1142/9789812839657_0001. [87] T. Moroi. “The Muon anomalous magnetic dipole moment in the minimal supersymmetric standard model”. In: Phys. Rev. D 53 (1996). [Erratum: Phys.Rev.D 56, 4424 (1997)], pp. 6565–6575. doi: 10.1103/PhysRevD.53.6565. [88] W. Altmannshofer and D. M. Straub. “New physics in b → s transitions after LHC run 1”. In: The European Physical Journal C 75.8 (2015). doi: 10.1140/epjc/s10052-015-3602-7. url: http://dx.doi.org/10.1140/epjc/s10052-015-3602-7. 74 BIBLIOGRAPHY

[89] Q.-Y. Hu, Y.-D. Yang, and M.-D. Zheng. “Revisiting the B-physics anomalies in R-parity violating MSSM”. In: The European Physical Journal C 80.5 (2020). doi: 10.1140/epjc/s10052-020-7940-8. url: http://dx.doi.org/10.1140/epjc/s10052-020-7940-8.

[90] W. Altmannshofer, P. S. B. Dev, A. Soni, and Y. Sui. “Addressing RD(∗) , RK(∗) , muon g − 2 and ANITA anomalies in a minimal R-parity violating supersymmetric framework”. In: Physical Review D 102.1 (2020). doi: 10.1103/physrevd.102.015031. url: http://dx.doi.org/10.1103/PhysRevD. 102.015031. [91] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Müller. “Measurement of the fine-structure constant as a test of the Standard Model”. In: Science 360.6385 (2018), pp. 191–195. doi: 10.1126/science. aap7706. url: http://dx.doi.org/10.1126/science.aap7706. [92] D. Hanneke, S. Fogwell, and G. Gabrielse. “New Measurement of the Electron Magnetic Moment and the Fine Structure Constant”. In: Physical Review Letters 100.12 (2008). doi: 10.1103/physrevlett. 100.120801. url: http://dx.doi.org/10.1103/PhysRevLett.100.120801. [93] M. Veltman. Diagrammatica: The Path to Feynman rules. Vol. 4. Cambridge University Press, 2012. [94] F. Staub. “Exploring New Models in All Detail with SARAH”. In: Advances in High Energy Physics 2015 (2015), pp. 1–126. doi: 10.1155/2015/840780. url: http://dx.doi.org/10.1155/2015/ 840780. [95] B. Allanach and T. Cridge. “The calculation of sparticle and Higgs decays in the minimal and next-to- minimal supersymmetric standard models: SOFTSUSY4.0”. In: Computer Physics Communications 220 (2017), pp. 417–502. doi: 10.1016/j.cpc.2017.07.021. url: http://dx.doi.org/10.1016/j. cpc.2017.07.021. [96] H. Bahl et al. Precision calculations in the MSSM Higgs-boson sector with FeynHiggs 2.14. 2018. url: https://arxiv.org/abs/1811.09073. [97] W. Porod and F. Staub. “SPheno 3.1: extensions including flavour, CP-phases and models beyond the MSSM”. In: Computer Physics Communications 183.11 (2012), pp. 2458–2469. doi: 10.1016/j. cpc.2012.05.021. url: http://dx.doi.org/10.1016/j.cpc.2012.05.021. [98] A. Djouadi, J.-L. Kneur, and G. Moultaka. “SuSpect: A Fortran code for the Supersymmetric and Higgs particle spectrum in the MSSM”. In: Computer Physics Communications 176.6 (2007), pp. 426– 455. doi: 10.1016/j.cpc.2006.11.009. url: http://dx.doi.org/10.1016/j.cpc.2006.11.009. [99] H. Baer, F. E. Paige, S. D. Protopescu, and X. Tata. ISAJET 7.69: A Monte Carlo Event Generator for pp, pp¯ , and e=e− Reactions. 2003. url: https://arxiv.org/abs/hep-ph/0312045. [100] A. Vicente. Computer tools in particle physics. 2015. url: https://arxiv.org/abs/1507.06349. [101] W. Porod, F. Staub, and A. Vicente. “A flavor kit for BSM models”. In: The European Physical Journal C 74.8 (2014). doi: 10.1140/epjc/s10052-014-2992-2. url: http://dx.doi.org/10. 1140/epjc/s10052-014-2992-2. [102] F. Staub. SARAH (part III): SPheno interface. Lecture at the Dartmouth-TRIUMF HEP Tools Bootcamp. 2017. url: https : / / indico . cern . ch / event / 656211 / contributions / 2756818 / attachments/1544315/2423185/sarah_part3.pdf.