Renormalization Group Invariants in the Minimal Supersymmetric Standard Model

Home , Electroweak interaction, Quartic interaction, Unitarity gauge

Master’s Thesis September 2014

Supervisor: Prof. dr. W.J.P. Beenakker

Department of Theoretical High Energy Physics Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen

Contents

1 Introduction 5 1.1 Conventions...... 5

2 The Standard Model 7 2.1 IntroductiontotheStandardModel ...... 7 2.2 TheStandardModelﬁeldcontent...... 7 2.3 The Standard Model Lagrangian ...... 8 2.4 Flavormixing...... 10

3 Symmetries of the Standard Model 15 3.1 Noether’stheorem ...... 15 3.2 Poincarésymmetries ...... 16 3.3 Gaugesymmetries ...... 16 3.4 Flavorsymmetries ...... 17 3.4.1 Flavor group without flavor mixing ...... 17 3.4.2 Flavor group with flavor mixing ...... 21

4 Supersymmetry 25 4.1 Introductiontosupersymmetry ...... 25 4.2 Motivation for supersymmetry ...... 26 4.3 Supersymmetric Lagrangians ...... 29 4.3.1 Chiralsupermultiplets ...... 29 4.3.2 Gaugesupermultiplets...... 30 4.3.3 Chiral and gauge supermultiplets combined ...... 31 4.4 Supersymmetrybreaking...... 32

5 The Minimal Supersymmetric Standard Model 33 5.1 TheMSSMﬁeldcontent...... 33 5.2 R-parity...... 34 5.3 The MSSM Lagrangian ...... 34 5.4 Softsupersymmetrybreaking ...... 37 5.5 Countingindependentparameters ...... 38 5.6 SymmetriesoftheMSSM ...... 42

6 Renormalization group techniques 45 6.1 What is renormalization? ...... 45 6.2 Renormalization group equations ...... 46 6.3 E↵ectiveﬁeldtheories ...... 46 6.4 Renormalization group invariants ...... 47

7 One-loop -functions for MSSM scalar masses 51 7.1 The propagator for scalar fields ...... 51 7.1.1 Higgs doublets ...... 52 7.1.2 Scalarfields...... 52 7.2 One-loop corrections to scalar propagators ...... 53 7.2.1 Type 1 diagrams ...... 54 7.2.2 Type 2 diagrams ...... 55 7.2.3 Type 3 diagrams ...... 56 7.2.4 Type 4 diagrams ...... 58 7.3 Renormalization of scalar fields ...... 59 7.3.1 Field renormalization ...... 59 7.3.2 Mass renormalization ...... 60 7.4 Results...... 61

3 8 RG invariants in the MSSM 62 8.1 Constructing RG invariants in the canonical way ...... 62 8.2 RG invariants from a di↵erentperspective ...... 65

9 Conclusion 68

AIndices 69

B Unitary matrices 70

C Spinors 72 C.1 Diracspinors ...... 72 C.2 Weylspinors ...... 72

D Superpotential contributions to the MSSM Lagrangian 74

E Integrals 78

F One-loop mass counterterms for MSSM scalar ﬁelds 83

List of abbreviations 87

Acknowledgements 88

Bibliography 90

4 1 Introduction

The experiments at the Large Hadron Collider (LHC) at CERN, among other experiments, are searching for physics beyond the Standard Model of particle physics, since we know that the Standard Model cannot be the final story. Experimentalists are looking for signals that do not fit well in the Standard Model framework. One job for theorists, on the other hand, is to come up with new models that describe physics at even smaller length scales (or, equivalently, higher energy scales) that solve the current problems with the Standard Model. Over the last decades, many new theories have been described in literature, all coming with both pros and cons. In this thesis we will focus on one particular beyond the Standard Model theory: the so-called Minimal Supersymmetric Standard Model (MSSM). This is the most minimal extension to the Standard Model that respects an additional spacetime symmetry, called supersymmetry. In short, the MSSM predicts that each Standard Model particle has a supersymmetric counterpart with exactly the same quantum numbers and mass (except for the spin that di↵ers by half a unit). We already know that even the MSSM could not be the “theory of everything” though. Namely, since supersymmetric particles have not been found yet, this means that either supersymmety is simply not implemented in nature, or that supersymmetric particles are much heavier (at least beyond the reach of the LHC) than their Standard Model counterparts, i.e. supersymmetry is a broken symmetry of nature. In this thesis, of course, we will assume the latter. At energy scales many orders of magnitude higher than the typical LHC collider scale of (10 TeV), even beyond the range of validity of the MSSM, new physics would provide boundary conditionsO to the MSSM. Say that supersymmetric particles are discovered at the LHC and that some relevant new parameters are measured, would there then be a way to check which of those boundary conditions could be realized in nature? The answer is yes: quite recently, a new method has been devised that could probe these boundary conditions by making clever use of so-called renormalization group (RG) invariants using low energy data only (i.e. the new data coming from particle accelerators). Constructing these RG invariants as probes for high scale physics models can be a very tedious and complicated job though, and this has not been done yet for (the scalar sector of) the MSSM, but only for heavily constrained supersymmetric models (in these models typically about 80% of the MSSM parameters are “thrown away”). In this thesis we will attempt to construct RG invariant quantities for the full MSSM, i.e. without any simplifications. In this thesis a thorough analysis of symmetries of the MSSM will be performed. Most of this analysis will be carried out for the Standard Model though (as this theory is a bit simpler) and the results will then be generalized to the MSSM. Finally, an attempt will be made to unify symmetries of the MSSM with the concept of RG invariant quantities. If such a relation would turn out to exist, then it would be much easier to tell for a given (supersymmetric) theory if RG invariants could exist at all, and, if the answer is positive, how to construct them. After all, simplifying a certain theory (i.e. setting parameters equal to each other or to zero) generally means that symmetry is added to its Lagrangian. Could this in turn mean that also the number of RG invariants increases? In the first section we will discuss the Standard Model, mostly focussing on its Lagrangian and mixing of fermionic flavors. In section 2 we will present a thorough analysis of symmetries of the Standard Model. The obtained results for family space will be most important for the remainder of this thesis. Subsequently, the basic concepts of supersymmetry will be discussed and a general supersymmetric Lagrangian will be constructed. In the next section we will consider the MSSM and perform a careful count of its independent parameters. Also the symmetry analysis for the Standard Model will then be generalized to the MSSM. In the subsequent section, the concept of renormalization and the utility of RG invariants will be discussed. Section 7 will be the most technical part of this thesis. Here the one-loop RG equations for the MSSM scalar masses will be derived. Finally, an attempt will be made to construct RG invariants in the MSSM from these obtained RG equations and we will consider the role of symmetries in this context.

1.1 Conventions Throughout this thesis we will make use of the following conventions:

We adopt natural units where ~ = c = 1. With this choice, mass is the only dimension that •

5 is left, i.e.: 1 1 [mass] = [energy] = [length] =[time] , (1) where the square brackets denote the mass dimension. We make use of the Einstein summation convention, which means that all repeated indices • are summed over (unless explicitly stated otherwise). The various types of indices that are used in this thesis are deﬁned in appendix A. For the Minkowski metric ⌘ we choose the convention ⌘ = diag(1, 1, 1, 1). • µ⌫ µ⌫ We deﬁne the contravariant spacetime and energy-momentum four-vectors to be • xµ =(t, ~x) ,pµ =(E,p~ ) , (2)

and the (ordinary) derivative is deﬁned as

@ @ . (3) µ ⌘ @xµ

The n n identity matrix will be denoted by In, and the most important matrices in family • space will⇥ be indicated in boldface (to make the text more readable).

It is important to be aware of the fact that the sections on the Standard Model make use • of Dirac spinors to describe the fermionic ﬁelds, while the sections on supersymmetry and renormalization group techniques use Weyl spinors instead. The di↵erences between Dirac and Weyl spinors are explained in appendix C. The abbreviation “h.c.” is short for “Hermitian conjugate”. •

6 2 The Standard Model

The Standard Model of particle physics incorporates all known fundamental particles and their (non-gravitational) interactions and forms the heart of particle physics. In this section we brieﬂy review the main aspects of the Standard Model. We will discuss its particle content and provide the most important mathematical structures. Much more on the Standard Model can be found in any textbook on quantum ﬁeld theory or particle physics, e.g. in [1].

2.1 Introduction to the Standard Model The Standard Model is a quantum field theory that describes the dynamics of all known fundamental particles up to an incredible accuracy. It provides the framework for three out of the four (known) fundamental forces of nature: the electromagnetic, weak and strong interactions. The Standard Model was completed in the early 1970s, but not fully experimentally verified until the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 [2, 3]. Despite the great success of the Standard Model, there are also a couple of (significant) shortcomings. One of them being the complete absence of gravity. It is expected that at the Planck scale gravity becomes comparable in strength to the other forces, which implies that quantum gravity e↵ects can no longer be ignored1. In other words, the Standard Model breaks down and a reconciliation of general relativity and quantum mechanics is needed. This is why the Standard Model cannot be the theory of everything; it provides an excellent e↵ective framework for “low” energy physics, but for higher energies (i.e. shorter distance scales) we simply need a new theory. An important beyond the Standard Model candidate theory is supersymmetry. In section 4 the ideas behind supersymmetry are elaborately discussed, as well as the shortcomings of the Standard Model and how supersymmetry claims to solve these.

2.2 The Standard Model field content In this subsection we outline the Standard Model field content. The Standard Model contains both fermions (the matter particles) and bosons. The fermionic sector consists of three families of quarks and leptons. The quarks come in two di↵erent types: charge2 +2/3, the up-type quarks (up, charm and top); and charge 1/3, the down-type quarks (down, strange and bottom). Likewise, also the leptons come in two di↵erent types: charge 1, the charged leptons (electron, muon and tau); and the neutral neutrinos (electon neutrino, muon neutrino and tau neutrino). The three fundamental interactions described by the Standard Model are mediated by spin- 1 gauge bosons. Gauge bosons arise from the theory by requiring local invariance under gauge transformations corresponding to a certain gauge group. The full gauge group of the Standard Model is SU(3)C SU(2)L U(1)Y. The Standard Model gauge groups are Lie groups and the number of gauge bosons⇥ equals⇥ the number of generators. The gauge bosons “live” in the adjoint representation of the gauge group, while the fermions that couple to the gauge bosons “live” in the fundamental representation. The various representations are referred to with their dimensions indicated in boldface. The particles that are in the fundamental representations 2 and 3 are called doublets and triplets respectively. A fermion that is uncharged under a certain gauge group is in 1 (the trivial representation) and is called a singlet. The first gauge group, SU(3)C, forms the underlying symmetry group of the strong interaction. Its corresponding eight gauge bosons are the massless gluons. The “C” stands for color, the quantum number that indicates sensitivity to the strong interaction. The colored particles in the Standard Model, i.e. the particles that participate in the strong interaction, include quarks (color triplets) and gluons (color octets). The second group, SU(2)L U(1)Y, describes the electroweak sector of the Standard Model. The electroweak interaction is⇥ mediated by the W1,2,3 and B bosons. The W1,2,3 bosons, that come from the SU(2)L part, only couple to particles with a non-zero weak isospin (the weak isospin doublets), or di↵erently stated, only to particles that have a left-handed chirality (hence

1One of these e↵ects is that around the Planck scale the wavelength of a particle typically becomes shorter than its Schwarzschild radius due to Heisenberg’s uncertainty principle. This implies that studying these short distance scales by performing high energy particle collisions, inevitably results in black hole production. 2When we write just “charge”, we mean electrical charge.

7 3 the “L” in the name of the group) . The B boson corresponds to the U(1)Y part and couples to all particles with a non-vanishing weak hypercharge (Y ). From experiment we know that certain particles are massive while others are massless. This means that we need to add mass terms to the Standard Model Lagrangian. If we do this in the simplest possible way we run into trouble. Namely, these mass terms spoil the gauge invariance of the theory. As the principle of local gauge invariance provides the basis of the theory, we better respect that. The easiest way to add mass terms to the Lagrangian in a gauge invariant way, is by introducing a complex SU(2) doublet scalar field. If this spin-0 field has a non-zero vacuum (i.e. ground state) expectation value, then after nature has chosen a ground state, the electroweak symmetry is spontaneously broken and the Standard Model particles have acquired a mass. This is called the Higgs mechanism, first described by Higgs, Brout and Englert (cf. [4,5]). In terms of groups, electroweak symmetry breaking is described as follows:

SU(3) SU(2) U(1) SU(3) U(1) (4) C ⇥ L ⇥ Y ! C ⇥ Q After electroweak symmetry breaking, the complex scalar doublet gives rise to one (massive) physical state called the Higgs ﬁeld. This breaking also mixes the electroweak gauge boson eigenstates to yield the familiar mass eigenstates that we observe in experiments: the massless photon, the mediator of the electromagnetic interaction; and the massive W± and Z bosons, the media- tors of the weak interaction. The electromagnetic interaction is characterized by the group U(1)Q, where “Q” stands for the electrical charge. Its mediator, the photon, couples to all particles that carry an electrical charge. This is the Standard Model in a nutshell. All Standard Model ﬁelds (before electroweak symmetry breaking) and their gauge group representations are given in table 1. Note that we have included the right-handed neutrino. In this thesis we assume that the right- handed neutrino completely decouples from the other particles.

Name Field SU(3) SU(2) U(1) C ⇥ L ⇥ Y L =(⌫ e )(1, 2, 1 ) L L L 2 leptons ⌫R (1, 1, 0) e (1, 1, 1) R 1 QL =(uL dL)(3, 2, 6 ) quarks 2 uR (3, 1, 3 ) d (3, 1, 1 ) R 3 + 0 1 Higgs boson =( )(1, 2, 2 ) Ga (8, 1, 0) gauge bosons W a0 (1, 3, 0) B (1, 1, 0)

Table 1: The ﬁelds of the Standard Model and their gauge group representations. The ﬁrst two numbers in the last column (those indicated in boldface) refer to the dimension of the gauge group representations, while the last number denotes the eigenvalue of the weak hypercharge generator Y .

2.3 The Standard Model Lagrangian Any quantum field theory is usually described by a Lagrangian. A Lagrangian generally involves several fields and contains kinetic and interaction terms. The Euler-Lagrange equations – originating from the principle of stationary action – give the equations of motion for the fields. Now let

3In appendix C we deﬁne the nomenclature for the left- and right-handed fermionic ﬁelds.

8 us consider the Lagrangian that describes the Standard Model. We will do this for the unbroken case and piece by piece. The names of the ﬁelds are given in table 1.

Kinetic terms for the gauge fields and interactions among themselves: • 1 1 1 L = Ga Gaµ⌫ W a0 W a0µ⌫ B Bµ⌫ , (5) gauge 4 µ⌫ 4 µ⌫ 4 µ⌫ where the field strength tensors are defined as follows:

Ga = @ Ga @ Ga g f abcGb Gc , (6a) µ⌫ µ ⌫ ⌫ µ s µ ⌫ W a0 = @ W a0 @ W a0 g✏a0b0c0 W b0 W c0 , (6b) µ⌫ µ ⌫ ⌫ µ µ ⌫ B = @ B @ B . (6c) µ⌫ µ ⌫ ⌫ µ The coupling constants pertaining to the strong, weak and electromagnetic interactions are denoted a0b0c0 abc respectively by gs,g,g0. The symbols ✏ and f are the completely antisymmetric structure constants of SU(2) and SU(3) respectively.

Kinetic terms for the fermions and gauge interactions: •

Lfermions = iLLDL/ L + i⌫R@⌫/ R + ieRDe/ R + iQLDQ/ L + iuRDu/ R + idRDd/ R , (7) where all fermionic ﬁelds are multiplets in family space (more on this in the next subsection). The gauge covariant derivatives are given by:

1 a0 a0 1 D L = @ + igW ig0B L , (8a) µ L µ 2 µ 2 µ L ✓ ◆

D e = @ ig0B e , (8b) µ R µ µ R ✓ ◆

1 a a 1 a0 a0 1 D Q = @ + ig G + igW + ig0B Q , (8c) µ L µ 2 s µ 2 µ 6 µ L ✓ ◆ 1 a a 2 D u = @ + ig G + ig0B u , (8d) µ R µ 2 s µ 3 µ R ✓ ◆ 1 a a 1 D d = @ + ig G ig0B d . (8e) µ R µ 2 s µ 3 µ R ✓ ◆ Here a0 (the Pauli matrices) and a (the Gell-Mann matrices) are twice the generators of the SU(2) and SU(3) Lie algebras respectively. The conventions for the left- and right-handed projections of the Dirac spinors are found in appendix C.

Yukawa couplings: • 2 2 L = iL y ⇤⌫ L y e iQ y ⇤u Q y d +h.c., (9) Yukawa L ⌫ R L e R L u R L d R where y⌫ , ye, yu, yd are general complex 3 3 matrices in family space, called the Yukawa coupling matrices. These couplings are discussed in⇥ the next subsection in much more detail. Note that we are taking the neutrinos massive here by including a right-handed neutrino multiplet ⌫R.This means that we simply treat the neutrinos the same as the other fermions4. In the sections on supersymmetry, however, the neutrinos are taken massless.

Kinetic term for the Higgs ﬁeld and Higgs self-interactions: • µ L =(D )† D V, (10) Higgs µ where the gauge covariant derivative for the Higgs doublet is given by:

1 a0 a0 1 D = @ + igW + ig0B . (11) µ µ 2 µ 2 µ ✓ ◆ 4Within the Standard Model the right-handed neutrino would be a Dirac particle, while in some beyond the Standard Model theories it also carries a Majorana component.

9 The Higgs potential V = V ( ) is a function of the Higgs doublet and given by: | | V ( )= µ2 2 + 4 , (12) | | h| | | | 2 where , µh > 0. The minimum of this potential is located at µ v = h = , (13) | | p2 p2 where v µh/p represents the vacuum expectation value of the Higgs doublet (i.e. 0 0 = v/p2). The⌘ groundstate of the Higgs doublet is actually degenerate and each state leads toh || the|| samei physics. Once nature has chosen one of these vacua, the electroweak symmetry is spontaneously broken. The Higgs doublet then takes the following form in the unitarity gauge:

1 0 = , (14) p2 v + h ✓ ◆ with h denoting the real physical Higgs scalar field for which 0 h 0 = 0. The sum of (5), (7), (9) and (10) gives the full Lagrangianh | describing| i the Standard Model, except for one part: the so-called gauge fixing terms. The gauge fields cannot be identified with physical objects as they contain redundant degrees of freedom; that is to say, applying gauge transformations to these fields does not change the physics. Hence we need to choose a certain gauge5 to be able to describe physics in a unique way, and this is called gauge fixing. This means that additional terms, that are arbitrary to some extent, must be added to the Standard Model Lagrangian. These gauge fixing terms, however, do not change the symmetry analyses in this thesis and can therefore be safely ignored. So far we have mainly considered the Standard Model before breaking of the electroweak symmetry and the parts of the Standard Model Lagrangian have been written in terms of gauge eigenstates only. After electroweak symmetry breaking, however, it is convenient to rewrite the Lagrangian in terms of mass eigenstates. The U(1)Y and SU(2)L gauge fields mix and yield the following electroweak vector boson mass eigenstates:

+ 1 1 2 1 1 2 W = W iW ,W = W + iW , µ p2 µ µ µ p2 µ µ A =sin✓ W 3 + cos ✓ B ,Z= cos ✓ W 3 sin ✓ B , (15) µ w µ w µ µ w µ w µ where the weak mixing angle ✓w is defined as g tan ✓ 0 . (16) w ⌘ g Besides the electroweak vector bosons, it also turns out to be convenient to redefine the fermionic fields. This is the topic of the next subsection. Table 2 provides an overview of the Standard Model parameters before electroweak symmetry breaking. In the next subsection we will see that not all of these parameters are physical; some degrees of freedom can be gauged away by redefining the fields.

2.4 Flavor mixing As has been discussed briefly in de previous section, the Standard Model fermions come in three di↵erent families (or generations). The first family of quarks and leptons is constituted by the up and down quarks and, respectively, by the electron and its corresponding neutrino. The next family consists of fermions with exactly the same quantum numbers, only their masses are larger. The third family is formed by the heaviest fermions6. Some Standard Model interactions involve fermions of di↵erent families, this is called flavor mixing. In this subsection we will see where flavor mixing comes from and how it can be parametrized.

5For certain applications, of course, some gauges prove handier than others. 6Although the mass hierarchy of the neutrinos is as yet unknown.

10 Name Physicaldescription Mathematicaldescription #Parameters

gs,g,g0 gauge couplings real numbers 3 y , y , y , y Yukawacouplingmatrices complex3 3matricesinf.s. 72 ⌫ e u d ⇥ µh Higgs symmetry breaking parameter real number 1 Higgs self-coupling strength real number 1 Total: 77

Table 2: Naive counting of parameters in the unbroken Standard Model. The abbreviation “f.s.” stands for “family space”.

The so-called Yukawa terms in the Standard Model Lagrangian couple the Higgs doublet to the fermionic ﬁelds. In the unbroken Standard Model, the part of the Lagrangian that describes these couplings reads (cf. eq. (9)):

2 2 L = iL y ⇤⌫00 L y e00 iQ y ⇤u00 Q y d00 +h.c., (17) Yukawa L00 ⌫ R L00 e R L00 u R L00 d R where the primes refer to particular choices of basis (we will come back to this in a bit). Note that also in this subsection the fermionic fields (LL00 ,⌫R00 ,eR00 etc.) actually denote family multiplets. In fact, all fermionic fields in this section are family multiplets. So with eR00 , for example, we denote (with a slight abuse of notation) the family multiplet constituted by the right-handed electron, muon and tau: eR00 e00 µ00 . (18) R ⌘ 0 R 1 ⌧R00 The electroweak symmetry of the Standard@ ModelA is broken once we expand the Higgs doublet around a chosen vacuum state (i.e. ground state). Eq. (14) gives the Higgs doublet after electroweak symmetry breaking in the unitarity gauge. Plugging this particular form of the Higgs doublet in eq. (17) gives us the Yukawa terms in the broken Standard Model: 1 LYukawa = (v + h) ⌫00y⌫ ⌫00 + e00 yee00 + u00 yuu00 + d00 ydd00 +h.c. . (19) p2 L R L R L R L R ⇣ ⌘ A priori, there is no reason to assume that no mixing takes place between the di↵erent families and thus the Yukawa couplings y (with = ⌫, e, u, d) are general complex 3 3 matrices. The components of the diagonalized Yukawa coupling matrices are proportional to⇥ the masses of the fermions. To diagonalize these matrices, the fermionic fields undergo unitary basis transformations, i.e. we switch from a description in terms of fermionic gauge (“double primed”) eigenstates to mass (“single primed”) eigenstates. The other terms in the Lagrangian do not change their form under these changes of basis, except for the terms that couple the fermions to the W boson. Here the unitary transformations on the fermionic fields result in the unitary quark and lepton mixing matrices. Before we derive the form of these mixing matrices, let us first find the basis transformations that diagonalize the Yukawa coupling matrices. To diagonalize the Yukawa coupling matrices y (we will suppress the label in this derivation), we start by considering the Hermitian matrices yy† and y†y. Any Hermitian matrix can be diagonalized by a unitary similarity transformation and the resulting diagonal matrix has real entries only. Let U, V U(3), then yy† and y†y are diagonalized as follows: 2

yy† U †yy†U D, y†y V †y†yV D, (20) ! ⌘ ! ⌘ where D is a diagonal matrix with non-negative real entries. The latter can be seen as follows:

D = U †yy†U

= U †y†U † U †y†U. (21)

11 For convenience we deﬁne A U †y†U,then ⌘

[D]jj = [A]ij⇤ [A]ij i X = [A] 2 0 . (22) | ij| i X From the transformations given by (20), it follows that

yy† = UDU† , y†y = VDV† . (23)

From the ﬁrst equation of (23) we infer that

y = UfW † , (24) where f is a diagonal matrix with [f] [D] and W U(3). Using eq. (24) yields jj ⌘ jj 2 p y†y = WDW† . (25)

From the second equation of (23), it follows that we can pick W = V . Now we arrive at the conclusion: U †yV = f . (26)

This means that the Yukawa coupling matrices y are diagonalized by a so-called bi-unitary transformation and the diagonalized matrices f have non-negative real entries only. The diagonalization of the Yukawa coupling matrices amounts to applying basis transformations to the fermionic ﬁelds. From eqs. (19) and (26) it follows that it is useful to transform a fermionic ﬁeld in the following way:

00 U † 00 0 , 00 V † 00 0 . (27) L ! L ⌘ L R ! R ⌘ R Writing eq. (19) in terms of the new fermionic bases (indicated by a single prime), yields: 1 LYukawa = (v + h) ⌫0f⌫ ⌫0 + e0fee0 + u0fuu0 + d0fdd0 . (28) p2 The way that we have combined here the left- and right-handed ﬁelds will be illustrated later (cf. eq. (59)). The Yukawa terms are now written in terms of the fermionic mass (“single primed”) eigenstates rather than the gauge (“double primed”) eigenstates. We can write the rest of the Standard Model Lagrangian in terms of these new bases too. For most terms this simply amounts to “removing one prime”. Let us for example consider the part of the Lagrangian that describes the interaction between right-handed up-type quarks and the Z boson (ignoring constant prefactors):

µ µ ZµuR00 uR00 = ZµuR0 Vu† VuuR0 µ = ZµuR0 uR0 . (29) So this term in the Lagrangian retains its form when we write it in terms of the new fermionic bases. The only part of the Lagrangian (besides the Yukawa terms) that does change form, is the coupling of fermions to the W boson. Here the unitary transformations that act on the fermionic ﬁelds result in the unitary ﬂavor mixing matrices:

g + µ + µ LW = W ⌫00 e00 + W u00 d00 +h.c. 0 p2 µ L L µ L L ⇣ ⌘ g + µ + µ = W ⌫0 U † Uee0 + W u0 U † Udd0 +h.c. p2 µ L ⌫ L µ L u L ⇣ ⌘ g + µ + µ = W ⌫0 V 0† e0 + W u0 V 0d0 +h.c. , (30) p2 µ L ` L µ L q L ⇣ ⌘ where we have deﬁned the matrices V`0 and Vq0 as

V 0 U †U , V 0 U †U . (31) ` ⌘ e ⌫ q ⌘ u d

12 Now let us try to ﬁnd explicit parametrizations for these matrices.

How many parameters are needed to parametrize V`0 and Vq0? As both matrices are unitary, we ﬁrst consider a general unitary matrix (cf. appendix B for more information about unitary matrices). The dimension of the unitary group of degree n is n2, so an arbitrary U U(3) is 2 parametrized by nine real parameters, including three rotation angles: ✓12, ✓23, and ✓13; and six phases: ↵i (i =1, 2), j (j =1, 2, 3), and . Without loss of generality, an arbitrary U U(3) can be written as [6]: 2 U = AV B , (32) where 10 0 ei1 00 A = 0 ei↵1 0 ,B= 0 ei2 0 (33) 0 1 0 1 00ei↵2 00ei3 and @ A @ A i c12c13 s12c13 s13e V = s c c s s ei c c s s s ei s c , (34) 0 12 23 12 23 13 12 23 12 23 13 23 13 1 s s c c s ei c s s c s ei c c 12 23 12 23 13 12 23 12 23 13 23 13 with c cos ✓ and@ s sin ✓ . A ij ⌘ ij ij ⌘ ij Any 3 3 unitary matrix can be written in the form given by eq. (32), thus also V 0 and V 0: ⇥ ` q

V 0 A V B , V 0 A V B . (35) ` ⌘ ` ` ` q ⌘ q q q

However, some degrees of freedom in V`0 and Vq0 are not physical and can be removed. The phase factors that appear in the matrices A`,B`,Aq,Bq can be completely “absorbed” by the quark and lepton ﬁelds without changing the physics. The absorption of phases by the lepton multiplets goes as follows: ⌫0 B ⌫0 ⌫ ,e0 A†e0 e , (36) L ! ` L ⌘ L L ! ` L ⌘ L and for the quark ﬁelds we have:

u0 A†u0 u ,d0 B d0 d . (37) L ! q L ⌘ L L ! q L ⌘ L

As a consequence, the matrices A`,B`,Aq,Bq that are a priori contained in V`0 and Vq0 are completely absorbed by the fermionic fields. Instead of nine real parameters, the resulting flavor mixing matrices are parametrized by only four parameters each. We define the lepton mixing matrix, also called the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, as7:

i` c12c13 s12c13 s13e V = s c c s s ei` c c s s s ei` s c , (38) ` 0 12 23 12 23 13 12 23 12 23 13 23 13 1 s s c c s ei` c s s c s ei` c c 12 23 12 23 13 12 23 12 23 13 23 13 @ A ` ` with cij cos ✓ij and sij sin ✓ij. The standard parametrization of the quark mixing matrix, also called the⌘Cabibbo–Kobayashi–Maskawa⌘ (CKM) matrix, reads:

iq c12c13 s12c13 s13e V = s c c s s eiq c c s s s eiq s c , (39) q 0 12 23 12 23 13 12 23 12 23 13 23 13 1 s s c c s eiq c s s c s eiq c c 12 23 12 23 13 12 23 12 23 13 23 13 q @ q A with cij cos ✓ij and sij sin ✓ij. The CKM⌘ and PMNS⌘ matrices are unitary 3 3 matrices that describe the mixing between the di↵erent quark and lepton families respectively.⇥ The coupling of leptons and quarks to the W boson is the only part of the Standard Model that exhibits mixing of fermionic families. In terms of these new fermionic bases (without primes), eq. (30) becomes:

g + µ + µ LW = W ⌫LV † eL + W uL VqdL +h.c. (40) 0 p2 µ ` µ ⇣ ⌘ Naturally, the terms in the Lagrangian that do not involve mixing of ﬂavors should be invariant under the transformations given by (36) and (37), as otherwise new parameters would enter the

7Assuming that neutrinos are Dirac particles rather than Majorana particles.

13 Standard Model. To this end, the right-handed projections of the fermionic ﬁelds have to transform the same as their left-handed counterparts, as can be inferred from the Yukawa terms given by eq. (28)8. In terms of the “unprimed” bases, the Yukawa terms take the form: 1 LYukawa = (v + h) ⌫f⌫ ⌫ + efee + ufuu + dfdd . (41) p2 A summary of the unitary transformations that act on the fermionic ﬁelds is given by table 3.

Fermionic ﬁelds

⌫00 B U †⌫00 ⌫ L ! ` ⌫ L ⌘ L ⌫00 B V †⌫00 ⌫ R ! ` ⌫ R ⌘ R e00 A†U †e00 e L ! ` e L ⌘ L e00 A†V †e00 e R ! ` e R ⌘ R u00 A†U †u00 u L ! q u L ⌘ L u00 A†V †u00 u R ! q u R ⌘ R d00 B U †d00 d L ! q d L ⌘ L d00 B V †d00 d R ! q d R ⌘ R Table 3: Transformations of the fermionic ﬁelds.

Let us briefly summarize what has happened to the degrees of freedom in the Yukawa sector of the Standard Model. A priori, the Yukawa matrices y (with = ⌫, e, u, d) are general complex 3 3 matrices, thus giving rise to 4 18 = 72 independent parameters. However, many parameters can⇥ be absorbed by redefinitions of⇥ the fermionic fields. After applying appropriate basis transformations to these fields (cf. table 3), we end up with diagonalized (real) Yukawa matrices f (with = ⌫, e, u, d) and two flavor mixing matrices V`, Vq that are both described by four parameters. This means that the number of independent physical parameters in the Yukawa sector of the Standard Model is 4 3+2 4 = 20. Table 4 summarizes the parameter counting in the Standard Model after having carried⇥ out⇥ all the redefinitions of the fields. From this table we infer that the Standard Model has 25 actual free parameters9. Electroweak symmetry breaking, of course, does not change the number of independent parameters.

Name Physicaldescription Mathematicaldescription #Parameters

gs,g,g0 gauge couplings real numbers 3 f , f , f , f diag. Yukawa coupling matrices real diag. 3 3matricesinf.s. 12 ⌫ e u d ⇥ V , V ﬂavor mixing matrices unitary 3 3matrixinf.s. 8 ` q ⇥ µh Higgs symmetry breaking parameter real number 1 Higgs self-coupling strength real number 1 Total: 25

Table 4: All parameters of the unbroken Standard Model. The abbreviations “diag.” and “f.s.” stand for “diagonal” and “family space” respectively.

8 The (diagonal) matrices Ai,Bi (with i = `, q)commutewiththediagonalizedYukawacouplingmatricesf (with = ⌫, e, u, d). 9Note that we are excluding the strong CP violating angle. For simplicity and good phenomenological reasons, we have set this parameter to zero.

14 3 Symmetries of the Standard Model

Symmetries of nature are transformations on a physical system that leave the physics unchanged. Symmetry principles play a very important role with respect to the laws of physics. More precisely, all continuous symmetries of a physical system give rise to conserved quantities. In this section we study the continuous symmetries of the Standard Model gauge theory. These continuous symmetries can be divided into three types: Poincaré, gauge and flavor symmetries. The first two types form the cornerstones of the Standard Model, while the last one constitues a class of accidental symmetries. After a short discussion on the importance of symmetries in field theory, we will list the symmetries of the Standard Model and their corresponding conservation laws.

3.1 Noether’s theorem The relation between symmetries and conserved quantities in field theory is given by Noether’s theorem. This very important theorem was proved by Noether in 1915 and today this theorem and its derivation are found in any standard textbook on field theory, e.g. in [1]. What does this theorem say? Let us consider a set of fields i that satisfy the Euler–Lagrange equations for the Lagrangian L = L ( i , @µi ): { } { } @L @L = @ , (42) @ µ @(@ ) i ✓ µ i ◆ and let these fields undergo the following continuous transformations: + ↵ , (43) i ! i i where ↵ is infinitesimal. The equations of motion of the fields i do not change as long as the action corresponding to the Lagrangian is left invariant under these transformations. In that case, the field transformations form a symmetry of the system. For the Lagrangian this means that it can change at most by a four-divergence: L L + ↵@ Gµ , (44) ! µ which means that µ ↵@µG = ↵L @L @L = ↵ + (@ ) @ i @(@ ) µ i ✓ i µ i ◆ @L @L @L = ↵@ + ↵ @ . (45) µ @(@ ) i @ µ @(@ ) i ✓ µ i ◆  i ✓ µ i ◆ The second term vanishes due to the Euler-Lagrange equations for the fields i (cf. eq. (42)), so we are left with µ @µj =0, (46) where the conserved current jµ is given by

µ @L µ j = i G . (47) @(@µi) This is the mathematical form of Noether’s theorem and the conserved currents are called Noether currents. Moreover, a conserved current implies a locally conserved charge deﬁned by

C (t) d~x j 0(x) , (48) V ⌘ ZV as d d C (t)= d~x j 0(x) dt V dt ZV = d~x ~ ~j(x) r· ZV = d~s ~j(x) , (49) · Z@V

15 for any volume V with boundary @V . In the following subsections we list all continuous symmetries of the Standard Model and relate these to their conserved charges that follow from Noether’s theorem.

3.2 Poincarésymmetries Minkowski spacetime is invariant under certain transformations. These symmetries come in three di↵erent types: Translations in time and space. • Rotations in space. • Boosts connecting two uniformly moving bodies. • These symmetries, called Poincarésymmetries, are global and continuous. What are, according to Noether’s theorem, the corresponding conservation laws? Invariance under translations in time and space give conservation of energy and (linear) momentum respectively, and rotational invariance of space implies conservation of angular momentum. Invariance under boosts gives that the field’s center of energy travels at constant velocity (which is not very interesting from a physics’ point of view). The full symmetry group of Minkowski spacetime is called the Poincarégroup P and it contains the Lorentz group SO(1, 3) (rotations and boosts) and the group of translations R1,3. The Lorentz group is a subgroup of dimension six and the group of translations is a normal subgroup of dimension four and every element of the Poincarégroup can be written uniquely as a product of a Lorentz transformation and a translation. This means that P is the so-called semidirect product of R1,3 and SO(1, 3), or in mathematical notation:

1,3 P = R o SO(1, 3) . (50) The dimension of the Poincar´egroup is 4 + 6 = 10. An equation that is invariant under Poincar´etransformations is called relativistically invariant. All sensible descriptions of nature should be relativistically invariant, which is why this principle has been one of the cornerstones of the Standard Model.

3.3 Gauge symmetries Another type of symmetries is given by the gauge symmetries, which provide the basis for the Standard Model. As explained in the previous section, gauge symmetries are internal symmetries that, once we make them local and require invariance, give rise to gauge bosons and interactions mediated by these gauge bosons in a natural way. The full gauge group of the Standard Model is the twelve-dimensional Lie group SU(3)C SU(2)L U(1)Y. According to Noether’s theorem, all twelve generators must give rise to a conserved⇥ quantity.⇥ Each fundamental interaction of nature (that is described by the Standard Model) corresponds to a particular gauge group. The strong interaction is connected to the Lie group SU(3). Invariance under local SU(3) transformations give – through Noether’s theorem – conserved quantities called color (C1,...,C8). The electroweak interaction, on the other hand, is related to the Lie group SU(2) U(1). The corresponding conserved quantities again follow from Noether’s theorem and ⇥ are called weak isospin (T1,T2,T3) and weak hypercharge (Y ). They correspond to the groups SU(2) and U(1) respectively. After the electroweak symmetry has been broken (cf. eq. (4)), weak isospin and weak hypercharge have combined into a new quantum number called electrical charge (Q), given by Q = T3 + Y. (51) Electrical charge corresponds to a U(1) Lie group and is a conserved quantity of the broken Standard Model. Both Poincar´eand gauge symmetries have served as guiding principles for the construction of the Standard Model. Some other symmetries of the Standard Model are rather accidental; they just happen “to be there”. These symmetries are found in the family space of the Standard Model and are elaborately discussed in the next subsection.

16 3.4 Flavor symmetries In this subsection we consider the purely accidental (global) continuous symmetries of family space. Our aim is to find the symmetry group of family space, called the flavor group.Wewill do this in the context of the Standard Model, and, for simplicity, consider the broken rather than the unbroken version. At the end of this subsection we will comment on the implications of our results for the unbroken Standard Model and notice that nothing really changes. To determine the flavor group, we will distinguish between various cases: with and without flavor mixing, as well as di↵erent fermionic mass spectra10.

3.4.1 Flavor group without flavor mixing We will first consider the Standard Model without flavor mixing. This simply means that we omit the CKM and PMNS matrices from our description. Another way of looking at this is saying that we aprioriassume the Yukawa coupling matrices to be diagonal. To find the flavor group, we first have a look at the kinetic terms of the fermions. If denotes a fermionic field11,thenitskinetic term in the Lagrangian reads: free L = i @/ . (52)

Let M GL(3, C) be a (general) linear transformation in family space that acts on the fermionic ﬁelds as2 follows: M . (53) ! This means that eq. (52) transforms as

i @/ i M† @/M = i M † M @/ . (54) ! ⇣ ⌘ Requiring invariance of (54) under the transformation M , leads to the restriction:

M † M = I3 , (55) which means that M U(3). However, this is not the final story. Let us have a closer look at the field . The Standard Model2 is a chiral theory, which means that left- and right-handed fermions (transforming independently under Lorentz transformations) live in di↵erent representations of the Standard Model gauge group (cf. appendix C for more information on this and conventions for notation). In eq. (52) the field can be split into its left- and right-handed projections, as

µ µ = (PL + PR) µ 2 2 = PL + PR µ µ = PR PL + PL PR µ µ = L L + R R , (56) so that eq. (52) can be written as

free L = i L@/ L + i R@/ R . (57)

As a consequence, the left- and right-handed projections of the fermionic ﬁelds in eq. (57) can transform independently: let M L,MR U(3), then 2 M L , M R . (58) L ! L R ! R These transformations leave the kinetic term for invariant. This is not only true for the kinetic terms though, also the interactions between fermions and vector bosons are of this form. However, as can be inferred from eq. (40), the neutrinos, charged leptons, up- and down-type quarks cannot transform independently: the left-handed charged leptons and neutrinos form doublets, as well as the left-handed up- and down-type quarks. These doublets are independent of each other though.

10The masses of the Standard Model fermions are varied over this analysis rather than taken ﬁxed. 11Note that as before these ﬁelds denote family multiplets.

17 Not all interactions decouple left- and right-handedness completely. The (only) terms in the Standard Model Lagrangian that mix left- and right-handed ﬁelds are the Yukawa interactions given by eq. (41). As

= (PL + PR) 2 2 = PL + PR

= R L + L R , (59) the Yukawa interaction for a ﬁeld can be written as

Yukawa 1 L = (v + h) f R + f L . (60) p2 L R This means that the ﬂavor group must be such that L and R do not transform independently. Let L R M ,M U(3) be the transformation matrices that act on the left- and right-handed projections 2 L R 12 of , like in (58). Eq. (60) then transforms under M and M as :

f M L†f M R . (61) L R ! L R L R If we require invariance of this term under M and M , it follows that

M L†f M R = f M Lf f M R =0. (62) ()

Before we study the solutions of this equation, let us ﬁrst have a closer look at f . The diagonalized Yukawa coupling matrix f is of the form

1 00

f = 0 2 0 , (63) 0 1 003 @ A 13 where i > 0(withi =1, 2, 3) is proportional to the mass of an ith generation fermion of type . To determine the ﬂavor group, it is clear that we need to distinguish between three di↵erent cases, namely case 1 : all diagonal components of f are di↵erent, case 2 : two components are equal

(1 = 2 , 1 = 3 , or 2 = 3 ), and case 3 : all three components are equal (1 = 2 = 3 ). Now each case will be dealt with separately, and to avoid clutter the labels “ ” will be omitted in the remaining of this subsection.

Case 1: , , are all di↵erent: • 1 2 3 We ﬁrst consider the case where all diagonal components of f are di↵erent. Let us write M L,MR in the generic way

a11 a12 a13 b11 b12 b13 M L = a a a ,MR = b b b , (64) 0 21 22 23 1 0 21 22 23 1 a31 a32 a33 b31 b32 b33 @ A @ A where aij,bij C (with i, j =1, 2, 3). Now 2

1 (a11 b11) 2a12 1b12 3a13 1b13 L R M f fM = 1a21 2b21 2 (a22 b22) 3a23 2b23 , (65) 0 a b a b (a b ) 1 1 31 3 31 2 32 3 32 3 33 33 @ A so that with requirement (62) it follows that

a11 = b11 ,a22 = b22 ,a33 = b33 , (66a)

12For simplicity we only consider the ﬁrst term and ignore the constant prefactor. 13Note that we are excluding the possibility of massless fermions.

18 1b12 1b13 2b21 a12 = ,a13= ,a21 = , 2 3 1 2b23 3b31 3b32 a23 = ,a31= ,a32 = . (66b) 3 1 2

This means that all elements of M L can be expressed in those of M R:

b 1b12 1b13 11 2 3 L 2b21 2b23 M = b22 . (67) 0 1 3 1 3b31 3b32 b33 B 1 2 C @ A L R L L R R L Both M and M need to be unitary, i.e. M †M = M †M = I3. For M this requirement reads

2 2 2 2 2 2 2 2 2 2 2 2 1 b11 +2 b21 +3 b31 1b11⇤ b12+2b21⇤ b22+3b31⇤ b32 1b11⇤ b13+2b21⇤ b23+3b31⇤ b33 | | | 2 | | | 1 12 13 2 2 2 2 2 2 2 2 2 2 2 2 1b12⇤ b11+2b22⇤ b21+3b32⇤ b31 1 b12 +2 b22 +3 b32 1b12⇤ b13+2b22⇤ b23+3b32⇤ b33 0 | | | 2 | | | 1 = I3 . (68) 12 2 23 2 2 2 2 2 2 2 2 2 2 2 2 b⇤ b + b⇤ b + b⇤ b b⇤ b + b⇤ b + b⇤ b b + b + b 1 13 11 2 23 21 3 33 31 1 13 12 2 23 22 3 33 32 1| 13| 2| 23| 3| 33| B 2 C B 1 3 2 3 3 C @ A First, it is important to note that the solution we are interested in should also respect the unitarity requirement for M R. This means that the solution has to be independent of the components of f. If we look at the diagonal elements of M L†M L, it is easy to see that the only possible solution is given by: b 2 = b 2 = b 2 =1, (69a) | 11| | 22| | 33| b12 = b21 = b13 = b31 = b23 = b32 =0. (69b) With these restrictions M L and M R satisfy eq. (62) and are unitary. For convenience, we choose the following parametrization that is consistent with requirement (69a):

i↵ i i b11 = e ,b22 = e ,b33 = e , (70) so that ei↵ 00 M L = M R = 0 ei 0 U(1) U(1) U(1) . (71) 0 00ei 1 2 ⇥ ⇥ @ A So when 1,2,3 are all di↵erent, the symmetry groups of family space are of the form U(1) U(1) U(1). But what about the second case? ⇥ ⇥ Case 2: = : • 1 2

Let us now consider the case where two diagonal components of f are equal. We set 1 = 2 , so that14 ⌘ 00 f = 0 0 . (72) 0 1 003 @ A Naturally we can follow the same procedure as for the ﬁrst case. In analogy to eq. (65) we now end up with

(a11 b11) (a12 b12) 3a13 b13 L R M f fM = (a21 b21) (a22 b22) 3a23 b23 , (73) 0 a b a b (a b ) 1 31 3 31 32 3 32 3 33 33 @ A so that with requirement (62) it follows that

a11 = b11 ,a22 = b22 ,a12 = b12 ,a21 = b21 ,a33 = b33 , (74a)

14 We could equally well have chosen here 1 = 3 or 2 = 3.However,thischoiceisnotimportanthereaswe are ultimately only interested in the form of the ﬂavor group.

19 b13 b23 3b31 3b32 a13 = ,a23 = ,a31 = ,a32 = . (74b) 3 3 The matrix M L can be expressed in terms of the elements of M R:

b b b13 11 12 3 M L = b b b23 , (75) 21 22 3 0 3b31 3b32 1 b33 @ A and the unitarity requirement for M L is given by

2 2 2 2 2 2 2 2 2 ( b11 + b21 )+3 b31 (b11⇤ b12+b21⇤ b22)+3b31⇤ b32 (b11⇤ b13+b21⇤ b23)+3b31⇤ b33 | | | 2| | | 2 3 2 2 2 2 2 2 2 2 2 (b12⇤ b11+b22⇤ b21)+3b32⇤ b31 ( b12 + b22 )+3 b32 (b12⇤ b13+b22⇤ b23)+3b32⇤ b33 2 | | | 2| | | = I3 . (76) 0 3 1 2 2 2 2 2 2 2 2 2 (b⇤ b +b⇤ b )+ b⇤ b (b⇤ b +b⇤ b )+ b⇤ b ( b + b )+ b 13 11 23 21 3 33 31 13 12 23 22 3 33 32 | 13| | 23| 3| 33| B 2 C B 3 3 3 C @ A Again there is only one solution that does not depend on the components of f and thus also respects the unitarity requirement for M R:

b 2 + b 2 = b 2 + b 2 = b 2 =1, (77a) | 11| | 21| | 12| | 22| | 33|

b13 = b31 = b23 = b32 =0, (77b)

b11⇤ b12 + b21⇤ b22 =0. (77c) With these restrictions both M L and M R satisfy requirement (62) and are unitary. For convenience, we choose the following parametrization that is consistent with requirement (77a):

b = cos ✓ei↵ ,b= sin ✓ei(↵+) , 11 21 i i(+") i b12 =sine ,b22 = cos e ,b33 = e . (78)

Requirement (77c) gives = ✓ and " = , so that

cos ✓ei↵ sin ✓ei 0 M L = M R = sin ✓ei(↵+) cos ✓ei(+) 0 U(2) U(1) . (79) 0 1 2 ⇥ 00ei @ A So for 1 = 2 the symmetry groups are of the form U(2) U(1) (more on the group U(2) and the parametrization of its elements can be found in appendix⇥ B). Let us now consider the ﬁnal case.

Case 3: = = : • 1 2 3

What if all three diagonal components of f are equal, i.e. 1 = 2 = 3 ?Thenf takes the simple form ⌘ f = I3 , (80) which, through eq. (62), directly gives M L = M R U(3). Now that we have considered di↵erent scenarios for the fermionic mass spectrum, we are2 ready to consider the following two di↵erent cases: flavor mixing absent (FM)/ and flavor mixing included (FM). Let us first consider the Standard Model without flavor mixing. We have already studied all the terms in the Lagrangian that contain fermionic fields and have seen that leptons and quarks form two di↵erent sectors in family space and that left- and right-handed fields transform the same. Omitting flavor mixing from our description means that the CKM and PMNS matrices are removed from eq. (40). It is then straightforward to notice that the charged leptons and their corresponding neutrinos must transform in the same manner, as well as the up- and down-type quarks. For both the lepton and quark sectors the symmetry groups are unitary and depend on the masses of the leptons and quarks respectively. In case of no flavor mixing, the flavor group (denoted by F )is: F = G G , (81) FM/ L ⇥ Q

20 where “L” refers to the leptons and “Q” to the quarks and

e e e ⌫ ⌫ ⌫ U(3)L if 1 = 2 = 3,1 = 2 = 3 (case 3) G = U(2) U(1) if e = e,⌫ = ⌫ (case 2) (82) L 8 `1,2 `3 1 2 1 2 U(1) ⇥U(1) U(1) else (case 1) < `1 ⇥ `2 ⇥ `3 and : u u u d d d U(3)Q if 1 = 2 = 3 ,1 = 2 = 3 (case 3) G = U(2) U(1) if u = u,d = d (case 2) (83) Q 8 q1,2 q3 1 2 1 2 U(1) ⇥U(1) U(1) else (case 1) < q1 ⇥ q2 ⇥ q3 Making a di↵erent: choice for the case of two equal components of f would change the flavor group and the nomenclature in a straightforward way. Now let us investigate whether the flavor group changes once we include mixing of flavors.

3.4.2 Flavor group with flavor mixing If we allow for mixing of flavors, i.e. we insert the flavor mixing matrices into the Standard Model Lagrangian, the flavor group might change. The only terms in the Lagrangian that imply flavor mixing are the interactions between fermions and the W boson described by eq. (40). For simplicity, we consider just one term of eq. (40) and ignore the constant prefactor:

+ µ Wµ uL VqdL . (84) What group of transformations in family space leaves this term invariant? We already know that the transformation matrices have to be unitary and that they depend on the mass spectrum of the fermions. Let us start by applying transformations Mu and Md to the (left-handed) up- and down-type quark ﬁelds respectively15:

u M u ,d M d . (85) L ! u L L ! d L The term (84) then transforms as:

+ µ + µ + µ W u V d W u M † V M d = W u M †V M d . (86) µ L q L ! µ L u q d L µ L u q d L

If we require invariance of this term under Mu and Md, it follows that

M †V M = V M V V M =0. (87) u q d q () u q q d To solve this equation, we again need to distinguish between the di↵erent fermionic mass spectra:

Case 1: , , are all di↵erent: • 1 2 3 In case that all diagonal components of the diagonalized Yukawa coupling matrices are di↵erent, the symmetry groups of ﬂavor space (without ﬂavor mixing) are of the form U(1) U(1) U(1). Let us therefore parametrize M ,M U(1) U(1) U(1) as follows: ⇥ ⇥ u d 2 ⇥ ⇥ ei↵k 00 M = 0 eik 0 , (88) k 0 1 00eik @ A where k = u, d. Requirement (87) then gives

ei↵u ei↵d (V ) ei↵u eid (V ) ei↵u eid (V ) q 11 q 12 q 13 eiu ei↵d (V ) eiu eid (V ) eiu eid (V ) =0, (89) 0 q 21 q 22 q 23 1 eiu ei↵d (V ) eiu eid (V ) eiu eid (V ) q 31 q 32 q 33 @ A where (Vq)ij is entry (i, j) of Vq. From eq. (89) it follows that

↵ = ↵ = = = = ↵, (90) u d u d u d ⌘ 15Note that we omit the label “L”here.Aswasshownintheprevioussubsection,theYukawainteractionsmake the left- and right-handed ﬁelds transform in the same fashion.

21 which gives M = M = ei↵I U(1) . (91) u d 3 2 We see that in case 1,2,3 are all di↵erent, the symmetry group of the quark sector of family space is U(1).

Case 2: = : • 1 2 In the case of two equal diagonal components, the symmetry groups are of the form U(2) U(1), as long as mixing of ﬂavors is ignored. Let us therefore parametrize M ,M U(2) U(1) as⇥ follows u d 2 ⇥

i↵k ik cos ✓k e sin ✓k e 0 M = sin ✓ ei(↵k+k) cos ✓ ei(k+k) 0 , (92) k 0 k k 1 00eik @ A 16 where k = u, d. Imposing requirement (87) on Mu and Md gives two possible solutions .One solution reads:

✓ = ✓ =0,= ↵ ,= ↵ ,↵= = = ↵ ↵, (93) u d u u u d d d d u d u ⌘ so that M = M = ei↵I U(1) . (94) u d 3 2 The other solution gives the same form for Mu and Md, but with a minus sign. However, this minus sign can be absorbed in the free parameter ↵, which means that there is only one physically relevant solution.

Case 3: = = : • 1 2 3 What if all three diagonal components of the diagonalized Yukawa coupling matrices are equal? As we have shown in the previous subsection, the symmetry groups in the case of no flavor mixing are U(3) groups. Just as for the other two cases it can be shown that Mu = Md U(1). The very same procedure, of course, can be applied to the lepton sector of the2 Standard Model (with the PMNS matrix) and similar results are obtained. This means that the flavor group of the Standard Model including flavor mixing is:

F = U(1) U(1) . (95) FM L ⇥ Q This result is independent of the fermionic mass spectrum. We have found the flavor group of the Standard Model, with and without flavor mixing, and for various fermionic mass spectra. We chose to perform our analysis on the broken rather than the unbroken Standard Model Lagrangian. Would our result have been di↵erent if we had chosen to consider the unbroken theory instead? Well, breaking the electroweak gauge symmetry does not a↵ect family space in the slightest, which means that the flavor groups of the broken and unbroken Standard Model have to be the same. The unbroken Standard Model Lagrangian is usually written in terms of the fermionic gauge (“double primed”) rather than the mass (“unprimed”) eigenstates (cf. section 2 for the precise definitions of these bases). We have found the symmetry transformations of family space with respect to the “unprimed” fermionic bases. How can we translate the transformations found to the “double primed” bases? In case no mixing of flavors occurs, the Yukawa coupling matrices y simply coincide with the diagonalized matrices f (with = ⌫, e, u, d) and the “double primed” and “unprimed” bases are the same (as no diagonalization procedure is necessary). The symmetry transformations found for the broken Standard Model thus also apply to the unbroken theory. Introducing mixing of flavors means that the Yukawa coupling matrices are, a priori, general complex matrices. Diagonalizing these matrices gives rise to the unitary basis transformations given by eq. (27). These transformations, in turn, give rise to the quark and lepton mixing matrices. These matrices contain some phases that can be removed by, again, applying unitary transformations to the fermionic fields (cf. eqs. (36) and (37)). A summary of all transformations of the fermionic fields is given by table 3. We notice that the net transformations on both the left- and

16These solutions have been found with the help of the computer program Maple [7].

22 right-handed fermionic ﬁelds are all unitary. For a fermionic ﬁeld the “double primed” and “unprimed” bases are related as follows:

L00 = S L , R00 = T R , (96) where S and T represent the (net) unitary transformations given by table 3. Let M denote the transformation that acts on the left- and right-handed projections of :

M , M . (97) L ! L R ! R The “double primed” ﬁelds, in turn, undergo the following transformations:

00 S M S† 00 , 00 T M T † 00 . (98) L ! L R ! R However, we have seen that in the case of ﬂavor mixing being present, the symmetries of family space are simply complex phases (so M here is a complex phase). As phases commute with the matrices S and T , the transformations in (98) simplify to

00 M 00 , 00 M 00 , (99) L ! L R ! R which is similar to the transformations with respect to the “unprimed” ﬁelds given by (97). We conclude that for the “double primed” and “unprimed” bases the symmetry transformations in family space have exactly the same form. The results of this and the previous subsection are summarized in tables 5 and 6.

F Case 1 Case 2 Case 3

U(1)`1 U(1)`2 U(1)`3 U(2)`1,2 U(1)`3 FM/ ⇥ ⇥ ⇥ ⇥ ⇥ U(3)L U(3)Q U(1) U(1) U(1) U(2) U(1) ⇥ q1 ⇥ q2 ⇥ q3 q1,2 ⇥ q3 FM U(1) U(1) U(1) U(1) U(1) U(1) L ⇥ Q L ⇥ Q L ⇥ Q Table 5: The flavor group (F ) of the Standard Model with (FM) and without (FM)/ flavor mixing. The di↵erent cases are defined in eqs. (82) and (83). For simplicity, the mass spectra of the quark and lepton sectors have the same form here, and for “case 2” the masses of the first and second generation fermions have been set equal to each other.

dim(F ) Case 1 Case 2 Case 3

FM/ 6 10 18 FM 222

Table 6: The dimension of the flavor group (F ) of the Standard Model with (FM) and without (FM)/ flavor mixing. The di↵erent cases are defined in eqs. (82) and (83). For simplicity, the mass spectra of the quark and lepton sectors have the same form here.

We have found the symmetry groups of family space for various cases. Continuous symmetries of the Lagrangian imply conserved quantum numbers according to Noether’s theorem. In case the Standard Model mixes flavors (which it does), the symmetry groups of the lepton and quark sectors of family space are U(1)L and U(1)Q respectively. These phase symmetries, that are in fact accidental, are related to conservation of lepton number and baryon number respectively. When flavor mixing is omitted and the masses of the fermions are all di↵erent (which reflects reality), the symmetries of family space are phases that relate to conservation of electron, muon and tau number in the lepton sector and to conservation of analogous numbers in the quark sector.

23 Table 7 gives an overview of all continuous symmetries of the Standard Model (before electroweak symmetry breaking) and their associated conservation laws. In section 5.6 we will see that it is fairly straightforward to generalize the results of this section to supersymmetric theories. The reason for this is simply that supersymmetric theories share many important features with the Standard Model.

Symmetry Type Lie group Conserved quantity

1,3 Poincar´e global R o SO(1, 3) energy, momentum, angular momentum, center of energy gauge local SU(3) SU(2) U(1) color, weak isospin, weak hypercharge C ⇥ L ⇥ Y ﬂavor global U(1) U(1) lepton number, baryon number L ⇥ Q Table 7: Continuous symmetries of the (unbroken) Standard Model and their corresponding conserved quantities.

24 4 Supersymmetry

As was mentioned in section 2, the Standard Model of particle physics has certain shortcomings and can therefore not be the ﬁnal story; a more fundamental theory of elementary particles is needed. A promising extension to the Standard Model is supersymmetry. The ﬁrst ideas about supersymmetric theories date back to the early seventies and there has been a lot of attention for supersymmetry ever since. For the last decades, supersymmetry has been one of the most promising and well-established theories that goes beyond the Standard Model. After we give a short introduction to supersymmetry, the motivation for supersymmetry is discussed, and thereafter a general supersymmetric Lagrangian is constructed. In section 5 one particular supersymmetric model is considered in great detail.

4.1 Introduction to supersymmetry Inspired by the great success of the role of symmetries in the Standard Model, supersymmetric theories possess one additional symmetry, called supersymmetry. Supersymmetry is a spacetime symmetry that relates bosons to fermions. Its generator Q transforms a bosonic state into a fermionic one and vice versa, schematically:

Q boson = fermion ,Qfermion = boson . (100) | i | i | i | i From these relations it follows that Q must be a (fermionic) spinor, which leads to an algebra of anticommutation relations [8]:

µ µ Q ,Q† = 2( ) p , (101a) { ↵ } ↵ Q ,Q = Q† ,Q† =0, (101b) { ↵ } { ↵ } µ µ [p ,Q↵]=[p ,Q↵† ]=0, (101c)

µ 1 2 3 a0 where I2, , , involves the Pauli matrices (for a0 =1, 2, 3). In supersymmetric⌘ theories all elementary particles are grouped into boson-fermion pairs called supermultiplets. Each supermultiplet has an equal amount of bosonic and fermionic degrees of freedom and particles from the same supermultiplet are so-called superpartners of each other. There are two types of supermultiplets: chiral and gauge supermultiplets. A chiral supermultiplet consists of a Weyl spinor and a complex scalar field, and a gauge supermultiplet consists of a vector boson field and a (Majorana) Weyl spinor. From eq. (101c) it follows that Q commutes with p2, which implies that superpartners must have equal masses. Furthermore, Q also commutes with the generators of gauge transformations. This means that superpartners must have the same quantum numbers as well. A quick analysis of the Standard Model field content reveals that it is not possible to construct any supermultiplets with Standard Model fields only. It turns out that to realize supersymmetry we need to at least double the number of particles that we already know. These new supersymmetric particles have not been found in particle accelerators yet. This means that (if supersymmetry exists) their masses cannot equal those of their Standard Model superpartners. Hence supersymmetry cannot be an exact symmetry of nature; it must be broken somehow. The simplest extension of the Standard Model that realizes supersymmetry is called the Minimal Supersymmetric Standard Model (MSSM). This model is extensively discussed in section 5. To distinguish (in speech) supersymmetric particles from their Standard Model counterparts, a specific nomenclature has been devised. The names of the scalar superpartners of the Standard Model fermions start with an “s” (for scalar), so the fermions (quarks and leptons) form supermultiplets with the sfermions (squarks and sleptons). The fermionic superpartners of the Standard Model gauge bosons are called gauginos and all their names end in “ino” (gluino, wino, etc.), and the superpartners of the Higgs bosons are called Higgsinos. Before we go into a more technical description of supersymmetric theories, we first list a number of problems with the Standard Model that supersymmetry claims to solve, as well as other advantages of this novel spacetime symmetry.

25 4.2 Motivation for supersymmetry There are several reasons for supersymmetry being one of the best candidate theories for physics beyond the Standard Model. We will now list the main arguments in favor of supersymmetry:

The hierarchy problem could be solved: • One problem that could be solved with supersymmetry is the so-called hierarchy problem. This has been one of the principal reasons for the construction of supersymmetry. The hierarchy problem amounts to the fact that within the Standard Model quantum corrections to the Higgs boson propagator destabilize the mass spectrum of the Standard Model. Masses of fermions and gauge bosons are protected against quantum corrections by the chiral and gauge symmetries respectively, but there is no such protection by a symmetry for scalar fields like the Higgs field. When we calculate loop corrections in the Standard Model, we perform unbounded integrals over internal momenta. However, accounting for arbitrarily large momenta is not very realistic as we know that the Standard Model must break down at some point. Therefore one introduces a cuto↵⇤, which represents the scale at which new physics is supposed to kick in (i.e. there are fields present that have masses of (⇤) or larger). In section 6 we will go much deeper into this topic. For now it is just importantO to realize that ⇤is unknown, possible very large (perhaps even the Planck scale of (1019 GeV)), and that we do not want our theory to depend on it. The following discussion of theO hierarchy problem can be found in most textbooks on quantum field theory and supersymmetry. To see where the problem with the Higgs field arises, let us start by considering the Higgs potential: V ( )= µ2 2 + 4 , (102) | | h| | | | 2 with , µh > 0. The minimum of this potential is located at v = , (103) | | p2 where v µ /p represents the vacuum expectation value of the Higgs doublet. Note that among ⌘ h µh,,v only two parameters are independent. The Higgs field is sensitive to all massive fields and it couples strongest to the heaviest mass scale present (here ⇤forms a lower bound for that scale). The “-term” in eq. (102) represents a Higgs quartic interaction and, in case that new fields “live” at a mass scale ⇤, this interaction gives the following correction to the “µh-term” at one-loop order [9]:

C⇤2 2 + less divergent terms , (104) ! | | 2 2 where C is a constant. This means that the coecient µh of is replaced by the physical value µ2 ,with | | h,phys µ2 = µ2 C⇤2 , (105) h,phys h which alters the vacuum expectation value of the Higgs doublet accordingly: µ v = h,phys . (106) p To see what this implies, let us ﬁrst consider the masses of the W and Higgs bosons: gv m = ,m= p2v . (107) W 2 h The masses of these bosons can be measured and experiments indicate that m 80 GeV W ⇡ and mh 126 GeV [10]. This means that now all (measurable) parameters in the Higgs sector ⇡ 17 (µh,phys,,v) are ﬁxed through measurements: µ (102 GeV) , (1) ,v (102 GeV) . (108) h,phys ⇠O ⇠O ⇠O 17For this analysis it suces to only look at orders of magnitude.

26 The physical parameter µh,phys is sensitive to quantum corrections and depends on the unknown bare Lagrangian parameter µh and the unknown cuto↵scale⇤ (cf. eq. (105)). On top of everything, not only the W and Higgs boson masses depend on v, in fact all Standard Model masses are proportional to v. It thus follows from eqs. (105) and (106) that the whole particle spectrum depends on the unknown cuto↵scale! 2 If the cuto↵⇤is a very high energy scale, then the one-loop correction to µh in eq. (105) requires a remarkable cancellation. Say ⇤is the Planck scale (1019 GeV), then we somehow need to go down from (1038 GeV) to (104 GeV)! This meansO that a lot of fine-tuning is required O O through a clever pick of µh; for this particular example its value would have to be chosen precisely up to the 32nd decimal! As the mass of the Higgs boson turns out to be much smaller than the Planck scale, this problem is called the hierarchy problem. We have seen that because of the presence of a scalar field in the Standard Model – the Higgs field – the whole mass spectrum has become unstable. Unlike scalar fields, all the other quantum fields are protected by symmetries. Is there a way out of this problem? Could we for example introduce another symmetry that also protects scalars? Let us consider more corrections to the Higgs propagator. A fermionic one-loop correction to the “µh-term” in eq. (102) is given by [9]:

C⇤2 2 + less divergent terms , (109) ! | |

2 with >0. If somehow = , then the quadratic sensitivity of µh,phys to ⇤would not occur. In fact, relations between coupling constants of this form are exactly what is found in supersymmetric theories. A fermionic superpartner of the Higgs boson would exactly cancel the unwanted divergences! Of course all fermions in the Standard Model have a Yukawa coupling to the Higgs field and 2 2 thus give rise to ⇤ -terms in µh,phys. However, supersymmetry predicts that all these fermions have a scalar superpartner that exactly cancels these divergences. Also the massive gauge bosons couple to the Higgs field and likewise introduce quadratic divergences. These, in turn, are cancelled by their fermionic superpartners (the gauginos) as well. All these cancellations do not depend on the masses of the superpartners. This means that supersymmetry could solve the hierarchy problem in a natural way, as then no “miraculous” cancellations are needed. After the quadratic divergences (that are cancelled in supersymmetric theories), the next most 2 divergent contributions to µh,phys grow logarithmically with ⇤. A superpartner of a Standard Model 2 fermion (with mass mf ), say, that couples to the Higgs field, would give a contribution to µh,phys 2 2 proportional to (mh mf ) log ⇤. This means that supersymmetry solves the hierarchy problem provided that the superpartners are not too heavy (their masses should not exceed (1 TeV)). The latter, of course, is a matter of taste though. O

Gauge coupling unification is encouraged: • The gauge couplings, which set the strength for the various fundamental forces, are not constant but depend on the energy scale (this is elaborately discussed in section 6). An interesting question to ask is whether there could be an energy scale for which all three gauge couplings take the same value? This could mean that all forces are, in fact, di↵erent manifestations of a single underlying interaction. We know that within the context of the Standard Model there is no gauge coupling unification. In some supersymmetric theories, however, the gauge couplings do seem to unify at high energies. In the MSSM, the gauge couplings unify at an energy scale of (1016 GeV). Figure 1 illustrates the possible unification of the gauge couplings in the MSSM. O

A dark matter candidate is provided: • From astronomical observations we know that the visible matter in our universe does not have enough mass to explain the speed of moving galaxies within the framework of Einstein’s general theory of relativity. The invisible matter is called dark matter and sadly (or interestingly) the Standard Model fails to provide a possible dark matter particle.

27 2 Figure 1: Two-loop evolution of ↵a(µ) ga(µ) /(4⇡)(witha =1, 2, 3) for the Standard Model and the MSSM with respect to the energy⌘ scale µ. The mass thresholds of the supersymmetric particles are varied between 500 GeV (blue line) and 1.5 TeV (red line). Here g1 5/3 g0, g2 g, and g g (the coupling constant g is the properly scaled version of g in⌘ the context of⌘ an 3 s 1 0 p SU(5) or⌘ SO(10) grand uniﬁed theory). This ﬁgure has been taken from [8].

Dark matter might be made up of so-called Weakly Interacting Massive Particles (WIMPs) that, by deﬁnition, can only interact through the weak force and gravity. In phenomenologically viable supersymmetric theories, the lightest supersymmetric particle cannot decay due to conservation of R-parity (cf. section 5.2). As this particle does not carry any color or electrical charge, it is a WIMP and therefore an attractive candidate for dark matter.

Gravity could be incorporated: • The Standard Model only describes three out of the four known fundamental forces of nature; gravity is missing out. As yet, no one has succeeded in establishing a good quantum theory of gravity in a consistent fashion18. To this day gravity is explained by Einstein’s theory of general relativity and has no place (yet) in a theory of elementary particles. Could supersymmetry perhaps describe gravity? The answer is positive. To take into account gravity, supersymmetry must be promoted to a local symmetry. The relation between local supersymmetry and general relativity is clear from anticommutation relation (101c). Local supersymmetry, namely, would correspond to invariance under local coordinate transformations, which is the underlying principle of general relativity! The resulting quantum theory of gravity is called supergravity and it contains the graviton (a spin-2 particle) and its superpartner the gravitino (a spin-3/2 particle).

It completes the list of possible spacetime symmetries: • Supersymmetry is a special symmetry in the sense that it combines an internal symmetry with a spacetime symmetry. The supersymmetry generators, namely, transform bosons into fermions and vice versa, while from eq. (101c) it follows that the anticommutator of two generators yields a translation in spacetime (the momentum operator is the generator of spacetime translations). According to the Haag–Lopusza´nski–Sohnius theorem, there is only one non-trivial way to have an interplay between internal and spacetime symmetries, and that this is exactly supersymmetry [11].

18Perhaps because it simply does not exist.

28 Supersymmetry thus completes the list of possible spacetime symmetries.

Electroweak symmetry breaking might be explained: • In the Standard Model the introduction of the Higgs potential is rather ad hoc. At the scale 2 where electroweak symmetry breaking takes place, the parameter µh in the Higgs potential changes sign: it goes from being negative to being positive, which transforms the shape of the potential into a “Mexican hat”. In the MSSM this form of the potential is derived rather than assumed. The 2 parameter equivalent to µh starts out negative at the gauge coupling uniﬁcation scale and becomes positive as it runs down to the electroweak scale. The other parameter in the Higgs potential, , is derived from the gauge couplings. Thus supersymmetry can explain the origin of electroweak symmetry breaking.

4.3 Supersymmetric Lagrangians In this subsection we construct a general supersymmetric Lagrangian from the principles of renormalizability, gauge invariance and invariance under supersymmetry transformations (up to a four- divergence). We will construct this Lagrangian step by step, although many small steps will be left out as we are only interested here in the general ideas. The supersymmetry transformations of the ﬁelds are not speciﬁed in this thesis either. More elaborate and precise derivations of supersymmetric Lagrangians, including all supersymmetry transformations, can be found in e.g. [8,9]. The general supersymmetric Lagrangian that is constructed here will eventually (in section 5) lead to the full Lagrangian that describes the MSSM. Let us start by considering separately the general Lagrangian for chiral and gauge supermultiplets. An overview of all indices used in this subsection can be found in appendix A.

4.3.1 Chiral supermultiplets A chiral supermultiplet consists of a complex scalar field and a left-handed Weyl spinor (cf. appendix C for the definition of a Weyl spinor). Our goal for now is to construct a Lagrangian for an arbitrary set of chiral supermultiplets. For the fields and the free Lagrangians read:

free µ free µ L =(@ )† (@µ) , L = i † @µ . (110)

We require the supersymmetry algebra to close both classically (on-shell) and quantum mechan- ically (o↵-shell) to prevent anomalies. This can be established by matching the (real) degrees of freedom for the bosonic and fermionic fields in each chiral supermultiplet, both on-shell and o↵- shell19. Both on- and o↵-shell has two degrees of freedom. The complex two-component Weyl spinor , however, has four degrees of freedom o↵-shell and only two degrees of freedom on-shell (corresponding to its two spin polarizations). To make the bosonic and fermionic degrees of freedom match, we introduce a complex scalar field F that has two degrees of freedom o↵-shell and none on-shell. This auxiliary field has no kinetic term in the Lagrangian, so its equation of motion is simply F = 0. The degrees of freedom of the various fields for a chiral supermultiplet are summarized in table 8.

on-shell 220 o↵-shell 242

Table 8: The number of real degrees of freedom in a chiral supermultiplet.

Of course, every pair of complex scalars and left-handed Weyl spinors requires an auxiliary ﬁeld to make the degrees of freedom match. Thus the free Lagrangian of all chiral supermultiplets

19This means that the classical equation of motion is not satisﬁed.

29 (labeled by i) reads: free µ µ 2 L =(@ )† (@ )+i † @ + F . (111) chiral i µ i i µ i | i| The supersymmetric action should be invariant under supersymmetry transformations. But what exactly are the supersymmetry transformations? Well, the simplest possible transformation of the scalar ﬁeld i is i = ✏ i (the dot indicates that this is a spinor product, cf. appendix · 20 C), where ✏ is – like i – a Weyl spinor . If we start out with this simple transformation then all the other supersymmetry transformations can be derived by the requirement that the action is left invariant. These transformations in conjunction with the invariance requirement restrict the terms that can be added to a supersymmetric Lagrangian. Now let us add non-gauge interactions to the Lagrangian. To respect renormalizability, each term must have a total mass dimension (carried by derivatives and ﬁelds) not greater than four. This requirement, as well as invariance under supersymmetry transformations, heavily constrain the allowed non-gauge interactions. It can be shown that the only possible non-gauge interaction terms are of the form [8]: 1 L int = W ij + W iF +h.c., (112) chiral 2 i · j i

ij i where W and W are polynomials in the scalar ﬁelds i. It also follows that these polynomials are not completely independent of each other; both can be expressed as (functional) derivatives of the so-called superpotential W :

W 2W W i ,Wij , (113) ⌘ i ⌘ ij with21 1 1 W = M + y , (114) 2 ij i j 6 ijk i j k where Mij is a symmetric mass matrix for fermionic fields, and yijk a Yukawa coupling for a scalar field and two fermionic fields. To guarantee invariance under supersymmetry transformations, the superpotential must be a holomorphic function of the complex scalar fields (i.e. the superpotential cannot depend on both i and i⇤). The auxiliary fields Fi can be eliminated from our description by using their equations of free int motion. From the Lagrangian Lchiral + Lchiral, the following equations of motion for the Fi can be derived: i F = W ⇤ . (115) i Thus the auxiliary fields can be expressed completely in terms of the scalar fields. Now eqs. (111), (112) and (115) can be combined to yield

µ µ i 2 1 ij L =(@ )† (@ )+i † @ W W +h.c. . (116) chiral i µ i i µ i 2 i · j Eq. (116) is a general supersymmetric Lagrangian that describes chiral supermultiplets with the addition of all admissible non-gauge interactions.

4.3.2 Gauge supermultiplets Now we turn to the gauge supermultiplets, also in the most general sense. Each gauge group with corresponding infinitesimal generators T a (in the fundamental representation) gives rise to a gauge a supermultiplet. A gauge supermultiplet consists of real gauge boson fields Aµ and (Majorana) Weyl spinors a (the gauginos). From the Standard Model we know that for these fields the free Lagrangians read: free 1 a aµ⌫ free a µ a L = F F , L = i † @ . (117) A 4 µ⌫ µ 20 We only consider global supersymmetry transformations here, which means that @µ✏ =0. 21 Also the term “+ Lii”(whereLi are parameters) that describes gauge singlet fields can be added to the superpotential. However, as there are no gauge singlets in the MSSM, we will simply omit this term.

30 a The ﬁeld strength tensor Fµ⌫ is deﬁned by

F a @ Aa @ Aa gfabcAb Ac , (118) µ⌫ ⌘ µ ⌫ ⌫ µ µ ⌫ where g denotes the gauge coupling of the considered gauge group and f abc the algebra’s structure constant. Again we require the bosonic and fermionic degrees of freedom to match in each supermultiplet. On-shell, both ﬁelds have two (real) degrees of freedom. O↵-shell, however, a has four degrees a of freedom (it consists of two complex numbers) and Aµ only three (one degree of freedom can be removed by a gauge transformation). To account for this mismatch, we introduce real bosonic auxiliary ﬁelds Da for all gauge supermultiplets. Table 9 summarizes the number of degrees of freedom for a gauge supermultiplet.

Aa a Da

on-shell 220 o↵-shell 341

Table 9: The number of real degrees of freedom in a gauge supermultiplet.

Like the “F -fields” for the chiral supermultiplets, the “D-fields” initially have vanishing equations of motion. Only after we introduce interactions between the chiral and gauge supermultiplets, the equations of motion for the “D-fields” can be used to eliminate these fields from our description. To incorporate the gauge interactions, we require local invariance of the Lagrangian under the gauge transformations. As usual, this results in replacing the ordinary derivatives by gauge covariant derivatives. The resulting Lagrangian for a set of arbitrary gauge supermultiplets labeled by v, reads: 1 a aµ⌫ a µ a 1 a 2 L = F F + i † (D ) + (D ) , (119) gauge 4 vµ⌫ v v µ v 2 v with the gauge covariant derivative given by

(D )a = @ ac g f abcAb c . (120) µ v µ v v vµ v 4.3.3 Chiral and gauge supermultiplets combined Naturally, the ﬁelds in the chiral and gauge supermultiplets also interact with each other. The a interactions between the gauge ﬁelds Avµ and the scalars and Weyl fermions i, i in the chiral supermultiplets are introduced by replacing the ordinary derivatives in eq. (116) by gauge covariant derivatives:

@ D = @ + ig Aa T a , (121a) µ i ! µ i µ v vµ v i @ D = @ + ig Aa T a . (121b) µ i ! µ i µ v vµ v i Could the “D-ﬁelds” and the gauginos also interact with the ﬁelds in the chiral supermultiplets? Well, gauge invariance and renormalizability allow for the following interaction terms in the La- grangian (up to constant factors):

a a a a †T +h.c.,†T D . (122) i v i · v i v i v Requiring invariance of the action under supersymmetry transformations ﬁxes the coecients of these interaction terms. Putting together (116), (119), (121a), (121b), and (122) gives the full

31 renormalizable supersymmetric Lagrangian:

1 a aµ⌫ a µ a µ µ L = F F + i † (D ) +(D )† (D )+i † D SUSY 4 vµ⌫ v v µ v i µ i i µ i a a i 2 1 ij p2 g †T +h.c. W W +h.c. v i v i · v 2 i · j ⇣ 2 ⌘ 1 2 a g †T . (123) 2 v i v i ⇣ ⌘ Note that this Lagrangian does not contain any auxiliary ﬁelds anymore; the “D-ﬁelds” have been eliminated by using their equations of motion given by:

a a D = g †T . (124) v v i v i The Lagrangian given by eq. (123) is the most general Lagrangian satisfying renormalizability, gauge invariance and invariance under supersymmetry transformations. To fully determine a Lagrangian for a speciﬁc supersymmetric theory, we need to specify the gauge groups, the ﬁeld content and the superpotential. This is done in the next section for the MSSM.

4.4 Supersymmetry breaking The fact that supersymmetric particles have not been observed yet in experiments gives two possibilities: either supersymmetry does not exist, or supersymmetry is a broken symmetry of nature. In this thesis, of course, we assume the latter. We also assume that supersymmetric particles are heavier than their Standard Model counterparts. How could supersymmetry be broken? Let us briefly discuss this and take the MSSM as an example (although the following discussion is completely general and applies to any supersymmetric model). Supersymmetry breaking at some scale means that above that scale the theory is invariant under supersymmetry transformations, but not below it. From the Standard Model we know a way to break a symmetry without spoiling the characteristics of the theory, namely spontaneous symmetry breaking. However, when we try to break supersymmetry in the MSSM spontaneously, we run into trouble. It turns out that spontaneous breaking of supersymmetry requires us to extend the MSSM. It is expected that breaking terms arise from quantum corrections rather than from tree level couplings, which is referred to as radiative supersymmetry breaking.Insuchmodels supersymmetry is broken in a hidden sector. This sector, by definition, contains (heavy) fields that do not couple directly to the MSSM fields (the visible sector). The breaking of supersymmetry must somehow be mediated from the hidden sector to the MSSM. There are many radiative supersymmetry breaking models available and all give di↵erent constraints on the MSSM parameters at high scales. In section 6 we will see how (in principle) we could say something about the way supersymmetry breaking is mediated by making use of low energy data only. First, however, we will discuss the MSSM in great detail and show how supersymmetry breaking e↵ects can be parametrized.

32 5 The Minimal Supersymmetric Standard Model

In this thesis we mainly focus on the minimal supersymmetric extension of the Standard Model, the MSSM. This section starts with a discussion of the MSSM ﬁeld content and Lagrangian. Subsequently, we will perform a rigorous count of free parameters and discuss the symmetries of the MSSM.

5.1 The MSSM field content The MSSM has, by construction, the same gauge group as the Standard Model. It contains all Standard Model fields, an additional Higgs doublet, and all their superpartners. One of the reasons for introducing another Higgs doublet is to prevent gauge anomalies (we will encounter another reason in section 5.3). The field content of the (unbroken) MSSM, including the corresponding gauge group representations, is given by tables 10 and 11. To distinguish the notation of Standard Model particles from supersymmetric particles, the latter receive a tilde. In these tables the right- handed Weyl spinors have been conjugated to bring them in a left-handed form: for a Weyl spinor c 2 we define R i R⇤ (cf. also appendix C). Furthermore, note that we omit the right-handed neutrinos from our⌘ analysis22. This means that there is no PMNS matrix present and thus no mixing of flavors in the lepton sector.

Name Spin 0 Spin 1/2 SU(3) SU(2) U(1) C ⇥ L ⇥ Y 1 LL =(⌫L eL) LL =(⌫L eL)(1, 2, ) sleptons, leptons 2 c eR⇤ eR (1, 1, 1) e e e 1 QL =(uL dL) QL =(uL dL)(3, 2, ) e 6 c 2 squarks, quarks uR⇤ uR (3, 1, ) e e e 3 c 1 dR⇤ dR (3, 1, ) e 3 + 0 + 0 1 Hu =(H H ) Hu =(H H )(1, 2, ) Higgs, Higgsinos e u u u u 2 0 0 1 H =(H H) H =(H H)(1, 2, ) d d d ed ed ed 2 Table 10: Chiral supermultiplets of the unbrokene MSSMe e and the corresponding gauge group representations. The ﬁrst two numbers in the last column (those indicated in boldface) refer to the dimension of the gauge group representations, while the last number denotes the eigenvalue of the weak hypercharge generator Y .

Name Spin 1/2 Spin 1 SU(3) SU(2) U(1) C ⇥ L ⇥ Y gluinos, gluons ga Ga (8, 1, 0) winos, W bosons W a0 W a0 (1, 3, 0) e bino, B boson BB (1, 1, 0) f Table 11: Gauge supermultiplets ofe the unbroken MSSM and the corresponding gauge group representations. The ﬁrst two numbers in the last column (those indicated in boldface) refer to the dimension of the gauge group representations, while the last number denotes the eigenvalue of the weak hypercharge generator Y .

As we know from the Standard Model, the interaction eigenstates of the theory (cf. tables 10 and 11) do not necessarily coincide with the mass eigenstates. We have seen in section 2 that

22This is simply to be more consistent with existing literature on the MSSM.

33 electroweak symmetry breaking causes the gauge bosons W 1,2,3 and B to mix, yielding the mass eigenstates W ±,Z and A. Naturally, these particles also have superpartners (with obvious names). In the MSSM (for the same reason) more mixing of interaction states takes place: also the Higgsinos 0 0 and electroweak gauginos mix. The neutral Higgsinos Hu, Hd combine with the neutral gauginos 3 + B,W and give four neutral mass eigenstates called neutralinos. The charged Higgsinos Hu , Hd 1 e 2 e combine with the charged gauginos W ± =(W iW )/p2 (cf. also Wµ± in eq. (15) for the Standarde f Model) and yield two charge +1 and two⌥ charge 1 mass eigenstates called charginose e . In the unbroken MSSM, the two complexf Higgsf doubletsf have a total of eight degrees of freedom. After electroweak symmetry breaking, three of these have become longitudinal polarizations of the massive electroweak gauge bosons. We are thus left with ﬁve physical scalar bosons in the Higgs sector: two charged ones (+1 and 1) and three neutral ones (two CP-even and one CP-odd). The 0 0 physical Standard Model Higgs boson is a linear combination of Hu and Hd and is CP-even.

5.2 R-parity Another feature of the MSSM is the conservation of a quantum number called R-parity. Imposing R-parity conservation restricts the terms that are allowed in the superpotential. The MSSM superpotential could in principle contain terms that violate the conservation of baryon number (B) or lepton number (L). However, processes that violate these numbers have never been observed in experiments. An example of such a process would be the decay of the proton (the simplest decay channel would be p+ e+ + ⇡0, followed by ⇡0 2). As experiments have put very severe constraints on this, we! try to avoid the introduction! of B or L violating terms. On the other hand, we also know that B and L are not fundamental symmetries of nature as it is known that they are violated in some non-perturbative electroweak processes. For this reason we do not want to introduce these symmetries in the MSSM explicitly. With the construction of the Standard Model there was no need to assume that these quantum numbers are conserved, as B or L violating terms in the Lagrangian were simply not possible (due to the renormalizability condition). To solve this problem, a new symmetry called R-parity has been introduced in the MSSM. This new multiplicative quantum number is deﬁned as:

3(B L)+2s P ( 1) , (125) R ⌘ where s is the spin of the particle. This number equals +1 for the Standard Model particles and the additional scalars in the Higgs sector and equals 1 for the supersymmetric partner particles, and the MSSM is deﬁned to conserve it. Conservation of R-parity eliminates the possibility of having renormalizable B or L violating terms in the superpotential. Moreover, R-parity conservation directly implies that all supersymmetric particles are produced in pairs and that the lightest supersymmetric particle is absolutely stable as supersymmetric particles have to decay to an odd number (at least one) of supersymmetric particles. The latter is phenomenologically interesting, as that makes this particle an attractive dark matter candidate (cf. also section 4.2).

5.3 The MSSM Lagrangian We have already speciﬁed the gauge group and the ﬁeld content of the MSSM. To fully determine the corresponding Lagrangian, we also need to specify the gauge invariant superpotential. Including the requirement of R-parity conservation, the MSSM superpotential is given by [8]:

↵ ↵ ↵ ↵ W = e† y (L ) (H ) + u† y (Q ) (H ) d † y (Q ) (H ) + µ(H ) (H ) . (126) MSSM R e L d ↵ R u L u ↵ R d L d ↵ u d ↵ Note that in this equation all gauge and family indices have been suppressed (in appendix D the e e e e e e superpotential is written with all indices restored). The Yukawa couplings ye, yu, yd are general dimensionless complex 3 3 matrices in family space, and the “µ-term” is a supersymmetric version of the Higgs boson mass.⇥ As the superpotential needs to be holomorphic in the scalar ﬁelds (i.e. for any scalar ﬁeld , not both and ⇤ can occur in the superpotential), one Higgs doublet is not enough to give all the fermions mass. In the Standard Model we simply use complex conjugation to achieve this, but that is not allowed here. Therefore – as we already mentioned earlier – we require the existence of two Higgs doublets in the MSSM.

34 Now that we have also speciﬁed the MSSM superpotential (cf. eq. (126)) we are ready to write down the full Lagrangian of the MSSM. The derivation of the terms that originate from the superpotential is presented in appendix D. Analogous to the Lagrangian of the Standard Model, also here we in principle need to include gauge ﬁxing terms. As those terms are irrelevant for our further analyses though, we will simply omit them. We make use of the generic form of supersymmetric Lagrangians given by eq. (123):

1 a aµ⌫ a µ a µ µ L = F F + i † (D ) +(D )† (D )+i † D MSSM 4 vµ⌫ v v µ v i µ i i µ i  a a i 2 1 ij p2 g †T +h.c. W W +h.c. v i v i · v 2 i · j ⇣ 2 ⌘ 1 2 a gv i†Tv i . (127) 2 MSSM ⇣ ⌘ Let us now split this Lagrangian up in parts:

Kinetic terms for the gauge supermultiplets and gauge interactions: •

1 a aµ⌫ a µ a 1 a aµ⌫ a µ a 1 a0 a0µ⌫ F F + i † (D ) = G G + ig † (D g) W W 4 vµ⌫ v v µ v 4 µ⌫ µ 4 µ⌫  MSSM a0 µ a0 1 µ⌫ µ + iW † (D We) B e B + iB† @ B, (128) µ 4 µ⌫ µ where the gauge covariant derivatives for the gauginosf read:f e e

(D g)a = @ ac g f abcGb gc , (129a) µ µ s µ a0 D W = @ a0c0 g✏a0b0c0 W b0 W c0 . (129b) µ e µ µe ⇣ ⌘ ⇣ ⌘ The MSSM ﬁeld strength tensorsf equal those of the Standard Model:f

Ga = @ Ga @ Ga g f abcGb Gc , (130a) µ⌫ µ ⌫ ⌫ µ s µ ⌫ W a0 = @ W a0 @ W a0 g✏a0b0c0 W b0 W c0 , (130b) µ⌫ µ ⌫ ⌫ µ µ ⌫ B = @ B @ B . (130c) µ⌫ µ ⌫ ⌫ µ

Kinetic terms for the chiral supermultiplets and gauge interactions: •

µ µ µ µ µ (D i)† (Dµi)+i i† Dµ i =(D LL)†DµLL + iLL† DµLL +(D eR⇤ )†DµeR⇤ MSSM h i c µ c µ µ + ie † D e +(D Q ) D Q + iQ† D Q Re µ eR L † µ L eL µe L µ c µ c µ +(D u⇤ )†D u⇤ + iu † D u +(D d⇤ )†D d⇤ R µ R eR µ eR R µ R c µ c µ µ + id † D d +(D H )†D H + iH† D H R µ R u µ u u e µ ue µe e µ +(D H )†D H + iH† D H , (131) d µ d d µ d e e e e

35 where the gauge covariant derivatives for the leptons, quarks and Higgs doublets are given by:

1 a0 a0 1 D L = @ + igW ig0B L , (132a) µ L µ 2 µ 2 µ L ✓ ◆ c c DµeR = @µ + ig0Bµ eR , (132b) ✓ ◆

1 a a 1 a0 a0 1 D Q = @ + ig G + igW + ig0B Q , (132c) µ L µ 2 s µ 2 µ 6 µ L ✓ ◆ c 1 a a 2 c D u = @ ig G ⇤ ig0B u , (132d) µ R µ 2 s µ 3 µ R ✓ ◆ c 1 a a 1 c D d = @ ig G ⇤ + ig0B d , (132e) µ R µ 2 s µ 3 µ R ✓ ◆

1 a0 a0 1 D H = @ + igW + ig0B H , (132f) µ u µ 2 µ 2 µ u ✓ ◆

1 a0 a0 1 D H = @ + igW ig0B H . (132g) µ d µ 2 µ 2 µ d ✓ ◆ As superpartners have the same quantum numbers, the gauge covariant derivatives of the sleptons, squarks and Higgsinos are exactly the same:

1 a0 a0 1 D L = @ + igW ig0B L , (133a) µ L µ 2 µ 2 µ L ✓ ◆ e e DµeR⇤ = @µ + ig0Bµ eR⇤ , (133b) ✓ ◆

1 a a 1 a0 a0 1 D Qe = @ + ig G e + igW + ig0B Q , (133c) µ L µ 2 s µ 2 µ 6 µ L ✓ ◆ 1 a a 2 D ue⇤ = @ ig G ⇤ ig0B u⇤ , e (133d) µ R µ 2 s µ 3 µ R ✓ ◆ 1 a a 1 D ed⇤ = @ ig G ⇤ + ig0B de⇤ , (133e) µ R µ 2 s µ 3 µ R ✓ ◆ 1 1 e a0 a0 e D H = @ + igW + ig0B H , (133f) µ u µ 2 µ 2 µ u ✓ ◆

1 a0 a0 1 D He = @ + igW ig0B He . (133g) µ d µ 2 µ 2 µ d ✓ ◆ e e Chiral supermultiplets coupled to gauginos: •

1 1 p a a p a0 a0 T c 2 gvi†Tv i v +h.c. = 2 gLL† LLW g0LL† LLB + g0eReRB · MSSM 2 2 h ⇣ ⌘i ✓ 1 a a 1 a0 a0 1 + g Q† eQ g +fgQ† Qe W e+ g0eQ† QeB 2 s L L 2 L L 6 L L 1 T a c a 2 T c 1 T a c a g ue ⇤u g g0eu u B f g d ⇤ed g e 2 s R Re 3 R R 2 s R R

1 T c 1 a0 a0 1 + g0d d B + gH† H We + ge0H†H B 3 eR R e2 u e u 2 u u e

1 a0 a0 1 + gHe† He W g0He†HfB +h.c. e e (134) 2 d d 2 d d ◆ e f e e

36 Scalar interactions coming from the “F -ﬁelds”: •

i 2 ↵ W =(Hd⇤) (LL† )ye†ye(LL) (Hd)↵ eR† yeye†eRHd†Hd MSSM h i ↵ ↵ +(H ) (Q† ) y† y (Q ) (H ) +(H ) (Q† ) y†y (Q ) (H ) u⇤ e L u eu L u ↵e d⇤e L d d L d ↵ u† y y† u H†H d † y y†d H†H + u† y y†d H†H +h.c. R u ue R u u e R d d R d d e R u d R e d u ⇣ ⌘ L† y†e e† y L Q† y† u u† y Q Q† y†d d † y Q eL e R eR e L eL u R eR u L Le d R Re d L Q† y†dRe† yeLL µ⇤e† yeH†LL µ⇤u† yuH†QL µ⇤d † ydH†QL e L ed e R e e Re e u e eR e de e R u ⇣ 2 2 +h.c. µ H†H µ H†H (135) e e| |e u eu | |e d d e e e e e ⌘ Yukawa couplings: •

1 ij ↵ ↵ W +h.c. = e† y (L ) (H ) + e† y (L ) (H ) 2 i · j R e L d ↵ R e L d ↵  MSSM ↵ ↵ + e† ye(LL) (Hd)↵ u† yeu(QL)e (Hu)↵ R R ↵ ↵ u† y (Q ) (H ) u† y (Q ) (H ) R u L e u ↵ R u L u ↵ e ↵ ↵ + d† y (Q ) (H ) + d† y (Q ) (H ) R d eL ed ↵ R d L ed ↵ ↵ e ↵ + d † y (Q ) (H ) µ(H ) (H ) + h.c. (136) R d L d ↵ u e d ↵e e e e e 4-scalar interactions coming from the “D-ﬁelds”: •

2 2 1 2 a 1 2 a a a 1 2 a0 g †T = g Q† Q u† u d † d g L† L 2 v i v i 8 s L L R R R R 8 L L  MSSM ⇣ ⌘ ⇣ 2⌘ ⇣ a0e e a0 ea0 e 1 2 1 e e + Q† Q + H† eH +e H† H g0 L† L L L u u d d 2 2 L L ✓ 1 2 1 ⌘ 1 ee† e e Q† Q + u† u d † d H†H e e R R 6 L L 3 R R 3 R R 2 u u 1 2 + e He†H e e e e e e (137) 2 d d ◆ 5.4 Soft supersymmetry breaking The MSSM Lagrangian as considered so far respects supersymmetry (by construction) and thus leads to superpartners that have equal masses. We know, though, that if supersymmetry exists, supersymmetric particles must be heavier than their Standard Model superpartners; supersymmetry is a broken symmetry of nature (cf. section 4.4). As we know that supersymmetry must be broken at “low” energies, we would like to see its e↵ect directly in the Lagrangian. Is there a way to study the MSSM phenomenology without knowing exactly how supersymmetry is broken? To parametrize the e↵ects of supersymmetry breaking, we could extend the MSSM Lagrangian by an e↵ective piece that includes all possible breaking terms that respect the basic characteristics of the theory. For example, we do not want the supersymmetry breaking terms in the Lagrangian to give rise to additional quadratic divergences to the scalar masses. After all, supersymmetry was invented in the ﬁrst place to cancel these unwanted divergences! This means that the e↵ect of the breaking terms should be suppressed at high energies. In other words, the couplings should be superrenormalizable (i.e. have a positive mass dimension). This type of symmetry breaking is called soft supersymmetry breaking. Although adding soft supersymmetry breaking terms to the Lagrangian is very artiﬁcial, it is assumed that supersymmetry breaking occurs spontaneously in a hidden sector (cf. section 4.4) and that the original theory respects supersymmetry. The most

37 general soft supersymmetry breaking Lagrangian as an extension to the MSSM is given by [8]:

soft 1 aT a a0T a0 T ↵ L = M g g + M W W + M B B +h.c. + e† a (L ) (H ) MSSM 2 3 2 1 R e L d ↵ ⇣ ↵ ↵ ⌘ h 2 2 2 u† a (Q ) (H ) + d † a (Q ) (H ) +h.c. L† m L e† m e Q† m Q R u e L e u ↵ f R fd L ed ↵e Le L Le R e R L Q L 2 2 2 2 i ↵ u† m uR d † m dR m H†Hu m H†Hd [b(Heu) (Hd)↵ +h.c.]. (138)e eR u e R d e Hue u Hd d e e e ee e e These terms breake supersymmetry,e but respect gauge invariance and R-parity conservation. In the e e e e soft breaking Lagrangian (138) the masses of the bino, wino and gluino are denoted by M1,M2,M3 respectively. The trilinear couplings ae, au, ad are complex 3 3 matrices in family space (analogous to the Yukawa coupling matrices), and the sfermion mass⇥ matrices m2 , m2, m2 , m2 , m2 L e Q u d are Hermitian 3 3 matrices in family space. The parameters mHu ,mHd are Higgs masses and b is called the supersymmetry⇥ breaking Higgs mixing parameter. e e e e e The parameters of the MSSM including soft supersymmetry breaking are listed in table 12. In the next subsection we will see that – just like in the Standard Model – many degrees of freedom turn out to be unphysical as they can be absorbed by clever ﬁeld redeﬁnitions.

Name Physicaldescription Mathematicaldescription #Parameters

gs,g,g0 gauge couplings real numbers 3 y , y , y Yukawa coupling matrices complex 3 3matricesinf.s. 54 e u d ⇥ M1,M2,M3 gaugino masses complex numbers 6 a , a , a trilinear coupling matrices complex 3 3matricesinf.s. 54 e u d ⇥ m2 , m2, m2 , m2 , m2 sfermion mass matrices Hermitian 3 3matricesinf.s. 45 L e Q u d ⇥

mHeu ,meHd f e e Higgs masses real numbers 2 µ sup.p. Higgs mixing parameter complex number 2 b sup.b. Higgs mixing parameter complex number 2 Total: 168

Table 12: Naive counting of parameters in the MSSM including soft supersymmetry breaking. The parameters that are listed above the horizontal dashed line also occur in the Standard Model. The abbreviations “sup.p.” and “sup.b.” stand for “supersymmetry preserving” and “supersymmetry breaking” respectively, and “f.s.” stands for “family space”.

5.5 Counting independent parameters To be able to read o↵the physical content of the MSSM better, we perform clever (unitary) basis transformations on the (s)fermionic fields to get rid of unphysical degrees of freedom, exactly like we did for the Standard Model in section 2.4. To distinguish between objects (fields, matrices etc.) before and after the basis transformations, we denote the former with a prime (in this subsection only). The first part of the analysis is the same as in section 2.4: we start by considering the Yukawa coupling matrices and note that these matrices can be diagonalized by bi-unitary transformations as follows23: f V †y U , (139) ⌘ where = e, u, d. Again U ,V are unitary 3 3 matrices, and the f are diagonal matrices with ⇥ non-negative real entries only. The transformation matrices U ,V give rise to the unitary CKM matrix, defined by24: V 0 U †U . (140) q ⌘ u d 23Acarefulreaderwillnoteadi↵erencewitheq.(26).Thisdi↵erencesimplyarisesfromthefactthattheYukawa coupling matrices as defined in the MSSM are the Hermitian conjugates of the ones in the Standard Model. This is purely conventional and will not a↵ect any results in this thesis. 24Note that this time there is no leptonic analogy of this matrix present due to the absence of right-handed neutrinos in our model.

38 As we have seen in section 2.4, not all parameters of the CKM matrix are physical; some phases can be absorbed by redefinitions of the fermionic and sfermionic fields. We define

V A†V 0B† , (141) q ⌘ q q q where Aq,Bq are (diagonal) unitary matrices (cf. eq. (267)). Eqs. (139) and (141) lead us to redefine the fermionic and sfermionic fields in such a way as to absorb the matrices U ,V and Aq,Bq (in exactly the same way as in the Standard Model). Because of the absence of lepton mixing (parametrized by the PMNS matrix) in the MSSM, there is more transformation freedom in the (s)lepton sector than in the (s)quark sector. As a consequence, by redefining the lepton and slepton fields two phases can be removed of one of the following matrices in the (s)lepton sector: 2 2 2 a0 , m0 , or m0 . Let us take for this the Hermitian matrix m0 . To see why this is possible, we e L e L realize that any Hermitian 3 3 matrix H0 can be parametrized by 9 real parameters as follows: e e ⇥ e i↵ i d1 ae be i↵ i H0 = ae d2 ce = P †HP , (142) 0 i i 1 be ce d3 @ A where i i↵ d1 abe e 00 H ad2 c ,P 010 , (143) ⌘ 0 i 1 ⌘ 0 i 1 be cd3 00e @ A @ A 2 with ↵ . From eq. (142) we indeed infer that two phases of m0 can be absorbed L by redefinitions⌘ of the lepton and slepton fields without changing the physics. We call the corre- e sponding (diagonal) unitary transformation matrix PL. Table 13 summarizes the transformations that act on the fermionic and sfermionic fields. Note that chiral superpartners must transform identically. This follows immediately from the part ofe the MSSM Lagrangian that couples chiral supermultiplets to gauginos (cf. eq. (134)). These unitary transformations diagonalize the Yukawa 2 coupling matrices, absorb five phases of the CKM matrix, and get rid of two phases of m0 . L

e Fermionic ﬁelds Sfermionic ﬁelds

⌫0 P U †⌫0 ⌫ ⌫0 P U †⌫0 ⌫ L ! L e L ⌘ L L ! L e L ⌘ L eL0 Pe Ue†eL0 eL eL0 Pe Ue†eL0 eL ! L ⌘ e ! L e ⌘ e eR0 P eVe†eR0 eR eR0 P eVe†eR0 eR ! L ⌘ e ! L e ⌘ e uL0 Aeq†Uu†uL0 uL uL0 Aeq†Uu†uL0 uL ! ⌘ e ! e ⌘ e uR0 Aq†Vu†uR0 uR uR0 Aq†Vu†uR0 uR ! ⌘ e ! e ⌘ e dL0 BqUd†dL0 dL dL0 BqUd†dL0 dL ! ⌘ e ! e ⌘ e d0 B V †d0 d d0 B V †d0 d R ! q d R ⌘ R eR ! q d eR ⌘ eR Table 13: Unitary transformations of thee fermionic and sfermionice e ﬁelds.

Due to the (s)fermionic basis transformations summarized in table 13, it is convenient to redeﬁne the trilinear coupling matrices and the Hermitian sfermion mass matrices that occur in the soft supersymmetry breaking part of the MSSM25. We redeﬁne the trilinear coupling matrices as follows: a P V †a0 U P † , a A†V †a0 U A , a B V †a0 U B† , (144) e ⌘ L e e e L u ⌘ q u u u q d ⌘ q d d d q and the sfermion mass matrices as e e 2 2 2 2 2 2 m P U †m0 U P † , m P V †m0 V P † , m A†U †m0 U A , L ⌘ L e L e L e ⌘ L e e e L Q ⌘ q u Q u q 2 2 2 2 m Ae†V †m0 V A , m BeV †m0 V B† . (145) ue ⌘ q u ue u qe de ⌘ q d de d eq e e 25 If not, then thee (unphysical)e parameters thate have beene removed from the MSSM Lagrangian show up again in the soft supersymmetry breaking terms.

39 As it turns out, we can make use of two more phase transformations to absorb even more parameters. The corresponding groups, that we call U(1)A and U(1)B, are equivalent to the Peccei- Quinn (P ) and R symmetries in various papers (e.g. [12, 13]). These phase transformations are only symmetries of the MSSM Lagrangian for µ = 0. Let denote any ﬁeld contained in the MSSM, then the transformations under U(1)A and U(1)B are deﬁned as follows:

eiQA!A , eiQB !B . (146) ! ! The charges QA,QB depend on the ﬁelds and !A,!B are free parameters. Tables 14 and 15 list the values of QA,QB for the ﬁelds that make up the chiral and gauge supermultiplets respectively.

Spin 0 QA QB Spin 1/2 QA QB

LL 01 LL 10 e 01 ec 10 eR⇤ R QL 01 QL 10 e u 01 uc 10 eR⇤ R c dR⇤ 01 dR 10 e H 20 H 1 1 eu u H 20 H 1 1 d ed

Table 14: The charges QA,QB (pertaining to the U(1)e A, U(1)B transformations respectively) for the ﬁelds that make up the chiral supermultiplets.

Spin 1/2 QA QB Spin 1 QA QB

ga 11 Ga 00 W a0 11 W a0 00 e B 11 B 00 f

Table 15: The charges QA,QBe(pertaining to the U(1)A, U(1)B transformations respectively) for the ﬁelds that make up the gauge supermultiplets.

Let us apply the U(1)A and U(1)B transformations to the MSSM ﬁelds. Requiring invariance of the MSSM Lagrangian (including the soft supersymmetry breaking part) under these transformations, leads us to simultaneously redeﬁne the Higgs mixing parameters:

µ e2i(!A+!B )µ, b e4i!A b, (147) ! ! as well as the gaugino masses and trilinear coupling matrices:

2i(! ! ) 2i(! ! ) M e A B M , a e A B a , (148) k ! k ! where k =1, 2, 3 and = e, u, d. As the transformations under U(1)A and U(1)B are parametrized by two independent parameters (!A,!B), two objects in eqs. (147) and (148) can each get one of their phases removed by ﬁxing !A,!B in a clever way. Conventionally, the soft breaking parameters b and M3 are made real. This is e↵ectuated by picking 1 1 ! = arg(b) ,!= ! + arg(M ) . (149) A 4 B A 2 3 If we apply all transformations and redeﬁnitions of this subsection to the MSSM Lagrangian (i.e. we go from a description in terms of the “primed” objects to one in terms of the “unprimed”

40 objects), the actual physical degrees of freedom become apparent. To see how the MSSM La- grangian changes form, it is most instructive to look at the parts that involve the superpotential and the soft supersymmetry breaking terms. As we have seen in section 5.3, the superpotential features in two pieces. The ﬁrst piece describes the scalar interactions that come from the “F-ﬁelds”:

i 2 ↵ W =(Hd0⇤) (LL0†)ye†ye(LL0 ) (Hd0 )↵ eR0†yeye†eR0 Hd0†Hd0 MSSM h i ↵ ↵ +(H ) (Q0†) y† y (Q ) (H ) +(H ) (Q0†) y†y (Q ) (H ) u0⇤ e L u eu L0 u0 ↵e d0⇤e L d d L0 d0 ↵ u0†y y† u0 H0†H0 d0†y y†d0 H0†H0 + u0†y y† d0 H0†H0 +h.c. R u ue R u u e R d d R d d eR u d R ed u ⇣ ⌘ L0†y†e0 e0†y L0 Q0†y† u0 u0†y Q0 Q0†y†d0 d0†y Q0 eL e R eR e L Le u R eR u L Le d R Re d L Q0†y†d0 e0†yeL0 µ0⇤e0†yeH0†L0 µ0⇤u0†yuH0†Q0 µ0⇤d0†ydH0†Q0 e L ed eR R e L e Re e u eL e R e ed L e R u L ⇣ 2 2 +h.c.) µ0 H0†H0 µ0 H0†H0 e |e |e u e u | e| d d e e e e e 2 2 0 2 2 0 2 2 = H ⌫† f ⌫ H e† f e + (H )⇤He† f ⌫ +h.c. e† f e H†H | d | L e L | d | L e L d d L e L R e R d d 0 2 2 + 2 2 h + 0 2i H u† f u H d † V †f V d + (H )⇤H d † V †f u +h.c. | u| eL u eL | u | e L qe u q L ue eu L q u L e e 2 2 0 2 2 h 0 2 i H u† V f V †u H d † f d + (H )⇤Hd † f V †u +h.c. | d | e L qe d q L e| d | L d eL d d eL d q eL 2 2 h i u† f u H†H d † f d H†H + u† f V f d H†H +h.c. R u eR u u eR d R de d e R u q d R ed u e ⇣ ⌘ ⌫† f e e† f ⌫ e† f e e† f e u† f u u† f u eL e eR R e L eL e eR R e L eL u R R eu L dL† Vq†fuuRuR† fuVqdL uL† VqfddRdR† fdVq†uL dL† fddRdR† fddL e e e e e e e e e e e e + 0 u† V f d e† f ⌫ + d † f d e† f e µ⇤(H )⇤e† f ⌫ µ⇤(H )⇤e† f e e L q de Re R e L e Led R R ee eL e u eR e eL e e u R e L ⇣ 0 + µ⇤(H )⇤u† f u µ⇤(H)⇤u† f V d µ⇤(H )⇤d † f V †u e d Re eu L e e d e Re u qe L u e R ed q L e e 0 2 2 µ⇤(H )⇤d † f d +h.c. µ H†H µ H†H , (150) u eR d eL | e| u u e| | d d e e ⌘ and the second piece describese thee Yukawa couplings:

1 ij ↵ ↵ ↵ W +h.c. = e0†y (L0 ) (H0 ) + e0†y (L0 ) (H0 ) + e0†y (L0 ) (H0 ) 2 i j R e L d ↵ R e L d ↵ R e L d ↵  MSSM ↵ ↵ ↵ u0†yu(Q0 ) (H0 )↵ u0†yeu(Q0 e) (H0 )↵ u0†yu(Q0e) (H0 )↵ R L u R L u e R L u ↵ ↵ ↵ + d0†y (Q0 ) (H0 ) + d0†y (Q0 ) (H0 ) + d0†y (Q0 ) (H0 ) R d L d ↵ R d eL ed ↵ R d L ed ↵ ↵ e µ0(H0 ) (H0 ) +h.c. u d ↵ e e e e 0 0 = He† f ⌫ + H e† f e He† f ⌫ + H e† f e d eR e eL d R e L d R e L d R e L 0 0 + He† f ⌫ + H e† f e + H u† f u H u† f V d d R e L d R e L eu R u L e u R u q L 0 + 0 e + e + H u† f u H u† f V d + H u† f u H u† f V d eu R u L e u R u q L u R u L u R u q L e e 0 0 HddR† fdVq†uL + Hd dR† fddL HddR† fdVq†uL + Hd dR† fddL e e e e e e e e 0 ↵ HddR† fdV †uL + Hd dR† fddL µ(Hu) (Hd)↵ +h.c., (151) q e e e e e e e e e e

41 and the soft supersymmetry breaking part of the MSSM transforms as

soft 1 aT a a0T a0 T ↵ L = M 0 g0 g0 + M 0 W 0 W 0 + M 0 B0 B0 +h.c. + e0†a0 (L0 ) (H0 ) MSSM 2 3 2 1 R e L d ↵ ⇣ ↵ ↵ ⌘ h 2 2 u0†a0 (Q0 ) (H0 ) + d0†a0 (Q0 ) (H0 ) +h.c. L0†m0 L0 e0†m0 e0 R u e L e u ↵ fR df L d e↵ e L Le L e R e R 2 2 2 2 i 2 Q0†m0 Q0 u0†m0 u0 d0†m0 d0 m H0†H0 me H0†H0 e L Qe L R u eR Re d R Hu u ue Hde d ed e e ↵ [b0(Hu0 ) (Hd0 )↵ +h.c.] e e e e e e e e e 1 aT a a0T a0 T 0 = M g g + M W W + M B B +h.c. + H e† a e He† a ⌫ 2 3 2 1 d R e L d R e L ⇣0 + 0 ⌘ h 2 + H u† a u H u† a V d + H d † a d Hd † a V †u +h.c. ⌫† m ⌫ u R eu Le u fR u fq L d eR ed L d R de q Le e L e L L 2 2 2 2 2 i 2 e† m e e† m e u† m u d † V †m V d u† m u d † m d e L e L L e R e eR L eQ L eL qe Q q Le R eu R R de R e 2 2 ↵ m H†Hu m H†Hd [b(Hu) (Hd)↵ +h.c.]. (152) e Hu e eu e Hede d e e e e e e e e e e ee Table 16 summarizes the number of independent parameters of the MSSM including soft supersymmetry breaking. From this table we infer that the MSSM has 123 independent parameters26.

Name Physicaldescription Mathematicaldescription #Parameters

gs,g,g0 gauge couplings real numbers 3 f , f , f diag. Yukawa coupling matrices real diag. 3 3matricesinf.s. 9 e u d ⇥ V CKM matrix unitary 3 3matrixinf.s. 4 q ⇥ M1,M2,M3 gaugino masses real and complex numbers 5 a , a , a trilinear coupling matrices complex 3 3matricesinf.s. 54 e u d ⇥ m2 , m2, m2 , m2 , m2 sfermion mass matrices Hermitian 3 3matricesinf.s. 43 L e Q u d ⇥

mHeu ,meHd f e e Higgs masses real numbers 2 µ sup.p. Higgs mixing parameter complex number 2 b sup.b. Higgs mixing parameter real number 1 Total: 123

Table 16: All parameters of the MSSM including soft supersymmetry breaking. The parameters that are listed above the horizontal dashed line also occur in the Standard Model. The abbreviations “sup.p.”, “sup.b.”, “diag.” and “f.s.” stand for “supersymmetry preserving”, “supersymmetry breaking”, “diagonal” and “family space” respectively.

5.6 Symmetries of the MSSM In section 3 we analyzed the symmetries of the Standard Model. In this subsection we argue that most of those results can be easily generalized to the MSSM. The MSSM has (by construction) the same Poincar´eand gauge symmetries as the Standard Model, but what about the ﬂavor symmetries? Let us start by considering the kinetic terms for a general chiral supermultiplet. From eq. (110) we know that these terms are given by:

free µ µ L =(@ L)† (@µL)+(@ R⇤ )† (@µR⇤ ) , (153a) free µ c µ c L = i L† @µ L + i R† @µ R , (153b)

L R L R where we have explicitly included left- and right-handed ﬁelds. Let M ,M ,M ,M GL(3, C) be (general) linear transformations in family space that act on the fermionic and sfermionic2 ﬁelds

26The number 124 that is often encountered in literature also includes the strong CP violating angle. For simplicity and good phenomenological reasons, we have – just like we did for the Standard Model – set this parameter to zero.

42 as follows:

M L , M R , (154a) L ! L R ! R M L , M R . (154b) L ! L R ! R This means that eqs. (153a) and (153b) transform as:

µ µ µ L L (@ )† (@ )+(@ ⇤ )† (@ ⇤ ) (@ )† M †M (@ ) L µ L R µ R ! L µ L ⇣ T ⌘ µ R R⇤ +(@ R⇤ )† M M (@µR⇤ ) , (155a)

µ c µ c µ L⇣ L ⌘ i † @ + i † @ i † M †M @ L µ L R µ R ! L µ L ⇣ T ⌘ c µ R R⇤ c + i R† M M @µ R . (155b) ⇣ ⌘ Requiring invariance of (155a) and (155b) under the transformations (154a) and (154b) leads L R L R to the conclusion that M ,M ,M ,M must be unitary (this is analogous to eq. (54) for the Standard Model). Let us now have a look at the Lagrangian that describes the coupling of a (left-handed) chiral supermultiplet to gauginos that belong to a certain gauge group with corresponding inﬁnitesimal generators T a and coupling constant g (cf. eq. (123)):

gauginos a a L = p2g † T +h.c. . (156) chiral L L · ⇣ ⌘ This Lagrangian transforms under (154a) and (154b) as (dropping the constant prefactor as well as the Hermitian conjugate):

a a a L L a † T † T M †M (157) L L · ! L L · ⇣ ⌘ L L From (155a) and (155b) we know that M ,M must be unitary, so requiring invariance of (157) under these transformations gives: L L M = M . (158) Of course we could do the same thing for the right-handed chiral supermultiplets and arrive at a similar result. From this we can conclude that chiral superpartners transform identically. To specify the flavor group of the MSSM we could again distinguish between flavor mixing (only quark mixing actually, as lepton mixing is absent) included or not. The exclusion of flavor mixing implies that we can set the CKM matrix equal to the identity matrix. From eq. (151) we then infer that, with regards to flavor symmetries, those terms have the same form as the Standard Model Yukawa couplings. This means that again we could distinguish between di↵erent fermionic mass spectra (cf. section 3.4 for the various cases) and we would obtain the same results. Let us not go too fast here though; the soft supersymmetry breaking sector, namely, has introduced much more matrices in family space: the trilinear coupling matrices and sfermion mass matrices. By looking at eq. (152) we notice that the presence of these additional matrices certainly spoils all transformation freedom in family space. However, when the Yukawa coupling matrices are a priori taken diagonal (to exclude flavor mixing), then it would only be “fair” to also do the same thing to the family space matrices in the soft supersymmetry breaking sector. In fact, this is also what happens in phenomenologically viable constrained supersymmetric models like the phenomenological MSSM (pMSSM). In the pMSSM, the trilinear coupling matrices are taken proportional to the Yukawa coupling matrices, and the sfermion mass matrices are assumed to be diagonal. As long as we consider the same cases for (the diagonal elements of) the trilinear coupling matrices and sfermion mass matrices as for the Yukawa coupling matrices27, then the same analysis for the Standard Model can also be carried out for the MSSM and similar results are obtained. 27More restrictive cases for the trilinear coupling matrices and sfermion mass matrices are also possible. This is to be understood as follows: say we consider the case where all diagonal components of the Yukawa coupling matrices are di↵erent, then we can do the same thing for the trilinear coupling matrices and sfermion mass matrices, but we could also take two or three components of these (diagonal) matrices equal to each other. After all, the transformation freedom in family space (i.e. the amount of symmetry present) is determined by the least restrictive (most general) coupling or mass matrix.

43 What about the MSSM flavor group once flavor mixing is included? This means that we restore the CKM matrix and consider the MSSM (including soft supersymmetry breaking) in its most general form. How much transformation freedom in family space is now left? Well, not much, as the trilinear coupling matrices are completely general complex 3 3 matrices in family space. This means that the only symmetries possible are complex phases,⇥ regardless of the form of the mass spectrum. As the fermions and sfermions must transform identically (cf. eq. (158)) and the (s)lepton and (s)quark sectors are decoupled (in terms of field transformations), it follows that family space is equipped with only two U(1) symmetries. These symmetries correspond to lepton and baryon number conservation, which is the same as for the Standard Model. To summarize, the symmetry analyses that we performed for the Standard Model in section 3 are directly applicable to the MSSM and similar results are obtained. The MSSM and the Standard Model turn out to have exactly the same symmetries28. Thus if we consider the MSSM in full, then its symmetries are – just like for the Standard Model – given by table 7.

28Without the inclusion of the soft supersymmetry breaking sector, of course, the MSSM also respects invariance under supersymmetry transformations (by construction).

44 6 Renormalization group techniques

When we measure something in quantum ﬁeld theory, for example the charge of a particle, we notice that the answer depends on the momenta of the particles involved, or in other words, the energy scale of the observed interaction. This is due to the presence of virtual particles. Although these short-lived particles cannot be detected directly, they do have a measurable e↵ect29.The strength of these quantum ﬂuctuations depends on the energy scale, and this forms the basis of the concept of renormalization. In this section the concepts of renormalization and the renormalization group (RG) are discussed, as well as RG techniques to probe high scale physics given low energy data only.

6.1 What is renormalization? When we calculate an observable in quantum field theory (i.e. a cross section), we often make use of Feynman’s perturbative approach. In the language of Feynman diagrams, classical physics corresponds to tree level diagrams and the quantum corrections are represented by loops that describe virtual (o↵-shell) particles. All loop four-momenta are integrated over from negative to plus infinity, which often gives rise to divergences. By doing this, we tacitly assume the theory to be valid up to arbitrarily high energy scales, or equivalently, arbitrarily small length scales, which is not very realistic. How are we supposed to interpret these divergences and obtain finite, physically relevant results? A cross section can both be measured experimentally and determined theoretically. To obtain a cross section theoretically, we need to sum all possible Feynman diagrams (which is an infinite number), in accordance with the principle of superposition. A Lagrangian is a classical object and contains so-called bare parameters. Even though these parameters have names like “mass” or “charge”, these parameters are not directly measured in collider experiments and di↵er from the physical masses and charges. The renormalizability hypothesis is that a reparametrization of the theory in terms of physical (measurable) quantities, called renormalized parameters, turns the singular perturbative expansion into a well-defined one. This hypothesis implies that the divergences do not come from the perturbation expansion itself, but from the choice of parameters used to perform it. The first step of any renormalization is the regularization procedure. This is needed to quantify the singularities. Regularizing an expansion means that a limit is introduced. Well-known examples of regularization procedures include the introduction of a cuto↵scale or dimensional regularization. In the first case, the integration over loop momenta is not carried out all the way up to infinity but instead up to some (arbitrary) cuto↵scale ⇤. This procedure is called cuto↵regularization. The physical meaning of the cuto↵scale ⇤is the suppression of quantum fluctuations at distance 1 scales shorter than ⇤ . A second well-known regularization procedure, loosely stated, consists of carrying out the integration over loop momenta in 4 " dimensions (with " infinitesimal) rather than in 4 dimensions. This procedure is called dimensional regularization and one advantage of this method is that it respects Lorentz invariance, unlike cuto↵regularization. The idea of regularization is that after renormalization the original expansion is recovered when the limits (⇤ or " 0) are taken. When this is possible, the theory is said to be renormalizable. Di↵!1erent regularization! schemes can lead to di↵erent intermediate results, but must all lead to the same final result (provided that the regularization procedure has not broken any relevant symmetries). Renormalizing the theory means that we write the expansion in terms of physical (renormalized) quantities. To this end, the bare parameters are split into a renormalized part and so-called counterterms. The counterterms are tuned such that the divergences exactly cancel out. The divergences in the theory cannot disappear completely after the described renormalization procedure. In fact, the divergences reside in the relation between the bare parameters and the renormalized (measurable) parameters. As the bare parameters are non-physical quantities that are only used in intermediate calculations, their singular behavior is harmless. The renormalization procedure is performed order by order in perturbation theory and only works if the divergences possess a very precise structure.

29An example of this is the Casimir e↵ect.

45 6.2 Renormalization group equations The renormalized parameters are defined at some reference scale µ. If we fix both the bare couplings and the cuto↵scale, then varying µ means that the renormalized couplings must change. This phenomenon is called the running of the couplings and is described by the so-called RG equations. Naturally, physical observables should be independent of the arbitrary renormalization scale. This fact can be used to determine the dependence on µ of the various couplings. Say we have an observable (calculated at some loop order) that depends on some couplings gi(µ) and masses m (µ) . Requiring that cannot depend on µ, means that: { } { j } d @ dg (µ) @ dm (µ) @ = + i + j =0. (159) dµ @µ dµ @g (µ) dµ @m (µ) ✓ i j ◆ When we do this for several observables, we obtain a set of coupled di↵erential equations, the RG equations. As observables depend on the loop order, these equations do too. Often it is convenient to multiply the RG equations by a constant. These new equations are called -functions and are defined for a parameter p as follows: dp (p) 16⇡2 , (160) ⌘ dt where t log (µ/µ0), and µ0 is an (arbitrary) energy scale that makes the argument of the logarithm⌘ dimensionless. Thus -functions encode how parameters depend on the renormalization scale µ.

6.3 E↵ective field theories When the theory of renormalization was developed, the prevalent view used to be that the cuto↵ scale was merely a mathematical trick to tame the present divergences. After renormalization, the limit ⇤ was taken and, in case of renormalizable theories, the result would be finite. Wilson gave a di!1↵erent interpretation to renormalization and the cuto↵scale30. In his point of view every quantum field theory possesses a cuto↵scale that has some physical significance. According to Wilson, the cuto↵scale should be interpreted as the scale where new physics becomes relevant. As an example, let us consider the absence of gravity in the Standard Model. As gravitational e↵ects become comparable in strength to the other forces at an energy scale of (1019 TeV), the Planck scale, we expect that the Standard Model breaks down here and that newO physics comes into play; we clearly need a quantum theory of gravity. Of course, the Standard Model could very well break down at much lower energy scales already due to other new physics phenomena. Nowadays the most accepted interpretation of renormalization is that this scale of new physics is to be identified with ⇤. Although a fundamental theory of everything could in principle describe all aspects of nature, we would only use it to describe physics at the smallest length scales. As we are zooming out, particular high energy e↵ects become irrelevant and can be ignored. This is what we do in physics all the time. For example, we do not use quantum electrodynamics to describe the chemical bonds between atoms. For that a classical non-relativistic description of the electromagnetic force is much more e↵ective. E↵ects at high energy scales can often be omitted in low energy descriptions, or taken into account as small perturbations. This is the idea behind an e↵ective field theory (EFT). These theories are only well-defined up to a certain energy scale ⇤. At this scale new physics becomes relevant and the e↵ective “low” energy theory cannot be used anymore. In EFTs the high energy e↵ects (the dominating e↵ects at scales larger than ⇤) have been integrated out. Suppose we consider physics at an energy scale E within the context of some EFT with cuto↵ scale ⇤. Then the e↵ects coming from nonrenormalizable operators (i.e operators with a mass dimension larger than four) describing heavy particles of mass M (where M ⇤), are suppressed by (positive) powers of E/M. These e↵ects can either be neglected or used perturbatively in powers of E/M. For this reason, it is no problem to include nonrenormalizable interactions in an EFT without spoiling its predictive power. The description of nature can be regarded as a chain of EFTs. Each EFT gives a description of physics in a certain range of energy scales and

30For his ideas on renormalization he received the 1982 Nobel prize in physics.

46 has a particular particle content. Naturally, consecutive EFTs should match at the corresponding particle thresholds where both theories must give the same physical predictions. This is called the matching condition. Each theory below a certain threshold is considered as the low energy EFT of the theory above the threshold. This makes all quantum ﬁeld theories e↵ective theories, also the Standard Model. The shortcomings of the Standard Model indicate that there must be a more fundamental theory. An example of a potentially more fundamental EFT is the MSSM, which is a central topic in this thesis.

6.4 Renormalization group invariants If we discover supersymmetric particles in collider experiments, we would like to find out what supersymmetry breaking model is realized in nature. However, supersymmetry breaking (probably) takes place at a very high energy scale that may be (far) beyond experimental access in the nearby future. Is this then the end of the story? The answer is no, and the great rescue is provided by the renormalization group. Making clever use of the RG equations of the running couplings, there are in fact several ways to probe physics at high energy scales using low energy data only. The two most well-known RG techniques for probing high scale physics are the top-down and bottom- up methods. For both methods the running parameters have to be evolved numerically as the coupled RG equations (in general) cannot be solved analytically. In the context of supersymmetry, the high scale models that can be probed this way typically describe supersymmetry breaking. We now briefly discuss both methods and some limitations. A more elaborate overview of these methods is presented in [14]. The top-down method starts by choosing a particular high scale model and identifying the free running parameters. After picking a point in the parameter space, the running parameters can be evolved down to the collider scale (with their RG equations). The obtained values can then be compared to data and this could constrain the parameter space of the model used. Examples of severe limitations of this method include that the high scale physics model must be assumed beforehand and that scanning the entire parameter space is usually very time consuming. The second procedure, the bottom-up method, works in the opposite direction: the free running parameters of the beyond the Standard Model theory that are measured in experiments are evolved upwards to the threshold of new physics (with their RG equations). The results can then be compared to the matching conditions of the various high scale models and thus consistency is checked. Also this method has significant limitations though. It is for example required to know all values of the running parameters at the collider scale as the RG equations are coupled di↵erential equations. Furthermore, also experimental errors are evolved upwards and may even become too large to use the values found to draw any sensible conclusions about the physics at high scales. Moreover, we do not know the value of the new physics threshold; this is always a guess. Fortunately there is yet another way to probe high scale physics that circumvents most limitations listed of the top-down and bottom-up methods. This method makes use of so-called RG invariants. These invariants are combinations of RG equations that are independent of the renormalization scale (i.e. they have a vanishing -function). The insight that RG invariants can be used to probe high scale physics, in particular supersymmetry breaking models, is quite recent, and is elaborately discussed in [14–19]. To illustrate the utility of RG invariants as probes of high scale physics, we now work through an example using the Standard Model gauge couplings. First we derive two RG invariants using the -functions of the three gauge couplings, and then we use these invariants to test the hypothesis of gauge coupling unification at some high energy scale. To one-loop, the -functions of the Standard 31 Model gauge couplings g 5/3 g0, g g, and g g are given by : 1 ⌘ 2 ⌘ 3 ⌘ s p 41 19 (g )= g3 ,(g )= g3 ,(g )= 7g3 . (161) 1 10 1 2 6 2 3 3 To find renormalization scale independent combinations of these -functions, it is helpful to rewrite them as follows: 2 41 2 19 2 g = ,g = ,g = 14 . (162) 1 5 2 3 3 31 The coupling constant g1 is the properly scaled version of g0 in the context of an SU(5) or SO(10) grand unified theory.

47 Note that on the right-hand sides of these three -functions we simply have numbers. Hence we can construct two (independent) vanishing -functions, for example:

2 123 2 2 41 2 g + g =0,g + g =0. (163) 1 95 2 1 70 3 ✓ ◆ ✓ ◆ We have constructed two objects that do not depend on the renormalization scale anymore. This leads us to deﬁne the following two RG invariants:

SM 2 123 2 SM 2 41 2 I g + g ,I g + g . (164) 1 ⌘ 1 95 2 2 ⌘ 1 70 3 Now let us see how these invariants can be used to probe high scale physics. An example of a question that one can ask about physics at high energies is whether the gauge couplings unify at some (high) energy scale. Our claim is that such a prediction can be tested using the RG invariants in (164) and low energy data only. To test whether gauge coupling uniﬁcation is consistent with the Standard Model, we proceed as follows: say uniﬁcation of gauge couplings takes place at a certain energy scale µ = µgcu,then:

g (µ )=g (µ )=g (µ ) g . (165) 1 gcu 2 gcu 3 gcu ⌘ gcu

At the scale µ = µgcu, the RG invariants in (164) take the form:

SM 218 2 SM 111 2 I (µ )= g ,I(µ )= g . (166) 1 gcu 95 gcu 2 gcu 70 gcu From this equation we infer that the invariants are not independent of each other anymore as there is only one parameter left. They can be related as follows: 703 1526 ISM + ISM =0. (167) 82 1 123 2 Eq. (167) is called a sum rule for RG invariants. As RG invariants (by definition) do not depend on the renormalization scale, this sum rule does neither! It therefore applies to all energy scales where the Standard Model is a valid e↵ective field theory. In terms of the gauge couplings, eq. (167) reads: 23 2 111 2 109 2 g g + g =0. (168) 6 1 10 2 15 3 This sum rule only holds if the Standard Model gauge couplings unify at some high energy scale. This means that violation of this sum rule implies that gauge coupling unification is not consistent with the Standard Model. To show that this is indeed the case, we consider the values of the gauge couplings at for example µ = mZ [10]:

g (m )=0.46152 0.00004 , (169a) 1 Z ± g (m )=0.65170 0.00011 , (169b) 2 Z ± g (m )=1.219 0.004 . (169c) 3 Z ± Plugging these values into the sum rule described by eq. (168), yields:

23 2 111 2 109 2 g (m ) g (m ) + g (m ) = 3.25 0.03 . (170) 6 1 Z 10 2 Z 15 3 Z ± As this number lies many standard deviations from 0, we conclude that gauge coupling unification is not consistent with the Standard Model, which should not come as a surprise. Within the MSSM, however, the couplings could unify at a scale (1016 GeV) (cf. figure 1). The same strategy as for the Standard Model can be applied to test thisO hypothesis for the MSSM. To do this, of course, one first has to discover supersymmetric particles. After the parameters of the MSSM (or any other supersymmetric theory) have been determined, the values can be plugged into the RG invariants of the theory (using the MSSM -functions). Again, gauge coupling unification at some high energy scale can be translated into a sum rule for RG invariants. If the sum rule is satisfied, then gauge coupling unification is consistent with the MSSM.

48 In a supersymmetric context, the predictions about physics at high energy scales that can be tested with this method of RG invariants are matching conditions that come from various supersymmetry breaking models. Examples of matching conditions are gauge coupling unification, gaugino mass unification and scalar mass unification. By way of illustration, figure 2 shows the RG evolution of the scalar and gaugino masses satisfying mSUGRA32 boundary conditions (i.e. scalar and gaugino mass unification) imposed at the energy scale ⇤= 2 1016 GeV. In [14] various matching conditions are considered for which the corresponding one-loop⇥ RG invariant sum rules are constructed. That paper, however, considers a heavily constrained version of the MSSM (the pMSSM) with only 26 independent (running) parameters rather than the full MSSM with 123 parameters (cf. section 5.5). Within the pMSSM, a total number of 16 RG invariants have been derived.

Figure 2: RG evolution of the scalar and gaugino masses with mSUGRA boundary conditions 16 imposed at ⇤= 2 10 GeV. The masses of the gauginos are indicated by M1,2,3, and the labels H ,H represent⇥ the Higgs masses ( µ 2 + m2 )1/2 and ( µ 2 + m2 )1/2 respectively. The other u d Hu Hd lines represent the running squark and| | slepton masses with| | dashed lines indicating the (square roots of) the third family parameters m2 ,m2 ,m2 ,m2 ,m2 (from top to bottom) and solid lines d3 Q3 u3 L3 e3 indicating the first two generations of sfermions. The scalar and gaugino masses at the unification e e e e e scale are denoted by m0 and m1/2 respectively. This figure has been taken from [8].

Using RG invariants to probe high scale physics has many advantages over using the top-down or bottum-up methods. One of the advantages of the RG invariants method is the fact that not all running parameters of the theory have to be included (this as opposed to the bottom-up method), simply because not all parameters give rise to RG invariants. Furthermore, using RG invariants is purely algebraical and does not require any numerical analyses. This also means that there is no need to evolve any experimental errors upwards, as is the case for the bottum-up method. In contrast to the top-down method, in a relatively small amount of time many high scale models can be tested with the RG invariants method. Another nice feature of this method is that the value of the new physics threshold is not relevant at all. It is important though to note that di↵erent high scale physics models (that yield particular matching conditions) could in principle give rise to the same sum rules for RG invariants. In these cases of ambiguity, the method of RG invariants gives inconclusive results. This means that the true power of RG invariants is their falsifying potential.

32This is a particular supergravity model, short for “minimal supergravity”.

49 Of course, the use of the RG invariants method is not completely unlimited. As we have discussed in section 6.2, RG equations depend on the loop order. This implies that RG invariance only holds up to a certain order in perturbation theory. Already at two-loop order, the RG equations of the MSSM are very complicated and it is not even known whether RG invariants can be constructed at all. Is this something that could ruin the beauty of this method? In [16] it is shown that two-loop contributions to RG invariants in the pMSSM are smaller than the expected experimental uncertainties on the one-loop invariants in case we would be able to measure the supersymmetric couplings within 1% uncertainty at the collider scale. In other words (at least in case of the pMSSM), the RG invariants method already works well at one-loop order and for most purposes there will be no need to consider higher loop orders. A genuine limitation of the RG invariants method, however, is that other new physics phenomena could be present well below the considered threshold scale. This could irrevocably alter the RG trajectories and thus also the RG invariants. For this reason, we might sometimes draw wrong conclusions about high scale physics models. Hence, to use the RG invariants method one has to assume that all RG equations remain valid all the way up to the considered threshold scale. In other words, with the RG invariants method we cannot look further than the first threshold of new physics and we can only make statements about the specific theory that we have used to construct the RG invariants and sum rules. To summarize, the RG invariants method is an entirely algebraical method that can be used to probe physics at energy scales that are far beyond experimental access. For this method we do not even need to know the exact value of the scale at which new physics comes into play. We have seen that predictions about physics at high energy scales (the matching conditions) can be translated into sum rules for RG invariants. The RG invariants of the beyond the Standard Model theory can be evaluated once (some of) the parameters have been measured at the collider scale. Subsequently, all sum rules that have been constructed can then be evaluated as well. If a sum rule is violated, the corresponding prediction about the matching conditions is not consistent with the collected data and this falsifies the corresponding high scale model. This method has already been used to construct sum rules for the pMSSM and various supersymmetry breaking models. In section 8 we will discuss whether RG invariants can be constructed in the full MSSM (i.e. the MSSM without any simplifications), making use of the one-loop -functions for the soft supersymmetry breaking scalar masses that will be derived in the next section.

50 7 One-loop -functions for MSSM scalar masses

In this section we derive the one-loop -functions for the soft supersymmetry breaking scalar masses of the MSSM (the masses of the sfermions, of course, are represented by matrices). This has in principle been done already in [20], so why would we do it again? The main reason for this is that we want to express the -functions in terms of the weak hypercharges of the various fields. This will make it easier to unify symmetries with the concept of RG invariants, which we will try in the next section. How can we calculate these one-loop -functions for the soft scalar masses? Well, the -functions follow from the one-loop scalar mass counterterms. The mass counterterms, in turn, are found by calculating one-loop corrections to the scalar field propagators. For a scalar field , the one-loop corrections to its propagator are of the form:

where the blob in the middle represents any one-loop quantum correction. The whole calculation will be performed in Weyl spinor language and before the electroweak symmetry is broken. To 2 arrive at the -functions of the scalar masses m one has to go through several steps: Step 1: Identify all (interaction) terms in the MSSM Lagrangian that give rise to one-loop • corrections to the propagator of . These terms represent either a 3-point interaction between and other scalars, fermions or vector bosons, or a 4-point scalar interaction that involves the ﬁeld twice. Subsequently, group the terms according to the type of one-loop diagram (i.e. the type of integral) that they give rise to.

Step 2: Specify the Feynman rules for propagators of scalars, fermions (beware of the lan- • guage of Weyl spinors) and vector bosons. Then perform the various integrals (there will be four of them, cf. appendix E) that occur in the one-loop diagrams. As we are eventually only 1 interested in -functions, only the singular behavior of the integrals is relevant (i.e. the " terms, where " is the inﬁnitesimal parameter that comes from dimensional regularization).

Step 3: For each relevant term in the Lagrangian, read o↵the corresponding Feynman rule for • the vertex factor(s). Then determine the total factors that come from the vertex or vertices of the corresponding one-loop diagrams. Step 4: Calculate all diagrams by multiplying the total vertex factors by the corresponding • integrals. Adding up all diagrams gives the total one-loop contribution to the propagator of . This quantity is the quantum correction that alters the free (classical) propagator of . Step 5: The terms in the one-loop correction to the propagator of that are proportional • to “p2” a↵ect both the field renormalization of and its mass matrix counterterm. Per- form the renormalization of the field and its mass matrix by specifying the corresponding counterterms. The mass counterterm is proportional to the one-loop -function coecient 2 of m. In the first subsection we will derive the propagators for the scalar fields (this is non-trivial as the mass matrices mix the family multiplet components), and, subsequently, the core of the calculation is presented by explicity working out a few diagrams (one example is given for each type of diagram). The four di↵erent types of integrals that we encounter are calculated in appendix E. Next, it is explained how the renormalization procedure works for scalar fields. In the last subsection we provide the results: the one-loop -functions for the MSSM soft scalar masses.

7.1 The propagator for scalar fields In this subsection we derive the form of the propagator for the scalar fields (sfermions and Higgs doublets) in the MSSM. The propagator for a scalar field, in principle, is defined as the inverse of the corresponding Klein-Gordon operator (times a conventional factor of i), but for the MSSM scalars it is a bit tricky: the mass terms for the Higgs doublets mix the doublets, and the masses of

51 the sfermions are represented by Hermitian matrices that cause the sfermionic gauge eigenstates to mix. So how should we define a propagator for a Higgs doublet or sfermion? To answer this question, we will first cast the free Lagrangian describing the Higgs doublets into the same form as the free Lagrangian for sfermions, and then we will derive the form of the propagator that applies to all scalar fields in the MSSM.

7.1.1 Higgs doublets

The MSSM contains the two Higgs doublets Hu,Hd and their free Lagrangian is given by (cf. section 5.3):

free µ µ 2 2 2 2 L =(@ H†)(@µHu)+(@ H†)(@µHd) m + µ H†Hu m + µ H†Hd Higgs u d Hu | | u Hd | | d ↵ ↵ b(H ) (H ) + b(H⇤) (H ) u d ↵ d u ↵ µ µ 2 2 2 2 =(@ H†)(@µHu)+(@ H†)(@µHd) m + µ H†Hu m + µ H†Hd u d Hu | | u Hd | | d + 0 0 0 0 + + b H H H H (H )⇤(H )⇤ +(H)⇤(H )⇤ , (171) u d u d d u d u where on the second⇥ line we have written the Higgs doublets in terms⇤ of their complex components (cf. table 10 for the deﬁnitions of the Higgs doublets). As can be inferred from eq. (171), the (real) mass parameter b mixes the components of Hu and Hd. We would like to rewrite this Lagrangian and be able to identify a Hermitian mass matrix for the Higgs doublets, just like for the sfermions. To this end, let us introduce a new doublet that contains both Hu and Hd:

H H u⇤ . (172) ⌘ Hd ✓ ◆ Using the new doublet H, we can now rewrite eq. (171) as

free µ 2 L =(@ H†)(@ H) H†m H, (173) Higgs µ H 2 where the (Hermitian) matrix mH is given by

m2 + µ 2 00 b Hu | | 0 m2 + µ 2 b 0 m2 0 Hu | | 1 . (174) H ⌘ 0 bm2 + µ 2 0 Hd | | B b 00m2 + µ 2 C B Hd | | C @ A The free Lagrangian for the Higgs doublets has now been cast into the same form as the free Lagrangian for a sfermionic ﬁeld. This means that all MSSM scalar ﬁelds are now described by a similar free Lagrangian: there is a kinetic part and there is a mass term containing a Hermitian mass matrix.

7.1.2 Scalar fields Now that all mass terms of the MSSM scalar fields are described in the same way, let us denote any complex scalar (multiplet) field in the MSSM by (it thus could be H or a sfermionic field). The free Lagrangian of is given by:

free µ 2 L =(@ †)(@ ) †m , (175) µ 2 where m is the corresponding Hermitian mass matrix. To derive the propagator for ,weneedto 2 2 ﬁnd the eigenvalues (i.e. masses) of m. As m is Hermitian, it can be diagonalized by a unitary 2 transformation (cf. also section 2.4). Let us denote the transformation that diagonalizes m by U, then eq. (175) can be rewritten as

free µ 2 L =(@ †)(@ ) † U †U m U †U µ µ ⇣ 2 ⌘ ⇣ ⌘ =(@ †)(@ ) †U †d U , (176) µ

52 where 2 2 d U m U † (177) ⌘ is a (real) diagonal mass matrix. As the propagator for a scalar ﬁeld is generally derived from the Klein-Gordon operator, let us consider the Euler-Lagrange equation. With indices restored (and, for convenience, omitting the labels frow now on), eq. (176) reads:

free µ 2 L =(@ (⇤) )(@ () ) (⇤) (U †) (d ) (U) () , (178) k µ k k k` `m mn n where k,`, m, n are family indices (cf. appendix A for an overview of the indices that are used in this thesis) and represents any additional multiplet indices (possibly SU(2) or SU(3) indices). The right-hand side of the Euler-Lagrange equation (eq. (42)) for (⇤)k is given by:

free µ @L µ @ µ = @ @µ()k , (179) @ (@ (⇤)k)!

2 which equals p ( )k in momentum space (here the tilde denotes the Fourier transformed ﬁeld), and the left-hand side is given by: e free @L 2 = (U †)k`(d )`m(U)mn()n . (180) @(⇤)k The Euler-Lagrange equation in momentum space now gives:

2 2 0= p (U †) (d ) (U) ( ) kn k` `m mn n 2 2 =(⇥U †)k` p `m (d )`m (U)mn⇤ ( )n e 2 2 = U † p I⇥3 d U , ⇤ (181) e 2 2 where U † p I3 d U is the Klein-Gordon operatore for the field . The scalar propagator in quantum field theory is by definition the inverse of the Klein-Gordon operator, times a conventional factor of i. This means that the propagator P of the field is given by:

2 2 1 P i U † p I d U ⌘ 3 = U⇥†DU , ⇤ (182) where we have deﬁned the diagonal matrix D as

i p2 m2 00 1 2 2 1 i D i p I3 d = 0 0 p2 m2 0 1 . (183) ⌘ 2 00 i B p2 m2 C B 3 C @ A The quantity mi (with i =1, 2, 3) denotes the mass of the ith (family) component of .The propagator matrix for scalar ﬁelds (eq. (182)) is an important ingredient in the calculation of the one-loop -functions for the MSSM soft scalar masses.

7.2 One-loop corrections to scalar propagators In this subsection we derive the one-loop contributions to the propagators of the MSSM scalar fields. Remember that we are interested in the singular behavior only (the pole contributions). As it turns out, the one-loop diagrams come in four di↵erent types. Each type, by definition, gives rise to one of the four integrals that are listed in appendix E. The MSSM contains seven scalar (multiplet) fields: two Higgs doublets and five sfermionic fields (combined family and gauge multiplets), cf. also table 10. For each field we will calculate the one-loop pole contributions to its propagator. In this subsection, for simplicity, we will only consider quantum corrections to the free Lagrangian of QL though. As this field is charged under each gauge group, it represents the most general case. Furthermore, for each type of diagram we will consider only one interaction (involving e 53 QL) in exquisite detail and calculate the corresponding one-loop correction (the matrix element). Other interactions that give rise to the same type of diagram (for which no calculation is provided here)e are simply mentioned at the beginning of each of the following subsections. Furthermore, we adopt the following conventions regarding Feynman diagrams: The undetermined loop momentum is denoted by ` and the momenta of the external particles • are denoted by p. Vertex factors are denoted by V . • The total factor that comes from all vertices in a given diagram is denoted by W . • The matrix element is denoted by M and its pole contribution by M UV. • We will now list the four di↵erent types of diagrams and provide one example for each type, all corresponding to QL as the external field.

7.2.1 Type 1 diagramse The ﬁrst type of one-loop diagram is constituted by 4-scalar interactions and is given by: `

pp I . (184) / 1

The following interactions contribute to this type of diagram (with QL in the initial and ﬁnal state): e I. Scalar interactions coming from the “F”-ﬁelds:

I ↵ ↵ L1 =(Hu⇤) (QL† )yu† yu(QL) (Hu)↵ +(Hd⇤) (QL† )yd†yd(QL) (Hd)↵

Q† y† u u† y Q Q† y†d d† y Q (185) L ue R R u Le L d R R d L e e II. 4-scalar interactions coming from the “D”-ﬁelds: e e e e e e e e

II 2 1 † † † † † † L1 = YQ g0 QLQL YL LLLL + Ye⇤ eReR + YQ QLQL + Yu⇤ uRuR + Yd dRdR L L R 2 L R R⇤ ✓ e e 1 e e 1e e 2 a0 a0 a0 e ae0 + Y H†He e+ Y H†eH e g Qe †e Q L† e eL + Q†ee Q + H† H Hu u u Hd d d 4 L L L L 2 L L u u ◆ ✓

a0 1 2 a 1 a a a + H† H g Q† Q Qe† Q e u†e u e d† ed e (186) d d 4 s L L 2 L L R R R R ◆ ✓ ◆ e e e e e e e e Example ...

As an example, let us consider the following interaction:

2 1 2 a0 a0 L1 = Y YH g0 Q† QLH†Hu g Q† QLH† Hu (187) QL u L u 4 L u Vertex factors: e e e e e •

QL QL 2 V = iY Y g0 1A QL Hu ↵ k` pq e ,`,q ↵,k,p e (188) 8 e ! 1 2 a0 a0 <> V1B = ig ( )↵( )k`pq 4 H H u u :>

54 Total vertex factor: • Hu

2 W1A = iY YH g0 ↵k`pq QL u M1 QL QL 8 e (189) ⌘ ,`,q ↵,k,p ! 1 2 a0 a0 <> W1B = ig ( )↵( )k`pq 4 e e Matrix element: :> • d 4 d d ` M =(W + W ) µ (U † D U ) 1 1A 1B (2⇡)d H H H Z 2 1 2 a0 a0 = i Y YH g0 ↵ + g ( )↵( ) QL u 4 ✓ ◆ e d 4 d d ` i µ (U † ) (U ) ⇥ (2⇡)d H i `2 m2 + i✏ ij H j k` pq Z ✓ Hi ◆

2 1 2 a0 a0 = Y YH g0 ↵ + g ( )↵( ) QL u 4 ✓ ◆ e d 4 d d ` µ (U † ) (U ) , (190) ⇥ H i (2⇡)d `2 m2 + i✏ ij H j k` pq Z Hi ! where we identify the integral inside the brackets with I1(mHi ). The pole contribution to M1 thus becomes: im2 UV 2 1 2 a0 a0 Hi M = Y YH g0 ↵ + g ( )↵( ) (U † )i ij(UH )jk`pq 1 QL u 4 H 8⇡2" ✓ ◆ ✓ ◆ i e 2 2 = Y YH g0 (U † )i(d )ij(UH )j↵ 8⇡2 QL u H H h 1 2 e 2 a0 a0 1 + g (U † ) (d ) (U ) ( ) ( ) " 4 H i H ij H j ↵ k` pq

i 2 2 1 2 2 a0 a0 1 = Y YH g0 Tr [m ]UL ↵ + g Tr [m ]UL ( )↵ k`pq" 8⇡2 QL u H 4 H  ⇣ ⌘ i e 2 2 2 1 = Y YH g0 m + µ ↵k`pq" , (191) 4⇡2 QL u Hu | | where the subscript “UL”e refers to the upper left 2 2 submatrix. The second term on the fourth 2 a0 ⇥ line vanishes as Tr([mH ]UL ) = 0.

7.2.2 Type 2 diagrams The second type of one-loop diagram involves 3-scalar interactions and is given by: ` pp I . (192) / 2

p + `

The following interactions contribute to this type of diagram (with QL in the initial and ﬁnal state): e I. Scalar interactions coming from the “F”-ﬁelds:

I L2 = µ⇤uR† yuHd†QL + µ⇤dR† ydHu†QL + h.c. (193) II. Soft supersymmetry breaking interactions: e e e e II ↵ ↵ L = u† a (Q ) (H ) + d† a (Q ) (H ) + h.c. (194) 2 R u L u ↵ R d L d ↵ e e e e 55 Example ...

As an example, let us consider the following interaction:

↵ L2 = dR† ad(QL) (Hd)↵ + h.c. (195) Vertex factors: • e e

dR `,q e QL V2A = i✏↵(ad)`kqp (196) ↵,k,p ! e Hd

dR `,q e QL V2B = i✏↵(ad† )k`pq (197) ↵,k,p ! e Hd

Total vertex factor: •

m,r e W2 = ✏↵✏(ad† )km(ad)m`prrq M2 QL QL (198) ⌘ ,`,q ↵,k,p ! = (a† a ) ↵ d d k` pq e e Hd

Matrix element: • d 4 d d ` i i M = W µ 2 2 (2⇡)d `2 m2 + i✏ (p + `)2 m2 + i✏ Z 1 2 d 4 d d ` µ = (a† a ) , (199) (2⇡)d (`2 m2 + i✏)((p + `)2 m2 + i✏) ↵ d d k` pq "Z 1 2 # 33 where we identify the integral inside the brackets with I2(m1,m2). The pole contribution to M2 thus becomes: UV i 1 M = (a† a ) " . (200) 2 8⇡2 ↵ d d k` pq

7.2.3 Type 3 diagrams The third type of one-loop diagram is formed by interactions between one scalar and two fermions and is given by: ` pp I . (201) / 3

p + ` 33Note that we have not speciﬁed the masses here. This is because for type 2 integrals the masses do not matter in the high energy limit.

56 The following interactions contribute to this type of diagram (with QL in the initial and ﬁnal state): e I. Chiral supermultiplets coupled to gauginos:

I 1 a0 a0 1 a a L = p2 Y g0Q† QLB + gQ† QLW + gsQ† QLg +h.c. (202) 3 QL L 2 L 2 L ✓ ◆ e II. Yukawa couplings: e e e f e e II ↵ ↵ L = u† y (Q ) (H ) + d† y (Q ) (H ) + h.c. (203) 3 R u L u ↵ R d L d ↵

Example ... e e e e

As an example, let us consider the following interaction:

1 a a L = p2g Q† Q g + h.c. (204) 3 2 s L L Vertex factors: • e e e

QL ,`,q 1 p a QL V3A = 2igs↵kl( )pq (205) ↵,k,p ! 2

g e

QeL ,`,q 1 p a QL V3B = 2igs↵`k( )qp (206) ↵,k,p ! 2

g e

Total vertex factor:e • Q 1 L W = g2 (a) (a) 3 2 s ↵ km m` pr rq ,m,r 2 1 M3 QL QL = gs ↵k` pqrr 3 prrq (207) ⌘ ,`,q ↵,k,p ! 8 2 e e = g ↵k`pq g 3 s where on the second line wee used the following identity for generators T a of the Lie algebra corresponding to the group SU(N) [1]:34 1 1 (T a) (T a) = . (208) ij k` 2 i` jk N ij kl ✓ ◆ Matrix element: • d µ ⌫ 4 d d ` i`µ i(p + l)⌫ M = W µ Tr 3 3 (2⇡)d `2 + i✏ (p + `)2 M 2 + i✏ Z  3 8 dd` µ4 d` (p + l) Tr (µ⌫ ) = g2 µ ⌫ , (209) 3 s (2⇡)d (`2 + i✏)((p + `)2 M 2 + i✏) ↵ k` pq "Z 3 # 34For SU(3) we have T a = a/2andforSU(2)wehaveT a0 = a0 /2.

57 where we identify the integral inside the brackets with I3(0,M3). The pole contribution to M3 thus becomes:

8 i 2M 2 p2 UV 2 3 M3 = gs 2 ↵k`pq 3 " 8⇡ " # 2 igs 2 2 1 = 2M p " . (210) 3⇡2 3 ↵ k` pq 7.2.4 Type 4 diagrams The fourth type of diagram is constituted by gauge interactions and given by:

` pp I . (211) / 4

p + `

The following interactions contribute to this type of diagram (with QL in the initial and ﬁnal state): e I. Gauge interactions:

I µ 1 a0 a0 1 a a L = i(@ Q† ) Y g0Bµ + gW + gsG QL + h.c. (212) 4 L QL 2 µ 2 µ  e e e Example ...

As an example, let us consider the following interaction:

1 a0µ a0 L = igW (@ Q† ) Q + h.c. (213) 4 2 µ L L Vertex factor: e e •

QL p2 ,`,q p1 e µ 1 a0 QL V4 = ig( )↵k`pq(p1 + p2)µ (214) ↵,k,p ! 2 p3 W e

Total vertex factor: • 1 W = g2(a0 ) (a0 ) 4 4 ↵ km n` pr rq QL (p `)µ(p `)⌫ ,r ⇥ ⌫ n e m µ 1 2 1 M4 QL QL = g ↵ ↵ (215) ⌘ ,`,q ↵,k,p ! 2 2 ✓ ◆ e e kmn`pq(p `)µ(p `)⌫ W ⇥ 3 = g2 (p `) (p `) 4 ↵ km n` pq µ ⌫ where on the second line we made use of eq. (208).

58 Matrix element: • d µ⌫ 4 d d ` i⌘ M = µ W (U † D U ) 4 (2⇡)d 4 Q Q Q mn (p + `)2 + i✏ Z  d e e e 3 2 4 d d ` i † = g µ d (U )mi 2 2 ij(UQ)jn 4 (2⇡) Q ` m + i✏! Z Qi i(p l) ⌘µ⌫ (p le) e µ ⌫ e ⇥ (p + `)2 + i✏ ↵ km n` pq  d 4 d µ⌫ 3 2 d ` µ (p l)µ⌘ (p l)⌫ † = g (U )ki d 2 2 2 ij(UQ)j`↵pq , (216) 4 Q " (2⇡) (` m + i✏)((p + `) + i✏)# Z Qi e e where we identify the integral inside the bracketse with I (m , 0). The pole contribution to M 4 Qi 4 thus becomes: e i m2 +2p2 UV 3 2 Qi M = g (U † ) (U ) 4 4 Q ki 2 ⇣ 8⇡2" ⌘3 ij Q j` ↵ pq e e e 3i 2 4 2 5 2 1 = g (U † ) (d ) (U ) +2p " 32⇡2 Q ki Q ij Q j` k` ↵ pq h i 3i 2 2 e 2 e 1 = g (me ) +2p " . (217) 32⇡2 Q k` k` ↵ pq h i This concludes our list of examples fore the various types of diagrams. How are we supposed to deal with all these inﬁnite contributions to the scalar propagators? This problem can be solved by renormalization and is the topic of the next subsection.

7.3 Renormalization of scalar ﬁelds As we have seen before, the free (classical) Lagrangian of a complex scalar (multiplet) ﬁeld is given by free µ 2 L =(@ †)(@ ) †m , (218) µ 2 where m is the corresponding Hermitian mass matrix. In terms of momentum space Feynman rules, the terms in eq. (218) give rise to

µ 2 (@ †)(@ ) ip , (219a) µ ! 2 2 †m im . (219b) ! Switching on interactions means that both the kinetic and mass terms receive quantum corrections. We have seen in the previous subsection that these quantum corrections can be inﬁnite. How can we make sense of this? In section 6.1 we argued that the bare Lagrangian parameters can be split in a renormalized (physical) part and counterterms. The counterterms are chosen as to exactly cancel out the unwanted divergences. In this subsection we will see two di↵erent types of renormalization: ﬁeld and mass renormalization. For convenience we will omit the label in the rest of this subsection.

7.3.1 Field renormalization Let us first consider field renormalization by looking at corrections to the kinetic term of eq. (218). We denote the coecient of the divergent one-loop quantum corrections proportional to ip2 by A (this coecient is generally a matrix). Quantum corrections alter the kinetic term of the Lagrangian as follows: µ µ (@ †)(@ ) (@ †)(I + A)(@ ) , (220) µ ! µ where I denotes the identity matrix. To get rid of the divergent term, we would like to absorb A in the field . This is called field renormalization. To this end, let us define the renormalized field r as follows: 1/2 Z , (221) r ⌘

59 and we also deﬁne Z I + , (222) ⌘ Z where Z is the kinetic counterterm (also Z and Z are matrices here and can be assumed to be 1/2 Hermitian for our calculation). In the following we will need the expansion of Z around Z =0 to ﬁrst order: 1 Z1/2 = I + . (223) 2 Z To one-loop order, the renormalized kinetic part of the Lagrangian (eq. (220)) now reads:

µ µ 1/2 1/2 (@ †)(I + A)(@µ)=(@ r†)Z (I + A)Z (@µr) µ µ (@ †)(@ )+(@ †)( + A)(@ ) . (224) ⇡ r µ r r Z µ r To cancel the divergent one-loop contribution A, it is clear that the kinetic counterterm must be chosen as = A. (225) Z The ﬁeld has now been renormalized and the divergence has disappeared by picking the right kinetic counterterm. What about the quantum corrections that a↵ect the mass of the scalar ﬁeld?

7.3.2 Mass renormalization Also the mass term of eq. (218) changes under quantum corrections. Let us denote the divergent one-loop corrections to im2 by iB (this, again, is generally a matrix). Switching on interactions gives the shift 2 2 †m † m B . (226) ! In terms of the renormalized ﬁeld this becomes to one-loop order:

2 1/2 2 1/2 † m B = †Z m B Z r r 2 1 2 1 2 †m + † B m m . (227) ⇡ r r r 2 Z 2 Z r ✓ ◆ To get rid of the divergences we apply mass renormalization. Let us deﬁne the renormalized mass matrix as follows: m2 m2 m2 , (228) r ⌘ where m2 is the so-called mass matrix counterterm. Renormalizing the mass term (227), yields:

2 1 2 1 2 2 2 1 2 †m + † B m m = †m † m B + m r r r 2 Z 2 Z r r r r r 2 Z ✓ ◆ ✓ 1 + m2 . (229) 2 Z r ◆ From this equation it follows that to cancel out the divergences, the mass matrix counterterm should be tuned to become 1 m2 = B + Am2 + m2A , (230) 2 where we have also made use of eq. (225). Now all divergences have disappeared by ﬁeld and mass renormalization. By comparison with [20] we know that we obtain the one-loop -function for m2 from the mass matrix counterterm as follows:

(m2) = 16⇡2"m2 . (231)

In the next subsection we present the result of our calculation: the one-loop -functions for the soft supersymmetry breaking scalar mass parameters. The intermediate step, the calculation of the one-loop mass matrix counterterms, is found in appendix F.

60 7.4 Results In this subsection we list the one-loop -functions for the seven soft supersymmetry breaking scalar masses that result from our calculation. The pole contributions to the scalar propagators, as well as the mass counterterms, are given in appendix F. The quantity S in the -functions below, is deﬁned as (cf. also appendix F):

2 2 2 2 2 2 2 S 2YHu mHu +2YHd mHd +Tr 6YQ m +2YL m +3Yu⇤ mu +3Yd m + Ye⇤ me ⌘ L Q L L R R⇤ d R 2 2 2 ⇣2 2 2 2 ⌘ = m m +Tr m m e2m e+ m e+ me . e e e e e e (232) Hu Hd Q L u d e ⇣ ⌘ Through eq. (231) we go frome the countertermse e founde toe the following one-loop -functions:

2 2 2 2 2 2 32 2 2 2 2 2 † (m )= 8Y g0 M1 6g M2 gs M3 +2mHu yu† yu +2mHd ydyd + m yu† yu Q QL | | | | 3 Q 2 2 2 2 2 e + m ey†y + y† y m + y†y m +2y† m y +2y†m y +2a† a e Q d d u u Q d d Q u u u d d d u u 2 +2a† a +2Y g0 S, (233a) de d QL e e e e 2 2 2 2 2 2 2 2 2 2 (m )= 8Y g0 M1 e 6g M2 +2mHd ye†ye + m ye†ye + ye†yem +2ye†meye L LL | | | | L L 2 +2a†a +2Y g0 S, (233b) e e e LL e e e 32 2 2 2 e2 2 2 2 2 2 2 (m )= 8Yu g0 M1 gs M3 +4mH yuy† +2m yuy† +2yuy† m +4yum y† u R⇤ | | 3 u u u u u u Q u 2 +4a a† +2Y g0 S, (233c) e eu u uR⇤ e e e 2 2 2 2 32 2 2 2 2 2 2 (m )= 8Y g0 M e g M +4m y y† +2m y y† +2y y†m +4y m y† d d 1 s 3 Hd d d d d d d d d d Q d R⇤ | | 3 2 e +4adea† +2Y g0 S, e e e (233d) d dR⇤ 2 2 2 2 2 2 2 2 (m )= 8Ye g0 M1 e +4mH yey† +2m yey† +2yey†m +4yem y† +4aea† e R⇤ | | d e e e e e L e e 2 +2Y g0 S, (233e) e eR⇤ e e e 2 2 2 2 2 2 2 2 2 (m )= 8Ye g0 M 6g M +6Tr m y† y + m y† y + y† m y Hu Hu | 1| | 2| Hu u u Q u u u u u 2 ⇣ e e + au† au +2YHu g0 S, (233f)

2 2 ⌘2 2 2 2 2 2 2 (m )= 8Y g0 M1 6g M2 +2Tr 3m y†yd +3m y†yd + m y†ye Hd Hd | | | | Hd d Q d Hd e 2 2 2 ⇣ 2 + m y†y +3y†m y + y†m y +3a† a + a†a e+2Y g0 S. (233g) L e e d d d e e e d d e e Hd ⌘ The -functionse that we havee found are ine correspondence with the ones derived in [20]. The advantage of our results though, is that these -functions are expressed in terms of the weak hypercharges of the ﬁelds. This will prove useful in the next section. Note that all terms in the -functions are proportional to supersymmetry breaking parameters. This is not a coincidence! The reason for this is that setting all supersymmetry breaking parameters to zero should give a supersymmetry preserving theory, as otherwise quantum corrections would explicitly break supersymmetry (which would mean that the theory was not supersymmetric to begin with). Recall that the (supersymmetry preserving) Higgs mixing parameter µ entered our calculation, for example through the 4-scalar interactions (cf. eq. (191)). As can been seen in appendix F though, this parameter neatly canceled out, as it should. For our calculation it was also safe to ignore the 4-point interactions between two scalars and two gauge bosons. To one-loop, the 4-point interactions give rise to type 1 integrals which, in the high energy limit, give a proportionality to the mass of the ﬁeld inside the loop squared. As the masses of gauge bosons are supersymmetry preserving parameters, this type of interaction cannot contribute to the counterterm of a supersymmetry breaking parameter35. In the next section we will try to construct RG invariant quantities in the MSSM from the obtained -functions for the soft supersymmetry breaking scalar masses.

35Moreover, the gauge bosons are massless in our analysis as the electroweak symmetry has not been broken.

61 8 RG invariants in the MSSM

In this section we attempt to construct RG invariants in the MSSM from the -functions for the soft supersymmetry breaking mass parameters given in section 7.4. Is this even possible at all? Well, first we will use “standard” algebraical techniques (cf. [14]) to see if it is possible to construct any invariants in the first place. Subsequently, we will look at this question from a di↵erent perspective using arguments involving global U(1) symmetries of family space. It is important to be aware of the fact that all sums in this section over products of quantum numbers and fields inside family space traces are implicitly understood to be over all gauge degrees of freedom in the multiplets. So for QL, for example, such a sum would give a factor of 2 for SU(2) and a factor of 3 for SU(3), yielding a total multiplication by 6. Furthermore, for the sums we adopt the following notation: e Scalars are denoted by . • Sfermions are denoted by f. • Higgs doublets are denoted by H. • e Scalar weak isospin doublets are denoted by d. • Scalar color triplets are denoted by t. • Sums without any traces involved are implicitly understood to be over all gauge degrees of freedom and over all sfermionic families.

8.1 Constructing RG invariants in the canonical way Let us now try to construct RG invariants from the -functions of the soft supersymmetry breaking scalar masses using the “standard” techniques from [14]. The ﬁrst thing we note when we look at the -functions of the scalar sector of the MSSM in section 7.4, is that no invariants can be constructed for the o↵-diagonal terms of the sfermion mass matrices. The reason for this is that the order of the family space matrices is di↵erent for all ﬁve -functions of the sfermion mass matrices. In order to avoid this, we need to work with terms that are insensitive to the order of the family space matrices. We could therefore consider the -functions for the traces of the sfermion mass matrices. These -functions are given by:

2 2 2 2 2 2 2 2 2 Tr(m ) = 24Y g0 M1 18g M2 32gs M3 +6Y g0 S Q QL | | | | QL h i 2 2 2 e +2Tre m y† y + m y† y + y† m y + ae† a Hu u u Q u u u u u u u ⇣ 2 2 2 ⌘ +2Tr m y†y + m ey†y + y†m ey + a† a , (234a) Hd d d Q d d d d d d d 2 2⇣ 2 2 2 2 2 ⌘ Tr(m ) = 24Y g0 M1 18g e M2 +6Y ge0 S L LL | | | | LL h i 2 2 2 e +2Tre m y†y + m y†y + y†me y + a†a , (234b) Hd e e L e e e e e e e 2 2⇣ 2 2 2 2 2 ⌘ Tr(m ) = 24Yu g0 M1 32ges M3 +6Yu⇤ g0eS u R⇤ | | R 2 2 2 ⇥ ⇤ +4Tre m y† y + m y† y +ey† m y + a† a , (234c) e Hu u u Q u u u u u u u 2 2⇣ 2 2 2 2 2 ⌘ Tr(m ) = 24Y g0 M1 32g eM +6Y g0 Se d d s 3 d⇤ R⇤ | | R h i 2 2 2 e +4Tre m y†y + m y†y + ye†m y + a† a , (234d) Hd d d Q d d d d d d d 2 2⇣ 2 2 2 ⌘ Tr(m ) = 24Ye g0 M1 +6Ye⇤eg0 S e e R⇤ | | R 2 2 2 ⇥ ⇤ +4Tre m y†y + me y†y + y†m y + a†a . (234e) e Hd e e L e e e e e e e ⇣ ⌘ e e Together with the -functions for mHu and mHd (eqs. (233f) and (233g)), that already involved traces, we have seven equations. These equations contain three di↵erent trace structures. Elimi-

62 nating these trace structures yields four independent equations, we pick:

2 1 2 1 2 2 2 2 2 2 Tr m m m =6g0 M 18g M +2g0 S, (235a) Q 2 u 2 d | 1| | 2|  ✓ ◆ 2 e 3 e2 1 2e 2 2 2 2 2 2 2 m Tr m + m = 14g0 M1 6g M2 + 48g M 7g0 S, (235b) Hd 2 d 2 e | | | | s 3  ✓ ◆ e2 e2 2 2 2 2 2 Tr m 2m = 12g0 M + 36g M + 12g0 S, (235c) e L | 1| | 2| h ⇣ ⌘i 3 2e 2e 2 2 2 2 2 2 2 Tr m m = 14g0 M +6g M 48g M 7g0 S. (235d) 2 u Hu | 1| | 2| s 3  ✓ ◆ With these four -functionse we can construct three -functions where S has been eliminated, we choose:36

2 2 2 2 2 2 2 2 2 Tr 6m 2m +3m +3m + m = 48g0 M + 144g M , (236a) Q L u d e | 1| | 2| h ⇣ ⌘i e e e e e 2 2 3 2 3 2 1 2 2 2 2 2 2 2 m + m Tr m + m + m = 28g0 M1 12g M2 + 96g M , (236b) Hu Hd 2 u 2 d 2 e | | | | s 3  ✓ ◆ e e e 8m2 +8m2 +Tr 14m2 14m2 +5m2 19m2 +3m2 =0. (236c) Hu Hd Q L u d e h ⇣ ⌘i As -function (236c) vanishes, we have founde thee followinge RG invariant:e e

2 2 2 2 2 2 2 I1 8m +8m +Tr 14m 14m +5m 19m +3m ⌘ Hu Hd Q L u d e 2 ⇣ 2 2 2 2 ⌘ = 8S + 11 Tr 2m 2m e m me + me . e e (237) Q L u d e ⇣ ⌘ Let us now also consider the -functionse for thee squarese ofe the gauginoe masses [20]:

2 2 2 M = 44g0 M , (238a) | 1| | 1| M 2 =4g2 M 2 , (238b) | 2| | 2| M 2 = 12g2M 2 . (238c) 3 s 3 Now there are ﬁve equations left (eqs. (236a), (236b), (238a) - (238c)) with three variables 2 2 2 2 2 2 (g0 M1 ,g M2 ,gs M3 ). This means that we should be able to ﬁnd two more RG invariant quantities.| | Let| us| pick the following two vanishing -functions:37

2 2 2 2 1 2 2 2 m + m Tr 3m + m M1 15 M2 +8M =0, (239a) Hu Hd Q L 11| | | | 3  ⇣ ⌘ e e 12 Tr 6m2 2m2 +3m2 +3m2 + m2 + M 2 36 M 2 =0. (239b) Q L u d e 11| 1| | 2|  ⇣ ⌘ e e e This leads us to deﬁne the following twoe RG invariants:e

2 2 2 2 1 2 2 2 I2 m + m Tr 3m + m M1 15 M2 +8M , (240a) ⌘ Hu Hd Q L 11| | | | 3 ⇣ ⌘ 12 I Tr 6m2 2m2 +3me 2 +3em2 + m2 + M 2 36 M 2 . (240b) 3 ⌘ Q L u d e 11| 1| | 2| ⇣ ⌘ We could actually have guessede e this numbere ofe threee RG invariants right away. Namely, we started out with ten equations (seven -functions for the scalar masses and three for the gaugino masses) that only contained seven di↵erent structures (three trace structures, three gaugino masses and S). Having ten equations to eliminate seven di↵erent terms gives three invariants. These invariants have also been found by an independent (ongoing) numerical study [21].

36We construct these new -functions as follows: (236a) 6 (235a) + (235c), (236b) (235b) (235d) and (236c) 14 (235a) + 8 (235b) + 7 (235c) + 8 (235d).⌘ ⇥ ⌘ ⌘ ⇥ ⇥ ⇥ ⇥ 37These -functions are obtained as follows: (239a) 1 (236a) + (236b) 1 (238a) 15 (238b) + 8 (238c) ⌘ 2 ⇥ 11 ⇥ ⇥ ⇥ and (239b) (236a) + 12 (238a) 36 (238b). ⌘ 11 ⇥ ⇥

63 As it turns out, one more invariant can be constructed using soft supersymmetry breaking scalar masses. To this end, let us consider the quantity S deﬁned by eq. (232) in more detail. The -function for S follows directly from the -functions for the soft scalar masses in section 7.4 and is given by: 2 2 2 (S)=2 Y g0 S = 22g0 S. (241) X 2 As this -function contains the U(1) gauge coupling, let us also consider the -function for g0 , given by [20]: 2 2 4 4 (g0 )=2 Y g0 = 22g0 . (242) X Now it immediately follows from these two -functions that we can construct one RG invariant quantity from S and g0;wedeﬁne:

S 1 I = Y m2 +Tr Y m2 . (243) 4 2 2 H H f f ⌘ g0 g0 " H f !# X X e e In addition to these four RG invariants, ﬁve more invariantse can be constructed in the MSSM. These invariants are constituted by parameters in the gauge and gaugino sectors. Let us consider the -functions for the fourth powers of the gauge couplings [20]:

4 6 4 6 6 6 (g0 ) = 44g0 ,(g )=4g ,(g )= 12g . (244) s s When we compare these -functions to the ones for the gaugino masses (cf. eqs. (238a) - (238c)), we immediately note that we can construct three combined invariants in the gauge and gaugino sectors: 2 2 2 M1 M2 M3 I5 | 4| ,I6 | 4| ,I7 4 . (245) ⌘ g0 ⌘ g ⌘ gs Analogous to the construction of the RG invariants for the Standard Model gauge couplings in section 6.4, we can construct two RG invariants in the gauge sector of the MSSM, we deﬁne: 1 11 1 11 I8 2 2 ,I9 2 + 2 . (246) ⌘ g0 g ⌘ g0 3gs

The RG invariants I4,...,I9 also occur in the pMSSM [14, 16]. Say we would consider a constrained version of the MSSM where all matrices in family space (i.e. the sfermion mass matrices, the Yukawa matrices and the trilinear coupling matrices) are taken diagonal (this is done in for example the pMSSM), then we could apply the same method as described for the MSSM to find RG invariants for this simplified model38. In this particular model we would have twenty equations (fifteen -functions for the sfermion mass matrices, two for the Higgs masses, and three for the gaugino masses) containing thirteen di↵erent structures (nine structures coming from the diagonal components of the Yukawa matrices and trilinear coupling matrices, three gaugino masses and S). Eliminating only thirteen di↵erent terms using twenty equations would result in seven RG invariant quantities for this simplified model (on top of I4,...,I9). In the pMSSM, there are a couple of additional constraints: the first two generations of sfermions are mass degenerate, and the Yukawa matrices as well as the trilinear coupling matrices only have non-zero entries for the third generation sfermions. This means that for the pMSSM we would have fifteen equations (ten -functions for the sfermion mass matrices, two for the Higgs masses, and three for the gaugino masses) to eliminate seven di↵erent structures (three structures coming from the (33)-components of the Yukawa matrices and trilinear coupling matrices, three gaugino masses and S), yielding eight RG invariants (on top of I4,...,I9). Indeed, for the pMSSM eight RG invariant quantities have been constructed in [14, 16] that involve scalar masses only or combinations of scalar masses and gaugino masses. According to [14,16], the total number of (useful) RG invariants in the pMSSM is fourteen, while in the (full) MSSM this number, apparently, equals nine.39

38As diagonal matrices commute, we would again end up with only three di↵erent structures containing Yukawa matrices and trilinear coupling matrices in the -functions of the scalar masses. 39In the pMSSM two more invariants that involve the Higgs mass parameters µ and b can be constructed, but as argued in [14] these cannot be used to probe high scale physics models.

64 8.2 RG invariants from a di↵erent perspective Now let us have a look at the described procedure for deriving RG invariants from a di↵erent perspective. Where do the coecients in front of the scalar masses in the RG invariants come from? Is there any philosophy behind these numbers? The first thing we note is that to construct RG invariants from scalar masses in the MSSM, we need to get rid of the Yukawa matrices and the trilinear coupling matrices that occur in their -functions. From sections 5.3 and 5.4 we know that for the MSSM all interactions in family space are described by the Yukawa terms in the superpotential and the soft supersymmetry breaking trilinear terms. In case that the order of the family space matrices does not matter by taking a trace, then these interactions give rise to three di↵erent trace structures in the -functions for the soft scalar masses (cf. eqs. (233f), (233g), (234a) - (234e)). These trace structures each belong to one of the three Yukawa and trilinear interaction terms in the Lagrangian, which involve a unique set of three scalar fields. The coecients of these trace structures are di↵erent for the various scalar fields, simply because in the -functions no traces have been carried out over the gauge degrees of freedom of the external scalar fields (the external fields are the fields for which we have calculated the corrections to their propagators).

How is this to be understood? Say we consider the trilinear interaction between uR, QL and Hu and the one-loop corrections to the scalar propagators that it gives rise to, then these three fields can all occur as external fields or inside loops. The gauge degrees of freedom thate aree “closed” inside the loops (i.e. those gauge degrees of freedom that the external fields do not possess) are summed over. Say we take uR to be the external field, then there is an SU(2) doublet degree of freedom inside the loop that has to be “traced over”, giving a factor of 2. If we had taken Hu to be the external field, thene a trace over SU(3) degrees of freedom inside the loop would have resulted, giving a factor of 3, etc. Hence, if we multiply the trace structures in the -functions by the factors that result from summing over the gauge degrees of freedom of the external scalar fields, then all Yukawa and trilinear trace structures get exactly the same coecients! To cancel these structures, we thus need to construct a linear sum of the -functions of the “traced” scalar masses, such that the coecients of this sum add up to zero. Now we can use the fact that each trace structure corresponds to a unique combination of three scalar fields in the Lagrangian. So for the cancellation to take place we should assign quantum numbers to these three fields that add up to zero, which is equivalent to saying that the quantum number should be conserved in the interaction. This means that to cancel the Yukawa and trilinear contributions to the -functions, we have to consider (global) U(1) symmetries of the family sector of the MSSM. From sections 5.5 and 5.6 we know that for µ = 0 there are five di↵erent U(1) symmetries of the MSSM Lagrangian without 40 soft supersymmetry breaking terms, corresponding to the quantum numbers Y,B,L,QA,QB. However, when we look at eq. (148), we notice that the family sector including soft supersymmetry breaking terms is not invariant under U(1)A and U(1)B separately, but only under the combination U(1)A+B. Concluding, the U(1) symmetries that can be used to get rid of the trace structures in the -functions correspond to the quantum numbers Y,B,L,QA + QB. In other words, the quantity

2 Tr Qm , ! X where Q denotes the quantum number pertaining to the symmetry U(1)Q, does not contain the Yukawa and trilinear trace structures anymore. Table 17 provides the quantum numbers Y,B,L,QA + QB for all scalar ﬁelds in the MSSM (cf. table 14 for the quantum numbers QA and QB).

Now let us see how S can be eliminated from the -functions. From eqs. (233f), (233g), (234a) - (234e) it can be inferred that the elimination of S amounts to the following requirement:

YQ =0. (247) X 40Setting µ to zero does not have any consequences for this analysis anyway since it is a supersymmetry preserving parameter (cf. also our comments in section 7.4).

65 Spin 0 YBLQA + QB

1 1 QL 6 3 01 L 1 01 1 eL 2 2 1 u⇤ 01 eR 3 3 1 1 dR⇤ 01 e 3 3 e⇤ 10 11 eR 1 Hu 00 2 e 2 H 1 00 2 d 2

Table 17: The quantum numbers Y,B,L,QA + QB for the scalar ﬁelds of the MSSM.

2 2 2 2 2 2 How could we eliminate the quantities g0 M1 ,g M2 ,gs M3 from the -functions? For the 2 2 | | | | elimination of the “g0 M1 -terms” we note that from eqs. (233f), (233g), (234a) - (234e) the following requirement must| | be satisﬁed:

2 Y Q =0. (248) X 2 2 Naturally, the “g M2 -terms” only show up in the -functions for the masses of the scalar weak isospin doublets. This| | means that to eliminate these terms we must have

Qd =0. (249) Xd 2 2 The “gs M3 -terms”, in turn, only arise in the -functions for the masses of the scalar color triplets. To cancel these terms we need the following condition:

Qt =0. (250) t X All these sums over charges are reminiscent of mixed anomaly cancellations of the charge Q with the separate gauge groups that we know from for example the Standard Model. 2 2 Now that we have summed up the requirements needed to cancel the quantities S, g0 M1 , g2 M 2, and g2M 2 from the -functions given by eqs. (233f), (233g), (234a) - (234e),| let| us | 2| s 3 investigate which quantum numbers (Y,B,L or QA + QB) actually satisfy these requirements (eqs. (247) - (250)). This is summarized in table 18. From this table it follows that to eliminate S, suitable quantum numbers are 3B +L, 8Y +11(B L), and QA +QB. For the cancellation of the 2 2 2 2 “g0 M1 -terms” we could use Y , B L, and 16B + 3(QA + QB), while for the “g M2 -terms” | | 2 2 | | Y , B L, and 8B 3(QA + QB) are suitable. For the “gs M3 -terms” to cancel, we could pick Y , B or L. Naturally, linear combinations of these quantum numbers also do the job.

Now that we have found a relation between RG invariants and U(1) symmetries, let us have a look again at the RG invariants I2 (cf. eq. (237)), I3 (cf. eq. (240a)), and I4 (cf. eq. (240b)). They

66 YBLQA + QB

Y Q 11 2 60 2 3 3 P Y Q 0 8 2 2 P d Qd 0 6 6 16

P t Qt 0 0 0 36 P Table 18: The evaluation of various sums for the quantum numbers Y,B,L,QA +QB. The outcome 0 indicates that the corresponding quantum number is suitable to eliminate one of the quantities 2 2 2 2 2 2 S, g0 M ,g M ,g M (from top to bottom) from the -functions of the “traced” scalar masses. | 1| | 2| s 3

can be rewritten in terms of the quantum numbers Y,B,L,QA + QB as follows:

I = 8 Y m2 +Tr Y m2 + 11 Tr (B L) m2 , (251a) 1 H H f f " H f !# ! X X e e X 1 e I = (Q + Q ) m2 +Tr (3B + L + Q + Q ) m2 2 4 A B H H A B f f " H f !# X X e 1 e M 2 15 M 2 +8M 2 , e (251b) 11| 1| | 2| 3

2 12 2 2 I3 = Tr (3B + L)m + M1 36 M2 . (251c) ! 11| | | | X Note that the quantum numbers that appear in these RG invariants are in perfect correspondence with the information in table 18. Namely, the quantum number 8Y + 11(B L) that appears in I1 cancels S and all gaugino mass terms. The second invariant contains the combination (3B + L + QA + QB)/4 which only cancels S, and, lasty, the quantum number 3B + L that appears in 2 2 I3 eliminates both S and gs M3 . We have found a general recipe for the construction of combined RG invariants in the soft scalar and gaugino sectors of supersymmetric models:

Step 1: Make sure that the -functions of the soft scalar masses are insensitive to the order • of the family space matrices. This can be achieved by considering the “traced” -functions or by assuming that the family space matrices are diagonal. Step 2: Count the number of available equations (i.e. -functions) as well as the number of • di↵erent structures in the -functions. The number of (independent) RG invariants that can be constructed equals the number of equations minus the number of structures. Step 3: Identify the U(1) symmetries of family space. The quantum numbers pertaining to • those symmetries can be used to eliminate the structures that contain family space matrices. Subsequently, investigate which of those symmetries also eliminate one or more of the remaining structures in the -functions of the soft scalar masses.

Step 4: RG invariants can be constructed by considering linear sums of -functions of soft • scalar masses and gaugino masses, where the coecients of the scalar masses are quantum numbers pertaining to a particular U(1) symmetry of family space.

67 9 Conclusion

In this thesis we have performed a rigorous count of symmetries and parameters for both the Standard Model and the MSSM. We have found that for µ = 0 the MSSM has five di↵erent U(1) symmetries. The inclusion of soft supersymmetry breaking terms, brings this number down to four for the family sector. To use this knowledge about symmetries of the MSSM to say something about (one-loop) RG invariant quantities, we first had to calculate the one-loop -functions for the soft supersymmetry breaking scalar masses in the MSSM, expressed in terms of the weak hypercharges of the scalar fields. From the obtained -functions, together with those for the gaugino masses, we were able to construct three new RG invariants, which have also been found in an independent (ongoing) numerical analysis [21]. These three RG invariants are very interesting, as their form turns out to be related to U(1) symmetries of family space. We have formulated a general recipe for the construction of RG invariants in supersymmetric models. The six other RG invariants that we have found in di↵erent sectors of the MSSM are already known from simplified supersymmetric models like the pMSSM. In future work it would be interesting to consider various supersymmetric models and to construct their RG invariants, making use of the general recipe involving symmetries that we have found.

68 A Indices

In this thesis many di↵erent indices are used. All these di↵erent types, including their range and position, are summarized in table 19.

Index Range Use Upper Lower

µ, ⌫ 0, 1, 2, 3 spacetime components ⇥⇥ a, b, c 1, 2,...,8 generators of SU(3) ⇥ a0,b0,c0 1, 2, 3 generators of SU(2) ⇥ i, j 1, 2,...,7 chiralsupermultiplets ⇥⇥ v 1, 2, 3 gauge supermultiplets ⇥ p, q, r, s 1, 2, 3 color triplets ⇥ ↵,,, 1, 2 weak isospin doublets / Weyl spinors ⇥⇥ k,`, m, n 1, 2, 3 fermionic families ⇥ Table 19: An overview of the various types of indices that occur in this thesis. In the last two columns it is indicated whether the index is written as an upper or lower index.

There are a few exceptions to table 19. The indices a, b, c, for example, not only label the generators of the Lie algebra corresponding to SU(3); they are also used to label the infinitesimal generators of an arbitrary gauge group. The labels i, j of the chiral supermultiplets are in principle written as lower indices. However, when these labels denote di↵erentiation with respect to scalar fields i, they appear as upper indices. The ranges that are indicated for the chiral and gauge supermultiplet labels apply specifically to the MSSM. And, lastly, the indices i, j, k, ` are also used for miscellaneous situations. In those cases it should be clear from the context what the indices stand for. The weak isospin doublet indices are raised and lowered by the antisymmetric symbol ✏↵ with 12 21 non-zero components ✏ = ✏ = ✏12 = ✏21 = 1. These indices are, by convention, always contracted diagonally downwards and swapping two weak isospin doublets gives a minus sign. To illustrate the latter, let us for example consider the contraction of the weak isospin doublets LL and Hd (we make a singlet without using conjugation): e ↵ ↵ (LL) (Hd)↵ =(LL) ✏↵(Hd) =(H )✏ (L )↵ e e d ↵ L = (H )✏ (L )↵ d ↵e L = (H )(L ) . (252) d L e A regular inner product between these two weak isospine doublets would be written in this thesis as (LL)↵(Hd)↵ (with two lower indices). The antisymmetric symbol ✏↵ can also be used to make a spin-0 singlet out of two spin-1/2 Weyl spinors, which is discussed in appendix C. This operation is indicatede by a dot, but is often suppressed in this thesis.

69 B Unitary matrices

Unitary matrices are extremely important in physics and in this thesis in particular. In this appendix we list some of their properties. Let us start with the deﬁnition of a unitary matrix: a complex square matrix U is called unitary if

U †U = UU† = I, (253) where I is the identity matrix. The real analogue of such a matrix is called an orthogonal matrix. If the determinant of a unitary matrix equals one, then it is called a special unitary matrix. Unitary matrices form groups: the unitary group of degree n, denoted by U(n), is the group of n n unitary matrices with matrix multiplication as the group operation. The special unitary group⇥of degree n, denoted by SU(n), is a subgroup of U(n) and consists of n n unitary matrices with unit determinant. (Special) unitary groups are Lie groups as their elements⇥ are parametrized by continuous parameters. To see how many independent parameters are needed to parametrize a (special) unitary matrix, we ﬁrst note that any unitary matrix U can be written as the complex exponential of a Hermitian matrix H:41

U = eiH , (254) since iH iH iH iH e † e = e e = I. (255) It is clear that a Hermitian matrix must have real numbers on its diagonal and that any o↵-diagonal entry (i, j) is related to entry (j, i) simply by complex conjugation. An n n matrix has n diagonal elements and n2 n o↵-diagonal elements. The dimension of a general⇥ complex n n matrix is 2n2. This means that the dimension of an n n Hermitian matrix is 2n2 n (n⇥2 n)=n2. From eq. (254) it now follows that ⇥ dim (U(n)) = n2 . (256) With respect to a unitary matrix, a special unitary matrix has one additional constraint (i.e. its determinant equals one), which means that

dim (SU(n)) = n2 1 . (257) Now let us consider some unitary groups in more detail. We are especially interested in deriving the form of arbitrary group elements.

U(1): • The group U(1) is one-dimensional and the elements are complex numbers z that satisfy:

z =1, (258) | | which means that any z U(1) can be written as 2 z = ei↵ , (259) where ↵ R. 2 U(2): • The group U(2) is four-dimensional which means that its elements are parametrized by four parameters. We want to ﬁnd a parametrization for a general element of U(2). To this end, let us write U U(2) in the generic way 2 a a U = 11 12 , (260) a a ✓ 21 22 ◆ 41 AcomplexsquarematrixH is called Hermitian if H = H†.

70 where aij C (with i, j =1, 2). Then the unitarity requirement (eq. (253)) translates into 2 a 2 + a 2 =1, (261a) | 11| | 12| a11a21⇤ + a12a22⇤ =0, (261b)

a21a11⇤ + a22a12⇤ =0, (261c) a 2 + a 2 =1, (261d) | 21| | 22| as well as

a 2 + a 2 =1, (262a) | 11| | 21| a11⇤ a12 + a21⇤ a22 =0, (262b)

a12⇤ a11 + a22⇤ a21 =0, (262c) a 2 + a 2 =1. (262d) | 12| | 22| From eq. (261a) we infer that we can introduce an angle ✓ such that a = cos ✓ and a =sin✓. | 11| | 12| From eqs. (262a) and (262d) it then follows that a21 =sin✓ and a22 = cos ✓. This means that so far U is of the form | | | | cos ✓ei↵11 sin ✓ei↵12 U = . (263) sin ✓ei↵21 cos ✓ei↵22 ✓ ◆ Let us now have a closer look at the various phases inside this matrix. From eq. (261b) it follows that i(↵ ↵ ) i(↵ ↵ ) cos ✓ sin ✓e 11 21 +sin✓ cos ✓e 12 22 =0, (264) which gives i↵ i(↵ ↵ +↵ ) e 21 = e 11 12 22 , (265) from which we infer that we have three independent phases. Deﬁning ↵ ↵ , ↵ , and ⌘ 11 ⌘ 12 + ↵22 yields ⌘ cos ✓ei↵ sin ✓ei U = . (266) sin ✓ei(↵+) cos ✓ei(+) ✓ ◆ We thus conclude that any 2 2 unitary matrix is parametrized by one angle and three phases. ⇥ U(3): • Finding the general parametrization of unitary 3 3 matrices is similar to that of 2 2 matrices. However, as U(3) is a nine-dimensional group rather⇥ than only four-dimensional, this is⇥ much more tedious. As this derivation is not particularly illuminating, we will simply just state the result here. A full derivation is found in e.g. [6]. Any element of U(3) is parametrized by nine real parameters. These parameters include three rotation angles: ✓12, ✓23, and ✓13 (the subscripts refer to the plane of rotation); and six phases: ↵i (i =1, 2), j (j =1, 2, 3), and . Without loss of generality, any 3 3 unitary matrix U can be written as42: ⇥ U = AV B , (267) where 10 0 ei1 00 A = 0 ei↵1 0 ,B= 0 ei2 0 (268) 0 1 0 1 00ei↵2 00ei3 and @ A @ A i c12c13 s12c13 s13e V = s c c s s ei c c s s s ei s c , (269) 0 12 23 12 23 13 12 23 12 23 13 23 13 1 s s c c s ei c s s c s ei c c 12 23 12 23 13 12 23 12 23 13 23 13 @ A with cij cos ✓ij and sij sin ✓ij. This particular parametrization is often used to consider the CKM and⌘ PMNS matrices⌘ that describe mixing of ﬂavors in family space.

42Of course many parametrizations exist; this is just one of them.

71 C Spinors

As it turns out, nature violates parity. This means that left- and right-chiral states (colloquially also called left- and right-handed) are not treated the same way. The Standard Model, as well as supersymmetric theories, are therefore chiral theories. This appendix is mainly about notation regarding left- and right-handed fermions. In this thesis the Standard Model fermions are described by Dirac spinors, while supersymmetry makes use of Weyl spinors. In the following subsections we will see how we can convert from one notation to the other.

C.1 Dirac spinors In the Standard Model the fermions are usually mathematically described by Dirac spinors. Dirac spinors are four-component ﬁelds that transform under Lorentz transformations according to a four-dimenionsal representation of the Lorentz group. In the Weyl representation (or chiral representation) the Dirac matrices µ (for µ =0, 1, 2, 3) take the following form: 0 µ µ = , (270) µ 0 ✓ ◆ µ 1 2 3 µ 1 2 3 a0 where I2, , , and I2, , , involve the Pauli matrices (for ⌘ ⌘ 5 0 1 2 3 a0 =1, 2, 3). In this representation the matrix i reads: ⌘ I 0 5 = 2 . (271) 0 I ✓ 2 ◆ This matrix is used to deﬁne the chiral projection operators PL and PR (where “L” stands for “left” and “R” for “right”):

1 5 I2 0 1 5 00 PL I4 = ,PR I4 + = . (272) ⌘ 2 00 ⌘ 2 0 I2 ✓ ◆ ✓ ◆ For a four-component Dirac spinor we now deﬁne the left- and right-handed projections as P , P . (273) L ⌘ L R ⌘ R To construct a relativistically invariant scalar out of Dirac spinors, we need the Dirac adjoint operation which is deﬁned as follows: 0 † . (274) ⌘ The quantity is relativistically invariant. Dirac spinors and their Dirac adjoints are used in the sections about the Standard Model, while the sections about supersymmetry make use of Weyl spinors.

C.2 Weyl spinors Supersymmetric models are often constructed using two-component Weyl spinors rather than four- component Dirac spinors. In the Weyl representation a Dirac spinor splits up into two Weyl spinors L and R as follows: = L . (275) ✓ R ◆ The left- and right-handed chiral projection operators, deﬁned by eq. (272), can be used to project onto the left- and right-handed Weyl spinors respectively: 0 P = L ,P = . (276) L 0 R ✓ ◆ ✓ R ◆ To elucidate the connection between Dirac and Weyl spinors, let us for example consider the (massless) Dirac Lagrangian, both in terms of Dirac spinors and left- and right-handed Weyl spinors: µ LDirac = (i @µ) µ µ = L† (i @µ) L + R† (i @µ) R . (277)

72 In supersymmetric theories it is convenient to bring the right-handed Weyl spinors in a left- handed form so that all fermionic ﬁelds “live” in the same representation of the Lorentz group. This type of conjugation is deﬁned by

c 2 i ⇤ . (278) R ⌘ R As spinors describe fermionic particles, they must be anticommuting objects. Their indices 12 21 are raised and lowered by the antisymmetric symbol ✏↵ with non-zero components ✏ = ✏ = ✏ = ✏ = 1. By convention, only adjacent spinor indices are contracted and this always happens 12 21 diagonally downwards (for left-handed Weyl spinors). The antisymmetric symbol ✏↵ can be used to make a singlet out of two Weyl spinors (the analogy in the case of Dirac spinors is the use of the Dirac adjoint operation) and this is indicated by a dot. As an example, let us consider the contraction of two left-handed Weyl spinors ⇠ and :

⇠ ⇠↵ = ⇠↵✏ = ✏ ⇠↵ = ✏ ⇠↵ = ⇠ ⇠. (279) · ⌘ ↵ ↵ ↵ ↵ ⌘ · To avoid clutter, however, this dot is implicit in most parts of this thesis.

73 D Superpotential contributions to the MSSM Lagrangian

In this appendix we derive the contributions of the superpotential to the MSSM Lagrangian. A general supersymmetric Lagrangian contains two terms that involve the superpotential W (cf. eq. (123)). First we have the term 2 W i , (280) i where W is the functional derivative of W : W W i , (281) ⌘ i and secondly we have 1 W ij +h.c. , (282) 2 i · j where W ij is a double functional derivative of W : 2W W ij . (283) ⌘ ij The R-parity conserving superpotential for the MSSM is given by: ↵ ↵ ↵ ↵ W = e† y (L ) (H ) + u† y (Q ) (H ) d † y (Q ) (H ) + µ(H ) (H ) MSSM R e L d ↵ R u L u ↵ R d L d ↵ u d ↵ ↵ ↵ = (e⇤ ) (y ) (L ) (H ) +(u⇤ ) (y ) (Q ) (H ) R k e e k` L ` d ↵ e R kp u k` eL `p e u ↵ e ↵ e ↵ (dR⇤ )kp(yd)k`(QL)`p(Hd)↵ + µ(Hu) (Hd)↵ , (284) e e e e where in the second line we have written down all indices explicitly (cf. appendix A for the relevant e e conventions). To derive all contributions of the MSSM superpotential to the Lagrangian, we will now consider the terms originating from (280) and (282) separately:

The “single derivative” terms: • We start by considering the terms in the Lagrangian that originate from the superpotential term that contains a single derivative (term (280)):

⇤ WMSSM WMSSM =((e⇤ )k(ye)k`(Hd)↵) (eR)m(y⇤)m`(H⇤)↵ ↵ ↵ R e d (LL)` ! (LL)` ! =(ee⇤ )k(yey†)km(eR)m(eH⇤)↵(Hd)↵ e e R e d = eR† yeye†eRHd†Hd , (285a) e e

WMSSM WMSSM ⇤ e e ↵ = (y ) (L ) (H ) (y⇤) (L⇤ ) (H⇤) (e ) (e ) e k` L ` d ↵ e km L m d ✓ R⇤ k ◆✓ R⇤ k ◆ ⇣ ⌘⇣ ↵ ⌘ = (H⇤) (eL⇤ )m(y†ye)m`(LL) e(Hd)↵ d L e ` e e ↵ = (H⇤) (L† ) y†y (L ) (H ) , (285b) d eL e e L e d ↵

⇤ e e WMSSM WMSSM = (u⇤ )kp(yu)k`(Hu)↵ (d⇤ )kp(yd)k`(Hd)↵ ↵ ↵ R R (QL) ! (QL) ! `p `p ⇣ ⌘ e e(u ) (y⇤ ) (H⇤) (d ) (y⇤) (H⇤) e e · R mp u m` u ↵ R mp d m` d ↵ ⇣ ⌘ =(u⇤ )kp(yuy† )km(uR)mp(H⇤)↵(Hu)↵ Re u ue (uR⇤ )kp(yuyd† )km(dR)mp(Hd⇤)↵(Hu)↵ e e (dR⇤ )kp(ydyu† )km(uR)mp(Hu⇤)↵(Hd)↵ e e +(dR⇤ )kp(ydyd†)km(dR)mp(Hd⇤)↵(Hd)↵ e e = u† y y† u H†H u† y y† d H†H R eu u R u u e R u d R d u dR† ydyu† uRHu†Hd + dR† ydyd†dRHd†Hd , (285c) e e e e e e e e 74 WMSSM WMSSM ⇤ ↵ = (y ) (Q ) (H ) (y⇤ ) (Q⇤ ) (H⇤) (u ) (u ) u k` L `p u ↵ u km L mp u ✓ R⇤ kp ◆✓ R⇤ kp ◆ ⇣ ⌘⇣ ↵ ⌘ = (H⇤) (eQ⇤ )mp(y† yu)m`(QL) e(Hu)↵ u L u `p e e ↵ = (H⇤) (Q† ) y† y (Q ) (H ) , (285d) u eL u u L eu ↵ e e ⇤ WMSSM WMSSM ↵ = (yd)k`(QL)`p(Hd)↵ (yd⇤)km(QL⇤ )mp(Hd⇤) (d⇤ )kp ! (d⇤ )kp ! R R ⇣ ⌘⇣ ⌘ ↵ = (H⇤) (eQ⇤ ) (y†y ) (Q ) e(H ) e e d L mp d d m` L `p d ↵ ↵ = (H⇤) (Q† ) y†y (Q ) (H ) , (285e) d eL d d L ed ↵ e e WMSSM WMSSM ⇤ ↵ ↵ = (uR⇤ )kp(yu)k`(QL)`p µ(Hd) (Hu)↵ (Hu)↵ ✓ ◆✓ ◆ ⇣ ⌘ ↵ ↵ e(uR)mq(y⇤ )mne(QL⇤ )nq µ⇤(Hd⇤) · u ⇣ ↵ ⌘ ↵ =(Q⇤ ) (y⇤ ) (u ) (u⇤ ) (y ) (Q ) Lenq u mn R emq R kp u k` L `p ↵ ↵ ↵ ↵ µ⇤(u⇤ ) (y ) (Q ) (H⇤) µ(Q⇤ ) (H ) (y⇤ ) (u ) e R kp u k` L `p d e L nq d u mn R mq 2 ↵ e ↵ e + µ (Hd⇤) (Hd) | | e e e e = Q† y† u u† y Q µ⇤u† y H†Q µQ† H y† u L u R R u L R u d L L d u R 2 + µ Hd†Hd , (285f) e | | e e e e e e e

WMSSM WMSSM ⇤ ↵ ↵ ↵ = (eR⇤ )k(ye)k`(LL)` +(dR⇤ )kp(yd)k`(QL)`p µ(Hu) (Hd)↵ (Hd)↵ ✓ ◆✓ ◆ ⇣ ⌘ ↵ ↵ ↵ (e ) (y⇤) e (L⇤ ) +(e d ) (y⇤)e (Q⇤ ) µ⇤(H⇤) · e R m e mn L n R mq d mn L nq u ⇣ ↵ ↵ ⌘ =(L⇤ ) (y⇤) (e ) (e⇤ ) (y ) (L ) Len e mn R em R k ee k` L ` e ↵ ↵ +(Q⇤ ) (y⇤) (d ) (e⇤ ) (y ) (L ) e L nq d mn R mq R k ee k` L ` ↵ e e ↵ +(L⇤ ) (y⇤) (e ) (d⇤ ) (y ) (Q ) eL n e mn Re m R kp d k` eL `p ↵ e ↵ +(Q⇤ ) (y⇤) (d ) (d⇤ ) (y ) (Q ) e L nq d mn R mqe R kp d k`e L `p e ↵ ↵ ↵ ↵ µ⇤(e⇤ ) (y ) (L ) (H⇤) µ⇤(d⇤ ) (y ) (Q ) (H⇤) e R k e k` eL ` eu R ekp d k` L `p u ↵ ↵ ↵ ↵ µ(L⇤ ) (H ) (y⇤) (e ) µ(Q⇤ ) (H ) (y⇤) (d ) L n u ee mn R m eL nq u ed mn R mq 2e ↵ ↵ + µ (Hu⇤) (Hu) | |e e e e = LL† ye†eReR† yeLL + QL† yd†dReR† yeLL + LL† ye†eRdR† ydQL

+ Q† y†d d † y Q µ⇤e† y H†L µ⇤d † y H†Q e L d R Re d Le eR e ue L e R d e u Le e e e 2 e µLL† Huye†eR µQL† Huyd†dR + µ Hu†Hu . (285g) e e e e e e| | e e e e e e

75 Collecting the terms (285a) - (285g) gives:

i 2 ↵ W = e† y y†e H†H (H⇤) (L† ) y†y (L ) (H ) + u† y y† u H†H MSSM R e e R d d d L e e L d ↵ R u u R u u u† yuy†dRH†Hu d † ydy† uRH†Hd + d † ydy†dRH†Hd R d d R e u u e R d d e e ↵ e ↵e (H⇤) (Q† ) y† y (Q ) (H ) (H⇤) (Q† ) y†y (Q ) (H ) u eL u u eL u ↵ d e L de d L d ↵ e e 2 + Q† y† u u† y Q µ⇤u† y H†Q µQ† H y† u + µ H†H + L† y†e e† y L L u Re R u L e R u d L Le d u R e| | d d L e R R e L + Q† y†d e† y L + L† y†e d † y Q + Q† y†d d † y Q µ⇤e† y H†L eL d R R e eL L e R R de L eL d R R d L R e e u L e e e e e 2 e e µ⇤d † y H†Q µL† H y†e µQ† H y†d + µ H†H e R ed u Le eL u e eR e L ue d Re e| | eu u e e ↵ e e ↵ = (H⇤) (L† ) y†y (L ) (H ) + e† y y†e H†H (H⇤) (Q† ) y† y (Q ) (H ) ed L e e e eL d ↵ Re e e R ed d u L u u L u ↵ ↵ e (Hd⇤) (QL† )yd†yd(QL) (Hd)↵ + uR† yuyu† uRHu†Hu + dR† ydyd†dRHd†Hd e e e e e e u† yuy†dRH†Hu +h.c. + L† y†eRe† yeLL + Q† y† uRu† yuQL R ed d e L ee R e L ue R e ⇣ ⌘ + Q† y†d d † y Q + Q† y†d e† y L µ⇤e† y H†L µ⇤u† y H†Q eL d R Re d L L d Re R ee Le e R e e u Le e R eu d L ⇣ 2 2 µ⇤d † y H†Q +h.c. + µ H†H + µ H†H . (286) e R ed eu L e e | |e eu u e | | ed d e e e ⌘ e e The “double derivative” terms: • Now we consider the terms in the Lagrangian that originate from the superpotential term that contains a double derivative (term (282)):

2 WMSSM ↵ ↵ (e⇤ )k(LL) = (e⇤ )k(ye)k`L (Hd)↵ ↵ R ` R ` (eR⇤ )k(LL)` ! ↵ = eR† ye(LL) (Hd)↵ , (287a) e e

2 WMSSM ↵ (e⇤ ) (H ) = (e⇤ ) (y ) (L ) (H ) (e ) (H ) R k d ↵ R k e k` L ` d ↵ ✓ R⇤ k d ↵ ◆ ↵ e = eR† ye(LL) (Hed)↵ , e (287b) e e e 2 WMSSM ↵ ↵ (LL) (Hd)↵ = (e⇤ )k(ye)k`(LL) (Hd)↵ ↵ ` R ` (LL)` (Hd)↵ ! ↵ e = ee† ye(LL) (Hd)↵ , e (287c) e R e e 2 WMSSM ↵ ↵ (u⇤ )kp(QL) =(u⇤ )kp(yu)k`(QL) (Hu)↵ ↵ R `p R `p (uR⇤ )kp(QL)`p ! ↵ = uR† yu(QL) (Hu)↵ , (287d) e e

2 WMSSM ↵ (u⇤ ) (H ) =(u⇤ ) (y ) (Q ) (H ) (u ) (H ) R kp u ↵ R kp u k` L `p u ↵ ✓ R⇤ kp u ↵ ◆ ↵ e = uR† yu(QL) (Heu)↵ , e (287e) e e e 2 WMSSM ↵ ↵ (QL) (Hu)↵ =(u⇤ )kp(yu)k`(QL) (Hu)↵ ↵ `p R `p (QL)`p(Hu)↵ ! ↵ e = ue† y (Q ) (H ) , e (287f) e R u L u ↵

e e 76 2 WMSSM ↵ ↵ (d⇤ )kp(QL) = (d⇤ )kp(yd)k`(QL) (Hd)↵ ↵ R `p R `p (dR⇤ )kp(QL)`p ! ↵ = d† y (Q ) (H ) , (287g) e e R d L d ↵

2 WMSSM ↵ (d⇤ ) (H ) = (d⇤ ) (y ) (Q ) (H ) R kp d ↵ R kp d k` L `p d ↵ (dR⇤ )kp(Hd)↵ ! ↵ e = d† y (Q ) (He) , e (287h) e R d L d ↵ e e 2 WMSSM ↵ ↵ (QL) (Hd)↵ = (d⇤ )kp(yd)k`(QL) (Hd)↵ ↵ `p R `p (QL)`p(Hd)↵ ! ↵ e = de† y (Q ) (H ) , e (287i) e R d L d ↵ e e 2W MSSM (H )↵(H ) = µ(H )↵(H ) . (287j) (H )↵(H ) u d ↵ u d ↵ ✓ u d ↵ ◆ Collecting the terms (287a) - (287j) gives: e e e e

ij ↵ ↵ ↵ W =2 e† y (L ) (H ) e† y (L ) (H ) e† y (L ) (H ) MSSM i · j R e L d ↵ R e L d ↵ R e L d ↵ h ↵ ↵ ↵ + u† y (Q ) (H ) + u† y (Q ) (H ) + u† y (Q ) (H ) R u L u ↵ R u eL eu ↵ eR u L e u ↵ ↵ ↵ ↵ dR† yd(QL) (Hd)↵ dR† yd(QL) (Hd)↵ dR† yd(QL) (Hd)↵ e e e e + µ(H )↵(H ) , (288) u d ↵ e e e e i where the factor of 2 in front comese e from the fact that swapping i and j gives exactly the same contributions.

77 EIntegrals

In section 7 four di↵erent integrals are encountered that arise from four di↵erent types of one- loop corrections to scalar propagators in the MSSM. In this appendix all these di↵erent types of diagrams are listed and the corresponding integrals are performed “in d =4 " dimensions” using dimensional regularization (cf. also section 6.1), where " is inﬁnitesimal. For each integral a parameter µ is introduced to make the equation dimensionally correct. Furthermore, all vertex factors are denoted by .43 As we will only be interested in the divergent behavior of the one-loop corrections, we will only calculate the pole contributions to the corresponding integrals (i.e. the 1 coecients of " ). A contribution to the pole is indicated by the superscript “UV”.

The ﬁrst type: • The ﬁrst type of one-loop diagram is constituted by 4-scalar interactions and is given by: `

pp d 4 d d ` i = µ . (289) (2⇡)d `2 m2 + i✏ Z Now let us solve the following integral step by step: dd` µ4 d I (m) 1 ⌘ (2⇡)d `2 m2 + i✏ Z 4 d 0 µ d 1 1 d` ~ = d d ` . (290) (2⇡) `0 + ~`2 + m2 i✏ `0 ~`2 + m2 i✏ Z Z1 ⇣ p ⌘⇣ p ⌘ The second integral can be solved in the complex plane `0-plane by invoking the residue theorem. When we integrate counterclockwise along a semicircle in the upper half plane, we enclose the pole 0 2 2 ` = ~` + m i✏.Theresiduea 1 corresponding to this pole is given by p 1 a 1 = , (291) 2 ~`2 + m2 so that eq. (290) becomes: p

4 d d 1 iµ d ~` I1(m)= 2(2⇡)d 1 ~2 2 Z ` + m 4 d d 2 iµ p 1 ` = d⌦d 1 d` , (292) 2(2⇡)d 1 ~2 2 Z Z0 ` + m where on the second line we switched to spherical coordinatesp in d 1 dimensions and ` ~` .The angular integral is given by ⌘| | d 1 2⇡ 2 d⌦d 1 = , (293) d 1 Z 2 where (x) is the gamma function deﬁned by

1 x 1 t (x) dtt e . (294) ⌘ Z0 Eq. (292) now becomes:

d 3 4 d 2 iµ 1 2 (` ) 2 I1(m)= d` d 1 d 1 (4⇡) 2 0 ~2 2 2 Z 2 ` + m d 3 4 d 1 d 2 iµ m p z 2 = dz , (295) d 1 d 1 d (4⇡) 2 0 2(1 z) 2 2 Z 43Factors contained in that are relevant for the integration are automatically extracted from and explicity added to the integrand.

78 where on the second line we have used the substitution `2 z . (296) ⌘ `2 + m2 Let us consider the so-called beta function B(x, y)deﬁnedby

1 x 1 y 1 B(x, y) dtt (1 t) ⌘ Z0 (x)(y) = . (297) (x + y)

This function can be used to simplify eq. (295):

4 d d 2 1 d 1 d iµ m ( ) 1 (1 ) 1 I1(m)= dzz 2 (1 z) 2 d 1 d 1 2(4⇡) 2 0 2 Z 4 d d 2 d 1 d iµ m 1 = 2 2 d 1 d 1 1 2 2(4⇡) 2 2 4 d d iµ 1 2 m = d , (298) 2 2 µ (4⇡) m ✓ ◆ where it was used that (1/2) = p⇡. This expression blows up for d 4. To study this pole in more detail, we will expand d =4 " in powers of the inﬁnitesimal parameter! ".Tothisend,we need the following identity for the gamma function:

(x + 1) = x(x) , (299) as well as the digamma function (x) 0(x)/(x), where the prime denotes di↵erentiation with respect to x. Furthermore, we will we⌘ use that (1) = 1 and (1) = ,with 0.577 the Euler-Mascheroni constant. Now we expand the factors of eq. (298) that depend on ⇡d in powers of " as follows: d " 1 = 1+ 2 2 ✓ ◆ ⇣ 1 ⌘1 " = 1+ 1+"/2 "/2 2 ⇣ ⌘ 2 " +1 (1) + (1)(1) ⇡ " 2 ✓ ◆ h i 2 " = +1 1 , (300a) " 2 ✓ ◆ ⇣ ⌘

d 2+ " (4⇡) 2 =(4⇡) 2 2 " ln(4⇡) =(4⇡) e 2 1 " 1+ ln(4⇡) , (300b) ⇡ (4⇡)2 2 h i

d 4 " m m = µ µ ✓ ◆ ✓ ◆ 4 m " ln( m ) = e µ µ ✓ ◆ m 4 m 1 " ln . (300c) ⇡ µ µ ✓ ◆  ✓ ◆

79 This means that eq. (298) becomes (up to zeroth order in "):

iµ4 2 " 1 " m 4 m I (m) +1 1 1+ ln(4⇡) 1 " ln 1 ⇡ m2 " 2 (4⇡)2 2 µ µ ✓ ◆ ⇣ ⌘ h i ✓ ◆  ✓ ◆ im2 2 m2 = ln +1 . (301) 16⇡2 " 4⇡µ2  ✓ ◆ We are interested in the contributions to the pole only, thus we conclude:

2 UV im 1 I (m)= " . (302) 1 8⇡2 ✓ ◆

The second type: • The second type of one-loop diagram involves 3-scalar interactions and is given by: `

pp d 2 4 d d ` i i = µ . (303) (2⇡)d `2 m2 + i✏ (p + `)2 m2 + i✏ Z 1 2 p + ` Let us have a look at the following integral: dd` µ4 d I (m ,m ) . (304) 2 1 2 ⌘ (2⇡)d (`2 m2 + i✏)((p + `)2 m2 + i✏) Z 1 2 In the high energy limit it is fair to set m1 = m2 m and also to neglect the ﬁnite momentum p. This simpliﬁes eq. (304) to ⌘

d 4 d d ` µ @ I2(m1,m2) = I1(m) , (305) ! (2⇡)d (`2 m2 + i✏)2 @m2 Z which means that the contribution to the pole of I2 is given by: @ IUV(m ,m )= IUV(m) 2 1 2 @m2 1 i 1 = " . (306) 8⇡2 ✓ ◆

The third type: • The third type of one-loop diagram is formed by interactions between one scalar and two fermions, and is given by: `

pp d µ ⌫ 2 4 d d ` i`µ i(p + l)⌫ = µ Tr . (307) (2⇡)d `2 m2 + i✏ (p + `)2 m2 + i✏ Z  1 2 p + ` Note that the fermion propagators are written in Weyl spinor language here and that there is no additional minus sign to account for the closed fermion loop (cf. [22] for a derivation of the Feynman rules using the language of Weyl spinors). Let us now have a look at the following integral: dd` µ4 d` (p + l) Tr (µ⌫ ) I (m ,m ) µ ⌫ 3 1 2 ⌘ (2⇡)d (`2 m2 + i✏)((p + `)2 m2 + i✏) Z 1 2 d 4 d d ` ` (p + l) =2µ · , (308) (2⇡)d (`2 m2 + i✏)((p + `)2 m2 + i✏) Z 1 2

80 where on the second line it was used that Tr (µ⌫ )=2⌘µ⌫ . Eq. (308) can be simpliﬁed further as follows44:

d 2 2 2 2 2 2 2 4 d d ` ` m1 + i✏ + (p + `) m2 + i✏ p + m1 + m2 I (m ,m )=µ 3 1 2 (2⇡)d (`2 m2 + i✏)((p + `)2 m2 + i✏) Z 1 2 d 4 d d ` 1 1 = µ + (2⇡)d (p + `)2 m2 + i✏ `2 m2 + i✏ Z  2 1 p2 m2 m2 1 2 . (309) (`2 m2 + i✏)((p + `)2 m2 + i✏) 1 2

The ﬁrst two terms are of the I1 type (with masses m2 and m1), and the third term is of the I2 type (with masses m1 and m2):

I (m ,m )=I (m )+I (m ) p2 m2 m2 I (m ,m ) . (310) 3 1 2 1 1 1 2 1 2 2 1 2 The pole contribution to I3 is now given by: @ IUV(m ,m )=IUV(m )+IUV(m ) p2 m2 m2 IUV(m) 3 1 2 1 1 1 2 1 2 @m2 1 im2 im2 i p2 m 2 m2 1 2 1 2 1 = 2 + 2 2 " " 8⇡ 8⇡ 8⇡ # i 2m2 +2m2 p2 1 2 1 = 2 " . (311) " 8⇡ #

The fourth type: • The fourth type of diagram is constituted by gauge interactions and is given by: `

pp d µ⌫ 2 4 d d ` i(p `)µ i⌘ (p `)⌫ = µ . (312) (2⇡)d `2 m2 + i✏ (p + `)2 + i✏ Z p + ` Let us have a look at the following integral:

d 4 d µ⌫ d ` µ (p `) ⌘ (p `) I (m, 0) µ ⌫ 4 ⌘ (2⇡)d (`2 m2 + i✏)((p + `)2 + i✏) Z d 2 4 d d ` (p `) = µ (2⇡)d (`2 m2 + i✏)((p + `)2 + i✏) Z d 2 2 2 2 2 4 d d ` 2 ` m + i✏ (p + `) + i✏ +2 p + m = µ (2⇡)d (`2 m2 + i✏)((p + `)2 + i✏) Z d 4 d d ` 2 1 = µ (2⇡)d (p + `)2 + i✏ `2 m2 + i✏ Z  2 p2 + m2 + . (313) (`2 m2 + i✏)((p + `)2 + i✏) #

The ﬁrst two terms are of the I1 type (with masses 0 and m), and the third term is of the I2 type (with masses m and 0):

I (m, 0) = 2I (0) I (m)+2 p2 + m2 I (m, 0) . (314) 4 1 1 2 44The addition of “✏-terms” to the numerator does not have to be compensated for, as ✏ 0aftercontour integration in the `0-plane anyway. !

81 The pole contribution to I4 is now given by: @ IUV(m, 0) = 2IUV(0) IUV(m)+2 p2 + m2 IUV(m) 4 1 1 @m2 1 2 2 2 im 2i p + m 1 = 2 + 2 " " 8⇡ 8⇡ # 2 2 i m +2p 1 = 2 " . (315) " 8⇡ #

82 F One-loop mass counterterms for MSSM scalar ﬁelds

In this appendix we list all one-loop mass counterterms for the scalar fields of the MSSM, resulting from the calculation that is described in section 7. Before we list the counterterms, we give all one- loop pole contributions to the scalar propagators of the MSSM. The one-loop pole contributions (i) corresponding to a scalar field that come from “type i” diagrams are denoted by C (where i =1, 2, 3, 4), and the sum over all diagrams is denoted by C. As the MSSM Lagrangian is symmetric in many ways, the expressions that we obtain for the various scalar fields are very similar. For this reason we derive general expressions for groups of

fields. The following fields are grouped together: LL, QL (generically denoted by L), eR⇤ , uR⇤ , dR⇤ (generically denoted by R⇤ ), and Hu,Hd (generically denoted by Hi). Furthermore, we will make use of the following definition: e e e e e e e 2 2 2 2 2 2 2 S 2YHu mHu +2YHd mHd +Tr 6YQ m +2YL m +3Yu⇤ mu +3Yd m + Ye⇤ me ⌘ L Q L L R R⇤ d R 2 2 2 ⇣2 2 2 2 ⌘ = m m +Tr m m e2m e+ m e+ me . e e e e e e (316) Hu Hd Q L u d e ⇣ ⌘ This quantity arises from typee 1 tadpolee diagrams.e Ane examplee of a contribution to S is given by eq. (191). Now let us list the results from the calculation in section 7, ordered by groups of fields.

The sfermionic weak isospin doublets L , Q : • L L First we consider the sfermionic weak isospine doubletse of the MSSM: LL, QL, generically denoted by L (with = L, Q). The right-handed counterpart of the left-handed ith component e e of the doublet L is denoted by i⇤R (with 2 = e for = L, and 1 = u, 2 = d for = Q). e (i) (i) Naturally, some terms only show up in either LL or QL.Thedi↵erencesbetweentheCL and CQ e e basically arise from two things: the right-handed neutrino is absent in the MSSM, and only QL is e (ei) (i) charged under SU(3). This means that going from CQ to CL is straightforward: set gs to zero, as e well as all terms that have something to do with the ﬁrst component of the QL doublet (i.e. the up squark), and, lastly, replace the labels “d”by“e”. To be able to di↵erentiate between terms that contribute to only one of the ﬁelds (LL or QL), we introduce the numbere D (for “doublet”) with e e D = 1 for L and D = 2 for Q . • L L The one-loop pole contributions to the propagator of are given by: e e L

(1) i 1 2 2 1 2 2 C = D2 m + µ (y† y )ke` + m + µ (y† y )k` 4⇡2 2 Hu | | 1 1 2 Hd | | 2 2  1 2 1 2 1 2 † † + D2(y m y1 )k` + (y m y2 )k` + Y g0 Sk` 2 1 1 2 2 2 2 L 1 2 2 e3 2 2 2 2e e 1 + Y g0 + g + D2gs (m )k` ↵ (D1 + D2pq) " , (317a) 2 L 8 3 ✓ ◆ e e

(2) i 1 1 1 2 C = D2(a† a )k` + (a† a )k` + D2 µ (y† y )k` 4⇡2 2 1 1 2 2 2 2 | | 1 1  1 2 1 + µ (y† y )k` ↵ (D1 + D2pq) " , (317b) 2| | 2 2

(3) i 2 2 2 3 2 2 8 2 2 C = 2Y g0 M1 k` + g M2 k` + D2gs M3 k` 4⇡2 L | | 2 | | 3  2 e1 2 + µ p D2(y† y )k` +(y† y )k` | | 2 1 1 2 2 ✓ ◆ ⇣ ⌘ 2 2 3 2 4 2 2 1 Y g0 + g + D2gs p k` ↵ (D1 + D2pq) " , (317c) L 4 3 ✓ ◆ e 83 (4) i 1 2 2 3 2 2 2 2 C = Y g0 + g + D2gs (m )k` 4⇡2 2 L 8 3 ✓ ◆ 2 2 e 3 2 4 2 2 e 1 + Y g0 + g + D2gs p k` ↵ (D1 + D2pq) " . (317d) L 4 3 ✓ ◆ The sum of all these contributionse is given by:

i 2 2 2 2 2 32 2 2 C = 8Y g0 M1 k` 6g M2 k` D2gs M3 k` 16⇡2 L | | | | 3  2 2 2 2 +2 m e + p (y† y ) +2 m + p (y† y ) D2 Hu 1 1 k` Hd 2 2 k` 2 2 +2D2(y† m y1 )k` + 2(y† m y2 )k` +2D2(a† a1 )k` 1 1 2 2 1

e 2 e 1 + 2(a† a ) +2Y g0 S ( + ) " , (318) 2 2 k` L k` ↵ D1 D2 pq e from which it follows (through eq. (230)) that the one-loop mass counterterm for L is given by:

2 1 2 2 2 2 2 32 2 2 (m )k` = 8Y g0 M1 k` 6g M2 k` D2gs M3 k` e 16⇡2 L | | | | 3  2 2 2 e +2 m e(y† y ) +2m (y† y ) + (m y† y ) D2 Hu 1 1 k` Hd 2 2 k` D2 1 1 k` 2 2 2 +(m y† y ) + (y† y m ) +(y† y m ) 2 2 k` D2 1 1 k` 2 2 e k` 2 2 +2D2(y† m y1 )k` + 2(y† m y2 )k` +2D2(a† a1 )k` e 1 1 2 e2 e 1

e 2 1e + 2(a† a ) +2Y g0 S " . (319) 2 2 k` L k` e

The sfermionic weak isospin singlets e⇤ , u⇤ , d⇤ : • R R R

Let us now consider the sfermionic weak isospine e singletse eR⇤ , uR⇤ , dR⇤ , generically denoted by R⇤ (i) (i) (i) (with = e, u, d). The main di↵erences between the Ce , Cu and Cd arise from two facts: the e e fields eR⇤ , dR⇤ are “down-type” fields, while uR⇤ is an “up-type”e field,e and eR⇤ is not charged under the gauge group SU(3) (in contrast to uR⇤ , dR⇤ ). To be able to di↵erentiate between the various fields ine thee following expressions, we makee use of two di↵erent Kroneckere deltas. To this end, let us define the numbers T (for “type”) ande S e(for “strong”) as follows: T = 1 for “up-type” fields and T = 2 for “down-type” fields, • S = 1 for SU(3) triplets and S = 2 for SU(3) singlets. • The one-loop pole contributions to the propagator of R⇤ are given by:

(1) i 2 2 2 2 2 C = T 1m + T 2m + µ (yy† )k`e+ S2(ym y† )k` + S1(ym y† )k` 4⇡2 Hu Hd | | L Q h 1 2 2 2 1 2 2 2 e 1 e + Y g0 Sk` + S1g + Y g0 (m )k` (S2 + S1pq) " , (320a) ⇤ s 2 R 3 2 R⇤ ✓ ◆ e e e (2) i 2 1 C = (a a† ) + µ (y y† ) ( + ) " , (320b) 4⇡2 k` | | k` S2 S1 pq h i (3) i 2 2 2 8 2 2 2 2 C = 2Y g0 M + g M + 2 µ p (y y† ) 2 1 k` S1 s 3 k` k` 4⇡ R⇤ | | 3 | |  4 2e 2 2 2 1 g + Y g0 p ( + ) " , (320c) S1 s k` S2 S1 pq 3 R⇤ ✓ ◆ e (4) i 2 2 1 2 2 2 C = g + Y g0 (m ) 2 S1 s k` 4⇡ 3 2 R⇤ ✓ ◆ 4 2 2 2 e 2 e 1 + g + Y g0 p ( + ) " . (320d) S1 s k` S2 S1 pq 3 R⇤ ✓ ◆ e 84 Combining all these contributions, yields:

i 2 2 2 32 2 2 2 2 2 C = 8Y g0 M g M +4 m + m + p (y y† ) 2 1 k` S1 s 3 k` T 1 Hu T 2 Hd k` 16⇡ R⇤ | | 3  2 2 +4 (y me y† ) +4 (y m y† ) + 4(a a† ) S2 L k` S1 Q k` k`

2 e 1e +2Y g0 Sk` (S2 + S1pq) " , (321) R⇤ e which gives, through eq. (230), the following one-loop mass counterterm for R⇤ :

2 1 2 2 2 32 2 2 2 e 2 (m ) = 8Y g0 M g M +4 m + m (y y† ) k` 2 1 k` S1 s 3 k` T 1 Hu T 2 Hd k` 16⇡ R⇤ | | 3  2 2 2 2 e + 2(m y ye† ) + 2(y y† m ) +4 (y m y† ) +4 (y m y† ) k` k` S2 L k` S1 Q k`

e 2 e 1 e e + 4(aa† )k` +2Y g0 Sk` " . (322) R⇤ e The Higgs doublets H ,H : • u d

Now let us consider the Higgs doublets Hu,Hd, generically denoted by Hi (i = u, d). The main di↵erences between Hu and Hd arise from the fact that Hu primarily couples to up-type objects, whereas Hd primarily couples to down-type objects. The one-loop pole contributions to the propagator of Hi are given by:

(1) i 3 2 1 2 2 C = iu Tr m y† yu + id 3Tr m y†yd +Tr m y†ye Hi 4⇡2 2 Q u 2 Q d L e  ⇣ ⌘ ⇣ ⇣ ⌘ ⇣ ⌘⌘ 3 2 e 1 2 e 2 e 1 2 + Tr y† m y + 3Tr y†m y +Tr y†m y + Y g0 S 2 iu u u u 2 id d d d e e e 2 Hi ⇣ ⇣ ⌘ ⌘ 1 2 2 3e 2 2 2 e1 e + Y g0 + g m + µ " , (323a) 2 Hi 4 Hi | | ↵ ✓ ◆

(2) i 3 1 C = iu Tr a† au + id 3Tr a† ad +Tr a†ae Hi 4⇡2 2 u 2 d e  ⇣ ⇣ ⌘ ⌘ 1 2 1 + 3 Tr y† y + 3Tr y†y +Tr y†y µ " , (323b) 2 id u u iu d d e e | | ↵ ⇣ ⇣ ⇣ ⌘ ⌘⌘

(3) i 2 2 2 3 2 2 3 1 C = 2Y g0 M1 + g M2 iu Tr y† yu + id 3Tr y†yd Hi 4⇡2 Hi | | 2 | | 2 u 2 d  ✓ ⇣ ⇣ ⌘ 2 2 3 2 2 2 2 3 2 2 1 +Tr y†y + Y g0 + g p + 2Y g0 + g µ " , (323c) e e Hi 4 Hi 2 | | ↵ ◆ ✓ ◆ ⌘

(4) i 2 2 3 2 2 1 2 2 3 2 2 2 1 C = Y g0 + g p + Y g0 + g m + µ ↵" . (323d) Hi 4⇡2 Hi 4 2 Hi 4 Hi | | ✓ ◆ ✓ ◆ The sum of all these contributions is given by:

i 2 2 2 2 2 2 C = 8Y g0 M 6g M +2Tr 3 y† y + 3y†y + y†y p Hi 16⇡2 Hi | 1| | 2| iu u u id d d e e h 2 2 ⇣ 2⇣ 2 ⌘⌘ +2Tr 3 m y† y + y† m y + a† a + 3m y†y + m y†y iu Q u u u u u u u id Q d d L e e ⇣ 2 ⇣ 2 ⌘ ⇣ 2 +3y†m y + y†em y +3a† ae + a†a +2Y g0 S +2e 3 Tr ey† y d d d e e e d d e e Hi id u u 2 ⌘⌘2 2 2 ⇣ 1 + 3Tre y†y +Tre y†y 4Y g0 3g µ " , (324) iu d d e e Hi | | ↵ ⇣ ⇣ ⌘ ⌘ ⌘ i 85 from which it follows (using eq. (230)) that the one-loop mass counterterm for Hi is given by:

2 1 2 2 2 2 2 2 m = 8Y g0 M 6g M +2m Tr 3 y† y + 3y†y + y†y Hi 16⇡2 Hi | 1| | 2| Hi iu u u id d d e e h 2 2 ⇣ 2 ⇣ 2 ⌘⌘ +2Tr 3 m y† y + y† m y + a† a + 3m y†y + m y†y iu Q u u u u u u u id Q d d L e e ⇣ 2 ⇣ 2 ⌘ ⇣ 2 +3y†m y + y†em y +3a† ae + a†a +2Y g0 S +2e Tr 3y†ey d d d e e e d d e e Hi u u 2 2 2 2 ⌘⌘ 1 2 ⇣ ⇣ +3y†y e+ y†y e4Y g0 3g µ " µ d d e e Hi | | | | 1 2 2 ⌘ 2 2 2 ⌘ 2i = 8Y g0 M 6g M +2m Tr 3 y† y + 3y†y + y†y 16⇡2 Hi | 1| | 2| Hi iu u u id d d e e h 2 2 ⇣ 2 ⇣ 2 ⌘⌘ +2Tr 3 m y† y + y† m y + a† a + 3m y†y + m y†y iu Q u u u u u u u id Q d d L e e ⇣ 2 ⇣ 2 ⌘ ⇣ 2 1 +3y†m y + y†em y +3a† ae + a†a +2Y g0 S e" , e (325) d d d e e e d d e e Hi ⌘⌘ i where in the second stepe the counterterme for µ 2 exactly canceled the left-over “ µ 2-term” (the one-loop -function for µ can be found in [20]).| | | |

86 List of abbreviations

CKM Cabibbo-Kobayashi-Maskawa EFT e↵ective ﬁeld theory FM ﬂavor mixing LHC Large Hadron Collider MSSM Minimal Supersymmetric Standard Model mSUGRA minimal supergravity PMNS Pontecorvo-Maki-Nakagawa-Sakata pMSSM phenomenological Minimal Supersymmetric Standard Model RG renormalization group WIMP weakly interacting massive particle

87 Acknowledgements

These ﬁnal words mark the end of my study period. I have had a wonderful time in Nijmegen and I would like to thank several people for that. This thesis would not have been possible without the help of various others. I thank Frank Saueressig for useful discussions and Nicolo de Groot for wanting to be the second reader of my thesis. A special thank you goes to my supervisor Wim Beenakker for guiding me through this project. His thorough reading of my thesis and his insights have taught me a lot. I also would like to thank the people at the high energy physics department for providing a pleasant work environment. Furthermore, I would like to thank all my friends and family, especially my parents and sisters, for their unconditional support throughout my study. Lastly, I would like to thank Karen for always being there for me and for motivating me even further during the writing of this thesis.

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