Textures, Model Building, and Orbifold Gauge Anomalies: Research in Three Topics in Physics Beyond the Standard Model

Home , Building (mathematics), Gauge anomaly

TEXTURES, MODEL BUILDING, AND ORBIFOLD GAUGE ANOMALIES: RESEARCH IN THREE TOPICS IN PHYSICS BEYOND THE STANDARD MODEL

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Leslie J. Schradin III, B.S., M.S.

*****

The Ohio State University

2006

Dissertation Committee: Approved by

Professor Stuart Raby, Adviser Professor Richard Kass Adviser Professor Junko Shigemitsu Graduate Program in Professor Terrence Walker Physics ABSTRACT

We introduce the Standard Model, list a large sector of the low energy data, and present extensions to the Standard Model including grand uniﬁcation, supersymmetry, and orbifold extra dimensions. These foundations underly the research presented in this dissertation, which is from three separate projects.

Texture models are Ans¨atze for the undiagonalized Yukawa matrices in which some of the matrix elements have been chosen to vanish. Recent precise measurements of sin 2β from the B-factories (BABAR and BELLE) and a better known strange quark mass from lattice QCD make precision tests of predictive texture models possible.

We show that in a set of these models, their maximal sin 2β values rule them out at the 3σ level. While at present sin 2β and V /V are equally good for testing N-zero | ub cb| texture models, in the near future the former will surpass the latter in constraining power.

We construct a supersymmetric SO(10) D grand uniﬁed model with an orbifold × 3 1 extra dimension S /(Z Z′ ). The model uses 11 parameters to ﬁt the 13 independent 2 × 2 low energy observables of the charged fermion Yukawa matrices and predicts the values of two quark mass combinations, mu/mc and mdmsmb, to each be approximately

1σ above their experimental values. The remaining observables are successfully fit at the 5% level. This model is shown to have a gauge anomaly on one of the fixed points and we discuss the alterations in field content necessary to repair it.

ii Extra dimensional orbifold theories have gauge anomaly structures which are more complicated than those of Minkowski space. We review previous work done by von

Gersdorﬀ and Quiros to derive general expressions for orbifold gauge anomalies. These equations are applied to a supersymmetric 6D orbifold model with E6 gauge symmetry presented by Kobayashi, Raby, and Zhang in order to verify the gauge anomaly cancellations. From this illustration we conclude that the constraining power of orbifold gauge anomalies on the ﬁeld content of the theory is about as great as the usual case in Minkowski space and depends highly on the gauge groups and number of dimensions present.

iii For my parents.

iv ACKNOWLEDGMENTS

There are many people who contributed in many ways to my graduate education. In the area of research, I’d like to thank my advisor Stuart Raby and fellow collaborator Hyung Do Kim for giving me the opportunity to work with them. My research would not have been possible without their guidance. I am also indebted to the rest of Stuart’s high energy group over the last several years, particularly Radovan

Dermisek, Motomichi Harada, and Akin Wingerter for discussions over my research.

In the area of my dissertation, I owe thanks to those past graduate students who created the osudissert96 Latex template I used to create this document. I also thank Patrick Randerson and Ashish Saxena for their help with questions on the construction of my dissertation, Stuart Raby for input on content, Ed Smith for his services as a courier, and my dissertation committee for their time and eﬀort.

My understanding of physics and mathematics was formed in the classrooms of

Ohio State, Yale, Turpin, and Sherwood. Many ﬁne teachers contributed to my education and I would not be where I am today without their instruction.

I am grateful for the many friends and acquaintances I have made over the years of graduate school. In particular I am glad for my friendship with Patrick and Susan

Blanderson.

My parents and family are my foundation. Their constant love, encouragement, and belief in me through all of my years have created the person I am today.

v Most importantly, I am thankful to Meredith Howard. Her friendship, support, understanding, and love helped me to survive the black times and to celebrate the occasional successes that made up graduate school. She provided useful advice on most aspects of my written dissertation and defense presentation, including discussions over content and emphasis, Latex formatting, and the use of PowerPoint. I would not have made it without her.

vi VITA

October 2, 1975 ...... Born - Cincinnati, Ohio, USA

1998 ...... B.S., Physics, Yale University 2003 ...... M.S., Physics, The Ohio State University 1999-2002 ...... Graduate Fellow, The Ohio State University 2002-present ...... Graduate Teaching or Research Asso- ciate, The Ohio State University

PUBLICATIONS

Research Publications

H.D. Kim, S. Raby, and L. Schradin, “Quark Mass Textures and sin 2β”, Physical Review D 69, 092002 (2004).

H.D. Kim, S. Raby, and L. Schradin, “Quark and Lepton Masses in 5D SO(10)”, The Journal of High Energy Physics 0505, 036 (2005).

FIELDS OF STUDY

Major Field: Physics

Studies in Physics beyond the Standard Model: Professor Stuart Raby

vii TABLE OF CONTENTS

Page

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita ...... vii

ListofTables...... xii

ListofFigures ...... xiv

Chapters:

1. Introduction...... 1

1.1 Quantum Field Theory and Gauge Symmetry ...... 1 1.2 TheStandardModel...... 1 1.3 ExtensionstotheStandardModel ...... 3 1.4 OriginalWork ...... 5

2. StandardModel...... 7

2.1 TheStandardModelDeﬁned ...... 7 2.2 Deﬁciencies of the Standard Model ...... 14 2.2.1 NeutrinoMasses ...... 15 2.2.2 HierarchyProblem ...... 17 2.2.3 DarkMatter ...... 18 2.2.4 TheUnexplained ...... 18 2.2.5 OtherProblems...... 19

viii 3. Data...... 20

3.1 ChargedFermionMasses...... 20 3.2 CKMelements ...... 22 3.3 GaugeSector ...... 24

4. ExtensionstotheStandardModel ...... 25

4.1 GrandUniﬁedTheories(GUTS) ...... 25 4.1.1 Predictions ...... 26 4.1.2 Problems ...... 29 4.1.3 Groups ...... 30 4.2 Supersymmetry(SUSY) ...... 35 4.2.1 Motivation ...... 35 4.2.2 SomeFinerPoints ...... 36 4.2.3 Minimal Supersymmetric Standard Model ...... 39 4.3 FamilySymmetry ...... 41 4.4 ExtraDimensionsandOrbifolds ...... 42 4.5 Conclusion ...... 44

5. Texture Models and Sin 2 β ...... 45

5.1 Introduction ...... 45 5.2 Setup ...... 48 5.3 Analysis...... 51 5.3.1 2 2LightQuarkMatrices ...... 51 × 5.3.2 3 3QuarkMatrices ...... 51 × 5.4 Discussion ...... 57 5.4.1 ComparisonoftheTwoCases...... 57 5.4.2 FutureProspects ...... 58

6. 5D SUSY SO(10) D GUTModel...... 59 × 3 6.1 Introduction ...... 59 6.2 Setup ...... 66 6.2.1 Background ...... 66 6.2.2 YukawaMatrix ...... 73 6.3 Analysis ...... 82 6.3.1 AnalyticFitting ...... 82 6.3.2 NumericalFitting ...... 93 6.4 Justiﬁcationin6D ...... 97 6.5 Pati-SalamBraneGaugeAnomaly ...... 101

ix 6.6 SummaryandDiscussion ...... 102

7. OrbifoldAnomalies...... 106

7.1 IntroductionandMotivation...... 106 7.2 AnomalyBackground ...... 108 7.2.1 FujikawaMethod ...... 109 7.2.2 GaugeTheories ...... 113 7.2.3 ExtraDimensions ...... 117 7.3 OrbifoldAnomalies ...... 119 7.3.1 Orbifolds ...... 119 7.3.2 OrbifoldAnomalyEquations ...... 123 7.4 Applications ...... 131 7.4.1 Model A1 of Kobayashi, Raby, and Zhang ...... 131 7.4.2 5DModel ...... 151 7.5 Discussion ...... 158

8. Conclusion...... 163

Appendices:

A. LorentzSpinors...... 167

A.1 General Notation in d Dimensions ...... 167 A.2 d =4Dimensions...... 169 A.3 d =6Dimensions...... 172

B. Diagonalization of 2 2Matrices...... 177 × B.1 GeneralNotation...... 177 B.2 Approximate Diagonalization for Hierarchical Matrices ...... 178 B.3 Exact Diagonalization for 12 = 21 and11=0...... 179 | | | | C. LoopRephasingMethod ...... 184

D.D3 FamilySymmetry...... 193

E. Massless States and Wavefunctions ...... 196

x F. ALifted4DModel ...... 198

G. CalculationofSpatialTraces...... 209

H. CalculationofGroupTraces ...... 211

Bibliography ...... 217

xi LIST OF TABLES

Table Page

2.1 TheStandardModelﬁelds...... 9

2.2 The independent Standard Model parameters...... 15

4.1 The Minimal Supersymmetric Standard Model ﬁelds...... 40

5.1 The symmetric 5-zero textures classiﬁed in [58] ...... 52

6.1 Bulkmatterﬁelds...... 74

6.2 SO(10)Braneﬁelds...... 74

6.3 PSBraneﬁelds ...... 75

6.4 Observables used in the χ2 analysis...... 94

6.5 Observables, target values, best ﬁt values, and χ2 contributions. . . . 96

6.6 Minimum χ2 ﬁtYukawaparameters...... 96

7.1 Pattern of gauge breaking for the bulk ﬁelds...... 135

7.2 6D Chirality β and ak forthebulkﬁelds...... 138

7.3 Orbifoldparitiesofbulkﬁelds...... 139

7.4 Non-singlet SO(10) U(1) braneﬁeldsinKRZ...... 148 × Z 7.5 Non-singlet SU(6) SU(2) braneﬁeldsinKRZ...... 151 × R

xii 7.6 ak forthebulkﬁelds...... 154

7.7 Pati-Salam brane ﬁelds in the 5d model...... 157

F.1 Bulkﬁelds...... 199

F.2 PSBraneﬁelds...... 200

F.3 Best χ2 ﬁt...... 207

H.1 Fundamental Casimir orders for the simple Lie algebras...... 212

H.2 Some fundamental indices for selected groups...... 215

xiii LIST OF FIGURES

Figure Page

2.1 Unitaritytriangle...... 13

4.1 Gauge coupling uniﬁcation in the Standard Model...... 27

4.2 Gauge coupling uniﬁcation in the Minimal Supersymmetric Standard Model...... 37

5.1 Graphical representation of the maximum value of β = arg 1 reiφ . 54 − 6.1 Arepresentation ofthe6Dorbifoldspace...... 99

6.2 Arepresentation oftheSO(10)groupspace...... 100

7.1 Fundamental domain for T2 ...... 136

7.2 Fundamental domain for T2/ (Z Z )...... 137 2 × 2

D.1 Graphical representation of our choice of D3 group elements...... 194

xiv CHAPTER 1

INTRODUCTION

1.1 Quantum Field Theory and Gauge Symmetry

The twentieth century saw the discovery of two basic physical theories: quantum mechanics and special relativity. The former governs the behavior of the fundamental objects of our world, particles, at very short length scales. The latter provides the stage of space-time on which these particles play. Combination of quantum mechanics and special relativity leads to quantum ﬁeld theory, a theory which describes the particles we observe as excitations of quantum ﬁelds.

The requirement that a quantum field theory be invariant under local symmetry transformations, or gauge transformations, leads to interactions between particle excitations of the fields. The fact that such a theory, a quantum field theory satisfying a gauge symmetry, can describe all known particle behavior is nothing short of remarkable.

1.2 The Standard Model

The application of such a theory to nature results in the Standard Model of particle physics. Aside from the principles of quantum ﬁeld theory and gauge symmetry, the

1 other main inputs necessary to determine the Standard Model are the particular gauge

symmetry respected by nature, the space-time properties, or spin, of the ﬁelds, and

the gauge transformation charges of those ﬁelds. With these inputs, the Standard

Model is determined up to a small number of free parameters. Once the remaining

free parameters of the theory are determined by a set of experiments, almost all other

observed particle physics phenomena are predicted by this theory.

Despite its remarkable success, there are a few cracks in the armor of the Standard

Model. The most glaring is the presence of neutrino oscillations in nature which cannot be described in the theory as it is now formulated. However, this phenomenon can be explained by simply adding some new ‘sterile’ neutrino fields to the current model. The addition of these fields to the Standard Model is so necessary that in a few years the definition of the ‘Standard Model’ will likely change to incorporate them.

Other problems with the Standard Model are more subtle. The observed facts of gauge symmetry breaking and particle masses require the inclusion of an unobserved

ﬁeld in the Standard Model, the Higgs ﬁeld, with a mass near the masses of the other

fields. For the Standard Model to be a stable theory, quantum corrections due to high energy excitations of known fields and as yet unknown heavy fields should not disturb the field masses or interaction strengths. With the exception of the Higgs

field, there are symmetries which protect the masses of all of the other fields from such quantum correction effects. These quantum effects suggest that the Higgs mass should naturally be as heavy as the heaviest fields in nature, but as mentioned above, symmetry breaking requires the Higgs mass to be about as light as the other Standard

Model ﬁelds. This problem is called the hierarchy problem.

2 Observations of the rotation of galaxies indicate that there is signiﬁcantly more gravitational attraction present in galaxies than can be accounted for by visible matter alone. This implies that there is some form of ‘dark matter’, invisible to light, which makes up the diﬀerence. Standard cosmological theories suggest that none of the

Standard Model particles could make up this dark matter.

The formulation of the Standard Model gives us an accurate working description

of nature. It tells us what is happening. However, it does not tell us why. Why do we

measure the particular gauge symmetries, ﬁelds, masses, and interaction strengths?

What made nature decide on those and not others?

All of the deﬁciencies described above are addressed by a few extensions to the

Standard Model relevant to the work presented in this thesis. Other problems with

our understanding of particle physics exist, and there are proposed solutions to these

problems. Though interesting, these other problems and solutions are not relevant to

our work here.

1.3 Extensions to the Standard Model

The extensions to the Standard Model which will be important for our work here

are grand uniﬁcation, supersymmetry, family symmetry, and orbifold extra dimen-

sions.

The essence of grand uniﬁcation involves the enlargement of the gauge symmetry

of the Standard Model. This implies a uniﬁcation of the forces of nature and of some of

the ﬁelds we observe. Grand uniﬁed theories generically imply relationships between

the gauge interaction strengths, imply relationships between ﬁeld masses, predict

new interactions, and predict the existence of new ﬁelds outside of the Standard

3 Model. In particular, most grand uniﬁcation theories require sterile neutrino ﬁelds.

Although grand uniﬁcation does not exactly tell us why we have the ﬁelds and gauge

symmetry that we do, it does try to explain where those ﬁelds and symmetries may

have originated.

Supersymmetry is a global symmetry which relates bosons, fields of integral spin, and fermions, fields of half-integral spin. The existence of this symmetry near the mass scale of the Standard Model fermions leads to a solution to the hierarchy problem. A fermion field in the theory, whose mass is protected from large quantum corrections by chiral symmetry, is related to the Higgs boson by supersymmetry. The relationship extends to the masses of the fields, and so the Higgs mass obtains protection from quantum corrections via supersymmetry and is stabilized near the energy scale where supersymmetry is broken. If the supersymmetry-breaking scale is near the energy scale of the observed field masses, the Higgs field can fulfill its intended purpose in the Standard Model. Supersymmetry generically predicts many new fields, and it is possible that among these fields there could be one which has the correct properties to be a dark matter candidate.

Another global symmetry often used as an extension to the Standard Model is called family symmetry. In the Standard Model, some of the fields are grouped into sets called families, of which there are three. Family symmetry relates the three families of fields to one another in non-trivial ways, leading to predictions for the masses and interactions of the fields.

The last extension important to us here is extra dimensional orbifolds. Extra dimensions, if they exist, cannot be as large as the three spatial dimensions evident to us all. Rather, these dimensions must be curled up small enough so that they

4 are undetectable both to the eye and to other experiments. An orbifold is an extra

dimensional space which has been folded onto itself in some way. The folding produces

special points inside the space where ﬁelds can be localized. The interplay between

the localized ﬁelds and the ﬁelds traveling freely around in the extra dimnension can

lead to interesting new theoretical predictions.

There are many other ideas which have been proposed to extend our current theory of particle physics, but those mentioned above are the only ones used in the research presented in this dissertation.

1.4 Original Work

The original research presented in this thesis falls into three areas. The ﬁrst is an analysis of texture models, the second is a new model combining the physics ideas mentioned above, and the third involves the calculation of anomalies in the presence of orbifold extra dimensions.

Texture models address one of the deﬁciencies of the Standard Model by explaining to some extent why the ﬁelds obtain the masses that they do. A texture is a particular set of constraints which leads to relationships between the masses of some of the

Standard Model ﬁelds. We have performed an analysis on some of these texture models and have shown that a class of them can be ruled out by the present data.

Theories which lead to these particular textures are then also ruled out.

All of the extensions to the Standard Model described above can be combined together simultaneously. We have created a model out of these ingredients, speciﬁcally a grand uniﬁed, supersymmetric, orbifold theory with a particular family symmetry.

5 This theory can ﬁt a large sector of the Standard Model at about the 5% level, and makes two predictions between the masses of some of the ﬁelds.

An anomalous symmetry is one which is conserved by the theory classically but is broken by quantum eﬀects. Such a symmetry is said to have an ‘anomaly’. When a theory has a gauge symmetry which is anomalous, the theory acquires inconsistencies and must be altered of abandoned. It is therefore important to require that all gauge anomalies cancel in gauge theories. Such requirements generically put constraints on the ﬁeld content of the theory. The inclusion of extra dimensions in general, and orbifolds in particular, leads to more complicated gauge anomaly structure. With the proliferation of orbifold models in the literature in recent years, it is important to understand the new aspects of these orbifold gauge anomalies and to know how the

field content of new orbifold theories will be constrained by requirements of anomaly cancellation. We have reproduced work present in the literature to derive general formulae for these gauge orbifold anomalies, and have then applied these formulae to a specific model. The application illustrates that anomaly cancellation in the presence of orbifolds constrains the fields to an extent, but not a great deal more than in theories in regular space-time. The knowledge of the structure of such anomalies should assist in building orbifold field theory models in the future.

6 CHAPTER 2

STANDARD MODEL

The Standard Model (SM) refers to the extremely successful model explaining the

behavior of the elementary particles and their interactions through the electromag-

netic, weak, and strong forces. What follows in this chapter is an introduction to this

theory in Section 2.1 and a discussion of its deﬁciencies in Section 2.2. In Appendix

A we have collected the conventions used in deﬁning Lorentz rotations, speciﬁcally

as they apply to spinor ﬁelds.

The notation and logic used in this chapter are based primarily on the ﬁeld theory

text of Peskin and Schroeder [1]. Other useful ﬁeld theory and particle physics texts

include Ryder [2], Cheng and Li [3], and Halzen and Martin [4].

2.1 The Standard Model Deﬁned

The Standard Model is a quantum ﬁeld theory satisfying a gauge symmetry with

the gauge group SU(3) SU(2) U(1) , a particular set of fermion and scalar C × L × Y ﬁelds, and their renormalizable interactions. This theory is based on the Glashow-

Weinberg-Salam SU(2) U(1) theory of weak interactions [5, 6, 7] and the SU(3) L × Y C theory of quantum chromo dynamics (QCD) [8, 9, 10, 11, 12]. The theory is set in 4- dimensional Minkowski space-time, and our conventions relating thereto are collected

7 in Appendix A. The ﬁelds present in the Standard Model are the vector gauge bosons,

the three families of fermions, and the scalar Higgs. The vector bosons are the gluon

ﬁelds Gµ of SU(3)C, the W -boson ﬁelds Wµ of SU(2)L, and the hypercharge boson

ﬁeld Bµ of U(1)Y. Each fermion family, of which there are three, has a quark doublet

field Q, anti-quark fields uc and dc, lepton doublet field L, and anti-lepton field ec.

The Higgs field we will label as Hu, and is the only Standard Model field that has yet to be directly observed. Its addition to the theory is necessary for providing symmetry breaking and fermion masses. The fields are listed with their gauge group charges,

Baryon and Lepton numbers, and spins in Table 2.1. Baryon and lepton symmetries are global U(1) symmetries satisﬁed by Standard Model interactions.

The Standard Model Lagrangian by deﬁnition contains all possible renormalizable terms which can be formed from the ﬁelds that are consistent with gauge and Lorentz symmetries. The kinetic energy terms are

1 1 1 Tr (GµνG ) Tr (W µν W ) BµνB + LKE ≡ −2 µν − 2 µν − 4 µν µ c µ c c µ c +Qi†iσ DµQi + ui †iσ Dµui + di †iσ Dµdi

µ c µ c +Li†iσ DµLi + ei †iσ Dµei

µ +(D Hu)† (DµHu) (2.1)

with

A A a a D ∂ + ig T G + igT W + ig′Y B (2.2) µ ≡ µ 3 µ µ µ G ∂ G ∂ G + ig [G ,G ] (2.3) µν ≡ µ ν − ν µ 3 µ ν W ∂ W ∂ W + ig [W , W ] (2.4) µν ≡ µ ν − ν µ µ ν B ∂ B ∂ B . (2.5) µν ≡ µ ν − ν µ

8 Gauge Baryon Lepton Field Charge Number B Number L Spin

Gµ (8, 1, 0) 0 0 1

Wµ (1, 3, 0) 0 0 1

Bµ (1, 1, 0) 0 0 1

ui 1 1 1 Qi = 3, 2, 6 3 0 2 di ! uc 3¯, 1, 2 1 0 1 i − 3 − 3 2 dc 3¯, 1, 1 1 0 1 i 3 − 3 2 νi 1 1 Li = 1, 2, 2 0 1 2 ei ! − ec (1, 1, 1) 0 1 1 i − 2 + φ 1 Hu = 0 1, 2, 0 0 0 φ ! 2

Table 2.1: The Standard Model fields. The index i runs from 1 to 3, indicating the three different families of fermions. Each fermion field transforms as a left- handed Weyl spinor under Lorentz symmetry. The gauge charges are listed for the SU(3) SU(2) U(1) gauge groups. Hypercharge Y is defined such that the C × L × Y electromagnetic charge Q is Q = T3L + Y . B and L are the charges under the global Baryon and Lepton U(1) symmetries.

9 The gauge bosons are Gµ, Wµ, and Bµ and the gauge couplings are g3, g, and g′ of the

1 groups SU(3)C, SU(2)L, and U(1)Y respectively. The Lagrangian contains Yukawa

terms which will lead to the fermion mass terms

U c D c E c Y Q H u Y Q H∗d Y L H∗e + h.c. (2.6) LYukawa ≡ − ij i u j − ij i u j − ij i u j

The Yukawa matrices Y U,D,E are general complex 3 3 matrices in family space. The ij × Higgs potential is

2 2 µ H†H λ H†H . (2.7) LHiggs ≡ u u − u u There are two parameters here: µ and λ. Finally there is a relatively obscure La-

grangian term:

θ GαβGµνε (2.8) LθQCD ≡ QCD αβµν

Of the three Standard Model gauge groups only SU(3)C has a topology such that a θ

term cannot be removed from the theory by reparameterization.

The Lagrangian terms above, all of those renormalizable terms allowed by gauge

and Lorentz invariance, happen to also satisfy global U(1) baryon and lepton sym-

metries. The charges of the ﬁelds under these symmetries is given in Table 2.1. This

implies that the Standard Model interactions lead to an absolutely stable proton, the

lightest baryon in the theory.

Minimization of the Higgs potential leads to a vacuum expectation value (VEV)

1 0 for the Higgs ﬁeld. In the unitary gauge, Hu = √ , where h(x) is a real 2 v + h(x) !

1 ′ 5 ′ The gauge couplings g and g are sometimes replaced by the couplings g2 g and g1 g . ≡ ≡ 3 This amounts to changing the normalization of the hypercharges and is done in order to compareq the gauge groups in grand uniﬁed theory models. These models will be discussed later in Chapter 4.

10 µ2 scalar ﬁeld and v = λ is the VEV. In this gauge, the Higgs potential becomes q µ4 λ = µ2h2 λvh3 h4. (2.9) LHiggs 4λ − − − 4

There is one remaining real scalar ﬁeld h, commonly called the Higgs ﬁeld, and it has

a mass mh = √2µ.

When the Higgs gets a VEV, the SU(2)L and U(1)Y groups are spontaneously broken, leaving over an unbroken subgroup U(1)EM, electromagnetism. Three gauge bosons of SU(2) U(1) acquire mass, and they are parameterized as L × Y

1 1 2 v W ± W iW MW = g (2.10) µ ≡ √2 µ ∓ µ 2 0 1 3 2 2 v Z gW g′B M = g + g . (2.11) µ ≡ √g2 + g 2 µ − µ Z ′ 2 ′ q

The remaining combination, the photon, remains massless and U(1)EM remains unbroken:

1 3 A g′W + gB (2.12) µ ≡ √g2 + g 2 µ µ ′ After applying the Higgs VEV, the Standard Model Lagrangian is usually rewritten

in terms of these gauge bosons and their interactions.

The Higgs VEV also leads to mass terms for some of the fermions in the theory:

v U c v D c v E c Yukawa Y uiu Y did Y eie + h.c. (2.13) L ⊃ −√2 ij j − √2 ij j − √2 ij j

Note that the neutrino ﬁelds νi do not gain masses. It is customary to make a change

of basis in the fermions in order to make the fermion mass matrices real, positive,

and diagonal. This can be accomplished by making making unitary transformations

11 as follows

c c U v U u u′ (V ) u (U c ) u ′ m V Y U c i ≡ k U ki j ≡ U jl l Diag ≡ √2 U U c c D v D di dk′ (VD) dj (UDc ) dl ′ mDiag VDY UDc ≡ ki ≡ jl ≡ √2 (2.14) c c E v E e e′ (V ) e (U c ) e ′ m V Y U c i ≡ k E ki j ≡ E jl l Diag ≡ √2 E E ν ν′ (V ) . i ≡ k E ki These unitary transformations cancel out in the kinetic energy terms and many of the gauge interaction terms. The only place in the Lagrangian where these unitary transformations do not cancel is in the quark charged current interaction terms.

gW +J µ + + h.c. (2.15) LKE ⊃ µ W g + µ µ = W ν†σ ei + u†σ di + h.c. (2.16) −√2 µ j i g + µ T µ T = W ν′ †σ (V ∗) V e′ + u′ †σ (V ∗) V d′ + h.c. (2.17) −√2 µ k E ki E ij j k U ki D ij j g + µ µ T = W ν′ †σ e′ + u′ †σ V ∗V d′ + h.c. (2.18) −√2 µ j j k U D kj j

T The Cabbibo-Kobayashi-Maskawa matrix, V V ∗V is a unitary 3 by 3 matrix CKM ≡ U D which remains after diagonalization of the quark mass matrices.2 It is often written as

Vud Vus Vub VCKM  Vcd Vcs Vcb  . (2.19) ≡ V V V  td ts tb    In general, such a matrix has 3 real angles (the 3 real angles in a 3 by 3 orthogonal matrix), and 6 phases. Because we can still rephase the 6 quark fields ui and di without altering the rest of the Lagrangian, we can remove 5 phases (only phase differences between the quark fields can be used to eliminate phases from VCKM ) from the CKM matrix. This leaves us with 4 parameters in the CKM matrix: 3 real

2There is no corresponding matrix for the leptons in the Standard Model. The absence of a neutrino mass matrix implies that we are free to choose to rotate the neutrino ﬁelds νi in the same way as the electron ﬁelds ei. The rotation matrices then cancel in the charged currents and we retain the diagonal couplings.

12 α VudVub∗ VtdVtb∗

γ β

VcdVcb∗

Figure 2.1: Unitarity triangle with the interior angles α, β, and γ.

angles and 1 phase. The fact that there could be a non-zero phase angle in the CKM matrix implies that it could contain complex elements. This fact is important in that such complexity leads to CP violation, a phenomenon which has speciﬁc experimental signatures.

One of the unitarity conditions on the CKM matrix is

VudVub∗ + VcdVcb∗ + VtdVtb∗ =0. (2.20)

Plotting each of the terms in the equation as vectors in the complex plane leads to the ‘unitarity triangle’ in Figure 2.1. The interior angles of this triangle are deﬁned to be

V V α arg td tb∗ (2.21) ≡ " V V ∗ # − ud ub V V β arg cd cb∗ (2.22) ≡ " VtdV ∗ # − tb 13 V V γ arg ud ub∗ . (2.23) ≡ " V V ∗ # − cd cb Of these, the angle β is the most accessible to experimental study in the form of sin 2β. The Jarlskog parameter J is deﬁned to be

Im V V V ∗ V ∗ J ε ε (2.24) aj bk ak bj ≡ abc jkl h i Xc,l and is equal to twice the area of the unitarity triangle. The phase angle parameter of the CKM matrix determines the imaginary parts of the matrix, which in turn determines the height of the unitarity triangle. Clearly, either J or β can be used to

parameterize the phase angle of the matrix.

There are many conventions used to parameterize the three angles and phase in the

CKM matrix. A common choice we will utilize here is to use the three matrix elements

V , V , and V to determine the three real angles and the Jarlskog parameter J | us| | cb| | ub| to ﬁx the phase. The remaining elements are determined by the unitarity conditions and the freedom involved in rephasing the fermion ﬁelds.

We have now eliminated all of the unphysical parameters from the theory and are left with the 19 independent free parameters of the Standard Model. These are listed in Table 2.2. Measurements have been made to ﬁt 18 of these 19 parameters, the lone exception being the mass of the Higgs scalar mh. Subsequent measurements on the properties of elementary particles are ﬁt extremely well by the Standard Model.

2.2 Deﬁciencies of the Standard Model

Though successful, the Standard Model does have some deﬁciencies. Some are

obvious, such as the presence in nature of neutrino masses, while some less so, such

as the Higgs hierarchy problem. In this section, we will present details of those

14 Sector Parameters Number

Gauge Couplings g3,g,g′ 3

mu, mc, mt,

Charged Fermion Masses md, ms, mb, 9

me, mµ, mτ Quark Mixing Angles V , V , V ,J 4 | us| | cb| | ub| Higgs Sector v, mh 2

QCD Theta θQCD 1

Table 2.2: The independent Standard Model parameters.

Standard Model problems which will be important for the work presented in the bulk

of this thesis.

2.2.1 Neutrino Masses

Oscillations between neutrino species have been observed in recent years. For a

review of this phenomena and related references, see the Particle Data Group (PDG)

article by Kayser [13]. Neutrino oscillation implies that the neutrinos have unequal

masses, further implying that at least some of the neutrino species have mass. As

we have seen above, the Standard Model lacks the ability to provide neutrinos with

mass.

There is a fairly simple extension to the Standard Model which can accomodate the presence of neutrino masses and mixings: the addition of a new left-handed Weyl

c fermion ﬁeld νi , uncharged (1, 1, 0) under the SM gauge group, with baryon number

B = 0 and lepton number L = 1. Such a field is self-conjugate and is called a − Majorana spinor field. Aside from the obvious kinetic term, the presence of this field

15 allows us to write the new Lagrangian terms

N c c c c = Y L H ν + h.c. M ν ν (2.25) Lν − ij i u j − ij i j v N c c c Y νiν + h.c. Mijν ν . (2.26) ⊃ −√2 ij j − i j

In the second line we have kept the mass terms after the Higgs gets its vacuum

N expectation value. Yij and Mij are complex and symmetric matrices respectively in family space. Since M has dimensions of mass, we are introducing some energy scale into our theory. The naive way to think about this is to assume that M is some high

c scale, or cutoff scale, of our theory. Thus the νi fields can be expected to have a very large mass compared to the weak scale set by the Higgs VEV v. If this is true, we should not observe these fields, but should see some evidence of their presence at low energy.

Looking at just one generation of neutrinos for simplicity, combine the mass terms:

0 v y c √2 ∗ ν νc (ν∗, ν ) v c (2.27) L ⊃ − √2 y M ! ν ! v2 y 2 − | | ν c 2M 0 ′ (ν′ ∗, ν ′) c (2.28) ≃ − 0 M ! ν ′ !

We have made a unitary rotation on the ﬁelds (ν, νc) in order to diagonalize the

c matrix. The new ﬁeld ν ′ has a large mass and does not mix with the other ﬁelds so

v2 y 2 can be removed entirely from the theory, while the new ﬁeld ν′ has a tiny mass 2|M| . Given the small masses measured in nature for the neutrinos, this theory works quite

well. This eﬀect, where we have a large Majorana mass ﬁeld νc mixing with a small

Dirac mass ﬁeld ν such that after diagonalization we have a low energy ﬁeld with a

tiny mass, is called the See-Saw mechanism [14, 15, 16, 17].

16 2.2.2 Hierarchy Problem

The Higgs mechanism is the way in which the Standard Model breaks the group

SU(2) U(1) to U(1) and gives mass to the Z and W bosons. Fitting the model L × Y EM to the masses of these bosons, we need the VEV to be v = 246 GeV. Looking to the

Higgs potential, the mass of the low-energy Higgs scalar h is mh = √2µ = v√2λ

where λ is the Higgs self-coupling. In order for the theory to be perturbative, we

need λ (1). Therefore, the Higgs mass should be (100 GeV). ≤ O O In the Standard Model, there is no symmetry which protects the Higgs mass,

making it sensitive to the cutoﬀ scale Λ of the theory. 1-loop processes coupling

the Higgs scalar to the known light ﬁelds or as yet unobserved heavy ﬁelds lead to

quantum corrections to m2 which are δm2 Λ2. Therefore, the Standard Model h h ∼ needs a bare-mass m2 Λ2, and the observed m2 comes from a cancellation: h bare ∼ h (102 GeV)2 m2 m2 δm2 (Λ2) (Λ2). Assuming that the cutoﬀ scale ∼ h ≃ h bare − h ∼ O − O is large, for example at the Planck scale Λ 1019 GeV, then the observed m2 comes ∼ h about due to a large ﬁne-tuning.

To avoid this naturalness problem, the cutoﬀ scale should be near the observed

Higgs mass. Further, the new theory at that scale should provide some symmetry which will protect the Higgs scalar from getting large corrections above that scale.

There are several solutions in the literature, including the Little Higgs model, warped extra dimensions, and supersymmetry. We will be concerned only with the latter, as it is relevant to the work in this thesis.

Note that the solution mentioned gives only a partial solution to the hierarchy problem. The new symmetry stabilizes the Higgs mass but there is still a question of why the new symmetry breaks at 100 GeV and not at a higher scale (like the Planck

17 scale). To be truly a full solution to the hierarchy problem, our new theory must

explain the choice of the 100 GeV scale over any other scale. Much work has been

done in this area, but it is not relevant to this thesis and will not be investigated or

explained further.

2.2.3 Dark Matter

Light curve observations of galaxies indicate that there is signiﬁcantly more gravitational attraction present in galaxies than can be accounted for by visible matter alone. This implies that there is some form of ‘dark matter’, invisible to light, which makes up the diﬀerence. Other cosmological evidence, such as measurements on the cosmic microwave background, supports the existence of dark matter and further suggests that it is made up of non-baryonic matter. Solutions to this problem include the possible existence of primordial black holes or the addition of a new stable

(or suﬃciently long-lived) particle which has the correct couplings and cosmological abundance to account for the dark matter. For a summary on dark matter and for references, see the review by Drees found in the PDG [13]. The information given here is based primarily on this review.

2.2.4 The Unexplained

Once we take measurements to ﬁx the parameters of the Standard Model, the theory ﬁts the remaining measurements extremely well. However, questions remain concerning the ideas upon which the SM is built. On these topics the SM by necessity remains silent.

Why do the parameters in the Standard Model take the values that they do? •

18 Why the gauge group SU(3) SU(2) U(1) ? • C × L × Y

Why three families of fermions? •

Why the particular fermions in a given family? •

Why gauge theory, general relativity, or quantum mechanics in the ﬁrst place? •

While these issues do not make the Standard Model inconsistent with nature, a theory which can explain some or all of these would be a desirable advance in our knowledge. Extensions to the SM presented in Chapter 4 of this thesis address some of these points.

2.2.5 Other Problems

The above list by no means exhausts the problems with the Standard Model and particle physics. The Strong CP problem, the conﬂict between quantum ﬁeld theory and gravity, the dark energy problem, and the related vacuum energy problem in general are all areas where our understanding of particle physics fails to a greater or lesser extent. There are extensions to the SM which attempt to address these issues, but we will not be concerned with them as they do not impact the work done in this thesis.

19 CHAPTER 3

DATA

Many measurements have been made on the masses and interactions of the ele-

mentary particles. In this chapter we tabulate the data used in the research presented

in this thesis. This includes the quark and charged lepton masses, the quark mixing

matrix elements, the gauge coupling constants, and the gauge boson masses. Other

measurements, such as neutrino mass diﬀerences and limits on θQCD, are not included here as they do not impact the work done in this paper.

The data quoted here is that which was used in the original work presented in

Chapter 6, performed in 2004 [18]. The data has not changed signiﬁcantly enough since that time to warrant updating the analysis.

3.1 Charged Fermion Masses

Due to strong interactions, the masses of the light quarks are not known well.

A combination of masses which has particularly small theoretical uncertainties was deﬁned in [19]

ms Q md . (3.1) ≡ 2 1 mu − md r

20 The most precise estimates of mu/md and ms/md also come from [19] using chiral

perturbation theory. There is some disagreement in the literature on the sizes of the

errors of the light quark mass ratios [20]. For this reason, we take mu/md and ms/md from [19] with doubled errors. The reason for including a dependent measurement

(mu/md) is that it will be useful later to have all three of these quantities with the

smallest measurement errors possible.

Recent lattice QCD calculations with dynamical quarks have improved our knowl-

edge of ms, previously the least known of the light quarks. We use the unquenched

lattice QCD result with nf = 2 for ms [21] and double the error to account for the

discrepancy with the sum rule result. The central value in equation (3.5) is near the

low end of the range given by the PDG [20]. The preliminary result with nf =2+1

indicates that the strange quark might be even lighter.

For the charm quark mass we use a quenched lattice QCD result since an un-

quenched calculation is not yet available. Quenching errors are known to be about

25% for the strange quark mass and 1 to 2% for the bottom quark mass. Because of

the mass hierarchy, it is expected that the quenching error on mc will lie somewhere

between these two bounds. Thus we take the lattice QCD result with a (probably

conservative) 10 percent systematic (quenching) error as in [22] and double it.

We use the bottom quark mass from [23] and the top quark pole mass from the

CDF and D Collaboration [24]. ∅

Q = 22.7 0.8 (3.2) ± m u = 0.553 0.043 2 (3.3) md ± × m s = 18.9 0.8 2 (3.4) md ± ×

21 m (2GeV) = 89 11 2 MeV (3.5) s ± × m (m ) = 1.30 0.15 2GeV (3.6) c c ± × m (m ) = 4.22 0.09GeV (3.7) b b ± M (pole) = 178.0 4.3GeV (3.8) t ± m (m ) = 169 4GeV (3.9) t t ± All running mass parameters are deﬁned in the MS scheme. We note here that the

doubled errors we are using almost incorporate the bounds found in the PDG [20].

In the analysis contained within this thesis, we will calculate most of our observ-

ables at the energy scale MZ . Therefore we need to convert our data to values at MZ .

Deﬁne the running parameters: mi(MZ ) for i = c, b, t mi(mi) ηi  (3.10) ≡   mi(MZ ) for i = u, d, s. mi(2 GeV)  At two loops in QCD these values are

ηc =0.56, ηb =0.69, ηt =1.06 (3.11)

ηu = ηd = ηs = 0.65. (3.12)

Finally, the lepton masses at MZ from [25] are

m = 0.48684727 0.00000014MeV (3.13) e ± m = 102.75138 0.00033MeV (3.14) µ ± m = 1746.7 0.3 MeV. (3.15) τ ± 3.2 CKM elements

These CKM elements we take from [22]:

V = 0.2240 0.0036 (3.16) | us| ± 22 3 V = (41.5 0.8) 10− (3.17) | cb| ± × 4 V = (35.7 3.1) 10− (3.18) | ub| ± × V ub = 0.086 0.008 (3.19) Vcb ±

V and sin 2β we take from [26]: | td| V = 0.0082 0.0008 (3.20) | td| ± sin 2β = 0.739 0.048 (3.21) ±

We use [20] for J and εK.

5 J = (3.0 0.3) 10− (3.22) ± × 3 ε = (2.282 0.017) 10− (3.23) | K| ± ×

In order to relate εK, a measure of the indirect CP violation present in the K-

meson system, to CKM matrix elements, we will use the following formula, which is

independent of CKM parameterization:

2 2 Cε (2c) (2t) 2c2t εK = BK Im 2 Scσc + Im 2 Stσt + 2 Im 2 Sctσct (3.24) | | 2 " u ! u ! u # where

2 V V ∗ V V ∗ (3.25) i ≡ ud us is id The various functions S are Inami-Lim functions [27], and the σ are (1) factors i i O which have been calculated in [22]

σc =1.32 σt =0.57 σct =0.47. (3.26)

Cε is a ratio of well known low-energy observables. From Battaglia [22], this combination is

2 2 2 G f + m 0 M C = F K K W =3.837 104. (3.27) ε 2 6π √2∆mK × 23 We can neglect the error associated with Cε since it is small and will not aﬀect the theoretical error in εK. A large part of this theoretical error comes from BK for which we use [22].

B = 0.86 0.06 0.14 (3.28) K ± ± 0.86 0.15 (3.29) ≃ ±

In the last line we have added the statistical and systematic errors in quadrature.

3.3 Gauge Sector

From [25] we take the gauge couplings at MZ :

α (M ) = 0.016829 0.000017 (3.30) 1 Z ± α (M ) = 0.033493 0.000059 (3.31) 2 Z ± α (M ) = 0.118 0.003 (3.32) 3 Z ±

g2 The coupling constants α are deﬁned in terms of the g couplings: α i . g and i i i ≡ 4π 1 g2 are related to the couplings g′ and g usually deﬁned in the Standard Model by

5 g g′ and g g. 1 ≡ 3 2 ≡ q We use the mass of the Z-boson to set the weak energy scale: [20]

M = 91.1876 0.0021GeV (3.33) Z ±

In the analysis contained in Chapter 6, we do not calculate the eﬀects of electro-weak symmetry breaking. We assume that it happens properly and gives the correct weak scale masses. For this reason, we use a calculated Higgs VEV at MZ .

M v = Z (3.34) 3 √π 5 α1(MZ )+ α2(MZ ) = 246.q41 0.17GeV (3.35) ±

24 CHAPTER 4

EXTENSIONS TO THE STANDARD MODEL

In this chapter we detail those extensions to the Standard Model which are relevant to the work done in this thesis. These ideas are grand uniﬁcation, supersymmetry, family symmetry, and extra dimensional orbifolds.

4.1 Grand Uniﬁed Theories (GUTS)

A grand uniﬁed theory, or GUT, is a theory where at some high energy scale a new gauge symmetry exists which encompasses the gauge group of the Standard

Model. Some mechanism breaks this new gauge group at the uniﬁcation scale, giving masses to those gauge bosons and particles which are not present in the Standard

Model. More information and references on grand uniﬁed theories can be found in the corresponding PDG review by Raby [13].

A well known example of gauge unification exists inside the Standard Model itself: the gauge symmetry U(1) is enlarged to SU(2) U(1) at a sufficiently high EM L × Y energy scale (the weak scale, roughly 100 GeV). New gauge particles, W and Z, appear at that scale to join the low-energy gauge boson A, the photon. The vacuum expectation value of the Higgs field at the weak scale provides the mechanism by which the larger gauge group breaks to the smaller.

25 There are many gauge groups which are large enough to hold the Standard Model inside of them. Those necessary for the purposes of this thesis are: the Pati-Salam

(PS) gauge group SU(4) SU(2) SU(2) , named after the first authors to offer × L × R the group as a unified group [28], SU(5), first proposed by Georgi and Glashow [29],

SO(10), ﬁrst put forward as a uniﬁcation group by Georgi [30] and by Fritzsch and

Minkowski [31], and E6. References which the author found particularly helpful in understanding group theory include the texts by Georgi [32] and Cahn [33] and reviews by Slansky [34] and Raby [35].

The main motivation for introducing a uniﬁcation group is the fact that the Stan- dard Model gauge couplings, if run up to high energy scales, appear to approach a common value. The running gauge couplings are shown in Figure 4.1.

4.1.1 Predictions

There are several consequences of having grand uniﬁcation.

Gauge Coupling Uniﬁcation

If the GUT group is simple, at some high energy scale the gauge couplings must combine into a single coupling of the new gauge group. For this to happen, the couplings must equal each other at the GUT energy scale, deﬁned to be MG. As

mentioned above, the Standard Model gauge couplings come close to meeting as the

energy scale is increased, and this situation is shown in Figure 4.1. With the addition

of supersymmetry, described later, this uniﬁcation becomes much better.

26 60

1 50 α1

40 1 α2 30

20 1 α3 10

5 10 15 20 µ log GeV Figure 4.1: Gauge coupling unification in the Standard Model. The thicknesses of the lines represent the uncertainties in the coupling constants. Although the running coupling constants come close to one another near the energy scale µ 1014 GeV, they do not unify within experimental errors. This figure should be compared∼ to the more exact unification shown in Figure 4.2 when supersymmetry is included. The equations for the running gauge couplings α (µ) at energy scale µ are 1 = i αi(µ) 1 βi µ MZ ln(10) log log and are based on 1-loop effects. The βi αi(MZ ) − 2π GeV − GeV contain the 1-looph renormalization running i constants determined by the field content of the theory. In the SM, β = 41 , 19 , 7 . Note that log is the logarithm to base i 10 − 6 − 10 and ln is the logarithm to base e.

Field Uniﬁcation

In a GUT, the ﬁelds of the Standard Model and possibly new ﬁelds with mass near

MG come out of representations of the GUT group. This implies that the fields in the same GUT representations, and their couplings, are related to one another beyond what the Standard Model predicts. For example, in SU(5), the 15 fermionic fields of a single family fit inside two SU(5) representations: 5=(¯ L, dc) and 10=(Q, uc, ec).

27 As a consequence of this, in the simplest SU(5) GUT the masses of the down and

electron ﬁelds in a given family are equal at the GUT scale.

New Particles

The new gauge group, necessarily larger than the Standard Model group, must have more gauge vector bosons than the SM. In addition, depending on the group, new fermion and scalar ﬁelds with masses at the GUT scale may be present in the theory. Continuing the example of SU(5), the smallest representation of this group is

5 dimensional. The Standard Model Higgs doublet Hu, which contains two complex

degrees of freedom, must be in some representation of SU(5), and so we at least need

to add 3 more scalar ﬁelds to put the SM Higgs into a 5. In this case, the new Higgs

ﬁelds, often called a Higgs triplet T , transform as a 3 under SU(3)C. An example of a

new fermion in a GUT comes from SO(10). In that theory, all of the Standard Model

fermions of a single family are contained in a single 16 dimensional representation of

SO(10), with the addition of a neutrino νc ﬁeld. Many GUTS require a νc ﬁeld, and

are attractive in that we are required to add a ﬁeld which new observations suggest

must be present in nature.

New Processes

The parts of the GUT group that are outside of the Standard Model will provide

diﬀerent interactions between the Standard Model particles than have been observed.

If quarks and leptons are uniﬁed into the same representation by the GUT, or if new

color-triplet Higgs ﬁelds are required, then there will be interactions with new gauge

bosons or with Higgs triplets which will transform quarks into leptons and vice versa.

Such an interaction would violate baryon and lepton number and can lead to processes

28 by which protons can decay into leptons and mesons. The Standard Model does not

allow baryon-number- or lepton-number-violating processes at the perturbative level.

Hypercharge Normalization Explained

If the group is semi-simple (does not have U(1) factors), then the Standard Model hypercharge U(1)Y must come from a non-abelian group. This ﬁxes the normalization of the hypercharges of the Standard Model ﬁelds.

4.1.2 Problems

While they are nice in that they provide some explanations of ﬁeld content of the

Standard Model, GUTS do have their detractions.

GUT Symmetry Breaking

There must be a mechanism by which the GUT group breaks down to the SM group. Often this is done by a Higgs mechanism provided by new scalar Higgs fields to break the new symmetry. Not only are these new fields which have not been seen, but in practice the symmetry-breaking potential for these fields can be very complicated.

Many New Fields

Especially for the larger groups, in order to accomodate the Standard Model particles there must be an embarrassing number of new ﬁelds to ﬁll out representations of the new group.

Rapid Proton Decay

Uniﬁcation of quarks and leptons into the same multiplet of the GUT group implies that there are gauge boson-mediated and possibly Higgs triplet-mediated processes

29 which transform quarks into leptons and vice-versa. Depending on the processes

involved, it may be possible for protons to decay through these new interactions, and

such processes are suppressed by powers of the masses of the new GUT particles. The

measured lower bounds on the proton lifetime place lower bounds on the masses of

the new gauge bosons and the Higgs triplets. Some simple GUTS have already been

ruled out by the conﬂict imposed on the new particle masses: the measured proton

lifetime requires the new particle masses be so high that gauge coupling uniﬁcation

no longer occurs.

4.1.3 Groups

There are many groups which have been considered as grand uniﬁcation groups.

What follows are those which will be relevant to this work.

SU(4) SU(2)L SU(2)R or Pati-Salam × ×

With the addition of a neutrino ﬁeld νc, each family of fermions ﬁts into two

representations of the Pati-Salam (PS) group:

ψ (4, 2, 1)=(Q, L) (4.1) ≡ c c c u ν ψ 4, 1, 2 = c , c (4.2) ≡ d ! e !!

Left-right is the SU(4) direction, up-down is the SU(2)R direction, and the SU(2)L

direction is supressed. We can think of the leptons as being the fourth ‘color,’ after

packing them in with the quarks and expanding SU(3)C to SU(4). In the Higgs sector,

we add another 2 ﬁelds so that we have

H H (1, 2, 2) = u (4.3) ≡ Hd !

30 where H has the Standard Model quantum numbers (1, 2, 1 ). With these ﬁelds, d − 2 the simplest Yukawa potential leads to a single term for each family:

λψHψc λH (Quc + Lνc)+ λH (Qdc + Lec) (4.4) → u d

With the same Yukawa coupling λ and diﬀerent VEVs for Hu and Hd, we would expect that the down and electron fermions of each family to have the same masses at the

GUT scale. This can work for the 3rd family (known as b-τ unification), but not for the first two. In the first two families, the Yukawa sector must be more complicated than above in order to account for the masses. The up and neutrino masses for each family are also predicted to be the same here, which is clearly contrary to the data.

While the above naive Yukawa sector does not ﬁt all of the data, more complicated sectors can do better while still retaining some interesting mass relationships between the diﬀerent particle types.

Like the SM, Pati-Salam is a group made up of a product of three separate groups.

As such, there are still three unrelated couplings in PS, and so we do not have any simpliﬁcations or predictions in this sector. However, the assumption of an additional left-right symmetry, implying that the Lagrangian is invariant under interchange of

SU(2)L and SU(2)R, does give a relation between two of the three couplings.

SU(5)

SU(5) is the smallest simple group which can contain the Standard Model. The

SM fermions of a single family ﬁt into two representations:

5¯ = (dc, L) (4.5)

10 = (Q, uc, ec) (4.6)

31 The Standard Model Higgs Hu has two gauge degrees of freedom. Since the smallest

representation of SU(5) is 5 dimensional, at least 3 more ﬁelds must be added to the

Higgs in order that it be contained in an SU(5) representation. In this case, the new

Higgs ﬁelds, often called a Higgs triplet T , must have charge 3, 1, 1 under the − 3 Standard Model.

5Higgs = (T,Hu) (4.7)

Because supersymmetry requires an additional Higgs doublet H = 1, 2, 1 , SU(5) d − 2 is often formulated with this ﬁeld as well. It sits inside a 5¯ with its own triplet ﬁeld

1 T with charge 3, 1, 3 .

5¯Higgs = T,Hd (4.8) With these ﬁelds, the simplest Yukawa sector for a given family will be

10105 ucQH , ucecT, QQT (4.9) Higgs → u 10 5¯ 5¯ ucdcT, QdcH , QLT , ecLH (4.10) Higgs → d d

In the minimal Yukawa coupling sector, then, the masses of the down quarks and the electrons should coincide at the GUT scale, since the couplings for the two are the same. The ﬁrst two families do not exhibit this behavior, and so their Yukawa sectors must be more complicated. For the third family, however, bottom-tau mass uniﬁcation can be made to work, under certain conditions.

SO(10)

SO(10) is the group next up the ‘chain’ from SU(5). It contains SU(5) and the

Pati-Salam groups. One family of Standard Model fermions, along with an additional

32 νc ﬁeld, are contained in a 16 dimensional representation of SO(10).

16 = (Q, uc,dc, L, νc, ec) (4.11)

The smallest representation of SO(10) is a 10, which contains 5 and 5¯ ﬁelds of SU(5)

10 = Hu,Hd, T, T (4.12) The simple Yukawa coupling becomes

161016 QH uc, QH dc, LH νc, LH ec, andothers (4.13) ⊃ u d u d

The simplest Yukawa sector for a given generation leads to Yukawa couplings which are equal across the entire family. This does not work for the ﬁrst two families, but can work for the third, and is known as top-bottom-tau mass uniﬁcation.3 With equal

Yukawa couplings, the large ratio of the top to bottom or tau masses at the GUT scale must come from a large ratio between the two Higgs vevs vu and vd. This ratio has been given a name: tan β vu . ≡ vd In Chapter 6 we will construct a theoretical model with SO(10) as the GUT gauge

group. The speciﬁc breaking patterns SO(10) PS and SO(10) SU(5) U(1) → → × X will be relevant to that case. Under SO(10) PS we have →

16 (4, 2, 1)+(4, 1, 2) (4.14) → uc νc = (Q, L)+ c , c (4.15) d ! e !! 10 (6, 1, 1)+(1, 2, 2) (4.16) → H = T, T + u (4.17) Hd ! 3The Dirac τ-neutrino mass is also equal to the other masses in this theory. However, other ﬁelds which are chargeless under the Standard Model or even SO(10) could have majorana masses and mix with the νc in the low-energy eﬀective theory. This could produce a see-saw mechanism leading to the tiny τ-neutrino mass, as observed.

33 and under the breakdown SO(10) SU(5) U(1) , → × X

16 101 + 5¯ 3 +15 (4.18) → − c c c c = (Q, u ,d )1 +(d , L) 3 +(ν )5 (4.19) −

10 5 2 + 5¯2 (4.20) → −

= (T,Hu) 2 + T,Hd (4.21) − 2 E6

Even larger than SO(10) is E6. The smallest representation in this group is 27 dimensional, and under a breakdown of E to SO(10) U(1), the 27 splits into the 6 × SO(10) representations

27 16+10+1 (4.22) →

along with some U(1) charges. Not only can a single 27 hold all of the Standard

Model fermions, it also contains a 10 of SO(10), a set of ﬁelds with the same charges

as the Higgs sector of the Standard Model. These are fermions, however, and so

are necessarily extra ﬁelds that are outside of the Standard Model. On the other

hand, another extension to the Standard Model, supersymmetry, requires fermionic

counterparts to the Higgs scalars- the very ﬁelds contained in this fermionic 10 of

SO(10) inside of the 27 of E6.

In Chapter 7, we will consider a GUT model with E6 as the unifying gauge group.

E8 E8 ×

E8 contains E6, and is the largest of the exceptional groups. The product group

E E is a group which is naturally chosen by some formulations of string theory. 8 × 8 The reason it is mentioned here is that we will consider a theoretical model in Chapter

34 7 which is derived from E E heterotic string theory. Beyond this fact, the details 8 × 8 of the group are not important to us here.

4.2 Supersymmetry (SUSY)

Supersymmetry, or SUSY for short, is an extension to the usual Poincare spacetime

symmetry. The generators of supersymmetry are anti-commuting, and so their action

switches boson and fermion states. Thus, in a supersymmetric theory, the number

of bosonic and fermionic degrees of freedom are identical. In a theory with exact

supersymmetry, the bosonic and fermionic partners would have the same masses.

This is not observed in nature, and so if supersymmetry is a symmetry of nature, it

must be a broken one.

Information from Haber’s PDG review on supersymmetry [13] was used through-

out this section, and contains more references to the literature on the subject.

4.2.1 Motivation

There are three main motivations for considering supersymmetry as an extension

to the Standard Model.

Gauge Hierarchy Problem

The existence of supersymmetry near or just above the weak scale implies that the

1-loop contributions to the scalar Higgs mass are not quadratically but logarithmically

dependent on the UV scale (the energy scale of the theory which caps the SUSY

theory). This means that the Higgs mass is not sensitive to the UV scale and allows

the bare Higgs mass, the physical Higgs mass, and the quantum correction between

35 them to all be at the same scale. This alleviates the ﬁne-tuning problem present in

the Standard Model.

Gauge Coupling Uniﬁcation

The addition of new ﬁelds at the weak scale ( 1 TeV) necessary to make the ∼ Standard Model supersymmetric causes a change in the running of the gauge couplings such that the three couplings unify quite precisely at a high energy scale (order

1016 GeV). Figure 4.2 shows the running of the three gauge couplings in the presence of the Minimal Supersymmetric Standard Model (deﬁned below).

Dark Matter

If the lightest supersymmetric particle (LSP) is stable, which is possible if the

theory has what is called R-parity, then these particles could have the correct masses

( TeV) and couplings (SU(2)L) that they could provide the thermal relics necessary

to account for the dark matter in the universe.

4.2.2 Some Finer Points

There are a few speciﬁc points concerning supersymmetry which should be covered

before they are used in the next chapters.

Supermultiplets, Kahler potential K, and superpotential W

A ﬁeld and its superpartner are related by the supersymmetry transformation.

These ﬁelds together form what is called a supermultiplet, and there is a notation

available, used in Chapter 6, which combines these into a single ﬁeld called a super-

ﬁeld.

36 60

1 50 α1

40 1 30 α2

20 1 α3 10

5 10 15 20 µ log GeV Figure 4.2: Gauge coupling unification in the Minimal Supersymmetric Standard Model. The thicknesses of the lines represent the uncertainties in the coupling constants, and the coupling constants overlap within experimental error near the energy scale µ 1016 GeV. The equations for the running gauge couplings α (µ) at energy ∼ i scale µ are 1 = 1 βi ln(10) log µ log MZ and are based on 1-loop αi(µ) αi(MZ ) − 2π GeV − GeV effects. The βi contain the 1-loop renormalizationh running i constants determined by the field content of the theory. In the MSSM, β = 33 , 1, 3 . For this calculation, i 5 − the MSSM fields all contribute to the running above the Z mass. Note that log is the logarithm to base 10 and ln is the logarithm to base e. Compare to the unification in Figure 4.1 when supersymmetry is not included.

In superfield notation, the Kahler potential K for a superfield is a single La- grangian term which simultaneously takes into account the kinetic energy terms, gauge interaction terms, and some supersymmetric interaction terms of the component fields within the superfield. The superpotential W refers to a set of Lagrangian terms which involve one, two, or three superfields, and takes into account supersymmetric couplings between the various component fields of these superfields. Masses and Yukawa couplings among component fields come from the superpotential. Work in the next chapter uses some of this notation.

37 4D, 5D, and 6D Supersymmetry

Since supersymmetry is a symmetry relating bosons and fermions, the physical degrees of freedom of each in a single supermultiplet must be the same. In 4D, the smallest spinor available is a 2-component Weyl spinor ψ2. On shell, this ﬁeld has

2 physical degrees of freedom. Thus, the smallest supermultiplet with a fermion must contain 2 bosonic degrees of freedom. This can be accomplished by combining the spinor with a complex scalar φ to form a ‘chiral multiplet’. The corresponding superﬁeld can be written as χ = (ψ2,φ). We can also combine this 2-spinor with a vector gauge ﬁeld Aµ (which also only has 2 physical degrees of freedom) to form a

‘vector multiplet’ V =(Aµ, ψ2).

In 5D or 6D, the smallest spinor is a 4-component Weyl spinor ψ4 which has 4 physical degrees of freedom. The smallest 5D or 6D chiral multiplet is then constructed from this ﬁeld and 4 bosonic degrees of freedom in the form of two complex scalars φ1 and φ2. Such 5D or 6D chiral multiplets are commonly called ‘hypermultiplets’: H = (ψ4,φ1,φ2). In 5D, the vector boson AM has 3 degrees of freedom, and so a 5D vector multiplet requires an additional real scalar Σ to make up the 4 bosonic degrees of freedom. The 5D vector supermultiplet would be =(A , ψ , Σ). V M 4

Lastly, in 6D the vector boson AM has 4 degrees of freedom and can alone match the 4 fermionic degrees of freedom from a 6D Weyl spinor in the 6D vector multiplet

=(A , ψ ). V M 4 It is possible to have more than one supersymmetry in a theory. The number of supersymmetries is usually labeled by N, and the least amount of SUSY possible is

N = 1. The above ﬁelds correspond to the ﬁeld content for N = 1 SUSY in the various dimensions. It is the nature of supersymmetry that N = 1 SUSY in 5D or 6D

38 is equivalent to N = 2 supersymmetry in 4D. Each ψ4 contains two ψ2 ﬁelds within it. The 5D or 6D multiplets break down to 4D multiplets in the following way: each

5D or 6D hypermultiplet splits into two 4D chiral multiplets,

H =(ψ ,φ ,φ ) χ(1) = ψ(1),φ χ(2) = ψ(2),φ (4.23) 4 1 2 → 2 1 ⊕ 2 2 and each 5D or 6D vector multiplet splits into a 4D vector and a 4D chiral multiplet

=(A , ψ , Σ) V = A , ψ(1) χ = ψ(2), (A + iΣ) /√2 (4.24) V5D M 4 → µ 2 ⊕ 2 5 =(A , ψ ) V = A , ψ(1) χ = ψ(2), (A + iA ) /√2 (4.25) V6D M 4 → µ 2 ⊕ 2 5 6 This type of splitting will be important for the ﬁeld content considered in the next two chapters of this work.

4.2.3 Minimal Supersymmetric Standard Model

The Minimal Supersymmetric Standard Model (MSSM) is a commonly used extension to the Standard Model. As the name implies, the ﬁelds added to the Standard

Model are the minimum necessary to convert the SM into a supersymmetric theory.

The ﬁelds we must add ﬁrst include another scalar Higgs doublet Hd with charges

1, 2, 1 under the Standard Model gauge group. This new ﬁeld is necessary to − 2 provide Yukawa couplings and masses to the particles since the ﬁeld Hu† used in the

Standard Model for this purpose leads to couplings which do not satisfy supersym-

metry. After adding this scalar Hd, for every ﬁeld in the theory we must add a superpartner. These new superpartner ﬁelds have the same gauge charges as their

SM partners but have diﬀerent spin. For each SM scalar and vector ﬁeld we must add

1 a spin- 2 fermion, and for each SM fermion ﬁeld we must add a scalar. The original ﬁelds and their superpartners then form supermultiplets and transform into one another under the supersymmetry transformation. There are lower limits on the masses

39 Supermultiplets Boson Fields Fermionic Partners SM gauge charges gluon/gluino G G (8, 1, 0)

0 0 gauge boson/ W ±, W W ±e, W (1, 3, 0)

gaugino B f Bf (1, 1, 0) 1 squark/ Qi = ui, di Qi =(eui,di) (3, 2, 6 ) quark uc uc (3, 1, 2 ) e ie e i − 3 c c 1 dei di (3, 1, 3 ) 1 slepton/ Li =(eνi, ei) Li =(νi, ei) (1, 2, ) − 2 lepton ec ec (1, 1, 1) e ie e i 1 Higgs/ Heu Hu (1, 2, 2 ) 1 higgsino Hd Hfd (1, 2, ) − 2 f

Table 4.1: The Minimal Supersymmetric Standard Model fields. The index i runs from 1 to 3, indicating the three different families. All fermions are left-handed Weyl spinors. The gauge charges are listed for the SU(3) SU(2) U(1) gauge groups. C × L × Y Hypercharge Y is defined such that Q = T3L + Y .

of these new ﬁelds due to their non-observation at particle colliders. In order that

supersymmetry be a solution to the hierarchy problem, the masses of the superpart-

ners should be generically of order the weak energy scale 1 TeV. The ﬁelds of the ∼ MSSM are listed in Table 4.1, inspired by a similar table which appears in Haber’s

PDG supersymmetry review [13].

The MSSM Lagrangian by deﬁnition includes all renormalizable interactions which

preserve the gauge symmetry, preserve space-time and supersymmetry, and conserve

B L (Baryon number minus Lepton number). In addition to these, the most general − set of soft-supersymmetry-breaking terms are included. The exact deﬁnition of ‘soft’- supersymmetry-breaking will not be important here- it is enough to know that these

40 are renormalizable terms which break supersymmetry in such a way that the theory

can still solve the gauge hierarchy problem.

In Chapter 6 we will be using a generic MSSM theory. It will be assumed that the parameters of the MSSM are such that the theory is consistent with the low-energy data. There are portions of MSSM parameter space for which this is true. The exact nature of these parameter spaces will not be relevant to this work, and so will not be discussed in further depth.

This background on supersymmetry is by no means enough to reproduce the work that follows, but it should be enough to provide some understanding. For more information on supersymmetry, the reader is referred to reviews or primers in the literature and to textbooks [36, 37, 38, 39, 40].

4.3 Family Symmetry

A family symmetry is one which relates the three families of fermions of the

Standard Model in some way. The symmetry can be continuous or discrete, local or global. By transforming the families into one another, relationships between the families can be forged which lead to relationships between the fermion masses. Such mass relationships can decrease the number of independent parameters in the theory and can lead to new predictions. The three families of fermions do not obviously exhibit any family symmetry at the energy scales we can observe. Thus, like most symmetries which extend the Standard Model, this symmetry must be broken at some high energy scale.

The third family satisfies some simple GUT relationships, such as possible mass unification at high energies, which are not satisfied by either of the first two families.

41 For this reason, family symmetries in GUTS often relate the ﬁrst two families to each

other and leave the third family unchanged. This is the case in the next chapter,

where a discrete family symmetry D3 will be used, under which the ﬁrst two families transform as a doublet and the third transforms as a singlet. More information and references on family symmetries in grand uniﬁed theories can be found in the reviews

[41, 42, 43].

4.4 Extra Dimensions and Orbifolds

It is possible that there are more dimensions in nature than the 3+1 Minkowski space we observe. If these extra dimensions are compact, and the characteristic radii of the dimensions are small enough, we would not have knowledge of them. If gravity is the only force to propagate in these new dimensions, they would only have to be smaller than about 0.1mm for them to be invisible to us [13]. On the other hand, if the Standard Model fields move in these extra directions the extra dimensions must be much smaller. The 1 TeV energies of collider experiments would have already ∼ 1 19 detected any extra dimensions bigger than 1 TeV− 2 10− m. ∼ ≈ × From a 4D point of view, the most important experimental signature of extra dimensions is the presence of Kaluza-Klein towers of particles. The momentum modes of fields moving around a compact dimension are quantized. When the dimension is compactified small enough that it is difficult to detect, the first evidence of its existence is the discovery of a set new of fields which appear to be heavy copies of a relatively light or massless field. The masses of this new set of fields, called a

Kaluza-Klein tower of ﬁelds, are spaced in units of 1/R where R is the radius of

compactiﬁcation. These Kaluza-Klein masses are nothing more than the momentum

42 modes of the same 4D ﬁeld moving around the compactiﬁed dimension. From a 5D

standpoint, this whole tower including the massless 4D ﬁeld together form a single

5D ﬁeld.

The extra dimensions need not be smooth. They may have non-trivial boundary conditions due to orbifold compactiﬁcations.4 An orbifold is the space which results from modding out a smooth compact space by some discrete symmetry. This discrete symmetry acts non-trivially on the space in such a way as to single out special ‘ﬁxed- points’ of the space. Fixed-points, also commonly called branes, are those points of the compact space which are unchanged under the action of the discrete symmetry.

The expanse of the orbifold space away from the ﬁxed-points is often called the ‘bulk’.

If there is a gauge theory living on the initial compact space, the orbifold action can lead to a breaking of that gauge symmetry. This means that grand uniﬁed theories can be broken to subgroups without a Higgs mechanism and the often complicated Higgs sector which accompanies it. This idea was recently put forward by Kawamura [46,

47]. GUT representations living in the bulk are split by the orbifold naturally, as the different fields inside the representations have different orbifold boundary conditions.

Some will attain compactification-scale masses while others will remain massless. An important application of this effect involves the Standard Model Higgs. If this field lives in the bulk along with Higgs triplet fields inside a GUT group representation, it is then possible to split the unwanted Higgs triplet fields from the doublet fields

4Orbifold boundary conditions are a subset of the general set of conditions called Scherk-Schwarz boundary conditions. These are used in some models, including a model by Kobayashi, Raby, and Zhang [44, 45] considered in Chapter 7. However, it is possible with some models, including the one in KRZ, to reformulate the theory such that it is an orbifold theory. Therefore, we need not concern ourselves with the general Scherk-Schwarz boundary conditions here.

43 using the orbifold, thus solving the doublet-triplet splitting problem which plagues many GUT models.

With the extra-dimensional space available, different fields which live in this space can have their wavefunctions localized to different areas, thus producing different coupling strengths of interactions due to wavefunction overlap. This is true of fixed- points as well, on which fields can be completely localized. This freedom of field localization will be used in the next chapter in the construction of a 5D orbifold GUT model.

The PDG review on extra dimensions by Giudice and Wells [13] contains more information and references on orbifolds and extra dimensions in general.

4.5 Conclusion

Full understanding of the Standard Model extensions introduced here will be assumed in Chapters 6 and 7. However, it is hoped that most of the physics can be understood by non-experts after the brief introduction to these extensions contained in this chapter. The next topic, contained in Chapter 5, involves the analysis of texture models, and should be accessible to those acquainted only with the Stan- dard Model. Chapter 6 explores a 5D SUSY SO(10) GUT orbifold model with D3 family symmetry. Chapter 7 investigates anomalies on orbifolds in general, and the anomalies in a speciﬁc 6D SUSY E6 GUT orbifold model derived from string theory.

44 CHAPTER 5

TEXTURE MODELS AND SIN 2 β

5.1 Introduction

As we have seen in the Chapter 2, while the Standard Model can accomodate the presence of the charged fermion masses and the quark mixing matrix, the theory does not make any predictions for the values of those parameters. Some theories beyond the Standard Model attempt to gain insight into this sector, which has been an area of research for more than twenty years [48, 49, 50]. This can be done by assuming some additional symmetries to forbid or constrain certain elements of the undiagonalized Yukawa matrices. A set of Yukawa matrices which have a particular set of vanishing elements is called a ‘texture’. Diagonalization of these matrices then leads to the fermion masses and mixing angles along with some relations among them due to the constrained form of the original undiagonalized matrices. Hyung Do Kim,

Stuart Raby and I have published a paper [51] which analyzes some of these Yukawa textures, and shows that a class of textures cannot ﬁt the low-energy data. Hence theories which lead to those textures can be ruled out. What follows in this chapter is the work done in that paper.

45 Testing theories of fermion masses requires precision data, the accuracy of which

has been severely limited by theoretical uncertainties inherent in QCD. In particu-

lar, light quark masses and some CKM elements have been diﬃcult to measure with

precision. Recent results from B-factories combined with advances in the theory of

heavy quarks and in lattice QCD have reduced the errors considerably for these ob-

servables [22, 26]. Current measurements of V and V have errors at the 2% | us| | cb| level [22], and sin 2β is now known to 6.5% from experiments on the asymmetry

in B decays [26]. Even V /V , whose errors largely come from non-perturbative | ub cb| QCD eﬀects, has now been determined to about 10% [22]. In the mass sector, the

most important improvement has been in ms, whose uncertainty has decreased from

50% to 12% over the last ten years. Moreover, lattice QCD results with light dy-

namical quarks indicate that the strange quark mass is much lighter than previously

thought [21].

In a pioneering work, Hall and Rasin [52] showed that the relation

V /V = m /m (5.1) | ub cb| u c q

is obtained for any hierarchical texture with vanishing (1, 1), (1, 3), and (3, 1) elements. Roberts et al. [53] then re-analyzed these textures, using more recent data, and concluded that such textures were disfavored, i.e. disagreeing with data at about

1σ (see also [54]), whereas the addition of small (1, 3) (3, 1) elements gave good ﬁts ≈ to the data. They also studied non-hierarchical asymmetric textures, with vanishing

(1, 3), (3, 1) elements but satisfying the “lopsided” relation (3, 2) (3, 3). Good ∼ ﬁts to the data were also obtained in this case. The strongest constraint, in their analysis, came from the observable V /V , which at the time had an uncertainty | ub cb|

46 (22%).5 On the other hand, the value of sin 2β was not well known and its central O value was much lower than it is now. However, with symmetric textures with non- zero (1, 3) = (3, 1) elements, they predicted sin 2β to be near its present experimental value.

With the signiﬁcant improvement in the data, it is once again a good time to analyze quark mass textures. In this chapter we study hierarchical textures with elements

(1, 2) = (2, 1). We ﬁnd that such textures, with vanishing (1, 1), (1, 3), (3, 1) elements, are now excluded by 3σ. In the present study, constraints from both V /V | ub cb| and sin 2β are equally strong. However, in the near future sin 2β may provide the most stringent constraint. For example, it is expected that the experimental precision of sin 2β will greatly improve (by a factor of 2), perhaps very soon when an expected

1 500 fb− of data will have been tabulated by both BaBar and Belle [55]. On the other hand, the experimental uncertainties associated with V /V (limited by uncertain- | ub cb| ties in V ) may require a Super-B factory (perhaps by 2010 [56]) to obtain a similar | ub| factor of 2 reduction [57].

This chapter goes as follows. In Section 5.2 we present the relevant approximations used when diagonalizing hierarchical fermion mass textures. In Section 5.3.1 we consider the oldest successful texture describing the lightest two quark families. We then obtain our main result in Section 5.3.2 where we study hierarchical 3 3 quark × mass textures with elements satisfying (1, 2) = (2, 1) and (1, 1) = (1, 3) = (3, 1) 0. ≡ In Section 5.3.2 we show that good ﬁts to the data can be obtained with non zero

(1, 3), (3, 1) elements. In particular we focus on two 5-zero texture models considered by Ramond, Roberts, and Ross [58]. Finally, in Section 5.4 we conclude this chapter

5Although Roberts et al. assumed an uncertainty half this size [53].

47 with some discussion over the results and a view to future prospects as more data is

taken.

Appendix B contains information on diagonalization of matrices and Appendix

C gives a method by which the non-removable phases of the quark matrices can be placed. Both are relevant to the work done in this chapter.

5.2 Setup

As presented in Section 2.1, the quark Yukawa matrices Y U and Y D can be diagonalized by bi-unitary transformations of the quark ﬁelds.

U U D D Y V Y U c Y V Y U c (5.2) Diag ≡ U U Diag ≡ D D

These unitary rotations of the u and d quarks remain in the Lagrangian in the weak current sector, and lead to the CKM mixing matrix

T V V ∗V . (5.3) CKM ≡ U D

The Yukawa matrices we wish to address here are those which are ‘hierarchical.’

By deﬁnition this means [52]:

Y Y Y (23,32) 1, 22 1, (11,12,21) 1, (5.4) Y33 ≪ Y33 ≪ Y ≪ e 22

Y13 Y31 Y12Y21 e (5.5) Y33 ≪ Y 22

with e

Y23Y32 iφ22 Y22 Y22 Y22 e . (5.6) ≡ − Y33 ≡| | e e e We use Y22 in the analysis since it can simultaneously address the following cases:

e Y23Y32 Y23Y32 0= Y22 < and 0 = Y22 > (5.7) | | Y33 6 | | ∼ Y33

48 It is possible to ﬁt the data with Yukawa matrices which are not hierarchical, but the

relatively small CKM angles will then come from cancellations between two larger

terms. We will not address these types of Yukawa matrices here. For an analysis of

such models, see [53].

When the Yukawa matrices are hierarchical, it is possible to split up the diagonalization process into a series of 2 by 2 subprocesses. The mixing matrices VU and

VD then become

U U 1 s12 0 1 0 s13 10 0 U − − U V s ∗ 1 0 0 1 0 0 1 s (5.8) U  12     − 23  ≃ 0 0 1 sU 0 1 0 sU 1    13∗   23∗   D D   D    c12 s12 0 1 0 s13 10 0 D −D − D VD  s12∗ c12∗ 0   0 1 0   0 1 s23  (5.9) ≃ 0 0 1 sD 0 1 0 sD −1    13∗   23∗        U,D U,D iηU,D D where s sin θ e ij and cD cos θD eiρ12 . The structure of the down sector ij ≡| ij | 12 ≡| 12| D requires that we keep track of c12. The CKM matrix in this approximation becomes

D D U U c12 s¯12∗ + c12∗s13∗s23 s12∗s23∗ + s13∗ D D D − U VCKM  s¯12 c12s13s23∗ c12∗ s23∗ + s12s13∗  . (5.10) ≃ −sD s − cD s cD s sD s 1  12 23 12 13 12∗ 23 12∗ 13   − − −  We have neglected terms here which are (s2 ) smaller than the leading terms, and O ij have neglected all subleading terms on the diagonals as it does not impact the analysis.

The mixing angles are

U,D U,D U,D U,D Y12 U,D Y13 U,D Y23 s12 U,D s13 U,D s23 U,D (5.11) ≃ Y22 ≃ Y33 ≃ Y33 s¯ sD cD sU s sD sU s sD sU . (5.12) 12 ≡ 12 −e 12 12 13 ≡ 13 − 13 23 ≡ 23 − 23

D D s12 and c12 are special in that the above approximations do not work as well for

D D Y12 md md them. Under such an approximation, s12 D m 1+ m , and we miss | |≃| Y22 | ≃ s O s q a correction at the level of md 5%. For this sector only (the 1st and 2nd O ms ∼ e 49 family down mixing), we will use exact 2 by 2 diagonalization. The other sectors do

not need this level of accuracy, since the errors from our approximations are 1 % or

mu 3 below. (e.g., 2 10− ). mc ∼ × The Yukawa matrices we will consider here will have Y = 0 and Y = Y . | 11| | 12| | 21| As shown in Appendix B, an exact 2 by 2 diagonalization of the down quarks in such

a case leads to

D md D ms s12 = c12 = . (5.13) | | sms + md | | sms + md

U U Y12 U U mu Using s12 as above, for Y12 = 0, we have s12 , and ≃ Y22 6 | | ≃ mc

e D D U md iφ ms mu s¯12 s12 c12s12 e (5.14) | | ≡ − ≃ sms + md − sms + md s mc

m m m = s d eiφ u (5.15) sms + md s ms − s mc

with

D U c12s12 φ arg D . (5.16) ≡ s12 !

For Y U = 0, we have sU 0, and then 12 12 ≃

D md s¯12 s12 = . (5.17) | |≃| | sms + md

D U Further, s 0.04, and so V s¯∗ + c ∗s ∗s s¯∗ up to about 1 %. Therefore, | 23| ∼ us ≃ 12 12 13 23 ≃ 12 we will use V s¯∗ . us ≃ 12 We will also need the expression for β in terms of CKM matrix elements which was given in equation (2.22) of Chapter 2

V V β arg cd cb∗ . (5.18) ≡ − VtdVtb∗ !

50 5.3 Analysis

In this section we analyze a set of Yukawa matrix texture models and compare them to the data. We ﬁrst start with predictions from the 2 by 2 light quark matrices, and then move on to the full 3 by 3 matrices.

5.3.1 2 2 Light Quark Matrices × As examined in the previous section, for exact 2 by 2 diagonalization of the light quark sectors, we have

V s sD cD sU (5.19) | us| ≃ | 12|≡| 12 − 12 12| md for Y U =0 ms+md 12 (5.20)  q m m iφ m U ≃  s d e u for Y =0.  ms+md ms mc 12 − 6 q q q  In the ﬁrst case, a prediction of Weinberg [48] works quite well:

V = 0.2240 0.0036 (5.21) | us| ± m d = 0.224 0.009 (5.22) sms + md ±

The second case then also works because mu is so small. Using the central values mc for the masses, the best value for φ is roughly π/2 (or 3π/2). However, φ is not

signiﬁcantly constrained due to the large uncertainties in the quark masses.

In the remainder of our analysis, we will be assuming that Y U,D = 0, Y U,D = 11 | 12 | Y U,D , and Y D = 0. This implies that V is given by one of the two equations | 21 | 12 6 | us| above.

5.3.2 3 3 Quark Matrices × In the next two subsections, we will analyze two diﬀerent sets of 3 family quark

Yukawa textures. In the ﬁrst section, we consider case A: models with vanishing

51 Y U Y D 0 X 0 0 X 0 I  X X 0   X X X  0 0 X 0 X X      0 X 0   0 X 0  II  X 0 X   X X X  0 X X 0 X X      0 0 X   0 X 0  III  0 X 0   X X X  X 0 X 0 X X      0 X 0   0 X 0  IV  X X X   X X 0  0 X X 0 0 X      0 0 X   0 X 0  V  0 X X   X X 0  X X X 0 0 X        

Table 5.1: The symmetric 5-zero textures classiﬁed in [58]. The number of zeros (here, 5) refers to the total number of zero elements in the upper-right and diagonal portions of the up and down matrices.

(1, 1) and (1, 3) elements and symmetric (1, 2) and (2, 1) elements. We show that these textures, if hierarchical, are ruled out. In the second section, we look at case

B: those textures with non-zero (1, 3) elements. In particular we will study models

III and V of Table 5.1, the most constrained examples of these textures, to show that texture models of this type with 5 or fewer zeros can accomodate the data.

Case A: Models with (1, 1) = (1, 3)=0 and (1, 2) = (2, 1) =0 6

Consider those hierarchical texture models with Y U,D = Y U,D = 0 and Y U,D = 11 13 | 12 | Y U,D = 0. These include type I, II, and IV of [58] listed in Table 5.1. Appendix C | 12 | 6 presents a method by which we can determine the number and possible placement of the non-removable phases in these matrices. There are 2 such phases in type I and

52 IV matrices and 3 in type II matrices. The location of these non-removable phases

can be chosen as follows:

D U I φ22 φ22 D U U II φ22 φ23 φ32 (5.23) D U IV φ22 φ32

U,D U,D Y13 = 0 implies s13 = s13 = 0, and our CKM matrix simpliﬁes to

D U c12 s¯12∗ s12∗s23∗ D − VCKM =  s¯12 c12∗ s23∗  . (5.24) s−D s cD s 1  12 23 12∗ 23   − 

ms md iφ mu Vus s¯12 e (5.25) | | ≃ | | ≃ sms + md s ms − s mc

Vub U mu s12 (5.26) Vcb ≃ | | ≃ s mc

s¯ cD sU β arg 12 = arg 1 12 12 arg 1 reiφ (5.27) ≃ sD ! − sD ! ≃ − 12 12 with D U mums c12s12 r and φ arg D (5.28) ≡ s mcmd ≡ s12 !

Because r 1 < 1, β must be a small angle. The maximum value for β, β , leads ∼ 5 max to (see Figure 5.1)

sin βmax = r (5.29)

sin 2β = 2 sin β cos β =2r√1 r2. (5.30) max max max −

Comparing this result to the data, we have

r = 0.21 0.04 (5.31) ± sin 2β = 0.41 0.07 (5.32) max ± sin 2β = 0.739 0.048 (5.33) exp ± 53 r

βmax (0, 0) (1, 0)

Figure 5.1: Graphical representation of the maximum value of β = arg 1 reiφ . −

Combining the errors in quadrature and comparing to the diﬀerence in central values leads us to conclude that these texture models are ruled out by 3.8σ or more. We

also have a prediction for Vub . Comparing to the data: Vcb

V m ub u =0.048 0.009 (5.34) Vcb th ≃ s mc ±

V ub = 0.086 0.008 (5.35) Vcb exp ±

This is a diﬀerence of 3.1σ.

In this subsection, we have thus shown that hierarchical texture models with

(1, 2) = (2, 1) = 0 and (1, 1) = (1, 3) = 0 elements are ruled out by the data by | | | | 6 at least 3σ. This is under the conservative assumption of doubled light quark mass

errors. This set includes those hierarchical textures of type I, II, and IV considered

above.

54 Case B: Models with non-zero (1, 3) elements

If we allow Y U or Y D to become non-zero, this can allow an increase in V /V 13 13 | ub cb| and sin 2β to the point where the data can be accomodated. Here let us consider rather constrained models in this mold, those of type III and V in Table 5.1. In the analysis that follows, we will see that these models can be consistent with the data, implying that this is also true for less constrained models of this type. This result agrees with that found by Roberts et. al. [53].

The models of type III and V both have Y U = 0 and Y U = 0. Each has 2 13 6 12 non-removable phases, which we choose to place as follows:

U D III φ13 φ32 (5.36) U U V φ13 φ32

U U Y12 = 0 implies s12 = 0, and the CKM matrix simpliﬁes to

D D D U c12 s12∗ + c12∗s13∗s23 s13∗ D D D D VCKM  s12 c12s13s23∗ c12∗ s23∗  . (5.37) ≃ −sD s − cD s cD s sD s 1  12 23 12 13 12∗ 23 12∗ 13   − − −  We have V s∗ and V s∗ . This implies that we can drop the second term in ub ≃ 13 cb ≃ 23 V since V V V . Summarizing: us | ub cb|≪| us|

D D U D md Vus s12∗ + c12∗s13∗s23 s12∗ (5.38) ≃ ≃ ≃ sms + md

U m iφ13 u Vub s13∗ e− (5.39) ≃ ≃ s mt

V s∗ (5.40) cb ≃ 23

We have predictions for V and V in terms of quark mass ratios. Comparing to | us| | ub| data:

md Vus th =0.224 0.009 (5.41) | | ≃ s ms + md ±

55 V = 0.2240 0.0036 (5.42) | us|exp ±

mu Vub th =0.00307 0.00047 (5.43) | | ≃ s mt ± V = 0.00357 0.00031 (5.44) | ub|exp ±

These predictions match the experimental values.

Now turning to β:

D D s12s23 c12s13 ∗ β arg D D = arg 1 D (5.45) ≃ s s23 c s13 ! − s s23 ! ! 12 − 12 12 msmu 1 iφU iφU arg 1 e− 13 = arg 1 ρe− 13 (5.46) ≃ − s mdmt Vcb ! − with m m 1 ρ s u (5.47) ≡ s mdmt Vcb | | ρ and sin 2β in these models are:

ρ = 0.32 0.05 (5.48) ± sin 2β = 2ρ 1 ρ2 =0.61 0.08 (5.49) max − ± q

This is about 1.4σ away from the measured value of sin 2β,0.739 0.048, and is thus ± fairly consistent with the data. There are no other predictions to be made in these

models.

We have shown in this subsection that the hierarchical texture models of type III

and V can accomodate the data inside the 2σ level. Since these models are rather

constrained versions of a broader class of models, those hierarchical models with non-

D U zero Y13 or Y13 which have fewer than 5 zeros should also be consistent with the

data.

56 5.4 Discussion

5.4.1 Comparison of the Two Cases

The two cases yield diﬀerent predictions for Vus, Vub, and sin 2β. However, the diﬀerences in their predictions can be essentially contracted to one point. Both cases

mu give a good ﬁt to Vus and are basically the same, due to the small size of : | | mc q

ms md iφ mu md Vus e (5.50) | | ≃ sms + md s ms − s mc ≃ sms + md

For β, the two cases have

√ 2 mums Case A sin 2βmax 2r 1 r with r m m ≃ − ≡ c d (5.51) Case B sin 2β 2ρ√1 ρ2 with ρ q mums 1 . max mtm V ≃ − ≡ d | cb| q Forming the ratio of ρ to r, we have

ρ 1 m = c . (5.52) r Vcb s mt | | Assuming that the models ﬁt all of the other data well, we can put in the central

values of m , m , and V . This leads to ρ 1.5r. It is this enhancement of ρ in case c t | cb| ≃

B over r of case A which allows the models in case B to have sin 2βmax large enough

to accomodate the data.

Another way to see the diﬀerence in these two models is to manipulate r and ρ as

follows:

m m V A m r u s ub s (5.53) ≡ s mcmd ≃ Vcb smd

m m 1 V B m ρ u s ub s (5.54) ≡ s mtmd Vcb ≃ Vcb smd

| |

The diﬀerence between r and ρ can be thought to come from the diﬀerent predictions

for V in the two models. | ub| 57 5.4.2 Future Prospects

We have shown that V and sin 2β are at the moment roughly equal in con- | ub| straining power on the texture models considered. Most of the uncertainty on V | ub| is theoretical and is unlikely to diminish signiﬁcantly in the near future. On the

other hand, the precision of the sin 2β measurement is mainly statistical and is likely to shrink as more data is taken. In the future, then, sin 2β will lead to stronger constraints on the types of texture models considered here.

58 CHAPTER 6

5D SUSY SO(10) D GUT MODEL × 3

6.1 Introduction

As discussed in Chapter 4, supersymmetry and grand uniﬁcation can address many of the deﬁciencies of the Standard Model. Models which combine these new ideas in a 4D framework, 4D SUSY GUTS, have long been considered as extensions to the Standard Model [59, 60, 61, 62, 63, 64].

As experiments become more accurate, we are able to test whether GUT theories work in their minimal forms. There are several indications that the most simple GUT theories cannot work.

With the deﬁnition of the GUT scale M : α (M ) α (M ) α , the as- • G 1 G ≡ 2 G ≡ GUT

sumption of exact uniﬁcation (α3(MG)= αGUT) leads to a prediction for α3(MZ )

which is larger than the measurements. This indicates that a negative threshold

α3(MG) αGUT correction to α3 must be present at the GUT scale: ε3 − 0.04 ≡ αGUT ≃ − [65].

Since the fermions are combined into fewer multiplets of the unifying group, • GUT theories in general give relationships between the fermion masses at the

59 GUT scale. In SO(10), third family uniﬁcation can be accomodated (mτ (MG)=

mb(MG) = mt(MG)), but the ﬁrst and second family masses do not unify at

1 MG. The Georgi-Jarlskog relation: ms/mµ = 3 mb/mτ [66], which had been known to work very well, holds less well as measurements of the strange quark

mass decrease.

GUT models place the weak Higgs doublet(s) into larger representations which • include new color triplet Higgs ﬁelds. It is diﬃcult to make the color triplets

heavy enough (near GUT scale mass) to avoid rapid proton decay while keeping

the weak doublets light (weak scale mass). The doublet-triplet (DT) splitting

problem is related to the method by which the GUT symmetry is broken to the

Standard Model, and the models which give proper DT splitting can be quite

complicated.

An idea that provides answers to some of the above issues and has been considered extensively for the last several years is orbifold GUTS [18, 46, 47, 67, 68, 69, 70, 71,

72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95,

96, 97, 98]. If the compactification radius is slightly smaller than the cutoff scale, the heavy Kaluza-Klein (KK) modes can give compactification scale threshold corrections necessary for exact gauge unification at the cutoff scale [99]. If the Standard Model

Higgs is a bulk ﬁeld, and if the triplet Higgs has twisted boudary conditions under the orbifold it will acquire a mass at the compactiﬁcation scale, thus solving the DT splitting problem naturally. The same mechanism breaks the GUT gauge symmetry and gives mass to gauge bosons outside of the Standard Model.

5D SU(5) models have a Standard Model brane on which all of the interesting

GUT mass relations are absent. The ﬁrst and second family matter ﬁelds usually

60 feel this brane and so there are no relations between quarks and leptons in these

two families. This is unattractive since the models lose predictivity. For this reason,

we choose to concentrate on SO(10) models which give more possible avenues of

symmetry breaking to the Standard Model and allow for GUT mass relations.

5D SO(10) SUSY GUTS were ﬁrst considered in Dermisek and Maﬁ [74], and

a setup which gives gauge coupling uniﬁcation was constructed by Kim and Raby

1 [99]. The extra dimension is an S /(Z Z′ ) orbifold, a line segment with endpoints 2 × 2 which are ﬁxedpoints of the orbifold Z2 symmetries. 5D N=1 SUSY is broken to

4D N=1 by the ﬁrst Z , while the second Z breaks SO(10) to SO(6) SO(4) = 2 2 × SU(4) SU(2) SU(2) Pati-Salam gauge group. Further breaking to the Stan- × L × R ≡ dard Model is provided by the Higgs mechanism on one of the ﬁxed points. One

ﬁxed point, the SO(10) brane, is invariant under the ﬁrst Z2 and has SO(10) gauge

symmetry. The other ﬁxed point, the Pati-Salam (PS) brane, is invariant under the

second Z2 and only has Pati-Salam gauge symmetry. The bulk has the full SO(10)

gauge symmetry. Gauge coupling uniﬁcation works if the compactiﬁcation scale is

approximately 1014 GeV, the cutoﬀ scale is approximately 1017 GeV, and if the Higgs

multiplet lives in the bulk. A virtue of the model is that SO(10) or Pati-Salam exists

after the orbifold breaking and we retain some of the Yukawa GUT relations.

Attempts to explain the mass hierarchy between the diﬀerent families of fermions

have often centered on ﬂavor symmetries. There are models based on abelian hori-

zontal U(1) symmetries where the smallness of certain couplings is explained by the

suppression given by high powers of U(1) breaking ﬁelds. The explanations can at

most be qualitative as there are order one coeﬃcients which are undetermined from

61 the U(1) symmetry. We therefore choose to concentrate on non-abelian ﬂavor sym-

metry. With three families, the largest possible symmetry would be SU(3). However,

the order one top Yukawa coupling would badly break the SU(3) symmetry, and so in-

stead we choose to concentrate on an SU(2) symmetry between the ﬁrst two families.

This symmetry can explain the absence of ﬂavor changing neutral currents in super-

symmetric theories and can relate unknown order one coeﬃcients between diﬀerent

families. In string theory, global symmetries are thought to be generally broken by

quantum gravity eﬀects [100, 101, 102], and so our ﬂavor symmetry should be a gauge

symmetry. However, the breaking of a continous gauge symmetry like SU(2) gives

unwanted ﬂavor changing neutral currents from the D-term contributions [103, 104].

For this reason, we assume a ﬂavor symmetry relating the ﬁrst and second families

based on D3, a discrete subgroup of SU(2). As a discrete gauge symmetry, D3 has all the virtues of SU(2) and can avoid the problem related to continuous gauge symmetry

[105].

In SUSY SO(10) models with flavor symmetries, the most difficult thing to explain is why m < m while m m . In SO(10) models, we usually have m /m = u d t ≫ b t b Y /Y tan β tan β, and the heaviness of the top quark is explained by a large value t b ∼ for tan β 50. In these models, it is natural that Y /Y Y /Y , which leads to ∼ u d ∼ t b m /m 50. To fit the data, an unusually small Yukawa coupling is needed for the up u d ∼ quark compared to the down quark. This is difficult to implement in SO(10) models.

Extra-dimensional theories provide a nice tool to suppress or enhance fermion masses, as the size of a mass can be inﬂuenced by the localization of the matter and Higgs ﬁelds within the extra dimension. Others, notably Arkani-Hamed and Schmaltz in [106], have used localization in an extra dimension to explain the fermion mass hierarchy.

62 Our model, on the other hand, retains some of the attractive Yukawa relations present

in 4D GUT models which are not present in [106].

We explain the ratios m /m < 1 and m /m 1 simultaneously by the quasi- u d t b ≫ localization of the Higgs ﬁeld in the bulk by a kink mass.6 We give a vacuum ex-

pectation value (VEV) in the U(1)X direction to a scalar adjoint ﬁeld Σ contained in

the 5D vector multiplet. Σ is odd under both Z2 parities, and we choose the VEV so

that its contribution to the D-term VEV is only at the two branes. Supersymmetry

can be preserved if there are ﬁelds on each brane which get VEVs in the right-handed

neutrino direction to counter the contribution of Σ to the D-term VEV. The VEV h i

of Σ acts to give a U(1)X-dependent mass to the hypermultiplet ﬁelds in the bulk, in

particular the Higgs ﬁeld. Because Hu and Hd carry opposite U(1)X charges, they are

localized towards diﬀerent branes. Thus if the 1st and 3rd families get their Yukawa

couplings on opposite branes, we can naturally have mu/md < 1 and mt/mb > 1.

The easiest way to ensure that the 1st and 3rd families get their Yukawa couplings

on opposite branes is to restrict them to these branes. By proton decay constraints,

the 1st family cannot reside on the SO(10) brane. Thus we choose the 1st (and by

family symmetry, the 2nd) family to be on the PS brane, and the 3rd family to be

on the SO(10) brane. We then choose the sign of the kink mass VEV to be such

that Hu is localized towards the SO(10) brane and Hd is localized towards the PS

brane.7 Communication between the ﬁrst two families and the third is provided by

6 The extreme limit is discussed in [107] where Hu is on the brane and Hd is in the bulk such that mt mb for order one tan β. ≫ 7These arguments do not rule out the case of all 3 families in the bulk, or some fields on branes with others in the bulk. However, if the 3rd family is a brane field, then the 2nd family must also be a brane field or the ratio mµ is made too small by volume suppression to fit the data. It is possible mτ to decrease the suppression from the volume factor by either placing only half of the 2nd family in the bulk and/or by altering the GUT threshold correction ε3. We made some attempts with these approaches but were unable to produce viable theories.

63 mixing with bulk ﬁelds. After requiring that the 1st and 3rd families be brane ﬁelds,

the location of the Higgs and all matter ﬁelds is entirely determined from gauge

coupling uniﬁcation, proton decay constraints, D3 family symmetry between the ﬁrst two families, and the action of the kink mass to explain 1st and 3rd family mass ratios.

To summarize, in this chapter we construct a 5D SO(10) supersymmetric orbifold

GUT model in which the 1st and 2nd families reside on the PS brane and transform as a doublet under the D3 family symmetry, the 3rd family is located on the SO(10) brane, and a kink mass localizes the MSSM Higgs doublets to opposite sides of the bulk. The model explains the 13 independent quark and charged lepton masses and

mu mixing angles in terms of 11 parameters. The two predictions are (MZ )=0.0037 mc ± 0.0006 and m m m (M ) = (10.7 5.0) 105 MeV3, both of which are approximately d s b Z ± × 1σ above the experimental values. This model was originally constructed by Hyung

Do Kim, Stuart Raby, and myself in [18].

We note that there is a problem with the ﬁve dimensional formulation of our

theory. In order for the gauge couplings to unify at the cutoﬀ scale, the PS brane

ﬁelds breaking Pati-Salam down to the Standard Model must have cutoﬀ-scale VEVs.

Therefore, in general one would expect all aspects of PS symmetry to be broken at

this high scale, and that the lower-energy theory should not be expected to exhibit

PS symmetry. Nevertheless, we wish to retain some of the PS symmetry below the

cutoﬀ scale in order to preserve the PS Yukawa coupling relations. This is possible

if there is an additional extra dimension in which the PS brane is separated from

the PS-breaking VEV ﬁelds. This 6th direction will solve this problem if its length

scale is only slightly larger than the cutoﬀ scale. The eﬀective 5D theory below the

64 energy scale of this 6th direction will not be aﬀected in any other way. For this

reason we choose to perform our analysis in the 5D theory, and will assume that PS

Yukawa relations are still valid below the cutoﬀ-scale. We discuss the 6D ﬁx to the

PS-breaking problem in Section 6.4.

A more serious problem exists in this theory. Orbifold anomaly calculations, presented in Chapter 7, show that this theory suﬀers from a gauge anomaly on the

PS brane. This fact was not known to the authors when the model was originally proposed in [18]. Gauge anomalies lead to inconsistencies in gauge theories, implying that we must either alter or abandon the theory as it was laid out in [18]. We have chosen to present the model here in its unaltered form since the exact alterations in

ﬁeld content necessary to ﬁx the theory are not yet known. We discuss this problem in

Section 6.5 and give a list of possible changes in ﬁeld content which would cancel the anomaly. Further analysis will be given in Chapter 7 in which the gauge anomalies for this theory are calculated.

We will continue this chapter by setting up our theory in Section 6.2. This includes defining the orbifold and its symmetry-breaking properties, describing the gauge coupling unification in 5D, and giving the field content and vacuum expectation values

(VEVs) of the theory. The remainder of Section 6.2 involves the removal of the massive ﬁelds and the derivation of the Yukawa matrices. In Section 6.3, through renormalization group running we will analytically and numerically ﬁt the Yukawa matrix parameters to the data and make two predictions in the quark mass sector.

The possibilities and advantages of placing our model in a 6D framework are given in

Section 6.4, and the discussion of the Pati-Salam brane anomaly is given in Section

6.5. Finally, in Section 6.6 we close with some discussion of the model.

65 The Appendices relevant to this chapter are Appendix D on D3 family symmetry,

Appendix E on the determination of massless states, and Appendix F which presents

a 5D orbifold version of a 4D SO(10) SUSY GUT model by Dermisek and Raby [108].

6.2 Setup

We will start by defining our theory and then move on to finding the Yukawa matrices of the MSSM effective field theory.

6.2.1 Background

We consider a ﬁve dimensional supersymmetric SO(10) GUT compactiﬁed on an

1 1 S /(Z Z′ ) orbifold where S is described by y [0, 2πR). The ﬁrst orbifold, Z , 2 × 2 ∈ 2 under which y y, breaks 5D N=1 supersymmetry (4D N=2) to 4D N=1. The → −

other orbifold, Z′ , under which y y + πR, breaks SO(10) down to the PS gauge 2 → − group SU(4) SU(2) SU(2) . The fundamental domain of the y direction is the C × L × R line segment y [0,πR/2]. SO(10) gauge symmetry is present everywhere except ∈ at the point y = πR/2, which only has Pati-Salam gauge symmetry. Hence, we

call the two inequivalent ﬁxed points the “SO(10)” (y = 0) and “Pati-Salam” branes

(y = πR/2) where each ﬁxed point is a three-brane (3+1 dimensional spacetime). The

Higgs mechanism on the PS brane completes the breaking of the PS gauge symmetry

to the SM gauge group.

The ﬁelds which live in the ﬁve dimensional space between the branes (known as

the “bulk”) are even or odd under the orbifold parities. We denote the parity of a given

field under these orbifolds by two subscripts on the field with the first corresponding ±

to Z2 and the second corresponding to Z2′ . This part of the setup, including the

66 orbifold structure, ﬁeld parities, and supersymmetry and gauge symmetry breaking

are based on work done in [74] and [99]. For more details, please see these references.

We wish to relate some of our fields by a family symmetry, D3, under which the first and second family fields form a doublet. Other fields within our model will be in various representations of D3 which will affect the structure of the Yukawa matrices we generate. We take the family symmetry to be independent of the orbifold symmetries. A brief summary of the D3 group is provided in Appendix D, where we give the information necessary to understand the family representations and couplings used in our model. For more information, see the Appendix presented in [108].

The 5D supersymmetric vector multiplet =(A ,λ ,λ , Σ) contains a 4D vector V M 1 2 multiplet V = (Aµ,λ1) and a 4D chiral adjoint φ = ((Σ+ iA5)/√2,λ2). For a

generic hypermultiplet =(h, h, ψ, ψ) which breaks up into the 4D chiral multiplets H Φ=(h, ψ) and Φ=(h, ψ), we use Arkani-Hamed et. al. [109] for the 5D action:

4 4 V V S d xdy d θ Φe Φ† + Φ†e− Φ ⊃ Z Z h i 2 1 + d θ Φ(m + ∂y φ)Φ+h.c. (6.1) "Z − √2 #) The mass m refers to the mass of the 5D ﬁelds, which we set to zero for all hyper-

multiplets. We take the ﬁeld Σ ( φ) to get a VEV in the X direction of SO(10). ⊂ Because of the coupling in the action:

4 2 1 S d xdy d θ Φ(∂y φ)Φ (6.2) ⊃ Z Z − √2 VEV 4 2 d xdy d θ Φ(∂y mX )Φ (6.3) → Z Z −

this VEV will generate X-dependent masses mX for the bulk chiral ﬁelds coming

from hypermultiplets in the theory. The U(1)X of SO(10) is contained inside Pati-

Salam, and the chiral adjoint for PS has ( ) parity. Therefore the mass m we −− X 67 have generated also has this odd-odd parity under the orbifold. For this reason we

call it a kink mass.

The D-term for this theory is

D = (∂ Σ+ i [Σ, A ]), (6.4) − y 5 hence a VEV of Σ is potentially dangerous since it could create a D-term VEV. To avoid this, we take Σ to be ﬂat in the bulk with discontinuities at the branes to be h i consistent with its ( ) parity. These discontinuities generate D-term VEVs on the −− c branes, but we can choose to add brane ﬁelds (16kink on the SO(10) brane, χkink and

c c c χ kink on the PS brane) with VEVs in the ν and ν directions to cancel these eﬀects

from ∂ Σ , leaving us with a D-ﬂat theory.8 More concretely, if the sterile sneutrino yh i c component of 16kink (νkink) and the charge conjugate sterile sneutrino component of

c c χ kink (ν kink) get their VEVs, the D-ﬂat condition for the U(1)X subgroup is given by

2 c 2 c 2 πR ∂y Σ = g 5 νkink δ(y) 5 ν kink δ(y ) . (6.5) h i |h i| − |h i| − 2

c c As νkink and ν kink carry only U(1)X charges, they do not appear in the D-term of the

Standard Model gauge group. The D-ﬂat condition is satisﬁed for

Σ = Σ0ε (y), (6.6) h i −− 2Σ νc 2 = νc 2 = 0 . (6.7) |h kinki| |h kinki| 5g2

ε + and ε are step functions on the orbifold. Both of these will be used later in − −− our analysis.

1 for y [ πR, 0] ε +(y) − ∈ − (6.8) − ≡ +1 for y [0, πR] ( ∈ 8This technique is described in [109] and used in [110].

68 πR +1 for y [ πR, 2 ] 1 for y ∈ [− πR , 0]− ε (y)  2 (6.9)  − ∈ − πR −− ≡  +1 for y [0, 2 ]  ∈ πR 1 for y [ 2 , πR]  − ∈  We choose to parameterize the kink mass as follows:

Σ0 ζ mX (y) = ε (y)X = ε (y)X (6.10) −− 2 −− πR Σ πR ζ 0 . (6.11) ≡ 2

ζ will turn out to be a useful dimensionless parameter in later analysis.

If our U(1) breaking ﬁelds were to get VEVs as large as the cutoﬀ scale, we X− could not keep SO(10) or Pati-Salam as our symmetries on the branes. However, it

turns out that ζ 2 is needed to ﬁt the observed physical quantities, and so the ∼ corresponding VEVs are

c 8 16 1 νkink 2 = 2 2 2 . (6.12) h i ∼ s5πRg s5π g4R ∼ R

Therefore, all the VEVs necessary to give a kink mass are around the compactiﬁcation

2 3 scale and the breaking eﬀects are suppressed at least by Mc/M 10− or 10− [99]. ∗ ∼

For the most part, the U(1)X breaking eﬀects come from the kink proﬁles of bulk

ﬁelds which are calculable and are exponentially proportional to the U(1)X charges.

It is a novel example of obtaining large (order one) symmetry-breaking with a very

c small symmetry-breaking order parameter Σ. We stress that the VEVs of Σ, νkink

c and ν kink do not spoil the symmetries on the branes.

On the other hand, on the Pati-Salam brane gauge coupling uniﬁcation requires

c c c c c χ and χ to get VEVs of order the cutoﬀ scale (χ and χ are diﬀerent from χ kink

c and χkink). This is a serious problem for our theory. There are order one corrections

in the Kahler potential and Pati-Salam symmetry is badly broken in the canonical

69 basis:

c c χ †χ K = (1+ C )ψ†ψ (6.13) M 2 ∗

c c where C is of order one, χ = χ † M , and where ψ is a generic Pati-Salam brane h i h i ∼ ∗ ﬁeld. This problem can be solved in a geometric way if there is a sixth dimension along

c which χ and ψ are separated. If this sixth dimension has a length scale R2 slightly larger than the cutoﬀ length scale, we can simultaneously retain the desired PS-brane

Yukawa relations and gauge coupling unification. The 6D setup will be discussed in detail in section 6.4. Until then, we choose to concentrate on the effective 5D theory below this 6D scale. The 5D analysis of the Yukawa matrices will not be quantitatively affected by the addition of this extra dimension.

We now brieﬂy summarize the gauge uniﬁcation results given by Kim and Raby

[99]. A 5D gauge theory is nonrenormalizable and gets large corrections at the cutoﬀ.

Corrections to gauge couplings, however, will be the same for all couplings unified into the larger gauge group. These corrections will affect the absolute values of the gauge couplings, but not the differences. Further, if the gauge symmetry is broken only by orbifolding or by Higgs mechanism on the branes, the differences in the couplings will have a logarithmic, calculable running.

The states which affect the differential running are the bulk vector multiplet and V the bulk Higgs hypermultiplet .9 The placement and number of complete matter H multiplets does not affect gauge coupling unification, since matter multiplets (16s of

9When considering differential running of the gauge couplings, a Higgs hypermultiplet in the bulk is effectively the same as a 4D 10 of SO(10) with light Higgs MSSM doublets and heavy Higgs triplets of mass Mc. This setup admits gauge coupling unification as shown by Kim and Raby [99]. In particular, see the calculations leading to equation (3.13) of that paper. Effects from brane Higgs doublets would be felt up to M∗ and would tend to inhibit unification since they drive the couplings apart rather than together.

70 SO(10)) act equally across the three gauge couplings and cannot aﬀect the coupling

diﬀerences. Those states in the theory outside of the MSSM have twisted orbifold

boundary conditions and so have masses at the compactiﬁcation scale (Mc). For

energies below Mc, the theory is the MSSM. The eﬀects of running between Mc and

M (the cutoﬀ scale), including the Kaluza-Klein (KK) towers, are taken as threshold ∗

corrections at Mc. Without these threshold corrections, it is known that the MSSM

unifies around M 3 1016 GeV with a coupling of α 1/24 and a GUT-scale G ∼ × GUT ∼ threshold correction for α of ε 0.04. Assuming unification in the orbifold theory 3 3 ∼ − at the cutoff scale M and that the PS breaking Higgs VEV is of order the cutoff scale, ∗

we can solve for M and Mc in terms of the 4D GUT parameters MG, αGUT, and ε3. ∗ 17 14 This leads to M 10 GeV and Mc 10 GeV [99]. ∗ ∼ ∼ The matter ﬁeld locations are constrained by proton decay:

Matter ﬁelds on the SO(10) brane • There are gauge bosons within SO(10) which mediate baryon (B) and lepton (L)

number-violating interactions. All of these are outside of the Standard Model

and hence have masses of order Mc or higher. After integrating out these ﬁelds

(and their KK modes), we get dimension six operators which violate B and L

for any matter multiplets on the SO(10) brane. These operators are suppressed

by 1/M 2. Given M 1014 GeV, current bounds on proton decay rule out c c ∼ models which have these operators for the ﬁrst and second families. Thus only

the third family can reside on the SO(10) brane [111].10 Dimension 5 proton

10With dimension six operators for the third family, mixing between this family and the first two can induce proton decay. Assuming that the mixing is of order Vub or Vcb and that the gauge | | | | bosons have mass at Mc, naive calculations using formulae in [111] put the proton lifetime many orders of magnitude above current limits, since this leads to an effective gauge boson mass of order 17 Mc/(VcbVub) 6 10 GeV. ≈ × 71 decay operators vanish since the color triplet Higgs states obtain off-diagonal

mass with triplets in 10 and these states do not couple to matter.

Matter ﬁelds on the PS brane or in the bulk • Pati-Salam gauge symmetry does not relate the left-handed ﬁelds ψ ((4, 2, 1) in

PS) to the right-handed ﬁelds ψc ((4, 1, 2) in PS), and we do not get baryon

number-violating dimension six operators after integrating out the heavy gauge

bosons. Therefore, any matter ﬁelds can be on the PS brane as long as the PS

breaking scale is not extremely low. This scale in our theory is M 1017 GeV, ∗ ∼ and proton decay is not a problem here.

In principle, we can consider higher dimensional operators with derivative inter-

actions ∂5 = ∂/∂y. Because the coeﬃcients of the higher dimensional operators

are not determined from the theory we cannot calculate the proton decay rate

from these operators accurately. However, we can get a bound that is consistent

with our setup by assuming unknown coeﬃcients to be order one. See Kim and

Raby for more details [99].

Proton decay constrains the ﬁrst and second families to reside either in the bulk or on the PS brane, but does not constrain the location of the third family. We choose to place the third family on the SO(10) brane. Given this placement, the second family is forced to reside on the PS brane. If it were in the bulk, volume suppression would

3 give m /m 10− , which is far too small to fit the data. Our D flavor symmetry µ τ ∼ 3 places the first and second families together into a doublet, so we place both on the

PS brane.

Let us summarize the basic setup.

72 Gauge symmetry : SO(10) in the bulk and at y = 0, Pati-Salam at y = πR/2. •

Higgs ﬁelds come from 10 dimensional hypermultiplets in the bulk. •

3rd family matter ﬁelds are on the SO(10) brane. •

1st and 2nd family matter ﬁelds, a doublet under D , are on the PS brane. • 3

Kink mass localizes the hypermultiplets through their X-dependence. • 6.2.2 Yukawa Matrix

This section introduces the ﬁelds and superpotential of our model. Here we cal-

culate the Yukawa matrices associated with the massless ﬁelds corresponding to the

Standard Model fermions.

First we introduce the matter content of the theory. In the bulk, we have two

5D N=1 hypermultiplets which transform as 16s under SO(10). These ﬁelds form a doublet (2A) under the D3 family symmetry. In our Lagrangian, we show the D3 doublet structure of these ﬁelds by a subscript a, with a = 1, 2. For information on how doublets and other D3 objects couple, please see Appendix D. We also have a hypermultiplet 10 of SO(10) which is a singlet (1A) under the family symmetry. These

ﬁelds are listed in Table 6.1 with their parities, PS gauge symmetries, and family symmetries. On the SO(10) brane we place a single 16 of SO(10), invariant under

D3, and an SO(10) gauge singlet φ which is a doublet under the family symmetry, in addition to the 16kink needed for D-flatness of the theory. These fields are listed in Table 6.2. The PS brane has a more complicated set of fields. We have already

c c mentioned χkink and χ kink which are needed to cancel the D-term on the Pati-Salam

brane, and the ﬁelds χc and χ which provide the Higgs ﬁelds necessary to break

73 Field PS Symm D3 Symm ψ (4, 2, 1) 16 = −−c 2A ψ + ! (4, 1, 2) ! − ψ++ (4, 2, 1) 16 = c 2A ψ + ! (4, 1, 2) ! − H++ (1, 2, 2) 10 = c 1A H+ ! (6, 1, 1) ! H − (1, 2, 2) 10 = c−− 1A H + ! (6, 1, 1) ! −

Table 6.1: Bulk matter ﬁelds.

Field PS Symm D3 Symm ψkink (4, 2, 1) 16kink = c 1A ψkink ! (4, 1, 2) ! ψ3 (4, 2, 1) 163 = c 1A ψ3 ! (4, 1, 2) ! φ (1, 1, 1) 2A

Table 6.2: SO(10) Brane ﬁelds

Pati-Salam to the Standard Model. These ﬁelds are all D3 singlets. There are the

c ﬁelds ψ and ψ , each doublets under D3, which transform as the two halves of an

SO(10) 16. N c and N c are also family doublets and carry charge under PS in order

c c c to allow mixing between ψ and N . There are various other ﬁelds φ, A, A3R, ω , and

ωc which will get vacuum expectation values. These ﬁelds are listede in Table 6.3 with their PS charges and D3 family symmetries.

74 Field PS Symm D3 Symm c χ (4, 1, 2) 1A χc (4, 1, 2) 1A c χkink (4, 1, 2) 1A c χ kink (4, 1, 2) 1A ψ (4, 2, 1) 2A c ψ (4, 1, 2) 2A c N (4, 1, 2) 2A N c (4, 1, 2) 2A φ (1, 1, 1) 2A A (1, 1, 1) 1B e A3R (1, 1, 3) 1A c ω (4, 1, 2) 1A ωc (4, 1, 2) 1A

Table 6.3: PS Brane ﬁelds

We choose to break the superpotential into two pieces: W = W1 + W2. The ﬁrst contains terms leading to interactions between the matter and Higgs ﬁeld (H H ): ≡ ++ 1 W1 = λ316310163 δ(y) 2 c c c πR + λαψaHψaA + λβψaHNaφa +2λγψaHA3R ψ + δ(y ) (6.14) − a − 2 h i e

W2 contains mass terms:

W = 16 (∂ m )16 + 10(∂ m )10 2 a y − X a y − X

+ 2σφa16a163 δ(y) h i c c c c c πR + 2η ψ χa + N a (aψ + b0ω ω N ) δ(y ) (6.15) ++ a a a − 2 h i Suﬃcient factors of the cutoﬀ scale M should be placed in the superpotential terms ∗ so that the couplings λ3, λα, λβ, λγ, σ, η, a, and b0 are dimensionless. We assume that at this stage our Lagrangian possesses a CP-symmetry and that these couplings

75 are all real. All CP-violation will come from spontaneous symmetry breaking. The

c c ﬁelds which get VEVs are φ1, φ2, φ2, A, A3R, ω , and ω . The ﬁrst 4 break D3

symmetry, while A = A0 T breakse SU(2) symmetry. All of these VEVs we h 3Ri 3R 3R R take to be real. ωc and ωc are ﬁelds which get VEVs in the right-handed neutrino

direction. These VEVs we take to be complex, and furthermore take the combination

to be in a particular direction: ωcωc ceiθ1 + X eiθ2 . More will be said later about h i ∝ R this choice of direction. All of the above mentioned VEV values are taken to be of order M rather than Mc, but are allowed to lie enough below M to be able to give ∗ ∗ the desired hierarchies in the Yukawa matrices. We also assume that the ﬁelds listed

here, which get VEVs, obtain mass near the cutoff scale M . This lifts these fields ∗ high enough that they cannot adversely affect the gauge coupling unification results

we wish to preserve.

Our superpotential allows 9 extra U(1) symmetries which we take to be symmetries

of the theory in order to forbid problematic superpotential terms.11 We choose to

c c parameterize these symmetries by the following 9 fields: H+ , ψ3, ψ3, ψa, A, A3R, − ωc, and ωc. We choose each of these fields to have an arbitrary charge under one and only one U(1). After specifying these charges, the charges of the remaining fields are determined by the terms in the superpotential. In addition to these U(1)s, the superpotential can have Z2 symmetries which are extensions of the first orbifold Z2.

The transformations of the bulk ﬁelds under this symmetry have already been given.

To have such a symmetry, we can choose A3R odd and the remaining 7 independent

c fields (H+ is already determined) even under this Z2. The transformations of the − 11We can avoid unwanted flavor changing neutral currents from the breaking of these extra symmetries by instead assuming only Zn subgroups of the U(1)s sufficient to forbid the unwanted superpotential terms.

76 remaining ﬁelds can then be determined from the superpotential terms. Another

possible choice is ψa odd with the remaining ﬁelds even. It can be shown that it

is not possible to similarly extend the second Z2 orbifold to the brane terms of the

superpotential.

Our 5D bulk Higgs ﬁeld H(xµ,y) can be represented as a 4D massless state and

a KK tower of 4D massive states. In order to ﬁnd the eﬀective Yukawa matrices, we

need to know the overlap between the 5D bulk ﬁeld H(xµ,y) and the eﬀective 4D massless Higgs H0(xµ).12 This overlap is

X ζy µ H 0 µ H(x ,y) e πR nH ρ M H (x ) (6.16) ⊃ ∗ q where

Xζ nX (6.17) ≡ seXζ 1 − 2M ρ c . (6.18) ≡ sπM ∗ The kink mass dependence gives the massless Higgs ﬁeld an exponential proﬁle, local-

izing it to an end of the extra dimension. We will use nX , nH , nL, nR, etc. to stand

for a normalization as above with the particular value for the X quantum number

substituted for the X in equation (6.17). In the same manner, at times we will use

n1 or n 3 to refer to these normalization factors with the indicated X value inserted. − We can see explicitly in equation (6.16) that the up- and down-Higgs wavefunctions

πR are localized towards opposite ends of the y direction domain 0, 2 due to their h i opposite X quantum numbers.

12When addressing the overlap between 5D and 4D ﬁelds, to avoid confusion we will explicitly show the dependence of the ﬁelds on the spacetime coordinates.

77 The light ﬁelds which will correspond to our observed three families of particles

come from various mixings of the ﬁelds in the superpotential. Consider the following

terms contained in W2:

πR W N c (aψc + b ωcωcN c) δ(y ) (6.19) 2 ⊃ a a 0 a − 2 h i With the factors of M explicit: ∗

c c ω ω VEV iθ1 iθ2 b0 b0′ (ce + XRe ) b (6.20) M 2 ! → ≡ ∗

c c One combination of ψa and Na is massive while the other combination leads to a

c 0 massless ﬁeld. In terms of a and b, the overlap between the massless ﬁeld ψa and

the original ﬁelds is:

ψc ncψc 0 (6.21) a ⊃ a a N c ncψc 0 (6.22) a ⊃ − b a 1 nc (6.23) ≡ a2 1+ b 2 | | q c 0 The light ﬁelds ψa form a D3 doublet and correspond to the 1st and 2nd family

a c c 0 right-handed particles. The factor of b in the overlap between Na and ψa will lead to

1 1 iθ iθ in the (2,2) elements of the Yukawa matrices. This factor is important b ∝ ce 1 +XRe 2 in ﬁtting inter-family fermion mass ratios and in providing us with nontrivial phases in the Yukawa matrices.

Now we turn to a diﬀerent subset of W2:

W 16 (∂ m )16 2 ⊃ a y − X a πR + 2σφa16a163 δ(y)+ 2η ψ χa δ(y ) (6.24) ++ a − 2 h i h i

78 µ µ These mass terms lead to mixing between the brane ﬁelds 163(x ), χa(x ) and the

µ bulk ﬁelds 16a(x ,y). The overlap between the resulting left-handed massless ﬁeld

0 µ (ψ3(x )) and those ﬁelds in the superpotential:

ψ (xµ) n ψ0(xµ) (6.25) 3 ⊃ L 3 XLζy φa µ e πR 0 µ (ψ )a (x ,y) ε (y)e σ M nLψ3 (x ) (6.26) −− ⊃ − −− M ! ∗ ∗ q XLζ φa σ e µ 2 0 µ χa(x ) e nLψ3(x ) (6.27) ⊃ − M ! η ∗ e The overlap in the right-handed ﬁelds is

ψc(xµ) n ψc 0(xµ) (6.28) 3 ⊃ R 3 XRζy φa c µ e πR c 0 µ ψ + (x ,y) ε +(y)e σ M nRψ3 (x ). (6.29) − a ⊃ − − M ! ∗ ∗ q e 0 c 0 The definitions of ε +(y) and ε (y) have been given in equation (6.8). ψ3 and ψ3 − −− correspond to the left- and right-handed 3rd family fields. Other definitions follow:

1 nL (6.30) ≡ 2 2 1 ρ XLζ 1+ r n2 + η2 e e s L 1 nR (6.31) 2 1 ≡ 1+ r 2 nR e q σ2 φ 2 φ 2 r2 1 + 2 (6.32) ≡ ρ2  M ! M !  ∗ ∗   For more information on our treatment of these overlaps and normalizations, please see Appendix E.

The left-handed 1st and 2nd family fields are equal to ψa, which do not mix with any other fields. We replace all fields within W1 with their massless components.

This yields the X-dependent Yukawa matrices for the massless ﬁelds with left-handed

79 doublets on the left:

c n 2T3R 0 α0 LR ε0 L c b′ c Y = n 0 n 2T3R λ. (6.33)  α0 LR β0 b LR γ0 L  − 0 0 1     Deﬁnitions follow:13

XLζ L e 2 n (6.34) ≡ L XRζ R e 2 ne (6.35) ≡ R λ λ ρnen n (6.36) ≡ 3 L R H λα eA e α0 h i (6.37) ≡ λ3 M ∗ λβ φ2 a β0 D E (6.38) ≡ − λ3 M b0′ e ∗ 0 λγ A3R φ1 γ0 h iσ (6.39) ≡ −λ3 M M ∗ ∗ φ2 ε0 γ0 (6.40) ≡ φ1

The Yukawa matrices may be simpliﬁed by looking at the normalization constant nc. We assume that a b , and so we can approximate nc 1. This is not ≪ | | ∼ incompatible with our deﬁnition of β , as we also expect β 1. 0 0 ≪

An approximation may also be made within nL.

1 e nL (6.41) ≡ 2 2 1 ρ XLζ 1+ r n2 + η2 e e s L 1 (6.42) 2 1 ≃ 1+ r 2 nL q We assume that η > (1), and since ρ 1/40 for reasonable values of the compact- ∼ O ∼ 2 ρ XLζ 1 ification and cutoff scales derived from gauge coupling unification, η2 e n2 for ≪ L

13 We use the fact that for all Yukawa terms, XL + XR + XH = 0.

80 10 <ζ< 10, which easily encompasses the ζ-range which has a chance of ﬁtting − the data.

It is convenient to reparameterize r2 in terms of other variables. With the deﬁni-

tion

λ3 M ∗ κ 0 (6.43) ≡ λγ A3R !

r2 may be rewritten as

κ2 r2 = (γ2 + ε2). (6.44) ρ2 0 0

With this redeﬁnition the normalization constants nL and nR become

1 e e nX = . (6.45) 2 2 2 κ γ0 +ε0 1+ ρ2 n2 e r X

It is this deﬁnition for the nX which we will use throughout the rest of this work.

Under the approximatione of nc 1, the Yukawa matrices simplify to: ∼

1 2T3R 0 α0 LR ε0 L b′ Y = α 1 β 0 1 γ 2T3R λ. (6.46)  − 0 LR 0 b LR 0 L   0 0 1    These are the Yukawa matrices which we will analyze in the next section.14

The lowest order diagrams which contribute to the Yukawa matrices follow. For

each diagram we give the Yukawa element(s) to which it contributes.

14We could have included terms in our superpotential allowing (3,1) and (3,2) elements in the Yukawa matrices. However, as long as such matrix elements are hierarchical, they cannot aﬀect the theoretical values of the low energy observables. Such terms are also not necessary in our theory (e.g., we do not have a left-right symmetry which requires their presence) and so we have left them out of our superpotential.

81 H (3,3) c ψ3 ψ3

H A3R φa X (1,3) (2,3) c c c ψa ψ + ψ + ψ3 − a − a

c c H φa ω ω

e X (2,2) c c c ψa Na N a ψa

H A (1,2) (2,1) c ψa ψa

6.3 Analysis

We take two routes in the analysis of our model. In the ﬁrst subsection we use analytic methods to extract predictions from our theory. Relations between the Yukawa parameters and the observables are given but are not solved due to the complexity of the equations. A more precise numerical analysis, in which a full ﬁt is achieved, is then given in the following subsection.

6.3.1 Analytic Fitting

The starting point for our analysis is the Yukawa matrix of equation (6.46). In our model, this Yukawa matrix is defined at the compactification scale Mc. We have taken into account the effects of integrating out the heavy fields but have neglected the running effects on the Yukawa matrix elements between M (where the superpotential ∗

82 of equations (6.14) and (6.15) is deﬁned) and Mc. We assume that these running eﬀects are small enough to ignore at the level of a few percent.

To make the analysis easier, we deﬁne αu, αd, etc. below.

eXH ζ/2 e−XLζ/2 0 α0 ε02T3R nLnR nL 2 2 2 Y (X) =  eXH ζ/ β0 eXH ζ/ e−XLζ/  λ (6.47) α0 iθ1 iθ2 γ02T3R x nLnR ce +XRe nLnR nL  − e e e     0 0 1   e e e e e  0 αx εx  αx βx γx  λx (6.48) ≡ −0 0 1     After inputting the proper X quantum numbers: ζ 1 ζ 1 ζ 1 αu α0e− 2 αd α0e αe α0e ≡ (n1) ≡ n1n−3 ≡ n1n−3 1 ζ 1 1 ζ 1 1 ζ 1 β iθ iθ β e− 2 β iθ iθ β e β iθ iθ β e u ce 1 +e 2 0 (n1) d ce 1 3e 2 0 n1n 3 e ce 1 +e 2 0 n1n 3 ≡ ≡ − − ≡ − ζ/e2 1 ζ/e 2e1 3ζ/e2e1 γ γ e− γ γ e− γ γ e u 0 n1 d 0 n1 e 0 n 3 (6.49) ≡ − e ≡ e e ≡ − e e ζ/2 1 ζ/2 1 3ζ/2 1 ε ε e− ε ε e− ε ε e u 0 n1 d 0 n1 e 0 n 3 ≡ − e ≡ e ≡ e− ζ n−3 ζ n−3 λt λb = λte− λτ = λte− e ne1 ne1 and e e e e 1 nX = (6.50) 2 2 2 κ γ0 +ε0 1+ ρ2 n2 e r X There are eleven parameters associated with the fermion masses derived from the

Yukawa matrices: ζ, α0, β0, γ0, ε0, c, θ1, θ2, κ, λt, and tan β. Because there are 13

independent observables in the quark and charged lepton sectors: 9 masses, 3 quark

mixing angles, and the CP-violating phase in the CKM matrix, we can expect two

predictions in this model.

We assume that both of the quark Yukawa matrices are hierarchical [52]. As

explained within [51], or Chapter 5 of this work, this allows us to use a simple set of

rotations to diagonalize our quark mass matrices and leads to a simple CKM matrix: U U 1 s12∗ + s13∗s23 s12∗s23∗ + s13∗ D − U V = s s s∗ 1 s∗ + s s∗ , (6.51) CKM  − 12 − 13 23 23 12 13  sD s s s sD s 1  12 23 13 23 12∗ 13   − − −  83 where

U αu D αd D U s12 , s12 , s12 s12 s12, ≃ βu ≃ βd ≡ − U D D U s13 εu, s13 εd, s13 s13 s13, (6.52) ≃ ≃ ≡ − sU γ , sD γ , s sD sU . 23 ≃ u 23 ≃ d 23 ≡ 23 − 23 The eigenvalues of the diagonalized quark Yukawa matrices lead to the quark masses:

2 vu mc mu αu m λ β 2 t t √2 mt u mc βu ≃ ≃| | ≃ | | 2 (6.53) vd ms md αd m λ β 2 b b √2 m d ms β ≃ b ≃| | ≃ | d| We will take the charged lepton Yukawa matrix to be lopsided to some extent, with γ (1) and the rest of the matrix following a hierarchy.15 This choice leads e ∼ O to the charged lepton masses:

2 vd 2 mµ βe me αe 2 mτ λτ 1+ γe | |2 2 1+ γe (6.54) ≃ √2 mτ ≃ 1+γe mµ ≃ βe q | | q We want to use the kink mass, here parameterized by ζ, to localize the two Higgs wave functions to opposite branes. If Hu is localized towards the SO(10) brane and

Hd is localized towards the PS brane, then there is a natural reason why mt is larger than mb while mu is smaller than md. As can be seen from the αx (equation (6.49)),

which play a large part in determining the masses of the ﬁrst family fermions, if

ζ ζ ζ > 0 we have mu suppressed by e− and md enhanced by e . Similarly, the third

ζ n−3 family masses have the opposite dependence as shown in the relation: λ = λ e− . b t n1 Looking at γ , we see that ζ > 0 enhances γ while suppressing γ and γ , so it ise not x e u d e unreasonable to assume that γ 1 while γ ,γ 1. e ∼ u d ≪ With a few approximations within the CKM matrix the mixing angles and CP-

violating angle β are:

U αd αuβd Vus s12∗ + s13∗s23 s12 1 (6.55) | | ≃ | |≃| | ≃ β − β α d u u

15 The lopsided eﬀect with order one γe is crucial to achieve correct b-τ uniﬁcation [112].

84 U V s∗ + s s∗ s γ γ (6.56) | cb| ≃ | 23 12 13|≃| 23|≃| d − u|

U αu (γd γu) Vub s13∗ s12∗s23∗ εd εu 1 − (6.57) | | ≃ | − |≃| − | − β (ε ε ) u d u −

VcdV ∗ αuβd βd (εd εu) β arg cb arg 1 arg 1 − (6.58) ≡ − VtdV ∗ ! ≃ − βuαd ! − − αd (γd γu)! tb − With hierarchical structure in the up and down quark matrices it is the case that

(2,3)D (2,3)U Vcb . Here this is reﬂected in the relation Vcb γd γu . To have | | ∼ (3,3)D − (3,3)U | |≃| − |

a nonzero Vcb, we need to have some dependence on right-handed quantum numbers

within γx. We have found that in our setup, regardless of the placement of ﬁelds, the

kink mass eﬀects (which come with ζX) can only bring XL into the Yukawa element

(2,3) ratio (3,3) . For this reason we have added the field A3R to the superpotential, and it is the VEV of this field (proportional to T3R) which gives us the necessary difference between γd and γu to generate a nonzero Vcb.

1 The structure of the (2, 2) Yukawa elements, proportional to iθ iθ , helps ce 1 +XRe 2

us to ﬁt several observables. The XR dependence gives diﬀerent phases between the

up and down elements, both of which contribute to the CKM matrix CP-violating

phase β. The phase of β is also important in fitting the size of V . In addition, u | ub| the magnitude of these (2, 2) elements helps us to fit the first to second and second to third family mass ratios.

We will now show the predictions of this model in the form of relations between observables at the compactiﬁcation scale. Consider the fermion mass combinations

mτ and mu / me : mb mc mµ m λ τ (M ) τ 1+ γ2 = 1+ γ2 (6.59) m c ≃ λ e e b b q q 2 2 mu me αu βe 1 1 / (Mc) 2 | |2 = (6.60) mc mµ ! ≃ α βu 2 2 e | | 1+ γe 1+ γe q q 85 One prediction of this model is then

mu me mb / (Mc) (Mc). (6.61) mc mµ ! ≃ mτ Consider also the determinants of the down quark and charged lepton mass matrices.

3 3 vd 2 vd mdmsmb(Mc) = det Yd = αdλb (6.62) | | √2! √2! 3 3 vd 2 vd memµmτ (Mc) = det Ye = αeλτ (6.63) | | √2! √2!

Because αd = αe and λb = λτ , we have the (exact) prediction:

mdmsmb(Mc) = memµmτ (Mc) (6.64)

These two predictions hold regardless of the actual values taken on by the Yukawa

parameters.

As for the Yukawa parameter values themselves, these must be determined from

the eleven remaining independent observables, if possible. The dependence of these

observables on the parameters is complicated enough that we ﬁnd it impossible to

make a ﬁt nonnumerically. Actual ﬁt values for the Yukawa parameters must wait

until the numerical analysis in Section 6.3.2.

Our predictions are relations at the compactiﬁcation scale, while our data has been

taken at the weak scale. In order to determine the extent to which our predictions are

reasonable, we need to estimate the renormalization eﬀects on the masses between

the two scales. In this analysis, we choose to diagonalize the Yukawa matrices at the

compactiﬁcation scale, and use the RG formalism of Barger, Berger, and Ohmann

[113] to relate the observables at MZ (or at mt in the case of the top quark) to their

values at Mc with simple scaling relations.

diag Mu (Mc)=

86 Su(MZ , Mc)mu(MZ)0 0  0 Su(MZ , Mc)mc(MZ ) 0  (6.65) 0 0 S (m , M )m (m )  t t c t t    diag Md (Mc)=

Sd(MZ , Mc)md(MZ )0 0  0 Sd(MZ , Mc)ms(MZ ) 0  (6.66) 0 0 S (M , M )m (M )  b Z c b Z    diag Me (Mc)= Se(MZ , Mc)me(MZ )0 0  0 Se(MZ , Mc)mµ(MZ ) 0  (6.67) 0 0 S (M , M )m (M )  τ Z c τ Z    V 2(M )= | | c V 2(M ) V 2(M ) S(M , M ) V 2(M ) | ud| Z | us| Z Z c | ub| Z V 2(M ) V 2(M ) S(M , M ) V 2(M ) (6.68)  | cd| Z | cs| Z Z c | cb| Z  S(M , M ) V 2(M ) S(M , M ) V 2(M ) V 2(M )  Z c td Z Z c ts Z tb Z   | | | | | |  The scaling factors S can be found in [113]. In deriving these scaling factors, the authors have used 2-loop running and have included running effects from the gauge couplings and the third family Yukawa couplings. We make a further approximation in that we keep only 1-loop effects and neglect all running effects from the Yukawa sector except for the top Yukawa coupling. With these approximations

3 S (M , M ) y− (m , M )G (M , M ) (6.69) u Z c ≃ t t c u Z c 6 S (m , M ) y− (m , M )G (m , M ) (6.70) t t c ≃ t t c u t c S (M , M ) G (M , M ) (6.71) d Z c ≃ d Z c 1 S (M , M ) y− (m , M )G (M , M ) (6.72) b Z c ≃ t t c d Z c S (M , M ) G (M , M ) (6.73) e Z c ≃ e Z c S (M , M ) G (M , M ) (6.74) τ Z c ≃ e Z c S(M , M ) y2(m , M ) (6.75) Z c ≃ t t c

87 with

Mc 1 2 yt(mt, Mc) exp 2 λt (µ) d(ln µ) (6.76) ≡ "−16π Zmt # Mc 1 x 2 Gx(MZ , Mc) exp 2 ci gi (µ) d(ln µ) (6.77) ≡ "−16π MZ # Z Xi cx i α (M ) 2Bi = i Z (6.78) αi(Mc) ! Yi . The cx and B govern the MSSM 1-loop running eﬀects due to the gauge couplings:

13 16 cu = , 3, (6.79) 15 3 7 16 cd = , 3, (6.80) 15 3 9 ce = , 3, 0 (6.81) 5 33 B = , 1, 3 (6.82) 5 −

We estimate that the neglect of the running due to λb and λτ introduces 5% error in the down and charged lepton sectors and around 1% error in the up sector. We further estimate the neglect of 2-loop gauge running to be 1% error in G and G , ∼ u d and 4% error in G . ∼ e

Because we use MSSM running from Mc down to MZ , errors are introduced when we include particles in the running below their mass scales. These eﬀects can be taken into account by weak-scale threshold corrections. However, in our analysis we do not keep track of the supersymmetric particle masses and so these corrections cannot be calculated. We can, nevertheless, estimate these eﬀects. The SUSY thresholds for all of the down sector quarks were calculated in [114]. There are tan β-enhanced

diagrams which contribute to mb, ms and md. Although the Higgsino contribution to the d, s threshold corrections are Yukawa-suppressed, the dominant correction comes from gluino and Wino loop diagrams which aﬀect all three of these quarks equally

88 (up to some small diﬀerences due to unequal squark masses). We therefore make the

simplifying approximation that the SUSY threshold corrections for all of the down

quarks are the same. The same approximation can be made separately for the up and

charged lepton sectors where the diﬀerences in thresholds among the same particle

type arise only from squark mass diﬀerences, and are small.

By assuming some range of SUSY parameters, Pierce et. al. [115] have estimated the SUSY threshold corrections to the 3rd family masses. Assuming a value of tan β ∼

30 and µ> 0, we estimate from plots in [115] the SUSY threshold eﬀects (at mt for top and MZ for bottom and tau) in terms of % shifts and errors. The shifts for these particles, and by approximation the shifts for the 1st and 2nd family particles, are m , m , m (+2.0 3)% F 1.02 u c t ± t ≡ md, ms, mb ( 10.0 10)% Fb 0.90 (6.83) m , m , m −(+1.5 ± 2)% F ≡ 1.015. e µ τ ± τ ≡ The factors Fx will be used to keep track of these corrections.

SUSY threshold corrections to the CKM matrix elements were also calculated in

[114]. As those authors state, to a good approximation the only CKM elements which shift their sizes are V , V , V , and V , and these shifts are the opposite of the | ub| | cb| | td| | ts| chargino-induced shift in mb. In addition, J shifts approximately twice the amount of the CKM elements. The contribution from charginos to the mb shift is plotted separately in Pierce et. al. [115], and we estimate it to be 5 10%. For those CKM ± observables which have large SUSY threshold corrections:

V , V , V , V ( 5.0 10)% F 0.95 | ub| | cb| | td| | ts| − ± V ≡ (6.84) J ( 10.0 20)% F 0.90. − ± J ≡ As explained in [114], each side of the unitarity triangle has one element which has a large SUSY threshold. This leads to the threshold for J, and also implies that the angles of the triangle are unaﬀected. Thus sin 2β does not get a correction. We

89 assume that the correction to ε is approximately the same size as the correction | K| for J.

The mutual dependence of yt(mt, Mc) and λt means that there is no simple analytic

way of integrating to ﬁnd yt(mt, Mc). We choose, therefore, to ﬁt mt(mt) by adjusting

λt(Mc) and using the 1-loop equations with λt and gi only to run down to mt. The

1-loop RG equation is:

dλt λt 2 u 2 6λt ci gi (6.85) dt ≃ 16π2 " − # Xi with t ln µ where µ is the energy scale. The relation between the experimental ≡

value of mt(mt) and λt(mt): v v mt(mt) = λt(mt)Ft sin β λt(mt)Ft (6.86) √2 ≃ √2 We have already assumed tan β 30, which means that sin β 1, and we have ∼ ≃

included the thresholds for mt(mt) in the scale factor Ft.

The PS-breaking VEVs of χc and χc are of order the cutoﬀ scale. As in [99], we

choose to parameterize these VEVs with a dimensionless parameter ζbrane:

c 4M c ∗ χ = χ 2 (6.87) h i h i ≡ sπg5ζbrane

Naive dimensional analysis leads to a value for ζbrane:

ζbrane =0.27 (6.88)

Using this value and using the assumption of 5D gauge uniﬁcation as in [99] with the

4D GUT scale inputs

M = (2.5 0.5) 1016 GeV (6.89) G ± × α = 1/(24 1) (6.90) GUT ± ε = 0.035 0.005 (6.91) 3 − ± 90 leads to knowledge of M , Mc, and αi(Mc): ∗

M = 2.3 1017 GeV (6.92) ∗ × M = 2.4 1014 GeV (6.93) c ×

α1(Mc) = 0.035 (6.94)

α2(Mc) = 0.040 (6.95)

α3(Mc) = 0.044. (6.96)

The uncertainties in the 4D GUT scale parameters (especially α ) introduce 9% GUT ± errors into G and G and 2% error into G and y . The errors listed here on G u d ± e t u and Gd have a high correlation and can be mostly neglected when considering ratios.

Considering all the sources of errors discussed above, we choose to parameterize our theoretical errors as 9% G and G (with the errors highly correlated), 4.5% ± u d ± on G , and 2% on y . In addition, in our expressions, the masses themselves should e ± t have extra theoretical errors from both threshold corrections and the neglect of the

contribution to Yukawa running due to λb and λτ . We choose the up-type quarks to

have 3% error, down-type 11% error, and the charged leptons 5%. Each of these ± ± ± errors is correlated to some degree within each particle type.

Starting from the αi(Mc) and using 1-loop running, we ﬁnd the gauge coupling scale factors to be

G (M , M ) = 0.33 0.03 (6.97) u Z c ± G (M , M ) = 0.33 0.03 (6.98) d Z c ± G (M , M ) = 0.70 0.03. (6.99) e Z c ±

91 Fitting the central value of m (m ) = 169 4 GeV, allowing for the m 1σ range, and t t ± t including the 3% threshold error on the top quark leads to

λ (M ) = 0.59 0.12 (6.100) t c ± y (m , M ) = 0.903 0.018. (6.101) t t c ±

We have included the 2% theoretical error on yt.

We are now in a position to analyze our predictions. First, consider

mu me mb / (Mc) (Mc). (6.102) mc mµ ! ≃ mτ We choose to make a prediction for mu (M ). The prediction and corresponding mc Z experimental values are:

mu me mb Fb Gd 1 (MZ ) (MZ ) (6.103) mc th ≃ " mµ ! mτ Fτ Ge yt !# 0.0037 0.0006 (6.104) ≃ ± mu (MZ ) = 0.0023 0.0010 (6.105) mc exp ± Our prediction falls outside of the experimental bounds and leads to a 1.2σ discrepancy with the data. Our model favors a larger value for mu , indicating possibly a mc larger mu and/or smaller mc than currently measured.

Consider the second prediction:

mdmsmb(Mc) = memµmτ (Mc) (6.106)

After using the scale factors to get down to MZ , this becomes

3 3 Fτ Ge (mdmsmb)th(MZ ) (memµmτ ) 3 3 yt (MZ ) (6.107) ≃ " Fb Gd #

(10.7 5.0) 105 MeV3 (6.108) ≃ ± ×

(m m m ) (M ) = (5.2 2.6) 105 MeV3. (6.109) d s b exp Z ± × 92 The uncertainty of 50% in the predicted value comes mostly from the 9% errors in the gauge running scale factors Ge and Gd and the 11% error associated with the down-type masses. The experimental error is dominated by the uncertainty in ms

(and hence md). The discrepancy in the two values is about 1σ, and we predict a slightly larger scale for the down-type masses than measured.

6.3.2 Numerical Fitting

Within this section we apply numerical methods to fit our model to the data. By using automated techniques, we can find all of the Yukawa parameters which fit the data, including those parameters which were difficult to determine analytically. We can also easily include 2-loop gauge running and (1-loop) running due to the whole

Yukawa matrices, not just λt.

Our fitting procedure starts from the Yukawa matrix in equation (6.46) taken at the compactification scale. We determine the gauge couplings at that scale assuming unification and some usual values for the unification parameters as listed in equations

(6.89) to (6.91). Using 2-loop gauge and 1-loop Yukawa MSSM running, the Yukawa matrices are run from the compactification scale down to MZ and diagonalized to find the fermion masses and quark mixing angles. On the way to MZ , the top running mass mt(mt) and the running due to λt below mt (yt(MZ , mt)) are determined. The observables at MZ are shifted by SUSY threshold effects, listed in equations (6.83) and

(6.84). To counteract the inclusion of the incorrect top running between MZ and mt, we multiply the observables at MZ by appropriate powers of yt(MZ , mt). The correct powers are determined from the scale factors listed in the previous section equations

93 Experimental Theoretical Total Observable Error % Error % Error % Value Used Q 3.5 6.4 7.3 22.7 1.7 ± mc 23.1 3.6 23.4 0.73 0.17 GeV m (m ) 2.4 3.2 4.0 169 ±7 GeV t t ± ms 8.5 2.5 8.8 18.9 1.7 md ± ms 24.8 2.2 24.9 0.0199 0.0049 mb ± m 2.1 12.2 12.4 2.91 0.36 GeV b ± me 0.0003 2.0 2.0 0.004738 0.000095 mµ ± mµ 0.02 2.0 2.0 0.0588 0.0012 mτ ± m 0.02 2.0 2.0 1.747 0.035 GeV τ ± V 1.6 0.0 1.6 0.2240 0.0036 | us| ± Vcb 1.9 10.0 10.2 0.0415 0.0042 |V |/V 9.3 0.0 9.3 0.086 ±0.008 | ub cb| ± V 9.8 10.0 14.0 0.0082 0.0011 | td| ± sin 2β 6.5 0.0 6.5 0.739 0.048 ± J 105 10.0 20.0 22.4 3.0 0.7 ε× 0.7 26.6 26.6 0.00228± 0.00061 | K| ±

Table 6.4: Observables used in the χ2 analysis. The theoretical errors are combinations of estimates of weak threshold eﬀects, GUT parameter uncertainties, and in the case of εK an additional theoretical uncertainty from BK. All observables are at the MZ energy scale, except for the top mass, which is at mt.

(6.69) to (6.75). The observables are then compared to data in a χ2 function, which we minimize by altering the Yukawa input parameters.

The sources of error in this section are the following. First, as in the analytic section, the threshold effects on the observables at the weak scale due to the SUSY particle spectrum are not calculated. We estimate these effects in the same way as in the previous section in equations (6.83) and (6.84). Second, the uncertainties in our canonical 4D GUT parameters still introduce theoretical errors in our low energy data. By altering these inputs and studying their effects on the calculated observables

94 numerically, we estimate the theoretical errors on our observables from these eﬀects.

These estimates are in general a bit less than the errors assigned in the analytic section because some correlations are automatically included numerically which were not included before. Third, we have the usual experimental errors. We choose to combine errors in quadrature and assign a combined theoretical and experimental error to each observable. These observables and errors are listed in Table 6.4. Most theoretical errors cancel in the same-family mass ratios, up to a few percent. Hence, we have assigned a minimum 2% error on these ratios due to threshold eﬀects. This is especially important for the lepton ratios, whose experimental errors are negligible.

Taken from Martin and Vaughn [116], we use 2-loop gauge and 1-loop Yukawa renormalization group equations. As explained before, our boundary conditions are at M : Given a set of choices of the Yukawa parameters, we have the 3 3 complex c ×

Yukawa matrices at Mc; with the assumption of gauge uniﬁcation and with the 4D

GUT parameters (equations (6.89) to (6.91)), we have αi(Mc). Running due to the neutrino sector of the theory is neglected. We ﬁt the 11 Yukawa parameters to the

16 observables listed in Table 6.4 by minimizing a χ2 function N (Xi Xi )2 2 Calc − Exp χ = i 2 . (6.110) (σX ) Xi=1 Our fit to the data is in Table 6.5 and the corresponding Yukawa parameters for this fit can be found in Table 6.6. A true χ2 function assumes gaussian errors for its observables, something which we have implicitly assumed but which is not true for some of the observables used. Our χ2 is more of an indication of how good the fit is, and the minimization of the function is a method by which we can find a “best” set of parameters for the fit. The fit value for our χ2 is around 5, with the majority of the contribution coming from the down-type quark sector.

95 Observable Target Value Fit Value χ2 Contribution Q 22.7 1.7 23.5 0.21 ± mc 0.73 0.17GeV 0.59GeV 0.67 m (m ) 169 ±7GeV 167GeV 0.11 t t ± ms 18.9 1.7 17.0 1.28 md ± ms 0.0199 0.0049 0.0238 0.64 mb ± m 2.91 0.36GeV 2.46GeV 1.57 b ± me 0.004738 0.000095 0.004729 0.01 mµ ± mµ 0.0588 0.0012 0.0588 0.00 mτ ± m 1.747 0.035GeV 1.757GeV 0.08 τ ± V 0.2240 0.0036 0.2237 0.01 | us| ± Vcb 0.0415 0.0042 0.0412 0.00 |V |/V 0.086 ±0.008 0.090 0.21 | ub cb| ± V 0.0082 0.0011 0.0084 0.03 | td| ± sin 2β 0.739 0.048 0.720 0.15 ± J 105 3.0 0.7 3.0 0.00 ε× 0.00228± 0.00061 0.00210 0.09 | K| ± Total: 5.06

Table 6.5: Observables, target values, best ﬁt values, and χ2 contributions. All observables are at the MZ energy scale, except for the top mass, which is at mt.

Parameter Value ζ 2.152 α0 0.0002007 β0 0.003274 γ0 0.02187 ε0 0.0009816 c 1.492 θ1 5.090 θ2 2.718 κ 1.495 λt(Mc) 0.6057 tan β 25.42

Table 6.6: Minimum χ2 ﬁt Yukawa parameters.

96 The value for mu/mc in the numerical ﬁt (0.0040) is about 0.5σ away from the

value found in the analytic section (0.0037 0.0006), and is consistent with our ±

prediction of a larger mu and/or smaller mc than measured. In addition, the best ﬁt

value for m m m in the χ2 analysis (5.0 105 MeV3) is consistent with the value from d s b × the previous section ((10.7 5.2) 105 MeV3) (within 1.1σ) and with the measured ± × value ((5.2 2.6) 105 MeV). As for the free parameters, in the analytic section we fit ± × only one of these: λ (M )=0.59 0.12. The numerical fit value found here, 0.6057, t c ± falls well within the range of the analytic fit value. Thus, the results of the numerical

(Section 6.3.2) and analytic (Section 6.3.1) analyses are consistent.

6.4 Justiﬁcation in 6D

In this section we discuss a 6D version of our theory which naturally justiﬁes our

setup given in this paper. Our 5D analysis has assumed that the PS brane keeps its

symmetry even though χc and χc get their VEVs near the cutoﬀ scale and break the

PS symmetry entirely. As mentioned in Section 6.2, there are generic large corrections in the Kahler potentials of PS-brane ﬁelds ψ

c c χ †χ K = (1+ C )ψ†ψ. (6.111) M 2 ∗ This problem can be solved in a geometric way by the addition of a sixth dimension

c along which χ and ψ are separated. Even for a tiny sixth dimension, 1/R2 M /3, ∼ ∗ 3 PS breaking eﬀects are suppressed by e− which is enough suppression to keep our

Yukawa relations. All of the numerical analysis given in the paper remains the same

as long as the sixth dimension is not too large compared to the cutoﬀ scale.

There are two distinct ways of constructing 6D models. First, a 6D N=1 (4D

2 N=2) theory with T /Z2 gives four ﬁxed points (3+1 dimensional spacetime). All

97 the ﬁelds in the bulk in the 5D theory are now in 6D in the 6D theory, while all the

ﬁelds living on branes still live on branes. As we have one 10 dimensional hypermulti-

plet and two 16 dimensional hypermultiplets, we need one additional 16 dimensional

hypermultiplet in order to cancel the 6D irreducible gauge anomaly of SO(10) [117].

2 Second, a 6D N=1 theory with T /(Z Z′ ) also gives a 4D N=1 theory below the 2 × 2 5 5 6 compactiﬁcation scale. There are four ﬁxed lines along x = 0, x = πR1/2, x = 0,

6 and x = πR2/2 and four ﬁxed points at the corners. The gauge sector is extended

to 6D, but all of the other bulk ﬁelds can still be on 5D ﬁxed lines. This setup is

anomaly free without introducing additional states.

Gauge coupling uniﬁcation restricts the possible sizes of R1 and R2 in both

cases. However, we can easily recover the 5D theory used in this paper in the limit

1/R2 M . Note, as long as 1/R2 is close to, but somewhat less than, M we simul- → ∗ ∗ taneously have gauge coupling uniﬁcation (as described in the 5D formulation) and

PS symmetry relations for Yukawa couplings.

2 We choose to focus on a 6D N=1 theory with T /Z2. Such a theory has been

studied by Asaka, Buchmuller and Covi [75] where the SO(10) gauge group in the

bulk is broken down to SM U(1) by two diﬀerent Wilson lines. One breaks × X

SO(10) down to the Pati-Salam gauge group along x5, and the other breaks SO(10)

down to SU(5) U(1) along x . Therefore, we end up with four ﬁxed points; × X 6 πR1 SO(10) brane at (x5, x6)=(0, 0), Pati-Salam brane at ( 2 , 0), Georgi-Glashow brane

πR2 πR1 πR2 (SU(5) U(1) ) at (0, ) and ﬂipped SU(5) brane (SU(5)′ U(1)′ ) at ( , ). × X 2 × X 2 2 We have provided a schematic of the extra-dimensional space in Figure 6.1. The 5D

setup we have considered so far can be easily lifted up to this conﬁguration. The

procedure is the following.

98 SU(5) U(1)X SU(5)′ U(1)X′

i × × 2 2 πR , 0 h ∈ 6 x SO(10) PS

x 0, πR1 5 ∈ 2 h i

Figure 6.1: A representation of the 6D orbifold space. The orbifold ﬁxed points are at the corners of the space. At each corner a diﬀerent subgroup of SO(10) is preserved. πR1 x5 runs along the horizontal direction from 0 to 2 , while x6 runs along the vertical πR2 direction from 0 to 2 .

5D bulk states are extended to 6D bulk states. Extra hypermultiplets are • introduced to cancel the 6D anomaly.

Fields on the 5D SO(10) brane are located on the 6D SO(10) brane. •

Fields on the 5D Pati-Salam brane are located on the 6D Pati-Salam brane • except χc and χc. We no longer need χc and χc as the gauge group is broken

down to SU(3) SU(2) U(1) U(1) by the two Wilson lines. U(1) C × L × Y × X X c broken near the compactiﬁcation scale by 16kink and χ kink, the ﬁelds which

generate the nonzero kink mass.

The 6D model is simple and economical. The additional states introduced to cancel the 6D anomaly are hypermultiplets and can become heavy by themselves. In

99 (+ ) − PS SM U(1)X ×

SM U(1) × X ( +) ( ) − (++) −− SU(5) SU(5)′ U(1)′ × X SM SM U(1)X ×

SO(10)

Figure 6.2: A representation of the SO(10) group space. The space has been broken into four subspaces by the orbifold symmetry, under which each subspace has a diﬀerent parity.

addition they do not affect the differential running of the three gauge couplings. Let us focus on gauge coupling unification. The spectrum of massive Kaluza-Klein vector multiplets are given by

2n 2 2m 2 M++ = ( ) +( ) (SM U(1)X) (6.112) s R1 R2 ×

2n 2 2m +1 2 PS M+ = ( ) +( ) (6.113) − s R1 R2 SM U(1)X ! × 2n +1 2 2m 2 SU(5) M + = ( ) +( ) (6.114) − s R1 R2 SM !

2n +1 2m +1 SU(5)′ U(1)′ M = ( )2 +( )2 × X . (6.115) −− s R1 R2 SM U(1)X ! ×

100 The breakdown of SO(10) into these subgroups is illustrated in Figure 6.2. If we send

R 0, we recover a 5D theory in which SU(5) U(1) is broken to the SM U(1) . 2 → × X × X The R 0 limit gives the same result as in Hall and Nomura [69] (and also in Kim 2 → 14 17 and Raby [99]). Thus we ﬁx Mc 1/R1 at around 10 GeV with M 10 GeV. If ∼ ∗ ∼

R2M 2 or 3, we get a tiny correction from extra + states and the result would ∗ ∼ − 1 R2 16 be proportional to (bPS bSM) ln( ) which is neglegibly small. The elongated 3 − M∗ 6D rectangular conﬁguration gives a perfect setup for the construction of a realistic

SO(10) model.

The only remaining question is the applicability of bulk ﬁeld localization in 6D which was possible with a kink mass in 5D. Though the quantitative results of 6D localization are diﬀerent from the 5D case, all of the qualitative aspects remain the same [118]. More interestingly, 4D N=1 supersymmetry is preserved.

6.5 Pati-Salam Brane Gauge Anomaly

This model suffers from a gauge anomaly localized on the Pati-Salam brane, the calculation of which is done later in this work in Chapter 7. The anomaly is due to the bulk fields, and was not then known to the authors of the paper in which the theory was originally proposed [18]. We have chosen to present the theory here as it was put forward in that paper since the exact alterations necessary to fix the theory have not yet been discovered.

The presence of a gauge anomaly implies that the theory is inconsistent and must be altered in some way or abandoned. The simplest way to ﬁx the theory is to alter the ﬁeld content in order to cancel the anomaly. Possible alterations include the

16The notation used is deﬁned in Kim and Raby [99].

101 addition of another pair of bulk 16s to the theory with opposite Z2′ parity to the pair

of 16s already present, the addition of another Pati-Salam brane ﬁeld with charge

(4, 2, 1) or (4, 1, 2), or the removal of one of the Pati-Salam brane ﬁelds transforming

as (4, 1, 2). These are only a few of the many possible ways the ﬁeld content could be

changed to cancel the anomaly.

The most diﬃcult aspect of altering the theory is that the ﬁelds must be added

or subtracted in such a way that we do not generate more massless ﬁelds and we

retain the Yukawa matrix structure of the low energy theory given in equation (6.46).

However the ﬁelds are arranged in the orbifold, if the theory results in this Yukawa

matrix structure, the subsequent analysis in Section 6.3 will be unaﬀected. Such

changes to the theory have not been investigated, and are left to future work.

6.6 Summary and Discussion

Extra dimensions provide a nice framework for understanding how grand uniﬁed

theories may be realized in nature. In this paper we have constructed a 5D SO(10)

model which accommodates the quark and charged lepton masses and the CKM

matrix. The model uses 11 parameters to ﬁt the 13 independent observables of

the quark and charged lepton Yukawa sectors, allowing us to make two predictions:

mu 5 3 (MZ )=0.0037 0.0006 and mdmsmb(MZ ) = (10.7 5.0) 10 MeV , both of mc ± ± × which are roughly 1σ larger than the experimental values. The kink mass localizes the

bulk Higgs ﬁelds according to their U(1)X quantum numbers, giving some explanation

for the hierarchy m m and m > m . Our 5D SO(10) model can be considered b ≪ t d u to be an eﬀective theory coming from 6D SO(10) with one small and one large extra

dimension. If the size of the sixth dimension is very small (i.e. if the inverse of its

102 characteristic length is near the cutoﬀ scale) then all of the calculations done here for

the 5D model can be regarded as good approximations for the 6D case.

Unfortunately, this theory is inconsistent as it has been presented because the

ﬁelds of the theory lead to a gauge anomaly on the Pati-Salam brane. We have

described the nature of the alterations necessary to ﬁx the theory in Section 6.5,

and more discussion and calculations will follow when orbifold gauge anomalies are

investigated in Chapter 7 and applied to this model in Section 7.4.2.

In a theory in which there are additional (heavy) vector-like states, commonly called Froggatt-Nielsen (FN) fields, which have the same quantum numbers as the light states, the light and heavy states can mix. The extra dimension naturally gives us a chance to unify Froggatt-Nielsen fields with ordinary matter fields if both come from the same hypermultiplet in higher dimensions. In our model, however, we chose to place the 1st and 3rd family fields on opposite branes to easily take advantage of the effects of the kink mass, and so our model does not achieve the unification of the massless and FN states into bulk fields. Nevertheless, it may be possible to place all of the matter fields in the bulk and still restrict the Yukawa terms of those fields to opposite branes to gain the desired hierarchy from the kink mass. We leave this possibility for further research.

Further work is needed in seven main areas in order to make our model complete.

First, it is necessary to find the alterations to the field content of the theory which would cancel the Pati-Salam brane anomaly while retaining the same low energy theory. If new fields are added, this would involve ensuring that the new fields lead neither to significant changes within the charged fermion Yukawa matrices nor to extra massless degrees of freedom. Second, the neutrino sector should be included to

103 explain neutrino oscillation experiments. We have already ﬁxed the neutrino Dirac

masses in terms of the Yukawa matrix between the left and right-handed neutrinos,

but the heavy Majorana masses of the right-handed neutrinos have not yet been

determined. It would be very interesting to expand the model to include Majorana

masses and to investigate the resulting neutrino masses and mixings. Third, the weak

scale supersymmetry breaking mechanism should be speciﬁed. As there is as yet no

such universally accepted mechanism, we chose not to specify how SUSY is broken

in our model. As a consequence, we could not calculate the electroweak threshold

corrections resulting from superparticle spectrum. Extra dimensions provide new

interesting channels for the understanding of supersymmetry breaking and its medi-

ation and can give entirely diﬀerent superparticle spectra. We leave the weak scale

supersymmetry breaking physics to future work. Fourth, our analysis is somewhat

incomplete since we have taken values for αGUT, MG and ε3 from other sources. Hence our treatment of gauge and Yukawa coupling RG running is not completely self consistent. We have however accounted for this shortcoming by including a theoretical uncertainty obtained by varying the gauge coupling parameters at MG. While a more complete unification treatment would be preferable, this would require knowledge of supersymmetry breaking and the sparticle spectrum. Fifth, we have neglected the effects of running on the Yukawa matrices between the compactification and cutoff scales. Our assumption is that such effects would contribute at the level of a few percent. Further research is necessary to investigate these effects. Sixth, the UV completion of the higher dimensional gauge theory could give us a better understanding of the model. String theory does not allow arbitrary matter configurations and the

104 constraints coming from it are usually stronger than those from field theory. There- fore, it would be interesting to see if the 5D model considered here can be derived as an effective field theory from a string theoretic starting point [44, 45]. Finally, we have assumed that many fields in our theory obtain vacuum expectation values at certain scales and in particular directions in group space. A complete theory would need to justify these VEVs by the minimization of the appropriate potentials. Such work would be most easily done after finding a successful UV completion for the theory, as many of these VEVs are of the order the cutoff energy scale.

105 CHAPTER 7

ORBIFOLD ANOMALIES

7.1 Introduction and Motivation

Given a symmetry of a classical theory, quantization of that theory can lead to aﬀects which can break the symmetry. Such a symmetry, one that is conserved classically but broken quantum mechanically, is called an anomalous symmetry. In addition, it is said that the symmetry ‘has an anomaly’.

The present chapter is devoted to gauge anomalies, anomalies which can break the gauge symmetry of a theory. Such anomalies were ﬁrst studied by Adler [119] and Bell and Jackiw [120]. Broken gauge symmetries lead to divergent masses and unphysical states for the gauge bosons, violations of the Ward identities of the gauge theory, and non-renormalization of the theory [121]. The gauge anomalies must cancel for the theory to be consistent. The requirement of gauge anomaly cancellation in general leads to restrictions on the number and type of charged fermions present in the quantized gauge theory.

There have been many theories proposed lately which have additional dimensions, and some of these have extra dimensions which are orbifold compactiﬁcations [18, 46,

47, 74, 75, 69, 67, 68, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,

106 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98]. Orbifolds are compactiﬁcations in which

the extra dimensional space can be considered a smooth manifold modded out by

some discrete group. In general this type of compactiﬁcation leads to special lower-

dimensional subspaces (called ﬁxed-points or branes) on which there can be localized

ﬁelds and gauge anomalies.

Extra dimensions in general, and orbifold dimensions in particular, lead to more complicated anomaly structure than in the usual 4D Minkowski space. The work presented in this chapter, based on the Fujikawa method of anomaly calculation [122, 123] and on the orbifold anomaly calculations done in [117] and [124], builds up a notation from the simplest 4D chiral anomalies to more complicated extra-dimensional orbifold anomalies. The purpose of this work is to understand extra-dimensional orbifold gauge anomalies in general, to test this understanding by applying the anomaly equations derived here to a particular orbifold theory, and if possible to find constraints on the fermionic field content of orbifold field theory models.

This work, done by the author with guidance from Stuart Raby, did not lead to

publishable results. However, its inclusion here should prove useful for future students

in building and testing extra dimensional orbifold models.

Section 7.2 gives background information explaining the Fujikawa method of cal-

culating anomalies, based on path-integral quantization of the theory, for global chiral

transformations, for gauge theories, and in the presence of smooth extra dimensions.

In Section 7.3, we present orbifolds in general and build on the work of Section 7.2

to ﬁnd general equations for the gauge anomalies in the presence of orbifold extra

dimensions. Section 7.4 is devoted to the application of the anomaly equations to a

particular theory of Kobayashi, Raby, and Zhang [45] and to the 5D orbifold model

107 presented in Chapter 6. The former theory is known to be anomaly free since it is

derived from an anomaly free string theory, but in applying the equations derived in

Section 7.3 to this particular theory, we can illustrate some of the important aspects

of orbifold anomalies. The latter theory of Chapter 6 is shown to have a ﬁxed point

gauge anomaly. These issues are then discussed in Section 7.5 along with general

thoughts on orbifold anomalies.

Three of the Appendices will be useful for this chapter. Appendix A gives general deﬁnitions and conventions for the Dirac spinor space in d dimensions, and gives the choices for the Dirac gamma matrices and conventions in 4 and 6 dimensions. Ap- pendix G contains a short calculation of spatial integrals necessary for calculations of path integral anomalies. Finally, Appendix H explains how gauge group traces, always present in anomaly expressions, can be calculated using the fundamental Casimir operators and fundamental indices of the given gauge group.

7.2 Anomaly Background

In this Section, we will discuss the Fujikawa method for calculating anomalies in path integral quantized theories, and apply it in turn to global chiral symmetries and gauge symmetries, and then gauge symmetries in the presence of smooth extra dimensions. The work done in this section will be used as a base for deriving anomaly equations for orbifold theories in the next section.

108 7.2.1 Fujikawa Method

Let S and be the classical action and Lagrangian density of a generic Dirac L

fermion ﬁeld ψ coupled to a gauge ﬁeld Aµ:

4 4 µ 1 µν S d x d x ψiγ Dµψ TrFµν F (7.1) ≡ Z L ≡ Z − 2 with D ∂ + igA . The global chiral transformation µ ≡ µ µ

iαγ5 ψ ψ′ e ψ (7.2) → ≡

0 iαγ5 0 iαγ5 ψ (ψ′)† γ = ψ†e− γ = ψe (7.3) → is a symmetry of the Lagrangian, and by Noether’s theorem there is a conserved current. Applying Noether’s method for ﬁnding the conserved current of a symmetry, we allow the parameter of the transformation to be space-time dependent:

ψ eiα(x)γ5 ψ (7.4) → ψ ψeiα(x)γ5 (7.5) →

The Lagrangian is no longer invariant under this transform:

(∂ α(x))ψγµγ ψ (7.6) L → L− µ 5 = + α(x)∂ ψγµγ ψ +boundaryterm (7.7) L µ 5 Let jµ ψγµγ ψ. The action S should be independent of α(x), and so 5 ≡ 5

δS δ 4 µ 0 = = d x ( + α(x)∂µj5 + boundary term) (7.8) δα δα Z L µ = ∂µj5 (7.9)

assuming that the boundary term vanishes. So we have our current conservation

µ ∂µj5 = 0.

109 Quantization of this theory leads to the destruction of the classical chiral symme-

try, and hence the symmetry is anomalous. A common method of determining the

anomalous eﬀects is to calculate 1-loop Feynman diagrams. For this work, however,

a more useful way of calculating the anomaly eﬀects is through the path integral

formalism. Such calculations go by the name of the Fujikawa method of calculating

anomalies, so named after the pioneer of this technique [122, 123]. Using path inte-

grals allows us to make a more general calculation, one which can apply to diﬀerent

numbers of dimensions with diﬀerent topologies.

Let us quantize the above theory using path integral formalism:

i d4x Z [DA ] DψDψe L (7.10) ≡ M Z R

[DAM ] contains the Faddeev-Popov factor whenever necessary. Perform the inﬁnites- imal version of the space-time dependent chiral transform considered above

ψ (1 + iα(x)γ )ψ (7.11) → 5 ψ ψ(1 + iα(x)γ ). (7.12) → 5

From the point of view of the path integral, such a transformation is simply a shift

of the integrated ﬁelds ψ and ψ, and so Z should be unchanged under this transformation.

The Lagrangian transforms as above

+ α(x)∂ ψγµγ ψ + boundary term. (7.13) L→L µ 5 The fermionic measure DψDψ changes non-trivially under this transformation, and provides the anomalous term which destroys this chiral symmetry.

110 Under the chiral transformation described in equations (7.11) and (7.12), the

fermionic measure changes as follows:17

1 1 DψDψ [det (1 + iαγ )]− Dψ [det (1 + iαγ )]− Dψ (7.14) → 5 5 2 = [det (1 + iαγ5)]− DψDψ (7.15)

We must convert this determinant factor into an exponential so that we can relate

it directly to the Lagrangian in the path integral. Using ln det A = Tr ln A and ln(1 + A) A (for small A), we have ≃

2 [det (1 + iαγ )]− = exp 2tr ln(1+ iαγ ) (7.16) 5 − 5 exp 2itr [αγ ] (7.17) ≃ − 5 exp iA (7.18) ≡ where we will call A the anomaly.18

A 2tr[αγ ] (7.19) ≡ − 5 The trace includes a sum over Dirac, gauge, and internal indices as well as a diagonal integration over all space-time. Separating out the spatial integral explicitly, we have

4 (D,G) A = 2 d xα(x) x tr [γ5] x (7.20) − Z h | | i where the trace is explicity over Dirac space and the gauge group generator space of group G. With this information, our measure transforms as

Dψ Dψ Dψ Dψ exp iA (7.21) → 4 (D,G) Dψ Dψ exp 2i d x α(x) x tr [γ5] x (7.22) ≃ − Z h | | i 17To produce these determinants, we can expand the fermionic degrees of freedom in terms of left- and right-eigenmodes of (iD/ ), and then perform the Grassman integrals. For more information, see the text by Peskin [1] or the original papers by Fujikawa [122, 123]. 18 Note that a diﬀerent quantity, namely x tr [γ5] x and expressions like it, is often termed the ‘anomaly’ elsewhere in the literature. h | | i

111 and the full transformation of the path integral is

i d4x Z [DA ] DψDψe L (7.23) ≡ M Z R 4 µ [DAM ] DψDψ exp i d x ( + α(x)∂µj5 (x)) + iA (7.24) → Z Z L i d4x 4 µ Z + [DA ] DψDψe L i d xα(x)∂ j (x)+ iA (7.25) ≃ M µ 5 Z R Z i d4x 4 = Z + [DA ] DψDψe Li d xα(x) M × Z R Z ∂ jµ(x) 2 x tr(D,G) [γ ] x . (7.26) µ 5 − h | 5 | i h i As stated previously, the path integral does not change under the transformation, and so the α(x) term must vanish. This leads to the anomalous non-conservation

equation

∂ jµ(x)=2 x tr(D,G) [γ ] x . (7.27) µ 5 h | 5 | i The chiral symmetry has been broken as promised by quantum eﬀects, calculable by

the path integral method as shown.

We need to know how to calculate x tr(D,G) [γ ] x in order to ﬁnd A. Naively, h | 5 | i

this appears to be zero, since the Dirac trace of γ5 is zero, but the expression needs

to be regulated. We can do this by using the operator (iD/ ):

(D,G) (D,G) (iD/ )2/M 2 x tr [γ5] x = lim x tr γ5e x (7.28) h | | i M h | | i →∞ h i The operator (iD/ )2 can be re-expressed:

(iD/ )2 = γµγνD D (7.29) − µ ν 1 = [ γµ,γν + [γµ,γν]] D D (7.30) −2 { } µ ν 1 = [2ηµν + [γµ,γν]] D D (7.31) −2 µ ν 1 = D2 γµγν [D ,D ] (7.32) − − 2 µ ν ig = D2 γµγνF (7.33) − − 2 µν 112 where we’ve used [Dµ,Dν]= igFµν . Our expression becomes

2 ig µ ν 2 (D,G) (D,G) ( D 2 γ γ Fµν )/M x tr [γ5] x = lim x tr γ5e − − x . (7.34) M h | | i →∞ h | | i Because we are taking a large M limit, we can concentrate on the large momentum part of the spectrum and can expand in powers of the gauge ﬁeld. The leading term

µ is one in which we have four γ factors to give a non-zero trace with γ5. Neglecting all other terms in the gauge ﬁeld leads to

(D,G) x tr [γ5] x h | | i ≃ 2 ∂2/M 2 (D,G) 1 ig µ ν lim x e− x tr γ5 γ γ Fµν(x) (7.35) M 2 ≃ →∞ h | | i " 2! −2M # 2 g ∂2/M 2 (D) α β µ ν (G) = lim − x e− x tr γ5γ γ γ γ tr [FαβFµν(x)] . (7.36) M 8M 4 ! h | | i →∞ h i αβµν ∂2/M 2 The Dirac trace is 4iε from equation (A.17) and the term x e− x can − h | | i iM 4 be evaluated to be (4π)2 using a Wick rotation to Euclidean space as in Appendix G equation (G.13). This leads to the ﬁnal form for x tr(D,G) [γ ] x : h | 5 | i 2 4 (D,G) g iM αβµν (G) x tr [γ5] x = lim − ( 4i)ε tr [FαβFµν (x)] (7.37) M 4 2 h | | i →∞ 8M ! (4π) − g2 = εαβµνtr(G) [F F (x)] (7.38) −32π2 αβ µν

The anomalous non-conservation equation is then

g2 ∂ jµ = εαβµνtr(G) [F F (x)] . (7.39) µ 5 −16π2 αβ µν

This is the ABJ anomaly, so-named for Adler, Bell, and Jackiw, who ﬁrst studied

this type of anomaly in [119] and [120].

7.2.2 Gauge Theories

The fact that a global chiral symmetry is anomalous is not a big problem. However,

it is possible for chiral gauge theories to have anomalies, and this is a big problem.

113 An anomaly in a gauge theory essentially implies that the gauge symmetry is not

a good symmetry of the theory. The gauge bosons are no longer protected by the

gauge symmetry and can attain mass. Therefore, if a ﬁeld theory is a gauge theory,

the anomalies of the gauge theory should cancel in some fashion.

Let us consider a theory similar to the one in the previous section. Let us have

a non-abelian gauge theory in 4 space-time dimensions with gauge group G and a

single fermion ψ(x) in representation ρ. Let the gauge theory transform of the ﬁeld

ψ be

ψ(x) eiθ(x)ψ(x) (1 + iθ(x)) ψ(x) (7.40) → ≃ iθ(x) ψ(x) ψ(x)e− ψ(x)(1 iθ(x)) (7.41) → ≃ − where θ(x) θa(x)T a. If ψ(x) is a Dirac fermion, then this transformation is non- ≡ chiral and the anomaly automatically cancels. Instead, let us have in our fermionic

0 sector a single left-handed Weyl fermion P ψ(x). The ﬁelds ψ and ψ (P ψ)† γ = L L L ≡ L

ψPR transform as

ψ (1 + iθ(x)) P ψ(x)=(1 iθ(x)γ ) ψ (x) (7.42) L → L − 5 L ψ ψP (1 iθ(x)) = ψ (1 iθ(x)γ ) . (7.43) L → R − L − 5

This transformation is similar to that studied in the previous section in that both ψL and ψL transform with the same sign in the γ5 term, and so the anomalies from each will add.

The fermionic Lagrangian of our theory is

ψ iD/ψ . (7.44) L ≡ L L If we want to ﬁnd the conservation equations for the gauge current, we must allow

the fermions to transform without transforming the gauge ﬁelds. We then have, after

114 some manipulation:

+ θa(x)Dacjµc(x)+boundaryterm (7.45) L→L µ

where Dac δac∂ +igAb (T b)ac is the covariant derivative acting on the adjoint repre- µ ≡ µ µ sentation gauge current jµc ψ γµT cψ . Classically then we have the conservation ≡ L L ac µc equation Dµ j (x)=0.

To ﬁnd the anomaly, we must again look to the transformation of the fermionic

measure in the path integral. Because we have the ﬁelds ψ P ψ and ψ ψP , L ≡ L L ≡ R the determinants (and later the traces) must be over only the physical modes. Hence

we need to add the appropriate projectors to the determinants to pick out only those

modes:

1 1 DψDψ [det P (1 iθ)]− Dψ [det P (1 + iθ)]− Dψ (7.46) → R − L 1 = [det P (1 iθ) det P (1 + iθ)]− DψDψ (7.47) R − L

1 The manipulations on the [det . . .]− term are the same as done previously in section

7.2.1

1 [det P (1 iθ) det P (1 + iθ)]− exp itr [θ (P P )] (7.48) R − L ≃ R − L

= exp itr [θγ5] (7.49)

exp iA (7.50) ≡ where we have deﬁned A

A tr [θγ ] (7.51) ≡ 5 4 a (D,G) a = d x θ (x) x tr [T γ5] x . (7.52) Z h | | i

115 Combining the transformation of the measure with the transformation of the La-

grangian and requiring that the path integral be invariant leads to the anomalous

non-conservation equation of the gauge current.

Dacjµc(x)= x tr(D,G) [T aγ ] x (7.53) µ −h | 5 | i

The calculation of x tr(D,G) [T aγ ] x here is identical to that done in section 7.2.1, h | 5 | i except that here we have a gauge group generator T a which comes along for the ride.

g2 x tr(D,G) [T aγ ] x = εαβµνtr(G) [T aF F (x)] (7.54) h | 5 | i −32π2 αβ µν

and the non-conservation equation becomes

g2 Dacjµc(x)= εαβµν tr(G) [T aF F (x)] . (7.55) µ 32π2 αβ µν

It will be useful later to consider A rather than x tr(D,G) [T aγ ] x . Including the h | 5 | i prior calculations, the anomaly A becomes

2 g 4 a αβµν (G) a A = 2 d xθ (x)ε tr [T FαβFµν (x)] . (7.56) −32π Z

a αβµν (G) a The expression θ (x)ε tr [T FαβFµν (x)] can be simpliﬁed by some manipulation

and by some deﬁnitions, which are described below the equations.

a αβµν (G) a αβµν a b1 b2 (G) a b1 b2 θ (x)ε tr [T FαβFµν(x)] = ε θ FαβFµν (x)tr T T T (7.57) h i = cab1b2 (x)tr(G) T aT b1 T b2 (7.58) h i = dab1b2 (x)tr(G) T aT b1 T b2 (7.59) h i L(G)(ρ, x) (7.60) ≡ 3

In the first line, we have used the definition F (x) F b1 (x)T b1 to take the wave- αβ ≡ αβ function part of the field strengths out of the group trace. In the second line, we

116 have made the deﬁnition cab1b2 (x) εαβµν θaF b1 F b2 (x). The anti-symmetry property ≡ αβ µν of ε implies that cab1b2 is symmetric in its last two indices. This symmetry, along

with the cyclicity of the trace, is enough to completely symmetrize the indices a,

b1, and b2. We have made use of this in the third line, along with another deﬁni-

tion dab1b2 (x) 1 cab1b2 (x)+ cb1ab2 (x)+ cb2b1a(x) . In the last line, we have used the ≡ 3 h i deﬁnition of an ‘index’ of order 3 in the representation ρ of group G:

L(G)(ρ, x) dab1b2 (x)tr(G) T aT b1 T b2 (7.61) 3 ≡ h i

(G) Indices Ln (ρ) and their calculation are described in Appendix H. With this information, we have for our anomaly A

2 g 4 (G) A = 2 d xL3 (ρ, x). (7.62) −32π Z

In general, each fermion ﬁeld in a gauge theory will contribute to A. Unless all

of these contributions cancel, the anomaly ruins the gauge symmetry. Therefore, for

a theory to be a gauge theory, we must require that A = 0 after summing over all

contributions.

7.2.3 Extra Dimensions

The main advantage of using a path integral calculation for anomalies is it easily

generalizes to any number of dimensions. When the number of dimensions is odd,

the anomaly automatically vanishes for each fermion ﬁeld because the fermions are

always non-chiral. The case of even number of space-time dimensions d is more interesting. Let us consider this case, let the gamma matrices be ΓM and the chiral gamma matrix be Γd+1. Let us have a Weyl fermion Pdβψ where β is the d-dimensional

chirality of the fermion and the projector is P 1 (1 + βΓ ). Let our fermion dβ ≡ 2 d+1 117 be in representation ρ of the non-abelian gauge group G. Paralleling the previous section, under the gauge transformation

P ψ(x) (1 + iθ(x)) P ψ(x) (7.63) dβ → dβ

ψPd( β)(x) ψ(x)Pd( β) (1 iθ(x)) (7.64) − → − − we have a gauge anomaly A of

A βtr [θΓ ] (7.65) ≡ − d+1

and an anomalous non-conservation equation

DacjMc(x)= β x Tr(D,G) [T aΓ ] x (7.66) M h | d+1 | i

for the gauge current jM a ψΓM T cP ψ. The same sort of calculation as found in ≡ dβ section 7.2.1 on A leads to

d a (D,G) a A = β d xθ (x) x tr [T Γd+1] x (7.67) − Z h | | i iφ d ie− d ig 2 = β − ddxθa(x)εM1N1...Md/2Nd/2 − d 4π × 2 ! Z (G) a tr T FM1M2 ...FMd/2Nd/2 (x) . (7.68) h i

∂2/M 2 iM d We have used the fact that in d dimensions x e− x = 2 from Appendix h | | i 2dπd/

M1 Md iφd d/2 M1...Md G equation (G.12) and Tr Γd+1Γ . . . Γ = e− 2 ε from equation (A.10) h i in Appendix A. Deﬁning quantities analogous to the previous section

b 2 cab1...bd/2 (x) εM1N1...Md/2Nd/2 θa(x)F b1 (x) ...F d/ (x) (7.69) ≡ M1N1 Md/2Nd/2 1 dab1...bd/2 (x) cab1...bd/2 + cb1ab2...bd/2 + . . . (7.70) ≡ d 2 +1 h i

(G) ab1...bd/2 (G) a b1 bd/2 L d +1(ρ, x) d (x)tr T T ...T (7.71) 2 ≡ h i 118 allows us to write A in a compact form

d iφd 2 ie− ig d (G) A = β − d xL d (ρ, x). (7.72) − d 4π 2 +1 2 ! Z 7.3 Orbifold Anomalies

In theories with orbifolds, not only are there extra dimensions, but those extra

dimensions are compactiﬁed in such a way that they have ﬁxed points. Depending

on the orbifold structure and the ﬁeld content of the theory, anomalies can arise both

on the ﬁxed points and in the bulk of the extra dimensions. If the theory is a gauge

theory, all such anomalies must separately cancel in order that the gauge symmetry

is everywhere a good symmetry.

For this section, we will follow the work done and notation used in a paper by von

Gersdorﬀ and Quiros [124].

7.3.1 Orbifolds

Let us consider a gauge theory with gauge group G in d space-time dimensions.

LetC be a d 4 dimensional compact manifold, and let G be a discrete group of order − N by which C should be modded out. Our space is thene M4 C/G. Parameterize × 4 M µ 5 d µ 1 2 d 4 µ ˆi M C by coordinates x = x , x ,...,x = x ,y ,y ,...,y − e = x ,y . × The action of G on the space C is to identify space-time positions x such that

e x P x (7.73) ∼ k where the P are a representation of the group elements k G. k ∈ By deﬁnition, an orbifold is a compactiﬁcation which hase points which are un-

changed under the action of non-trivial elements of the group G. Therefore, given

e 119 k G, there are points y such that P y = y , and these points are called fixed ∈ kf k kf kf e points. ykf will signify a fixed point that is specifically fixed under some k, while yf will signify a generic fixed point without specifying k.

The representation of the action of the group G on a generic ﬁeld of spin σ is given by e

1 φ(x) a λ P φ P − x . (7.74) → k k ⊗ kσ k ak is a multiplicative factor, λk acts on internal ﬂavor or gauge indices, and Pkσ acts on spin indices. All three of ak, λk, and Pkσ must form representations of the group

G. The orbifold identiﬁcation leads to

e 1 φ(x)= a λ P φ P − x k G. (7.75) k k ⊗ kσ k ∀ ∈ e Let Pˆ be the unitary operator acting on the functions space x : k | i

1 Pˆ x P x x Pˆ P − x (7.76) k | i≡| k i h | k ≡ k D

Our constraints on the ﬁeld φ(x) x φ in equation (7.75) can then be rewritten as ≡h | i

1 a λ P Pˆ φ =0 k G. (7.77) − k k ⊗ kσ ⊗ k | i ∀ ∈ e We can take into account all of these constraints simultaneously if we can ﬁnd a projector Qφ for the ﬁeld φ that has the property

1 a λ P Pˆ Q =0 k G. (7.78) − k k ⊗ kσ ⊗ k φ ∀ ∈ e This projector is

1 Q = a λ P Pˆ (7.79) φ N k k ⊗ kσ ⊗ k Xk 120 2 19 and has the properties Qφ = Qφ and Qφ† = Qφ.

If the ﬁeld is a fermion ψ, the projector will be denoted Qψ and the constrained

ﬁeld is Qψψ. The conjugate of this ﬁeld is

0 0 0 0 0 (Q ψ)† Γ = ψ†Q Γ = ψ†Γ Γ Q Γ ψQ (7.80) ψ ψ ψ ≡ ψ

where Q Γ0Q Γ0 is the conjugate projector. ψ ≡ ψ It is useful to study the transformation of the gauge bosons, for which λ Λ k ≡ k

and Pk1 = Pk:

1 A(x) a Λ P A P − x . (7.81) → k k ⊗ k k When written with the adjoint gauge index a and space-time vector index M, this is

a ab N b 1 A (x) a Λ (P ) A P − x . (7.82) M → k k k M N k

ab The choice of the Λk leads to the pattern of gauge breaking in the orbifold. The Λk can be diagonalized, and from this point on we will be using such a basis. At each

ﬁxed-point y , there is a subgroup G G whose elements k G leave y constant. f f ⊂ ∈ f f Furthermore, at each y , there is ane unbrokene gauge group H e G with generators f f ⊂ T a such that Λaa = 1 for all k G . When considering the gauge group H associated k ∈ f f e with the ﬁxed point yf , a gauge index af will signify that the corresponding generator

T af H while the gauge index a will signify that T af / H . The gauge group H in ∈ f f ∈ f e the four-dimensional eﬀective theorye is then the intersection of these subgroups Hf ,

and H has the generators T a such that Λaa = 1 for all k G. k ∈ 19 e † −1 This follows from the fact that the ﬁnite group G has unitary representations: ak = ak , † −1 † −1 ˆ† ˆ−1 λk = λk , Pkσ = Pkσ , and Pk = Pk , and from the basic group property that each element k has a unique inverse k−1 in the group. e

121 Invariance of the Lagrangian under the action of the discrete group G leads to

e some constraints on the transformation matrices λk, P 1 in the presence of gauge k 2 bosons and charged fermions:

0 0 M N N ab b a 1 1 Γ P †1 Γ Γ Pk (Pk)M =Γ Λk T = λkT λk− (7.83) k 2 2

a It will be useful later to know how the gauge bosons AM behave at the fixed- points. By equation (7.82) above, we know how this field transforms under the action of the group G, and further by equation (7.75) that the transformation must leave the field invariant.e Thus we have

a ab N b 1 a A (x) a Λ (P ) A P − x A (x). (7.84) M → k k k M N k ≡ M Let us consider a specific fixed point y of the element k G as our space-time point kf ∈ µ î e x. That is, x = x ,ykf . Pk acts non-trivially only on the extra-dimensional space ν ν î µ represented by y. This means that (Pk)µ = δµ, and (Pk)µ =0=(Pk)î . Further, if ykf

1 is a ﬁxed-point of the group element k, then Pkykf = ykf and Pk− ykf = ykf . Since we

ab b aa a are working in a basis where Λk is diagonal, we can say that Λk AM =Λk AM where

there is no sum on a. Putting this together, we have

a µ ˆi aa N a µ ˆi AM (x ,ykf )= akΛk (Pk)M AN (x ,ykf ). (7.85)

Let us restrict our attention to those vector bosons with indices in the 4-D Minkowski

subspace (M = µ only). The restrictions on the other bosons will in general depend

on the particulars of the orbifold and so we will consider those separately in the

speciﬁc applications later. Since Pk acts non-trivially only on the extra dimensional

space, we have for the 4-D gauge bosons

a µ ˆi aa a µ ˆi Aµ(x ,ykf )= akΛk Aµ(x ,ykf ) (7.86)

122 which implies

Aa (xµ,yˆi ) [1 a Λaa]=0. (7.87) µ kf − k k

a The gauge group index a can be af or af such that T is either in Hf or outside of

Hf respectively, Hf being the gauge groupe preserved by the orbifold at the ﬁxed-point

af af af ykf . For the former, we have Λk = 1 and the restriction on Aµ is

ˆ Aaf (xµ,yi ) [1 a ]=0. (7.88) µ kf − k

We have assumed that the gauge symmetry Gf is preserved at the ﬁxed-point ykf , and

af µ ˆi so we must not require that Aµ (x ,ykf ) vanish. Therefore, we need ak = 1 in order to satisfy the restriction. For a = a , we have Λaf af = 1, and so our transformation f k 6 e e equation implies (after using ak =e 1)

af µ ˆi af af Aµ (x ,ykf ) 1 Λk =0 (7.89) − e e e which means that

af µ ˆi Aµ (x ,ykf )=0. (7.90) e Those gauge bosons with indices in the Minkowski space-time directions which have gauge indices outside of the group Hf at the ﬁxed-point ykf must vanish at that

a ﬁxed-point. It then follows that the ﬁeld-strengths FMN (x) have the same behavior:

af µ ˆi Fµν (x ,ykf )=0 (7.91) e 7.3.2 Orbifold Anomaly Equations

We will now ﬁnd the general gauge anomaly equations for theories with orbifolds.

We will allow the space-time dimension d to be even or odd, since odd dimensional orbifolds can still have even dimensional ﬁxed points and thus gauge anomalies.

123 Let us consider our orbifold gauge theory with a single fermion. In even space-time

dimension d, let the fermion be a Weyl spinor Pdβψ where Pdβ is the Weyl projector

P 1 (1 + βΓ ) with β = 1. For d odd, there is no notion of chirality, and so dβ ≡ 2 d+1 ± the fermion is a Dirac fermion. However, the case of odd d can be simultaneously taken into account with even d if for odd d we let β = 0. Let ψ be in a representation

ρ of the gauge group G. Consider the inﬁnitesimal gauge transformation

ψ(x) (1 + iθ(x)) ψ(x) (7.92) → ψ(x) ψ(x)(1 iθ(x)) (7.93) → −

with θ(x) θa(x)T a. The fermionic measure DψDψ provides the anomaly. In the ≡ transformation of this measure, we must sum over those fermionic modes which are

present in the theory. For this reason we need to insert the projectors PdβQψ and

Pd( β)Qψ to pick out the physical fermion modes left over after the orbifold projection. −

1 − 1 DψDψ det (1 iθ) Pd( β)Qψ Dψ [det (1 + iθ) PdβQψ]− Dψ (7.94) → − − h i 1 − = det (1 iθ) Pd( β)Qψ det (1 + iθ) PdβQψ DψDψ (7.95) − − h i 1 The manipulations on the [det . . .]− term are the same as done previously in sections

7.2.1, 7.2.2, and 7.2.3

1 − det (1 iθ) Pd( β)Qψ det (1 + iθ) PdβQψ (7.96) − − h i

1 − exp tr ln (1 iθ) Pd( β)Qψ exp tr ln [(1 + iθ) PdβQψ] (7.97) ≃ − − h h i i

exp tr iθPd( β)Qψ exp tr [iθPdβQψ] (7.98) ≃ − − − − h i

= exp i tr θPd( β)Qψ tr [θPdβQψ] (7.99) − − h h i i exp iA (7.100) ≡ 124 And so the anomaly A is

A = tr θ PdβQψ Pd( β)Qψ . (7.101) − − − h i Following the same paths as in section 7.2.1 for the calculation of A:

(iD/ )2/M 2 A = lim tr θ PdβQψ Pd( β)Qψ e (7.102) − M − − →∞ h i 2 ig M N 2 ( D 2 Γ Γ FMN )/M = lim tr θ PdβQψ Pd( β)Qψ e − − (7.103) − M − − →∞ ∂2/M 2 ig ΓM ΓN F 2M2 MN lim tr θ PdβQψ Pd( β)Qψ e− e− (7.104) ≃ − M − − →∞ = lim tr [θ M − →∞ × 1 1 1 0 0 ˆ akλk (1 + βΓd+1) Pk 1 (1 βΓd+1)Γ Pk 1 Γ Pk N ⊗ 2 2 − 2 − 2 ⊗ ! × Xk r ∂2/M 2 1 ig M N e− 2 Γ Γ FMN (7.105) r r! −2M # X 1 1 ig r = lim aktr [θλkFM1N1 ...FM N − M N r! −2M 2 r r ⊗ →∞ Xkr 1 0 0 0 0 M1 N1 Mr Nr P 1 Γ P 1 Γ + βΓd+1 P 1 +Γ P 1 Γ Γ Γ . . . Γ Γ 2 k 2 − k 2 k 2 k 2 × ∂2/M 2 Pˆ e− (7.106) ⊗ k i r 1 1 ig (G) a b1 br = lim aktr T λkT ...T − M N r! −2M 2 →∞ Xkr h i (D) 1 0 0 0 0 M1 N1 Mr Nr tr P 1 Γ P 1 Γ + βΓd+1 P 1 +Γ P 1 Γ Γ Γ . . . Γ Γ 2 k 2 − k 2 k 2 k 2 2 2 d ˆ ∂ /M a b1 br d x x Pke− x θ (x)FM1N1 (x) ...FMrNr (x) (7.107) Z h | | i

The sum over the k elements of the group G includes the identity (akλk P 1 Pˆk = ⊗ k 2 ⊗ 1 1 1). This term leads to the bulk (d-dimensional)e anomaly. Consider just this ⊗ ⊗ term:

r 1 1 ig (G) a b1 br A lim 2 tr T T ...T ⊃ − M N r r! −2M →∞ X h i (D) M1 N1 Mr Nr tr βΓd+1Γ Γ . . . Γ Γ h i 2 2 d ∂ /M a b1 br d x x e− x θ (x)FM1N1 (x) ...FMrNr (x) (7.108) Z h | | i 125 d The r = 2 term is the only one to survive. The others vanish either by the Dirac trace or by the M limit.

1 1 ig d/2 A lim tr(G) T aT b1 ...T bd/2 ⊃ − M N d −2M 2 →∞ 2 ! h i

(D) M1 N1 Md/2 Nd/2 tr βΓd+1Γ Γ . . . Γ Γ h d i iM b ddx θa(x)F b1 (x) ...F d/2 (x) (7.109) d d/2 M1N1 Md/2Nd/2 Z 2 π iφ d/2 β ie− d ig = tr(G) T aT b1 ...T bd/2 −N d −4π 2 ! h i b 2 εM1N1...Md/2Nd/2 ddxθa(x)F b1 (x) ...F d/ (x) (7.110) M1N1 Md/2Nd/2 Z iφ d/2 β ie− d ig = tr(G) T aT b1 ...T bd/2 ddxdab1...bd/2 (x) (7.111) −N d −4π 2 ! h i Z iφ d d/2 β ie− ig d (G) = d xL d (ρ, x) (7.112) −N d −4π 2 +1 2 ! Z In the last line we have used the deﬁnitions given in equations (7.69) to (7.71) and

(G) the logic described at the end of section 7.2.2. Indices Ln (ρ) are discussed and calculated in Appendix H. Note that the bulk anomaly is proportional to β. In odd dimensions d the bulk anomaly must vanish, which is taken into account by our notation with β = 0 for that case.

When k does not correspond to the identity element of G, there may be lower-

e dimensional ﬁxed points ykf associated with the operator Pˆk acting on the extra-

∂ ∂i/M 2 dimensional space. When this is the case, the spatial trace factor y Pˆ e− i y h | k | i will lead to delta-functions localizing the possible anomaly onto those ﬁxed-points.

For example, let us assume that the group element k G leads to 4-dimensional ﬁxed ∈ ∂ ∂i/M 2 points. The spatial factor y Pˆ e− i y is then ﬁnitee in the large M limit: h | k | i

2 2 µ 2 i 2 ∂ /M µ ∂µ∂ /M µ ∂ ∂ /M x Pˆ e− x = x e− x y Pˆ e− i y (7.113) h | k | i h | | ih | k | i iM 4 = y Pˆ y (7.114) 24π2 h | k | i 126 iM 4 1 = 4 2 δ (y ykf ) (7.115) 2 π νk − Xf

This result is quoted from [124]. νk is the number of ﬁxed points of the operator

Pˆk on the orbifold, and the delta function is over all of the extra-dimensional space

D 4 i δ (y) − δ (y ). ≡ i=1 Q Considering this case further, let us assume that P 1 contains a term that is k 2 proportional to the 4-dimensional chiral operator Γ iΓ0Γ1Γ2Γ3 deﬁned in Appendix 5 ≡ 20 A equation (A.47). That is, P 1 pΓ5. This will allow non-zero anomaly terms on k 2 ⊃ the 4-dimensional ﬁxed-points, since the Dirac trace will lead to

(D) 1 0 0 0 0 M1 N1 Mr Nr tr P 1 Γ P †1 Γ + βΓd+1 P 1 +Γ P †1 Γ Γ Γ . . . Γ Γ k 2 k k 2 k 2 − 2 2 ptr(D) Γ ΓM1 ΓN1 . . . ΓMr ΓNr . (7.116) ⊃ 5 h i For this to be non-zero, four insertions of ΓM are necessary, and so the r = 2 term is the only one to survive. (For r < 2 the Dirac trace vanishes and for r > 2 the terms are suppressed by factors of 1/M.) Looking speciﬁcally to the r = 2 term, the Dirac trace becomes21

(D) M1 N1 M2 N2 iα [d/2] µ1ν1µ2ν2 M1 M2 N1 N2 tr Γ5Γ Γ Γ Γ = e 2 ε δµ1 δµ2 δν1 δν2 (7.117) h i

20 1 1 Depending on the definition of Pk 2 , it may be possible that other terms in the expansion of Pk 2 1 1 could lead to 4D anomalies. For example, if Pk 2 pΓd+1Γ5, this part of Pk 2 would lead to a 4D anomaly proportional to the 6D chirality β through⊃ a different part of the Dirac trace. The choice we have made in the text is just one possibility, and is made in particular with an eye toward an application later in section 7.4. To capture all parts of the anomaly properly, one must use equation (7.107) after the specific choices of the orbifold operators have been made.

21Dirac traces all are equal to sums of factors of the metric η, that is tr ΓA1 . . . ΓAn = Perm ( 1) ηA1A2 ηA3A4 ...ηAn−1An where n is some even number and the sum has some Perms − permutation factor which depends on the ordering of the Ai’s. In the case at hand, we have Pitr Γ0Γ1Γ2Γ3ΓM1 ΓN1 ΓM2 ΓN2 ( 1)Perm η0M1 η1N1 η2M2 η3N2 . This means that the vec- ∝ Perms − tor indices M1 = µ1, N1 = ν1, etc. in order to give non-zero values in the trace. Further, the resulting terms can be shown to beP completely anti-symmetric in the indices, and so the result is proportional to the usual anti-symmetric tensor εµ1ν1µ2ν2 .

127 where eiα is some phase depending on the deﬁnitions of the gamma matrices. The

factor [d/2] is included to account for both even and odd space-time dimensions d.

The brackets [. . .] in this case mean to take the integer part of the expression. For

example, this means that [d/2] = 2 for both 4D and 5D spaces, as it should since

both of these space-time dimensions have 22 22 dimensional Dirac gamma matrices. × Continuing on, the anomaly A contains the term

2 1 1 ig (G) a b1 b2 (D) M1 N1 M2 N2 A lim aktr T λkT T ptr Γ5Γ Γ Γ Γ M 2 ⊃ − →∞ N 2! −2M 4 h i h i d iM 1 a b1 b2 d x 4 2 δ (y ykf ) θ (x)FM1N1 (x)FM2N2 (x) (7.118) 2 π νk − Z Xf ipeiα2[d/2]g2 = a tr(G) T aλ T b1 T b2 128π2Nν k k k h i ddx εµ1ν1µ2ν2 θaF b1 F b2 (x) δ (y y ) . (7.119) µ1ν1 µ2ν2 − kf Z Xf

b1 As explained previously, there are restrictions on the field strengths Fµν at the fixed- points ykf depending on their gauge index. Specifically, by equation (7.91), we have

b1f Fµν = 0 at that ﬁxed point. Therefore, we need only sum over b1 = b1f and b2 = b2f , e those indices which lead to non-zero results in A.

ipeiα2[d/2]g2 (G) a b1f b2f A 2 ak tr T λkT T ⊃ 128π Nνk × Xf h i d µ1ν1µ2ν2 a b1f b2f d x ε θ Fµ1ν1 Fµ2ν2 (x)δ (y ykf ) (7.120) Z −

bf Using the fact that λk, T = 0 and the cyclicity of the trace, it can be shown h i (G) af b1f b2f that when the gauge index is a = af the trace vanishes: tr T λkT T = 0. h i Incorporating this, we have e e

ipeiα2[d/2]g2 (G) af b1f b2f A 2 ak tr λkT T T ⊃ 128π Nνk × Xf h i d µ1ν1µ2ν2 af b1f b2f d x ε θ Fµ1ν1 Fµ2ν2 (x)δ (y ykf ) (7.121) Z − 128 ipeiα2[d/2]g2 (G) af b1f b2f d af b1f b2f = 2 ak tr λkT T T d x d (x)δ (y ykf ) (7.122) 128π Nνk − Xf h i Z

In general, the λk will split the gauge group representation of the bulk ﬁeld by the fact that it will have diﬀerent values on the various representations ρkf of Gf into which the original representation ρ of G must split. We must sum over these ρkf ,

each with its separate trace and λk. Also, we know that the λk commute with the

af T , and so the λk must be proportional to the identity in this subspace. For the

(ρkf ) representation ρkf of Gf , let the action of λk be λk = λk 1. We then have

(G) af b1f b2f (ρkf ) (Gf ) af b1f b2f tr λkT T T = λk tr T T T (7.123) ρ h i Xkf h i and the anomaly becomes

iα [d/2] 2 ipe 2 g (ρ ) A a λ kf tr(Gf ) T af T b1f T b2f 2 k k ρkf ⊃ 128π Nνk ρ × Xf Xkf h i d af b1f b2f d x d (x)δ (y ykf ) (7.124) Z − iα [d/2] 2 ipe 2 g (ρkf ) d (Gf ) 2 ak λk d xδ (y ykf ) L3 (ρkf , x). (7.125) ⊃ 128π Nνk ρ − Xf Xkf Z

This expression gives 4 dimensional anomalies localized on the ﬁxed points ykf arising

purely from bulk fermions.

It is often the case that a theory will have not only bulk ﬁelds contributing to

brane anomalies, it will have fields confined to those fixed points (commonly called

brane ﬁelds) which will contribute to the 4D anomalies on their respective branes.

Gauge invariance requires the cancellation of the sum of the anomaly contributions

from all ﬁelds. Therefore we must also know the brane ﬁeld contributions in addition

to the bulk ﬁeld contributions.

In an orbifold theory, a charged fermionic field confined to a 4 dimensional fixed

point will have 4D chirality and couplings to the bulk gauge bosons. Let us consider

129 a 4D Weyl fermion ψb(x, yf ) on the ﬁxed point yf with chirality determined by the

4D projector P 1 (1 + αγ ), and let this field transform under the representation 4α ≡ 2 5 ρ of the group G. The gauge transformation of this brane field is no different from

that of the bulk ﬁelds, except that the transformation parameter is also restricted

to be on the ﬁxed point y . ψ (1 + iθ(x, y )). The anomaly calculation here is f b → f identical to that of section 7.2.2, and we have

αg2 A = tr(Gf ) T af T b1f T b2f ddxdaf b1f b2f (x)δ(y y ) (7.126) 32π2 − f 2 h i Z αg d (Gf ) = 2 d xδ(y yf )L3 (ρ, x). (7.127) 32π Z − Restating the important results, the anomaly A due to bulk ﬁelds on an orbifold in general is

r 1 1 ig (G) a b1 br A = lim aktr T λkT ...T − M N r! −2M 2 →∞ Xkr h i (D) 1 0 0 0 0 M1 N1 Mr Nr tr P 1 Γ P 1 Γ + βΓd+1 P 1 +Γ P 1 Γ Γ Γ . . . Γ Γ 2 k 2 − k 2 k 2 k 2 2 2 d ˆ ∂ /M a b1 br d x x Pke− x θ (x)FM1N1 (x) ...FMrNr (x). (7.128) Z h | | i There are two special cases at which we have looked more closely. The ﬁrst is the bulk

ﬁeld case leading to the bulk anomaly (when k is the identity element, and r = d/2

in the expansion)

iφd d/2 β ie− ig d (G) A = d xL d (ρ, x). (7.129) −N d −4π 2 +1 2 ! Z The second case is for 4 dimensional ﬁxed point anomalies. Here, the bulk ﬁeld

contribution comes when k is a non-trivial element of the group G, P 1 pΓ5, and k 2 ⊃ r = 2 in the expansion. e

iα [d/2] 2 ipe 2 g (ρkf ) d (Gf ) A 2 ak λk d x δ (y ykf ) L3 (ρkf , x) (7.130) ⊃ 128π Nνk ρ − Xf Xkf Z 130 Our results match those presented in the paper by von Gersdorﬀ and Quiros [124] which we have followed in this section.

7.4 Applications

We will apply the orbifold anomaly equations we have derived to two models. The

ﬁrst is the case of a string-theory derived model of Kobayashi, Raby, and Zhang [45], and second is the 5d model presented in Chapter 6. We will ﬁnd that in the former theory, the gauge anomalies cancel as they should, but in the latter theory, there is a non-zero brane anomaly which must be addressed.

7.4.1 Model A1 of Kobayashi, Raby, and Zhang

In this section we will apply the orbifold gauge anomaly equations to a model of

Kobayashi, Raby, and Zhang [45], hereafter referred to as KRZ. Speciﬁcally, we will be investigating model A1 in their paper.

Setup of KRZ Theory

KRZ have taken E E heterotic string theory and compactified it on an orb- 8 × 8 ifold to get a 4D effective field theory. Partway through the compactification, the theory can be considered as a 6D N = 1 SUSY field theory with gauge group E6 with various fields in the bulk and on the fixed points. It is to this theory, and its subsequent compactification down to the effective 4D level, that we will apply the anomaly equations developed in section 7.3. The original string theory is anomaly free, which implies that the low energy effective theory we will consider here must also be anomaly free. Therefore, the application of our orbifold anomaly equations is a check on the theory. In addition, it is an illustration of how the equations are

131 applied and will allow us to see in a particular case how anomaly cancellation can

restrict possible ﬁeld theories.

The theory is in 6 dimensions, with 6D N = 1 SUSY, and gauge group E6. The extra dimensions will be a torus T2 compactiﬁed by Z Z .22 The compactiﬁca- 2 × 2 tion breaks the 6D N = 1 SUSY to 4D N = 1, and breaks the gauge group to

SO(10) U(1) on one set of branes and SU(6) SU(2) on another set of branes. × Z × R The intersection of these two groups, SU(4) SU(2) SU(2) U(1) (Pati-Salam × L × R × Z with an extra U(1)), is the 4D eﬀective theory gauge group. In terms of the formal-

ism of the previous section, our compact space C is T2, the compactiﬁcation group

G=Z Z , our gauge group is E , the brane gauge groups H are SO(10) U(1) 2 × 2 6 f × Z ande SU(6) SU(2) , and the 4D gauge group is H = PS U(1) . × R × Z

The group elements of Z2 are the identity 1 and a parity operator P . The elements

of the product group Z Z are then k = 1 1,P 1, 1 P,P P , which means 2 × 2 { × × × × } that the order of the group Z Z is N = 4. The choice of operators P determine 2 × 2 k how the orbifold twists the extra dimensional space, the Λk and λk determine the pattern of gauge breaking, and the P 1 aﬀect how the fermions transform under the k 2 action of the orbifold. The choices which determine these operators have been made

by KRZ.

22 2 The theory described in KRZ is actually a T /Z2 compactification with a Wilson line which helps to break the gauge symmetry. As explained in [124], theories with ZN compactifications with p Wilson lines on T tori with tori radii Ri are equivalent to theories with a larger compactification ′p ′ 2 group on a larger torus T with radii Ri = NRi without Wilson lines. In the case at hand, a T /Z2 2 theory with a Wilson line in one direction along the torus is equivalent to a T /Z2 Z2 theory with a torus that is twice as large and with no Wilson lines. It is possible to apply the× formalism introduced in section 7.3 to the case when Wilson lines are involved, by taking into account the non-trivial boundary conditions of the fields in the projectors Qφ. However, it is easier here to consider the theory containing no Wilson lines.

132 The action of the operators associated with k = (1, 1) must leave their respec-

tive spaces unchanged, and thus must be trivial. Because the operators must form

representations of the group, the action of the operators with k = (P,P ) must be

the product of the operators for k = (P, 1) and (1,P ). Therefore the k = (P, 1)

and (1,P ) operators are the only interesting ones. We will determine these operators

now.

Let the non-trivial action of the ﬁrst Z2, corresponding to k = (P, 1), act on the extra coordinates y, break the gauge group as E representations to SO(10) U(1) 6 × Z representations, and split the Dirac space in the following way:

ˆ y1 y1 P(P,1) : y = − (7.131) y2 → y2 ! − ! 450 450 10 10 λ(P,1) :78=     (7.132) 16 3 → 16 3  −   − −   16   16   3   3     −  10 2 10 2 − − λ(P,1) :27=  14   14  (7.133) 16 → 16  1   1     −  102 102 λ(P,1) : 27 =  1 4   1 4  (7.134) − → − 16 1 16 1  −   −     −  ψ4L ψ4L P(P,1) 1 : ψ = − (7.135) 2 ψ4R ! → ψ4R !

The non-trivial action of the second Z2, corresponding to k = (1,P ), will be chosen to act on the y coordinates, the representations of E SU(6) SU(2) , and 6 → × R the Dirac fermion space as:

y1 y1 + πR1 Pˆ(1,P ) : y = − (7.136) y2 → y2 ! − ! (35, 1) (35, 1) λ(1,P ) :78=  (1, 3)   (1, 3)  (7.137) (20, 2) → (20, 2)        −  133 (15, 1) (15, 1) λ :27= (7.138) (1,P ) (6, 2) → (6, 2) ! − ! (15, 1) (15, 1) λ : 27 = (7.139) (1,P ) (6, 2) → (6, 2) ! − ! ψ4L ψ4L P(1,P ) 1 : ψ = − (7.140) 2 ψ4R ! → ψ4R !

Each Z2 breaks the 6D N = 1 SUSY by diﬀerentiating between the 4D chirality

of the fermions. This action, listed above, can be taken into account by setting

P 1 = P 1 = Γ5. The above choices for the operators lead to the theory (P,1) 2 (1,P ) 2 described by KRZ.

The action of the λk, listed in equations (7.132) through (7.134) and equations

(7.137) through (7.139), has been summarized in Table 7.1.

The fundamental domain of the extra-dimensional torus T2 is

y ( πR , πR ] (7.141) 1 ∈ − 1 1 y ( πR , πR ] (7.142) 2 ∈ − 2 2 and is shown in Figure 7.1. The actions of P(P,1) and P(1,P ) on the coordinates y of

T2 lead to the ﬁxed points on the torus:

y = (0, 0), (0, πR ), (πR , 0), (πR , πR ) (7.143) (P,1)f { 2 1 1 2 } πR1 πR1 πR1 πR1 y(1,P )f = ( , 0), ( , πR2), ( , 0), ( , πR2) (7.144) − 2 − 2 2 2

Each Z2 identiﬁes half of the torus to itself, resulting in a fundamental domain for the orbifold that is one quarter of the original space of the torus. We will choose the particular fundamental domain of the orbifold as:

πR1 y1 0, for y2 =0, πR2 (7.145) ∈ 2 y [0, πR ) for y (0, πR ) (7.146) 1 ∈ 1 2 ∈ 2 134 78 [450]+ [10]+ [16 3] 163 − − h i− [(35, 1)]+ [(15, 1, 1)0 + (1, 3, 1)0]++ [(1, 1, 1)0]++ [(4, 2, 1) 3] + (4, 2, 1)3 − − + h i− [(1, 3)]+ [(1, 1, 3)0]++

[(20, 2)] [(6, 2, 2)0]+ (4, 1, 2) 3 [(4, 1, 2)3] − − − −− h i−− 27 [10 2]+ [14]+ [161] − − [(15, 1)]+ [(6, 1, 1) 2]++ [(1, 1, 1)4]++ [(4, 2, 1)1] + − −

(6, 2) [(1, 2, 2) 2]+ (4, 1, 2)1 − − h i− h i−− 27 [102] [1 4] 16 1 + − + − h i− (15, 1) [(6, 1, 1)2] [(1, 1, 1) 4] (4, 2, 1) 1 + ++ − ++ − + h i h i− [(6, 2)] [(1, 2, 2)2]+ [(4, 1, 2) 1] − − − −−

Table 7.1: Pattern of gauge breaking for the bulk fields. The left/right direction describes how the fields split under E SO(10) U(1) while the up/down direction 6 → × Z describes the split under E6 SU(6) SU(2)R. The subscripts refer to the parities of these fields under the operators→ λ×. When two ±are present, the first refers to k ± the transform of the field under λ(P,1) while the second refers to the transform under λ(1,P ).

135 y2

πR2

πR2 2

πR1 πR1 πR1 πR1 − − 2 2

πR2 − 2

πR − 2

Figure 7.1: Fundamental domain for T2.

136 y2

πR2

πR2 2

πR1 πR1 πR1 πR1 − − 2 2

πR2 − 2

πR − 2

Figure 7.2: Fundamental domain for T2/ (Z Z ). The ﬁxed points are represented 2 × 2 by circles for k =(P, 1) and by boxes for k =(1,P ). Note that while there are four ﬁxed points of each kind in the domain of the torus, there are only two of each kind in the domain of the orbifold.

The fundamental domain of the orbifold contains half of the fixed points contained on the torus. Figure 7.2 shows the orbifold space T2/Z Z and the fixed points. 2 × 2 Because the orbifold has two fixed points each for k = (P, 1) and k = (1,P ), the parameters νk used in the anomaly equations are ν(P,1) = ν(1,P ) = 2. For more information on the spaces T2 and T2/(Z Z ), including wavefunctions and calculations 2 × 2 of spatial traces, see the appendix of [117].

137 Field β a(P,1) a(1,P ) 78 +1 1 1 − − 27 1 +1 1 1 − − 272 1 +1 +1 27 −1 +1 +1 3 − 27 1 1 1 4 − − −

Table 7.2: 6D Chirality β and ak for the bulk ﬁelds.

Each field propagating in the 6D space is a 6D N = 1 SUSY multiplet. Listing the fields by their E6 representation numbers, the theory described in KRZ has a single 78 and four 27s. Each bulk field will have its own 6D chirality β23 and phase a associated with each of the Z Z actions. Since a must form a representation k 2 × 2 k of the group, for each of the fields a = 1, and a = 1 and a = a a . k ± (1,1) (P,P ) (P,1) (1,P ) 24 25 The ak chosen in KRZ for the bulk fields are listed in Table 7.2. Applying all of the phases ak and operators λk, P 1 , and Pˆk to each field, we can find the parities of k 2 each individual field under the action of (P, 1) and (1,P ). These are listed in Table

7.3 and match those listed in KRZ.

23In 6 dimensions, supersymmetry requires that the fermions charged under the gauge group have 6D chirality opposite that of the gaugino fields. This is a consequence of the supersymmetry transformation and the fact that in 6D the right- and left-handed Weyl spinors are self-conjugate. This leads to the opposite choices of β in Table 7.2 for the 78 gaugino fields and the 27 charged fermion fields. 24We are concentrating on the fermionic fields because they are responsible for anomalies. The other fields are related to the fermions by 4D N = 1 SUSY which is preserved by the orbifold. Consequently, the bosonic fields must all be partners of the fermions, and the phases and operator choices made for the fermions in turn determine the phases and operators for the bosons. 25 The ak for the gauginos inside of the 78 is -1, which appears to contradict the statement made in section 7.3 after equation (7.88) that ak = 1 for the gauge bosons, the superpartners of the gaugino fields. The situation is consistent, however, since we are requiring that the gauginos are 4D 1 left-chiral and so pick up an extra sign from the action of Pk 2 = Γ5.

138 Field PS U(1) (P,1) (1,P) PS U(1) (P,1) (1,P) × Z × Z (15, 1, 1)0 + + (15, 1, 1)0 (1, 3, 1) + + (1, 3, 1) − − 0 0 − − (1, 1, 3)0 + + (1, 1, 3)0 (1, 1, 1) + + (1, 1, 1) − − 0 0 − − 78 (6, 2, 2) + (6, 2, 2) + 0 − 0 − (4, 2, 1) 3 + (4, 2, 1) 3 + − − − − (4, 1, 2) 3 (4, 1, 2) 3 + + − − − − (4, 2, 1)3 + (4, 2, 1)3 + (4, 1, 2) − (4, 1, 2) + +− 3 − − 3 (4, 2, 1)1 + + (4, 2, 1) 1 − − − (4, 1, 2)1 + (4, 1, 2) 1 + − − − 271 (6, 1, 1) 2 + (6, 1, 1)2 + − − − (1, 1, 1)4 + (1, 1, 1) 4 + − − − (1, 2, 2) 2 (1, 2, 2)2 + + − − − (4, 2, 1)1 + (4, 2, 1) 1 + − − − (4, 1, 2)1 + + (4, 1, 2) 1 − − − 272 (6, 1, 1) 2 (6, 1, 1)2 + + − − − (1, 1, 1)4 (1, 1, 1) 4 + + − − − (1, 2, 2) 2 + (1, 2, 2)2 + − − − (4, 2, 1)1 + (4, 2, 1) 1 + − − − (4, 1, 2)1 + + (4, 1, 2) 1 − − − 273 (6, 1, 1) 2 (6, 1, 1)2 + + − − − (1, 1, 1)4 (1, 1, 1) 4 + + − − − (1, 2, 2) 2 + (1, 2, 2)2 + − − − (4, 2, 1)1 + (4, 2, 1) 1 + − − − (4, 1, 2)1 (4, 1, 2) 1 + + − − − 274 (6, 1, 1) 2 + + (6, 1, 1)2 − − − (1, 1, 1)4 + + (1, 1, 1) 4 − − − (1, 2, 2) 2 + (1, 2, 2)2 + − − −

Table 7.3: Orbifold parities of bulk fields. The indicate the parity of these fields ± about the fixed points associated with the Z2 operators (P, 1) and (1,P ). This table matches the states presented in KRZ [45] Table 1.

139 We can apply the anomaly equations derived in section 7.3 to ﬁnd the anomalies

associated with this orbifold theory.

KRZ Bulk Anomaly

The diﬀerent k group elements of Z Z lead to diﬀerent anomaly terms. k = 2 × 2

(1 1) has a trivial action on the bulk fermions, and so we have akλk P 1 Pˆk = × ⊗ k 2 ⊗ 1 1 1 for all of the bulk ﬁelds. This leads to a possible anomaly in 6 dimensions, ⊗ ⊗ iφ6 given by equation (7.129) with N = 4, d = 6, and e− = 1 (the last from Appendix −

A equations (A.10) and (A.40)). We must sum over all of the 6D Weyl fermions ψi

with chirality βi, each in representation ρi of the gauge group E6

iφd d/2 1 ie− ig d (G) A βi d xL d (ρi, x) (7.147) ⊃ − N d ! −4π 2 +1 Xi 2 Z g3 = d6x β L(E6)(ρ , x). (7.148) −44π33! i 4 i Z Xi This is the bulk anomaly of our theory. Using Appendix H, which deﬁnes the indices

Ln(ρ), we can go further with this expression. Consider the term

(E6) βiL4 (ρi, x) (7.149) Xi to which the bulk anomaly is proportional. Appendix H equation (H.8) allows us to

write L4(ρi, x) in terms of fundamental indices Dn(ρi) of E6:

(E6) (E6) (E6) (E6) (E6) D2 (ρi) 1 D2 (ρ0) L4 (ρi, x)= c4(x)D4 (ρi)+ c4′ (x)D2 (ρi) (7.150)  d(ρi) − 6 d(ρ0)    As explained more fully in Appendix H, the c4(x) and c4′ (x) depend only on the

ab1b2b3 tensor d (x) in the deﬁnition of L4(ρi, x) given in equation (7.71), while the

ab1b2b3 indices Dn(ρi) are independent of d (x). Here ρ0 is by deﬁnition the adjoint representation of the group (for E6, ρ0 = 78), and d(ρ) is deﬁned to be the dimension

140 (E6) of the representation ρ. The ﬁrst term, c4(x)D4 (ρi), is known as the irreducible

anomaly, while the rest is known as the reducible anomaly. As listed in Table H.1,

(E6) (E6) D4 (ρ) = 0 identically for any representation ρ of E6. Our index L4 (ρi, x) thus

becomes

(E6) (E6) (E6) (E6) D2 (ρi) 1 D2 (ρ0) L4 (ρi, x)= c4′ (x)D2 (ρi) (7.151)  d(ρi) − 6 d(ρ0)    and our expression from equation (7.149) becomes

(E6) (E6) (E6) (E6) D2 (ρi) 1 D2 (ρ0) βiL4 (ρi, x)= c4′ (x) βiD2 (ρi) . (7.152)  d(ρi) − 6 d(ρ0)  Xi Xi   Performing the sum over the bulk ﬁelds listed in Table 7.2 leads to

(E6) (E6) (E6) (E6) D2 (78) 1 D2 (78) β L (ρ , x) = c′ (x) D (78) i 4 i 4  2  d(78) − 6 d(78)  Xi  (E6) (E6)   D (27) 1 D (78) 4D(E6)(27) 2 2 (7.153) − 2  d(27) − 6 d(78)   4  1 4 1 1 4   = c4′ (x) 4 4 (7.154) 78 − 6 78 − 27 − 6 78 20 = c′ (x) . (7.155) 4 351

(E6) (E6) In the second equality we have used D2 (78) = 4 and D2 (27) = 1 from Appendix

H Table H.2.

The fact that β L(E6)(ρ , x) = 0 implies that the bulk anomaly in equation i i 4 i 6 P (7.148) is not equal to zero. This poses a potential problem. There is, however,

another ﬁeld which can be present in the theory that can give an anomaly of just this

type which is not listed in KRZ nor mentioned here thus far. The ﬁeld is an anti-

symmetric tensor ﬁeld B, and its coupling to the gauge ﬁeld allows it to contribute to

the bulk anomaly with the same coeﬃcient c4′ (x) as above. Field theories, especially

those which are derived from string theories, in 6 or more dimensions often invoke the

141 presence of this ﬁeld and assume that it leads to an anomaly which exactly cancels

the reducible anomaly from the bulk fermions [117, 73]. This technique has been

named the Green-Schwarz mechanism, after the authors of the ﬁrst paper to invoke

it [125]. It is not possible to use this mechanism to cancel the irreducible anomaly

in equation (7.150), which can only get contributions from the bulk fermion ﬁelds.

Thus it is well that our irreducible anomaly is zero already.

The element k =(P,P ) leads to a zero term in the anomaly equation. This is be-

∂ ∂i/M 2 cause the spatial integration term y Pˆ e− i y vanishes. Pˆ = Pˆ Pˆ h | (P,P ) | i (P,P ) (P,1) (1,P ) by the properties of the orbifold group and because Pˆk forms a representation of that group. By equations (7.131) and (7.136) for the transformation of y under the operators Pˆ(P,1) and Pˆ(1,P ), the action of Pˆ(P,P ) on the extra dimensional space is then

y1 y1 πR1 Pˆ(P,P ) : y = − . (7.156) y2 ! → y2 !

This is a translation in the y1 direction halfway around the torus. Applying the

deﬁnition of Pˆ on y from equation (7.76), we then have for the spatial trace (P,P ) h |

∂ ∂i/M 2 1 ∂ ∂i/M 2 y Pˆ e− i y = P − y e− i y (7.157) h | (P,P ) | i (P,P ) | i D y1 + πR1 ∂ ∂i/M 2 y1 = e− i (7.158) * y2 ! y2 !+

5 2 6 2 ∂5∂ /M ∂6∂ /M = y + πR e− y y e− y (7.159) h 1 1| | 1ih 2| | 2i M = 0 (7.160) ∗ 2√π = 0. (7.161)

5 2 ∂5∂ /M In y πR e− y , the derivative operator can do nothing to take y to h 1 − 1| | 1i 1 6 2 ∂6∂ /M M y πR , and so this term must vanish. Although y e− y = and 1 − 1 h 2| | 2i 2√π becomes large in the large M limit, this cannot compete with the exact 0 we obtain

from the other term. The whole trace then vanishes.

142 This calculation can also be done using mode functions deﬁned on the orbifold.

These are given in the appendix of [117].

KRZ Brane Anomalies

Next we must move to the brane anomalies. First we will consider the brane anomalies generated by the element k =(P, 1). Equation (7.130) contains the proper

expression for this part of the anomaly. In the case at hand, we need to use N = 4,

d = 6, p = 1, eiα = i, ν = 2, and we must sum over the 6D fermionic ﬁelds − (P,1)

ψi each in the representation ρi of E6. These representations are then split by the

operator λ into representations ρ = (r ) of SO(10) U(1) , where r is a (P,1) i,(P,1)f i qi × Z i

representation of SO(10) and qi is the charge under U(1)Z.

iα [d/2] 2 ipe 2 g i (ρi,kf ) d (Gf ) A 2 ak λk d x δ (y ykf ) L3 (ρi,kf , x) (7.162) ⊃ 128π Nνk ρ − Xi Xf Xi,kf Z g2 = d6x δ y y 32π2 − (P,1)f Xf Z 1 i ((ri)q ) (SO(10) U(1)Z ) i × a(P,1)λ(P,1) L3 ((ri)qi , x) (7.163) 4 i (r ) X Xi qi In an orbifold theory, there can also be fields localized on the fixed points. The fields on a given SO(10) U(1) brane can contribute to an anomaly on that brane as given × Z by equation (7.127), the normal anomaly equation in 4 dimensions. We therefore have on each SO(10) U(1) brane (a fixed point of k =(P, 1), y ) the anomaly from × Z (P,1)f

each 4D Weyl brane fermion ψi with 4D chirality αi in representation ρi = (ri)qi of

SO(10) U(1) : × Z α g2 A i ddxδ(y y )L(Gf )(ρ , x) (7.164) ⊃ 32π2 − f 3 i Xi Z 2 g 6 (SO(10) U(1)Z ) = d xδ(y y ) α L × (ρ , x) (7.165) 32π2 − (P,1)f i 3 i Z Xi

143 Combining the two anomaly equations due to the bulk and brane ﬁelds on a given

SO(10) U(1) brane gives us × Z g2 A d6x δ y y ⊃ 32π2 − (P,1)f Z 1 i ((ri)q ) (SO(10) U(1)Z ) a λ i L × ((r ) , x)  (P,1) (P,1) 3 i qi 4 i (r ) X Xi qi  (SO(10) U(1)Z ) × + αiL3 ((ri)qi , x) . (7.166) # Xi It is the factor [. . .] in the square brackets which must cancel in order that the gauge theory be consistent. By Appendix H equation (H.12), we can rewrite the index

(SO(10) U(1)Z ) × L3 ((ri)qi , x) as the sum of products of indices of the two separate groups

SO(10) and U(1)Z.

3 (SO(10) U(1)Z ) (SO(10)) (U(1)Z) × L3 ((ri)qi , x)= Lm (ri, x)L3 m (qi, x) (7.167) − mX=0 Further calculation can be done with the following information, found in Appendix

L(G)(ρ, x) = c (x)D(G)(ρ) for n 0, 1, 2, 3 (7.168) n n ∈ D(G)(ρ) d(ρ) (7.169) 0 ≡ D(SO(10))(ρ) D(SO(10))(ρ) 0 (7.170) 1 ≡ 3 ≡ D(U(1))(q) qn (7.171) n ≡

Applying these to our expression gives

(SO(10) U(1)Z ) × L3 ((ri)qi , x)= (7.172)

3 (SO(10)) (U(1)Z) = Lm (ri, x)L3 m (qi, x) (7.173) − mX=0 144 3 (SO(10)) (U(1)Z) = cm(x)c3 m(x)Dm (ri)D3 m (qi) (7.174) − − mX=0 (SO(10)) (U(1)Z) = c0(x)c3(x)D0 (ri)D3 (qi)

(SO(10)) (U(1)Z) +c2(x)c1(x)D2 (ri)D1 (qi) (7.175)

3 (SO(10)) = c0(x)c3(x)d(ri)(qi) + c2(x)c1(x)D2 (ri)(qi) (7.176) and the expression in brackets in equation (7.166) becomes

1 ((r ) ) [. . .] = c (x)c (x) ai λ i qi d(r )(q )3 + α d(r )(q )3 0 3  (P,1) (P,1) i i i i i  4 i (r ) i X Xi qi X   1 ((r ) ) +c (x)c (x) ai λ i qi D(SO(10))(r )(q ) 1 2  (P,1) (P,1) 2 i i 4 i (r ) X Xi qi  (SO(10)) + αiD2 (ri)(qi) .(7.177) # Xi

ab1b2 The cn(x) originate from the tensor d (x), which in turn is made up of symmetrized

a gauge ﬁeld strengths Fµν (x). The diﬀerent cn(x) are independent functions, since each

a is made up of fields Fµν(x) with different numbers of gauge indices a in the different parts of the subgroup SO(10) U(1) . Because the c (x) are independent, each piece × Z n

of equation (7.177) proportional to diﬀerent cn(x) must cancel separately in order that we have a good gauge symmetry. We then have

1 i ((ri)qi ) 3 3 0 = a(P,1)λ(P,1) d(ri)(qi) + αid(ri)(qi) (7.178) 4 i (r ) i X Xi qi X

1 i ((ri)qi ) (SO(10)) (SO(10)) 0 = a(P,1)λ(P,1) D2 (ri)(qi)+ αiD2 (ri)(qi). (7.179) 4 i (r ) i X Xi qi X 3 2 The ﬁrst line contains the U(1)Z anomaly, the second the SO(10) U(1)Z anomaly.

Let us ﬁrst calculate the bulk contributions to these equations. Consider the bulk

3 contribution to the U(1)Z anomaly:

1 i ((ri)qi ) 3 a(P,1)λ(P,1) d(ri)(qi) = 4 i (r ) X Xi qi 145 1 = ( 1) d(45)(0)3 + d(1)(0)3 d(16)( 3)3 d(16)(3)3 4 − − − − h +(+1+1+1 1) d(10)( 2)3 + d(1)(4)3 d(16)(1)3 (7.180) − − − 1 i = 2 (10( 8)+64 16) (7.181) 4 − − = 16 (7.182) −

i We have used Table 7.2 to get the a(P,1) for each bulk ﬁeld and Table 7.1 to deter-

i mine how each bulk multiplet ρ of E6 breaks down to the various (ri)qi pieces of

((r ) ) SO(10) U(1) , and what values of the λ i qi are on each of those representations. × Z (P,1) Note that the anomaly contribution from the bulk gauginos in the 78 representation

of E6 vanished on the brane. This is a consequence of the general statement

L (ρ)=( 1)nL (ρ) (7.183) n − n which follows from the definition of L (ρ) ba1...an tr [T a1 ...T an ], the definition of the n ≡ complex conjugate representation ρ whose generators are defined to be T a (T a)T , ρ ≡ − ρ and the properties of the trace. In particular, when a real representation is considered,

we have ρ ρ, and those indices L (ρ) with odd n must vanish. The brane anomaly ≡ n on the SO(10) U(1) brane was proportional to L (ρ, x), and so the brane anomaly × Z 3 contribution had to vanish from the bulk gauginos with the real representation ρ = 78.

The SO(10) U(1) therefore has a brane anomaly in the U(1)3 direction due to the × Z Z bulk 27 ﬁelds, and its value we have calculated to be

1 i ((ri)qi ) 3 a(P,1)λ(P,1) d(ri)(qi) = 16. (7.184) 4 i (r ) − X Xi qi 2 Next we consider the bulk ﬁeld contribution to the SO(10) U(1)Z anomaly. To

calculate this, we will need the second order indices for SO(10) from Table H.2

D2(1)=0 (7.185)

146 D2(10)=1 (7.186)

D2(16)=2 (7.187)

D2(16)=2 (7.188)

D2(45) = 8. (7.189)

3 The same logic applies as in the case of the U(1)Z anomaly above:

1 i ((ri)qi ) (SO(10)) a(P,1)λ(P,1) D2 (ri)(qi)= 4 i (r ) X Xi qi

1 = ( 1) D (45)(0) + D (1)(0) D (16)( 3) D (16)(3) 4 − 2 2 − 2 − − 2 h +(+1+1+1 1)(D (10)( 2) + D (1)(4) D (16)(1))] (7.190) − 2 − 2 − 2 1 = 2(1( 2) 2) (7.191) 4 − − = 2 (7.192) −

So the bulk 27 ﬁelds contribute a SO(10)2U(1) anomaly on the SO(10) U(1) brane Z × Z such that

1 i ((ri)qi ) (SO(10)) a(P,1)λ(P,1) D2 (ri)(qi)= 2. (7.193) 4 i (r ) − X Xi qi Using the results in (7.184) and (7.193), the contraint equations (7.178) and

(7.179) become

3 αid(ri)(qi) =16 (7.194) Xi (SO(10)) αiD2 (ri)(qi) = 2. (7.195) Xi The brane ﬁelds need to satisfy these constraints.

In KRZ, the UV string theory dictates the type, number, and location of the

fixed point fields. Table 7.4 lists the set of non-singlet fields residing on each of the

147 α ρ n 1 16 1 1 − − 1 100 3 −1 1 12 − 2 1 1 2 12 − −

Table 7.4: Non-singlet SO(10) U(1)Z brane fields in KRZ. All brane fields have 4D-left chirality α. ρ is the SO(10)× U(1) representation, written as r where r is × Z q the representation of SO(10) and q is the U(1)Z charge. n is the multiplicity of the field. Note that these fields are present on each SO(10) U(1) brane in the theory. × Z

SO(10) U(1) branes. Using these ﬁelds, we can calculate the brane anomalies. × Z 3 First the U(1)Z anomaly:

3 αid(ri)(qi) = Xi

= ( 1) d(16)( 1)3 +3d(10)(0)3 + 12d(1)(2)3 + 12d(1)( 2)3 (7.196) − − − h i = 16 (7.197)

2 and second the SO(10) U(1)Z anomaly

(SO(10)) αiD2 (ri)(qi)= Xi

= ( 1) [D (16)( 1)+3D (10)(0) + 12D (1)(2) + 12D (1)( 2)] (7.198) − 2 − 2 2 2 −

= D2(16) (7.199)

= 2. (7.200)

Both constraint equations (7.194) and (7.195) are satisﬁed by this set of brane ﬁelds,

as they should be. Note that the result would have been the same if there were only

the 16 1 ﬁeld on the SO(10) U(1)Z brane. The rest of the ﬁelds, 3 100, 12 12, − × ∗ ∗ 148 and 12 1 2 together form a set of representations which are self-conjugate. Such a ∗ −

set must have a vanishing index Ln for n odd by equation (7.183), which is the case here since we are calculating the third order index of SO(10) U(1) on the brane. × Z The other branes, the SU(6) SU(2) branes, can also have anomalies from the × R bulk and brane ﬁelds. The same analysis used for the SO(10) U(1) branes can be × Z used here.

On the SU(6) SU(2) branes, the bulk ﬁelds contribute to the anomaly through × R the k = (1,P ) element of the orbifold. The bulk ﬁeld representations ρi of E6 split into

i i i i ρi =(r1,r2) where r1 and r2 are representations of SU(6) and SU(2)R respectively.

g2 A d6x δ y y ⊃ 32π2 − (1,P )f Xf Z i i 1 i (r1,r2) (SU(6) SU(2)R ) i i a(1,P )λ(1,P ) L3 × ((r1,r2), x) (7.201) 4 i i i X (rX1,r2) Given one of the SU(6) SU(2) fixed points, the anomaly on that fixed point due × R to the fixed point fields is

2 g 6 (SU(6) SU(2)R ) i i A d xδ(y y ) α L × ((r ,r ), x). (7.202) ⊃ 32π2 − (1,P )f i 3 1 2 Z Xi Combining the two anomaly equations gives the anomaly for a single SU(6) SU(2) × R brane

g2 A d6x δ y y ⊃ 32π2 − (1,P )f Z i i 1 i (r1,r2) (SU(6) SU(2)R ) i i a λ L × ((r ,r ), x) 4 (1,P ) (1,P ) 3 1 2 i (ri ,ri )  X X1 2  (SU(6) SU(2)R ) i i + αiL3 × ((r1,r2), x) . (7.203) # Xi

149 We can split the third order index of SU(6) SU(2) into indices of each of the × R separate product groups

3 (SU(6) SU(2)R ) i i (SU(6)) i (SU(2)R) i L3 × ((r1,r2), x) = Lm (r1, x)L3 m (r2, x) (7.204) − mX=0 (SU(6)) i (SU(2)R ) i = L3 (r1, x)L0 (r2, x) (7.205)

(SU(6)) i i = c0(x)c3(x)D3 (r1)d(r2). (7.206)

We have used the following information from Appendix H: L1(r, x) = 0 for both

(SU(2)R ) groups, L3 (r, x)=0, L3(ρ, x)= c3(x)D3(ρ), and L0(ρ, x)= c0(x)d(ρ). There is

only one non-trivial anomaly term, the SU(6)3 anomaly, and so the anomaly cancel-

lation constraint becomes

i i 1 i (r1,r2) (SU(6)) i i (SU(6)) i i 0= a(1,P )λ(1,P ) D3 (r1)d(r2)+ αiD3 (r1)d(r2). (7.207) 4 i i i i X (rX1,r2) X The bulk contribution to this anomaly is

i i 1 i (r1,r2) (SU(6)) i i a(1,P )λ(1,P ) D3 (r1)d(r2)= 4 i i i X (rX1,r2)

1 = ( 1) D(SU(6))(35)d(1) + D(SU(6))(1)d(3) D(SU(6))(20)d(2) 4 − 3 3 − 3 h +( 1+1+1 1) D(SU(6))(15)d(1) D(SU(6))(6)d(2) (7.208) − − 3 − 3 i = 0. (7.209)

i We have made use of Tables 7.2 and 7.1 for the a(1,P ) and the gauge breaking pattern,

(SU(6)) and Table H.2 for the values of the fundamental indices D3 (r). The anomaly from the bulk 78 of E6 again must vanish since 78 is a real representation, and an

index Ln(ρ)=0for n odd over a real representation ρ. The bulk 27s do not contribute

to this anomaly simply because the sum of the a(1,P ) for these ﬁelds is zero.

150 ρ n (6, 1) 2 (6, 1) 2 (1, 2) 6

Table 7.5: Non-singlet SU(6) SU(2) brane fields in KRZ. ρ is the SU(6) SU(2) × R × R representation, written as (r1,r2) where r1 and r2 are the representations of SU(6) and SU(2)R respectively. n is the multiplicity of the field. Note that these fields are present on each SU(6) SU(2) brane in the theory. × R

Using this result, equation (7.207) becomes a constraint on the SU(6) SU(2) × R brane ﬁelds:

(SU(6)) i i 0= αiD3 (r1)d(r2) (7.210) Xi In the KRZ theory, the brane ﬁelds on each SU(6) SU(2) brane are known, × R and have been listed in Table 7.5. Using these ﬁelds in the brane anomaly equation leads to

(SU(6)) i i αiD3 (r1)d(r2)= Xi

= ( 1) 2D(SU(6))(6)d(1)+2D(SU(6))(6)d(1)+6D(SU(6))(1)d(2) (7.211) − 3 3 3 = ( 1)(2(1)(1) + 2( 1)(1)+6(0)(2)) (7.212) − − = 0. (7.213)

This satisﬁes the constraint in equation (7.210).

7.4.2 5D Model

We can apply the orbifold gauge anomaly equations to the 5D model presented in

Chapter 6. We will ﬁnd that, unfortunately, the model suﬀers from a gauge anomaly

151 on the Pati-Salam brane due to the bulk ﬁelds of the theory. This can be remedied by

the addition of more ﬁelds either in the bulk or on that brane. The logic followed in

this section mirrors that of Section 7.4, and so we will be more brief in our descriptions

here.

Setup of 5D Model

Chapter 6 has already described this model in some detail. We will present brieﬂy here the model in terms of the notation created in this chapter.

The model is in 5 dimensions with the extra dimension a circle S1 modded out

by the orbifold group Z Z . The elements of the orbifold group are then k = 2 × 2 1 1,P 1, 1 P,P P . The operators corresponding to k = (1, 1) are trivial { × × × × } and those corresponding to k = (P,P ) are determined once the k = (P, 1) and

k = (1,P ) operators are known.

k =(P, 1) is associated with the action of the ﬁrst Z2 on the space. This Z2 breaks

half of the supersymmetry in the theory and leaves the gauge group SO(10) intact.

This ﬁxes the action of the (P, 1) operators:

Pˆ : y y (7.214) (P,1) → − (15, 1, 1) (15, 1, 1) (1, 3, 1) (1, 3, 1) λ :45=     (7.215) (P,1) (1, 1, 3) → (1, 1, 3)          (6, 2, 2)   (6, 2, 2)      (4, 2, 1) (4, 2, 1) λ(P,1) :16= (7.216) (4, 1, 2) ! → (4, 1, 2) ! (4, 2, 1) (4, 2, 1) λ(P,1) : 16 (7.217) (4, 1, 2) ! → (4, 1, 2) ! (1, 2, 2) (1, 2, 2) λ(P,1) :10= (7.218) (6, 1, 1) ! → (6, 1, 1) !

ψ4L ψ4L P(P,1) 1 : ψ = − (7.219) 2 ψ4R ! → ψ4R !

152 The second Z2 breaks the gauge symmetry from SO(10) to Pati-Salam and is

associated with the element k = (1,P ). The operators are

Pˆ : y y + πR (7.220) (1,P ) → − (15, 1, 1) (15, 1, 1) (1, 3, 1) (1, 3, 1) λ :45=     (7.221) (1,P ) (1, 1, 3) → (1, 1, 3)          (6, 2, 2)   (6, 2, 2)     −  (4, 2, 1) (4, 2, 1) λ :16= (7.222) (1,P ) (4, 1, 2) → (4, 1, 2) ! − ! (4, 2, 1) (4, 2, 1) λ : 16 (7.223) (1,P ) (4, 1, 2) → (4, 1, 2) ! − ! (1, 2, 2) (1, 2, 2) λ :10= (7.224) (1,P ) (6, 1, 1) → (6, 1, 1) ! − ! ψ4L ψ4L P(1,P ) 1 : ψ = − . (7.225) 2 ψ4R ! → ψ4R !

Both Z2 symmetries diﬀerentiate between the two 4D Weyl spinors in the original 5D

spinor. The operators on this space can then be taken to be P 1 = P 1 = Γ5. (1,P ) 2 (P,1) 2 The fundamental domain for the circle is

y [0, 2πR) . (7.226) ∈

1 The actions of P(P,1) and P(1,P ) on this space lead to the ﬁxed points on S

y = 0, πR (7.227) (P,1) { } πR 3πR y(1,P ) = , . (7.228) 2 2

The fundamental domain of the orbifold is one quarter the size of S1

πR y 0, (7.229) ∈ 2 and contains only one ﬁxed point of each kind at either end of the space. This implies that the νk used in the anomaly equations are ν(P,1) = ν(1,P ) = 1.

153 Field a(P,1) a(1,P ) 45 1 1 − − 161 +1 +1 162 +1 +1 10 1 1 − −

Table 7.6: ak for the bulk ﬁelds.

Each bulk ﬁeld in the 5D model is a 5D N = 1 SUSY supermultiplet. The

ﬁeld content in the bulk for the theory, with the ﬁelds listed by their SO(10) representations, is a single 45 vector multiplet, a pair of 16 hypermultiplets related by

D3 symmetry, and a single 10 hypermultiplet. The phases ak associated with each

ﬁeld are listed in Table 7.6. Application of the operators and phases associated with k =(P, 1) and (1,P ) leads to the parities of the ﬁelds listed in Table 6.1 in Chapter

5D Model Bulk Anomaly

In odd dimension space-times, as discussed in Section 7.2, anomalies will vanish due to the non-chiral nature of those spaces. Thus, the 5D bulk anomaly for this theory must be zero.

5D Model Brane Anomalies

The brane anomaly on the fixed point of the first Z2 must also vanish. This is because the fixed point associated with k = (P, 1) has SO(10) gauge symmetry and

(SO(10)) is a 4D space. The brane anomaly will then be proportional to the index D3 (ρ) which vanishes identically for any SO(10) representation ρ according to Appendix H.

154 The Pati-Salam brane anomaly is the only one which requires calculation. This

fixed point is associated with the element k = (1,P ). Equation (7.130) contains the expression for a fixed point anomaly due to bulk fields. For the 5D orbifold model, we have N = 4, d = 5, p = 1, eiα = i, ν = 1, and we must sum over the − (1,P )

5D fermionic ﬁelds ψi each in the representation ρi of SO(10). These representa-

tions are split by the operator λ(1,P ) into representations ρi,(1,P )f = (r4,r2L,r2R) of

PS = SU(4) SU(2) SU(2) . × L × R

iα [d/2] 2 ipe 2 g i (ρi,kf ) d (Gf ) A 2 ak λk d x δ (y ykf ) L3 (ρi,kf , x) (7.230) ⊃ 128π Nνk ρ − Xi Xf Xi,kf Z 2 g 5 = 2 d x δ y y(1,P )f 32π Z − 1 ai λ((r4,r2L,r2R))L(PS)((r ,r ,r ), x) (7.231) 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R) The anomaly at the Pati-Salam brane due to the brane ﬁelds is from equation (7.127).

We must sum over each 4D Weyl brane fermion ψi with 4D chirality αi in represen-

tation r r r of Pati-Salam 4 × 2L × 4L α g2 A i ddxδ(y y )L(Gf )(ρ , x) (7.232) ⊃ 32π2 − f 3 i Xi Z g2 = d5xδ(y y ) α L(PS)((r ,r ,r ), x). (7.233) 32π2 − (P,1)f i 3 4 2L 2R Z Xi Combining the two anomaly equations due to the bulk and brane ﬁelds on the Pati-

Salam brane gives

g2 A d5x δ y y ⊃ 32π2 − (P,1)f Z 1 ai λ((r4,r2L,r2R))L(PS)((r ,r ,r ), x) 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R)  (PS) + αiL3 ((r4,r2L,r2R), x) . (7.234) # Xi

155 (PS) Making use of Appendix H, we can rewrite the product group index L3 in terms

of the sum of products of indices of the individual subgroups. Due to the fact that

(SU(4)) (SU(2)) (SU(2)) 3 L1 = L1 = L3 = 0, the only combination which survives is the SU(4) index

(PS) (SU(4)) (SU(2)L) (SU(2)R) L3 ((r4,r2L,r2R), x) = L3 (r4, x)L0 (r2L, x)L0 (r2R, x) (7.235)

2 (SU(4)) = c0(x)c3(x)D3 (r4)d(r2L)d(r2R). (7.236)

The requirement that the brane anomaly cancels leads to

1 0 = ai λ((r4,r2L,r2R))D(SU(4))(r )d(r )d(r ) 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R) (SU(4)) + αiD3 (r4)d(r2L)d(r2R). (7.237) Xi First let us calculate the bulk ﬁeld contribution to the brane anomaly.

1 ai λ((r4,r2L,r2R))D(SU(4))(r )d(r )d(r )= 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R)

1 = [( 1)(D (15)d(1)d(1)+2D (1)d(3)d(1) D (6)d(2)d(2)) 4 − 3 3 − 3 +(+1 + 1) D (4)d(2)d(1) D (4)d(1)d(2) 3 − 3 +( 1)(D (1)d(2)d(2) D (6)d(1)d(1))] (7.238) − 3 − 3 1 = 2 (2 ( 1)2) (7.239) 4 − − = 2 (7.240)

The bulk ﬁelds contribute a SU(4)3 brane anomaly on the Pati-Salam brane:

1 ai λ((r4,r2L,r2R))D(SU(4))(r )d(r )d(r )=2 (7.241) 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R) The brane ﬁelds of the 5d model are listed in Table 6.3 in Chapter 6. We retabulate

them here only by 4D chirality, PS representation, and multiplicity in Table 7.7. Using

156 α ρ n 1 (4, 2, 1) 2 − 1 (4, 1, 2) 7 −1 (4, 1, 2) 5 −1 (1, 1, 3) 1 − 1 (1, 1, 1) 5 −

Table 7.7: Pati-Salam brane fields in the 5d model. All brane fields have 4D-left chirality α. ρ is the Pati-Salam SU(4) SU(2)L SU(2)R representation and n is the multiplicity of the field. × ×

these ﬁelds, we can calculate the brane ﬁeld contribution to the PS anomaly:

(SU(4)) αiD3 (r4)d(r2L)d(r2R)= Xi = ( 1) 2D (4)d(2)d(1)+7D (4)d(1)d(2)+5D (4)d(1)d(2) − 3 3 3 h +5D3(1)d(1)d(1) + D3(1)d(1)d(3)] (7.242)

= ( 1)[2(1)2 + 7( 1)2+5(1)2] (7.243) − − = 0 (7.244)

The brane ﬁelds yield zero anomaly.

(SU(4)) αiD3 (r4)d(r2L)d(r2R)=0 (7.245) Xi The anomaly cancellation constraint in equation (7.237) is not satisﬁed, since we

have

1 ai λ((r4,r2L,r2R))D(SU(4))(r )d(r )d(r ) 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R) (SU(4)) + αiD3 (r4)d(r2L)d(r2R) = 2 (7.246) Xi and the 5d model suﬀers from a brane anomaly on the Pati-Salam brane. The theory

is then untenable as it has been presented in Chapter 6, but can be saved by making

157 changes to the ﬁeld content of the theory to cancel this anomaly. This is discussed in

the next section.

7.5 Discussion

We have seen that the gauge anomalies due to bulk ﬁelds in an extra dimensional orbifold theory are equal to that given in equation (7.107)

r 1 1 ig (G) a b1 br A lim aktr T λkT ...T ⊃ − M N r! −2M 2 →∞ Xkr h i (D) 1 0 0 0 0 M1 N1 Mr Nr tr P 1 Γ P 1 Γ + βΓd+1 P 1 +Γ P 1 Γ Γ Γ . . . Γ Γ 2 k 2 − k 2 k 2 k 2 2 2 d ˆ ∂ /M a b1 br d x x Pke− x θ (x)FM1N1 (x) ...FMrNr (x). (7.247) Z h | | i

In particular, the trivial element k of the orbifold group G leads to an anomaly in the d-dimensional space. As we saw in the case of the theorye of KRZ, in 6 dimensions

(and higher even dimensions) it is possible for the index L d to have irreducible and 2 +1 reducible parts. As discussed in section 7.4, there are no other contributions to the

irreducible part and so it must cancel among the bulk fermions. The reducible parts,

however, can be cancelled by anomaly contributions of an anti-symmetric tensor ﬁeld

B in the theory. Invokation of such a ﬁeld in order to cancel a reducible anomaly in

a ﬁeld theory is known as the Green-Schwarz mechanism [125].

Non-trivial k elements can lead to lower-dimensional anomalies in the orbifold

theory. We have focussed in particular on those k which lead to four-dimensional

ﬁxed points, and therefore possibly to four-dimensional anomalies proportional to

third order indices L3. For consistency of the gauge theory, these ﬁxed-point or brane

anomalies must cancel after summing over the contributions from the bulk and brane

ﬁelds. Though the requirement of anomaly cancellation places some constraints on

158 the ﬁelds present in a ﬁeld theory, these constraints are not nearly strong enough to

determine the exact ﬁeld content.

To illustrate, consider again the KRZ theory, and the SO(10) U(1) brane × Z anomalies. Given our bulk ﬁeld content, there were constraints on the brane ﬁelds in terms of anomaly cancellation equations (7.194) and (7.195)

3 αid(ri)(qi) =16 (7.248) Xi (SO(10)) αiD2 (ri)(qi) = 2. (7.249) Xi The ﬁelds given by the string theory for the KRZ theory were (on each SO(10) brane) the 4D left-handed ﬁelds: 16 1 +3 100 +12 (12 +1 2). The contribution − × × − to the anomalies came only from the 16 1 and were exactly those necessary to cancel − the bulk anomalies and satisfy the contraint equations. Instead of this 16 1, there −

could have been instead 16 1, 10 2 +14, or some more complicated set of 1s, 10s, − −

16s, or higher dimensional SO(10) ﬁelds with some U(1)Z charges. The other ﬁelds,

3 100 +12 (12 +1 2), formed a set of representations that were real (vector-like) × × − and so their third order index (and therefore their anomaly contributions) had to

vanish. As far as anomaly cancellation is concerned, these latter ﬁelds did not have

to be present, nor was there any restriction on the presence of still more ﬁelds as

long as together their anomalies vanished. This includes chiral sets of ﬁelds, such as

r0 with any representation r of SO(10) with U(1)Z charge zero, and the sets of ﬁelds

10 2q +14q + 16q or 10 2q +14q + 16q for any q, for example. Any ﬁeld 1q1 can be − − 3 3 3 replaced by two singlet SO(10) ﬁelds 1q2 +1q3 as long as q1 = q2 + q3 . As we allow larger SO(10) representations, the number and complexity of the anomaly-free sets of representations will increase.

159 As we can see, there are clearly an inﬁnite number of sets of SO(10) brane ﬁelds

which can satisfy the anomaly constraints. Similar freedom exists for the choice of

ﬁelds on the SU(6) SU(2) branes in the KRZ theory. Because the bulk contribution × R to the anomalies on these branes vanished, the brane ﬁelds had to satisfy equation

(7.210)

(SU(6)) i i αiD3 (r1)d(r2)=0. (7.250) Xi The particular set of ﬁelds in KRZ was 2 (6, 1)+2 (6, 1)+6 (1, 2). This set happens × × × to form a real representation of SU(6) SU(2) and so must have vanishing L . Any × R 3 set of representations with vanishing anomalies could have been on the SU(6) branes:

for example, combinations of fields such as (6, 2)+(15, 1), 2 (6, 1)+(15, 1), (20, 2), × 3 or no fields at all. The SU(2)R part of the group contributes to the SU(6) anomaly only by the dimension of the SU(2)R representation, and so, as far as anomalies are concerned, there is no distinction between the multiplicity of the field and the representation of SU(2)R. This means that the anomaly from a generic field (r1,r2) is the same as the anomaly from r2 fields with representation (r1, 1).

In KRZ, the anomaly constraints do restrict the possible ﬁeld content on the branes, but these constraints are not very strong. Many diﬀerent sets of fermion

fields could lead to brane anomaly cancellation. The particular fields that are present were determined by the string theory which completes the effective field theory at the 6D level. As far as field theory model-building is concerned, it is likely that the anomaly constraints in most orbifolds will not lead to strong restrictions on the fields present.

It is of course possible to restrict the bulk ﬁelds rather than the brane ﬁelds when working toward anomaly cancellation. For example, in the KRZ theory, the bulk

160 ﬁelds produced anomalies on the SO(10) branes but not on the SU(6) branes. The

SU(6) anomaly contribution vanished because the sum over the k = (1,P ) element

i i intrinsic parities a(1,P ) of the bulk 27 hypermultiplets vanished: 27i a(P,1) =0. A P change in the bulk parities for the k = (P, 1) element could also be used to remove

the anomalies on the SO(10) branes due to the bulk ﬁelds.

We also analyzed the gauge anomaly structure of the 5d model presented in Chap-

ter 6. There we found that the bulk and SO(10) brane anomalies vanished, but that

the gauge anomaly on the Pati-Salam brane did not. The anomaly there was propor-

tional to

1 ai λ((r4,r2L,r2R))D(SU(4))(r )d(r )d(r ) 4 (1,P ) (1,P ) 3 4 2L 2R Xi (r4,rX2L,r2R) (SU(4)) + αiD3 (r4)d(r2L)d(r2R) = 2. (7.251) Xi The anomaly was due to the two 16 hypermultiplets of SO(10) in the bulk. To ﬁx this

theory, more ﬁelds could be added, such as two more bulk 16s with a = 1, or the (1,P ) − brane ﬁeld content on the Pati-Salam brane could be altered. For example, the addi-

tion of another (4, 2, 1) brane ﬁeld would give the correct contribution to completely

cancel the anomaly, or the removal of one (4, 1, 2). There are many combinations of

brane fields which could work. The more difficult problem is to alter the fields of the

theory in such a way that the low energy Yukawa structure is unchanged and that

there are no extra massless ﬁelds. Such work we leave to the future.

There were topics important to the theory of orbifold anomalies which could have

been included in this work but were not. A derivation of the anomaly due to anti-

symmetric tensor ﬁelds would lead to a better understanding of the Green-Schwarz

mechanism and the cancellation of bulk reducible anomalies. Another important

161 topic is the study of mixed gauge-gravity anomalies, which leads to the well-known

constraint in the Standard Model that the sum of the electromagnetic charges must

vanish. In orbifold theories, mixed gauge U(1) gravity anomalies should appear on − the ﬁxed points of the theory in some way, which would lead to further constraints on bulk and brane fermions charged under this U(1) due to the requirements of anomaly cancellation.

Another topic which is not included here is a conjecture by Stuart Raby: the possible equality of two brane anomaly calculation techniques. It appears that in some particular orbifold theories, at a given brane, the brane anomalies due to bulk

fields as calculated in Section 7.3 are equivalent to those brane anomalies calculated by only including bulk fields which are non-vanishing at that brane. An investigation of this conjecture should be possible with the formalism presented in section 7.3 coupled with some new definitions of projector fields in order to pick out only those parts of the bulk fields which are non-vanishing at a particular brane. The investigation of this conjecture we leave to future research.

162 CHAPTER 8

CONCLUSION

In this work we have presented an introduction to the Standard Model of particle physics, some of its deficiencies, and some of the extensions which address these deficiencies. The extensions described were grand unification, supersymmetry, family symmetry, and orbifold extra dimensions. This discussion was necessary as the foundation of the research presented in this work.

We analyzed a class of texture models, those hierarchical textures with symmetric

(1, 2) and (2, 1) elements and vanishing (1, 1) elements. For those classes with the further constraint of vanishing (1, 3) elements, we showed that the maximal value of sin 2β in those models was 3 to 4σ below the experimental value, thus ruling these models out at that level. We additionally showed that relaxation of the constraint on the (1, 3) elements leads to models with maximal sin 2β approximately 1.5σ below the experimental value, allowing such models to still accomodate the data. At the moment, sin 2β and V have roughly equal constraining power on texture models. | ub| The precision of the sin 2β measurement is largely statistical, and the precision of the V measurement is largely theoretical. As more data is taken, sin 2β will gain | ub| in precision, while it is unlikely that precision on V will improve signiﬁcantly in | ub|

163 the near future. Therefore, we expect that sin 2β will become the dominant probe of texture models.

In the second of the three research topics, we built a supersymmetric SO(10) D × 3 1 model with an orbifold extra dimension S /(Z Z′ ). The orbifold itself breaks half 2 × 2 of the supersymmetry and the gauge symmetry SO(10) to SU(4) SU(2) SU(2) , × L × R and vacuum expectation values of various fields break the family symmetry D3 and the remaining gauge symmetry to the SM gauge group SU(3) SU(2) U(1) . C × L × Y The low energy effective field theory below the compactification scale of the orbifold is the Minimal Supersymmetric Standard Model with a particular set of Yukawa matrices for the charged fermions. These Yukawa matrices depend on 11 parameters,

2 fewer than in the Standard Model, allowing us to make two predictions among the low energy data. The predictions were that the quark mass combinations mu/mc and

mdmsmb were about 1σ above the experimental values. The remaining data was ﬁt at the 5% level.

We noted that this model has a potential problem. The ﬁelds χc and χc on the

Pati-Salam brane get vacuum expection values which break the gauge symmetry at approximately the cutoﬀ scale of the theory. This implies that the gauge symmetry below the cutoﬀ scale is the Standard Model. However, we have used Pati-Salam symmetry to restrict the mass terms which lead to our Yukawa matrices.

We gave a solution to this conﬂict by adding an additional dimension to our theory through which we can separate the ﬁelds χc and χc from the other Pati-Salam brane

ﬁelds. The small separation in the sixth dimension implies that though Pati-Salam gauge symmetry is broken, the symmetry protecting the mass terms remains at a low enough energy scale to still be useful.

164 This model unfortunately suﬀers from a gauge anomaly localized to the Pati-Salam

brane. Such an anomaly renders the theory inconsistent, and it must be altered or

abandoned. To save the theory, the ﬁeld content in the bulk or on the Pati-Salam

brane must be changed in order to cancel the anomaly. Fixes include the addition

of another D3 doublet of SO(10) 16s in the bulk, the addition of a Pati-Salam brane

ﬁeld transforming as (4, 2, 1) or (4, 1, 2), or the removal of a Pati-Salam brane ﬁeld

with charge (4, 1, 2). While there are many combinations of fields which could lead to a vanishing Pati-Salam brane anomaly, the real difficulty lies in changing the fields in such a way which does not alter the low energy Yukawa matrix and does not leave over superfluous massless fields.

The 5D model is far from complete. Future work could include the addition of

fields to produce neutrino masses, specifying the mechanism by which supersymmetry is broken at the weak scale, a more complete analysis of gauge unification, an analysis of the running of the Yukawa matrices between the cutoff and compactification scales, an ultra-violet theory, such as a string theory, which gives us the fields and symmetries we have assumed at the 5D level, potentials which lead to the vacuum expectation values assumed in our theory, and the alteration of the field content of the theory in order to cancel the Pati-Salam brane gauge anomaly while retaining the Yukawa structure and fields present at low energies.

As the ﬁnal research topic we investigated orbifold gauge anomalies. We derived

gauge anomaly equations present in the literature and then applied them to a partic-

ular 6D orbifold model. We conﬁrmed that the gauge anomalies cancel in this theory,

and illustrated the extent to which anomaly cancellation restricts the ﬁeld content.

165 We concluded that in general, while gauge anomaly cancellation puts constraints on the fields in an orbifold theory, it does not completely fix the field content.

166 APPENDIX A

LORENTZ SPINORS

First some general notation issues. For indices: large Roman letters run over all of

space-time: A,B,... =0, 1, 2, 3,...; small Roman letters at the front of the alphabet

will run over all of space: a, b, . . . = 1, 2, 3,...; small Greek letters will as usual run

over 4D space-time: µ,ν,... =0, 1, 2, 3; small Roman letters starting from i will run

over the ﬁrst three spatial directions: i, j, . . . = 1, 2, 3. 1n will describe the identity

matrix in n n dimensional spinor space. × A.1 General Notation in d Dimensions

The spinor representations are built up from the Pauli-sigma matrices, and so we

will deﬁne them here with their most important properties:

0 1 0 i 1 0 σ1 σ2 − σ3 (A.1) ≡ 1 0 ≡ i 0 ≡ 0 1 ! ! − ! These satisfy

σiσj = δij12 + iεijkσk (A.2) which implies

σ , σ = 2δ 1 (A.3) { i j} ij 2

[σi, σj] = 2iεijkσk. (A.4)

167 We will use a space-time metric with negative spatial signature

η diag [1, 1, 1, 1,...] . (A.5) ≡ − − −

Following Peskin [1], the Lorentz transformation generators satisfy the commutation

relations

[JAB,JCD]= i (ηAC JDB + ηADJBC + ηBC JAD + ηBDJCA) . (A.6)

The vector representation is

(J )C i δC η δC η (A.7) AB D ≡ A BD − B AD and the Cliﬀord algebra is

ΓM , ΓN 2ηMN 1 (A.8) ≡ n o M d/2 (d 1)/2 where the Γ are square matrices of dimension 2 or 2 − for even or odd space-

time dimension d respectively. With an even number of dimensions d, an additional

anti-commuting gamma matrix can be formed from the product of all of the ΓM :

Γ eiφd Γ0Γ1Γ2Γ3Γ5 . . . Γd (A.9) d+1 ≡

4 Note that by convention Γ is never used. The phase angle φd can be chosen such

2 that Γd†+1 =Γd+1 and (Γd+1) = 1. To give a non-zero trace with Γd+1, the smallest number of other gamma matrices necessary is d.

A1 A2 Ad iφd d/2 A1A2...Ad Tr Γd+1Γ Γ . . . Γ = e− 2 ε (A.10) h i where ε is a totally anti-symmetric tensor with ε01235...d +1. ≡ From the ΓM , the Σ matrices can be formed,

i Σ [Γ , Γ ] (A.11) MN ≡ 4 M N

which are the Dirac spinor representation of the Lorentz symmetry.

168 A.2 d =4 Dimensions

We now specify to the well-known case of 4 spacetime dimensions. The choices here parallel those made in Peskin [1], whereever possible. Choose the chiral basis for the Dirac spinors:

0 0 12 γ 12 σ1 = (A.12) ≡ × 12 0 !

i 0 σi γ σi iσ2 = (A.13) ≡ × σi 0 − ! These can be combined

µ µ 0 σ γ µ (A.14) ≡ σ 0 !

σµ (1 , σ ) σµ (1 , σ ) . (A.15) ≡ 2 i ≡ 2 − i

Let γ5 be deﬁned as follows:

5 0 1 2 3 12 0 γ5 γ iγ γ γ γ = 12 σ3 = − (A.16) ≡ ≡ − × 0 12 !

µ ν µν 5 µ It follows from these deﬁnitions that γ ,γ = 2η 1 , γ ,γ = 0, γ† = γ , { } 4 { } 5 5 2 (γ5) = 14, and

Tr γ γαγβγµγν = 4iεαβµν . (A.17) 5 − h i The spinor representation generators in this basis are

4 i i σµσν σν σµ 0 Σµν [γµ,γν]= − . (A.18) ≡ 4 4 0 σµσν σν σµ − ! This is a reducible representation. Deﬁne the two irreducible dimension-2 spinor representations

Σ2 σ σ σ σ Σ2 σ σ σ σ (A.19) µν ≡ µ ν − ν µ µν ≡ µ ν − ν µ 169 so that

2 4 Σµν 0 Σµν 2 . (A.20) ≡ 0 Σµν !

µ 2 2 Because the σ are hermitian, we have Σµν† = Σµν . The independent elements of the

Σ4 matrices are

4 i Σ0i = σi σ3 2 × (A.21) Σ4 = 1 ε σ 1 ij 2 ijk k × and the independent elements of the 2-dimensional counterparts are

2 i 2 i Σ0i = σi Σ0i = σi 2 − 2 (A.22) 2 1 2 1 Σij = 2 εijkσk Σij = 2 εijkσk.

Let ψ4 be a Dirac spinor, transforming under Lorentz transformations with the

4 2 2 generators Σ , and let ψ2 and ψ2 be Weyl spinors, transforming with Σ and Σ .

0 Further, let ψ ψ†γ . 4 ≡ 4

iωµν Σ2 ψ ψ′ e µν ψ (A.23) 2 → 2 ≡ 2 iωµν Σ2 ψ ψ′ e µν ψ 2 → 2 ≡ 2 iωµν Σ4 ψ ψ′ e µν ψ 4 → 4 ≡ 4 0 iωµν Σ4 ψ (ψ′ )† γ = ψ e− µν 4 → 4 4

The ωµν ωνµ are the real parameters of the Lorentz transformation. The last ≡ − 0 4 0 4 transformation follows from the fact that γ Σµν†γ = Σµν . The structure of the spinor

ψ2 generators makes it clear that ψ4 contains a ψ2 and ψ2: ψ4 = . ψ2 ! C Consider the transformation of the spinor ψ C ψ∗, the conjugate spinor to ψ . 2 ≡ 2 2 2

µν 2 µν 2 ∗ µν 2 ∗ −1 C iω Σµν iω Σµν 1 iω [ C2Σµν C2 ] C ψ C e ψ ∗ = C e− C− C ψ∗ = e − ψ (A.24) 2 → 2 2 2 2 2 2 2

170 With a proper choice of the matrix C2, this spinor must transform either as a 2 or 2.

It is easily shown that C cσ , with a non-zero constant c, leads to 2 ≡ 2

2 C 2 1 2 2 Σ C Σ ∗C− = σ Σ ∗σ = Σ (A.25) µν ≡ − 2 µν 2 − 2 µν 2 µν

C 2 C 2 and so ψ2 transforms like ψ2. There is no choice of C2 such that Σ = Σ . Similar logic applied to ψ leads to the same conclusions, that with C cσ , 2 2 ≡ 2

2 C 2 1 2 2 Σ C Σ ∗C− = σ Σ ∗σ = Σ . (A.26) µν ≡ − 2 µν 2 − 2 µν 2 µν

Because 2 and 2 representations can be interchanged in this way, it is possible to

formulate a general theory of fermions in 4 dimensions using only 2 or 2 Weyl spinors.

This has been done in this work, choosing 2 (or left-handed) representations of Weyl

fermions to represent the 4D ﬁelds of the Standard Model.

The useful Dirac and Weyl spinor invariants of the theory, which follow from the transformation properties above in equations (A.23) and (A.7), are

Dirac Weyl

ψ4ψ4 ψ2†ψ2

† ψ4γ5ψ4 ψ2ψ2 (A.27) µ µ ψ4γµψ4p ψ2†σµψ2p µ µ † ψ4γµγ5ψ4p ψ2σµψ2p . Here, pµ is some Lorentz vector. The relationships between the above invariants:

† † ψ4ψ4 = ψ2ψ2 + ψ2ψ2 (A.28)

ψ γ ψ = ψ†ψ ψ†ψ (A.29) 4 5 4 2 2 − 2 2 µ µ µ † † ψ4γµψ4p = ψ2σµψ2p + ψ2σµψ2p (A.30)

µ µ µ ψ γ γ ψ p = ψ†σ ψ p + ψ†σ ψ p (A.31) 4 µ 5 4 − 2 µ 2 2 µ 2

171 A.3 d =6 Dimensions

Let the 6D Dirac spinor matrices be the following:

Γ0 1 1 σ (A.32) ≡ 2 × 2 × 1 Γi σ σ iσ (A.33) ≡ i × 3 × 2 Γ5 1 σ iσ (A.34) ≡ 2 × 1 × 2 Γ6 1 σ iσ (A.35) ≡ 2 × 2 × 2

M N MN These clearly satisfy the Cliﬀord algebra Γ , Γ =2η 18. The gamma matrices n o can be combined as follows

M M 0 A Γ M (A.36) ≡ A 0 ! AM (1 1 , σ σ , 1 σ , 1 σ ) (A.37) ≡ 2 × 2 i × 3 2 × 1 2 × 2 AM (1 1 , σ σ , 1 σ , 1 σ ) . (A.38) ≡ 2 × 2 − i × 3 − 2 × 1 − 2 × 2

Deﬁne Γ7:

Γ Γ7 Γ0Γ1Γ2Γ3Γ5Γ6 = 1 1 σ (A.39) 7 ≡ ≡ − − 2 × 2 × 3

M 2 Γ7 satisﬁes the properties it should: Γ7, Γ = 0, Γ7† = Γ7, and (Γ7) = 18. The n o simplest non-zero trace of Γ7 is

Tr Γ ΓM ΓN ΓP ΓQΓRΓS = 8εMNPQRS. (A.40) 7 − h i

Γ7 defines 6D chirality. Those fields which are odd under Γ7 (such as ψ4 defined below)

are termed left-handed, while those which are even under Γ7 (ψ4) are right-handed.

The Dirac spinor representation generators in this basis are

8 i i AM AN AN AM 0 ΣMN [ΓM , ΓN ]= − (A.41) ≡ 4 4 0 AM AN AN AM − ! 4 ΣMN 0 4 . (A.42) ≡ 0 ΣMN ! 172 M 4 4 This basis is 6D chiral. Because the A are hermitian, we have ΣMN† = ΣMN . The

independent elements of the Σ matrices are

Σ8 = i σ σ σ 0i 2 i × 3 × 3 Σ8 = i 1 σ σ 05 2 2 × 1 × 3 Σ8 = i 1 σ σ 06 2 2 × 2 × 3 Σ8 = 1 ε σ 1 1 (A.43) ij 2 ijk k × 2 × 2 Σ8 = 1 σ σ 1 i5 2 i × 2 × 2 Σ8 = 1 σ σ 1 i6 − 2 i × 1 × 2 Σ8 = 1 1 σ 1 56 2 2 × 3 × 2

Σ4 = i σ σ Σ4 = i σ σ 0i 2 i × 3 0i − 2 i × 3 Σ4 = i 1 σ Σ4 = i 1 σ 05 2 2 × 1 05 − 2 2 × 1 Σ4 = i 1 σ Σ4 = i 1 σ 06 2 2 × 2 06 − 2 2 × 2 Σ4 = 1 ε σ 1 Σ4 = 1 ε σ 1 (A.44) ij 2 ijk k × 2 ij 2 ijk k × 2 Σ4 = 1 σ σ Σ4 = 1 σ σ i5 2 i × 2 i5 2 i × 2 Σ4 = 1 σ σ Σ4 = 1 σ σ i6 − 2 i × 1 i6 − 2 i × 1 Σ4 = 1 1 σ Σ4 = 1 1 σ . 56 2 2 × 3 56 2 2 × 3 4 4 Note that the Σ0i and Σij here are the same as those deﬁned in the previous section in equations (A.21) and (A.21) for the 4 dimensional space. This means that we have a basis in which the 6D and 4D components are simultaneously chiral.

Let us deﬁne the 6D Dirac spinor ψ8 and the 6D Weyl spinors ψ4 and ψ4 by their

0 transformation properties, and let ψ ψ†Γ . 8 ≡ 8

iωMN Σ4 ψ ψ′ e MN ψ (A.45) 4 → 4 ≡ 4 iωMN Σ4 ψ ψ′ e MN ψ 4 → 4 ≡ 4 iωMN Σ8 ψ ψ′ e MN ψ 8 → 8 ≡ 8 0 iωMN Σ8 ψ (ψ′ )† Γ = ψ e− MN 8 → 8 8

173 The ﬁeld ψ8 is made up of a ψ4 and ψ4. Further, if we consider a compactiﬁcation down to 4 dimensions, each of the 6D Weyl spinors ψ4 and ψ4 are made up of 4D

Weyl spinors ψ2 and ψ2. This gives us

ψ2 ψ4 ψ2 ψ8   . (A.46) ∼ ψ ! ∼ ψ 4  2     ψ2    The ordering of the 4D Weyl spinors in ψ8 can be seen by considering the form of γ5

in 6D:

5 Γ Γ iΓ0Γ1Γ2Γ3 = 1 σ σ = γ σ (A.47) 5 ≡ ≡ − 2 × 3 × 3 5 × 3

When considering 6D ﬁelds, this is the gamma matrix which deﬁnes the 4D chirality

of those ﬁelds.

C On to conjugate spinors. Consider the transform of the spinor ψ C ψ∗: 4 ≡ 4 4

MN 4 ψC C eiω Σµν ψ ∗ (A.48) 4 → 4 4 MN 4 ∗ iω ΣMN 1 = C4e− C4− C4ψ4∗ (A.49)

MN 4 ∗ −1 iω [ C4ΣMN C4 ] C = e − ψ4 (A.50)

The choice of C cσ σ leads to 4 ≡ 2 × 1

4 C 4 1 4 4 Σ C Σ ∗ C− = (σ σ ) Σ ∗ (σ σ ) = Σ . (A.51) MN ≡ − 4 MN 4 − 2 × 1 MN 2 × 1 MN

C 4 C 4 The spinor ψ4 transforms like ψ4. There is no choice of C4 such that Σ = Σ .

Similar logic applied to ψ leads to the same conclusions. With C cσ σ , 4 4 ≡ 2 × 1 4 C 4 ΣMN = ΣMN . The Weyl spinors in 6D are self-conjugate. This is diﬀerent than in

4D, where conjugacy switches the chirality of the 4D Weyl spinors.

We can also consider the conjugacy of the 6D Dirac spinor: ψ C ψ∗. Because 8 ≡ 8 8 the 4 and 4 Weyl spinor parts of the Dirac spinor are separated, and because the 4 and

174 12 4 are self-conjugate with the same C4, ψ8 will be self-conjugate with C8 C4 , ≡ × σ3 ! where we can take either choice in the last part of the tensor product.

The useful invariants of the theory, following from the transformation properties of the spinors, are Dirac Weyl

ψ8ψ8 ψ4†ψ4

† ψ8Γ7ψ8 ψ4ψ4 (A.52) M M ψ8ΓM ψ8p ψ4†AM ψ4p M M † ψ8ΓM Γ7ψ8p ψ4AM ψ4p . The relationships between the above invariants:

† † ψ8ψ8 = ψ4ψ4 + ψ4ψ4 (A.53)

ψ Γ ψ = ψ†ψ ψ†ψ (A.54) 8 7 8 4 4 − 4 4 M M M † † ψ8ΓM ψ8p = ψ4AM ψ4p + ψ4AM ψ4p (A.55)

M M M ψ Γ Γ ψ p = ψ†A ψ p + ψ†A ψ p (A.56) 8 M 7 8 − 4 M 4 4 M 4

Deﬁne the following projection operators, which project onto various 4D and 6D chiralities:

1 P 1 + Γ (A.57) 4R ≡ 2 8 5 1 P 1 Γ (A.58) 4L ≡ 2 8 − 5 1 P (1 +Γ ) (A.59) 6R ≡ 2 8 7 1 P (1 Γ ) (A.60) 6R ≡ 2 8 − 7

More compact deﬁnitions are

1 P 1 + αΓ (A.61) 4α ≡ 2 8 5 1 P (1 + βΓ ) (A.62) 4β ≡ 2 8 7 175 where α, β = 1. ± Here are some useful commutation relations, easily proven in the basis given in this section:

C8, Γ5 =0 (A.63) n o [C8, Γ7]=0 (A.64)

Γ5, Γ7 =0 (A.65) h i µ Γ , Γ5 =0 (A.66) n o 5,6 Γ , Γ5 =0 (A.67) h o These lead to the following facts about the chiral projection operators

C8P4α = P4( α)C8 C8P6β = P6βC8 − µ µ M M Γ P4α = P4( α)Γ Γ P6β = P6( β)Γ − − 5,6 5,6 Γ P4α = P4αΓ (A.68) Γ5P4α = αP4α Γ7P6β = βP6β

[Γ7,P4α] = 0 Γ5,P6β = 0 h i [P4α,P6β] = 0.

176 APPENDIX B

DIAGONALIZATION OF 2 2 MATRICES ×

B.1 General Notation

Let Y be a general 2 by 2 complex matrix:

a b Y (B.1) ≡ c d !

This matrix can be diagonalized by a bi-unitary transformation

Y V Y U (B.2) Diag ≡ where V and U are 2 by 2 unitary matrices. We can parameterize such matrices with

1 real rotation angle and 3 phases:

iρ1 iφ1 iα1/2 cos θ1e sin θ1e iα1/2 c1 s1 V e iφ1 − iρ1 e − (B.3) ≡ sin θ1e− cos θ1e− ! ≡ s1∗ c1∗ !

iρ2 iφ2 iα2/2 cos θ2e sin θ2e− iα2/2 c2 s2∗ U e iφ2 iρ2 e . (B.4) ≡ sin θ2e cos θ2e− ≡ s2 c∗ − ! − 2 ! V and U have the correct properties for U(2) = SU(2) U(1) representations: ×

det V = eiα1 det U = eiα2 (B.5) 1 0 V †V = VV † = U †U = UU † = (B.6) 0 1 !

177 Written out explicitly, the expression for our diagonalized matrix is

Y V Y U (B.7) Diag ≡ c s a b c s∗ eiα1/2 1 − 1 eiα2/2 2 2 (B.8) ≡ s∗ c∗ c d s2 c∗ 1 1 ! ! − 2 ! i = e 2 (α1+α2) × ac c cs c bc s + ds s ac s∗ cs s∗ + bc c∗ ds c∗ 1 2 − 1 2 − 1 2 1 2 1 2 − 1 2 1 2 − 1 2 . (B.9) as∗c + cc∗c bs∗s dc∗s as∗s∗ + cc∗s∗ + bs∗c∗ + dc∗c∗ 1 2 1 2 − 1 2 − 1 2 1 2 1 2 1 2 1 2 ! B.2 Approximate Diagonalization for Hierarchical Matrices

For this section, assume that our matrix is hierarchical: a, b, c d. There will ≪ be a small angle (θ1 and θ2) solution to the diagonalization. Requiring that the oﬀ- diagonal elements of the diagonalized matrix (B.9) vanish, and keeping only lowest-

a b c order factors of si, d , d , and d , we have

0 = ac s∗ cs s∗ + bc c∗ ds c∗ bc c∗ ds c∗ (B.10) 1 2 − 1 2 1 2 − 1 2 ≃ 1 2 − 1 2

i(ρ1 ρ2) iρ2 i(ρ1 ρ2) iρ2 = b c c e − ds c e− be − ds e− (B.11) | 1|| 2| − 1| 2| ≃ − 1 implies b s eiρ1 (B.12) 1 ≃ d

0 = as∗c + cc∗c bs∗s dc∗s cc∗c dc∗s (B.13) 1 2 1 2 − 1 2 − 1 2 ≃ 1 2 − 1 2

i( ρ1+ρ2) iρ1 i( ρ1+ρ2) iρ1 = c c c e − d c s e− ce − ds e− (B.14) | 1|| 2| − | 1| 2 ≃ − 2 implies c s eiρ2 . (B.15) 2 ≃ d

Under these approximations, the diagonal elements of (B.9) become

i 11 = e 2 (α1+α2) (ac c cs c bc s + ds s ) (B.16) 1 2 − 1 2 − 1 2 1 2

178 i bc e 2 (α1+α2)ei(ρ1+ρ2) a (B.17) ≃ − d !

i 2 (α1+α2) 22 = e (as1∗s2∗ + cc1∗s2∗ + bs1∗c2∗ + dc1∗c2∗) (B.18)

i (α1+α2) i(ρ1+ρ2) c∗ b∗ e 2 e− d + c + b (B.19) ≃ d∗ d∗ ! i (α1+α2) i(ρ1+ρ2) e 2 e− d. (B.20) ≃

We can use the phase combinations (α1 + α2) and (ρ1 + ρ2) to make the diagonalized

elements real. This leaves one combination of ρ1 and ρ2 free, and so we can use it

to make ρ1 = 0. This makes s1 slightly simpler. Summarizing, this approximate

diagonalization for the case of hierarchical matrices leads to

b s c eiρ1 1 (B.21) 1 ≃ d 1 ≃ ≡ c s eiρ2 c eiρ2 (B.22) 2 ≃ d 2 ≃ 1 bc (α1 + α2)+(ρ1 + ρ2) arg a (B.23) 2 ≃ − − d ! 1 (α + α ) (ρ + ρ ) arg(d) (B.24) 2 1 2 − 1 2 ≃ − bc a d 0 YDiag − . (B.25)

≃ 0 d ! | | B.3 Exact Diagonalization for 12 = 21 and 11=0 | | | |

0= λ2 λT + D (B.27) −

179 where T and D are the trace and determinant of the matrix, respectively. The

solutions are

1 T 4D λ = T √T 2 4D = 1 1 . (B.28) ± 2 ± − 2  ± s − T 2  h i   D We can see that if T 2 is real, our eigenvalues are somewhat simpliﬁed. It will be easier to diagonalize Y in two steps. In the ﬁrst, we will use the unitary

matrices

eiη1 0 eiη2 0 iη1 and iη2 (B.29) 0 e− ! 0 e− !

to simplify the eigenvalues of the diagonalized matrix.

This leads to eigenvalues for this new matrix

i T 4D d e 2 (β+γ+πz) 4 b 2 λ = 1 1 = 1 1+ (B.34) 2 | | v | |2 ± 2  ± s − T  2  ± u d  u | |    t  λ = b y + y2 +1 . (B.35) | ±| | | ± q

After inverting these equations, we have

d λ+ λ b = λ+ λ d = λ+ λ y | | = | |−| −| . (B.36) | | | || −| | | | |−| −| ≡ 2 b q 2 λ+ λ | | | || −| q

180 Now for the second part of the diagonalization. We can use the unitary matrices

V and U deﬁned in equations (B.3) and (B.4) to diagonalize the new matrix Y ′.

Y VY ′U (B.37) Diag ≡ c s 0 b′ c s∗ eiα1/2 1 − 1 eiα2/2 2 2 (B.38) ≡ s∗ c∗ c′ d′ s2 c∗ 1 1 ! ! − 2 ! i = e 2 (α1+α2) × c′s c b′c s + d′s s c′s s∗ + b′c c∗ d′s c∗ − 1 2 − 1 2 1 2 − 1 2 1 2 − 1 2 (B.39) c′c∗c b′s∗s d′c∗s c′c∗s∗ + b′s∗c∗ + d′c∗c∗ 1 2 − 1 2 − 1 2 1 2 1 2 1 2 ! i (α1+α2) Isolate the 12 and 21 elements (ignoring the overall phase term e 2 , which cannot play any part in setting the 12 and 21 elements to 0):

12 = c′s s∗ + b′c c∗ d′s c∗ (B.40) − 1 2 1 2 − 1 2 ′ ′ ′ i(γ +φ1 φ2) i(β +ρ1 ρ2) i(δ +φ1 ρ2) = b s s e − + b c c e − d s c e − (B.41) −| || 1|| 2| | || 1|| 2| −| || 1|| 2| ′ ′ i(β +ρ1 ρ2) i(σ2 σ1) i(δ σ1) e − b c c b s s e − d s c e − (B.42) ≡ | || 1|| 2|−| || 1|| 2| −| || 1|| 2| h i 21 = c′c∗c b′s∗s d′c∗s (B.43) 1 2 − 1 2 − 1 2 ′ ′ ′ i(γ ρ1+ρ2) i(β φ1+φ2) i(δ ρ1+φ2) = b c c e − b s s e − d c s e − (B.44) | || 1|| 2| −| || 1|| 2| −| || 1|| 2| ′ ′ i(γ ρ1+ρ2) i(σ1 σ2) i(δ σ2) e − b c c b s s e − d c s e − (B.45) ≡ | || 1|| 2|−| || 1|| 2| −| || 1|| 2| h i where

σ β′ + ρ φ (B.46) 1 ≡ 1 − 1

σ γ′ + ρ φ . (B.47) 2 ≡ 2 − 2

YDiag being diagonal implies that 12=21=0. Consider the following cases:

b =0= d no constraints (B.48) | | | | ⇒ b =0 = d c s =0= c s (B.49) | | 6 | | ⇒ | 1|| 2| | 2|| 1| b =0= d σ σ =2πZ and cos(θ + θ )=0 (B.50) | | 6 | | ⇒ 1 − 2 1 2 181 The most general case, with b = 0 = d leads immediately to the conclusion that | | 6 6 | | all of the trigonometric functions are non-zero. This allows us to further manipulate

our equations by dividing by cosines and leaving us with tangents:

′ ′ i(β +ρ1 ρ2) i(δ σ1) i(σ2 σ1) 12 = b c c e − 1+2y t e − + t t e − (B.51) −| || 1|| 2| − | 1| | 1|| 2| h i ′ ′ i(γ ρ1+ρ2) i(δ σ2) i(σ1 σ2) 21 = b c c e − 1+2y t e − + t t e − (B.52) −| || 1|| 2| − | 2| | 1|| 2| h i 12 = 21 = 0 then implies

′ i(δ σ1) i(σ2 σ1) 0 = 1+2y t e − + t t e − (B.53) − | 1| | 1|| 2| ′ i(δ σ2) i(σ1 σ2) 0 = 1+2y t e − + t t e − . (B.54) − | 2| | 1|| 2|

′ i(δ σ1) Comparing the ﬁrst equation with the conjugate of the second leads us to t e − = | 1| ′ i(δ σ2) t e− − , which implies t = t and σ + σ = 2δ′ +2πZ. Combining this | 2| | 1| | 2| 1 2 information into one equation, we have

′ ′ i(δ σ1) 2 i2(δ σ1) 2 0= 1+2y t e − + t e − 1+2yt + t . (B.55) − | 1| | 1| ≡ − 1 1

The solutions:

t = y y2 +1 (B.56) 1 − ± q ′ i(δ σ1) We have t t e − , and the our solutions for t and t are 1 ≡| 1| | 1| | 2| λ 2 ± t1 = t2 = t y + y +1= v| | (B.57) | | | | | ±| ≡ ± u λ q u| ∓| with t

′ ′ i(δ σ1) i(δ σ2) e− − = e− − = 1. (B.58) ∓

The tangent solutions lead to solutions for the sine and cosine functions:

t 1 λ s1 = s2 = s = | ±| = = ± (B.59) | | | | | ±| 2 λ∓ sλ+ + λ 1+ t 1+ − | ±| λ± q r 182 1 1 λ c1 = c2 = c = = = ∓ (B.60) | | | | | ±| 2 λ± sλ+ + λ 1+ t 1+ − | ±| λ∓ q r Applying these solutions leads to

i (α1+α2) Y = e 2 Diag × c′s c b′c s + d′s s c′s s∗ + b′c c∗ d′s c∗ − 1 2 − 1 2 1 2 − 1 2 1 2 − 1 2 (B.61) c′c1∗c2 b′s1∗s2 d′c1∗s2 c′c1∗s2∗ + b′s1∗c2∗ + d′c1∗c2∗ ! − − ′ 1 1 i i(γ +φ +ρ ) (α1+α2) λ e 0 2 ′ = e −| ±| i(β φ1 ρ2) (B.62) 0 λ e − − ! | ∓| λ 0 | ±| . (B.63) ≡ 0 λ ! | ∓|

183 APPENDIX C

LOOP REPHASING METHOD

The purpose of this Appendix is to present a method to easily count and place the non-removable phases of the quark mass matrices. Although Kusenko and Shrock

[126] have already given a procedure for this task, we feel that our method is in practice easier and faster to use.

For ease of reference, we present our method in a concise form here. Explanation and deﬁnitions of terms will follow.

The loop rephasing method: Form the combined quark mass matrix and ﬁnd all of its loops. The number of loops in a set of basis loops is equal to the number of non-removable phases. The non-removable phases may be placed on any non-zero elements subject to the constraint that every loop has at least one corner with a phase.

A more thorough description follows, including deﬁnitions of the combined quark matrix, loops, corners, and sets of basis loops. Throughout this description we will be considering as an example model III from Table 5.1:

0 0 X 0 X 0 U D Y =  0 X 0  Y =  X X X  (C.1) X 0 X 0 X X         We deﬁne the combined quark matrix Y as the 3 by 6 matrix formed when ad- joining the up and down quark matrices. The 3 rows correspond to the 3 families

184 of SU(2) doublet ﬁelds Qi, and the 6 columns correspond to the SU(2) singlet ﬁelds

c c (u ,d )k.

c u1 uc XXXXXX  2  uc uc Q Y = (Q , Q , Q ) XXXXXX  3  (C.2) i ij dc 1 2 3    dc  !j  1  XXXXXX  c     d     2   dc   3    Although we are concerned speciﬁcally with the rephasing of combined quark matri-

ces with three generations, the method presented here can be applied to any ﬁnite

complex matrix whose ﬁelds can be rephased independently.

For model III, the combined quark matrix is

U D 0 0 X 0 X 0 0 0 Y13 0 Y12 0 U D D D Y III =  0 X 0 X X X  =  0 Y22 0 Y21 Y22 Y23  . (C.3) X 0 X 0 X X Y U 0 Y U 0 Y D Y D    31 33 32 33      Consider horizontal and vertical lines which connect non-zero matrix elements.

We deﬁne a loop to be a set of such lines which traces a closed path, provided that no two lines are on top of one another. We further deﬁne that a corner element of a loop be an element where the loop path makes a 90◦ turn.

In the combined quark matrix Y III, we may form 3 loops:

0 0 X 0 X 0 0 0 X 0 X 0 0 0 X 0 X 0  0 X 0 X X X   0 X 0 X X X   0 X 0 X X X  (C.4) X 0 X 0 X X X 0 X 0 X X X 0 X 0 X X             loop1 loop2 loop3

For each loop there is a corresponding rephase-invariant combination of elements.

At each corner of the loop there is a non-zero matrix element. Multiply these corner elements together with the prescription that as the loop is traversed the elements should be taken as alternating with and without a complex conjugate. The phase of

185 this product is the phase of the loop. Our convention is that we start on the left end

of the upper row of the loop and take this element without a complex conjugate.

We now prove that the phase corresponding to a loop is invariant under rephasing.

Consider a matrix M which contains loops. Consider further one of those loops, and further still a row (call it the ith row) which contains corners of that loop. Along this row, the corners of the loop must be connected pair-wise by horizontal lines. This follows from the deﬁnition of a loop and of loop corners. Let θ be the phase associated with the loop:

θ = arg(...MijMik∗ ...MilMim∗ . . .) (C.5) where we have shown only those corner pairs in the ith row. Rephase the ith row

iφ iφ ﬁeld by e so that the new matrix has the ith row elements: M ′ e M for all j. ij ≡ ij The phase of the loop in the new matrix will be:

θ′ arg(...M ′ M ′∗ ...M ′ M ′∗ . . .) (C.6) ≡ ij ik il im

iφ iφ ∗ iφ iφ ∗ = arg(. . . e Mij e Mik . . . e Mil e Mim . . .)

= arg(...MijMik∗ ...MilMim∗ . . .)

= θ.

These same arguments clearly hold for columns as well. Therefore, rephasing any row or column cannot change θ and so the phase of the loop is a rephase-invariant quantity.

For model III the phases corresponding to the three loops are:

U D D U θ1 = arg(Y13 Y12 ∗ Y32 Y33∗) (C.7)

D D D D θ2 = arg(Y22 Y23 ∗ Y33 Y32 ∗)

186 U D D D D U θ3 = arg(Y13 Y12 ∗ Y22 Y23 ∗ Y33 Y33∗).

D Note that Y32 does not appear in the expression for θ3 since the element does not lie

on a corner of the loop.

The phases of the loops of a matrix are not necessarily independent. This can

be seen in our example: θ1 + θ2 = θ3. There is a parallel between adding phases and combining loops. To add a loop, take the loop in a clockwise direction starting from the left end of the upper row of the loop. To subtract a loop, take the loop in the counter-clockwise direction starting from the same place. When adding or subtracting loops, if two loops have congruent lines with opposite directions, cancel these lines out. The resulting loop will correspond to a phase which is the sum (or diﬀerence) of phases of the input loops.

For the speciﬁc case of model III, we can see that loop1 + loop2 = loop3:

0 0 X 0 X 0 0 0 X 0 X 0  0 X 0 X X X   0 X 0 X X X  (C.8) X 0 X 0 X X → X 0 X 0 X X        

We deﬁne a set of basis loops for a matrix as the minimal set of loops from which all loops of that matrix can be made by addition and subtraction. For the following, let the number of loops in a basis be equal to N. We will now prove claims relating

N to the number and placement of the non-removable phases.

First, we claim that the number of phases necessary to parameterize the invariant phases of a matrix is equal to N. Our proof: Given a basis of loops for a matrix, all possible loops of that matrix can be made from linear combinations of these basis loops. Similarly, all invariant phases associated with the loops can be parameterized by the basis loop phases. As proven later in this Appendix, any loop-free matrix can

187 be made real by rephasing. Thus all invariant phases are associated with loops, and

all invariant phases in a matrix are parameterized by the N phases of the basis loops.

Our second claim is that it is always possible to place the N non-removable phases into N diﬀerent elements of the matrix while choosing the rest of the matrix elements to be real. As proof we give a procedure by which we place these N phases into a

conﬁguration reachable by rephasing a matrix of complex elements. For the following,

we assume N 3. The procedure clearly works for N =1 or 2. ≥ 1) Choose any loop, then choose any corner of that loop. Place a phase on that

corner and choose all other corners of that loop to be real.

2) Choose another loop, independent of the ﬁrst, which does not contain the corner

where we have placed the ﬁrst phase.26 Because this loop is independent from the

ﬁrst, it must contain new corners not contained in the ﬁrst loop. Among these new

corners, choose one to contain a phase and choose the rest to be real.

3) Choose another loop, independent of the previous loops, which contains no

phased corners. Again, it is possible to choose such a loop by combining any new,

independent loop with the previous loops in such a way that we achieve a new loop

which contains no previously phased corners. Once we have our desired new loop,

among the new corners reached by this loop give one corner a phase and choose the

rest to be real.

4) Repeat step 3) until there have been N loops chosen and N phases placed.

5) Choose as real all elements which are not corners of any loops.

26It is always possible to choose such a loop because: a) it is possible to choose a loop that is independent from the first (because N > 1), and b) if this second loop did contain the phased corner, we could find a third loop (independent from the first) which did not contain that corner by adding/subtracting the first and second loops.

188 Since they are independent of each other, the chosen set of loops will form a basis.

Further, the set will be such that each loop in the basis has exactly one corner with a phase. This implies that each basis phase is in one to one correspondence with an arbitrary matrix element phase. Therefore, by altering these arbitrary matrix phases, we can independently alter the basis phases.

All invariant loop phases θi can be parameterized by these basis phases θα:

N b θi = aiαθα. (C.9) αX=1 b The a have values 1 or 0 and depend only on the geometry of the matrix loops. iα ±

For any given θi, we can alter the independent and arbitrary θα to give θi any value we choose. This implies our phase choices have resulted in ab matrix in which every loop contains a corner with a phase. We show later that such a situation can be achieved by rephasing the ﬁelds of the matrix starting from arbitrary complex entries.

Therefore, starting from a matrix of arbitrary complex elements, we have shown that it is possible to place the N non-removable phases in N diﬀerent elements of the matrix, while making all other elements real.

In our example involving model III, any two of the loops may be considered as a basis set and so there are two non-removable phases. As a parallel, it is also clear that any two of the θi may combine to give the third.

Now to determine where the non-removable phases can be placed. Our claim here is that to have a correct rephasing of the matrix it is necessary and suﬃcient to place phases in the matrix in such a way that all loops have at least one corner with a phase. As stated above, the minimal number of phases necessary to do this is equal to N, the number of loops in a set of basis loops.

189 Argument for the necessity of our claim: If after placement of phases there is a loop with no phases on corner elements, then the invariant phase corresponding to that loop is zero. However, starting from a complex matrix with arbitrary phases there is no way to rephase the ﬁelds to set an invariant phase to a speciﬁc value (zero here). Therefore, for the rephasing to be correct, each loop must have at least one corner with a phase.

Argument for the suﬃciency of our claim: Let our matrix be A. Place phases in

A as described above so that each loop has at least one corner with a phase. Form a new matrix B from A by setting to zero those elements where phases were just placed.

By deﬁnition, B must have no loops. There exists a rephasing procedure, described below, by which all elements in a loop-free matrix such as B may be made real. The same rephasing procedure on a matrix of the form A with arbitrary phases in all elements will lead to the placement of phases we achieved by our method. Therefore, the phase placement chosen for A results in a conﬁguration which can be achieved by rephasing a generally complex matrix A.

The rephasing procedure for a loop-free matrix is as follows. First draw all possible horizontal and vertical lines between the non-zero elements of the matrix. Define the term tree to refer to each set of connected paths. Choose a tree, and choose a row or column of the matrix which contains an element of that tree. For simplicity of language, we imagine starting with a row. For all non-zero elements in the row (all must be connected in this single tree), use the corresponding column field phases to make these elements real. Now find all other elements within the columns just used in rephasings. Note that each of these elements are connected to the previous elements by vertical lines. Because their column fields have been used already, these elements

190 must be rephased by using their individual row ﬁelds. This is possible since no two

of these elements may be in the same row, a fact which follows from the loop-free

structure of the matrix and is proven later.

Now ﬁnd all other elements in the rows just used. Note again that these newly found elements are connected to those just rephased by horizontal lines. Rephase these new elements by their column ﬁelds. Continue this procedure, alternating between rows and columns, until all of the elements within the tree are real.

Now we show that during our procedure, when rephasing in rows (columns), no two elements to be rephased can be in the same row (column). For simplicity, we take the case of rephasing in rows. The argument clearly also applies to columns.

Suppose that when rephasing by rows, two of the elements to be rephased lie in the same row. Because our procedure rephases elements connected by lines to those we have rephased before, any element we reach in the procedure has a path back to the original (connected) row. Our two elements we are considering are connected to each other through two paths: the horizontal line between them, and the path which goes through the starting row of the procedure. The existence of two paths connecting two elements implies the existence of a loop in our supposed loop-free matrix. The contradiction leads us to conclude that our supposition is false. Therefore, when rephasing by rows, no two elements to be rephased will lie in the same row.

Our procedure reaches the whole tree in a finite number of steps because (by assumption) the matrix is finite which means the tree is finite, and because each step necessarily reaches more of the tree. Other trees within the matrix, by definition, have no elements on the rows or columns used by any other trees. Therefore, the

191 trees of a matrix may be rephased independently, one after another, by the procedure

described above.

As for our model III example, in section 5.3.2 we chose to place the non-removable

U D phases in Y13 and Y32 . This places a phase on at least one corner of each loop. Removal

of the elements at which we have placed phases leads us to

D 0 0 0 0 Y12 0 0 0 0 0 X 0 U D D D  0 Y22 0 Y21 Y22 Y23  =  0 X 0 X X X  . (C.10) Y U 0 Y U 0 0 Y D X 0 X 0 0 X  31 33 33       

There is one tree. We now go through our rephasing procedure for which we choose

U U D to begin with the bottom row. Use columns 1, 3 and 6 to make Y31 , Y33 and Y33 real.

D The only other non-zero entry in these three columns is Y23 , which we rephase by

U D D row 2. In this row, there are three other non-zero entries: Y22 , Y21 and Y22 . Rephase

D these by columns 2, 4 and 5 respectively. There is one other element, Y12 , in these

columns. Rephase it by row 1. There are no other elements in this row, and we have

reached the end of the tree.

When performed on the combined quark matrix Y III starting from an arbitrary

set of phases on each element, this same rephasing procedure will lead to the desired

U D situation of having phases on only Y13 and Y32 . Note that this phase choice is not

U D unique. For example, we could instead have placed phases on Y33 and Y23 .

192 APPENDIX D

D3 FAMILY SYMMETRY

In this Appendix, we deﬁne the D3 group, give its character table, and give other information necessary to understand the couplings and representations used in our model. Our presentation is an abbreviation of a more complete description given in

[108].

D3 is the group of all rotations in three dimensions which leave an equilateral

2 triangle invariant. The group contains six elements in three classes: E; C3, C3 ; Ca,

2 Cb, Cc. E is the identity element. C3 and C3 signify 120◦ and 240◦ rotations about

an axis through the center of and perpendicular to the triangle. Ca, Cb, and Cc

signify 180◦ rotations about the three diﬀerent axes which run from the center to the three vertices of the triangle. We choose orientations such that Cb = CaC3 and

2 Cc = CaC3 . Figure D.1 contains a graphical representation of our conventions for the group elements. The group has two inequivalent singlet representations: 1A and 1B,

and one doublet representation: 2A.

The character table is

D3 E C3 Ca 1 1 1 1 A (D.1) 1 1 1 1 B − 2 2 1 0 A −

193 Ca

Cb Cc

Figure D.1: Graphical representation of our choice of D3 group elements. The solid (black) lines represent the triangle. The 3 dashed (blue) lines represent the axes about which we take the Ca, Cb, and Cc rotations. The center (red) circle and dot represent the axis about which we take the C3 rotation (it points out of the page). Our rotation convention is right-handed.

We choose our representations as follows. When acting on a one dimensional

representation, the elements are the characters in the character table. When acting

on the two dimensional representation, the elements are

1 0 ǫ 0 0 1 E = C3 = 1 Ca = , (D.2) 0 1 ! 0 ǫ− ! 1 0 !

with ǫ e2πi/3. The remaining elements can be found by group multiplication. ≡

Our Lagrangian of Section 6.2.2 contains terms with various combinations of D3

ﬁelds. It is understood that the Lagrangian only contains theD3 singlet (1A) part of these combinations. Here we list the combinations of ﬁelds we have used, and their singlet projections.

194 Let θi be 2A (doublet) ﬁelds under D3, and let the internal degrees of freedom be

xi represented by xi and yi: θi . Also, let φ be a 1B (“anti-symmetric” singlet) ≡ yi ! ﬁeld, with internal degree of freedom α: φ α. Then the combinations we need are: ≡

θ θ 1 = x y + y x (D.3) 1 × 2| A 1 2 1 2

θ θ φ 1 = (x y y x )α 1 × 2 × | A 1 2 − 1 2

θ θ θ 1 = x x x + y y y 1 × 2 × 3| A 1 2 3 1 2 3

Of course, multiplication by a “true” singlet 1A has no eﬀect concerning the D3 structure of the Lagrangian terms.

195 APPENDIX E

MASSLESS STATES AND WAVEFUNCTIONS

This appendix shows how we determine the overlap between the original La- grangian states and the massless states of Chapter 6. We choose to illustrate this through a speciﬁc example: the determination of the overlap between the Lagrangian

ﬁelds and the 3rd family left-handed states in equations (6.25) of section 6.2.2.

The relevant superpotential terms are (from equation (6.15)):

W2 ψ++ (∂y mX )(ψ ) ⊃ a − −− a πR + 2σφa ψ ψ3 δ(y)+ 2η ψ χa δ(y ) (E.1) ++ a ++ a − 2 h i h i The equations of motion are:

∂W = 0= (E.2) ∂ ψ ⇒ ++ a φa πR 0 = (∂y mX )(ψ ) +2δ(y)σ M ψ3 +2δ(y )η M χa. − −− a M ∗ − 2 ∗ ∗ q q We have added factors of M where necessary to ensure that we have unitless couplings ∗ σ and η. This equation is satisﬁed for the massless projections of these ﬁelds. Away from the branes, (ψ ) emX y. Because (ψ ) is odd under both orbifold parities −− a ∝ −− a

mX y πR and because e is not an odd function about either y = 0 or y = 2 , the proﬁle of (ψ ) must have discontinuities at both branes. These discontinuities can cancel −− a 196 the brane terms in the equations of motion. Deﬁne the overlap between ψ3 and the

massless ﬁeld ψ0 as ψ n ψ0. We can then solve for the other two ﬁelds: 3 3 ⊃ L 3 e XLζy φa πR 0 (ψ )a ε (y)e σ M nLψ3 (E.3) −− ⊃ − −− M ! ∗ ∗ q XLζ φa σ e 2 0 χa e nLψ3 (E.4) ⊃ − M ! η ∗ e What remains is to determine the normalization nL. We do this by requiring that the

0 effective 4D field ψ3 have a canonical Kahler potentiale term. Listing the fields which

contribute to this term (we have suppressed the gauge ﬁeld factors here):

πR 2 πR dy ψ3†ψ3δ(y)+ (ψ )a† (ψ )a + χa† χaδ(y ) ( a −− −− a − 2 ) Z0 X X πR 2 2 2 2 2 0 2 φa XLζy σ φa 0 2 πR XLζ ψ3 †ψ3 nL 1+ σ M dy e + e  (E.5) ∗ 2 ⊃ | |  a M ! Z η M !   X  ∗ 0 ∗  e      The integral is  

πR 2 2XLζy 1 πR dy e = 2 2 (E.6) M ρ nL Z0 ∗ which implies that for canonical normalization:

2 2 2 σ φa 1 ρ 1 n 2 1+ + eXLζ . (E.7) L  2 2 2  ≡| | ρ a M ! "nL η #  X ∗  e   Choose nL to be real:

e 1 nL (E.8) ≡ 2 1 ρ XLζ 1+ r n2 + η2 e e s L The other mixings and normalizations listed in Section 6.2.2 are calculated in the

same way.

197 APPENDIX F

A LIFTED 4D MODEL

We present here a 5D version of a 4D model by Dermisek and Raby in [108] which itself was based on prior works [127, 128, 129, 130]. Our purpose is to illustrate how such a 4D model could be placed into a 5D context and what advantages can be gained from the use of the extra direction.

In using the extra dimension, we have separated the ﬁelds on the PS brane from the important mass parameter Mχ = M0(1+αX) which gives much of the distinction

between the particle types. There are matter ﬁelds on the PS brane, and bulk matter

ﬁelds which mediate between these brane ﬁelds and the SO(10) Mχ VEV in such a

way to give us the desired Yukawa matrix elements.

The setup for this model is largely the same as that presented in Section 6.2, save

for the following points. This model does not have a kink mass in the bulk; the small

up quark mass comes instead from an approximate left-right symmetry in the model.

Here we use a B L VEV on the PS brane instead of T . Other differences lie in the − 3R placement of the matter fields and the extra VEV fields. As with the model presented

in Chapter 6, for the current model under discussion we also have in mind a setup

which includes a small 6th dimension (as detailed in Section 6.4) in order to separate

198 Field PS Symm D3 Symm ψ++ (4, 2, 1) 16 = c 2A ψ+ ! (4, 1, 2) ! ψ − (4, 2, 1) 16 = c−− 2A ψ + ! (4, 1, 2) ! −

ψ+′ (4, 2, 1) 16′ = c − 2A ψ++′ ! (4, 1, 2) ! ψ′ + (4, 2, 1) 16′ = c − 2A ψ ′ ! (4, 1, 2) ! −−

ψ + (4, 2, 1) 16 = − 2A ψc ! (4, 1, 2) ! e −− f ψ+ (4, 2, 1) 16 = f − 2A  c  (4, 1, 2) ! ψe ++ f   fψ′ (4, 2, 1) 16 = 2A ′ c −− ψ ′ + ! (4, 1, 2) ! f − g gψ′++ (4, 2, 1) 16′ = 2A  c  (4, 1, 2) ψ ′+ ! f − g   Hg++ (1, 2, 2) 10 = c 1A H+ ! (6, 1, 1) ! H − (1, 2, 2) 10 = c−− 1A H + ! (6, 1, 1) ! −

Table F.1: Bulk ﬁelds.

ﬁelds on the PS-brane from the PS-breaking VEVs. This allows us to preserve the

PS Yukawa relations on the PS-brane below the cutoﬀ scale.

The states are placed as follows. In the bulk, we have 8 matter hypermultiplets, forming 4 doublets under D3 and transforming as 16s under SO(10). Each of these

hypermultiplet doublets has a diﬀerent parity under the orbifold. The Higgs, as

before, is contained inside a 10 hypermultiplet of SO(10). These ﬁelds are listed in

Table F.1. On the SO(10) brane there is only the mass parameter Mχ, which is a

199 Field PS Symm D3 Symm ψ (4, 2, 1) 2A c ψ (4, 1, 2) 2A ψ3 (4, 2, 1) 1A c ψ3 (4, 1, 2) 1A φ (1, 1, 1) 2A φ (1, 1, 1) 2A A (1, 1, 1) 1B e A15 (1, 1, 3) 1A ΦL (1, 1, 1) 1A ΦR (1, 1, 1) 1A

Table F.2: PS Brane ﬁelds.

c singlet under D and is a mix of singlet and X under SO(10): M = λ χ χc = 3 χ χ M∗ M0(1 + αX). On the Pati-Salam brane, we have three sets of left- and right-handed matter fields. Two of these sets form a doublet under D3. In addition, there are several extra fields: φa, φa, A, A15, ΦL, and ΦR. Of these extra fields, those with

e subscript a are D3 doublets. The rest are 1A save A which is 1B under D3. All of these extra fields get nonzero VEVs except for φ1. All extra fields are SO(10) singlets, save A which is an SU(4) adjoint and gets ae VEV A = A0 (B L). The PS 15 h 15i h 15i − brane fields are listed in Table F.2.

We choose the following superpotential:

πR c c c W1 = δ(y ) λ1ψ3Hψ + λ2ψaH ψ ′ ΦR + λ2 (ψ++) Hψ ΦL (F.1) − 2 3 ++ a a a n o

W2 = 16a∂y16a + 16′a∂y16a′ + 16a∂y16a + 16′a∂y16′a + 10∂y10

f f g g +δ(y) Mχ16a16a′ + Mχ16′a16a πRf g +δ(y ) ψ′++ λ3A15φaψ3 + λ4A15φaψa + λ5Aψa ΦL − 2 a h i fc c e c c + ψ ++ λ3A15φaψ3 + λ4A15φaψa + λ5Aψa ΦR (F.2) a h i f e 200 There are 12 U(1) symmetries associated with our superpotential. We choose to

parameterize these by allowing the following 12 ﬁelds to be charged each under diﬀer-

c c c ent U(1)s: ψ3, ψ3, H+ , Mχ, ψ′ , ψ ′ + , ψ + , ΦL, A15. After specifying − −− a − a − a the U(1) symmetries of the abovef fields,g the U(1) transformate ions of all other fields are uniquely defined. There are in addition 2 Z2 symmetries. First is the Z2 orbifold parity under y y. The transformations of the bulk fields under this symmetry → − c have already been defined. In addition, let the rest of the independent fields: ψ3, ψ3,

Mχ, ΦL, A15 be even under this symmetry. The second Z2 involves a sign ambiguity in the transformation of Φ . Under a given symmetry, if Φ eiαΦ , then the R L → L superpotential terms imply that Φ einπeiαΦ with n =0, 1. We choose to require R → R that n = 1 here in order to forbid terms created by the replacement of ΦL by ΦR or

2 vice-versa. We also assume that ΦL /M 1 so that replacements like ΦL ΦR h i ∗ ≪ → are negligible.27 Let the 12 independent ﬁelds be uncharged under this symmetry.

We require all of these symmetries just listed to be symmetries of the theory so as to

forbid unwanted extra terms in the superpotential.

The superpotential has a left-right symmetry, under which ψ ψc, ψ ψc, 3 ↔ 3 ↔ 16 16′ 16 16′ , , ΦL ΦR. This symmetry, which commutes 16 ! ↔ 16′ ! 16 ! ↔ 16 ! ↔ f g′ with the family D symmetry, is broken spontaneously by the VEVs of Φ and Φ , 3 f g L R which we require to be slightly diﬀerent. This diﬀerence is encapsulated by a small

parameter η: Φ = Φ (1 + η). h Ri h Li We now turn to the equations of motion for the left-handed states:

∂W =0 = ∂y (ψ++)a =0 (F.3) ∂y ψ ⇒ −− a 27 2 2 Such terms given by the substitution ΦL ΦR or ΦR ΦL would lead to a Yukawa matrix structure diﬀerent from the one desired and so→ should be forb→idden by some symmetry.

201 ∂W =0 = ∂y ψ+′ =0 (F.4) − a ∂y ψ′ + ⇒ − a ∂W =0 = ∂y ψ + + δ(y)Mχ ψ+′ =0 (F.5) ⇒ − a − a ∂y ψ+ −a e ∂We =0 = ∂y ψ′ + δ(y)Mχ (ψ++)2 (F.6) ⇒ −− 2 ∂y ψ′++ 1 f f πR +δ(y ) λ A φ ψ + λ A φ ψ + λ Aψ Φ =0 − 2 3 15 2 3 4 15 1 1 5 2 L h i ∂W e =0 = ∂y ψ′ + δ(y)Mχ (ψ++)1 (F.7) ⇒ −− 1 ∂y ψ′++ 2 f f πR +δ(y ) λ A φ ψ + λ A φ ψ λ Aψ Φ =0 − 2 3 15 1 3 4 15 2 2 − 5 1 L h i e Solving these equations leads to knowledge of the overlap between the massless ﬁelds

and the original ﬁelds in the Lagrangian. The following relationships only have the

massless components on the right hand sides of the equations. (We have replaced the

brane ﬁelds by their VEVs)

ψ+′ 0 (F.8) − a ⊃ ψ + 0 (F.9) − a ⊃ e 1 ψ′ ε(y) λ3 A15 φ1 ψ3 + λ4 A15 φ2 ψ2 λ5 A ψ1 ΦL (F.10) −− 1 ⊃ 2 h ih i h i − h i h i h D E i f 1 e e ψ′ ε(y) [λ3 A15 φ2 ψ3 + λ5 A ψ2] ΦL (F.11) −− 2 ⊃ 2 h ih i h i h i 1 (fψ ) e λ A φ ψ + λ A φ ψ λ A ψ Φ (F.12) ++ 1 ⊃ −M 3 h 15ih 1i 3 4 h 15i 2 2 − 5 h i 1 h Li χ h D E i 1 e (ψ++)2 [λ3 A15 φ2 ψ3 + λ5 A ψ2] ΦL (F.13) ⊃ −Mχ h ih i h i h i

The equations for the right-handed states can be obtained from the left-right symmetry present in the model. We list the most important of these equations:

c 1 c c c ψ ′ λ3 A15 φ1 ψ + λ4 A15 φ2 ψ λ5 A ψ ΦR (F.14) ++ 1 ⊃ −M h ih i 3 h i 2 − h i 1 h i χ h D E i e 202 c 1 c c ψ ′ [λ3 A15 φ2 ψ + λ5 A ψ ] ΦR (F.15) ++ 2 ⊃ −M h ih i 3 h i 2 h i χ

The three ψi fields span the space of the left-handed massless states. Because the other fields which we’ve integrated out have massless components, the kinetic energy and gauge interaction terms for the ψi fields are no longer orthonormal. The same

c is true for the ψi states by the left-right symmetry. We will discuss the eﬀects of rotating and rescaling these ﬁelds to an orthonormal basis later.

c We replace the (ψ++) and ψ ′ ﬁelds in W1 by their corresponding massless a ++ a parts in order to get the low energy Yukawa matrices. The result:28

0 ε′ρ rεκTuc Yu =  ε′ρ ερ rεTuc  λ (F.16) rεκT− rεT 1  Q Q    0 ε′ rεσκTdc Yd =  ε′ ε rεσTdc  λ (F.17) rεκT− rεT 1  Q Q    0 ε′ rεκT c − e Ye =  ε′ 3ε rεTec  λ (F.18) rεσκT rεσT 1  L L    Deﬁnitions follow for these variables, where we have used Mχ = M0(1 + αX),

Φ = Φ (1 + η), and have added factors of the cutoﬀ scale in order to make the h Ri h Li couplings λ all unitless. We have also assumed η 1 and α (1). T represents i ≪ ∼ O f the (B L) quantum number for the ﬁeld f. − 0 2 λ2λ4 A15 φ2 ΦL 4α ε h i h i (F.19) D3 E ≡ 3λ1M eM0 (1 + α)(1 3α) ∗ 2 − λ2λ5 A ΦL 4α ′ ε h 2ih i (F.20) ≡ − λ1M M0 (1 + α)(1 3α) ∗ − 2η(1 3α) ρ − (F.21) ≡ 4α λ φ 3(1 3α) r 3 h 1i − (F.22) ≡ −λ4 φ2 4α D E 28We have chosen to use the notatione found in [108] to ease comparison between prior works and our own.

203 φ κ h 2i (F.23) ≡ φ h 1i 1+ α σ (F.24) ≡ 1 3α − 2Mc λ λ1 (F.25) ≡ sπM ∗ These Yukawa matrices are the same as those found in [108], except for the (1,3) and

(3,1) elements. It has been shown in [51] that (1,3) elements are needed in models of this kind in order to ﬁt sin 2β.

Here we list the lowest-order diagrams which give the Yukawa matrices. Each diagram is followed by the element(s) to which it contributes.

204 H (3,3) c ψ3 ψ3

φa H ΦR A15 ΦR Mχ X (1,3) (2,3) c c c ψa ψ++′ ψ ++ ψ3 a a f φa A15 ΦL H ΦL Mχ X (3,1) (3,2) c ψ3 ψ′++ (ψ++)a ψa a f φa H ΦR A15 ΦR Mχ X f (2,2) c c c ψa ψ++′ ψ ++ ψa a a f φa A15 ΦL H ΦL Mχ f X (2,2) c ψa ψ′++ (ψ++)a ψa a f

H ΦR A ΦR Mχ X (1,2) (2,1) c c c ψa ψ++′ ψ ++ ψa a a f

A ΦL H ΦL Mχ X (1,2) (2,1) c ψa ψ′++ (ψ++)a ψa a f We performed a χ2 analysis, the same as that used in [127], save that here we have nonzero (1,3) and (3,1) terms and a new parameter κ. The ﬁt parameters follow.

205 First the GUT parameters:

1 = 25.12 (F.26) αGUT M = 2.54 1016 GeV (F.27) G × ε = 3.61% (F.28) 3 −

Next the (modulus) Yukawa sector:

λ = 0.70 (F.29)

r = 24.8 (F.30)

σ = 1.19 (F.31)

ε = 0.0090 (F.32)

ρ = 0.061 (F.33)

ε′ = 0.0034 (F.34)

κ = 0.15 (F.35)

Next the Yukawa phase information (φ arg(x)) in radians: x ≡

φσ = 0.51 (F.36)

φ = 1.87 (F.37) ρ −

φκ = 0.89 (F.38)

The other Yukawa parameters are assumed to be real. Finally some SUSY breaking scales and tan β:

µ =234GeV (F.39)

M1/2 =606GeV (F.40)

206 Observable Target Value Fit Value χ2 Contribution 1 137.04 0.14 137.0 0.08 αEM 5 ± Gµ 10 1.1664 0.0012 1.166 0.11 α × 0.1172 ± 0.0020 0.1167 0.06 s ± MTop 178.0 4.3 176.5 0.12 m 4.220 ± 0.090 4.243 0.07 b ± M M 3.40 0.20 3.35 0.06 b − c ± ms 0.089 0.011 0.104 1.86 1 3 ± 2 10 2.03 0.20 2.00 0.02 Q × ± md 0.050 0.015 0.074 2.54 ms ± Mτ 1.7770 0.0018 1.777 0.00 M 0.10566± 0.00011 0.1057 0.13 µ ± M 103 0.51100 0.00051 0.5110 0.00 e × ± Vus 0.2230 0.0040 0.2216 0.12 V 0.0402 ± 0.0019 0.0390 0.40 cb ± Vub 0.0860 0.0080 0.0863 0.00 Vcb 3 ± εK 10 2.28 0.23 2.35 0.10 M × 91.188± 0.091 91.19 0.00 Z ± M 80.419 0.080 80.41 0.01 W ± mc 1.30 0.15 1.15 1.00 (b sγ) 103 0.334± 0.038 0.335 0.00 → × ± sin 2β 0.727 0.036 0.700 0.57 V 103 8.20 ±0.82 8.35 0.03 td × ± Total: 7.30

Table F.3: Best χ2 ﬁt. Mass dimensions are in GeV.

m16 =4160GeV (F.41)

A = 7736GeV (F.42) 0 − m 2 10 = 1.80 (F.43) m16 m 2 D = 0.128 (F.44) m16 tan β = 52.2 (F.45)

207 The fit itself can be found in Table F.3. As explained in Section 6.3.2, we are using our χ2 function as a vehicle to find the best possible fit rather than as a true statistical χ2 function. Our best χ2 is 7.30, indicating that we are not fitting some of

the data. Specifically, as shown in Table F.3, the observables m , md , and m have s ms c χ2 contributions of 1 or greater and make up the majority of the χ2 fit value. The fit

values of the down and strange quark masses are on the large side, while the charm

mass fit value is smaller than the data. We present this fit as a first attempt at this

kind of model. More work is needed to alter the model to obtain a better ﬁt.

In our analysis we have used a basis in which the massless matter ﬁelds are not or-

thonormal. Rotation and rescaling to a canonical orthonormal basis would in general

introduce changes to the Yukawa matrices. We have neglected eﬀects from this change

of basis, and our justiﬁcation for this is the following. Were a ﬁt to be done with these

eﬀects included, the input Yukawa parameters would compensate by changing their

values. We assume that the input parameters could compensate to the extent that

we would obtain essentially the same ﬁt in this case as in the case we have presented

here in the appendix without these extra eﬀects. We leave it to further research to

explore whether this assumption is a good one.

208 APPENDIX G

CALCULATION OF SPATIAL TRACES

∂2/M 2 In this Appendix, we show how to calculate the flat-space trace x e− x h | | i ∂2/M 2 in d space-time dimensions. In order to calculate x e− x , we will insert an h | | i d integration over momentum space: 1 = d k k k , use the bracket definition (2π)d | ih | ik x R x k e− · , and make the Wick rotation to Euclidean space with the definitions h | i ≡

kM k0,ki ik0 ,ki (G.1) ≡ ≡ E E 2 2 2 2 k2 k0 ki k0 ki k2 . (G.2) ≡ − ≡ − E − E ≡ − E x2 We will also use the result ∞ dxe− = √π. −∞ R

d ∂2/M 2 d k ∂2/M 2 x e− x = x d k k e− x (G.3) h | | i h | (Z (2π) | ih |) | i d d k ∂2/M 2 = d x k e− k x (G.4) Z (2π) h | i h | i d d k ik x ∂2/M 2 ik x = d e− · e− e · (G.5) Z (2π) d d k k2/M 2 = d e (G.6) Z (2π) d d kE 2 2 kE /M = i d e− (G.7) Z (2π) d ∞ dkE k2 /M 2 = i e− E (G.8) " 2π # Z−∞ 209 d M ∞ k′ 2 = i dk′ e− E (G.9) 2π E Z−∞ M d = i √π (G.10) 2π iM d = (G.11) 2dπd/2

We therefore have in d dimensions

d ∂2/M 2 iM x e− x = . (G.12) h | | i 2dπd/2

Speciﬁcally in d = 4,

4 ∂2/M 2 iM x e− x = . (G.13) h | | i (4π)2

210 APPENDIX H

CALCULATION OF GROUP TRACES

In Chapter 7, calculations showed that the anomalies were proportional to symmetrized gauge group traces

A ba1...an tr [T a1 ...T an ] (H.1) ∝ where b is totally symmetric in its indices. It is important to know how to calculate these traces given the group G, the irreducible representation ρ, and the number n of insertions of the generators T a. A paper by Okubo and Patera [131] allows us to determine these expressions up to unimportant constant factors.

Let us deﬁne

L (ρ) ba1...an tr [T a1 ...T an ] (H.2) n ≡ ρ where b is completely symmetric in its indices and the T ai are generators of the simple

Lie group G. The value of Ln(ρ) can be expressed as a sum of basis ‘indices’, which can be chosen to be traces of the fundamental Casimir operators Jn of the group

G. The number of these fundamental Casimirs is equal to the rank of the group, and the number of insertions of T a in each fundamental Casimir (called the order) is known for each simple Lie algebra. Table H.1 lists the orders of the fundamental

211 Lie Group Fundamental Casimir Order

An =SU(n+1) 2, 3,..., n+1

Bn =SO(2n+1) 2, 4,..., 2n

Cn = Sp(2n) 2, 4,..., 2n D =SO(2n) 2, 4,..., 2n 2, n n − G2 2, 6

F4 2, 6, 8, 12

E6 2, 5, 6, 8, 9, 12

E7 2, 6, 8, 10, 12, 14, 18

E8 2, 8, 12, 14, 18, 20, 24, 30

Table H.1: Fundamental Casimir orders for the simple Lie algebras, quoted from [131].

Casimirs for each simple Lie algebra. For example, the fundamental order 3 Casimir is J ga1a2a3 T a1 T a2 T a3 . This Casimir exists for the groups SU(n) for n 3, but 3 ≡ ≥ does not for all other groups. Here, ga1a2a3 is an appropriately chosen completely symmetric tensor. We will not need to know exactly how the Jn are chosen. The fact that their existence is restricted by group and that their traces determine Ln(ρ) is enough here.

Let d(ρ) be the dimension of the representation ρ, and let Dn(ρ) be the nth order fundamental index deﬁned as follows:

D (ρ) d(ρ)J (ρ)=tr J (H.3) n ≡ n ρ n

The special case of n = 0 is deﬁned to be D (ρ) d(ρ). Those D (ρ) with n > 0 0 ≡ n for which the Jn do not appear in Table H.1 are deﬁned to be zero. As explained in

[131], any Ln(ρ) can be expressed as a sum of products of these Dn(ρ), provided that

212 each term in the sum has the same total order n. In particular, we are interested in

the cases for n =0 to 4.

For n = 0, it follows directly from the deﬁnition of Ln(ρ) that

L0(ρ)= c0d(ρ). (H.4)

Here, c0 is what is left of b once the indices are taken away. Because there are no fundamental Casimirs of order 1,

L1(ρ)=0 (H.5)

for all simple Lie groups. There is only one term each in the expansions of L2(ρ) and

L3(ρ):

L2(ρ) = c2D2(ρ) (H.6)

L3(ρ) = c3D3(ρ) (H.7)

a1a2 a1a2a3 c2 and c3 are constants which depend only on the tensors b and b deﬁning

the Ln(ρ)s. Speciﬁcally, the ci are independent of the representation ρ. Since it is possible for J2 and J4 to be non-zero, L4(ρ) can contain two terms. The expansion is derived in [131]

D2(ρ) 1 D2(ρ0) L4(ρ) = c4D4(ρ)+ c4′ D2(ρ) . (H.8) " d(ρ) − 6 d(ρ0) #

ρ0 is deﬁned to be the adjoint representation of the group. For higher order Ln(ρ),

it is possible to ﬁnd the appropriate expansions. However, they are not necessary for

our work here.

In order to calculate the Dn(ρ), we will relate them (up to a proportionality

constant) to a particular Ln(ρ) which lends itself to calculation. Use the Cartan-

Weyl basis for the generators T a, and order them with the Cartan matrices T i ﬁrst,

213 α α a i α α then the raising and lowering generators T and T − , so that T = (T , T , T − ).

Let the vector va have non-zero values only in the Cartan subalgebra space: va =

(vi, 0, 0). Let ba1...an va1 ...van and form L (ρ) with that b tensor: L (ρ) ≡ n n ≡

a1 an a1 an v ...v tr [T ...T ]. It then follows that Ln(ρ)= ln′ (ρ) where

n l′ (ρ) (v, M) . (H.9) n ≡ XM The sum is over the weights M of the representation ρ and (v, M) is the inner product in weight space. These ln′ (ρ) are easily calculated and their only dependence on v is as an overall constant. Because the ln′ (ρ) are a particular kind of Ln(ρ), they are related to the Dn(ρ) as above. After some calculation, or using tables present in the literature [34, 132], we can form a table of the Dn(ρ) up to proportionality constants

present for each group and order. The resulting indices useful for this work are

tabulated in Table H.2.

Our formalism can be easily extended to include U(1) factors which are not treated

above with the simple algebras. There are only singlet representations for U(1), and

the inner product (v, M) with that state is proportional to the charge q. We therefore

have for U(1) groups

n n ln′ (q)= v q (H.10)

Let us have an index Dn(q) for U(1) groups proportional to this ln′ (q). Choose the

proportionality so that

D (q) qn (H.11) n ≡

Finally, we can consider the case of direct product gauge groups, G = G G 1 × 2 with representation ρ = ρ ρ . From the deﬁnition of L(G)(ρ) (where the trace is 1 × 2 n 214 Group ρ D2(ρ) D3(ρ) D4(ρ) 1 0 0 0 27 1 0 0 E 6 27 1 0 0 78 4 0 0 1 0 0 6 1 1 6 1 1 − SU(6) 15 4 2 15 4 2 20 6 −0 35 12 0 1 0 0 10 1 0 SO(10) 16 2 0 16 2 0 45 8 0 1 0 0 4 1 1 SU(4) 4 1 1 − 6 2 0 15 8 0 1 0 0 SU(2) 2 1 0 3 4 0

Table H.2: Some fundamental indices for selected groups. Dn(ρ) is the fundamental index of order n in the irreducible representation ρ. For those indices which are not identically zero, the unknown proportionality constant present for each set of indices given the group and order have been chosen so that the lowest dimensional fundamental representation has index equal to 1. We have included those groups, representations, and indices which will be needed in Chapter 7.

215 deﬁned explicitly over the generator space of group G), the symmetry of the tensor b, and the direct product properties of the group, we have

n (G) (G1) (G2) Ln (ρ)= Lm (ρ1)Ln m(ρ2) (H.12) − mX=0

(G) The trace Ln (ρ) breaks up into a sum over products of traces in the subgroup spaces.

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