Accidental Supersymmetry and the Naturalness of Codimension-2 Branes

ACCIDENTAL SUPERSYMMETRY AND THE NATURALNESS OF CODIMENSION-2 BRANES

MATTHEW R. WILLIAMS, M.Sc., B.Sc.

A Thesis Submitted to the School of Graduate Studies in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy

McMaster University ©Copyright by Matthew R. Williams, August 2013. DOCTOR OF PHILOSOPHY (August 2013) McMaster University (Physics) Hamilton, Ontario

TITLE: Accidental Supersymmetry and the Naturalness of Codimension-2 Branes AUTHOR: Matthew R. Williams, M.Sc. (McMaster), B.Sc. (Windsor) SUPERVISOR: Professor Cliﬀord P. Burgess NUMBER OF PAGES: xv, 235

ii Abstract

This thesis addresses two separate naturalness issues which generically come to bear on physical theories with large extra dimensions, and so a gravity scale much lower than the Planck scale. The first is related to the observed stability of the proton, wherein we determine the relevant constraints on an additional gauge boson which conserves baryon number. Although several such proposals have been previously considered, our analysis is distinctive in its interest in lighter gauge boson masses (which naturally arise in such models), and in its focus on the dependence of constraints due to kinetic mixing effects. The second is related to the main purpose of large extra dimensions—namely, to address the smallness of the observed vacuum energy—wherein we compute the leading-order quantum corrections to the four-dimensional (4D) vacuum energy resulting from loops of extra- 1 dimensional fields. We compute the contributions from bulk scalars (spin 0), fermions (spin 2 ), and gauge fields (spin 1) in a flux-stabilized, spheroidal extra-dimensional geometry whose rugby- ball shape is due to two codimension-2 branes—one at each pole. (We also obtain the corresponding beta functions for both bulk and brane operators.) These results are then combined to obtain the net contribution from various multiplets in the context of a particular supersymmetric extra-dimensional model that has been shown to give a vanishing result for the 4D vacuum energy at the classical level. Surprisingly, we find that supersymmetry can be preserved dynamically at one loop in the case of identical branes, without arranging any particular relationship between the brane parameters. Perturbing away from the case of identical branes is shown to give a positive 1-loop contribution to the 4D vacuum energy whose size is set by the radius of the extra dimensions.

iii Acknowledgements

I would like to thank my supervisor, Cliff Burgess, for suggesting these research topics; for his guidance throughout my studies; and for his countless insightful comments, particularly those regarding this thesis. I really appreciate his support, as it made my experience as a graduate student an enjoyable one. Thanks to the McMaster Physics and Astronomy department and the OSAP OG(S) scholarship program for their financial support. I would also like to thank the McMaster Physics and Astronomy department, as well as the Perimeter Institute for Theoretical Physics, for providing a stimulating and beneficial research environment. Thanks to my collaborators, Anshuman Maharana, Susha Parameswaran, Fernando Quevedo, Alberto Salvio, and Leo van Nierop for their help in preparing the research articles presented in this thesis. To the particle physics group at McMaster, including (at various times) Itay Yavin, Hyun Min Lee, Michael Lennek, Leo van Nierop, Allan Bayntun, Ross Diener, Michael Horbatsch, Andrew Louca, Matthew McCreadie, Joey Sham, and Robin Tunley: it has been a pleasure working with you all and I appreciate the many hours of fruitful discussion, particularly during our long rides in the PIPB. Special thanks to Leo for his patience and friendship throughout our time together at McMaster. Thanks to my family, for encouraging and inspiring my interest in the physical sciences. And to my wife, Nicole: thank you for always putting up with me. You are an angel and I am thankful for every day we spend together. Sophie and Rémy are blessed to have you as their mother.

iv Preface

Chapters 2 through 4 of this thesis are original papers written by the me (Matthew R. Williams) and are published in the Journal of High Energy Physics. The journal references are:

• Chapter 2—JHEP 1108, 106 (2011) [arXiv:hep-ph/1103.4556];

• Chapter 3—JHEP 1301, 102 (2013) [arXiv:hep-th/1210.3753];

• Chapter 4—JHEP 1302, 120 (2013) [arXiv:hep-th/1210.5405].

These works are collaborative; my coauthors are Drs. C.P. Burgess (all chapters), A. Maharana (Chapter 2), S. Parameswaran (Chapter 4), F. Quevedo (Chapter 2), A. Salvio (Chapters 3&4), and L. van Nierop (Chapters 3&4). My contribution to these collaborative works involved: participating in discussions to formulate research goals and strategies for their obtention; performing the requisite calculations (with the exception of parts of the work presented in sections 2.2.2, 2.2.3, 3.5 and 4.5); determining the relative successfulness of various research strategies; developing novel computational tools (particularly for the loop calculations presented in Chapter 3, which I obtained autonomously); creating a draft which lays out the main results and their context; providing feedback to other collaborators regarding their contributions to the draft; submitting the drafts for publication; responding to referee inquiries. All previously published material has been reformatted to conform to the required thesis style. I grant an irrevocable, non-exclusive license to McMaster University and the National Library of Canada to reproduce this material as part of this thesis.

v vi Contents

1 Introduction and Motivation 1 1.1 Three Postulates ...... 1 1.1.1 (Technical) Naturalness ...... 1 1.1.2 Large Extra Dimensions ...... 5 1.1.3 The Stability of the Proton ...... 6 1.2 Constraints on a Gauge Boson Conserving Baryon Number ...... 8 1.3 Codimension-2 Casimir Energies and the Eﬀective 4D Vacuum Energy ...... 8 1.4 Accidental Supersymmetry in 6D Gauged Chiral Supergravity ...... 9

2 New Constraints (and Motivations) for Abelian Gauge Bosons in the MeV–TeV Mass Range 15 2.1 Introduction and summary of results ...... 15 2.2 Theoretical motivation ...... 19 2.2.1 Low-energy gauge symmetries, consistency and anomaly cancellation . . . . . 20 2.2.2 Anomaly cancellation ...... 22 2.2.3 Motivations from UV physics ...... 26 2.3 Gauge boson properties ...... 30 2.3.1 The mixed lagrangian ...... 31 2.3.2 Physical couplings ...... 32 2.4 High-energy constraints ...... 35 2.4.1 Eﬀects due to modiﬁed W, Z couplings ...... 36 2.4.2 Processes involving X-boson exchange ...... 42 2.5 Constraints at intermediate energies ...... 45 2.5.1 Neutrino-electron scattering ...... 45 2.5.2 Neutrino-nucleon scattering ...... 49 2.6 Low-energy constraints ...... 51 2.6.1 Anomalous magnetic moments ...... 51 2.6.2 Upsilon decay ...... 51 2.6.3 Beam-dump experiments ...... 54 2.6.4 Neutron-nucleus scattering ...... 58

vii viii CONTENTS

2.6.5 Atomic parity violation ...... 59 2.6.6 Primordial nucleosynthesis ...... 61 2.A Appendix: Diagonalizing the gauge action ...... 63

3 Running with Rugby Balls: Bulk Renormalization of Codimension-2 Branes 81 3.1 Introduction ...... 81 3.2 Bulk field theory and background solution ...... 84 3.3 General features of bulk loops ...... 88 3.4 Results for low-spin bulk fields ...... 98 3.4.1 Scalars ...... 98 3.4.2 Spin-half fermions ...... 102 3.4.3 Gauge fields ...... 106 3.5 The 4D vacuum energy ...... 111 3.5.1 Classical bulk back-reaction ...... 112 3.5.2 Higher derivative corrections on the brane ...... 115 3.6 Conclusions ...... 117 3.A Heat kernels and bulk renormalization ...... 119 3.B Sums and zeta functions ...... 125 3.C Spectra and mode sums ...... 134

4 Accidental SUSY: Enhanced Bulk Supersymmetry from Brane Back-reaction 163 4.1 Introduction ...... 163 4.2 Bulk ﬁeld theory and background solution ...... 168 4.2.1 6D gauged, chiral supergravity ...... 168 4.2.2 Rugby-ball compactiﬁcations ...... 171 4.2.3 Supersymmetry of the solutions ...... 175 4.3 Mode sums and renormalization ...... 178 4.4 Supermultiplets ...... 182 4.4.1 Hypermultiplet ...... 185 4.4.2 Massless gauge multiplet ...... 189 4.4.3 Massive matter multiplet ...... 193 4.5 The 4D vacuum energy ...... 197 4.5.1 Classical bulk back-reaction ...... 198 4.5.2 Application to supersymmetric renormalizations ...... 199 4.5.3 Loop-corrected 4D cosmological constant ...... 201 4.6 Conclusions ...... 202 4.A Heat kernels and bulk renormalization ...... 206 4.B Results for spins zero, half and one ...... 212 CONTENTS ix

4.C Complete results for the massive multiplet ...... 219

5 Conclusion and Outlook 231 x CONTENTS List of Figures

2.1 Summary of the constraints presented herein. Each plot shows the bound on the

new gauge coupling, αX , as a function of MX for various values of the kinetic-mixing

parameter, sh η, assuming a vector coupling Xf L = Xf R := X, with X = B − L (X = B) drawn as sparse (dense) cross-hatching...... 18

2.2 Summary of the constraints on kinetic mixing relevant in the MeV-GeV mass range.

Each plot shows the bound on the kinetic mixing parameter sh η as a function of MX , −10 −8 for αX = 0, 1 × 10 and 1 × 10 . The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 20

2.3 Plot of the bounds on z as a function of MX and sh η. The blue crosses limit the region in which z is real, and the red squares limit the region in which z 1. The hatched regions are excluded...... 37

2.4 Plot of the EWWG bound on the S and T oblique parameters, showing how T is more tightly constrained given prior knowledge that S = 0...... 38

2.5 Constraint obtained from limiting the inﬂuence of kinetic mixing on the SM value of the W mass. The hatched regions are excluded...... 39

+ − 2 2.6 The constraint arising from Z → ` ` decay on the coupling αX = gX /4π as a

function of MX , for various values of sh η. The parameters agreeing with the positive

bound (∆Γ = +∆Γexp) are marked with blue crosses, while those agreeing with the

negative bound (∆Γ = −∆Γexp) are marked with red squares. The plot assumes a

coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 40

2.7 Plot of the constraint arising from considering Z decay into leptons in the limit where

MX MZ . The upper bound (∆Γ = +∆Γexp) is marked with blue crosses; the lower

bound (∆Γ = −∆Γexp) is marked with red squares. Hatched regions are excluded. . 41

2.8 Relevant tree-level Feynman diagrams corresponding to electron-positron annihilation into fermion-antifermion pairs...... 42

xi xii LIST OF FIGURES

√ 2.9 The constraint obtained from σhad evaluated for s = MZ , as a bound in the αX −MX plane for various values of sh η. Blue crosses (red squares) indicate parameters where predictions diﬀer by 2 σ from experiment on the upper (lower) side. The hatched regions are excluded, while diagonal shading indicates a region excluded by global ﬁts to oblique parameters...... 44

2 2.10 Plot of the constraint from σhad s = MZ in the region where MX MZ . The parameters agreeing with the positive bound are marked with blue crosses, while those agreeing with the negative bound are marked with red squares. The hatched regions are excluded...... 45

2 − 2.11 Bound obtained on αX = gX /4π by limiting the inﬂuence of the X boson on the ν −e

cross section ratio R, obtained as a function of MX for various values of sh η. The vertical line indicates the region ruled out by electroweak oblique ﬁts when η = 1. . . 48 2.12 Plot of the constraint from R− (neutrino-nucleon scattering) assuming X = B − L.

2.13 Plots of the constraint on the gauge coupling αX arising from the electron and muon

AMM as a function of MX , for various values of sh η. The electron AMM bound is marked with blue crosses; the muon AMM bound is marked with red squares. The

plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 52 2.14 Plots of the constraint on the kinetic mixing, sh η, arising from the electron and muon

AMM as a function of MX , for various values of the gauge coupling αX . The electron AMM bound is marked with blue crosses; the muon AMM bound is marked with red

squares. The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 53

2.15 Plots of the constraint on the gauge coupling αX arising from Υ(3s) decay as a function

of MX , for sh η = 0, 0.001. The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 54 2.16 Plots of the constraint on the kinetic mixing, sh η, arising from Υ(3s) decay as a −8 function of MX , for αX = 0, 1 × 10 . The plot assumes a coupling XeL = XeR = −1, such as would be true if X = B − L. Hatched regions are excluded...... 55

2 2.17 The constraint arising from beam dump experiments on the coupling αX = gX /4π as

a function of MX , for sh η = 0, 0.001. The E774 bound is marked with red squares; the E141 bound is marked with blue crosses; the E137 bound is marked with black circles.

The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 56 LIST OF FIGURES xiii

2.18 The constraint arising from beam dump experiments on the kinetic mixing sh η as a −15 −12 −9 function of MX , for αX = 0, 1 × 10 , 1 × 10 , and 1 × 10 . The E774 bound is marked with red squares; the E141 bound is marked with blue crosses; the E137

bound is marked with black circles. The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded...... 57

2.19 Plot of the constraint on the gauge coupling αX due to neutron-nucleus scattering as

a function of the X-boson mass MX . The hatched regions are excluded...... 59

2.20 Plot of the constraint on the gauge coupling αX due to the weak charge of cesium as

a function of the X-boson mass MX , for various values of sh η. The hatched regions are excluded...... 61 2.21 Constraint on the gauge coupling of the X due to its eﬀect on nucleosynthesis, as a function of the X-boson mass. The red squares indicate the bound due to Xν→Xν scattering; the blue crosses indicate the bound due to X→νν decay...... 62

3.1 Schematic plot of the potential as a function of Φb...... 101 xiv LIST OF FIGURES List of Tables

2.1 Parameter values for the E774, E141, and E137 beam dump experiments...... 55 2.2 Charge assignments for the “right-handed” U (1)...... 60

xv xvi LIST OF TABLES Chapter 1

Introduction and Motivation

The work presented in this thesis aims to answer the following questions:

1. If low-scale gravity models predict new—potentially baryon number-violating—physics just above the TeV scale, under what conditions could a gauge symmetry be introduced to preserve baryon number? How would the properties of the corresponding gauge boson be constrained by experiments to date? (Chapter 2)

2. If the dominant contribution to the observed vacuum energy arises from loops of extra-dimensional matter ﬁelds (as it would in a particular toy model for the SLED proposal [1, 2, 3], namely 6D chiral gauged supergravity), how would the vacuum energy depend on the properties of these ﬁelds and the extra-dimensional geometry? (Chapters 3&4)

As we shall see, the impetus to ask these questions relies on three primary postulates: technical naturalness, large extra dimensions, and the stability of the proton. We begin by introducing these postulates, and then discussing the approach taken herein to address the questions posed above.

1.1 Three Postulates

Although there are many concepts and techniques which are relevant to the work presented in this thesis, the three postulates discussed in this section have been selected to provide a succinct introduction to the relevant physical issues. As we shall see, these postulates are also interrelated: the postulate of technical naturalness—when applied to the cosmological constant problem—leads (almost uniquely) to the postulate of at least one large extra dimension; these together lead necessarily to the existence of some mechanism to explain the stability of the proton. It is for this reason that the correlations between these concepts are emphasized in the following discussion.

1.1.1 (Technical) Naturalness

The term ‘naturalness’ is conventionally used in two diﬀerent contexts:

1 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

1. a physical quantity is natural if its value is roughly set by the other parameters in the theory. < 4 For example, the pressure at the centre of a neutron star is natural as it is roughly pNS ∼ mN ' (1 GeV)4 (in units where ~ = c = 1, which are used throughout) [4];

2. a physical quantity is technically natural if its smallness can be explained at any energy scale. For example, the smallness of the electron mass (as compared to the weak scale) can be understood at low energies as being due to broken chiral symmetry: all quantum corrections to the electron mass must be proportional to the electron mass because they must vanish as

we take me → 0. Similarly, at high energies, we understand that the smallness of the electron mass is due to its Yukawa-type coupling to the SM (Standard Model) Higgs boson: in this case, the relevant coupling constant is a dimensionless parameter which only receives small, logarithmic quantum corrections.

This last example is a situation where the technical distinction is apt, as the Yukawa-type Higgs- −6 electron coupling in the SM is only ye ' 2 × 10 1. There is no reason for this dimensionless physical parameter to be so small. Nevertheless, it is technically natural because its quantum corrections are perturbatively small, and don’t require any detailed cancellation in order to find agreement with the observed value of the electron mass. Since several such counterexamples exist— where the first naturalness condition is not satisfied but the second one is—there is no reason to think the first condition ought to be a criterion according to which a physical theory should be judged. However, it seems the second condition is satisfied by all parameters in particle physics, except for two important special cases.

The Higgs mass

In the case where one takes the SM to be valid up to some high energy scale then there is generally some large contribution to the the Higgs mass,

Λ2 δm2 ∼ , (1.1) H (4π)2 resulting from new physics (e.g. a yet-to-be-disovered particle which couples to the Higgs) at the scale > Λ. Now, if Λ were much larger than the Higgs mass (let’s say Λ ∼ 10 TeV mH ' 125 GeV), then we would expect some incredible cancellation between contributions from the high energy theory and radiative corrections in the eﬀective theory below Λ:

2 2 2 mH = mhe(Λ) + δmH (Λ) . (1.2)

(Such a cancellation would be particularly tuned if Λ were at the Planck scale, i.e. Λ ∼ Mpl := (8πG)−1/2 = 2.43 × 1018 GeV.) The Higgs mass parameter in such a theory would thus be deemed 2 unnatural because the quantum corrections would not be perturbatively small compared to mH , and would required a detailed cancellation in order to ﬁnd agreement with the observed value of the Higgs mass. This is canonically referred to as the ‘hierarchy problem’.

2 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

There are three primary ways to avoid such large, ﬁnely-tuned quantum corrections:

1. Impose some symmetry in the high energy theory (e.g. supersymmetry) that forbids such large contributions;

2. Impose a new structure above the scale Λ (e.g. string theory) with a gravity scale, Mg ' 10–

100 × mH . In this context, the Planck scale would no longer be fundamental; its size would be set by the size of the extra dimensions;

3. Impose some composite structure to the Higgs particle, so that its structural dynamics would become relevant at scales higher than Λ.

Although the MSSM (Minimally Supersymmetric Standard Model) has some redeeming qualities— such as gauge coupling uniﬁcation—the absence of superpartners in the LHC data [6] has limited its eﬀectiveness as a viable solution to the hierarchy problem. It is the second of these approaches, namely the existence of a string theory describing physics above the TeV scale, which will of interest to us in what follows.

The cosmological constant

From a ﬁeld theory perspective, a cosmological constant is the unique way of modifying the physics of a massless spin-2 particle (a.k.a. the graviton) at long wavelengths [7]. The corresponding action takes the form Z 4 √ 1 S = d x −g − 2 (R + 2λ) + L (1.3) 2κ4 µν µν σ 1 2 where gµν is the four-dimensional metric, R = g Rµν = g Rµσν is the Ricci scalar, κ4 = 8πG is the gravitational coupling strength, λ is the cosmological constant, and L is a lagrangian which determines how the graviton couples to matter. The corresponding equations of motion are

1 R − R g − λg = −κ2T , (1.4) µν 2 µν µν 4 µν where √ 2 ∂( −g L) ∂L T µν := √ = gµν L + 2 (1.5) −g ∂gµν ∂gµν is the covariantly conserved stress-energy tensor which sources the spin-2 ﬁeld. Naturalness is a concern in this circumstance because: a) a non-zero cosmological constant is equivalent to a constant vacuum energy density; and b) because the latter receives very large quantum corrections due to vacuum polarization. To elaborate on part a), the following are two equivalent ways of describing the same physics:

1. λ1 6= 0, L1 = L

2 2. λ2 = 0, L2 = −ρV + L with ρV = λ1/κ4. 1Throughout, we use Weinberg’s Riemann curvature convention [8] (which diﬀers from that of MTW [9] by an σ σ λ σ ρ 1 ρλ overall sign): Rµνρ = ∂µΓνρ + ΓµρΓλν − (µ ↔ ν) where Γµν = 2 g (∂µgλν + ∂ν gλµ − ∂λgµν ).

3 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

µν µν µν µν (In the second case, T = TV + Tm where TV = −ρV gµν .) Therefore, it is redundant to have non-zero λ if we allow L to have some constant value, LV = −ρV when the matter ﬁelds are set to zero. (We make this allowance in what follows by dropping λ, and instead tracking the value of the 2 cosmological constant in terms of ρV ↔ λ/κ4.) The naturalness discussion related to part b) then proceeds in much the same way as for the Higgs mass: the observed physical value [10],

4 ρV = (2.3 meV) , (1.6) results as a sum of two terms, he ρV = ρV (Λ) + δρV (Λ) (1.7) where an explicit loop calculation of vacuum polarization shows

Λ4 δρ ∼ . (1.8) V (4π)2

The main diﬀerence between the unnaturalness of the vacuum energy and the unnaturalness of the Higgs mass is that there is no impetus to explain the smallness of the Higgs mass in a theory below the weak scale, since the Higgs is not a propagating ﬁeld at low energies and can be integrated out. Therefore, an explanation for the smallness of the Higgs mass resides solely in new physics energies above the TeV scale, whereas our current understanding of the renormalization of vacuum energies says even the lightest known charged particle, the electron (with mass me = 511 keV), would contribute an unnaturally large vacuum energy:

δρ (m ) V e ∼ 1031 . (1.9) ρV

Clearly, contributions to the cosmological constant from vacuum loops of heavier, weak scale particles— such as the Higgs—only exacerbate the problem of unnaturalness:

δρ (m ) V H ∼ 1053 . (1.10) ρV

Since a natural resolution to the cosmological constant problem requires modifying the long- wavelength behaviour of well-understood physical theories (such as quantum electrodynamics), there are strong experimental constraints which must be satisﬁed, and very few viable modiﬁcations. How- ever, since cosmologists measure vacuum energy only indirectly through the corresponding curvature of spacetime, it’s reasonable to think that progress could be made by relaxing the direct connection between the observed curvature of our (four-dimensional) spacetime and the smallness of the vacuum energy. One way to achieve this is by introducing large extra dimensions—the topic of the next section. Under the appropriate circumstances, extra dimensions cause a large vacuum energy to preferentially curve those dimensions which are transverse to our own2, and allow for our

2This occurs, for example, in the case of cosmic strings: having some energy localized on the string will deform the geometry of spacetime in the directions perpendicular to the string, and leave the longitudinal direction ﬂat.

4 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy four-dimensional spacetime to be relatively ﬂat.

1.1.2 Large Extra Dimensions

Extra spatial dimensions have been a fruitful avenue of theoretical consideration ever since the original proposals of Kaluza & Klein [11] in the 1920’s. However, the idea that particles comprising the Standard Model may be forbidden from propagating in the extra dimensions—and that only gravity may sense them—did not arise until a paper by Rubakov and Shaposhnikov the 1980’s [12]. This notion was later taken more seriously in the mid 1990’s in the context of D-branes in string theory [13]. These developments led to the ideas of Arkani-Hamed, Dimopoulos and Dvali [14], where they pointed out that the existence of two (or more) extra dimensions at the sub-millimeter scale can address one or both naturalness issues discussed in section 1.1.1 above. Some time thereafter Carroll and Guica [15] proposed that the four-dimensional vacuum energy might be cancelled entirely in the case of two extra dimensions, and in [2] Aghababaie et al. argue that extra-dimensional quantum eﬀects are the dominant source of the vacuum energy in the case of a supersymmetric 6D model. This argument proceeds as follows: if the observed vacuum energy arises from loops of massless

ﬁeld(s) propagating in n := D − 4 compact extra dimensions—with volume Vn and thus some 1/n characteristic size R := (Vn) —then the size of the extra dimensions is ﬁxed by

1 ρ = (2.3 meV)4 ∼ → R ∼ 25 µm . (1.11) V (4π)2R4

Furthermore, the attractive gravitational force between two point-like masses, m1 and m2, at small separations r is determined from Gauss’ law:

κ2 m m F (r) = D 1 2 , (1.12) g D−2 2 SD−2r

(k+1)/2 k+1 3 2 where Sk := 2 · π /Γ 2 is the surface area of the unit k-sphere and κD := 8πGD = D−2 1/(Mg) is the gravitational coupling strength. However, as r grows larger than R the integral of the compact dimensions saturates and—in the case where the extra dimensions are spherical (for simplicity)—the force law then becomes the usual one, i.e.

2 κD m1m2 GD m1m2 Fg(r) = n 2 = n 2 , (1.13) 2 (SnS2)(R r ) SnR r

so long as we ﬁx GD such that

1 2 ! 2+n GD Mpl G = n → Mg = n . (1.14) SnR SnR

The reason why this particular construction can address both naturalness issues is because, when

3 R ∞ z−1 −t The deﬁnition of Sk uses the Gamma function, Γ(z) = 0 dt t e .

5 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy the codimension n is equal to 2 (i.e. D = 6), the gravity scale sits just above the TeV scale:

25 µm1/2 M = 2.3 TeV × . (1.15) g R

Hence, a resolution to the hierarchy problem can be achieved as a result of the extra-dimensional gravity theory becoming non-perturbative above the TeV scale.

One immediate obstacle to introducing a gravity scale Mg Mpl is that, generally, proton decay occurs much more quickly than seen observationally. This occurs because new eﬀective interactions, mediated by the universal brane couplings to the graviton, become strong at this energy scale. Therefore—in the absence of some symmetry forbidding such interactions—the theoretical prediction for the proton decay rate becomes unacceptably large. It is for this reason that the stability of the proton is our third postulate.

1.1.3 The Stability of the Proton

Our understanding of the stability of the proton in the SM is one that arises for ‘free’; it just happens that, once the SM matter content and charges are inputted, no such decay is allowed. As it turns out, there are four such ‘accidental’ symmetries: baryon number (B); and electron, muon, tau lepton number (Le, Lµ, Lτ ). All of the tree-level interactions in the SM conserve these charges. However, in the absence of any symmetry principle, there is no guarantee that either loop corrections or physics above the TeV scale should preserve the same accidental symmetries.

From an effective field theory perspective, we can parametrize the effects of baryon number non-conservation with a number of non-renormalizable, dimension-6 operators, e.g.

C O = Q γ L D γ U . (1.16) abcd Λ2 a L b c R d

({a, b, c, d} are ﬂavour indices.) Given such interactions, we can compute the resulting proton decay rate, Γp, at tree-level. Schematically, this gives

m5 Γ ∼ C2 p . (1.17) p Λ4

Experimentally, no such decay has been observed; proton decay search experiments ﬁnd [16] that −1 the proton lifetime, τp = (Γp) , must satisfy

29 Γp −61 τp > 2.1 × 10 yr → < 1.1 × 10 . (1.18) mp

(Some decay modes—such as p → π0e+—are more tightly constrained [5]; the value quoted here applies to the total lifetime, summing over all decay modes.) Therefore, we ﬁnd that the eﬀective

6 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

operators Oabcd—and others like it—appropriately describe what is observed as long as

15 1/2 Λ ∼> 10 GeV × C . (1.19)

The naturalness arguments of Section 1.1.1 posit the coupling strength of such an interaction is set roughly by the energy scale Λ at which new physics—not included in the eﬀective theory—is expected to arise. (More explicitly, the notion that the constant C could be tuned so that Λ 1015 GeV is one that is unstable under radiative corrections.) Therefore a low-scale gravity theory, with Λ 1015 GeV and which couples to matter universally, exerts a tension with the notion that baryon number is an accidental symmetry.

The main consequence of this line of reasoning is that there must be a symmetry associated with baryon number innate in any low-scale gravity theory. This alone does not dictate the existence of a gauge boson, since this symmetry needn’t be a gauge symmetry, and could instead be either a global or discrete symmetry. In chapter 2, we argue that the latter would be unsatisfactory because their existence would immediately result in odd paradoxes [17]. For example: if matter charged under a global symmetry falls into a black hole, how would an external observer verify that the corresponding charge is conserved?

Therefore, we presume that any theory with large extra dimensions would only be satisfactorily compatible with the observed stability of the proton if there were some additional gauge symmetry (beyond that needed for the SM) under which baryons are charged.

There are many ways in which such additional gauge symmetries may arise. In supersymmetric extra-dimensional theories (like the one discussed in Chapter 4 herein), one can ﬁnd that extra gauge bosons may arise as R-symmetries. Extra gauge bosons can also arise from oﬀ-diagonal components of the bulk metric [11]: (i) (i) Aµ ζm ↔ gµm , (1.20)

(i) where ζm is a Killing vector corresponding to some extra-dimensional isometry. Such gauge bosons become massive in the case where the corresponding extra-dimensional isometry is broken, and the corresponding Killing vector ceases to satisfy the Killing equation

(i) (i) ∇mζn + ∇nζm = 0 . (1.21)

Nevertheless, the current to which they couple remains conserved.

Having introduced the three main postulates upon which most of this thesis is based, let’s continue with a brief overview of the material to be covered in subsequent chapters.

7 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

1.2 Constraints on a Gauge Boson Conserving Baryon Num- ber

The aim of the research presented in Chapter 2 is to examine the relevant constraints on an extra gauge boson with a mass in the MeV-TeV range, which gauges either baryon number (B) or baryon- minus-lepton number (B −L) with a relatively small gauge coupling. The motivation for considering such a gauge boson is twofold: i) Many extra-dimensional gravity models [18] predict a 4D gauge coupling g that is inversely proportional to the volume of the extra dimensions—possibly as small as g ∼ 10−5—and this region of parameter space is sometimes neglected; ii) A small gauge coupling allows for TeV-scale physics (parametrized by v) to give rise to a MeV-scale gauge boson mass m, since m ∼ gv. Although we restrict our analysis to speciﬁc charge assignments, our constraints are applied to the full remaining space of renormalizable couplings, including a kinetic mixing with the hypercharge gauge boson of the form

χ L = X Bµν . (1.22) mix 2 µν

Herein, we demonstrate that, although the argument for anomaly cancellation becomes necessarily more subtle, many constraints can be avoided (at least when there is not much kinetic mixing) by coupling only to baryon number. We also show that the presence of a non-vanishing gauge coupling can drastically alter previously-derived bounds for so-called “dark force” models, which are relevant to the phenomenology of dark matter. This work draws upon constraints from cosmology, neutrino physics, nuclear physics, and electroweak precision physics. In many instances, it is the ﬁrst time in the literature that a given bound has been considered in terms of the gauge coupling, the gauge boson mass, and the kinetic mixing. As such, it is likely to be a valuable reference for those interested in the use of Z0 physics to constrain proton decay in theories with large extra dimensions.

1.3 Codimension-2 Casimir Energies and the Eﬀective 4D Vacuum Energy

The research found in Chapter 3 aims to answer the following question: assuming that large extra dimensions exist, how do vacuum fluctuations in these extra dimensions—and their resulting Casimir energy—influence our four-dimensional world? Dimensional analysis brings us a long way towards the answer: if n extra dimensions exist and are populated by fields with mass m, then the leading 1- loop contributions to the 4D vacuum energy (in order from most to least UV-sensitive) are generally O(m4+n rn), O(m2+n rn−2), O(mn rd−n), . . . , and O(r−4), where r is a characteristic size of the extra dimensions. The reason why it is reasonable to be considering loop contributions in the context of six dimensions is because a particular 6D supergravity model has been shown in [3] to dynamically relax

8 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy into a configuration with a net zero classical contribution to the 4D vacuum energy, so long as the 4D branes do not couple to the extra-dimensional dilaton. (More on this model in section 1.4, and details in Chapter 4.) We perform a 1-loop calculation of the Casimir energy—for fields with spin-zero, -one-half and -one—in the case of two extra dimensions that are flux-stabilized and in the shape of a rugby ball. (This geometry belongs to a set of known supersymmetric solutions [20].) Our analysis allows the fields to be charged under the background flux field, and includes arbitrary brane-localized fluxes at each end of the rugby ball. We use the result of our calculation to identify local counterterms that renormalize the UV divergences. Necessarily, some counterterms are localized at each end of the rugby ball (where the four-dimensional branes reside). We also read off contributions to various beta functions, both on and off the brane. The main technical complication of our analysis is that, from the 4D perspective, every field in the Kaluza-Klein tower contributes non-trivially to the Casimir energy. Using heat kernel techniques and Poisson resummation, we present a way to sum over Kaluza-Klein modes that is mindful of this complication.

1.4 Accidental Supersymmetry in 6D Gauged Chiral Super- gravity

Given the results found in Chapter 3, we consider a conﬁguration in which the extra-dimensional

field content is supersymmetric and the background flux field gauges the U(1)R symmetry (thereby fixing the charges of individual fields in a supermultiplet). The magic of supersymmetry then ensures that the net extra-dimensional contribution to the Casimir energy from an entire supermultiplet is identically zero [21]. This occurs because the extra-dimensional supersymmetry is partially preserved in the case of a sphere [22], and because extra-dimensional fluctuations are locally unaware of the rugby ball’s conical singularities. Remarkably, we show herein that supersymmetry is similarly preserved in the case of the rugby- ball geometry, so long as there is equal brane-localized flux at each end of the rugby ball. This is remarkable because the brane lagrangian we use is an effective one—it is the most general lagrangian that can be written down, up to the one-derivative level—so any number of non-supersymmetric lagrangians (such as the Standard Model lagrangian) could have been integrated out to yield it at low energies. So long as the brane lagrangians are identical at each end of the rugby ball, an extra-dimensional supermultiplet makes no net contribution to the Casimir energy at 1-loop. This is surprising because we are not adopting any specific relationship between the brane- localized tension and flux; so long as the branes are identical, then supersymmetry is preserved at 1-loop for arbitrary choices of the brane couplings. This occurs because there is a flat direction which develops a potential once brane-localized fluxes are included, and which dynamically tracks the supersymmetric minimum. This result relies on there being two distinct complications that arise due to brane back-reaction:

9 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy brane-localized ﬂuxes aﬀect not only the shape of the rugby ball’s conical singularities, but also the

flux quantization condition satisfied by the background U(1)R gauge field. As such, it provides an interesting contrast to the lore of pure tension branes, indicating that an understanding of brane- localized flux and back-reaction is necessary when evaluating naturalness issues in the context of flux-stabilized extra dimensions. In the case of unequal fluxes (∆Φ 6= 0), we compute the contribution to the 4D vacuum energy while including the effect of brane back-reaction. The latter is necessary because, in the effective theory, the bulk adiabatically adjusts to the running of the brane-localized counterterms which give the non-zero value. The net result is an expression of the form

|∆Φ| ρ ∼ , (1.23) V (4πr2)2 which is of the right size based on the arguments laid out in section 1.1.1.

10 Bibliography

[1] Y. Aghababaie, C. P. Burgess, J. M. Cline, H. Firouzjahi, S. L. Parameswaran, F. Quevedo, G. Tasinato and I. Zavala, “Warped brane worlds in six-dimensional supergravity,” JHEP 0309, 037 (2003) [hep-th/0308064].

C. P. Burgess, “Supersymmetric large extra dimensions and the cosmological constant: An Up- date,” Annals Phys. 313, 283 (2004) [hep-th/0402200]; “Towards a natural theory of dark energy: Supersymmetric large extra dimensions,” AIP Conf. Proc. 743, 417 (2005) [hep- th/0411140];

[2] Y. Aghababaie, C. P. Burgess, S. L. Parameswaran and F. Quevedo, “Towards a naturally small cosmological constant from branes in 6-D supergravity,” Nucl. Phys. B 680, 389 (2004) [hep-th/0304256];

[3] C. P. Burgess and L. van Nierop, “Technically Natural Cosmological Constant From Supersym- metric 6D Brane Backreaction,” arXiv:1108.0345 [hep-th].

[4] N. Arkani-Hamed, PSI Lecture: “Research Skills” (2009), http://pirsa.org/09080035.

[5] J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012).

[6] V. A. Mitsou [ATLAS Collaboration], “Highlights from SUSY searches with ATLAS,” arXiv:1210.1679 [hep-ex].

S. Chatrchyan et al. [CMS Collaboration], “Search for supersymmetry in hadronic ﬁnal states √ using MT2 in pp collisions at s = 7 TeV,” JHEP 1210, 018 (2012) [arXiv:1207.1798 [hep-ex]].

[7] S. Weinberg, “The Quantum theory of ﬁelds. Vol. 1: Foundations,” Cambridge, UK: Univ. Pr. (1995) 609 p.

[8] S. Weinberg, “Gravitation and Cosmology,” Wiley (1973).

[9] C. W. Misner, K. S. Thorne and J. A. Wheeler, San Francisco 1973, 1279p

[10] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. I. Overview of products and scientiﬁc results,” arXiv:1303.5062 [astro-ph.CO].

11 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

[11] T. Kaluza, “On the Problem of Unity in Physics,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921, 966 (1921).

O. Klein, “Quantum Theory and Five-Dimensional Theory of Relativity. (In German and En- glish),” Z. Phys. 37, 895 (1926) [Surveys High Energ. Phys. 5, 241 (1986)].

[12] V. A. Rubakov and M. E. Shaposhnikov, “Do We Live Inside a Domain Wall?,” Phys. Lett. B 125, 136 (1983).

[13] For a review see, for example: J. Polchinski, “TASI lectures on D-branes,” [arXiv:hep- th/9611050].

[14] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “The hierarchy problem and new dimensions at a millimeter,” Phys. Lett. B 429 (1998) 263 [arXiv:hep-ph/9803315];

N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity,” Phys. Rev. D 59, 086004 (1999) [hep-ph/9807344].

[15] S. M. Carroll and M. M. Guica, “Sidestepping the cosmological constant with football shaped extra dimensions,” hep-th/0302067.

[16] S. N. Ahmed et al. [SNO Collaboration], “Constraints on nucleon decay via ’invisible’ modes from the Sudbury Neutrino Observatory,” Phys. Rev. Lett. 92, 102004 (2004) [hep-ex/0310030].

[17] R. Kallosh, A. D. Linde, D. A. Linde and L. Susskind, “Gravity and global symmetries,” Phys. Rev. D 52, 912 (1995) [hep-th/9502069].

[18] See, for example: V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, “Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactications,” JHEP 0503 (2005) 007 [arXiv:hep- th/0502058];

J. P. Conlon, F. Quevedo and K. Suruliz, “Large-volume ﬂux compactiﬁcations: Moduli spectrum and D3/D7 soft supersymmetry breaking,” JHEP 0508 (2005) 007 [arXiv:hep-th/0505076].

[19] M. Williams, C. P. Burgess, L. van Nierop and A. Salvio, “Running with Rugby Balls: Bulk Renormalization of Codimension-2 Branes,” JHEP 1301, 102 (2013) [arXiv:1210.3753 [hep-th]].

[20] G. W. Gibbons, R. Gueven and C. N. Pope, “3-branes and uniqueness of the Salam-Sezgin vacuum,” Phys. Lett. B 595, 498 (2004) [arXiv:hep-th/0307238].

[21] D. Hoover and C. P. Burgess, “Ultraviolet sensitivity in higher dimensions,” JHEP 0601, 058 (2006) [arXiv:hep-th/0507293].

[22] A. Salam and E. Sezgin, “Chiral Compactiﬁcation on Minkowski x S2 of N=2 Einstein-Maxwell Supergravity in Six-Dimensions,” Phys. Lett. B 147, 47 (1984).

12 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

[23] C. P. Burgess, L. van Nierop, S. Parameswaran, A. Salvio and M. Williams, “Accidental SUSY: Enhanced Bulk Supersymmetry from Brane Back-reaction,” JHEP 1302, 120 (2013) [arXiv:1210.5405 [hep-th]].

13 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

14 Chapter 2

New Constraints (and Motivations) for Abelian Gauge Bosons in the MeV–TeV Mass Range

— M. Williams, C.P. Burgess, A. Maharana, F. Quevedo

JHEP 1108 (2011) 106 [arXiv:hep-ph/1103.4556]

2.1 Introduction and summary of results

New particles need not have very large masses in order to have evaded discovery; they can also be quite light provided they couple weakly enough to the other particles we do see. This unremarkable observation has been reinforced by recent dark matter models, many of which introduce new particles at GeV or lower scales in order to provide dark-matter interpretations for various astrophysical anomalies [1]. This model-building exercise has emphasized how comparatively small experimental eﬀorts might close oﬀ a wide range of at-present allowed couplings and masses for putative new light particles [2, 3].

Light spin-one bosons

Spin-one gauge bosons are particularly natural kinds of particles to seek at low energies, since (unlike most scalars) these can have light masses in a technically natural way. Furthermore, their couplings are reasonably restrictive, allowing only two kinds of dimensionless interactions with ordinary Stan- dard Model particles: direct gauge couplings to ordinary matter and kinetic mixing [4] with Standard Model gauge bosons. Most extant surveys of constraints on particles of this type assume the existence of one or the other of these couplings, with older studies studying only direct gauge-fermion

15 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy interactions [5, 6] and later studies (particularly for dark-matter motivated models) [7, 8, 9, 10, 11] usually allowing only kinetic mixing. In this paper we have both motivational and phenomenological goals. On the phenomenological side, we analyze the constraints on new (abelian) gauge bosons, including both direct gauge-fermion couplings and gauge-boson kinetic mixing. In this way we include all of their dimensionless couplings, which (if all other things are equal) should dominate their behaviour at low energies. We can follow the interplay of these couplings with one another, and how this changes the bounds that can be inferred concerning the allowed parameter space. In particular we find in some cases (such as beam dump experiments) that bounds derived under the assumption of the absence of the other coupling can sometimes weaken, rather than strengthen, once the most general couplings are present. Our motivational goal in this paper is twofold. First, we argue that the existence of gauge bosons directly coupled to ordinary fermions is very likely to be a generic and robust property of any phenomenologically successful theory for which the gravity scale is much smaller than the GUT scale [12, 13, 14]. Next, we argue that these gauge bosons often very naturally have extremely weak gauge interactions within reasonable UV extensions of the low-energy theory, such as extra-dimensional models [15] and low-energy string vacua [16]. Besides motivating the otherwise potentially repulsive feature of having very small couplings, the smallness of these couplings (together with the low value for the fundamental gravity scale) also naturally tends to make the corresponding gauge bosons unusually light. The remainder of this paper is organized as follows. The rest of this section, §1, briefly summarizes the basic motivational arguments and phenomenological results. §2 then provides a more detailed theoretical background that motivates the sizes and kinds of couplings we consider, which may be skipped for those interested only in the bounds themselves. In §3 we briefly summarize the basic properties of the new gauge boson, with details given in an Appendix. By diagonalizing all kinetic terms and masses we identify the physical combination of couplings that are bounded in the subsequent sections. The next three sections, §4, §5 and §6, then explore the bounds on these couplings that are most restrictive for successively lighter bosons, starting at the weak scale and working down to MeV scales.

Motivational summary

Why consider light gauge bosons that couple directly to ordinary fermions? And why should their couplings be so small? We here brieﬂy summarize the more lengthy motivations given below, in §2.

Low-scale gravity and proton decay

Weakly coupled gauge bosons are likely to be generic features of any (phenomenologically viable) UV physics for which the fundamental gravity scale is systematically small relative to the GUT 15 scale, MGUT ∼ 10 GeV. Such bosons arise because of the diﬃculty of reconciling a low gravity scale with the observed stability of the proton. After all, higher-dimension baryon- and lepton- violating interactions that generically cause proton decay are not adequately suppressed if they arise

16 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

accompanied by a gravity scale that is much smaller than MGUT. Similarly, global symmetries cannot themselves stop proton decay if the present lore about the absence of global symmetries in quantum gravity [17] should prove to be true (as happens in string theory, in particular [18, 19]). This leaves low-energy gauge symmetries as the remaining generic mechanism for suppressing proton decay. Indeed, extra gauge bosons are often found in string vacua, and when the string scale is much smaller than the GUT scale, Ms MGUT, these bosons typically play a crucial role in protecting protons from decaying. Furthermore, very weak gauge couplings appear naturally in such string compactiﬁcations, once modulus stabilization is included. In these systems the gauge couplings can be small because they are often inversely proportional to the volume of some higher- dimensional cycle, whose volume gets stabilized at very large values [20]. Similar things can also occur in non-stringy extra-dimensional models [21].

Unbroken gauge symmetry without unbroken gauge symmetry

We believe there is a generic low-energy lesson to be drawn from how proton decay is avoided in phenomenological string constructions. This is because in these models, even though proton decay is forbidden by conservation of a gauged charge, the gauge boson that gauges this symmetry is not massless [16]. This combines the virtues of an unbroken symmetry (no proton decay), with the virtues of a broken symmetry (no new forces mediated by a massless gauge boson).1 Usually this happy situation arises in the string examples because the gauge symmetry in question is anomalous, if judged solely by the light fermion content, with anomaly freedom restored through Green-Schwarz anomaly cancellation. But in four dimensions Green-Schwarz anomaly cancellation relies on the existence of a Goldstone boson, whose presence also ensures that the gauge boson acquires a nonzero mass. For these constructions the eﬀective lagrangian obtained just below the string scale from matching to the stringy UV completion is invariant under the symmetry apart from an anomaly-cancelling term that breaks the symmetry in just the way required to cancel the fermion loop anomalies. §2 argues that this property remains true (to all orders in perturbation theory) as one integrates out modes down to low energies. Leading symmetry breaking contributions arise non-perturbatively, exponentially suppressed by the relevant gauge couplings. Consequently they remain negligibly small provided only that the gauge groups involved in the anomalies are weakly coupled. Although supersymmetry also plays a role in the explicit string examples usually examined, our point here is that this is not required for the basic mechanism that allows massive gauge bosons to coexist with conservation of the corresponding gauge charge.

Phenomenological summary

We next summarize, for convenience of reference, the combined bounds obtained from the constraints examined throughout the following sections.

1Superconductors are similar in this regard: the photon acquires a mass without implying gross violations of charge conservation.

17 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.1: Summary of the constraints presented herein. Each plot shows the bound on the new gauge coupling, αX , as a function of MX for various values of the kinetic-mixing parameter, sh η, assuming a vector coupling Xf L = Xf R := X, with X = B − L (X = B) drawn as sparse (dense) cross-hatching.

Mass vs coupling

2 Fig. 2.1 presents a series of exclusion plots in the αX − MX plane, where αX = gX /4π is the gauge-

18 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

fermion coupling and MX is the gauge boson mass. Each panel shows these bounds for different fixed values of the kinetic mixing parameter, sh η (for details on the definition of variables, see §2.3). The figure shows the collective exclusion area of all of the different bounds considered in this paper. For concreteness they are calculated for a vector-like charge assignment, Xf L = Xf R, with the choice X = B − L denoted by a lighter shading and the choice X = B denoted with a heavier shading. Comparison of the cases X = B and X = B − L shows how much the bounds strengthen once direct couplings to leptons are allowed. For η = 0, the dominant bounds are from neutrino scattering, upsilon decay, anomalous magnetic moments, beam-dump experiments, neutron-nucleus scattering and nucleosynthesis. Once kinetic mixing is introduced, many of these bounds improve, with the exception of the beam-dump bounds. Once sh η & 0.06, kinetic mixing becomes sufficiently strong that the W -mass bound prevails over < any other bounds in the MX ∼ MZ region. For sh η = 1, we discard the region where the oblique

T parameter is large (for details, see §2.4), and focus on the region where MX > 385 GeV. In this region, it is the neutrino-electron scattering bound and the W -mass bound that dominate.

Mass vs Mixing angle

It is useful to show these same bounds as exclusion plots in the mixing-angle/boson-mass plane, for

ﬁxed choices of the gauge-fermion coupling, αX . This allows contact to be made with similar bounds obtained in the context of dark matter-inspired U(1) models [9, 2, 3, 10], which correspond to the

αX → 0 limit of the bounds we ﬁnd here. This version of the plots is shown in Figure 2.2, restricted to the MeV-GeV mass range (in order to facilitate the comparison with earlier work).

−10 For small, but non-zero, gauge coupling (αX ∼ 10 ) the bounds from beam dump experiments weaken significantly. However, another strong bound from neutrino-electron scattering also begins > −7 to take effect. This bound dominates for larger αX , and once αX ∼ 10 the entire MeV−GeV mass range is excluded. Since the bounds in Figure 2.2 all rely on coupling to leptons, in the case where X = B the constraints arise through the kinetic mixing and are independent of αX . The resulting plot for X = B is therefore the same as is shown in the figure for αX = 0. However, as the gauge coupling is increased the neutron-nucleus scattering bound — discussed in §2.6.4 — eventually becomes important, first −8 being visible as an exclusion in the sh η – MX plane in the panel for αX ' 10 in Fig. 2.2.

2.2 Theoretical motivation

This section elaborates the motivations for weakly coupled, very light gauge bosons alluded to above. This is done both by summarizing the consistency conditions they must satisfy within the low-energy eﬀective theory relevant to experiments, and by describing how such bosons actually arise from several representative UV completions in string theory and extra-dimensional models.

19 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.2: Summary of the constraints on kinetic mixing relevant in the MeV-GeV mass range.

Each plot shows the bound on the kinetic mixing parameter sh η as a function of MX , for αX = 0, −10 −8 1 × 10 and 1 × 10 . The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

2.2.1 Low-energy gauge symmetries, consistency and anomaly cancellation

Very general arguments [22, 23] indicate that the interplay between unitarity and Lorentz invariance require massless gauge bosons only to couple to conserved charges that generate exact symmetries of the matter action. Consequently we normally expect the direct couplings of very light gauge bosons to be similarly restricted. This section reviews these arguments, emphasizing how they can break down [24, 25, 26] if the energy scale, Λ, of any UV completion is suﬃciently small compared with the gauge boson mass, M, and coupling, g:Λ ∼< 4πM/g. For the present purposes it suﬃces to restrict our attention to abelian gauge bosons (see however [26] for some discussion of the nonabelian case). The upshot of the arguments summarized here is that massive spin-one bosons can couple in an essentially arbitrary way if their mass, M, lies within a factor g/4π of the scale of UV completion. But once M becomes smaller than gΛ/4π, then the corresponding boson must gauge an honest-to-God,

20 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy linearly realized exact symmetry. In particular this symmetry must be anomaly free. However any anomalies that Standard Model fermions give a putative new gauge charge needn’t be cancelled by adding new, exotic low-energy fermions; they can instead be cancelled by the Goldstone boson whose presence is in any case required if the gauge boson has a mass. But this latter sort of cancellation also requires the UV completion scale to satisfy Λ ∼< 4πM/g. Notice that for any given M the condition Λ ∼< 4πM/g need not require Λ to lie below the TeV scale if the coupling g is small enough. For instance, if M ' 1 MeV then Λ lies above the TeV −5 scale provided g ∼< 10 (an upper limit often required in any case by the strong phenomenological bounds we ﬁnd below). And, as subsequent sections argue, such small couplings can actually arise in a natural way from reasonable UV completions.

Massless spin-one bosons

What goes wrong if a spin-one particle is not coupled to matter by gauging an exact symmetry of the matter action? If the spin-one particle is massless, then the problem is that one must give up either Lorentz invariance or unitarity (provided the particle has non-derivative, Coulomb-like couplings that survive in the far infrared). Lorentz invariance and unitarity ﬁght one another because the basic

ﬁeld, Aµ(x), cannot transform as a Lorentz 4-vector if Aµ creates and destroys massless spin-one particles [22, 23]. Instead it transforms as a 4-vector up to a gauge transformation, Aµ → Aµ + ∂µω, and so interactions must be kept gauge invariant in order to be Lorentz invariant [27].

Massive spin-one bosons

For massive spin-one particles the argument proceeds differently, as is now described. The difference arises because a 4-vector field, Aµ, can represent a massive spin-one particle [22]. To examine the relevance of symmetries, it is worth first considering coupling massive spin-one particles to other matter fields, ψ, in some arbitrary non-gauge-invariant way, with lagrangian density

L(Aµ, ψ). The first observation to make is that any such a lagrangian can be made gauge invariant for free, by introducing a Stückelberg field, φ, according to the replacements Aµ → Aµ := Aµ − ∂µφ and ψ → Ψ := exp[−iφ Q]ψ, where Q is a hermitian matrix acting on the fields ψ. With this replacement the lagrangian L(Aµ, Ψ) is automatically invariant under the symmetry Aµ → Aµ + ∂µω, φ → φ + ω and ψ → exp[iω Q]ψ, since both Aµ and Ψ are themselves invariant under these replacements. The original non-symmetric formulation corresponds to the specific gauge φ = 0. For gauge symmetry, absence of gauge invariance is evidently equivalent to nonlinearly realized gauge invariance (similar arguments can also be made in the nonabelian case [26]). But this gauge invariance is obtained at the expense of introducing a new scale. Since φ is dimensionless, its kinetic term involves a scale, v,

1 v2 L = − F F µν − (∂ φ − A )(∂µφ − Aµ) . (2.1) kin 4g2 µν 2 µ µ

In φ = 0 gauge the scale v is seen to be related to the gauge boson mass by the relation M = gv. In

21 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy a general gauge the scale v controls the size of couplings between the canonically normalized ﬁeld, ϕ = φ v, and other particles. For instance the coupling

∂ ϕ L = −i(ψγµQψ) A − µ , (2.2) coupling µ v

µ shows that the (ψγ Qψ)∂µϕ coupling is dimension-five, being suppressed by the scale v = M/g. Lagrangians with nonrenormalizable couplings like this must be interpreted as effective field theories, whose predictive power relies on performing a low-energy expansion in powers of E/Λ, for some UV scale Λ. The interpretation of the scale v then generically depends on the how high Λ is relative to 4πM/g = 4πv. We consider each case in turn.

Light spin-one bosons: M gΛ/4π

If the gauge boson is very light compared with the UV scale, then its low-energy interactions should be describable by some renormalizable theory. But renormalizability is only consistent with a dimension- 2 µ five interaction like the (ψγ Qψ)∂µϕ coupling of eq. (2.2) if this coupling is a redundant interaction, such as it would be if it could be removed by a field redefinition. A sufficient condition for an µ interaction of the form J ∂µϕ to be redundant in this way is if the field equations for ψ were to µ µ imply the quantity J (ψ) satisfies ∂µJ = 0 [28]. This shows that if the gauge boson is to be arbitrarily light relative to Λ, its low-energy, renormalizable couplings must be to a (dimension- three) conserved current. This is the usual prescription for obtaining these couplings by gauging a linearly realized matter symmetry, for which J µ is the usual Noether current.

More generic massive spin-one bosons: M ∼> gΛ/4π

µ If, on the other hand, the dimension-ﬁve coupling (ψγ Qψ)∂µϕ is not redundant, then there must be an upper bound on the UV scale: Λ ∼< 4πv = 4πM/g. Sometimes this may be seen from the energy- dependence predicted for the cross section of reactions in the low energy theory: if σ(E) ∝ 1/(4πv)2 2 then this would be larger than the unitarity bound σ ∼< 1/E for energies E ∼> Λ ' 4πv, indicating the failure at these energies of the low-energy approximation. If so, the full UV completion must intervene at or below these energies to keep the theory unitary. The upshot is that spin-one particles can couple fairly arbitrarily to matter provided they are massive, and provided the energy scale, Λ, of any UV completion satisﬁes Λ ∼< 4πM/g, where M is the gauge boson mass and g is its coupling strength. (Everyday examples of spin-one particles of this type include the ρ meson or spin-one nuclei.) It is only spin-one particles with M < gΛ/4π that must gauge linearly realized symmetries.

2.2.2 Anomaly cancellation

Any new gauge symmetry — henceforth denoted U(1)X — must be an exact symmetry (though possibly spontaneously broken), and in particular must be anomaly free. This is true regardless of

2The careful reader will recognize that this argument assumes negligible anomalous dimensions, and so needs re-examination for strongly coupled theories.

22 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy whether the symmetry is the linearly realized symmetry of a light gauge boson, or the nonlinearly realized symmetry of a massive gauge boson. Of particular interest in this paper are models where the new symmetry acts on ordinary fermions, because a robust motivation for thinking about light gauge bosons is the avoidance of proton decay in models with a low gravity scale (more about which below). In this case these ordinary fermions usually contribute gauge anomalies for the new symmetry, and an important issue is how these anomalies are ultimately cancelled. The two main anomaly-cancellation scenarios then divide according to whether or not anomalies cancel among the SM ﬁelds themselves, or require the addition of new particles.

Anomaly cancellation using only SM ﬁelds

The simplest situation is where the new gauge symmetry is simply a linear combination of one or more of the SM’s four classical global symmetries — baryon number B, electron number Le, muon number Lµ and tau number Lτ . In this situation there are only two independent combinations of these symmetries that are anomaly free3 [29], corresponding to arbitrary linear combinations of the anomaly-free symmetries Le − Lµ and Lµ − Lτ :

X = a(Le − Lµ) + b(Lµ − Lτ ) . (2.3)

Of course, evidence for neutrino oscillations [30] make it unlikely that these symmetries are unbroken in whatever replaces the Standard Model in our ultimate understanding of Nature.

Anomaly cancellation using the Green-Schwarz mechanism

If more general combinations of B, Le, Lµ and Lτ are to be gauged, it is necessary to introduce new particles that can cancel their Standard Model anomalies. For a new U(1)X symmetry the minimal way to do this is to add only the Goldstone boson, which must in any case be present if the corresponding gauge boson has a mass (as it typically must to avoid mediating a macroscopic, long-range new force, whose presence is strongly disfavoured by observations [31]). For a U(1)X symmetry this can always be done using the 4D version [32] of the Green-Schwarz mechanism [33]. Besides its intrinsic interest, this is a way of cancelling anomalies that actually arises from plausible UV physics, such as low energy string models. In principle, there are four types of new anomalies that can arise in 4D once the SM is supplemented by a new gauge symmetry, U(1)X . These are proportional to Tr[XXX], Tr[XXY ], Tr[XYY ] and Tr[XGaGa], where the trace is over all left-handed fermions and X denotes the new symmetry generator, Y is Standard Model hypercharge, and Ga represents the generators of the Standard

Model nonabelian gauge groups, SU(2)L × SU(3)c, as well as the generators of Lorentz transformations. In four dimensions CPT invariance implies the absence of pure gravitational anomalies,

3Notice that B − L carries a Standard Model anomaly in the absence of sterile right-handed neutrinos (see below).

23 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy and anomaly cancellation within the Standard Model ensures the absence of anomalies of the form Tr[YYY ] and Tr[GaGbGc]. It is always possible to redefine the new symmetry generator, V := X + ζ Y , to remove one of the two mixed anomalies. For instance, Tr[VVY ] = Tr[XXY ] + 2 ζ Tr[XYY ] can be made to vanish by choosing ζ appropriately (provided Tr[XYY ] does not vanish). It suffices then to consider only the case of nonzero anomalies of the form Tr[VVV ] and Tr[VGaGa], where Ga now includes also the generator Y . The anomaly then can be written in the Ga- and Lorentz-invariant form4 Z 4 n o δΓ = − d x ω cX FV ∧ FV + caTr[Fa ∧ Fa] − cLTr[R ∧ R] (2.4) Z 4 n a a o = − d x ω cX FX + ζ FY ) ∧ (FX + ζ FY ) + caTr[G ∧ G ] − cLTr[R ∧ R] , where Γ is the ‘quantum action’ (generator of 1PI correlations), the symmetry parameter is normalized by δXµ = ∂µω and the coefficients, cX , ca and cL, are calculable. Here FV = dV = FX + ζ FY is the gauge-boson field strength for the generator X + ζ Y , while Fa is the same for the Standard Model gauge bosons and R is the gravitational curvature 2-form.

Given the coefficients cX , ca and cL, here is how 4D Green-Schwarz anomaly cancellation works [32]. Consider the gauge kinetic lagrangian, including the Stückelberg field φ,

2 1 X µν 1 a µν v µ µ L = Linv − 2 Fµν FX − 2 Tr[Gµν Ga ] − (∂ φ − X )(∂µφ − Xµ) 4g 4ga 2 n a a o +φ cX FX + ζ FY ) ∧ (FX + ζ FY ) + caTr[G ∧ G ] − cLTr[R ∧ R] . (2.5)

Here Linv denotes those parts of the lagrangian that are invariant under all of the gauge symmetries that are not written explicitly. The second line is not invariant under gauge transformations because φ is not; its variation precisely cancels the fermion anomaly, eq. (2.4). An important observation is that the anomaly cancelling term is dimension-five, and so is not renormalizable. For instance, in terms of the canonically normalized field, ϕ = φv, the first anomaly cancelling term is Lanom = (ϕ/f)FX ∧FX +··· , where f = v/cX . As before, this implies the existence of a UV-completion scale, Λ, above which the low-energy effective description breaks down [25]. For < weakly coupled theories typically Λ ∼ 4πv ' 4πM/g ' 4πcX f marks the scale where the fields arise that are required to extend the Goldstone boson to a linear representation of the symmetry. Perhaps the most interesting feature of cancelling anomalies with the Green-Schwarz mechanism in this way is that the lagrangian remains invariant under the U(1) symmetry, apart from the anomaly-cancelling term. This is interesting because it means that the corresponding charge still appears to be conserved in the low-energy theory, despite the gauge field being massive. This opens up interesting phenomenological possibilities for the gauging of symmetries like U(1)B and U(1)B−L, which appear to be conserved in Nature but which are also ruled out as sources of the new long-range force that a massless gauge boson would imply.

4A similar formulation can be made using the anomaly in its ‘consistent’ form, rather than the ‘covariant’ form used in the text.

24 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

One might worry that arbitrary symmetry-breaking interactions might be generated by embed- ding the anomaly cancelling interactions (or the fermion triangle anomaly graph) into a quantum fluctuation. For instance if X = B, so the new gauge boson couples to baryon number, then why can’t some complicated loop generate a ∆B = ±1 interaction, O±1, that can mediate proton decay? ∓iφ After all, this can be U(1)B invariant if it arises multiplied by a factor e , which carries baryon number ∆B = ∓1. The difficulty with generating this kind of interaction is that it must involve φ undifferentiated.

But if we restrict Lanom to constant φ conﬁgurations, it becomes a total derivative. For constant φ, the dependence of observables on φ is similar to the dependence of observables on the vacuum angle, θ. Consequently it arises at best only non-perturbatively, proportional at weak coupling to a power of ∼ exp[−8π2/g2], where g is the anomalous gauge coupling. As a result the only potentially dangerous contribution of this type comes from the mixed X-QCD-QCD anomaly, which can generate nontrivial < φ-dependence once we integrate down to scales ∼ ΛQCD . This is not dangerous in particular for the classical symmetries, B, Le, Lµ and Lτ , since these do not have mixed QCD anomalies [29].

Anomaly cancellation using new fermions

More complicated possibilities for new gauge bosons emerge if new, light exotic fermions are allowed that also carry the new X charge (and so can also take part in the anomaly cancellation). We briefly describe some features involving such new exotic particles, although they do not play any role in our later phenomenological studies. The simplest example along these lines is X = B − L, which is anomaly-free provided only that the SM spectrum is supplemented by three right-handed neutrinos (one for each generation). Furthermore, conservation of L is consistent with all evidence for neutrino oscillations, although it would be ruled out should neutrinoless double-beta decay ever be witnessed. A practical way in which such new fermions can arise at TeV scales is if the UV theory at these scales is supersymmetric. In this case the plethora of new superpartners can change anomaly cancellation in one of two ways (or both). They can either directly contribute to the anomalies themselves, and possibly help anomalies cancel without recourse to the Green-Schwarz mechanism. Alternatively, they can modify the details of how the Green-Schwarz mechanism operates if the UV scale, v, associated with it is larger than the supersymmetry breaking scale, Msusy. In particular, supersymmetry typically relates the kinetic term for the Stückelberg field, (2.5), with a Fayet-Iliopoulos term in the scalar potential [35],

2 1 Z X S = − d4x τ − q φ†φ , (2.6) FI g2 i i i i where τ is a dynamical field whose vev acts as the low-energy Fayet-Iliopoulos parameter; the qi are the charges of the fields φi under the U(1) in question. In string examples the field τ corresponds to a modulus of the compactification, which controls the size of a cycle in the internal geometry on which some branes wrap. We note that the vanishing of the D-term is consistent with vanishing vevs

25 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy of the charged ﬁelds if τ = 0, i.e. the symmetry survives as an exact global symmetry when the cycle size vanishes (the singular locus). Small values of the vev are obtained if the cycle size is small.

2.2.3 Motivations from UV physics

The above summary outlines some of the theoretical constraints on coupling ordinary fermions to very light gauge bosons. This section shows how very small couplings can naturally appear in well- motivated ultraviolet physics, such as extra-dimensional models or string vacua. In particular, they often arise due to considerations of proton stability in constructions for which the gravity scale is small compared with the Planck scale, as we now explain.

Proton decay in low-scale gravity models

One of the surprises of the late 20th century was the discovery that the scale, Mg, of quantum −1/2 18 gravity could be much smaller than the Planck scale, Mp = (8πG) ' 10 GeV [12]. From the point of view of particle physics this possibility is remarkable for several reasons. Most obvious is the potential it allows for experimental detection if it should happen that Mg is in the vicinity of the TeV scale [13, 14].

But there is a potentially more wide-reaching consequence that Mg Mp has for the low-energy sector: the suppression by powers of Mg/Mp it allows for otherwise UV-sensitive radiative corrections [36]. This suppression arises because the contribution of short-wavelength degrees of freedom can saturate at Mg, allowing their effects to be suppressed by powers of the gravitational coupling. The most precise examples of this are provided by string theory, in the regime where the string scale is low, Mg := Ms Mp [12]. String theory makes the suppression of UV-sensitive contributions precise by providing an explicit stringy ultraviolet completion within which the effects of the full UV sector can be explored. Large-volume (LV) models [20] are particularly useful laboratories for these purposes, since these systematically exploit the expansion of inverse powers of the extra-dimensional 6 volume (in string units), V := (Vol)/`s 1, and it is ultimately these kinds of powers that enforce −1/2 the suppressions of interest since Ms/Mp ∝ V . Proton decay — that is, its experimental absence — turns out to impose a very general constraint on any fundamental theory of this type, with Mg Mp. It does so because having Mg very small removes two of the standard ways of keeping the proton stable in specific models. On one hand quantum gravity, and string theory in particular [18, 19], seems to preclude the existence of global symmetries, and this forbids ensuring proton stability by simply using a conserved global charge (such as baryon number).

If Mg is too small then it is also unlikely that such a symmetry simply emerges by accident for the lowest-dimension interactions in the low-energy eﬀective theory. The problem in this case is that we know that generic higher-dimensional interactions,

X ciOi Leﬀ = , (2.7) di−4 i Mg

26 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy eventually do arise in the low-energy eﬀective theory, such as the standard baryon-number violating

4-quark operators arising at dimension di = 6 [37] in the low-energy limit of grand-uniﬁed theories (GUTs) [38, 39]. But a dimension-six interaction of the form O/M 2 generically contributes a proton- 5 4 decay rate of order Γ ' mp/M , where mp is the proton mass, which is too large to agree with 16 observations once M falls below MGUT ' 10 GeV.

The way theories with Mg Mp usually evade proton decay is through the appearance of a gauged U(1), whose conservation forbids the decay. Of course, to be useful the gauged U(1) that appears must couple to the proton or its decay products in order to forbid its decay. But because this means ordinary particles couple to the new gauge boson, it potentially introduces other phenomenological issues. If the gauge symmetry is conserved, why isn’t the gauge boson massless? If the gauge boson is light, why isn’t the new boson seen in low-energy observations? If the gauge boson is heavy, the corresponding symmetry must be badly broken and so how can it help with proton decay? Interestingly, extant models can naturally address both of these issues, and often the low-energy mechanism that is used is Green-Schwarz anomaly cancellation with gauge boson mass generated through the St¨uckelberg mechanism described above. Sometimes this mechanism is also combined with supersymmetry to suppress the dangerous decays. The existence of these gauge bosons, their properties, and the way they evade the above issues, may be among the few generic low-energy consequences of viable theories with a low gravity scale:

Mg MGUT .

Sample symmetries:

The simplest proposals for new low-energy gauge groups that forbid proton decay are either baryon or lepton number, X = B or X = L. If the anomalies for these symmetries due to Standard Model fermions are cancelled through the Green-Schwarz mechanism, then no new light particles are required besides the massive gauge boson itself. More complicated examples are possible if the low-energy theory at TeV scales is supersymmetric. In this case symmetries like B − L, that in themselves cannot forbid proton decay, can help suppress proton decay if taken together with supersymmetry [16]. (For instance, the parity R = (−)F +3(B−L) that is usually used to suppress proton decay in the MSSM is a combination of fermion number and B − L.) More general combinations of B and L can also suppress proton decay in supersymmetric theories. Ref. [16] provides a list of the kinds of symmetries of this type that can be relevant to proton decay, as well as the conditions they must satisfy in order to have their anomalies be cancelled through the Green-Schwarz mechanism. The general form for the low-energy charge may be written

X = mTR + nA + pL , (2.8)

where TR is right-handed isospin; A is an axionic PQ symmetry; and L is lepton number, with the charge assignments given in the Table. The coeﬃcients m, n and p are subject to (but not over-constrained by) several anomaly cancellation conditions [16]. In particular B and L violating

27 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy interactions can be forbidden up to and including dimension six for some choices of these symmetries in the supersymmetric limit, as can the µ-terms of the superpotential – W ' µLLH and W ' µHH – if n 6= 0.

QUDLEH H

TR 0 1 −1 0 −1 1 −1 A 0 0 1 1 0 −1 0 L 0 0 0 1 −1 0 0 X 0 m n − m n + p −m − p m − n −m

Very light and weakly coupled gauge bosons from extra-dimensional models

For the phenomenological discussions of later sections we consider gauge bosons in the MeV to TeV mass range, whose direct couplings to Standard Model fermions are much smaller than those arising within the Standard Model itself. This section and the next one describe several way that very light and weakly coupled bosons can arise from reasonable UV physics. Extra-dimensional supergravity provides a simple way to obtain very light gauge bosons that are very weakly coupled. A concrete example is six-dimensional chiral gauged supergravity [40], for which the bosonic part of the gravity multiplet contains the metric, gMN , a Kalb-Ramond 2-form potential, BMN , and a scalar, φ. Because it is chiral this supergravity potentially has anomalies, whose cancellation imposes demands on the matter content. In six dimensions Green-Schwarz anomaly cancellation is not automatic, because cancellation of the pure gravitational anomalies requires the existence of a speciﬁc number of gauge multiplets [41]. Given these multiplets, mixed gauge-gravity anomalies can be cancelled through the Green-Schwarz mechanism using the couplings of the ﬁeld

BMN . The resulting supergravity admits simple solutions for which the extra dimensions are a sphere [42], whose moduli can be stabilized by a combination of background fluxes in the extra dimensions [15], and branes coupling to the 6D dilaton [43, 21]. An important feature of this stabilization is that the value of the dilaton field becomes related by the field equations to the size of the extra dimensions: φ 1 e = 2 , (2.9) (M6r) where M6 denotes the 6D Planck scale. This ensures these models are a rich source of U(1) gauge bosons, some of whom can have massless modes that survive to low energies below the Kaluza-Klein scale. Some of these gauge modes also naturally acquire masses through the Stückelberg mechanism

[15] (with the Stückelberg field arising as a component of the Kalb-Ramond field, BMN ). Besides having light gauge bosons, these models also naturally furnish them with very small coupling constants. This is because the loop-counting parameter for all bulk interactions turns out to be the value of the 6D dilaton, φ, with g2 ' eφ. But modulus stabilization, eq. (2.9), ensures that this coupling can be extremely small because it scales inversely with the size of the extra dimensions (measured in 6D Planck units).

28 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Very light and weakly coupled gauge bosons from low-energy string vacua

A related mechanism also often arises in low-scale string models. In early heterotic models the role of the Goldstone boson is played by a member of the dilaton super-multiplet: a ' Im S [35], while in later Type I and Type II models it is twisted closed string multiplets that instead play this role [44, 16]. Although the universal couplings of the dilaton restrict the kinds of symmetries that can arise in heterotic constructions of this type, the same is not true for Type I and II models. There is a simple reason why additional U(1) gauge groups often arise. The basic building blocks for constructing models of particle physics in type IIB and IIA string theory are D-branes. Generically, the gauge group associated with a stack of N D-branes is U(N), but the Standard Model gauge group involves special unitary groups, SU(3) × SU(2) × U(1). Typical GUT models also involve special unitary groups, like SU(5), SU(3)×SU(2)×SU(2)×U(1) (Left-Right symmetric models) or SU(4) × SU(2) × SU(2) (Pati-Salam models). It is the additional U(1)s that distinguish the Standard Model SU(N) factors from the U(N) factors arising from the D-branes, that give new low-energy gauge symmetries. Furthermore, anomaly cancellation in string theory typically demands the presence of additional D-brane stacks, in addition to those providing the Standard Model gauge group factors. These stacks also lead to extra U(1)s under which Standard Model particles are charged. Extra U(1)s also appear naturally in F-theory models (for a recent discussion see [45]). In many concrete examples these additional gauge ﬁelds correspond to U(1)B or U(1)B−L, hence can be relevant for the stability of the proton [46, 16] (see also [47] for a recent discussion).

Masses and couplings

For string vacua the masses and couplings of any gauged U(1)s can be computed, as we now briefly describe. Consider first the U(1)s associated with the same stack of D-branes as gives rise to the Stan- dard Model gauge group. As discussed earlier, such gauge bosons often acquire masses from the Stückelberg mechanism. The size of the mass generated in this way is the string scale when the U(1) is anomalous [48, 47], but it is the smaller Kaluza-Klein scale for non-anomalous U(1)s. For models with the compactification volume not too much larger than the string scale, these U(1) gauge bosons are very heavy. On the other hand, for large-volume models the string scale can be quite low, leading to additional U(1)s potentially as light as the TeV scale. The latter can have interesting low energy phenomenology (see for instance [46, 49, 50, 51, 52]). In these models the strength of the gauge coupling for the additional U(1)s is roughly the same as for the Standard Model gauge couplings (evaluated at the string scale), because both have the same origin: the world-volume theory of the stack. Hence they cannot be extremely small. The masses and couplings of the extra U(1) gauge bosons vary more widely when they arise from D-brane stacks whose SU(N) factors are not part of the Standard Model gauge group. For instance, the case of additional U(1)s associated with D7 branes wrapping bulk four cycles of the compactification is discussed in detail in [19]. The value of the gauge coupling in this case is inversely

29 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy proportional to the volume (in string units) of the cycle, Σ, that the D7 brane wraps,

4π g2 ≈ . (2.10) VΣ

In the context of the large volume scenario (LVS) of modulus stabilization [20], the size of the bulk 9 cycle associated with the overall volume of compactiﬁcation can easily be approximately VΣ ∼> 10 in string units, set by the requirement that one generate TeV-scale soft terms. Thus one can obtain −4 gauge couplings as low as g ∼< 2 × 10 [19, 53] (couplings larger than this can be obtained if the D7 brane wraps a cycle diﬀerent from the one associated with the overall volume). With couplings this < small, the gauge boson mass can be MX ' gv ∼ 100 MeV even if v is a TeV.

2.3 Gauge boson properties

With the above motivation, our goal in the remainder of the paper is to work out various constraints on the parameters of a massive (yet comparatively light) gauge boson, the X boson, that couples to a new U(1)X symmetry. Since the lowest dimension interactions dominate in principle at low energies, we include in our analysis all of the dimensionless couplings that such a boson could have with Standard Model particles: i.e. both direct fermion-gauge couplings and gauge kinetic mixing. We see how these are constrained by present data as a function of the gauge boson mass. More speciﬁcally, we consider an eﬀective lagrangian density below the supersymmetry breaking scale of the form

L = LSM + LX + Lmix (2.11)

where LSM is the usual Standard Model lagrangian; LX describes the X boson, including its couplings to the SM fermions; and Lmix is the kinetic-mixing interaction between the X boson and that of the

SM gauge factor U(1)Y [4]. Explicitly

1 m2 L = − X Xµν − X X Xµ + iJ µX , (2.12) X 4 µν 2 µ X µ

µ where Xµν := ∂µXν − ∂ν Xµ is the curl of the appropriate gauge potential, Xµ, and JX is the current for the U(1)X gauge symmetry involving the SM fermions. Similarly, Lmix has the form,

µν Lmix = χ Bµν X (2.13)

where Bµ is the SM gauge boson for the gauge factor U(1)Y . The analysis we provide complements and extends earlier studies of extra gauge boson phenomenology. In the lower part of the mass range we may compare with [5], who some time ago considered the special cases X = B − L and χ = 0. Contact is also possible in this mass range with µ more recent Dark Matter models [9, 2, 3] in the absence of direct matter couplings, gX J = 0. At masses much lower than those considered here other constraints on kinetic mixing have also been

30 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy studied, from the cosmic microwave background [11], and from the absence of new long-range forces [31] or milli-charged particles [4, 7]. There is also a broad literature on the phenomenology of gauge bosons at the upper end of the mass range, largely done in the context of a Z0 ﬁeld and often motivated by GUTs [38, 54, 55, 56, 57, 58]. Until recently, most did not include the kinetic mixing term. Constraints including kinetic mixing arising from precision electroweak experiments are considered in [59, 60, 61]; more recent bounds are found in [62, 63, 64, 8]. Many of these analyses overlap parts of our parameter space. For instance Z0 searches, such as [65], give bounds on the mass of the Z0 that apply in the regime that the couplings to fermions are identical to that of the Z. Others [6, 66] derive bounds for a Z0 coupled only to baryon number. One diﬀerence between the models examined here and those usually considered for Z0 phenomenology at the weak scale, such as those of ref. [67], is the absence in L of mixing between the X 2 µ and the Z bosons in the mass matrix (i.e. a term of the form Lmix = δm ZµX ). We do not consider this type of mixing because we imagine the models of interest here to break the X symmetry with a SM singlet. Notice that because the SM Higgs is uncharged under the X symmetry, the strong bounds as found, for example, in [68] don’t apply.

2.3.1 The mixed lagrangian

In this section we diagonalize the gauge boson kinetic mixing terms (and SM mass terms) and identify the physical combination of parameters relevant for phenomenology within the accuracy to which we work. Our goal in so doing is to follow ref. [69, 70] and identify once and for all how the gauge boson mixing contributes to fermion couplings and to oblique parameters [71] modified by the gauge-boson mixing. This allows an efficient identification of how observables depend on the mixing parameters. We begin by writing the lagrangian of interest more explicitly, after spontaneous symmetry breaking. Because it is the Z and photon that potentially mix with the X boson, we also focus on these sectors of the SM lagrangian. In order to distinguish the fields before and after mixing, where appropriate we denote the still-mixed fields with carets, e.g. Xˆµ, reserving variables like Xµ for the final, diagonalized fields. With this notation, the lagrangian of interest is

L = Lgauge + Lf + Lint , (2.14) where

Lgauge = Lkin + Lmass (2.15) with

1 1 1 χ L = − Wˆ 3 Wˆ µν − Bˆ Bˆµν − Xˆ Xˆ µν + Bˆ Xˆ µν (2.16) kin 4 µν 3 4 µν 4 µν 2 µν 1 m2 L = − m Wˆ 3 − m Bˆ m Wˆ µ − m Bˆµ − X Xˆ Xˆ µ , (2.17) mass 2 3 µ 0 µ 3 3 0 2 µ

31 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy and

X Lf = − f (/∂ + mf ) f (2.18) f X n µ ˆ 3 µ ˆ Lint = i g2 fγ T3f γLf Wµ + g1 fγ (Yf LγL + Yf RγR) f Bµ (2.19) f µ ˆ o +gX fγ (Xf LγL + Xf RγR) f Xµ . (2.20)

Here T3f , Yf L and Yf R denote the usual SM charge assignments, while Xf L and Xf R are the fermion charges under the new U(1)X symmetry. The SM masses, m3 and m0, are defined as usual [29] in 1 terms of the standard model gauge couplings, g1 and g2, and the Higgs VEV, v: m3 = 2 g2v and 1 m0 = 2 g1v. γL and γR are the usual left- and right-handed Dirac projectors. Defining the gauge-field-valued vector Vˆ to be

  Wˆ 3   Vˆ =  Bˆ  , (2.21)   Xˆ the above lagrangian can be written in matrix form

1 1 L + L = − Vˆ T Kˆ Vˆ µν − Vˆ T Mˆ Vˆ µ + iJˆT Vˆ µ (2.22) gauge int 4 µν 2 µ µ where    2  1 0 0 m3 −m3m0 0 ˆ   ˆ  2  K := 0 1 −χ and M := −m3m0 m 0  , (2.23)    0  2 0 −χ 1 0 0 mX and  3    Jµ g2 fγµT3f γLf ˆ  Y  X   Jµ := J  =  g1 fγµ (Yf LγL + Yf RγR) f  . (2.24)  µ    ˆX f Jµ gX fγµ (Xf LγL + Xf RγR) f

The oﬀ-diagonal elements of Mˆ ensure it has a zero eigenvalue, and the condition that the matrix Kˆ be positive deﬁnite requires χ2 < 1.

2.3.2 Physical couplings

In order to put this lagrangian into a more useful form we must diagonalize the kinetic and mass terms, and then eliminate the SM electroweak parameters in terms of physically measured input 2 quantities like the Z mass, MZ , the ﬁne-structure constant, α = e /4π, and Fermi’s constant, GF , as measured in muon decay.

32 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

The diagonalization is performed explicitly in the Appendix, leading to the diagonalized form

1 M 2 M 2 L = − VT Vµν − Z Z Zµ − X X Xµ + iJT Vµ , (2.25) 4 µν 2 µ 2 µ µ where the physical masses are

2 m q 2 M 2 = Z 1 +s ˆ2 sh2η + r2 ch2η + ϑ 1 +s ˆ2 sh2η + r2 ch2η − 4r2 ch2η (2.26) X 2 W X X W X X and

2 m q 2 M 2 = Z 1 +s ˆ2 sh2η + r2 ch2η − ϑ 1 +s ˆ2 sh2η + r2 ch2η − 4r2 ch2η . (2.27) Z 2 W X X W X X

2 1 2 2 2 In these expressions mZ := 4 g1 + g2 v , g g cˆ := cos θˆ := 2 ands ˆ := sin θˆ := 1 , (2.28) W W p 2 2 W W p 2 2 g1 + g2 g1 + g2 while χ 1 sh η := sinh η := and ch η := cosh η := . (2.29) p1 − χ2 p1 − χ2

Finally, the quantities rX and ϑX are deﬁned by

( mX +1 if rX > 1 rX := and ϑX := , (2.30) mZ −1 if rX < 1

which ensures MZ → mZ and MX → mX as η → 0.

The currents in the physical basis are similarly read oﬀ as

    Z ˇZ ˇZ ˇA ˇX Jµ Jµ cξ + −Jµ sˆW sh η + Jµ cˆW sh η + Jµ ch η sξ  A   A  Jµ := J  =  Jˇ  , (2.31)  µ   µ  X ˇZ ˇZ ˇA ˇX Jµ −Jµ sξ + −Jµ sˆW sh η + Jµ cˆW sh η + Jµ ch η cξ where       ˇZ ˆ3 ˆY 2 Jµ Jµ cˆW − Jµ sˆW ieˆZ fγµ T3f γL − Qf sˆW f  ˇA   ˆ3 ˆY  X   J  := J sˆW + J cˆW  =  ie fγµQf f  ,  µ   µ µ    ˇX ˆX f Jµ Jµ igX fγµ [Xf LγL + Xf RγR] f and e := g2sˆW = g1cˆW ,e ˆZ := e/(ˆsW cˆW ) and Qf = T3f + Yf L = Yf R. Finally, cξ := cos ξ and sξ := sin ξ with the angle ξ given by

−2ˆs shη tan 2ξ = W . (2.32) 2 2 2 2 1 − sˆW sh η − rX ch η

33 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Writing the resulting lagrangian as

L = LSM + δLSM + LX , (2.33) shows that the X boson has two kinds of physical implications: (i) direct new couplings between the X boson and SM particles; (ii) modiﬁcations (due to mixing) of the couplings among the SM particles themselves.

Modiﬁcation of SM couplings

The modiﬁcation to the SM self-couplings caused by Z − X mixing are given by

z X δL = − m2 Z Zµ + ieˆ fγµ (δg γ + δg γ ) f Z , (2.34) SM 2 Z µ Z f L L f R R µ f

2 2 2 with [70] z := (MZ − mZ )/mZ and 2 gX δgf L(R) = (cξ − 1)g ˆf L(R) + sξ sh η sˆW (Qf cˆW − gˆf L(R)) + ch η Xf L(R) . (2.35) eˆZ

The last step before comparing these expressions with observations is to eliminate the parameters sˆW and mZ (the second of which enters the interactions through rX ) from the lagrangian in favour of a physically deﬁned weak mixing angle, sW , and the physical mass, MZ . This process reveals the physical combination of new-physics parameters that is relevant to observables, and thereby provides a derivation [70] of the X-boson contributions to the oblique electroweak parameters [71].

To this end deﬁne the physical weak mixing angle, sW , so that the Fermi constant, GF , measured in muon decay is given by the SM formula,

G e2 √F := 2 2 2 . (2.36) 2 8sW cW MZ

But this can be compared with the tree-level calculation of the Fermi constant obtained from W - exchange using the above lagrangian, giving (see Appendix)

2 2 2 z cW sˆW = sW 1 + 2 2 , (2.37) cW − sW to linear order in z (which we assume is small — as is justiﬁed shortly by the phenomenological bounds).

Eliminatings ˆW in favour of sW in the fermionic weak interactions introduces a further shift in these couplings, leading to our ﬁnal form for the neutral-current lagrangian:

X µ SM SM LNC = ieZ fγ gf L + ∆gf L γL + gf R + ∆gf R γR f Zµ , (2.38) f

34 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

where eZ := e/sW cW and

z s2 c2 ∆g = − gSM − z W W Q + δg f L(R) f L(R) 2 2 f f L(R) 2 cW − sW αT s2 c2 = gSM + αT W W Q + δg . (2.39) f L(R) 2 2 f f L(R) 2 cW − sW

SM 2 SM 2 The SM couplings are (as usual) gf L := T3f − Qf sW and gf R := −Qf sW , while the oblique parameters [71] S, T and U are given by αS = αU = 0 , (2.40) and

αT = −z . (2.41)

Direct X-boson couplings

The terms explicitly involving the X boson similarly are

1 M 2 X L = − X Xµν − X X Xµ + i fγ (k γ + k γ ) fXµ , X 4 µν 2 µ µ f L L f R R f with

e 2 SM SM kf L(R) = cξ ch η gX Xf L(R) + cξ sh η (Qf cW − gf L(R)) − sξ eZ gf L(R) . (2.42) cW We are now in a position to compute how observables depend on the underlying parameters, and so bound their size. When doing so we follow [70] and work to linear order in the deviations,

∆gf L(R), of the SM couplings, since we know these are observationally constrained to be small.

2.4 High-energy constraints

This section considers the constraints on the X boson coming from its inﬂuence on various precision electroweak observables measured at high-energy colliders. There are two main types of observables to consider: those that test the changes that X-boson mixing induces in SM couplings; and those sensitive to the direct couplings of the X boson to SM fermions. We consider each type in turn. We begin with two well-measured observables that are sensitive only to changes to the SM self- + − couplings: the W boson mass, MW , and the Z-boson branching fraction into leptons, Γ(Z → ` ` ). Later subsections then consider reactions to which direct X exchange can contribute, such as the + − cross section, σres(e e → h), for electron-positron annihilation into hadrons evaluated at the Z resonance.

35 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Consistency limits on accessible parameter space

Since the SM is in such good agreement with experiment [72], it is useful to linearize corrections to the SM parameters as we have done in the previous section. To be consistent, we limit ourselves to considering the subset of parameter space which is consistent with this linearization procedure. In practice, we require that the following two conditions of z be satisﬁed:

1. z must be real (see the discussion in the Appendix), which amounts to demanding that it is

obtained by a physically allowed choice for the initial parameters mX and χ. This implies that

2 2 2 2 ∆X − RX sW sh η ≥ 0 , (2.43)

where ∆X is deﬁned in eq. (2.138). This simpliﬁes to

p |∆X − κ| ≥ κ(κ + 1) (2.44)

where 2 2 κ := sW sh η . (2.45)

2. z must be small: z 1. To quantify this statement, we assume that z (or, equivalently, αT ) will be at most within 2σ from its global ﬁt value [70]:

|z| ≤ 0.014 . (2.46)

This bound has been considered in [73] in the context of hidden sector dark matter models.

In ﬁgure 2.3, we show the regions in the MX − sh η parameter space that are excluded by each of these bounds. From this, we see that the ﬁrst condition is dominant when sh η < 3 × 10−2, whereas for greater values of sh η, it is the second condition that is dominant.

2.4.1 Eﬀects due to modiﬁed W, Z couplings

We start with several examples of constraints that probe the induced changes to the SM self- couplings.

The W mass

Mixing with the X boson modiﬁes the SM prediction for the W mass due to its contribution to the electroweak oblique parameter T , as follows [70, 71]:

2 2 2 2 MW = mW = mZ 1 − sˆW (2.47) 2 h 2 i 2 cW αT = MZ (1 + αT ) 1 − sW 1 − 2 2 (2.48) cW − sW 2 2 sW ' (MW )SM 1 + αT 1 + 2 2 , (2.49) cW − sW

36 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.3: Plot of the bounds on z as a function of MX and sh η. The blue crosses limit the region in which z is real, and the red squares limit the region in which z 1. The hatched regions are excluded.

2 2 2 2 where (MW )SM is the full SM prediction, including radiative corrections: (MW )SM = MZ (1 − sW )+ loops. Because both the SM radiative corrections and the oblique corrections are known to be small, we can neglect their product in the above expression. At this point one might ask why bother examine the W mass correction separately, since the W mass is one of the observables included in the global ﬁts to oblique parameters, and we have already assumed that z must be small enough to ensure that the oblique parameter T lies within its 2-σ range obtained from global electroweak ﬁts (as in Figure E.2 of [72]). The reason we re-examine the W mass is that it leads to a slightly stronger constraint, because the mixing between the Z and X bosons does not contribute to the S parameter, and this prior information leads to a slightly stronger limit on T (as is shown in Figure 2.4).

Using the result, eq. (2.41), αT = −z together with eq. (2.137) for z as a function of η and MX gives the desired expression for ∆MW as a function of η and MX . In the limit when the Z and X masses are very close to one another — i.e. when ∆X is such that the equality in eq. (2.44) holds — the expression for z becomes

κ − ∆X 2 2 3 z = = −ϑX sW |η| + sW η + O(η ) (near-degenerate Z and X masses) , (2.50) 1 + ∆X and so 2 cW sW |η| η |∆MW | ' MZ cW 2 2 ' 2.75 GeV . (2.51) cW − sW 2 0.1

37 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.4: Plot of the EWWG bound on the S and T oblique parameters, showing how T is more tightly constrained given prior knowledge that S = 0.

Moving away from degeneracy, we ﬁnd the expression for z can be simpliﬁed as follows:

r 2 κRX κ − ∆X 1 − 1 − ∆2 2 2 2 X κR s η z = = κ − X + O(κ2) = W + O(η4) , (2.52) r 2 2 2∆X 1 − R κRX X 1 + ∆X 1 − 1 − 2 ∆X

where RX := MX /MZ (c.f. eq. (2.132)). So when MX and MZ are very diﬀerent,

2 3 2 3 2 sW cW η MZ 5 η 3 ∆MW ' 2 2 2 2 ' 1.10 × 10 2 2 GeV . (2.53) 2(cW − sW ) MX − MZ MX − MZ

The large-MX limit of eq. (2.53) agrees with the result given in [64], which ﬁnds

η 2 250 GeV2 ∆MW ' (17 MeV) . (2.54) 0.1 MX

The experimental agreement of the measured W mass with the SM prediction implies ∆MW ≤

0.05 GeV [74] (2σ uncertainty), and the constraint this imposes on sh η as a function of MX is shown in Figure 2.5. Several points about the comparison given in the ﬁgure are of note:

• The W -mass bound on η is model-independent inasmuch as it relies only on the kinetic mixing and does not depend at all on the fermion quantum numbers to which X couples;

38 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.5: Constraint obtained from limiting the inﬂuence of kinetic mixing on the SM value of the W mass. The hatched regions are excluded.

−3 • The strongest constraints on η occur for MX nearest to the Z pole, where |ηpole| ≤ 1.8 × 10 ;

−2 • When MX MZ , the bound on η becomes approximately MX -independent: |η| ≤ 6.2 × 10 . This behaviour is also visible in the analytic expression, eq. (2.53);

> • When MX MZ , the W mass bounds the ratio η/MX : giving MX /η ∼ 1.5 TeV.

Z Decay

The Z decay rate has been measured with great accuracy at LEP and SLC (for details regarding their analysis, see [72]). The experimental value [74] for the decay Z → `+`−, where ` can be any of the charged leptons, is Γ`+`− = 83.984 ± 0.086 MeV (1 σ), and agrees well with the SM result [74] 83.988 ± 0.016 MeV. The modiﬁed Z-fermion couplings change the tree-level decay rate,

2 MZ eZ 2 2 Γ + − = g + g , (2.55) ` ` 24π `L `R

SM where the couplings gÌ = gÌ + ∆gÌ (with I = L, R) are defined by the interaction (2.38). The deviation from the SM prediction therefore is

2 SM MZ eZ X SM ∆Γ + − := Γ + − − Γ + − ' [2g + ∆g ] ∆g . (2.56) ` ` ` ` ` ` 24π Ì Ì Ì I=L,R

39 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

+ − 2 Figure 2.6: The constraint arising from Z → ` ` decay on the coupling αX = gX /4π as a function of MX , for various values of sh η. The parameters agreeing with the positive bound (∆Γ = +∆Γexp) are marked with blue crosses, while those agreeing with the negative bound (∆Γ = −∆Γexp) are marked with red squares. The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

SM + − Notice that this vanishes if ∆gÌ = 0 or when ∆gÌ = −2gÌ . It can therefore happen that ∆Γ` ` vanishes for two separate regions as one varies through parameter space.

To obtain bounds on η and MX we use eq. (2.39) to eliminate ∆gf L(R), giving

2 2 z 1 2 zcW sW 1 gX g`I = cξ − − δIL + sW + 2 2 − sξ − δIL + 1 sW sh η − X`I ch η (2.57) 2 2 cW − sW 2 eZ

1 2 SM Here − 2 δIL + sW is the SM contribution, g`I , where δIL denotes a Kronecker delta function. Re- quiring ∆Γ`+`− to be smaller than the experimental (2 σ) experimental error gives the desired bound 2 on the parameters gX , η and MX . Figure 2.6 shows the excluded values in the αX = gX /4π vs MX

40 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.7: Plot of the constraint arising from considering Z decay into leptons in the limit where

MX MZ . The upper bound (∆Γ = +∆Γexp) is marked with blue crosses; the lower bound (∆Γ = −∆Γexp) is marked with red squares. Hatched regions are excluded.

plane, with the leptonic X-boson charge assumed to be X`L = X`R = −1 (such as would apply if X = B − L). Each panel of the figure corresponds to a different choice for sh η. For the panel in which sh η = 1 bounds at lower mass scales than roughly 385 GeV are not plotted, since these would conflict with a z = −αT satisfying the global electroweak fit, as outlined in figure 2.3. In order to understand the features present in the plots it is useful to consider the small-η limit of z and ξ. As discussed above for the W mass bound, the small-η limit when MX and MZ are very similar or very different must be considered separately. The expressions when MX and MZ are very different are 2 2 sW η sW η z ' 2 and ξ ' 2 . (2.58) 1 − RX RX − 1 2 2 2 As might be expected, all terms in ∆gÌ are suppressed by a factor of 1/RX ' MZ /MX and so go to 2 2 4 2 2 zero when MX MZ . In the opposite limit, RX → 0, ∆gÌ ' XÌ ηsW (gX /eZ ) + η sW cW /(cW − sW ), which can pass through zero (if XÌ η < 0) when |XÌ |(gX /eZ ) 'O(η). Several features of these plots should be highlighted:

• The best bounds come for MX ' MZ , even for small couplings gX , because in this limit the Z − X mixing parameter ξ becomes maximal (tan 2ξ → ∞), leading to strong constraints.

• For a similar reason, once η is suﬃciently large (sh η ' 0.1 — see also ﬁgure 2.7) the regime of

41 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

vanishingly small αX remains excluded because ∆g`L(R) is dominated by the oblique corrections to the weak mixing angle.

• For MX MZ the excluded area approaches a straight line, corresponding to a bound on the 2 ratio gX /MX , as expected from the form of ∆g`L(R).

• The graph is more intricate for MX MZ , with slivers of allowed parameter space emerging

for a narrow, η-dependent but MX -independent, value of αX . This happens (for suﬃciently large η) because ∆Γ = 0 is a multiple-valued condition on the parameters, as discussed above.

Figure 2.7 provides a view of the bounds taken on a diﬀerent slice through the three-dimensional parameter space (η, αX , MX ). This ﬁgure plots the constraints on αX vs sh η, in the regime where

MX MZ , showing how a wider range of αX is allowed as sh η shrinks. Note that bounds are only shown for the region where z 1.

2.4.2 Processes involving X-boson exchange

In this section we consider precision electroweak observables, like the resonant cross section for e+e− → hadrons, that receive direct contributions from X-boson exchange, in addition to the mod- iﬁcations to SM Z-boson couplings.

The annihilation cross section

We again proceed by computing the leading change to the tree-level cross section for e+e− → ff at leading order in the new interactions. Interference terms between SM loops and X-boson contributions may be neglected under the assumption that their product is negligible [70]. The relevant Feynman diagrams are shown in Figure 2.8, where the exchanged boson is either a photon, Z or X boson.

Figure 2.8: Relevant tree-level Feynman diagrams corresponding to electron-positron annihilation into fermion-antifermion pairs.

Neglecting fermion masses the relevant spin-averaged squared matrix element for this process is

42 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

(see, e.g. [29] for a treatment of SM scatterings using similar conventions)

1 X 2 h 2 2 2 2 i |M| = N |A (s)| + |A (s)| u2 + |A (s)| + |A (s)| t2 , (2.59) 4 c LL RR LR RL where s, t and u are the usual Mandelstam variables and

2 2 e QeQf eZ geI gf J keI kf J AIJ (s) := + 2 + 2 . (2.60) s s − MZ + iΓZ MZ s − MX + iΓX MX

The total unpolarized cross section that follows from this is

N s σ e+e− → ff = c |A |2 + |A |2 + |A |2 + |A |2 . (2.61) 48π LL RR LR RL

The couplings gf I and kf I in these expressions are deﬁned in terms of η, gX and MX by eqs. (2.39) and (2.42). The quantities ΓZ and ΓX are only important near resonance, and denote the full decay widths for the Z and X boson, respectively:

2 e MZ X Γ = Z g2 + g2 N (2.62) Z 24π f L f R c 2mf ≤MZ

MX X and Γ = k2 + k2 N , (2.63) X 24π f L f R c 2mf ≤MX where Nc is the colour degeneracy for fermion f.

The Hadronic Cross Section at the Z Pole √ Summing the above over all quarks lighter than MZ and evaluating at s = MZ gives the leading 2 correction to the resonant cross section into hadrons, σhad s = MZ , which is well-measured to be 41.541 ± 0.037 nb [74]. Requiring the deviation from the SM to be smaller than the 2σ error gives the desired constraints. Figure 2.9 shows a number of exclusion limits for the coupling αX vs the 3 X-boson mass (for Xf L = Xf R = (B − L)f , and MX in the range of 10 − 10 GeV), with each panel corresponding to a diﬀerent choice for η. These plots reﬂect several features seen in the analytic expressions for the couplings:

• For MX MZ and when η is small enough, the mass dependence of the bound on αX com- < −2 pletely drops out, leaving αX ∼ 10 in this limit. For larger η small values of αX can still be ruled out because the contributions of mixing are already too large. This mixing also ensures

that the region near MX = MZ tends to give the strongest bounds.

2 > • The regime MX MZ similarly constrains only the combination MX /αX ∼ 800 GeV (when η is small).

• For η not too small and MX smaller than MZ , ﬁgure 2.9 shows a window of unconstrained

couplings, for the same kinds of reasons discussed above for Γ`+`− .

43 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

√ Figure 2.9: The constraint obtained from σhad evaluated for s = MZ , as a bound in the αX − MX plane for various values of sh η. Blue crosses (red squares) indicate parameters where predictions diﬀer by 2 σ from experiment on the upper (lower) side. The hatched regions are excluded, while diagonal shading indicates a region excluded by global ﬁts to oblique parameters.

44 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 Figure 2.10: Plot of the constraint from σhad s = MZ in the region where MX MZ . The parameters agreeing with the positive bound are marked with blue crosses, while those agreeing with the negative bound are marked with red squares. The hatched regions are excluded.

Figure 2.10 shows a sample slice of the constraint region in the αX vs sh η plane, in the limit

MX MZ . Once again, bounds are not plotted within regions of parameter space for which z is not > 1. This plot shows that the smallest η for which small αX can be ruled out is sh η ∼ 0.06. Once η is larger than this, mixing rules out the X boson even with arbitrarily small gauge couplings.

2.5 Constraints at intermediate energies

Better constraints on lower-mass X bosons can be obtained from low-energy scattering of muon neutrinos with electrons and nuclei. The purpose of this section is to quantify these bounds by identifying how the cross section depends on the parameters gX , η and MX . We consider electron and nuclear scattering in turn.

2.5.1 Neutrino-electron scattering

− The Feynman graphs relevant for νµe scattering are those of Fig. 2.8, with three changes: (i) the gauge bosons are exchanged in the t-channel rather than s-channel; (ii) there is no photon-exchange graph and (iii) omission of right-handed neutrino polarizations.

45 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Crossing to t-channel can be obtained by performing the following substitution

s → t, t → u, u → s (2.64)

1 P 2 among the Mandelstam variables in the invariant amplitude 2 |M| . With these replacements, − − the diﬀerential cross section for the process νµe → νµe is

dσ 1 h i ν e− → ν e− = − |A (t)|2 s2 + |A (t)|2 (s + t)2 , (2.65) dt µ µ 8πs2 LL RL where 2 geI gνJ keI kνJ AIJ (t) = eZ 2 + 2 . (2.66) t − MZ t − MX

In the rest frame of the initial electron s ' 2 meEν and t ' −2y meEν , where Eν is the incoming f f neutrino energy and y is the fractional neutrino energy loss, y := Ee /Eν where Ee is the energy of f the outgoing electron. (In such experiments [75, 76], Eν ,Ee ∼ 1 − 10 GeV so ratios of the form f me/Eν and me/Ee can be neglected.) In terms of these new variables the diﬀerential cross section is dσ m E 2 2 ν e− → ν e− = e ν A [t(yE )] + A [t(yE )] (1 − y)2 . (2.67) dy µ µ 4π LL ν RL ν The cross section for anti-neutrino scattering is easily found from the above by interchanging

ALL ↔ ARL.

Special case: Low-energy limit with η = 0

One case of practical interest is when the boson masses, MZ and MX , are much greater than the p invariant energy exchange in the process of interest (i.e. |t| MX ,MZ ). When this holds the amplitudes, AIL, can be simpliﬁed to

2 eZ geI gνL keI kνL AIL ' − 2 − 2 MZ MX 2 2 eZ gνL MZ kνLkeI = − 2 geI + 2 2 , (2.68) MZ MX eZ gνL allowing the eﬀects of X-boson exchange be interpreted as an eﬀective shift in the electron’s electroweak couplings. For Eν ' 1 GeV and y order unity this approximation remains good down to

MX ' 30 MeV. The resulting cross section is particularly simple in the case of no kinetic mixing, for which we 1 2 1 can substitute the SM values geI = − 2 δIL +sW and gνL = 2 and the X-boson couplings keI = gX XeI and kνJ = gX XνJ , and obtain

√ 1 g2 X X A ' −2 2 G − δ + s2 + √X eI νL , (2.69) IL F 2 IL W 2 2 2 GF MX √ 2 2 using the SM result 2 2 GF ' eZ /2MZ . We see that the X-boson contribution can be regarded as

46 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 an additional contribution to sW in this limit. This is convenient because it allows the simple use of 2 2 2 constraints on sW to directly constrain the ratio gX /MX . The bounds are usually taken from the following ratio [76] of total cross sections,

− − σ(νµe → νµe ) R := − − . (2.70) σ(νµe → νµe )

Given the diﬀerential cross section

dσ 2G2 m E ν e− → ν e− = F e ν g2 + g2 (1 − y)2 , (2.71) dy µ µ π eL eR the total cross section becomes

2G2 m E g2 σ ν e− → ν e− = F e ν g2 + eR , (2.72) µ µ π eL 3 and so 2G2 m E g2 σ ν e− → ν e− = F e ν eL + g2 . (2.73) µ µ π 3 eR

Specializing to SM couplings the result depends only on sW :

2 2 2 4 2 3geL + geR 3 − 12sW + 16sW 1 + κ + κ R = 2 2 = 2 4 = 2 , (2.74) geL + 3geR 1 − 4sW + 16sW 1 − κ + κ

2 where κ ≡ 1 − 4sW 1. Using the experimental limit [75] ∆s2 = 0.0166 (2σ error) with G = 1.1664 × 10−5 GeV−2 √ W F 2 2 2 [74] to constrain ∆sW = gX /2 2 GF MX (assuming the choice XeI XνL = 1, as would be true for X = B − L for example), gives [77]

MX & 4 TeV . (2.75) gX

General Case: η 6= 0

More generally, the couplings kf I also acquire contributions from Z −X mixing even when gX = 0, as the above calculations show. In this case the more general bounds on gX , η and MX can be extracted by demanding that these contribute within the experimental limit ∆R. Since the experimental limit 2 2 2 is often quoted in terms of sW [75], we translate using ∆R = dRSM /dsW ∆sW . In obtaining R, we p 2 p 2 integrate over y using |t| MZ , but without assuming that |t| MX . When evaluating R, we set Eν to a nominal value of 1 GeV. Figure 2.11 shows the resulting bound in the αX − MX plane, for several choices for η assuming XeI XνL = 1. The resulting curves inspire a few comments:

• For large MX the bound is independent of η due to the MX /MZ suppression of the mixing in 2 2 ∆g and ∆k. This allows the direct gX /MX term to dominate. The bounds in this regime are

47 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 − Figure 2.11: Bound obtained on αX = gX /4π by limiting the inﬂuence of the X boson on the ν − e cross section ratio R, obtained as a function of MX for various values of sh η. The vertical line indicates the region ruled out by electroweak oblique ﬁts when η = 1.

relatively strong, and compete with those found in direct searches (e.g., by CDF [78] in the case of a SM-like Z0).

• For smaller MX , it is the terms in the couplings that are linear in η that inﬂuence the deviation from the η = 0 result. To see this, note that

sW 2 ∆gνL = −η gX XνL + O(η ), (2.76) eZ

2 2 and so there will be a term in |AIL| that is linear in η with the parametric dependence gX /MX . 2 2 When gX 0, it is this term that is dominant compared to the gX /MX term from X-boson

exchange. However, when gX ∼ η, this new term is no longer dominant and the bound regresses back to its original slope from the η = 0 case at high masses.

< p • Once 1 ≤ MX ≤ 10 MeV and MX ∼ |t|, the bound loses its dependence on MX and levels out to some ﬁxed value. This is expected from the form of eq. (2.66).

• When sh η = 1, much of the parameter space is excluded due to the requirement that z 1.

Therefore, only a small region with MX > 385 GeV is bounded by electron-neutrino scattering in this case.

48 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2.5.2 Neutrino-nucleon scattering

For the bounds from neutrino-nucleon scattering it is worth ﬁrst recalling how the standard analysis is performed. In terms of the neutral-current quark couplings, the quark-level cross sections for neutral-current muon-neutrino scattering are

g2 g2 σ (ν u → ν u) = σ g2 + uR , σ (ν d → ν d) = σ g2 + dR (2.77) µ µ 0 uL 3 µ µ 0 dL 3 g2 g2 σ (ν u → ν u) = σ uL + g2 , σ (ν d → ν d) = σ dL + g2 µ µ 0 3 uR µ µ 0 3 dR while those for charged currents are

σ σ ν d → µ−u = σ and σ ν u → µ+d = 0 , (2.78) µ 0 µ 3

2 where σ0 := 2NcGF meEν /π and Nc = 3. These show that the quark neutral-current and charged-current cross sections are all proportional to one another. The resulting cross section for neutrino-nucleon scattering in the deep-inelastic limit is obtained by summing incoherently over the quark contributions, giving

2 − 2 + σ (νµN → νµX) = εL σ νµN → µ X + εR σ νµN → µ X 2 + 2 − σ (νµN → νµX) = εLσ νµN → µ X + εRσ νµN → µ X , (2.79) where 2 2 2 εL(R) := guL(R) + gdL(R) . (2.80)

The experimental bounds come from the following ratios:

ν σ (νµN → νµX) 2 2 R := − = εL + r εR σ (νµN → µ X) 2 ν¯ σ (νµN → νµX) 2 εR R := + = εL + (2.81) σ (νµN → µ X) r

+ − where r := σ (¯νµN → µ X) /σ (νµN → µ X). Most useful is the Paschos-Wolfenstein ratio [79], from which the comparatively uncertain ratio r cancels:

ν ν¯ − R − rR σ (νµN → νµX) − σ (¯νµN → ν¯µX) 2 2 R := = − + = εL − εR . (2.82) 1 − r σ (νµN → µ X) − σ (¯νµN → µ X)

Experiments measure the following values [80]

2 εL = 0.30005 ± 0.00137 (2.83) 2 εR = 0.03076 ± 0.00110 . √ To constrain the X-boson coupling parameters we work in the regime with −t MX , for which

49 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.12: Plot of the constraint from R− (neutrino-nucleon scattering) assuming X = B − L.

Here, we plot the bound on αX as a function of MX for various values of η. Blue squares (red crosses) indicate parameters whose predictions lie 2 σ above (below) the central experimental value. The vertical line indicates the region excluded by precision oblique fits. the effects of X-boson mixing and exchange can both be rolled into a set of effective neutral-current couplings. The cross sections for quark-level scattering are then given by integrating eqs. (2.67) using (2.68), leading to expressions identical with eqs. (2.77) but with

2 eﬀ MZ kqI kνL gqI → gqI := 2 gqI gνL + 2 2 , (2.84) MX eZ

2 SM eff 2 where q = u, d and I = L, R. Using these in eq. (2.82) gives constraints on εI = guI + ∆guI + SM eff 2 gdI + ∆gdI . 2 Figure 2.12 plots the constraint found by requiring ∆εL ≤ 0.00137, assuming that X = B − L. 2 The plots are cut off at low mass where the condition t/MX ≤ 0.01 breaks down. Notice that for

50 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

η = 0 the bound is similar to that found for neutrino-electron scattering, with stronger bounds on

αX at smaller MX . For nontrivial η the strength of X − Z mixing eventually provides the strongest constraint, leading to strong bounds even for small gX at suﬃciently low MX .

2.6 Low-energy constraints

We ﬁnally turn to constraints coming from lower-energy processes.

2.6.1 Anomalous magnetic moments

The accuracy of anomalous magnetic moment (AMM) measurements [74] produce a strong constraint on the parameters of an extra gauge boson. We consider the bound arising from both the electron and muon AMM on the X gauge coupling as a function of the mass MX , for various values of the kinetic mixing parameter sh η. The correction to the AMM of a lepton, `, is given by [3]

2 Z 1 2 2 2 3 2 2 m` k`V z(1 − z) − k`A z(1 − z)(3 + z) + 2(1 − z) m` /MX δa` = 2 2 dz 2 2 2 , (2.85) 4π MX 0 z + (1 − z) m` /MX where the vector and axial couplings to the X boson are of the form

k`L + k`R e 1 1 2 k`V := = cξ ch η gX X`V − sh η − + 1 − sξeZ − + sW (2.86) 2 cW 4 4 k`L − k`R e 1 1 k`A := = cξ ch η gX X`A − sh η − − sξeZ − . 2 cW 4 4

There is, however, some subtlety in comparing this shift with experiment [10]: since the electron

AMM, δae, is used to determine the ﬁne-structure constant, α. The best bound on X boson couplings therefore comes from the next most precise experiment that measures α, and not the errors from −10 the (g − 2) experiments themselves. Following [10] this leads to the constraints δae < 1.59 × 10 −9 and δaµ < 7.4 × 10 , which when compared with the above expression gives the bounds shown in

Figure 2.13. These plots reproduce the results found in [3] when sh η = 0. In particular, the MX values below which any gauge coupling is excluded are consistent with the bounds shown in [10]. Since these bounds are often considered (e.g. in [10], [9, 2]) in the context of a constraint on kinetic mixing, we also plot the constraint on sh η as a function of the X boson mass, for various values of the gauge coupling. This is shown in Figure 2.14.

2.6.2 Upsilon decay

The bound we present here is an extension of the result found in [2]. By looking at the decay rate of the Υ(3s) bb bound state, researchers from the BABAR collaboration were able to place a bound

51 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.13: Plots of the constraint on the gauge coupling αX arising from the electron and muon

AMM as a function of MX , for various values of sh η. The electron AMM bound is marked with blue crosses; the muon AMM bound is marked with red squares. The plot assumes a coupling

X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

52 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.14: Plots of the constraint on the kinetic mixing, sh η, arising from the electron and muon

AMM as a function of MX , for various values of the gauge coupling αX . The electron AMM bound is marked with blue crosses; the muon AMM bound is marked with red squares. The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

on the occurrence of a particular channel involving a light pseudoscalar A0 [81]:

+ − + − e + e → Υ(3s) → γ + A0 → γ + µ + µ . (2.87)

Their upper limit on the number of events

+ − + − N = σ(e + e → Υ(3s)) × L × Br(Υ(3s) → γ + A0) × Br(A0 → µ + µ ) , (2.88)

+ − places a bound on the quantity Q := Br(Υ(3s) → γ + A0) × Br(A0 → µ + µ ).

53 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

However, the reaction of interest to us is

e+ + e− → γ + X → γ + µ+ + µ− , (2.89) which would have an identical signature. So the measured bound can also be reinterpreted as applying to the quantity

σ(e+ + e− → γ + X) Q := × Br(X → µ+ + µ−) (2.90) X σ(e+ + e− → Υ(3s))

−6 The experimental limit [81] QX < 3 × 10 gives the plots found in Figure 2.15 over the range

2mµ < MX < Ecm(= 10.355 GeV). This bound is quite strong, as it eliminates the entire region for sh η & 0.002 (as is shown in [2]). For smaller sh η, the bound is roughly constant when MX Ecm.

Figure 2.15: Plots of the constraint on the gauge coupling αX arising from Υ(3s) decay as a function of MX , for sh η = 0, 0.001. The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

As in the case of the AMM bounds, we also plot the constraint on sh η as a function of the X boson mass for various values of the gauge coupling — as shown in Figure 2.16.

2.6.3 Beam-dump experiments

In the MeV−GeV mass range, small gX and η are constrained by several beam dump experiments. These bounds are considered in detail in [2]; we apply a simpliﬁed version of their analysis here.

In these experiments, a large number Ne of electrons with initial energy E are collided with a ﬁxed target made of either aluminum or tungsten. Many of the resulting collision products are absorbed either by the target or by some secondary shielding. (Here, we use t to denote the total thickness of both the target and the shielding.) The remaining products continue along an evacuated tube to the detector, located at some distance D away from the target. For a summary of values for

54 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.16: Plots of the constraint on the kinetic mixing, sh η, arising from Υ(3s) decay as a function −8 of MX , for αX = 0, 1 × 10 . The plot assumes a coupling XeL = XeR = −1, such as would be true if X = B − L. Hatched regions are excluded. these parameters, see Table 2.1.

Experiment Target Ne Beam Energy t D E774 W 0.52 × 1010 275 GeV 30 cm 7.25 m E141 W 2 × 1015 9 GeV 12 cm 35 m E137 Al 1.87 × 1020 20 GeV 200 m 400 m

Table 2.1: Parameter values for the E774, E141, and E137 beam dump experiments.

The bound arises from the non-observation of X decay products. The incoming electron emits an X boson as bremsstrahlung during photon exchange with the nucleon (N): e− + N → e− + N + X. The X can then decay into either an e+e− or µ+µ− pair. However, a decay that occurs too soon is absorbed by the shield while a decay that occurs too late occurs past the detector. Therefore, the number of lepton anti-lepton pairs observed at the detector can be computed by multiplying the number of X bosons produced, NX , by the probability for the X to decay between z = t to z = D:

Z D 1 −z/`0 Nobs = NX dz e . (2.91) t `0

2 −1/2 Here, we write the lab frame decay length as `0 := γcτ, where γ = (1 − v ) is the relativistic time-dilation factor and τ is the inverse of the X rest-frame decay rate: τ := 1/ΓX . In estimating the number of X’s produced, we use the following result from [2]:

2 2 NX ∼ Ne µ 2 , (2.92) MX

2 2 where = χcW and µ ' 2.5 MeV is an overall factor that contains information regarding the details

55 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 Figure 2.17: The constraint arising from beam dump experiments on the coupling αX = gX /4π as a function of MX , for sh η = 0, 0.001. The E774 bound is marked with red squares; the E141 bound is marked with blue crosses; the E137 bound is marked with black circles. The plot assumes a coupling

X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

of the nuclear interaction, and is shown in [2] to be roughly constant for MX between 1 and 100 MeV. There is, however, an obstacle in applying this result directly to our analysis: it was derived without µ including any coupling to JX . In order to introduce the keL(R)-dependence in this expression, we note from [2] that the -dependence above arises from the cross section σ(e−γ → e−X) under the assumption that the electron is massless. This means that the left- and right-handed helicity X − e interactions contribute equally to the cross section, allowing the substitution

1 k2 + k2 2 → eL eR , (2.93) 4πα 2

2 2 with the normalization chosen so that the above expression reduces to χ cW in the case where

XeL(R) = 0, sh η 1 and MX MZ .

All in all, we ﬁnd that the number of X’s we expect to observe is given by

N µ2 k2 + k2 e eL eR −t/`0 −D/`0 Nobs ∼ 2 e − e . (2.94) MX 8πα

Applying the experimental exclusions [2] Nobs < 17 events (E774), Nobs < 1000 events (E141), and

Nobs < 10 events (E137) gives the bounds shown in Figure 2.17.

The plot for sh η = 0 gives good agreement with a similar plot in [3] (in the region over which these results overlap). The lower bounds for each experiment are approximately ﬂat because, in the

56 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Figure 2.18: The constraint arising from beam dump experiments on the kinetic mixing sh η as a −15 −12 −9 function of MX , for αX = 0, 1 × 10 , 1 × 10 , and 1 × 10 . The E774 bound is marked with red squares; the E141 bound is marked with blue crosses; the E137 bound is marked with black circles.

The plot assumes a coupling X`L = X`R = −1, such as would be true if X = B − L. Hatched regions are excluded.

region where t D `0, the fraction of X’s that decay is just D/`0, which gives

2 2 2 Ne µ keL + keR D Nobs ∼ 2 . (2.95) MX 8πα `0

2 The leading MX -dependence then cancels since `0 ∼ 1/MX . The upper bound results from the situation where the X bosons decay too quickly, and the decay products do not escape the shielding. We have only included plots for the cases where sh η = 0 and 0.001 because the bounds become too weak to constrain any region of this parameter space whenever sh η > 0.007.

57 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

An interesting feature of these bounds is that, at any given value of sh η, the gauge coupling can be increased such that the bounds are evaded. This occurs because a stronger gauge coupling causes the X bosons to decay within the shielding. Therefore, any bound on kinetic mixing which results from these experiments can weaken if the direct coupling of electrons to the X is taken to be non-zero. To demonstrate this, consider the bounds shown in Figure 2.18, which plots the bound on kinetic > −6 mixing as a function of the X-boson mass, for various values of αX . Note that, for αX ∼ 1 × 10 , these bounds are satisﬁed for all values of sh η in the relevant mass range.

2.6.4 Neutron-nucleus scattering

Low-energy neutron-nucleus scattering is important because most of the other low-energy bounds evaporate if the new boson doesn‘t couple to leptons (such as if X = B). For neutron-nucleus scattering a bound is obtained by considering the effects of the new Yukawa-type potential that would arise from a non-zero vector coupling of the X to neutrons. For light X bosons this can be seen over the strong nuclear force because it has a longer range, and can affect the angular dependence of the differential cross section for elastic scattering, dσ(nN → nN)/dΩ. This bound is discussed in the context of a scalar boson in [82] and more generally in [83]. Following these authors we parameterize the differential cross section as

dσ σ = 0 (1 + ωEcosθ) , (2.96) dΩ 4π where σ0 and ω are to be taken from experiments. Then an interaction of the form

g2 e−MX r ∆V (r) = n (2.97) nN 4π r leads to a correction to the expected value of ω, which is measured experimentally in the energy range E ∼ 1–10 keV for neutrons scattering with 208Pb. Agreement with observations leads to the bound [82, 83]

2 knV −11 4 < 3.4 × 10 , (2.98) 4πMX

1 where knV = kuV + 2kdV with kf V := 2 (kf L + kf R), as above. Figure 2.19 shows a plot of this bound for the nominal case sh η = 0. For this combination of couplings, an interesting cancellation occurs. For small kinetic mixing, the correction ∆kf L(R) has the form

e 2 SM SM 2 ∆k L R = η (Qf c − g ) + ηsW eZ g + O(η ) f ( ) cW W f L(R) f L(R) 2 = ηecW Qf + O(η ) , (2.99) and so the leading correction in η vanishes for any electrically neutral particle, like a neutron. This

58 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy makes this bound relatively insensitive to changes in kinetic mixing, not varying appreciably over the range 0 ≤ sh η ≤ 1. A similar cancellation occurs in the case of nucleosynthesis, considered in §2.6.6.

Figure 2.19: Plot of the constraint on the gauge coupling αX due to neutron-nucleus scattering as a function of the X-boson mass MX . The hatched regions are excluded.

2.6.5 Atomic parity violation

The Standard Model predicts a low-energy eﬀective coupling between the electron axial current and the vector currents within a given nucleus. The so-called weak charge of a nucleus with Z protons and N neutrons is deﬁned (up to an overall constant) as the coherent sum of the Z-boson vector couplings over the constituents of that nucleus [84]:

QW (Z,N) := 4 [Z (2guV + gdV ) + N (guV + 2gdV )] . (2.100) where g + g g − g g := f L f R and g := f L f R . (2.101) f V f 2 f A 2 In terms of these the leading parity-violating eﬀective electron-nuclear interaction generated by Z boson exchange is √ µ Leff = − 2GF geAQW (eγµγ5e) Ψγ Ψ . (2.102)

59 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy where Ψ is the ﬁeld describing the nucleus. X-boson exchange adds an additional term to this eﬀective lagrangian of the form

X keAQX µ Leff = − 2 (eγµγ5e) Ψγ Ψ (2.103) MX where

QX := Z (2kuV + kdV ) + N (kuV + 2kdV ) . (2.104)

Therefore, the total shift in the QW due to the X boson is

√ geA SM 2 2 keAQX ∆QW = QW − QW − 2 (2.105) (−1/4) GF MX where

SM SM SM SM SM QW (Z,N) = 4 [Z (2guV + gdV ) + N (guV + 2gdV )] 2 = Z 1 − 4sW − N. (2.106)

Notice that the bracketed term in ∆QW goes to 0 as η → 0, whereas the second term does not as long as kAe does not vanish in the same limit. The total eﬀective lagrangian for this system can then be written as X GF SM µ Leff + L = √ (Q + ∆QW )(eγµγ5e) Ψγ Ψ . (2.107) eff 2 2 W It is expected that the second term in eq. (2.105) will be dominant, so it is useful to consider the 2 form of kAe/MX in the limit where MX MZ and η 1:

1 2 2 √ keA gX XeA 1 + 2 cW η sW 2 = 2 − 2 η . (2.108) GF MX GF MX eZ

Therefore, if XLe = XRe, then the constraint becomes signiﬁcantly less stringent at low masses since, 2 2 instead of bounding the ratio gX /MX , it is now the combination gX η that is bounded. In order to emphasize the strength of this bound when XAe 6= 0, we use the charge assignments as shown in Table 2.2.

SM Fermion Charge X uL, dL 0 uR −1/3 dR +1/3 νL, eL 0 eR −1/3

Table 2.2: Charge assignments for the “right-handed” U (1).

If the X boson is light enough the above eﬀective interaction eventually becomes inaccurate in describing the electron-nucleus interactions. In this case, rather than pursuing a detailed analysis of

60 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

the microscopic lagrangian, we follow ref. [84] and introduce a corrective factor K(MX ) to account for the non-locality caused by the small mass of the X boson. This modiﬁes our expression for ∆QW as follows: √ geA SM 2 2 keAQX ∆QW = QW − QW − 2 K(MX ) . (2.109) (−1/4) GF MX

In [84] a table is given for K for various values of MX in the range 0.1 MeV < MX < 100 MeV. In order to render the graphs shown here, we have interpolated values of K by doing a least squares ﬁt to the values in [84].

Figure 2.20: Plot of the constraint on the gauge coupling αX due to the weak charge of cesium as a function of the X-boson mass MX , for various values of sh η. The hatched regions are excluded.

As with the neutrino-electron scattering bounds, the slope of the bound changes for η 6= 0 due to the production of a new dominant term through cancellation with the modified Z-fermion coupling. Once again, we exclude the region below 385 GeV for the sh η = 1 plot in order to avoid conflict with the electroweak oblique fits that require z 1. Since this bound relies crucially on there being an axial vector coupling to the electron, we did not include it when compiling the summary of bounds given in figures in §1.

2.6.6 Primordial nucleosynthesis

We close with the study of constraints coming from cosmology, which for the mass range of interest in this paper consists dominantly of Big Bang Nucleosynthesis.

61 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Any X bosons light enough to be present in the primordial soup at temperatures below T ∼ 1 MeV can destroy the success of Big Bang Nucleosynthesis (BBN) if they make up a suﬃciently large fraction (∼< 10%) of the universal energy density, leading to potentially strong constraints. In particular, such a boson poses a problem if it is in thermal equilibrium at these temperatures. Quantitatively, measurements of primordial nuclear abundances forbid the existence of the number of additional neutrino species (beyond the usual 3 of the SM) to be [85] δNν ≤ 1.44 (at 95% 8 C.L.). But since each boson in equilibrium counts 7 times more strongly in the equilibrium abundance, and since a massive X boson carries 3 independent spin states, the corresponding bound on the number, NX , of new species of spin-1 particles in equilibrium at BBN is

NX ≤ 0.84 . (2.110)

Even just one additional massive spin-1 boson into relativistic equilibrium is excluded at the 95% conﬁdence level. In a universe containing only the X boson and ordinary SM particles at energies of order 1 MeV, this leads to two kinds of constraints: either the X boson’s couplings are weak enough that it does not ever reach equilibrium; or if the X boson is in equilibrium it must be heavy enough (∼> 1 MeV) to have a Boltzmann-suppressed abundance. Figure 2.21 sketches the regions in the coupling-mass plane that are excluded by these conditions. The vertical line corresponds to the situation where abundance is suppressed by Boltzmann factors.

Figure 2.21: Constraint on the gauge coupling of the X due to its eﬀect on nucleosynthesis, as a function of the X-boson mass. The red squares indicate the bound due to Xν→Xν scattering; the blue crosses indicate the bound due to X→νν decay.

The constraints on couplings arise only for suﬃciently light particles, and express the condition that the couplings be weak enough to avoid equilibrium, at least up until the freeze-out temperature

TF . There are two curves of this type drawn, which diﬀer by whether it is collision or decay processes

62 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy that are the dominant equilibration mechanisms. Qualitatively, the requirement that reactions like

Xν ↔ Xν not equilibrate the X bosons leads to a constraint on the couplings that is MX -independent in the limit where MX TF , because then the size of both the reaction rate and Hubble scale is set by the temperature. The same is not true for decay reactions, X ↔ ν ν, since the rate for this also depends on the X-boson mass. A few other comments are appropriate for Figure 2.21. First, because they are outside the main scope of this study, the bounds shown are derived assuming that MX T (rather than being evaluated numerically as a function of MX ) and so are drawn only up to the mass range within 0.5 MeV of the freeze-out temperature. Second, the resulting expressions depend only weakly on η, showing little diﬀerence over the range 0 < sh η < 1. As discussed in earlier sections, this is a consequence of the neutrino’s electrical neutrality, which ensures that the leading small-η limit of the kinetic mixing ﬁrst arises at O(η2) rather than O(η).

Acknowledgments

We would like to thank Brian Batell, Joseph Conlon, Rouven Essig, Sven Krippendorf, David Poland, Maxim Pospelov, Philip Schuster, Natalia Toro and Michael Trott for helpful discussions. AM and FQ thank McMaster University and Perimeter Institute for hospitality. CB and AM thank the Abdus Salam International Centre for Theoretical Physics (ICTP) for its kind hospitality while part of this work was done. The work of AM was supported by the EU through the Seventh Framework Programme and Cambridge University. CB and MW’s research was supported in part by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information (MRI).

Note Added

While preparing this paper for publication, we have learned of a beam dump analysis [86] that has enlarged5 the exclusion regions discussed in §2.6.3.

2.A Appendix: Diagonalizing the gauge action

This appendix provides the details of the diagonalization of the gauge boson kinetic and mass mixings. The starting point is eq. (2.22),

1 1 L = − Vˆ T Kˆ Vˆ µν − Vˆ T Mˆ Vˆ µ + JˆT Vˆ µ , (2.111) 4 µν 2 µ µ with Kˆ and Mˆ given in eqs. (2.23).

5We thank Johannes Bl¨umlein for bringing this to our attention

63 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Diagonalization

We begin by performing the usual weak-mixing rotation to diagonalize the mass term:

   ˇ cˆW sˆW 0 Z ˆ ˇ    ˇ V = R1V := −sˆW cˆW 0 A (2.112)     0 0 1 Xˇ where g g cˆ := cos θˆ := 2 ands ˆ := sin θˆ := 1 . (2.113) W W p 2 2 W W p 2 2 g1 + g2 g1 + g2 The lagrangian then becomes

1 1 L = − Vˇ T Kˇ Vˇ µν − Vˇ T Mˇ Vˇ µ + JˇT Vˇ µ , (2.114) 4 µν 2 µ µ with new matrices

   2  1 0 χsˆW mZ 0 0 ˇ T ˆ   ˇ T ˆ   K = R KR1 =  0 1 −χcˆW  and M = R MR1 =  0 0 0  (2.115) 1   1   2 χsˆW −χcˆW 1 0 0 mX

2 1 2 2 2 where mZ := 4 g1 + g2 v . Under the same transformation the currents become   ˆ3 ˆY Jµ cˆW − Jµ sˆW ˇ T ˆ  ˆ3 ˆY  Jµ = R Jµ = J sˆW + J cˆW  (2.116) 1  µ µ  ˆX Jµ     ˆ 2 ˇZ ieZ fγµ T3f γL − Qf sˆW f Jµ X    A  =  ie fγµQf f  := Jˇ  ,    µ  f ˇX igX fγµ [Xf LγL + Xf RγR] f Jµ

which deﬁnese ˆZ := e/(ˆsW cˆW ) and uses the standard SM relations g2sˆW = g1cˆW := e and Qf =

T3f + Yf L = Yf R.

The kinetic term is diagonalized by letting

   ˜ 1 0 −sˆW sh η Z ˇ ˜    ˜ V := LV := 0 1c ˆW sh η  A (2.117)     0 0 ch η X˜ with χ 1 sh η := sinh η := and ch η := cosh η := . (2.118) p1 − χ2 p1 − χ2

64 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

This gives, by construction   1 0 0 T   K˜ = L KLˇ = 0 1 0 (2.119)   0 0 1 and  2 2  mZ 0 −mZ sˆW sh η T   M˜ = L MLˇ =  0 0 0  , (2.120)   2 2 2 2 2 2 −mZ sˆW sh η 0 mX ch η + mZ sˆW sh η while the currents become   ˇZ Jµ ˜ T ˇ  A  Jµ := L Jµ =  Jˇ  . (2.121)  µ  ˇZ ˇA ˇX −Jµ sˆW sh η + Jµ cˆW sh η + Jµ ch η

(Notice that L and R1 satisfy LR1 = R1L, so it is immaterial whether we ﬁrst diagonalize the SM mass or the kinetic terms.)

Finally, the mass matrix is diagonalized by letting

    cξ 0 −sξ Z ˜     V = R2V :=  0 1 0  A (2.122)     sξ 0 cξ X where cξ := cos ξ and sξ := sin ξ with the angle ξ given by

−2ˆs shη tan 2ξ = W , (2.123) 2 2 2 2 1 − sˆW sh η − rX ch η where we deﬁne for convenience mX rX := . (2.124) mZ

The diagonalized lagrangian then is

1 M 2 M 2 L = − VT Vµν − Z Z Zµ − X X Xµ + JT Vµ , (2.125) 4 µν 2 µ 2 µ µ where the physical masses are

2 m q 2 M 2 = Z 1 +s ˆ2 sh2η + r2 ch2η + ϑ 1 +s ˆ2 sh2η + r2 ch2η − 4r2 ch2η (2.126) X 2 W X X W X X 2 m q 2 M 2 = Z 1 +s ˆ2 sh2η + r2 ch2η − ϑ 1 +s ˆ2 sh2η + r2 ch2η − 4r2 ch2η (2.127) Z 2 W X X W X X

65 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

with ϑX deﬁned such that MZ → mZ and MX → mX as η → 0:

( +1 if rX > 1 ϑX := . (2.128) −1 if rX < 1

The currents in the physical basis are similarly read oﬀ as

    ˇZ ˇZ ˇA ˇX Z Jµ cξ + −Jµ sˆW sh η + Jµ cˆW sh η + Jµ ch η sξ Jµ  A   A  Jµ =  Jˇ  := J  . (2.129)  µ   µ  ˇZ ˇZ ˇA ˇX X −Jµ sξ + −Jµ sˆW sh η + Jµ cˆW sh η + Jµ ch η cξ Jµ

Since we are eventually interested in obtaining bounds in terms of the physical masses MZ and 2 2 MX , it is useful to invert these mass equations to ﬁnd the input parameters mZ and mX as a function of the physical masses and η. This gives

M 2 q m2 = Z 1 + R2 + ϑ (1 + R2 )2 − 4 1 +s ˆ2 sh2η R2 (2.130) X 2ch2η X X X W X M 2 q m2 = Z 1 + R2 − ϑ (1 + R2 )2 − 4 1 +s ˆ2 sh2η R2 (2.131) Z 2 2 X X X W X 2 1 +s ˆW sh η

where RX is used to denote the ratio of the physical masses:

MX RX := . (2.132) MZ

Also, the sign ϑX is now +1 for RX > 1 and −1 for RX < 1.

Given this inversion, the angle ξ can now be written as a function of RX and η only:

2ˆs shη tan 2ξ(R , η) = − W , (2.133) X 2 2 2 2 1 − sˆW sh η − rX (RX , η)ch η where

q 2 2 2 2 2 2 2 2 1 +s ˆW sh η 1 + RX + ϑX (1 + RX ) − 4 1 +s ˆW sh η RX r2 (R , η) = . (2.134) X X q 2 2 2 2 2 2 2 ch η 1 + RX − ϑX (1 + RX ) − 4 1 +s ˆW sh η RX

Physical couplings

We are now in a position to read oﬀ the physical implications of the X boson. That is, we may write

L = LSM + δLSM + LX , (2.135)

66 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy where the modiﬁcation to the SM self-couplings are given by

z X δL = − m2 Z Zµ + ieˆ fγµ (δg γ + δg γ ) f Z , (2.136) SM 2 Z µ Z f L L f R R µ f with [70] q 2 2 2 2 2 2 2 2 M − m sˆW sh η − ∆X + ϑX ∆X − RX sˆW sh η z(R , η) := Z Z = , (2.137) X 2 q mZ 2 2 2 2 1 + ∆X − ϑX ∆X − RX sˆW sh η where 1 ∆ := (R2 − 1) . (2.138) X 2 X

p 2 (Note that the η → 0 limit of z is easily veriﬁed by implementing the identity ∆X = ϑX ∆X .)

Given the form of z, one might worry that, for some choice of the parameters MX and sh η, z would yield a complex value. However, any such choice does not correspond to a choice of real values for the original parameters of the lagrangian, mX , mZ , and χ. This happens because suﬃciently large kinetic mixing tends to preclude the existence of mass eigenvalues, MX and MZ , that are too close to one another. This is why this region of parameter space is excluded from the plots of §2.4. The fermion couplings are similarly

2 gX δgf L(R) = (cξ − 1)g ˆf L(R) + sξ sh η sˆW (Qf cˆW − gˆf L(R)) + chη Xf L(R) . (2.139) eˆZ

The terms explicitly involving the X boson are

1 M 2 L = − X Xµν − X X Xµ (2.140) X 4 µν 2 µ X µ +i fγµ (kf LγL + kf RγR) fX , f with e 2 kf L(R) = cξ ch η gX Xf L(R) + sh η (Qf cˆW − gˆf L(R)) − sξeˆZ gˆf L(R) . (2.141) cˆW

Notice that in this basis Xµ does not couple directly to the electroweak gauge bosons at tree-level, but has acquired modiﬁed fermion couplings due to the mixing.

Oblique parameters

The only remaining step is to eliminate parameters likes ˆW and mZ in the lagrangian in favour of a physically deﬁned weak mixing angle, sW , and mass MZ . This process reveals the physical combination of new-physics parameters that is relevant to observables, and thereby provides a derivation [70] of the X-boson contributions to the oblique electroweak parameters [71].

67 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

We have already seen how to do this for the Z mass, for which

z m ' M 1 − . (2.142) Z Z 2

For the weak mixing angle it is convenient to deﬁne sW so that the Fermi constant, GF , measured in muon decay is given by the SM formula,

G e2 √F := 2 2 2 . (2.143) 2 8sW cW MZ

But this can be compared with the tree-level calculation of the Fermi constant obtained in our model from W -exchange, G g2 e2 √F 2 = 2 = 2 2 2 , (2.144) 2 8mW 8ˆsW cˆW mZ to infer 2 2 2 2 sˆW cˆW = sW cW (1 + z) , (2.145) which, to linear order in z, implies that

2 2 2 z cW sˆW = sW 1 + 2 2 . (2.146) cW − sW

Eliminatings ˆW in favour of sW in the fermionic weak interactions introduces a further shift in these couplings, leading to our ﬁnal form for the neutral-current lagrangian:

ie X L = fγµ T γ − Q sˆ2 f NC sˆ cˆ 3f L f W W W f µ +fγ (δgf LγL + δgf RγR) f Zµ ie z X z c2 ' 1 − fγµ T γ − Q s2 1 + W f s c 2 3f L f W c2 − s W W f W W µ + fγ (δgf LγL + δgf RγR) f Zµ

X µ SM SM := ieZ fγ gf L + ∆gf L γL + gf R + ∆gf R γR f Zµ , (2.147) f

where eZ := e/sW cW and

z s2 c2 ∆g = − gSM − z W W Q + δg , (2.148) f L(R) f L(R) 2 2 f f L(R) 2 cW − sW

SM 2 SM 2 where (as usual) gf L := T3f − Qf sW and gf R := −Qf sW . It is assumed throughout that the corrections z, δgf L(R), and

∆kf L(R) := kf L(R) − gX Xf L(R) (2.149) are small, so that any expression can be linearized in these variables. In particular, this means that

68 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

one can replace hatted electroweak parameters (i.e.s ˆW , etc...) with unhatted ones in our previous expressions to give:

q 2 2 2 2 2 2 sW sh η − ∆X + ϑX ∆X − RX sW sh η z(RX , η) = q (2.150) 2 2 2 2 1 + ∆X − ϑX ∆X − RX sW sh η SM 2 SM gX δgf L(R) = (cξ − 1) gf L(R) + sξ sh η sW (Qf cW − gf L(R)) + chη Xf L(R) (2.151) eZ

e 2 SM SM ∆kf L(R) = (cξch η − 1) gX Xf L(R) + cξsh η (Qf cW − gf L(R)) − sξeZ gf L(R) . (2.152) cW

Alternatively, one can use the relationship between z and η to determine the contribution to the 2 oblique parameters [71] S = U = 0 and αT = −z, where (as usual) α := e /4π. In this case, ∆gf L(R) can be written as in [70]

αT s2 c2 ∆g = gSM + αT W W Q + δg . (2.153) f L(R) f L(R) 2 2 f f L(R) 2 cW − sW

69 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

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[55] Q. Shaﬁ, “E(6) as a Unifying Gauge Symmetry, Phys. Lett. B 79 (1978) 301; R. Barbieri and D. V. Nanopoulos, “An Exceptional Model for Grand Uniﬁcation,” Phys. Lett. B 91 (1980) 369; F. Gursey and M. Serdaroglu, “E6 Gauge Field Theory Model Revisited,” Nuovo Cim. A 65 (1981) 337; V. D. Barger, N. G. Deshphande and K. Whisnant, “Phenomenological Mass Limits on Extra Z of E6 Superstrings,” Phys. Rev. Lett. 56 (1986) 30; D. London and J. L. Rosner, “Extra Gauge Bosons in E6,” Phys. Rev. D 34 (1986) 1530; F. del Aguila, M. Quiros and F. Zwirner, “Detecting E(6) Neutral Gauge Bosons Through Lepton Pairs at Hadron Colliders, ” Nucl. Phys. B 287 (1987) 419. A. Font, L. E. Ibanez and F. Quevedo, “Does Proton Stability Imply the Existence of an Extra Z0?,” Phys. Lett. B 228 (1989) 79.

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[56] J. L. Hewett and T. G. Rizzo, “Low-energy Phenomenology of Superstring-inspired E6 Models,” Phys. Rep. 183 (1989) 193; K. R. Dienes, “String Theory and the Path to Uniﬁcation: A Review of Recent Developments,” Phys. Rept. 287 (1997) 447 [arXiv:hep-th/9602045].

[57] G. Senjanovic and R. N. Mohapatra, “Exact Left-Right Symmetry and Spontaneous Violation of Parity, Phys. Rev. D 12 (1975) 1502; X. Li and R. E. Marshak, “Low-Energy Phenomenology And Neutral Weak Boson Masses In The Su(2)-L X Su(2)-R X U(1)-(B-L) Electroweak Model,” Phys. Rev. D 25 (1982) 1886.

[58] M. S. Carena, A. Daleo, B. A. Dobrescu and T. M. P. Tait, “Z0 Gauge Bosons at the Tevatron,” Phys. Rev. D 70 (2004) 093009 [arXiv:hep-ph/0408098].

[59] B. Holdom, “Oblique Electroweak Corrections and an Extra Gauge Boson,” Phys. Lett. B 259 (1991) 329.

[60] K. S. Babu, C. F. Kolda and J. March-Russell, “Implications of Generalized Z − Z0 Mixing,” Phys. Rev. D 57 (1998) 6788 [arXiv:hep-ph/9710441].

[61] A. Leike, “The Phenomenology of extra neutral gauge bosons,” Phys. Rept. 317 (1999) 143-250. [arXiv:hep-ph/9805494].

[62] A. Hook, E. Izaguirre, J. G. Wacker, “Model Independent Bounds on Kinetic Mixing,” [arXiv:hep-ph/1006.0973].

[63] J. Erler, P. Langacker, S. Munir and E. R. Pena, “Improved Constraints on Z0 Bosons from Electroweak Precision Data,” [arXiv:hep-ph/0906.2435].

[64] J. D. Wells, “How to Find a Hidden World at the Large Hadron Collider,” in Kane, Gordon (ed.), Pierce, Aaron (ed.): Perspectives on LHC physics, 283-298 [arXiv:hep-ph/0803.1243].

[65] S. Abachi et al. [D0 Collaboration], “Search for Additional Neutral Gauge Bosons,” Phys. Lett. B 385 (1996) 471.

[66] C. D. Carone and H. Murayama, “Realistic Models with a Light U(1) Gauge Boson Coupled to Baryon Number,” Phys. Rev. D 52 (1995) 484 [arXiv:hep-ph/9501220].

[67] See, e.g., K. T. Mahanthappa and P. K. Mohapatra, “Limits on Mixing Angle and Mass of Z0 using ∆ρ and Atomic Parity Violation,” Phys. Rev. D 43 (1991) 3093, Erratum: ibid. 44 (1991) 1616; A. E. Faraggi and D. V. Nanopoulos, “A Superstring Z0 at O(1 TeV)?,” Mod. Phys. Lett. A 6 (1991) 61; J. L. Lopez and D. V. Nanopoulos, “Primordial Nucleosynthesis Corners the Z0,” Phys. Lett. B 241 (1990) 392.

78 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

[68] S. Cassel, D. M. Ghilencea, G. G. Ross, “Electroweak and Dark Matter Constraints on a Z-prime in Models with a Hidden Valley,” Nucl. Phys. B 827 (2010) 256-280. [arXiv:hep-ph/0903.1118].

[69] C. P. Burgess, S. Godfrey, H. Konig, D. London and I. Maksymyk, “Bounding anomalous gauge boson couplings,” Phys. Rev. D 50 (1994) 7011 [arXiv:hep-ph/9307223];

P. Bamert, C. P. Burgess, J. M. Cline, D. London and E. Nardi, “R(b) and new physics: A Comprehensive analysis,” Phys. Rev. D 54 (1996) 4275 [arXiv:hep-ph/9602438].

[70] C. P. Burgess, S. Godfrey, H. Konig, D. London and I. Maksymyk, “Model independent global constraints on new physics,” Phys. Rev. D 49 (1994) 6115 [arXiv:hep-ph/9312291].

[71] G. Altarelli and R. Barbieri, “Vacuum Polarization Eﬀects of New Physics on Electroweak Processes,” Phys. Lett. B 253 (1991) 161;

M. E. Peskin and T. Takeuchi, “Estimation of Oblique Electroweak Corrections,” Phys. Rev. D 46 (1992) 381;

I. Maksymyk, C. P. Burgess and D. London, “Beyond S, T and U,” Phys. Rev. D 50 (1994) 529 [arXiv:hep-ph/9306267];

C. P. Burgess, S. Godfrey, H. Konig, D. London and I. Maksymyk, “A Global ﬁt to extended oblique parameters,” Phys. Lett. B 326 (1994) 276 [arXiv:hep-ph/9307337].

[72] The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the SLD Electroweak and Heavy Flavour Groups, “Precision Electroweak Measurements on the Z Resonance,” Phys. Rept. 427 (2006) [arXiv:hep-ex/0509008].

[73] E. J. Chun, J. -C. Park, S. Scopel, “Dark matter and a new gauge boson through kinetic mixing,” JHEP 1102 (2011) 100 [arXiv:hep-ph/1011.3300].

[74] C. Amsler et al. (Particle Data Group), “Review of Particle Physics,” Phys. Lett. B 667 (2008) 1.

[75] P. Vilain et al. [ CHARM-II Collaboration ], “Precision measurement of electroweak parameters from the scattering of muon-neutrinos on electrons,” Phys. Lett. B 335 (1994) 246-252.

[76] See also L. A. Ahrens, S. H. Aronson, P. L. Connolly, B. G. Gibbard, M. J. Murtagh, S. J. Murtagh, S. Terada, D. H. White et al., “Determination of electroweak parameters from the elastic scattering of muon-neutrinos and anti-neutrinos on electrons,” Phys. Rev. D 41 (1990) 3297-3316.

For a review, see G. Radel, R. Beyer, “Neutrino Electron Scattering,” Mod. Phys. Lett. A8 (1993) 1067.

[77] A. de Gouvea, J. Jenkins, “What Can We Learn from Neutrino Electron Scattering?,” Phys. Rev. D 74 (2006) 033004, [arXiv:hep-ph/0603036].

79 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

[78] T. Aaltonen et al. [ CDF Collaboration ], “Search for new physics in high mass electron-positron √ events in pp¯ collisions at s = 1.96-TeV,” Phys. Rev. Lett. 99 (2007) 171802. [arXiv:hep- ex/0707.2524].

[79] E.A. Paschos, L. Wolfenstein, “Tests for Neutral Currents in Neutrino Reactions,” Phys. Rev. D 7 (1973) 91.

[80] G. P. Zeller et al. (NuTeV Collaboration), “A Precise Determination of Electroweak Parameters in Neutrino Nucleon Scattering,” Phys. Rev. Lett. 88 (2002) 091802, [arXiv:hep-ex/0110059].

[81] Bernard Aubert et al. (BABAR), “Search for Dimuon Decays of a Light Scalar Boson in Ra- diative Transitions Upsilon → gamma A0,” Phys. Rev. Lett. 103 (2009) 081803 [arXiv:hep- ex/0905.4539].

[82] R. Barbieri, T. E. O. Ericson, “Evidence Against the Existence of a Low Mass Scalar Boson from Neutron-Nucleus Scattering,” Phys. Lett. B 57 (1975) 270.

[83] V. Barger, C. -W. Chiang, W. -Y. Keung, D. Marfatia, “Proton size anomaly,” [arXiv:hep- ph/1011.3519].

[84] C. Bouchiat, P. Fayet, “Constraints on the Parity-violating Couplings of a New Gauge Boson,” Phys. Lett. B 608 (2005) 87 [arXiv:hep-ph/0410260].

[85] R.H. Cyburt et al., “New BBN Limits on Physics Beyond the Standard Model from 4He,” Astropart. Phys. 23 (2005) 313.

[86] J. Blumlein, J. Brunner, “New Exclusion Limits for Dark Gauge Forces from Beam-Dump Data,” Phys. Lett. B 701 (2011) 155-159 [arXiv:hep-ex/1104.2747].

80 Chapter 3

Running with Rugby Balls: Bulk Renormalization of Codimension-2 Branes

— M. Williams, C.P. Burgess, L. van Nierop, A. Salvio

JHEP 1301 (2013) 102 [arXiv:hep-th/1210.3753]

3.1 Introduction

Does the vacuum have energy? If so, does it gravitate? Much of what we do not understand about quantum field theory is contained in these deceptively simple questions because calculations robustly indicate the vacuum should have lots of zero-point energy, yet cosmological observations indicate that this energy gravitates very little. This disagreement is particularly sharp if there are only four dimensions because then the Lorentz invariance of the vacuum makes its energy equivalent to a cosmological constant, which acts as a homogeneous and isotropic obstruction to having the comparatively flat universe in which we appear to live. By contrast localized energy sources need not curve all directions equally; for example General Relativity predicts the world-sheet geometry of a cosmic string to be flat regardless of the value of its tension, whose main effect is to curve the dimensions transverse to the string world sheet (and in particular to produce a conical singularity at the string position) [1]. This observation suggests exploring whether extra dimensions can help understand how the vacuum energy gravitates, such as if we were to live on a four-dimensional analog of a cosmic string within a spacetime having a few relatively large dimensions. Fewer dimensions are better in this context since bulk fields fall off less quickly with distance, and so allow all branes to compete with the

81 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy bulk regardless of how far apart they are in the extra dimensions.1 This has sparked the construction of several such brane-world systems, within one [2] or two [3, 4, 5, 7, 8, 9] extra dimensions. The explicit solutions identified in this way have the property that their on-brane geometry is flat despite having large on-brane tensions. Of course this in itself does not provide a solution to the cosmological constant problem, which also requires an understanding of why these geometries should be robust against quantum corrections. In particular, although solutions with flat on-brane geometries exist, so too do solutions with curved on-brane geometries. What is required is an understanding of how the on-brane curvature depends on physical choices for the bulk and the branes, and whether these choices remain stable against renormalization as high-energy modes are integrated out. The most progress understanding these issues has been made for 6D models, for which fairly general yet explicit calculations can be made. Although no special magic is found for non-supersymmetric models [10], theories with a bulk described by 6D supergravity have very attractive features [5, 11, 12]. In particular, although explicit solutions with on-brane de Sitter geometries are known2 [13], a sufficient condition for the absence of on-brane curvature is the absence of a coupling between the branes and a particular bulk field: the scalar dilaton that is related to the graviton by 6D supersymmetry [8, 10]. This is attractive since it is the kind of condition that is stable against arbitrary loops involving only on-brane particles [12, 15]. Such a condition would not be stable against bulk loops, however, since the brane must couple to the metric and the metric couples to the dilaton. So bulk loops must provide an important part of any naturalness story, particularly tracking how loops of heavy bulk states contribute to the low- energy effective vacuum energy. A missing step in this story is an explicit calculation of the UV sensitivity of Casimir energy calculations on the 6D geometries of interest. (See, however, [16, 17] for an assessment of bulk UV sensitivity for Ricci-flat geometries – including also the gravity sector, but in the absence of branes.) This paper and its companion [18] close part of this ‘bulk gap’, by computing explicitly the UV-sensitive part of the corrections to both brane and bulk interactions obtained by integrating out low-spin (spins zero, half and one) bulk fields in an extra-dimensional spacetime sourced by two codimension-two branes. This paper presents general results for arbitrary low-spin fields, while the companion specializes the results to the case where the bulk matter comes from a 6D supergravity. Our restriction to low-spin fields is a temporary one due to the technical complications of diagonalizing the full gravity-sector spectrum in the geometries of interest, and we intend to report on calculations using the full spectrum at a later date.3 The background geometry of the extra dimensions with which we work is a rugby ball [4, 5, 6]. This is described in §4.2 below, and is basically a flux-stabilized sphere with conical singularities at

1More than two extra dimensions might exist, but would not be relevant at low energies if they were much smaller than the ones of interest here. 2These solutions are interesting in their own right as a counter-example to the many no-go theorems for de Sitter solutions to higher-dimensional supergravity [14]. 3See also [19] for a partial calculation of the Casimir energy of the gravity sector and [20] for the bosonic part of the spectrum on a rugby ball.

82 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy both of its poles corresponding to the back-reaction of codimension-two branes located there. In a nutshell, §3.3 argues that the Casimir energy obtained by integrating out a bulk ﬁeld of 6D mass m has the generic form F(mr, α) V (α, m, r) = , (3.1) (4πr2)2 where r is the rugby ball’s radius and α is related to its defect angle, δ, (see below for more precise deﬁnitions) by α = 1 − δ/2π. In the limit of large mr the dimensionless function, F, becomes

(mr)6 (mr)4 F(mr, α) 'F (α) + s (α) − s (α) + (mr)2s (α) − s (α) + ··· ln(mr) , (3.2) 0 6 −1 2 0 1 2 where F0(α) is m-independent and the sk(α)’s are calculated explicitly for spins zero, half and one in §3.4. These constants contain the dependence on α in this limit, and so also encode the dependence of the answer on the boundary conditions of the bulk fields near the branes situated at the poles. Our results reduce in special cases to results in the literature for spheres [21, 22, 23]. The logarithm appearing in eq. (3.2) is slightly more complicated than what normally arises for loop calculations with tori [17, 24], and §3.3 shows this ultimately can be traced to the existence of a number of effective interactions involving the curvature (or the background flux), that happen to vanish when evaluated for tori. These effective interactions arise both in the bulk and on the branes, and are renormalized by quantum loops of bulk fields. The resulting running produces the logarithmic coefficient, and because it can be traced to the renormalization of UV divergences this logarithmic running (and the power-law dependence on m that pre-multiplies it) captures the dominant sensitivity to very heavy bulk loops that we seek. Furthermore, in the special case that the renormalized interactions are in the bulk lagrangian, the coefficients si are known for arbitrary geometries using very general Gilkey-de Witt methods [25, 26], a result we summarize in Appendix 4.A. We check that our results reduce to these general results in the appropriate limit: α → 1. A well-known property of the Gilkey-de Witt coefficients is that their contributions to bulk counterterms never depend on boundary conditions. Physically this is because they capture the effects of very short-wavelength modes, and because these see only local properties of the fields they don’t ‘know’ about conditions imposed at the boundaries. More precisely, the only UV divergences that directly involve the boundary conditions are those that renormalize the brane action at which the boundary conditions are applied. For the rugby ball this implies the renormalization of bulk interactions is identical to that for the sphere, and once these universal bulk counter-terms are subtracted we can separately identify how the brane action is renormalized by bulk loops. This exposes how these renormalizations depend on boundary conditions, and how they contribute to the coefficients si. Again, because short-wavelength modes cannot know about conditions at distant boundaries, our results for the renormalization of brane-bulk interactions are not specific to rugby balls, and apply equally well to any codimension-2 brane situated within a 6D geometry, and give rise there to a conical singularity. Explicit results for the sk’s generated by loops of low-spin bulk fields are given in §3.4, where we also see that our results agree (when appropriate) with known divergence calculations for spacetimes with conical

83 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy singularities [28]. Finally, §3.5 identifies a subtlety that brane back-reaction introduces when using standard Casimir energy calculations to identify the effective 4D cosmological constant as seen by a low-energy observer on the brane. The quantity V computed above is the standard fare of Casimir energy calculations: it is the (negative of the) loop-corrected 1PI effective lagrangian density evaluated at the background classical geometry (in this case a rugby-ball). And in the absence of branes this quantity is also the effective cosmological constant seen by a 4D observer, at least for systems where the zeroth-order 4D geometry is flat. This is because the 1PI action of the low-energy 4D effective theory has the √ generic form, Leff / −g = −Λ + curvature terms, which becomes −Λ if evaluated at the classical 4D background (which was assumed to be flat). As §3.5 shows, what changes in this argument once brane back-reaction is included is that it is no longer sufficient to evaluate the 1PI action at the uncorrected classical spacetime, even at first order in the loop corrections. This is most clear for systems where back-reaction cancels a classical brane tension, since in this case renormalizations of the tension should also be cancelled in the same way. In summary, what we present here as new is: an explicit calculation of the divergent part of the Casimir energy obtained from loops of low-spin bulk fields in a 6D geometry compactified on a flux-stabilized 2D rugby ball. We explicitly show how these divergences are renormalized into bulk and brane counter-terms, and compute how the corresponding effective interactions run as a result. Finally, we show how these renormalizations can be matched to the low-energy 4D effective theory, to see how they feed through to the cosmological constant as seen by a low-energy 4D observer. While we think these calculations can have a variety of applications to loop effects in extra- dimensional spacetimes, our main application is to use them to examine how supersymmetry ame- liorates the UV sensitivity of the vacuum energy, as described in a companion paper [18]. Brief comments on the relevance to using codimension-2 branes to address the cosmological constant problem are summarized in §3.6.

3.2 Bulk ﬁeld theory and background solution

We begin by summarizing the field content and dynamics of the bulk field theory of interest: D- dimensional matter (with spins zero, half and one) coupled to gravity. We then describe two- dimensional compactifications of this system in the presence of d = D −2 dimensional brane sources. We do not work within the probe limit, and so explicitly include the back-reaction of these sources on the bulk geometry. In our explicit one-loop calculations we specialize to the case D = 6 and d = 4.

Field content and action

i The fields of interest consist of a metric gMN , plus a collection of scalar fields φ , gauge potentials a r AM , and spin-half fermions ψ . We imagine the scalars and spin-half fields to transform under the i r gauge group, respectively represented on these by hermitian generators (ta) j and (Ta) s.

84 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

The bosonic part of the classical lagrangian for these ﬁelds is, in the Einstein frame:4

L 1 1 1 √ B = − R − G (φ)D φi DM φj − H (φ) F a F bMN − U(φ) , (3.3) −g 2κ2 2 ij M 4 ab MN

a where DM denote gauge-covariant derivatives for the scalars, FMN is the ﬁeld strength for the gauge potentials, and the functions Gij(φ), Hab(φ) and U(φ) are to be speciﬁed.

Rugby-ball compactiﬁcations

In general some scalars carry gauge charge and so having these be nonzero in the background would give some gauge bosons masses. Since in what follows our main interest is in background configurations for the massless gauge fields (though we do include massive gauge fluctuations about these backgrounds), when solving the classical field equations we assume that all nonzero background scalars do not carry the charges of the nonzero background gauge fields. In this case, the equations of motion which follow from the lagrangian, eq. (4.8), are:

1 G (φ) φj − H (φ) F a F bMN − U (φ) = 0 , ij 4 ab, i MN , i

bMN DM Hab(φ) F = 0 , (3.4)

2 i j 2 a b P RMN + κ Gij(φ)DM φ DN φ + κ Hab(φ) FMP FN 2κ2 1 + −U(φ) + H (φ)F a F bPQ g = 0 , 2 − D 4 ab PQ MN where

j MN h j P j j k li φ := g ∂M ∂N φ − ΓMN ∂P φ + γkl(φ) ∂M φ ∂N φ , (3.5)

P j and ΓMN and γkl(φ) respectively denote the Christoffel symbols constructed from the spacetime metric, gMN , and the target-space metric, Gij(φ). The simplest compactifications [31] are found using the Freund-Rubin ansatz [32] for which φi is a constant and ! ! gµν (x) 0 0 0 gMN = and FMN = , (3.6) 0 gmn(y) 0 f mn(y) where gµν is a maximally-symmetric Lorentzian metric (i.e. de Sitter, anti-de Sitter or flat space), and gmn is the metric on the two-sphere, S2, whose volume form is mn. The quantity f appearing in the background gauge field — which could be any one of the gauge fields present in the theory — is a constant. All other fields vanish.

The gauge potential, Am, that gives rise to such a field strength, Fmn, is the potential of a magnetic monopole and so satisfies a flux-quantization condition in the presence of charged matter.

4Our metric is ‘mostly plus’ and we follow Weinberg’s curvature conventions [29], which diﬀer from those of MTW [30] only by an overall sign in the deﬁnition of the Riemann tensor.

85 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

We denote the background gauge coupling constant by

1 := H , (3.7) g˜2 with H the component of Hab corresponding to the nonzero background flux Fmn. With this definition, requiring gauge transformations be single-valued for charged matter fields with charge qg˜ for integer q gives the quantization condition Z 2πN = q F = 4πr2qf , (3.8) S2 where N = 0, ±1, ... is an arbitrary integer and r is the sphere’s radius, in our conventions satisfying 2 Rmn = −gmn/r . Quantization requires the normalization constant, f, to satisfy

N f = , (without brane sources) (3.9) 2qr2 for all matter fields in the theory. If all charged fields have the same charge it is conventional to defineg ˜ so that q = 1. When more than one nonzero charge is present we choose q = 1 for the smallest nonzero charge (say) and then for any second charged field with q 6= 1 eq. (4.15) requires there to exist another integer N such that N/q = N1 is also an integer, and so all charges are rational multiples of the smallest one. p 2 mn 2 With the above ansatz — and using the identities FmpFn = f gmn and FmnF = 2f — the field equations boil down to the following three conditions:

κ2 f 2 κ2 N 2 R = − 2U g = − 2U g µν D − 2 g˜2 µν D − 2 4 q2g˜2r4 µν 1 κ2 f 2 κ2 N 2 = 2U + (D − 3) = 2U + (D − 3) (3.10) r2 D − 2 g˜2 D − 2 4 q2g˜2r4 f 2 N 2 and 2U − 2 = 2U − 2 2 4 = 0 . g˜ , i 4 q g˜ r , i

With f ﬁxed by ﬂux quantization, eq. (4.15), these equations can be solved for φi — and so alsog ˜(φ) and U(φ) — as well as r and the curvature in the D − 2 directions spanned by the (µν) coordinates.

Brane sources

The solutions as outlined so far describe an extra-dimensional 2-sphere supported by ﬂux, with metric ds2 = r2 dθ2 + sin2 θ dϕ2 , (3.11) without the need for brane sources [31]. However a class of solutions with brane sources can be included very simply [5], just by allowing the angular coordinate to be periodic with period ϕ ' ϕ + 2πα with α not necessarily equal to unity. Geometrically, this corresponds to removing a wedge

86 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy from the sphere along two lines of longitude and identifying points on opposite sides of the wedge [4, 5]. This introduces a conical singularity at both the north and south poles, with defect angle δ = 2π(1 − α), a geometry called the rugby ball.5 Physically, such a construction corresponds to the introduction of two identical brane sources, one situated at each of the singularities, with Einstein’s equations relating the defect angle to the properties of the branes. Concretely, take the action of the brane to be6 Z d √ Sb = − d x −g Lb A with L = T − b mnF + ··· , (3.12) b b 2˜g2 mn where the ellipses denote other terms involving two or more derivatives, and the coeﬃcients Tb, Ab, i Bb, Cb and so on could depend on the extra-dimensional scalars φ . However, the existence of a rugby ball solution does require the derivative with respect to φ of the total brane Lagrangian to vanish at the background, because the near-brane boundary condition for the bulk scalars requires [15]

2 j κ δSb lim Gij ρ ∂ρφ = , (3.13) ρ→0 2π δφ i where ρ denotes proper distance from the brane. For conical singularities, the near-brane boundary conditions for the metric imply [1, 10, 15] the defect angle at the brane’s position is given by

2 δb = κ Lb . (3.14)

A rugby-ball solution requires identical branes at each pole,7 for which

κ2L 1 − α = ± = 4G L , (3.15) 2π 6 ± where L± is the lagrangian for either of the source branes. The presence of the brane sources complicates the flux quantization condition in two important ways. The first complication arises because the resulting defect angle changes the volume of the sphere, which appears in the flux-quantization condition when integrating over the bulk magnetic field, Z F = 4πα r2f . (3.16) S2(α) The second complication arises because the branes themselves can carry a localized flux, given by

A Φ = b . (3.17) 1b 2π

5‘Rugby ball’ is used rather than ‘football’ to avoid a cultural ambiguity in what the shape of a football is. 6 ? A more covariant way of writing the term linear in Fmn is as the integral of the Hodge dual, F , over the d-dimensional brane world-sheet [10]. 7See [7, 8, 9] for solutions with conical singularities that can diﬀer at the two poles.

87 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

P For two identical branes the total ﬂux localized in this way is Φ1 := b Φ1b, in terms of which the ﬂux-quantization condition becomes

2πN X Z 2πN = = A + F = 2π Φ + 4πα r2f . (3.18) 1 q b 1 b S2(α)

The condition on the normalization constant f is then

N f = , (with brane sources) (3.19) 2qr2 where N := ω(N − Φ) (3.20) and (for later convenience) ω := 1/α, Φ := qΦ1 (and also Φb := qΦ1b).

Control of approximations

Since our entire discussion takes place within the semi-classical approximation we must demand all fields vary slowly enough to trust the low-energy effective-field-theory approximation [33, 34] for whatever (possibly a string theory) ultimately provides its ultraviolet completion. In practice, without knowing the details of this UV completion, we ask fields to vary slowly relative to the length scale, `, set by the gravitational coupling: κ2 = `D−2. Since (barring unnatural cancellations) eqs. (3.10) imply 1/r2 ∼ κ2V ∼ κ2N 2/g˜2r4, we also see that r2 ∼ κ2N 2/g˜2 ∼ 1/κ2V and so r ` also impliesg ˜2 N 2`D−4 and V `−D. Finally, once brane sources are included we must also demand them not to curve excessively the background geometry, and for branes with tension T this requires κ2T 1. For rugby-ball geometries this ensures the defect angle satisfies δ 2π.

3.3 General features of bulk loops

We now turn to the size of one-loop quantum fluctuations about the rugby-ball background just discussed. In particular, we calculate the UV-divergent part of loops computed for the various fields i r a φ , ψ and AM expanded about this background. Notice for these purposes that we need not restrict ourselves to the loop contributions of fields that are nonzero in the background. The main assumption we use to compute a field’s one-loop contribution to divergences is to 8 2 suppose that its kinetic operator can be written in the form ∆ = − + X + m , for some choice of local quantity X (perhaps a curvature or background flux) and a squared mass, m2. This is sufficiently general to include most of the spin-zero, -half and -one particles of interest in later sections. As we shall argue in more detail below, because we restrict to the UV sensitive part of the calculation our results apply more generally than just to rugby-ball geometries. On the other hand,

8For one-loop purposes this can also be done for fermions by working with the square of D/.

88 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy our calculation is insensitive to eﬀective interactions that cannot be distinguished using only a rugby-

2 MN ball geometry, such as diﬀerences between R and RMN R interactions (which are indistinguishable

M for spheres), or those interactions like ∇M R∇ R involving gradients of the curvature (which vanish for spheres). The other main restriction to our calculations that emerges in subsequent sections is the need to avoid fields whose fluctuations mix nontrivially with those of the metric. These must be avoided because for them it is not straightforward to show that ∆ takes the desired form. In particular this precludes our computing the effects of those fields that are nonzero in the classical background. Finally, for simplicity our final expressions also specialize to D = 6 and d = 4. Although our methods work equally well for any D and d = D−2, our explicit evaluations only capture all one-loop divergences for D ≤ 6. For D > 6 they give only a subset (the most divergent) of UV divergences.

One-loop calculations

Writing the generator of 1PI correlators as Γ = S + Σ, then the UV-sensitive part of one loop quantum corrections, Σ, can be calculated using heat-kernel methods. For a real D-dimensional ﬁeld of mass m we have the 1-loop quantum action9

Z 1 − + X + m2 iΣ = −i ddx V = −(−1)F Tr Log D , (3.21) 1−loop 2 µ2 where (−1)F = +1 for bosons and −1 for fermions, X is a local combination of background ﬁelds

(such as curvatures and background fluxes) and V1−loop denotes the effective 1-loop scalar potential (or Casimir energy density) in d = D − 2 dimensions. We assume that for the compactification of interest the higher-dimensional d’Alembertian splits into the sum of d- and two-dimensional pieces:

D = d + 2. Anticipating the need to dimensionally regularize ultraviolet divergences we write the spacetime dimension as d = dˆ− 2 ε, where dˆ is an integer and ε → 0 at the end of the calculation, so that

Z d 2 2 2 ! 1 X d kE kE + m + mjn V = (−1)F µ2ε ln , (3.22) 1−loop 2 (2π)d µ2 jn

2 where mjn denote the eigenvalues of −2 + X in the compactiﬁed space, and we Wick rotate to d d R ∞ Euclidean signature using d k = i d kE . Using the identity ln X = − 0 (ds/s) exp(−sX) (which is valid up to an inﬁnite constant that is independent of X) we then have

Z d Z ∞ 1 X d kE ds V = −(−1)F µ2ε exp −s k2 + m2 + m2 , (3.23) 1−loop 2 (2π)d s E jn jn 0

9 2 We return below to how the kinetic operator for higher-spin ﬁelds can be put into the form − + X + m .

89 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 d where µ in the exponential is absorbed into a rescaling of s. Performing the d kE integral using

Z d d/2 d kE −sk2 1 e E = , (3.24) (2π)d 4πs gives

µ2ε X Z ∞ ds V = −(−1)F exp −s m2 + m2 1−loop 2(4π)d/2 s1+d/2 jn jn 0 2ε ∞ µ Z dt 2 = − e−t(mr) S(t) , (3.25) 2 d/2 1+d/2 2(4πr ) 0 t where t = s/r2 is dimensionless,

F X S(t) := (−1) exp [−tλjn] , (3.26) jn and the dimensionless quantities λjn are deﬁned by

λ m2 := jn . (3.27) jn r2

In the examples of interest 1/r is the generic Kaluza-Klein scale for the compactiﬁcation.

In the appendices, we show that the function S(t) has the following small-t limit: √ s−1 s−1/2 3/2 2 5/2 S(t) = + √ + s0 + s1/2 t + s1 t + s3/2t + s2 t + O(t ) , (3.28) t t where the coefficients si depend on the spectra, λjn, and so also on the spin of the particle involved, and in principle also on the boundary conditions used near any branes situated within the background geometry. Much of what follows is devoted to computing these coefficients explicitly for the fields and boundary conditions of interest.

Using this small-t expansion in eq. (3.25) gives

2ε ∞ µ Z 2 s s s s V = − dt e−t(mr) −1 + −1/2 + 0 + 1/2 1−loop 2 d/2 2+d/2 3/2+d/2 1+d/2 1/2+d/2 2(4πr ) 0 t t t t s s s + 1 + 3/2 + 2 + O(t−2+d/2) td/2 t−1/2+d/2 t−1+d/2 mdµ2ε X = − s (mr)−2i Γ(i − d/2) , (3.29) 2(4π)d/2 i i

ˆ ˆ which shows that the result diverges when d → d for all terms, si, for which i − d/2 is a non-positive integer.

ˆ At this point we specialize to d = 4 so that all divergences are captured by si with i ≤ 2. For

90 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy this case we have

mdµ2ε s s s s V = −1 (mr)2 − 0 + 1 − 2 Γ(−d/2) + (ﬁnite as d → 4) . (3.30) 1−loop (4π)d/2 6 2 (mr)2 (mr)4

4−d −1 Using x Γ(−d/2) = (4 − d) + ln x + ﬁnite, the divergent part of V1−loop emerges as

C 1 µ V = + ln + V , (3.31) 1−loop (4πr2)2 4 − d m f where Vf is ﬁnite and µ-independent in the limit d → 4, and

s s C := −1 (mr)6 − 0 (mr)4 + s (mr)2 − s . (3.32) 6 2 1 2

What is important in what follows is that the coeﬃcient C depends on the independent external variables that control the properties of the background geometry, like r, N,Φb and α. Subsequent sections use this dependence to extract more information about the eﬀective interactions that renormalize these divergences.

Renormalization

Ultraviolet divergences are renormalized, as usual, into counter-terms in the action, and in the setup of interest here this action has both bulk and brane contributions. The goal of the next subsections is to separate each of these types from one another.

Bulk counterterms

The crucial feature of bulk counterterms that allows them to be separated from brane counterterms is their insensitivity to the boundary conditions satisfied by the bulk fields in the vicinity of the branes. This is most easily seen if they are computed using Gilkey-de Witt heat-kernel techniques [25, 26] — see Appendix 4.A — since this calculation is explicitly boundary-condition independent (for bulk counterterms). Physically this arises because divergences are sensitive only to modes with wavelengths much shorter than the physical size of the spacetime, since this both ensures their effects are captured by local interactions and that they are too short to build correlations between points in the bulk and distant boundaries. Because they do not depend on the boundary information, bulk counterterms can be computed using only the bulk geometry, without making reference to brane properties. In particular, they may be obtained by specializing the coefficients si to the special case where the background geometry has no defect angle: α = 1. The interactions to be renormalized are found by writing the most general local bulk lagrangian consistent with the given field content and symmetries, organized into a derivative expansion: LB =

91 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

LB0 + LB2 + LB4 ··· . Restricting to terms that are nonzero in the background, this expansion gives

√ LB0 = − −g U √ 1 1 L = − −g R + F F MN (3.33) B2 2κ2 4˜g2 MN √ κζ ζ 2 2 L = − −g AR RF F MN + R R (3.34) B4 8˜g2 MN κ

√ 3 LB6 = − −g ζR3 R + ··· (3.35) and so on. Here we deﬁne

2 2 MN MNPQ R = aR R + 2bR RMN R + cR RMNPQR , (3.36)

2 2 with aR + bR + cR = 1 so that R = R when specialized to a spherical geometry (for which mnpq mn 2 4 3 RmnpqR = 2RmnR = R = 4/r ). A similar, but more elaborate, deﬁnition is used for R .

Calculations on a sphere can only track the overall renormalization ζR2 , ζR3 and not how the separate parameters such as aR, bR and cR renormalize, but — as summarized in Appendix 4.A — for the bulk contributions these separate renormalizations are known explicitly for general geometries from earlier work [26]. A similar story also holds for terms that involve gradients of the background scalar ﬁelds, which vanish for the rugby-ball conﬁgurations. Since this includes in particular kinetic terms like (∂φ)2, it represents an obstruction to computing wave-function renormalizations for the scalars. This limits the generality of our later formulae for the renormalizations of scalar couplings.

Evaluating the bulk action at the background rugby-ball solution and integrating over the compact directions gives

2 Z 1 f κζ 4ζ 2 8ζ 3 V = − d2x L = 4πα r2 U − + 1 − AR + R − R + ··· (3.37) B B κ2r2 2˜g2 r2 κ r4 r6 2 1 N κζ 4ζ 2 8ζ 3 = 4πα r2 U − + 1 − AR + R − R + ··· , κ2r2 8 q2g˜2r4 r2 κ r4 r6

2 2 0 −2 −4 showing that U, 1/κ , ζR2 , and ζR3 terms can be read off respectively from the r , r , r , and r 2 2 2 2 4 terms in VB , while the 1/g˜ and ζAR terms are identified as the coefficients of N /r and N /r , respectively. In particular, the power of N acts as a proxy for the power of f, and so does not appear at all if the particle in the loop does not carry the charge gauged by the background gauge field.

2 The divergences in V1−loop are absorbed by splitting all couplings – i.e. U, 1/κ etc. – into a renormalized and counter-term part, with the infinities of V1−loop that arise in the bulk canceling the divergences in the bulk counter-terms as d → 4. Once this is done the µ-dependence of the renormalized quantities also must cancel the explicit µ-dependence in the corresponding finite parts of V1−loop. And the bulk part of the divergences in si can be identified because they do not depend on the brane boundary conditions, and so are the same as they would have been for a calculation on sph a sphere. Explicitly, if we denote by si what the coefficients si would have been if evaluated on a

92 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

sphere (i.e. with α = 1 and Φb = 0), then the full result for si can be written

sph tot si := α si + δsi , (3.38)

tot sph which can be regarded as the definition of δsi . (The factor of α pre-multiplying si arises from the integration over the extra dimensions, as in eq. (4.59).) This split is useful because only the sph divergences in the first term, αsi , can be absorbed into renormalizations of the bulk interactions tot of VB , while those of δsi must be absorbed into brane interactions. With this definition the running of the renormalized bulk couplings satisfy

∂U m6 ∂ 1 m4 µ = − ssph, 0 , µ = − ssph, 0 , (3.39) ∂µ 6(4π)3 −1 ∂µ κ2 2(4π)3 0 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − ssph, 0 , µ R = − ssph, 0 . (3.40) ∂µ κ 4(4π)3 1 ∂µ 8(4π)3 2

10 sph, 0 sph, 0 sph, 0 sph, 0 and so on. Here the quantities s−1 , s0 , s1 and s2 are computed below by explicitly summing over the KK spectrum on a sphere, giving results that agree with those obtained in Appendix 4.A using general heat-kernel methods. We note at this point that since we derive the running of the couplings by assuming they cancel only the explicit ln(µ/m) dependence of V1−loop, our explicit formulae exclude the case where additional µ-dependence enters through the appearance of renormalized couplings and fields pre-multiplying the pole in 1/(d − 4) (though the formulae are easily generalized to include this more general case). 2 The renormalization of the gauge-field terms, 1/g˜ and ζAR, is similarly done by keeping track of those divergences involving f. This can be done, for example by comparing loops of particles that couple directly to the background flux with those that do not. The result is

∂ 1 2 m2 8 q2m2 µ = − ssph, 2 = − ssph, 2 , (3.41) ∂µ g˜2 (4π)3r4f 2 1 (4π)3N 2 1 and ∂ κζ 2 8 q2 µ AR = − ssph, 2 = − ssph, 2 (3.42) ∂µ g˜2 (4π)3r4f 2 2 (4π)3N 2 2

sph, 2 sph where the particle in the loop has charge qg˜, and si represents that part of si that is proportional to N 2.

Brane counterterms

A similar reasoning can be applied to brane-localized interactions, which (unlike the bulk counterterms) can depend on the boundary conditions used near the brane but should be independent of those boundary conditions imposed on distant branes. (For earlier treatments of divergences in the presence of conical singularities, see [28].)

10 sph, k Notationally, for si the “sph” emphasizes that these quantities are evaluated on the sphere, and the superscript ‘k’ denotes terms involving k powers of the gauge-ﬁeld normalization, N .

93 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

To identify the brane contributions we ﬁrst subtract the contributions of the universal bulk tot P counterterms found above, using eq. (3.38), and use δsi = b δsi(b) to extract how each individual brane-localized interaction renormalizes. This can be done as before, by distinguishing those contributions that come from couplings to the background gauge ﬁeld from those that do not.

An additional complication in the case of brane counterterms is the necessity to disentangle which contributions come from which branes. This complication arises because although the KK spectra encountered below depend explicitly the ﬂux, Φb, localized on each brane, they only depend on the brane tensions through the common defect angle α = 1/ω = 1 − δ/2π. We write

tot same X diﬀ δsi = δsi + δsi(b) , (3.43) b

same diﬀ where δsi receives equal contributions from each brane and where the δsi(b) depends on the explicit Φb that in general diﬀer on each brane. Since we are interested in tracking the renormalization of each brane separately, the quantity of interest is

δssame δs := i + δsdiﬀ , (3.44) i(b) 2 i(b)

tot P since this is the one for which we can write δsi = b δsi(b). To avoid any extra notational clutter, we choose to drop the subscript (b) when writing δsi(b) in what follows.

Writing the most general local brane lagrangian in a derivative expansion: Lb = Lb0 + Lb1 +

Lb2 + Lb3 + ··· , and dropping terms that vanish when evaluated at the rugby ball background, we have √ Lb0 = − −γ Tb (3.45) √ A L = −γ b mnF (3.46) b1 2˜g2 mn √ ζ κζ L = − −γ R b R + Ab F F MN (3.47) b2 κ 4˜g2 MN √ κζ L = −γ AR˜ b R mnF , (3.48) b3 2˜g2 mn

2 √ 2 κ ζAR b MN L = − −γ ζ 2 R + RF F , (3.49) b4 R b 8˜g2 MN

M N mn and so on, where γµν := gMN ∂µx ∂ν x is the induced metric on the brane and the Fmn terms arise covariantly as the integral of the Hodge dual, ?F , over the brane world-volume.

Evaluating these at the background solution gives a contribution from each brane of size

2 2 2 A f 2ζ κζ f 2 κζ f 4ζ 2 κ ζ f V = T − b − R b + Ab + AR˜ b + R b − AR b + ··· b b g˜2 κ r2 2˜g2 g˜2r2 r4 2g ˜2r2 2 2 2 A N 2ζ κζ N κζ N 4ζ 2 κ ζ N = T − b − R b + Ab + AR˜ b + R b − AR b + ··· , (3.50) b 2 qg˜2 r2 κ r2 8 q2g˜2r4 qg˜2r4 r4 8 q2g˜2r6

94 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

tot P to the 1PI potential, with the sum over branes giving V := b Vb. Requiring the counter-term parts of the sum of the two brane actions to cancel the remaining divergences implies the renormalized quantities satisfy

∂ T m4 ∂ A 2 qm2 µ b = δs0 , µ b = δs1 , ∂µ 2(4π)2 0 ∂µ g˜2 (4π)2N 1 ∂ ζ m2 ∂ κζ q µ Rb = δs0 , µ AR˜ b = δs1 , (3.51) ∂µ κ 2(4π)2 1 ∂µ g˜2 (4π)2N 2 2 ∂ζ 2 1 ∂ κζ 8 q µ R b = δs0 , µ Ab = δs2 , ∂µ 4(4π)2 2 ∂µ g˜2 (4π)2N 2 2 where both branes are assumed to be identical, δsi is the quantity deﬁned by eqs. (3.38), (3.43) and (3.44), for which the superscripts ‘0’, ‘1’ and ‘2’ respectively correspond (as above) to the terms independent of, linear in, or quadratic in the background gauge ﬁeld (or its proxy, N ).

What remains is to compute the coeﬃcients s−1 through s2 for the ﬁelds of interest. This is the aim of the next sections.

KK mode sums

This section now sketches how the coefficients si are computed, by performing the sum defining S(t) using a hypothetical eigenvalue spectrum, mjn, that is general enough to include most of the special cases of practical interest. We can obtain our later results by performing the sum over the mode labels, n and j, in either of two different ways. The most reliable (and more cumbersome) method first performs a Poisson resummation, which has the advantage of casting the sums in a way that converges more quickly for small t. The second (and easier) method avoids the complications of Poisson resummation, instead using zeta-function regularization to regularize the part of the mode sum that is non-singular as t → 0. We present both methods because although the zeta-function technique is much simpler to use, its validity ultimately relies on the more complicated calculation based on Poisson resummation. More details on the equivalence of these two techniques can be found in Appendix 3.B.

To explain these two techniques, consider the following expression for the KK spectrum, λjn, that is general enough to include many of the cases met in later sections,

ω ω 2 λ = j + |n + b | + |n − b | + a − τ . (3.52) jn 2 + 2 −

Here n is an arbitrary integer and j = 0, 1, 2,... is a whole number. The quantities a, b± and τ are real parameters that differ for different spin fields in the loop and for different background flux quantum number, N. For example, in the case of a complex scalar field that is charged under the 1 background flux we show in the Appendix 3.C that the appropriate choices are a = 2 , b+ = |N|, 1 11 b− = 0 and τ = 4 , where N is the background flux quantum.

11 These expressions work in a particular patch for the background monopole gauge potential, with b± interchanged in the opposite patch.

95 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Our interest is in tracking how the sum, S(t), deﬁned using this spectrum depends on the geometrical quantities N and r, as well as the rugby-ball defect angle that is encoded in the quantity

1 δ ω = where α = 1 − , (3.53) α 2π and δ is the rugby-ball defect angle described in §4.2. In particular, the limiting case ω = 1 corresponds to the sphere, for which a variety of results are known [21, 22, 23, 35] for the Casimir energy. For instance, in this limit and with no gauge ﬂux (N = 0) the scalar spectrum becomes

λjn = `(` + 1), where ` = j + |n|. In this case the sums can be re-ordered to give the usual form for scalars on a sphere: ∞ ∞ ∞ ∞ ∞ ` X X X X X X = = . (3.54) n=−∞ j=0 n=−∞ `=|n| `=0 n=−`

Poisson resummation technique

We ﬁrst sketch the calculation that best controls the convergence of the sums at intermediate steps. For clarity of explanation we do not work with the most general form for the spectrum, but specialize to the following special case

∞ ∞ X X S(ω, t) = eτt exp −t(j + ω|n| + a)2 . (3.55) j=0 n=−∞

The diﬃculty with this sum is that it converges poorly in the regime of interest: where t is small. To remedy this we use the Poisson resummation formula, which relates the sum over a function f(x) to the sum over its Fourier transform F(q). That is, if

Z ∞ F(q) = dx f(x) e−iqx , (3.56) −∞ then ∞ ∞ X X f(n) = F(2πk) . (3.57) n=−∞ k=−∞ To apply this in the case of interest deﬁne

2 fj(x) := exp −t(j + ω|x| + a) (3.58) and Poisson resum the n-sum, giving S(ω, t) as

∞ ∞ τt X X S(ω, t) = e Fj(2πk) (3.59) j=0 k=−∞ where r 1 π 2 F (q) = e−q¯ e2i¯jq¯ 1 − erf(¯j + iq¯) + c.c. , (3.60) j 2ω t

96 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy and √ q ¯j := t(j + a) , q¯ := √ . (3.61) 2 ω t

Now comes the important observation, that is justiﬁed in some detail in Appendix 3.B. Because 2 √ of the factor e−q¯ and the inverse power of ω t appearing inq ¯, all of the terms with k 6= 0 in the sum are ‘regular’, in the sense that their sum vanishes in the limit that either ω or t vanishes. Only the k = 0 term contributes to the singular part of the small-t limit of S(ω, t). That is,

S(ω, t) = Ssing(ω, t) + Sreg(ω, t) , (3.62) with ∞ τt r ∞ X e π X Ssing(ω, t) := eτt F (0) = 1 − erf(¯j) + c.c. , (3.63) j 2 ω t j=0 j=0 and ∞ ∞ reg τt X X S (ω, t) := 2 e Fj(2πk) . (3.64) k=1 j=0

This last equality uses Fj(−q) = Fj(q), which follows from fj(−x) = fj(x). Appendix 3.B evaluates the remaining sums explicitly in the small-t limit, giving results that agree with the somewhat simpler techniques we now describe.

A simpler zeta-function method

A somewhat simpler way to compute the small-t limit of S(ω, t) is to start with eqs. (3.62) and eq. (3.63), but not to perform the Poisson resummation for Sreg(ω, t). The regular part is instead computed by zeta-function regularizing the initial sum over n and j. To see how this works use the Euler-Maclaurin formula, which states that for any analytic function f(x), ∞ Z ∞ ∞ X X Bi f (j) = dx f (x) − f (i−1)(0) , (3.65) n n i! n j=0 0 i=1

1 1 where Bi denote the Bernoulli numbers (of which the relevant ones are B1 = − 2 , B2 = 6 , B3 = 0, 1 (i−1) B4 = − 30 , and B5 = 0) and where f (0) denotes the (i − 1)-th derivative of f(x) with respect to x, evaluated at x = 0. Applying this formula to the j-sums in S(ω, t), requires using the function

2 fn(x) = exp −t(x + an) with an = ωn + a , (3.66) and so ∞ r ∞ X 1 π √ X Bi S (ω, t) := f (j) = 1 − erf(a t) − f (i−1)(0) , (3.67) n n 2 t n i! j=0 i=1

97 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

(i−1) where the ﬁrst few fn (0) terms are

2 2 2 (0) −tan (1) −tan (2) 2 2 −tan fn (0) = e , fn (0) = −2tan e , fn (0) = (−2t + 4t an) e 2 2 (3) 2 3 3 −tan (4) 2 3 2 4 4 −tan fn (0) = (12 t an − 8 t an) e , fn (0) = (12 t − 48 t an + 16 t an) e (3.68)

(i) 3 and fn (0) ∼ O(t ) for all i ≥ 5. Using this in " ∞ # reg τt X S (ω, t) = e S0(ω, t) + 2 Sn(ω, t) , (3.69) n=1 and Taylor expanding the error function gives a series expression for Sreg(ω, t) that involves divergent P∞ k sums of the form n=1 n with k a non-negative integer. Remarkably, deﬁning these as ζR(−k), reg where ζR(s) is Riemann’s zeta-function, gives a ﬁnite expression, which agrees with S (ω, t) as computed using Poisson resummation. To this must be added Ssing(ω, t), computed using eq. (3.63). Once this is done the resulting small-t expansion for S(ω, t) can be compared with previous calculations of the small-t limit using Gilkey-de Witt heat-kernel expansions, when these are known. They are known in particular for the sphere, where ω = 1, and Appendix 4.A shows that they agree in this limit.

3.4 Results for low-spin bulk ﬁelds

We next collect results for the coeﬃcients si of the small-t limit for bulk ﬁelds with spins zero, half and one.

3.4.1 Scalars

Consider first the simplest case of a single minimally coupled real scalar field, satisfying φ = 0, that is coupled to the background gauge field with monopole number N and brane–localized fluxes Φb. In this case the scalar spectrum (as derived in Appendix 3.C, in the case where the north patch12 of the gauge potential is used) is given by

ω ω 12 1 + N 2 λs = j + |n − Φ | + |n − N + Φ | + − , (3.70) jn 2 + 2 − 2 4 where the superscript ‘s’ is meant to emphasize that this (and later formulae in this section) applies only for scalars. The Casimir sum becomes

∞ ∞ " 2# 2 X X ω ω 1 S (ω, t) = et(1+N )/4 exp −t j + |n − Φ | + |n − N + Φ | + . (3.71) s 2 + 2 − 2 j=0 n=−∞

12As Appendix 3.C also shows, an equivalent result is obtained if the south patch is instead used.

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Using the results of Appendix 3.C, and its notation

(n) X n (1) Fb := |Φb| (1 − |Φb|) ,F := Fb ,F := F,G(x) := (1 − x)(1 − 2x) , (3.72) b we ﬁnd the following small-t coeﬃcients:

1 ss = , (3.73) −1 ω 1 1 ω2 ss (ω, N, Φ ) = + (1 − 3F ) , (3.74) 0 b ω 6 6 " # 1 1 N 2 ω2 ω3N X ω4 ss (ω, N, Φ ) = − + (1 − 3F ) − Φ G(|Φ |) + (1 − 15F (2)) , (3.75) 1 b ω 180 24 18 12 b b 180 b " 1 1 11 N 2 1 N 2 ω3N X ss (ω, N, Φ ) = − − + − (1 − 3F )ω2 − Φ G(|Φ |) 2 b ω 504 720 90 144 24 b b b ω4(1 − N 2) ω5N X + (1 − 15F (2)) − Φ G(|Φ |)(1 + 3F ) (3.76) 360 120 b b b b ! # 1 F (2) F (3) + − − ω6 . 1260 120 60

When ω = 1 and Φb = 0, these become

1 1 N 2 ssph = 1 , ssph = , ssph, 0 = , ssph, 2 = − , (3.77) −1 0 3 1 15 1 24 4 N 2 ssph, 0 = , and ssph, 2 = − 2 315 2 40 in agreement with the result in [23], as well as with the Gilkey-de Witt coeﬃcients as computed for a 6D scalar on a sphere in Appendix 4.A, using the general results found in [26]. If the scalar couples to the background ﬁeld with strength qg˜, its contribution to the running of the leading bulk counterterms therefore is

∂U m6 ∂ 1 m4 µ = − , µ = − , ∂µ 6(4π)3 ∂µ κ2 6(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , (3.78) ∂µ κ 60(4π)3 ∂µ 630(4π)3 ∂ 1 2 q2m2 ∂ κζ 2 q2 µ = , µ AR = . ∂µ g˜2 3(4π)3 ∂µ g˜2 5(4π)3

It is straightforward to check that the above expression for the loop component of the renormalization of the gauge coupling agrees with the result obtained directly by evaluating the Feynman graphs for the vacuum polarization in 6D ﬂat space, following standard methods [27], despite its initially unfamiliar sign.

Returning to general ω and Φb, to identify the brane-localized renormalizations we must ﬁrst

99 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

subtract the contribution of the bulk counterterms to obtain δsi. As described earlier, because these counterterms do not depend on the boundary conditions at the branes for ω 6= 1 their contribution to V contains only the trivial proportionality to 1/ω = α due to the volume, 4πα r2, that appears when V is computed by integrating the counterterms over the rugby ball. This is consistent with the overall factor of 1/ω that pre-multiplies all of the si in eqs. [4.247–4.248]. We then identify δsi as prescribed by the discussion surrounding eq. (3.44) and ﬁnd that δs−1 = 0,

ω2 − 1 ωF 1 δω δω2 ω2F δω |Φ | δs = − b = + − b ' − b , (3.79) 0 12 ω 2 ω 6 12 2 6 2 1 ω2 − 1 ω4 − 1 ω2F ω4F 2 δs0 = + − b − b 1 ω 36 360 6 12 1 δω 2 δω2 δω3 δω4 ω2F ω4F 2 δω |Φ | = + + + − b − b ' − b , (3.80) ω 15 45 90 360 6 12 15 6 ω2N N Φ δs1 = − Φ G(|Φ |) ' − b , (3.81) 1 12 b b 12 N 2 δs2 = s2 − − = 0 , (3.82) 1 1 24 ω " # 1 ω2 − 1 ω4 − 1 ω6 − 1 F ω2F 2 ω4F 2 ω4F 3 δs0 = + + − ω2 b + b + b + b 2 ω 180 720 2520 30 24 120 60 " 1 2 δω 5 δω2 17 δω3 37 δω4 δω5 δω6 = + + + + + ω 105 252 1260 5040 420 2520 # F ω2F 2 ω4F 2 ω4F 3 2 δω |Φ | −ω2 b + b + b + b ' − b , (3.83) 30 24 120 60 105 30 ω2N ω4N N Φ δs1 = − Φ G(|Φ |) − Φ G(|Φ |)(1 + 3F ) ' − b , (3.84) 2 24 b b 120 b b b 20 N 2 ω2 − 1 ω4 − 1 ω2F ω4F 2 δs2 = − + − b − b 2 ω 288 720 48 24 N 2 δω 17 δω2 δω3 δω4 ω2F ω4F 2 δω |Φ | = − + + + − b − b ' −N 2 − b . (3.85) ω 80 1440 180 720 48 24 80 48

Here δω := ω−1, and the last, approximate, equalities give the leading dependence in the limit where

δω 1 and |Φb| 1. The corresponding contributions to the running of the brane counterterms are

∂T m4 δω δω2 ω2F m4 δω µ b = + − b ' − |Φ | , (3.86) ∂µ 2(4π)2ω 6 12 2 4(4π)2 3 b ∂ A qΦ ω2m2 q2m2 A µ b = − b G(|Φ |) ' − b , (3.87) ∂µ g˜2 6(4π)2 b 3(4π)3 ∂ ζ m2 δω 2 δω2 δω3 δω4 ω2F ω4F 2 µ Rb = + + + − b − b ∂µ κ 2(4π)2ω 15 45 90 360 6 12 m2 δω |Φ | ' − b , (3.88) 2(4π)2 15 6

100 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

∂ κζ qΦ ω2 ω2 q2 A µ AR˜ b = − b G(|Φ |) + G(|Φ |)(1 + 3F ) ' − b , (3.89) ∂µ g˜2 24(4π)2 b 5 b b 10(4π)3 " 2 3 4 5 6 ∂ ζ 2 1 2 δω 5 δω 17 δω 37 δω δω δω µ R b = + + + + + ∂µ 4(4π)2ω 105 252 1260 5040 420 2520 # F ω2F 2 ω4F 2 ω4F 3 −ω2 b + b + b + b 30 24 120 60 1 2 δω |Φ | ' − b , (3.90) 4(4π)2 105 30 ∂ κζ 8 q2 δω 17 δω2 δω3 δω4 ω2F ω4F 2 µ Ab = − + + + − b − b ∂µ g˜2 (4π)2ω 80 1440 180 720 48 24 q2 δω |Φ | ' − − b . (3.91) (4π)2 10 6

Figure 3.1: Schematic plot of the potential as a function of Φb.

The appearance in these expressions of non-analytic terms involving |Φb| bears some comment. As we see in Figure 3.1, the signs of these terms are such that they represent maxima of the potential at Φb = 0, which are cuspy in that derivatives with respect to Φb are discontinuous at this point. We believe this discontinuous derivative arises because of a level crossing of the ground state at these points, for the following reasons. On the one hand, the explicit calculations given above reveal

V1−loop as a polynomial in |Φb|, which must therefore grow without bound for large |Φb|. On the other hand, V1−loop should be periodic under the replacements Φb → Φb + 1 and N → N + 1, as can be seen from the invariance of the KK spectrum, eq. (3.70), under this shift. (When shifting

Φ+ → Φ+ + 1 both N and n must be shifted by unity, but the shift in n is lost in V1−loop once the KK mode sum is performed.) This shows that the energy is not minimized for the same value of N as Φb is varied to suﬃciently large values; instead a new vacuum with N → N + 1 is energetically preferred once Φb become larger than unity. Cuspy maxima in the potential can arise at the points where this crossover between vacua occurs.

101 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

3.4.2 Spin-half fermions

As shown in Appendix 3.C, the KK spectrum for a fermion that is charged under the U(1) whose ﬂux supports the background rugby ball (using the north patch of the gauge potential) is given by [36] ω σ ω σ 12 N 2 λfσ = j + n − Φ − + n − N + Φ + + − (3.92) jn 2 1/2 + 2ω 2 1/2 − 2ω 2 4 where n1/2 = n − σ/2 and where σ ∈ {±1} speciﬁes the 4D helicity of each component of the spinor, of which there are 2 (4) each in the case of a 6D Weyl (Dirac) spinor. By identifying

1 1 − α δ N := N − σ , Φfσ := Φ − σΦf , Φf := 1 − ω−1 = = (3.93) fσ b b 0 0 2 2 4π

(and so Nfσ := ω(Nfσ − Φfσ) = N − σ as well), we can relate these fermionic spectra to the scalar spectrum considered previously:

N 2 (1 + N 2 ) λfσ (ω, N, Φ ) + = λs (ω, N , Φfσ) + fσ . (3.94) jn b 4 jn fσ b 4

As discussed in Appendix 3.C, the corresponding small–t coefficients take different forms depend- f f ing on whether or not |Φb| ≤ Φ0. When |Φb| ≤ Φ0 (for both Φb’s), the mode sum over the above f spectrum yields the following small–t coefficients for a 6D Weyl spinor: s−1 = −4/ω, " ! # 1 1 1 X sf (ω, N, Φ ) = + − 2 Φ2 ω2 , (3.95) 0 b ω 3 3 b b " ! 1 7 ωN Φ N 2 1 1 X sf (ω, N, Φ ) = − − + − Φ2 ω2 1 b ω 360 2 3 36 6 b b ! # ω3N X 7 1 X − Φ (1 − 4Φ2) + − Φ2(1 − 2Φ2) ω4 , (3.96) 6 b b 360 6 b b b b " ! 1 31 ωN Φ 31 N 2 7 N 2 X 7 X sf (ω, N, Φ ) = − − + − 1 − 6 Φ2 − Φ2 ω2 2 b ω 10080 16 720 1440 72 b 240 b b b ! ω3N X 7 7 N 2 (1 − 2N 2) X − Φ (1 − 4Φ2) + − − Φ2(1 − 2Φ2) ω4 24 b b 1440 720 24 b b b b ! X 7 Φ2 Φ4 −ω5N Φ − b + b b 240 6 5 b ! # 31 X 7 Φ2 Φ4 + − Φ2 − b + b ω6 . (3.97) 10080 b 240 12 15 b

(Note that the above expressions would be the ones valid when considering the limit Φb → 0 while

102 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

f f holding ω ﬁxed at some value 6= 1.) When |Φb| ≥ Φ0, we ﬁnd that s−1 is unchanged, but that

1 2 2 ω2 sf (ω, N, Φ ) = − + 2F˜ + 2ω − , (3.98) 0 b ω 3 3 " 1 1 ρN X Φ˜ 2 N 2 F˜(2) sf (ω, N, Φ ) = − + − N Φ˜ F˜ + b − + 1 b ω 45 6 b b 3 3 3 b ! # 1 F˜ N Φ˜ ρN ω4 + − + − ω2 − , (3.99) 9 3 3 6 45

" ˜(2) ˜(3) ˜ 2 ˜ ˜ 4 1 1 F F ρN N X Φ Fb Φ sf (ω, N, Φ ) = − + + + − Φ˜ F˜2 + b + b 2 b ω 315 30 15 30 3 b b 2 10 b

1 F˜(2) 1 F˜(2) N X −N 2 + + − − Φ˜ G(|Φ˜ |) 45 6 90 6 6 b b b ! ! # 1 F˜ 1 F˜ ρN N Φ˜ N 2 ω6 −N 2 − ω2 + − − + + ω4 − (3.100) 18 6 90 30 30 30 90 315 where X ˜ f ρb := sgn(Φb) = Φb/|Φb| , ρ := ρb and Φb := ω(Φb − ρbΦ0) . (3.101) b

(We also use tilded versions of the notational contractions, such as F˜b := |Φ˜ b|(1 − |Φ˜ b|), in the same way as is done in the previous section on scalars.) As a check of these expressions, we can evaluate f ˜ them when |Φb| = Φ0 (or simply Φ = 0 in the second case), and ﬁnd they each give the same result.

sph In the limit ω → 1, Φb → 0 the above results agree that s−1 = −4,

2 1 N 2 1 N 2 ssph = , ssph, 0 = , ssph, 2 = − , ssph, 0 = and ssph, 2 = − . (3.102) 0 3 1 15 1 3 2 63 2 15

These also agree with the results found using Gilkey-de Witt methods in Appendix 4.A. The special case where q also vanishes then gives exactly one-half the result found in [23] for a fermion on a sphere, as is appropriate due to our use here of 6D Weyl (rather than Dirac) fermions.

For a fermion with charge qg˜, the corresponding contributions to the running of the bulk couplings are

∂U 2m6 ∂ 1 m4 µ = , µ = − , ∂µ 3(4π)3 ∂µ κ2 3(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , (3.103) ∂µ κ 60(4π)3 ∂µ 504(4π)3 ∂ 1 8 q2m2 ∂ κζ 8 q2 µ = , µ AR = . ∂µ g˜2 3(4π)3 ∂µ g˜2 15(4π)3

Subtracting the contribution of these bulk counterterms leaves the contributions to the Gilkey-de

103 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

f Witt coeﬃcients that renormalize the brane action. When |Φb| ≤ Φ0, we ﬁnd that δs−1 = 0 and

1 ω2 − 1 1 δω δω2 δω δs0 = − 2ω2Φ2 = + − 2ω2Φ2 ' − 2Φ2 , (3.104) 0 ω 6 b ω 3 6 b 3 b 1 ω2 − 1 7(ω4 − 1) ω2Φ2 ω4Φ2(1 − 2Φ2) δs0 = + − b − b b (3.105) 1 ω 72 720 6 6 1 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2(1 − 2Φ2) δω Φ2 = + + + − b − b b ' − b , ω 15 180 180 720 6 6 15 3 N Φ ω2N Φ 2N Φ δs1 = − b − b (1 − 4Φ2) ' − b , δs2 = 0 , (3.106) 1 2 6 b 3 1 " 1 7(ω2 − 1) 7(ω4 − 1) 31(ω6 − 1) 7 ω2Φ2 ω4 δs0 = + + − b − Φ2(1 − 2Φ2) 2 ω 2880 2880 20160 240 24 b b # 7 Φ2 Φ4 −ω6Φ2 − b + b b 240 12 15 " 1 δω 101 δω2 17 δω3 257 δω4 31δω5 31 δω6 = + + + + + ω 42 2520 420 10080 3360 20160 # 7 ω2Φ2 ω4 7 Φ2 Φ4 δω Φ2 − b − Φ2(1 − 2Φ2) − ω6Φ2 − b + b ' − b , (3.107) 240 24 b b b 240 12 15 42 10 N Φ ω2N Φ 7 Φ2 Φ4 2 N Φ δs1 = − b − b 1 − 4Φ2 − ω4N Φ − b + b ' − b , (3.108) 2 16 24 b b 240 6 5 15 N 2 ω2 − 1 7(ω4 − 1) ω2Φ2 ω4Φ2 δs2 = − + − b − b 1 − 2Φ2 2 ω 144 1440 12 12 b N 2 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2 = − + + + − b − b 1 − 2Φ2 ω 30 360 360 1440 12 12 b δω Φ2 ' −N 2 − b , (3.109) 30 6 where, as before, δω = ω − 1 and the approximate equalities give the leading result for δω, Φb 1. The brane counterterms therefore renormalize as follows:

∂T m4 δω δω2 m4 δω µ b = + − 2ω2Φ2 ' − Φ2 , (3.110) ∂µ 2(4π)2ω 3 6 b (4π)2 6 b ∂ A qΦ m2 ω2 8 q2m2 µ b = − b 1 + (1 − 4Φ2) ' − A , (3.111) ∂µ g˜2 (4π)2 3 b 3(4π)3 b ∂ ζ m2 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2(1 − 2Φ2) µ Rb = + + + − b − b b ∂µ κ 2(4π)2ω 15 180 180 720 6 6 m2 δω Φ2 ' − b , (3.112) 2(4π)2 15 3 ∂ ζ qΦ 1 ω2 7 Φ2 Φ4 µ AR˜ b = − b + 1 − 4Φ2 + ω4 − b + b ∂µ g˜2 (4π)2 16 24 b 240 6 5 4 q2 ' − A , (3.113) 15(4π)3 b

104 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

" 2 3 4 5 6 ∂ζ 2 1 δω 101 δω 17 δω 257 δω 31δω 31 δω µ R b = + + + + + ∂µ 4(4π)2ω 42 2520 420 10080 3360 20160 # 7 ω2Φ2 ω4 7 Φ2 Φ4 − b − Φ2(1 − 2Φ2) − ω6Φ2 − b + b 240 24 b b b 240 12 15 1 δω Φ2 ' − b , (3.114) 4(4π)2 42 10 ∂ ζ 8 q2 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2 µ Ab = − + + + − b − b 1 − 2Φ2 ∂µ g˜2 (4π)2ω 30 360 360 1440 12 12 b 4 q2 δω ' − − Φ2 . (3.115) 3(4π)2 5 b

f 2 When instead |Φb| ≥ Φ0, we ﬁnd that (as always) δs−1 = δs1 = 0 and

1 ω2 − 1 1 δω δω2 δω δs0 = (ω − 1) − + 2F˜ = − + 2F˜ ' + 2|Φ˜ | , (3.116) 0 ω 3 b ω 3 3 b 3 b ! 1 ω2 − 1 (ω4 − 1) F˜2 ω2F˜ δs0 = − + b − b 1 ω 18 90 3 3 ! 1 δω δω2 2 δω3 δω4 F˜2 ω2F˜ δω |Φ˜ | = − − − + b − b ' − b , (3.117) ω 15 90 45 90 3 3 15 3 " # N ρ (1 − ω2) Φ˜ 2 ω2Φ˜ ρ N δω N Φ˜ δs1 = b − Φ˜ F˜ + b + b ' − b + b , (3.118) 1 ω 6 b b 3 3 3 3 " # 1 (ω2 − 1) (ω4 − 1) (ω6 − 1) F˜2 F˜3 ω2F˜2 ω4F˜ δs0 = + − + b + b − b − b 2 ω 180 180 630 30 15 6 30

1 δω 19 δω2 δω3 23 δω4 δω5 δω6 = + − − − − ω 42 1260 105 1260 105 630 ! F˜2 F˜3 ω2F˜2 ω4F˜ δω |Φ˜ | + b + b − b − b ' − b , (3.119) 30 15 6 30 42 30 " # 1 ρ N (1 − ω4) N Φ˜ Φ˜ 2F˜ Φ˜ 4 ω2N Φ˜ ω4N Φ˜ δs1 = b − b F˜2 + b b + b − b G(|Φ˜ |) + b 2 ω 30 3 b 2 10 6 b 30 2ρ N δω 2N Φ˜ ' − b − b , (3.120) 15 15 " # N 2 ω2 − 1 (ω4 − 1) F 2 ω2F˜ δs2 = − + − b + b 2 ω 36 180 6 6 ! N 2 δω δω2 δω3 δω4 F 2 ω2F˜ = − + + + − b + b ω 30 180 45 180 6 6 δω |Φ˜ | 'N 2 − + b . (3.121) 30 6

105 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

f Therefore, when |Φb| ≥ Φ0, the brane counterterms renormalize as follows:

∂T m4 δω δω2 m4 δω µ b = − + 2F˜ ' + |Φ˜ | , (3.122) ∂µ 2(4π)2ω 3 3 b (4π)2 6 b " # ∂ A q m2 ρ (1 − ω2) Φ˜ 2 ω2Φ˜ µ b = b − Φ˜ F˜ + b + b ∂µ g˜2 (4π)2ω 6 b b 3 3 ! q m2 ρ δω Φ˜ ' − b + b , (3.123) (4π)2 3 3 ! ∂ ζ m2 δω δω2 2 δω3 δω4 F˜2 ω2F˜ µ Rb = − − − + b − b ∂µ κ 2(4π)2ω 15 90 45 90 3 3 ! m2 δω |Φ˜ | ' − b , (3.124) 2(4π)2 15 3 " ∂ ζ q ρ N (1 − ω4) N Φ˜ Φ˜ 2F˜ Φ˜ 4 ω2N Φ˜ µ AR˜ b = b − b F˜2 + b b + b − b G(|Φ˜ |) ∂µ g˜2 (4π)2ω 30 3 b 2 10 6 b # ω4N Φ˜ 2 q + b ' − ρ δω + Φ˜ , (3.125) 30 15(4π)2 b b

2 3 4 5 6 ∂ζ 2 1 δω 19 δω δω 23 δω δω δω µ R b = + − − − − ∂µ 4(4π)2ω 42 1260 105 1260 105 630 ! ! F˜2 F˜3 ω2F˜2 ω4F˜ 1 δω |Φ˜ | + b + b − b − b ' − b , (3.126) 30 15 6 30 4(4π)2 42 30 ! ∂ ζ 8 q2 δω δω2 δω3 δω4 F 2 ω2F˜ µ Ab = − + + + − b + b ∂µ g˜2 (4π)2ω 30 180 45 180 6 6 4 q2 δω ' − − |Φ˜ | . (3.127) 3(4π)2 5 b

3.4.3 Gauge ﬁelds

We next state the results for the Casimir coefficient for a gauge field, provided this gauge field is not the field whose flux stabilizes the background 2D geometry. We consider in turn the cases where the 6D gauge field is massless or massive (in the 6D sense).

Massless 6D gauge ﬁelds

We begin with the massless case. Picking an appropriate gauge (such as light-cone gauge) allows the 6D gauge ﬁeld to be decomposed into four components,13 each with a spectrum (when evaluated using the north patch of the gauge potential) given by

2 2 gfξ ω ξ ω ξ 1 (1 + N ) λ = j + n − Φ+ + + n − N + Φ− − + − (3.128) jn 2 ω 2 ω 2 4

13Ghosts also do not contribute to the one-loop result in this gauge.

106 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy where ξ ∈ {0, 0, +1, −1} for each of the four components. (For more details see Appendix 3.C and [39].) From this, we see that two modes have the exact same spectrum as scalars (i.e. the ξ = 0 modes), and two modes have almost the same spectrum as scalars (i.e. the ξ = ±1 modes). Similar to before, we can identify

δ N := N, Φgfξ := Φ − ξΦgf , Φgf = ω−1 = α = 1 − (3.129) gfξ b b 0 0 2π

(and so Ngfξ := ω(Ngfξ − Φgfξ) = N + 2ξ) to relate these modes to the scalar spectrum:

(1 + N 2) (1 + N 2 ) λgfξ(ω, N, Φ ) + = λs (ω, N , Φgfξ) + gfξ . (3.130) jn b 4 jn gfξ b 4

However, since the resulting spectrum for the ξ = ±1 modes ends up being very similar to that of the scalars, we shall just write gf s gf si = 4si + ∆si (3.131)

gf where ∆s−1 = 0, " # 1 X ∆sgf (ω, N, Φ ) = −2ω + ω2 |Φ | , (3.132) 0 b ω b b " # 1 ω2 X ω3N X ω4 X ∆sgf (ω, N, Φ ) = N 2 + ωN Φ + |Φ | − Φ |Φ | − |Φ |3 , (3.133) 1 b ω 3 b 2 b b 3 b b b b " 1 ωN 2 1 N 2 X ω3N X ∆sgf (ω, N, Φ ) = − + ω2 − |Φ | − Φ |Φ | 2 b ω 4 15 24 b 4 b b b b # ω4(1 − N 2) X ω5N X ω6 X − |Φ |3 + Φ |Φ |3 + |Φ |5 , (3.134) 6 b 4 b b 10 b b b b

gf gf so long as |Φb| ≤ Φ0 . As it turns out, the |Φb|–dependent terms seen here in ∆si serve to exactly s cancel any corresponding terms in 4si in eq. (3.131) that are odd in |Φb| (for the ξ = ±1 modes only).

Even though only s2 contributes to the Casimir energy for massless ﬁelds, we nonetheless also follow s−1, s0 and s1 since these are useful as intermediate steps when assembling the contributions of a massive vector ﬁeld. The corresponding bulk quantities therefore are

2 4 5 N 2 ssph = 4 , ssph = − , ssph, 0 = , ssph, 2 = , −1 0 3 1 15 1 6 16 7 N 2 ssph, 0 = , ssph, 2 = − (3.135) 2 315 2 20 and after these are subtracted the brane renormalizations are obtained from

1 (ω2 − 1) 1 δω δω2 δs = −(ω − 1) + − 2ω2F + ω2|Φ | = − + − 2ω2F + ω2|Φ | 0 ω 3 b b ω 3 3 b b

107 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

δω ' − − |Φ | , (3.136) 3 b 1 (ω2 − 1) ω4 − 1 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 δs0 = + − b + b − b − b 1 ω 9 90 3 3 3 3 1 4 δω 8 δω2 2 δω3 δω4 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 = + + + − b + b − b − b ω 15 45 45 90 3 3 3 3 4 δω |Φ | ' − b , (3.137) 15 3 ω2N ω2N 2 N Φ δs1 = − Φ G(|Φ |) + N Φ − Φ |Φ | ' b , (3.138) 1 3 b b b 2 b b 3 1 (ω2 − 1) ω4 − 1 ω6 − 1 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 δs0 = + + − b + b − b − b 2 ω 45 180 630 15 15 6 6 ω6F 2 ω6F 3 ω6|Φ |5 − b − b + b 30 15 10 1 8 δω 5 δω2 17 δω3 37 δω4 δω5 δω6 2 ω2F ω2|Φ | ω4F 2 = + + + + + − b + b − b ω 105 63 315 1260 105 630 15 15 6 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 8 δω |Φ | − b − b − b + b ' − b , (3.139) 6 30 15 10 105 15 ω2N ω4N ω2N Φ ω4N Φ δs1 = − Φ G(|Φ |) − Φ G(|Φ |)(1 + 3F ) − b |Φ | + b |Φ |3 2 6 b b 30 b b b 4 b 4 b N Φ ' − b , (3.140) 5 N 2 ω − 1 ω2 − 1 ω4 − 1 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 δs2 = − + + − b + b − b − b 2 ω 8 72 180 12 24 6 6 N 2 7 δω 17 δω2 δω3 δω4 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 = − + + + − b + b − b − b ω 40 360 45 180 12 24 6 6 7 δω |Φ | ' −N 2 − b , (3.141) 40 24

2 along with δs−1 = δs1 = 0 (as usual).

4−2k Because the renormalizations coming from sk are proportional to m , where m is the 6D mass, for massless ﬁelds we need only follow the contributions of s2, ensuring the only nonzero renormalizations are

2 ∂ζ 3 2 ∂ κζ 14 q µ R = − and µ AR = , (3.142) ∂µ 315(4π)3 ∂µ g˜2 5(4π)3 in the bulk, and

2 3 4 5 6 ∂ζ 2 1 8 δω 5 δω 17 δω 37 δω δω δω µ R b = + + + + + ∂µ 4(4π)2ω 105 63 315 1260 105 630 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 − b + b − b − b − b − b + b 15 15 6 6 30 15 10 1 2 δω |Φ | ' − b , (3.143) (4π)2 105 60

108 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

∂ κζ q ω2 ω4 ω2Φ ω4Φ µ AR˜ b = − Φ G(|Φ |) + Φ G(|Φ |)(1 + 3F ) + b |Φ | − b |Φ |3 ∂µ g˜2 (4π)2 6 b b 30 b b b 4 b 4 b q Φ 2 q2 ' − b = − A , (3.144) 5(4π)2 5(4π)3 b ∂ ζ 8 q2 7 δω 17 δω2 δω3 δω4 ω2F ω2|Φ | µ Ab = − + + + − b + b ∂µ g˜2 (4π)2ω 40 360 45 180 12 24 ω4F 2 ω4|Φ |3 q2 7 δω |Φ | − b − b ' − − b , (3.145) 6 6 (4π)2 5 3 on the brane.

Massive 6D gauge ﬁelds

Let us now turn to massive gauge-field fluctuations, corresponding to those gauge directions that acquire mass because of the nonzero (but constant) values taken by some of the scalar fields. By assumption, these gauge fields are vanishing in the background, both because this would require a more complicated ansatz than assumed here for the rugby-ball backgrounds [40], and because it would complicate the diagonalization of the metric and gauge-field fluctuations. In this case the linearized theory simplifies [37] if we choose light-cone gauge, as described in more detail in Appendix 3.C. The result is that a massive gauge field leads to the 4D spectrum of a massless gauge field, given in eq. (3.128), plus that of a scalar provided earlier. It follows that the si coefficient of a massive gauge field are

mgf gf s s gf si = si + si = 5si + ∆si , (3.146)

s where the si are the corresponding quantities for a 6D scalar, those given in eqs. (4.247)–(4.248), and we use N ∈ {0, ±1} to ensure stability (see Appendix 3.C). Let us now give the contribution of the massive gauge ﬁeld to the running of the bulk couplings. Since in the sphere limit we have

1 1 19 N 2 ssph, 0 = 5 , ssph, 0 = − , ssph, 0 = , ssph, 2 = , −1 0 3 1 3 1 24 4 3 N 2 ssph, 0 = , and ssph, 2 = − (3.147) 2 63 2 8 we obtain

∂U 5 m6 ∂ 1 m4 µ = − , µ = , ∂µ 6(4π)3 ∂µ κ2 6(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , ∂µ κ 12(4π)3 ∂µ 126(4π)3 ∂ 1 19 q2m2 ∂ κζ 3 q2 µ = − , µ AR = . ∂µ g˜2 3(4π)3 ∂µ g˜2 (4π)3

109 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

The running of the brane couplings is similarly obtained by computing the δsi coeﬃcients:

1 δω 5 δω2 5 ω2F δω 3|Φ | δs = − + − b + ω2|Φ | ' − − b , (3.148) 0 ω 6 12 2 b 6 2 1 δω 2 δω2 δω3 δω4 5 ω2F ω2|Φ | 5 ω4F 2 ω4|Φ |3 δs0 = + + + − b + b − b − b 1 ω 3 9 18 72 6 3 12 3 δω |Φ | ' − b , (3.149) 3 2 5 ω2N ω2N 7 N Φ δs1 = − Φ G(|Φ |) + N Φ − Φ |Φ | ' b , (3.150) 1 12 b b b 2 b b 12 1 2 δω 25 δω2 17 δω3 37 δω4 δω5 δω6 ω2F ω2|Φ | δs0 = + + + + + − b + b 2 ω 21 252 252 1008 84 504 6 15 5 ω4F 2 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 2 δω |Φ | − b − b − b − b + b ' − b , (3.151) 24 6 24 12 10 21 10 5 ω2N ω4N ω2N Φ ω4N Φ δs1 = − Φ G(|Φ |) − Φ G(|Φ |)(1 + 3F ) − b |Φ | + b |Φ |3 2 24 b b 24 b b b 4 b 4 b N Φ ' − b , (3.152) 4 N 2 3 δω 17 δω2 δω3 δω4 5 ω2F ω2|Φ | 5 ω4F 2 ω4|Φ |3 δs2 = − + + + − b + b − b − b 2 ω 16 288 36 144 48 24 24 6 3 δω |Φ | ' −N 2 − b . (3.153) 16 16

These give

∂T m4 δω 5 δω2 5 ω2F µ b = − + − b + ω2|Φ | ∂µ 2(4π)2ω 6 12 2 b m4 δω 3|Φ | ' − + b , (3.154) (4π)2 12 4 ∂ A 2 qm2 5 ω2 ω2 µ b = − Φ G(|Φ |) + Φ − Φ |Φ | ∂µ g˜2 (4π)2 12 b b b 2 b b 7 qm2Φ 7 q2m2 ' b = − A , (3.155) 6(4π)2 3(4π)3 b ∂ ζ m2 δω 2 δω2 δω3 δω4 5 ω2F ω2|Φ | µ Rb = + + + − b + b ∂µ κ 2(4π)2ω 3 9 18 72 6 3 5 ω4F 2 ω4|Φ |3 m2 δω |Φ | − b − b ' − b , (3.156) 12 3 (4π)2 6 4 ∂ κζ q 5 ω2 ω4 µ AR˜ b = − Φ G(|Φ |) + Φ G(|Φ |)(1 + 3F ) ∂µ g˜2 (4π)2 24 b b 24 b b b ω2Φ ω4Φ q Φ q2 + b |Φ | − b |Φ |3 ' − b = − A , (3.157) 4 b 4 b 4(4π)2 2(4π)3 b 2 3 4 5 6 2 2 ∂ζ 2 1 2 δω 25 δω 17 δω 37 δω δω δω ω F ω |Φ | µ R b = + + + + + − b + b ∂µ 4(4π)2ω 21 252 252 1008 84 504 6 15 5 ω4F 2 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 − b − b − b − b + b 24 6 24 12 10

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1 2 δω |Φ | ' − b , (3.158) (4π)2 84 40 ∂ ζ 8 q2 3 δω 17 δω2 δω3 δω4 5 ω2F ω2|Φ | µ Ab = − + + + − b + b ∂µ g˜2 (4π)2ω 16 288 36 144 48 24 5 ω4F 2 ω4|Φ |3 q2 3 δω |Φ | − b − b ' − − b . (3.159) 24 6 (4π)2 2 2

3.5 The 4D vacuum energy

The previous sections show how to compute the 1PI potential, V1−loop = V∞ + Vf , obtained by integrating out low-spin bulk fields, and how these divergences are renormalized into various bulk P and brane interactions, VB and Vb, so that V := VB + b Vb +Vf is finite. In this section we compute the implication of the above renormalizations for the effective 4D cosmological constant, Λ, and on-brane curvature as seen by a low-energy 4D observer. In the 4D theory Λ would be obtained as the value of the low-energy 4D effective potential, V , after minimizing over any light scalar fields in the 4D effective theory. If no branes had been present, a standard result for the low-energy potential would have been

V = VB +Vf , so it may come as a surprise that once branes are included the potential V is not simply given by V := V, suitably renormalized. Instead we must also recompute the classical contribution to the low-energy cosmological constant coming from integrating out KK bulk modes, keeping track of how the bulk back-reacts to the renormalization of the source branes, along the lines of refs. [10]. Neglecting this back-reaction would be inconsistent, since for codimension-two systems it is known to be of the same size as the direct effects of the changes to the brane lagrangians themselves [5, 11, 12]. Indeed, it is this back-reaction that allows flat solutions to exist at all at the classical level, despite the large classical positive tensions carried by each brane. The logic to determining the cosmological constant of the low-energy effective theory is to compute within the 6D theory how perturbations to brane and bulk interactions change the predicted value for the curvature, Rµν , along the brane directions, and then to ask what cosmological constant in the effective 4D theory would give this same curvature. This is a special case of a ‘matching’ calculation between the effective theory and its UV completion [33, 34]. It can be carried out fairly explicitly for small changes about a known background solution, as we do below following refs. [10]. An alternative route to the same end is compute the value taken by the loop-corrected (1PI) action, Γ := S + Σ, evaluated at the background configuration that solves the loop-corrected field equations, δΓ/δψ = 0. This lends itself well to the present purposes because the quantity V1−loop as computed in earlier sections is precisely such a contribution to the loop-corrected action evaluated at the background solution. For maximally symmetric geometries, the value of Λ in the 4D low-energy effective theory can be found by comparing the result for Γ obtained using the full 6D theory with the result for Γ computed using the effective 4D theory.

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3.5.1 Classical bulk back-reaction

It is useful first to review how back-reaction works at the classical level. For the 6D theory we solve the bulk field equations, eqs. (4.10), using the renormalized bulk couplings. The renormalized brane couplings enter through the near-brane boundary conditions they imply for the bulk fields [15]. Our interest is in starting with a classical solution for which Rµν = 0, and then ask how Rµν changes as the various renormalized couplings change by small amounts. Of particular interest is how Rµν responds to changes of the couplings in the brane action. Recall that the rugby ball is sourced by identical brane actions, which to first order in a derivative R 4 √ expansion couple to the background fields by Sb = − d x −γ Lb with

A L = T − b mnF + ··· , (3.160) b b 2˜g2 mn

M N and where (as before) γµν := gMN ∂µx ∂ν x is the induced metric on the brane, and the ellipses denote terms involving two or more derivatives. As shown in detail in [10], the nominally subdominant Ab-term can play an important role in understanding the low-energy effective 4D curvature in those situations [4, 5] where the stabilization of the extra dimensions arises as a competition between brane and bulk flux. In this case the influence of the A term gets enhanced by the volume of the extra dimensions through its effect on the flux-quantization condition. At the classical level, the source branes back-react onto the background solutions in two distinct ways. First, they change the boundary conditions of the bulk fields, schematically relating the near- brane limit, limρ→0(ρ ∂ψ/∂ρ), to the (appropriately renormalized) derivative of the brane action,

δSb/δψ, for any bulk field ψ. In the special case that the functions Tb and Ab defining the brane action are independent of any bulk scalar fields, then these boundary conditions boil down to the familiar statement that the brane induces14 a conical defect angle at each brane position, of size (see 2 2 2 eq. (3.14)): δ = κ Lb = κ Tb − Abf/g˜ . The second way back-reaction influences the background is through the flux quantization condition, eq. (4.25), which depends on Tb and Ab because: (i) the defect angle changes the volume of integration for the flux, and (ii) because the Ab term directly contributes as flux localized on the branes. Once both effects are included [10], the flux-quantization 2 P condition generalizes from eq. (4.15) to 2πN/q = 4πα r f + b Ab , as before.

What must be done is to track how the bulk solutions react to loop-induced changes, δTb, δAb (and others), to see how the changes in the bulk solutions appear in the low-energy eﬀective theory.

A straw man

Before doing so, it is worth first putting to rest a common misconception that can confuse issues at this point. Schematically, our goal is to compute the loop-corrected action, Γ = S + Σ, evaluated at the loop-corrected background field configuration, ψ = ψ0 + δψ, with being the small loop-counting

14 More generally, if Tb or Ab depend on bulk scalars, back-reaction leads to a bulk curvature singularity at the brane positions [15, 10].

112 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy parameter. We do so in order to compare the result when performed in 6D and in 4D. Working to order gives

2 Γ[ψ] ' S[ψ0 + δψ] + Σ[ψ0] + O( ) ! δS 2 ' S[ψ0] + δψ + Σ[ψ0] + O( ) , (3.161) δψ ψ0 and it is tempting to argue that the first of the O() terms vanishes because ψ0 satisfies the classical 2 field equations: (δS/δψ)ψ0 = 0. If so, then Γ[ψ] = S[ψ0] + Σ[ψ0] + O( ).

Although this argument is often true, if it were true in the present instance then there would be no need to know how bulk fields back-react — i.e. to compute δψ — in response to loop corrections to the action — Σ — since it would suffice to evaluate both S and Σ at the uncorrected classical solution, ψ0. If so, then the renormalized potential, V, computed in previous sections could be directly interpreted as the effective vacuum energy.

To see why this argument fails it is useful to examine a concrete example. To see why Seﬀ [ψ0 +

δψ] 6= Seﬀ [ψ0] at O(), consider the case

Z ! 4 √ R Seﬀ = − d x −g Λ0 + 2 2κ4,0 Z 4 √ R and Γ = − d x −g Λ + 2 , (3.162) 2κ4

2 2 2 where Λ = Λ0 + δΛ and 1/κ4 = 1/κ4,0 + δ(1/κ4), with both corrections of order . For simplicity, suppose further that Λ0 = 0.

With these choices the unperturbed classical background satisfies R0 = 0 while the background 2 solving the full loop-corrected equations is R1 = −4κ4,0δΛ. Consequently the classical action evaluated at the classical background is Seff [g0] = 0, which disagrees with its evaluation at the loop- corrected solution: Z Z 4 √ R1 4 √ Seff [g0 + δg] = − d x −g 2 = +2 d x −g0 δΛ . (3.163) 2κ4,0

Formally, the reason Seff [g0 +δg] can differ from Seff [g0] is because Seff is actually proportional to the volume of spacetime for any constant nonzero curvature, and so diverges. Consequently precise statements must be regularized, such as by cutting the geometry off at a large radius. But then the action also contains a boundary, Gibbons-Hawking, term at this radius, and it is this boundary term that need not be stationary when evaluated at a solution to Einstein’s equations. A similar phenomenon was noticed for the on-shell gravitational action in another context in ref. [41].

113 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Loop-corrected back-reaction

We now turn to how the bulk solutions respond to the presence of the loop-generated changes to brane and bulk interactions. Since these changes are small within the semiclassical approximation we can analyze this by following ref. [10] and solving the bulk field equations linearized about the classical background, using the brane-bulk boundary conditions to relate integration constants to properties of the brane action. There are two qualitatively different kinds of loop-generated effects. First, renormalizations of the bulk action captured by VB can change the features of the bulk classical solution, such as (but not restricted to) the generation of new terms in the bulk cosmological constant. Second, the renormalizations of the brane action included in Vb change the bulk solutions in the two ways mentioned above: changing the boundary conditions and modifying the flux-quantization condition. The calculation in [10] can be used to include the influence of the brane renormalizations, once generalized to the loop corrections to the brane action found earlier in this paper. We now briefly describe the result of such a calculation. We first examine corrections to the lowest-derivative terms in the action, and return to the effects of the higher-derivative terms in the next section. It suffices to work in a simple case to make the point that back-reaction is required to properly infer the implications of V1−loop for the 4D curvature. We therefore assume that all gauge fields that are zero in the classical background remain zero at loop level. The calculation is performed, without loss of generality, using Gaussian normal coordinates in the scalar-field target space centered at the background solution, so that the kinetic terms are canonical at this point. Finally, to avoid the issues of ref. [42] we imagine the branes only to couple to bulk scalars with vanishing bulk masses, in order not to have the brane and bulk compete in their implications for φi, and thereby preclude the existence of constant background configurations.15 We then imagine the bulk and brane actions to be chosen to admit a rugby-ball solution, and consider the corrections to this solution due to generic corrections to the coefficients

κ2 → κ2 + δκ2 ,U(φ) → U(φ) + δU(φ) , g˜2(φ) → g˜2(φ) + δg˜2(φ) , (3.164) in the bulk and

Tb → Tb + δTb(φ) and Ab → Ab + δAb(φ) , (3.165) on the branes. We ignore corrections to the target space metric, δGij(φ), since at linear order these do not contribute to the effective cosmological constant. We discuss higher-derivative corrections to the brane action below, but the bottom line is that they are suppressed by powers of the gravitational scale κ. Following the steps laid out in ref. [10], we first compute the on-brane curvature generated by these perturbations, by solving the linearized 6D field equations (assuming the initial rugby ball to be flat in the brane directions). The the effective 4D cosmological constant, Λ, is read off as that 2 constant that would produce the same curvature using the 4D equations of motion: i.e. R4 = −4κ4Λ.

15Both examples of [10] fall into this class

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This gives16

παr2 παr2 2 δf δU 2 δg˜2 Λ := − gµν R = − − − κ2 µν κ2 r2 f U g˜2 φ? fδA πα 4 δκ2 3δU δg˜2 = δL − tot + + + , (3.166) tot g˜2 κ2 κ2 U g˜2 φ? P P where f is defined (as before) by Fmn = f mn, while Atot := b Ab and Ltot := b Lb with Lb is as given in (3.160). Generically the right-hand-side of eq. (3.166) is evaluated at the loop-corrected i i i stationary point for the bulk scalar field: φ? = φ0 + δφ . When this is fixed from the classical bulk i i field equations then it suffices to evaluate eq. (3.166) at the classical solution, φ0, since keeping δφ would be subdominant in the loop expansion. A more complicated situation arises if the bulk scalars parameterize flat directions along which there is no scalar potential at all in the absence of the perturbed bulk-brane couplings. This can occur if the brane couplings break an approximate symmetry of the bulk equations of motion, and can be used to generate a Goldberger-Wise [45] type stabilization of the bulk geometry [15]. In this case it is the perturbed brane-bulk couplings that stabilize these flat directions, with the stabilized i point, φ?, satisfying κ2δL U δf δU (δg˜2) tot,j + ,j + ,j − ,j = 0 . (3.167) 4πα Uf δU g˜2 φ? The first term here corresponds to the case studied in ref. [10]. Because we work perturbatively in i i δφ , some care is required if φ0 labels a flat direction of the zeroth-order equations. In this case (as is standard for degenerate perturbation theory) δφi need only be small if the arbitrary unperturbed point is chosen near the minimum of the loop-corrected potential. Although reasonably complicated in the general case, it is clear that the above result only agrees with V as computed earlier in the special case that the first term, δLtot, dominates eq. (3.166). This is not true, in particular, in the supersymmetric examples that are the subject of a companion paper [18] (for which the above formulae simplify considerably).

3.5.2 Higher derivative corrections on the brane

The above discussion is restricted to a tension and a localized flux, but in general loop corrections to the bulk and brane actions also include higher derivative terms, with the dimensions made up by factors of κ. Given our experience with the flux, which contributes at the same level as the tension despite being down one derivative, we pause to check that the other terms in the derivative expansion on the brane are indeed suppressed with respect to the tension and flux terms. The two ways in which higher derivative terms can contribute is by modifying the brane flux, and by modifying the matching condition that sets the defect angle at the branes. The matching condition that relates the defect angle to the on-brane Lagrangian is more compli-

16The ﬁrst two terms in the last equality are ref. [10]’s result, since the corrections to δκ, δU and δg˜ were not considered.

115 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy cated if higher derivative corrections are taken into account. The precise statement of the matching condition in the general case is [15]

2 B µν µν κ δS e ∂ρB g = g + √ , (3.168) π −γ δgµν where the metric near the brane is assumed to take the form ds2 = dρ2 + e2B(ρ)dθ2. For most ﬁelds, maximal symmetry in the brane directions guarantees that the right hand side of this matching √ condition only gets contributions from the variation of −γ in the action. An important exception to this are curvature terms on the brane, which we now explore. A brane with the structure

Z √ ζ S = − d4x −γ L + Rb R (3.169) b b κ has a metric variation

1 δS ζ R L ζ R L √ b = Rb Rµν − gµν − b gµν = − Rb + b gµν , (3.170) −γ δgµν κ 2 2 4κ 2 where the last equality uses the maximal symmetry of the on-brane directions. Eﬀectively this means that in eq. (3.166) we should replace δLtot → δLtot + (ζRbR4)/(2κ), and re-solve for R4. The result is

κζ κ2 fδA 4 δκ2 3δU δg˜2 1 + Rb R = tot − δL − − − . (3.171) 2παr2 4 πα r2 g˜2 tot r2κ2 2U 2˜g2 φ?

From this we see that the correction that is associated with a brane curvature term is of higher order in the expansion in κ/r2. The other contribution higher derivative terms can have is to modify the brane localized flux. In particular, the brane localized Maxwell term has this effect. Following the regularization of the brane flux in [10] we find a divergent contribution to the gauge potential at a small distance, δ, from the brane. Such divergences are normal for codimension-two matching [15, 45], and the limit δ → 0 is to be taken after renormalization of the brane-bulk couplings. The divergent near-brane form for the bulk gauge field is in this case

2 2 2 δ fg˜ + Ab/πδ Aθ(δ) = 2 2 1 + κζAb/πδ δ2fg˜2 A − fg˜2κζ κ A ζ ≈ + b Ab − b Ab . (3.172) 2 2π δ2 2π2

The ﬁrst term vanishes as δ → 0 and so can be ignored. The other correction terms are also small because consistency of the semiclassical approximation requires both κg˜2 1 and κδ−2 1. Again, we ﬁnd corrections to the cosmological constant due to the higher derivative corrections to the brane are subleading in κ.

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3.6 Conclusions

To repeat the summary of the introduction, in this paper we present an explicit calculation of the divergent part of the Casimir energy obtained from loops of low-spin bulk fields in a 6D geometry compactified on a flux-stabilized 2D rugby ball. We explicitly show how the UV divergences can be disentangled in order to identify separately how both bulk and brane interactions renormalize, and we compute the renormalization-group beta functions for each of the corresponding couplings. Although our results are general, the first applications we envisage are to the cosmological constant problem within the framework of supersymmetric large extra dimensions [18]. There are several features about this paper’s results that are noteworthy for such applications.

• The ﬁrst observation is that extra dimensions are not a free lunch in themselves. For large m, 6 6 2 in general all positive powers of m up to m appear in the bulk loops, V1−loop ∝ m r . This is as expected on dimensional grounds for a contribution from a 6D cosmological constant. Although the m6 term cancels whenever there are equal numbers of bosons and fermions, in 4 general the subdominant V1−loop ∝ m term is still present. In the absence of something special (like unbroken supersymmetry [16]), the UV sensitivity we ﬁnd is precisely what would have been expected generically on dimensional grounds.

• Second, although loops of heavy particles are dangerous in the bulk, in a world where all Standard Model particles live on a brane nothing really requires there to be any heavy bulk particles. If all bulk masses were generically of order 1/r then a single bulk loop need not be so dangerous, since V is of generic order V ∼ C/(4πr2)2. In such a case it would be two-loop and higher contributions that would instead be worrisome, since these could introduce the larger brane-localized masses of the Standard Model back into the result. What is required is a mechanism that suppresses higher loops, and here again supersymmetry is likely to be relevant, as described in Refs. [5, 12].

• Should a mechanism ensure that the 4D vacuum energy is indeed of order 1/(4πr2)2 (as we believe to be the case for a bulk supergravity17 [18]), then it is the Kaluza-Klein scale, 1/r, that would set the size of the cosmological constant. This could be acceptably small if r is chosen not to be far below its current upper limit, r ∼< 45 microns, arising from tests of Newton’s laws at short distances [43]. (Modiﬁcations of Newton’s law from supersymmetric large extra dimensions have been studied in [44].) Notice that the numerical factor of 16π2 in the vacuum energy is an important part of why these scales can be compatible.

• Finally, even should the numerical size of the Casimir energy be right, its sign is also important. It is in this context that the discussion of §3.5 is most important, since it shows that the sign 17Although it is tempting to ask what the 4D picture is for this mechanism, it is not clear that such a picture must exist given the central role played by back reaction (whose dynamics occurs above the KK scale [9], where the low energy 4D eﬀective theory breaks down). What the 4D eﬀective theory should be able to do, however, is to explain why the 4D curvature remains precisely zero when working purely at the classical level for the bulk, regardless of the energy density on the branes. A proper description of how this happens is under development, but goes well beyond the scope of the present article.

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of the low-energy cosmological constant need not simply be the sign of V as computed in §3.3.

In any event, there is no substitute for a real calculation for which we regard the results of this paper as a crucial ﬁrst step.

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Acknowledgements

We thank Riccardo Barbieri, Gregory Gabadadze, Hyun-Min Lee, Susha Parameswaran, Oriol Pu- jol`as,Fernando Quevedo, Seifallah Randjbar-Daemi, George Thompson and Itay Yavin for useful discussions. Various combinations of us are grateful for the support of, and the pleasant environs provided by, the Abdus Salam International Center for Theoretical Physics, and AS thanks the Perimeter Institute and McMaster University for its hospitality, while thinking about these problems. CB’s research was supported in part by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information (MRI). The work of AS was supported by the EU ITN “Uni- ﬁcation in the LHC Era”, contract PITN-GA-2009-237920 (UNILHC) and by MIUR under contract 2006022501.

3.A Heat kernels and bulk renormalization

In this appendix we collect for convenience the explicit expressions for the heat-kernel coeﬃcients for manifolds without singularities and boundaries, and specialize the results to the case where the two extra dimensions are a 2-sphere. The small-t expansion for the heat-kernel representation of the one-loop vacuum energy described in the main text can be evaluated for a broad class of geometries in great generality [26]. Such generality is possible because the small-t limit physically corresponds to the coincidence limit of the corresponding propagator, and this does not ‘know’ about the boundary conditions and topology of the space if the coincident points are taken far from any boundaries or branes.

Gilkey-de Witt coeﬃcients

This section collects the results for the ultraviolet-divergent parts of the one-loop action obtained by integrating out various kinds of particles in 6 dimensions. To this end, consider a collection of

N fields, assembled into a column vector, Ψ, and coupled to a background spacetime metric, gMN , i a scalars, ϕ , and gauge fields, AM . We suppress the gauge and/or Lorentz indices to which these fields couple, leading to a background-covariant derivative, DM , of the form

a DM Ψ = ∂M Ψ + ωM Ψ − iAM taΨ , (3.173)

where ωM is the matrix-valued spin connection, and the gauge group is represented by the hermitian matrices ta. For real ﬁelds the ta are imaginary antisymmetric matrices, which (for canonically- normalized gauge bosons) include a factor of the corresponding gauge coupling, ga. The commutator of two such derivatives deﬁnes the matrix-valued curvature, YMN Ψ = [DM ,DN ]Ψ, which has the following form: a YMN = RMN − iFMN ta . (3.174)

119 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Here RMN is the curvature built from the spin connection ωM , which is also related to the Riemann curvature of the background spacetime in a way which is made explicit in what follows. Integrating out the ﬁelds Ψ at one loop often leads to the following contribution to the quantum action 1 iΣ = −(−)F Tr log − + X + m2 , (3.175) 2

F MN where (−) = + for bosons and − for fermions. As before, = g DM DN and the quantity X is some local quantity built from the background ﬁelds. The mass matrix, m2, are constants. Our interest is in that part of Σ for which the functional trace is ultraviolet divergent. To identify the divergent part we work within dimensional regularization and so continue the spacetime dimension to complex values, n, which are slightly displaced from the actual integer spacetime dimension, 6, which is of interest: n = 6 − 2. We then follow the poles in Σ as → 0, in the usual fashion. Notice that this continuation to d = 6 diﬀers from the regularization used for the Casimir energy calculation, for which the limit d → 4 was taken. For 6D spaces without boundaries and singularities the resulting divergent terms are simply characterized. They can be written as [26]

3 1 X Z √ Σ = (−)F Γ(k − 3 + ) d6x −g tr [m6−2k a ] (3.176) ∞ 2(4π)3 k k=0 where Γ(z) denotes Euler’s gamma function. The divergence as → 0 is contained within the gamma function, which has poles at non-positive integers of the form Γ(−r + ) = (−)r/(r!) + ··· , for an infinitesimal and r a non-negative integer. The coefficients, ak, are known matrix-valued local quantities constructed from the background fields, to which we return below. The trace is over the 0 matrix indices of the ak, of which there are N = N d with N counting the number of fields and d being the dimension of the relevant Lorentz representation. The above expression shows that for massless fields (m = 0) in compact spaces without boundaries and singularities in 6 dimensions the divergent contribution is proportional to tr [a3], so the problem reduces to the construction of this coefficient. By contrast, for massive fields there are also divergences 6 4 2 2 proportional to tr [m a0], tr [m a1], tr [m a2] and tr [a3]. (Notice that the freedom to keep m within 2 or separate from X implies that the divergence obtained from computing just a3 using Xm = X +m gives the same result as computing a0 through a3 using only X.)

An algorithm for constructing the coefficients ak is known for general X and DM , involving and the result for the first few has been computed explicitly [26] and can be given as a closed form in terms of X, background curvatures and the generalized curvature YMN . The first few coefficients are given explicitly by [26]:18

a0 = I (3.177)

18In comparing with this reference recall that our metric is ‘mostly plus’ and we adopt Weinberg’s curvature conventions [29], which for the Riemann tensor agree with those of ref. [26], but disagree with this reference by a sign for the Ricci tensor and scalar.

120 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

1 a = − (R + 6X) (3.178) 1 6 1 a = 2R RABMN − 2R RMN + 5R2 − 12 R 2 360 ABMN MN 1 1 1 1 + RX + X2 − X + Y Y MN (3.179) 6 2 6 12 MN 1 a = −18 2R + 17D RDM R − 2D R DLRMN − 4D R DN RML 3 7! M L MN L MN K MNLP MN M L N +9DK RMNLP D R + 28RR − 8RMN R + 24R N D D RML 35 14 14 +12R RMNLP − R3 + RR RMN − RR RABMN MNLP 9 3 MN 3 ABMN 208 64 16 + RM R RNL − RMN RKL R + RM R RNKLP 9 N ML 3 MKNL 3 N MKLP 44 80 − RAB R RMNKL − RA M R RBKNP 9 MN ABKL 9 B N AKMP 1 + 8D Y DM Y NK + 2DM Y D Y NK + 12Y MN Y (3.180) 360 M NK NM K MN M N K MNKL M NK −12Y N Y K Y M − 6R YMN YKL + 4R N YMK Y

MN 2 M 3 −5RY YMN − 6 X + 60XX + 30DM XD X − 60X

MN MN M 2 −30XY YMN + 10R X + 4R DM DN X + 12D RDM X − 30X R 2 MN ABMN + 12X R − 5XR + 2XRMN R − 2XRABMN R .

Here I is the N × N identity matrix and YMN is the matrix-valued quantity deﬁned above in terms of the commutator to two covariant derivatives.

Bulk Counterterms for Spheres

Since our applications are to compactifications on spaces which are spheres, it suffices to specialize the general results of the appendix to these simpler background field configurations. Consider therefore 6D spacetime geometries which are the product of 4D Minkowski space with a maximally-symmetric 2D manifold:

2 µ ν m n ds = ηµν dx dx + gmn dy dy , (3.181)

1 1 where maximal symmetry for the 2D metric implies Rmnpq = 2 R (gmpgnq −gmqgnp), Rmn = 2 R gmn mnpq mn 2 and DmR = 0, and so RmnpqR = 2RmnR = R . We also take any background scalars to be constants, and allow only a single background gauge ﬁeld to be nonzero, and take it to be proportional to the 2D volume form: Fmn = f mn, for some scalar f.

With these choices the only nonzero components of YMN lie in the 2 dimensions, Ymn, and all of the curvatures are covariantly constant. The coeﬃcients a1 through a3 simplify considerably, reducing to [26]:

a0 = I (3.182)

121 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

1 a = − R − X (3.183) 1 6 1 1 1 1 a = R2 + RX + X2 + Y Y mn (3.184) 2 60 6 2 12 mn 1 1 1 1 a = − R3 − Y m Y n Y l − RY Y mn − XY Y mn 3 630 30 n l m 40 mn 12 mn 1 1 1 − X3 − X2 R − XR2 . (3.185) 6 12 60

0 0 0 0 Here I is the N × N identity matrix, and X and Ymn are the N × N matrix-valued quantities deﬁned above. These expressions may be used to compare with the bulk part of the ultraviolet divergences encountered in the explicit calculations of the main text.

Scalars, spinors and gauge ﬁelds

This section collects the results for X and YMN , and so also for the ultraviolet-divergent parts of the one-loop action, for several kinds of particles in 6 dimensions. Attention is restricted to those fields that do not mix appreciably with the gravity sector, and this means in particular that any gauge fields considered cannot be those whose background flux stabilizes the extra dimensions.

Scalars

For spinless ﬁelds we begin with the following general scalar-ﬁeld action involving two derivatives or less,

Z √ 1 1 1 S = − d6x −g gMN G D ΦiD Φj + V + UR + H F a F MN , (3.186) 2 ij M N 2 4 MN a

i for a collection of N real scalar ﬁelds, Φ , coupled to a background metric, gMN , and gauge ﬁelds, a AM . The functions U, V , W and the target-space metric, Gij, are imagined to be known functions of the Φi. The background-covariant derivative appropriate to this case is:

i i a i j DM Φ = ∂M Φ − iAM (ta) jΦ , (3.187)

i where the matrices (ta) j represent the gauge group on the scalars. To compute the one-loop quantum effects of scalar fluctuations we linearize this action about a i i i i particular background configuration, ϕ , according to: Φ = ϕ + φ , where ∂M ϕ = 0. Expanding i i the classical action to quadratic order in φ reveals the kinetic operator ∆ j, which is given by

i i i ∆ j = −δ j + X j , (3.188)

i with X j given by

h 1 1 i Xi = Gik V (ϕ) + RU (ϕ) + F a F MN H (ϕ) . (3.189) j kj 2 kj 4 MN a kj

122 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

In this last expression the subscripts on U, V and H denote differentiation with respect to the background field ϕi. Specializing to the simple 2D geometries and Maxwell fields discussed earlier, i ik 1 1 2 these simplify to X j = G [Vkj + 2 RUkj + 2 f Hkj] and Ymn = −igf˜ Q mn, whereg ˜ is the gauge coupling andgQ ˜ = ta is the hermitian, antisymmetric charge matrix for the background gauge field. mn 2 2 2 m n l Notice that these imply YmnY = −2˜g f Q and Y n Y l Y m = 0. 0 Since scalars transform trivially under Lorentz transformations, d = 0 and so N = N. The ak are then N × N matrices, with the trace of a0 through a3 given by

N tr a = N, tr a = − R − tr X, (3.190) 0 1 6 and

N 1 1 1 tr a = R2 + R tr X + tr X2 − g˜2f 2 tr Q2 (3.191) 2 60 6 2 6 N 1 1 tr a = − R3 + R g˜2f 2 tr Q2 + g˜2f 2 tr (XQ2) 3 630 20 6 1 1 1 − tr X3 − R tr X2 − R2 tr X. (3.192) 6 12 60

These give explicit functions of ϕ once the above expression for X is used.

Fermions

For N 6D massless spinors, ψa with a = 1, ..., N, we take the following action

Z √ 1 a S = − d6x −g G (ϕ) ψ Dψ/ b , (3.193) 2 ab

M A whereD / = eA γ DM with

1 D ψa = ∂ ψa − ωAB γ ψ − iAa t ψ , (3.194) M M 4 M AB M a

A M 1 with γ being the 6D Dirac matrices and eA the inverse sechsbein, γAB = 2 [γA, γB ], and ta denotes the gauge-group generator acting on the spinor ﬁelds. Since 6D Weyl spinors have 4 complex components their representation of the 6D Lorentz group has d = 8 real dimensions.

M A The diﬀerential operator which governs the one-loop contributions is in this caseD / = eA γ DM and so in order to use the general results of the previous section we write (assuming there are no 1 2 gauge or Lorentz anomalies) log detD / = 2 log det(−D/ ), which implies

1 1 iΣ = Tr logD / = Tr log −D/ 2 1/2 2 4 1 1 1 = Tr log − − R + γAB F a t . (3.195) 4 4 4 AB a

This allows us to adopt the previous results for the ultraviolet divergences, provided we divide the

123 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy result by an overall factor of 2 (and so eﬀectively d = 4 instead of 8), and use

1 1 X = − R + γAB F a t . (3.196) 4 4 AB a

Similarly, we ﬁnd i Y = − R γAB − iF a t , (3.197) MN 2 MNAB MN a and so19

N Tr [Y Y MN ] = −4 tr (t t ) F a F bMN − R RABMN 1/2 MN 1/2 a b MN 2 ABMN N = −8g ˜2f 2 tr (Q2) − R2 , (3.198) 1/2 2 where the second line specializes to the simple spherical 2D geometry and background gauge ﬁelds discussed earlier. Keeping explicit the sign due to statistics, and dropping terms which vanish when traced, this leads to the following expressions for the divergent contributions of N 6D Weyl fermions:

N (−)F Tr [a ] = −4N, (−)F Tr [a ] = − R (3.199) 1/2 0 1/2 1 3 N 4 (−)F Tr [a ] = R2 − g˜2f 2 tr (Q2) (3.200) 1/2 2 60 3 1/2 N 2 (−)F Tr [a ] = − R3 + g˜2f 2 R tr (Q2) . (3.201) 1/2 3 504 15 1/2

Gauge bosons

a a For N gauge bosons, AM , with field strength FMN and a = 1, ..., N, we use the usual Yang-Mills action Z √ 1 S = − d6x −g H(ϕ) F a F MN , (3.202) 4 MN a a a a expanded to quadratic order about the background fields: AM = AM + δAM . For an appropriate choice of gauge the differential operator which governs the loop contributions becomes

aM a M aM ∆ bN = −δ b δ N + X bN , (3.203) with

aM M a a cM X bN = −R N δ b + 2i(tc) bF N , (3.204) where tc here denotes a gauge generator in the adjoint representation. The dimension of the 6-vector representation of the Lorentz group is in this case d = 6. We

19We adopt the convention of using Tr [...] to denote a trace which includes the Lorentz and/or spacetime indices, while reserving tr [...] for those which run only over the ‘ﬂavor’ indices which count the ﬁelds of a given spin.

124 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

therefore ﬁnd Tr V I = 6N, Tr V (X) = −NR, and

2 MN a MN Tr V (X ) = NRMN R + 4 C(A) FMN Fa (3.205)

MN ABMN a MN Tr V (YMN Y ) = −NRABMN R − 6 C(A) FMN Fa , (3.206) where C(A) is the Dynkin index for N ﬁelds in the adjoint representation, deﬁned by tr (tatb) =

C(A) δab. The subscript ‘V ’ in these expressions is meant to emphasize that the trace has been taken over a vector ﬁeld (as opposed to the physical spin-1 ﬁeld, including ghosts).

These expressions suffice to compute Tr V [ak], for the vector field. Once specialized to the spherical geometries and background gauge field of interest we find

F F (−) Tr V [a0] = 6N, (−) Tr V [a1] = 0 (3.207) N (−)F Tr [a ] = R2 + 3˜g2f 2 tr (Q2) (3.208) V 2 10 N 4 (−)F Tr [a ] = − R3 + g˜2f 2 R tr (Q2) . (3.209) V 3 105 5

To this we must add the ghost contribution, which consists of N complex scalar fields having fermionic statistics and transforming in the adjoint representation of the gauge group. The contributions to the ak may be read off from our previously-quoted expressions for scalar fields in the special case X = 0. For such fields we have

N (−)F Tr [a ] = −2N, (−)F Tr [a ] = R (3.210) gh 0 gh 1 3 N 1 (−)F Tr [a ] = − R2 + g˜2f 2 tr (Q2) (3.211) gh 2 30 3 N 1 (−)F Tr [a ] = R3 − g˜2f 2 R tr (Q2) . (3.212) gh 3 315 10

Summing the contributions of eqs. (3.207) and (3.210) gives the contribution of N physical 6D massless gauge bosons:

N (−)F Tr [a ] = 4N, (−)F Tr [a ] = R (3.213) 1 0 1 1 3 N 10 (−)F Tr [a ] = R2 + g˜2f 2 tr (Q2) (3.214) 1 2 15 3 1 2N 7 (−)F Tr [a ] = − R3 + g˜2f 2 R tr (Q2) . (3.215) 1 3 315 10 1

3.B Sums and zeta functions

This appendix has two purposes. The ﬁrst section provides a more secure theoretical foundation for many of the zeta-function calculations of the main text; while the second section provides formulae for several useful sums encountered in the calculations.

125 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Justifying zeta function regularization

The purpose of this section is to compute the sum

∞ reg X X d (ω, t, a0) := G(2πk) , (3.216) j=0 k6=0 where Z ∞ 2 −iqx G(q) := dx exp[−t(j + ω|x| + a0) ] e , (3.217) −∞ without resorting to the use of zeta function regularization. Pursuing this exercise will show that the zeta function approach used later is valid. Computing this Fourier transform gives

r 1 π 2 G(q) = e−q¯ e2i¯jq¯ 1 − erf(¯j + iq¯) + c.c. (3.218) 2ω t where √ q ¯j := t(j + a0) , q¯ := √ . (3.219) 2ω t Since G(−q) = G(q), we can write

∞ ∞ reg X X d (ω, t, a0) = 2 G(2πk) . (3.220) j=0 k=1

When evaluating G(q) for non-zero q, it is important to note that the overall factor of exp(−q2/(4ω2t)) in eq. (3.218) will exponentially suppress any quantity multiplying it that does not diverge in the t → 0 limit. Since we are only interested in the t → 0 limit on this sum, we can therefore approximate G(q) as follows:

r 1 π 2 h i G(q) ' − e−q¯ e2i¯jq¯ erf(¯j + iq¯) + c.c. 2ω t " ¯j ¯j+iq¯ ! # 1 2 Z 2 Z 2 = − √ e−q¯ e2i¯jq¯ dx e−x + dx e−x + c.c. ω t 0 ¯j q¯ 1 2 Z 2 ' − √ e−q¯ i e2i¯jq¯ du e−(iu+¯j) + c.c. . (3.221) ω t 0

(Here, we have taken u = −i(x − ¯j).) It is also helpful to make the following substitutions:

q¯ 2 2 Z 2 2 G(q) = √ e−q¯ du eu −¯j sin[2¯j(¯q − u)] ω t 0 1 2¯q 2 Z 2 2 2 = √ e−q¯ dy eq¯ (1−y) e−¯j sin(2¯jq¯ y) (3.222) ω t 0 where y := (¯q−u)/q¯. Since all of the j-dependence is in the bracketed term within the above integral,

126 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy we can consider ﬁrst carrying out the j-sum and then performing the integral over y:

  ∞ ¯ Z 1 ∞ X 2k ¯2 ¯2 2 X ¯2 √ G(2πk) = e−k /t dy ek (1−y) /t e−j sin(2¯jk¯ y/ t) (3.223) ωt   j=0 0 j=0 | {z } :=R(ky,t¯ ) where we have introduced the notation k¯ = πk/ω. Calculating R(ky,¯ t) can be done as before, using the Euler-Maclaurin formula in eq. (3.65). Deﬁning

2 ¯ −t(x+a0) ¯ g(x + a0; ky, t) := e sin 2(x + a0)ky , (3.224) this gives Z ∞ ∞ X Bi R(ky,¯ t) = dx g(x + a ; ky,¯ t) − g(i−1)(a ; ky,¯ t) (3.225) 0 i! 0 0 i=1 (i−1) ¯ ¯ where g (a0; ky, t) denotes the (i − 1)-th derivative of g(x; ky, t) with respect to x evaluated at x = a0. The integral can be simpliﬁed as follows:

Z ∞ Z −a0 Z ∞ ¯ ¯ ¯ dx g(x + a0; ky, t) = dx g(x + a0; ky, t) + dx g(x + a0; ky, t) 0 0 −a0 Z ∞ Z a0 = dx g(x; ky,¯ t) − dx g(x; ky,¯ t) 0 0 √ ! 1 ky¯ Z ta0 √ = √ D √ − dx g(x; ky/¯ t, 1) (3.226) t t 0 where ∞ Z 2 D(w) := dx e−x sin(2xw) (3.227) 0 is known as Dawson’s integral – a real, analytic function with known series expansions about w = 0 and w → ∞. The second integral can be reliably Taylor expanded in its upper limit, since the t-dependence of the integrand is contained within the sine function (which is guaranteed not to √ diverge). Furthermore, since the integrand is an odd function of x, only even powers of ta0 will appear. The coeﬃcients in such an expansion will be related to derivatives of g(x; ky,¯ t) (with respect to x): √ ∞ √ Z t a0 √ X g(2i−1)(0; ky/¯ t, 1) dx g(x; ky/¯ t, 1) = (t a2)i . (3.228) (2i)! 0 0 i=1

Given all of this, we see that the sum over G(2πk) can be divided into three contributions:

∞ X 2 G(2πk) = G (k,¯ t) + G (k,¯ t) + G (k,¯ t) (3.229) ωt 1 2 3 j=0

127 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy where

¯ Z 1 ¯ ¯ k −k¯2/t k¯2(1−y)2/t ky G1(k, t) = √ e dy e D √ (3.230) t 0 t ¯ ∞ 2 i Z 1 k X (t a ) ¯2 ¯2 2 √ G (k,¯ t) = −√ 0 e−k /t dy ek (1−y) /tg(2i−1)(0; ky/¯ t, 1) (3.231) 2 (2i)! t i=1 0 ∞ Z 1 X Bi ¯2 ¯2 2 G (k,¯ t) = −k¯ e−k /t dy ek (1−y) /tg(i−1)(a ; ky,¯ t) . (3.232) 3 i! 0 i=1 0

The small-t limit of these expressions can be obtained by making extensive use of the deﬁnition of the Dawson integral in eq. (3.227), as well as its asymptotic expansion

1 1 3 15 1 D(w) ' + + + + O . (3.233) 2w 4w3 8w5 16w7 w9

(This is most easily done using computing software.) And, of course, any term in the ﬁnal result that is suppressed by a factor of exp(−k¯2/t) can be safely dropped in this limit. Performing this expansion yields

t t2 3t3 G (k,¯ t) ' + + + ... (3.234) 1 4k¯2 8k¯4 16k¯6 ta2 t 3t2 t2a4 3t G (k,¯ t) ' − 0 × + + ... − 0 × − + ... − ... (3.235) 2 2! 2k¯2 4k¯4 4! k¯2 B a t2 a3 3a G (k,¯ t) ' − 1 0 + − 0 + 0 t3 + ... 3 1! 2k¯2 2k¯2 4k¯4 B t2 3a2 3 − 2 + − 0 + t3 ... 2! 2k¯2 2k¯2 4k¯4 B 3a t3 B 3t3 − 3 − 0 ... − 4 − ... − ... (3.236) 3! k¯2 4! k¯2

3 In the above, the ... indicate terms that will contribute to S0(ω, t) at O(t ) or higher, and so will give no contribution to the Casimir energy.

Finally, we can substitute this result into eq. (3.220) and, with the use of

∞ X 1 ω n π2 π4 π6 = ζ(n) with ζ(2) = , ζ(4) = , ζ(6) = k¯n π 6 90 945 k=1 ∞ ∞ ∞ X 1 1 X 1 1 X 1 1 → = ω2 , = ω4 , = ω6 , (3.237) k¯2 6 k¯4 90 k¯6 945 k=1 k=1 k=1 we ﬁnd that

∞ X ω2t ω4t2 ω6t3 4 G (k,¯ t) ' + + + ... (3.238) 1 6 180 1260 k=1

128 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

∞ X a2ω2t2 a2ω4t3 a4ω2t3 4 G (k,¯ t) ' − 0 − 0 + 0 + ... (3.239) 2 6 60 12 k=1 ∞ 2 2 3 2 3 4 3 X 1 a0ω t a ω t a0ω t 4 G (k,¯ t) ' − − − 0 + 3 2 3 3 30 k=1 1 ω2t2 a2ω2t3 ω4t3 − − 0 + 6 6 2 60 1 ω2t3 −(0) − − − − ... (3.240) 30 12 and so

∞  ∞  reg X X d (ω, t, a0) = 2 ×  G(2πk) k=1 j=0 ∞ 4 X h i = G (k,¯ t) + G (k,¯ t) + G (k,¯ t) ωt 1 2 3 k=1 ( 1 1 1 1 1 1 = ω2 + − + a − a2 ω2 + ω4 t ω 6 36 6 0 6 0 180 1 1 1 1 + − + a2 − a3 + a4 ω2 360 12 0 6 0 12 0 ) 1 1 1 1 + − + a − a2 ω4 + ω6 t2 (3.241) 360 60 0 60 0 1260 as found later, using the more eﬃcient zeta–function regularization method.

Some useful sums

Next, we will explicitly evaluate the small-t limit of two often-encountered sums.

The sum c(t, a0)

This sum is deﬁned as follows:

∞ X 2 c(t, a0) := exp[−t(j + a0) ] . (3.242) j=0

The Euler-Maclaurin formula states that, for any analytic function f(x), we can write

∞ Z ∞ ∞ X X Bi f(j) = dx f(x) − f (i−1)(0) (3.243) i! j=0 0 i=1

(i−1) where f (x) denotes the (i−1)-th derivative of f(x) and where the Bi are the Bernoulli numbers.

(The ﬁrst few are B1 = −1/2, B2 = 1/6, B4 = −1/30, and B2i+1 = 0 for i ≥ 1.) Identifying

129 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 f(x) = exp[−t(x + a0) ], the ﬁrst few derivative terms are

2 2 2 (0) −ta0 (1) −ta0 (2) 2 2 −ta0 f (0) = e , f (0) = −2ta0 e , f (0) = (−2t + 4t a0) e 2 2 (3) 2 3 3 −ta0 (4) 2 3 2 4 4 −ta0 f (0) = (12t a0 − 8t a0) e , f (0) = (12t − 48t a0 + 16t a0) e , (3.244) and f (5)(0) ∼ O(t3). (We only need powers of t smaller than 3, for reasons discussed in the text.) The integral can be written in terms of the error function, and has the following series expansion:

∞ r Z 2 1 π √ −t(x+a0) dx e = 1 − erf(a0 t) 0 2 t 1rπ 1 1 = − a + a3 t − a5 t2 + O(t3) . (3.245) 2 t 0 3 0 10 0

Upon Taylor expanding the derivative terms, we ﬁnd that

c−1/2 2 3 c(t, a0) = √ + c0 + c1 t + c2 t + O(t ) (3.246) t where √ π 1 1 1 1 c = , c := − a , c := a − a2 + a3 , −1/2 2 0 2 0 1 6 0 2 0 3 0 1 1 1 1 c := a − a3 + a4 − a5 . (3.247) 2 60 0 6 0 4 0 10 0

We will usually need a “tilded” version of the above sum. This is deﬁned in the following way:

√ τt c˜−1/2 3/2 2 5/2 c˜(t, a0, τ) := e c(t, a0) = √ +c ˜0 +c ˜1/2 t +c ˜1 t +c ˜3/2 t +c ˜2 t + O(t ) . (3.248) t

In terms of components,

c˜−1/2 = c−1/2 , c˜0 = c0 , c˜1/2 = c−1/2 τ , c˜1 = c1 + c0 τ , c c c˜ = −1/2 τ 2 , c˜ = c + c τ + 0 τ 2 . (3.249) 3/2 2 2 2 1 2

The double sum d(ω, t, a0)

This sum is deﬁned as follows:

∞ ∞ X X 2 d(ω, t, a0) := exp[−t(j + ω|n| + a0) ] . (3.250) j=0 n=−∞

The j-sum can be performed as before, but the n-sum will pose a diﬃculty in the ω, t → 0 limit, where it converges very slowly. This is because the sum in fact diverges in this limit. However, by using Poisson resummation, we can compute these divergent contributions eﬃciently.

130 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

With this in mind, let’s write

sing reg d(ω, t, a0) = d (ω, t, a0) + d (ω, t, a0) (3.251)

reg where we require that d (ω, t, a0) contain all terms that vanish as ω, t → 0. Since the calculation sing of d (ω, t, a0) is more subtle (for the above-mentioned reasons), we will compute it ﬁrst and then reg apply a more cavalier approach to d (ω, t, a0).

sing Poisson resummation and d (ω, t, a0)

sing A method for ﬁnding the behaviour of d (ω, t, a0) as ω, t → 0 is to use Poisson resummation. 2 2 Deﬁning g(n) := exp[−ω t(|n| + bj) ] where bj := (j + a0)/ω, we can exchange

∞ ∞ X X g(n) = G(2πk) (3.252) n=−∞ k=−∞ where Z ∞ G(q) := dx g(x)e−iqx (3.253) −∞ is the Fourier transform of g(x). For example, in the special case where bj = 0 (i.e. j = a0 = 0), we ﬁnd that g(n) = exp(−ω2t n2) and so

∞ r ∞ X 2 2 π X 2 2 2 e−ω t n = e−π k /(ω t) (3.254) ω2t n=−∞ k=−∞ since Z ∞ r −ω2t x2−iqx π −q2/(4ω2t) G(q) = dx e = 2 e . (3.255) −∞ ω t This special case is suﬃcient to illustrate the value of Poisson resummation: a sum that converges slowly in the ω, t → 0 limit is transformed into one that converges quickly in this same limit. In fact, the part of the n–sum that diverges in this limit gets mapped entirely onto the k = 0 term in the k–sum. Therefore, we can write

∞ sing X d (ω, t, a0) = G(0) (3.256) j=0 where now, for bj 6= 0,

∞ r Z 2 2 π √ −ω t(|x|+bj ) G(0) = dx e = 2 [1 − erf(ω tbj)] . (3.257) −∞ ω t

131 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

This sum can easily be evaluated in the same way as in section 3.B using the Euler-Maclaurin formula. Integrating and diﬀerentiating G(0) as prescribed gives

r ∞ 1 π X h √ i dsing(ω, t, a ) = 1 − erf t(j + a ) (3.258) 0 0 ω t 0 j=0 sing dsing d = −1 + −√1/2 + dsing + dsing t + dsing t2 + O(t3) t t 0 1 2 in the small–t limit, where √ 1 π 1 1 1 dsing := , dsing := − a , dsing := − a + a2 , (3.259) −1 ω −1/2 ω 2 0 0 ω 6 0 0 1 1 a2 a3 a4 1 1 a2 a4 a5 a6 dsing := − 0 + 0 − 0 , dsing := − 0 + 0 − 0 + 0 . 1 ω 180 6 3 6 2 ω 1260 60 12 10 30

As promised, each of these contributions diverges in the limit where ω, t → 0.

reg Calculating d (ω, t, a0) using a zeta function approach

First, we should mention that the use of zeta function method here is not necessary. In fact, we reg could continue as before, and compute d (ω, t, a0) using Poisson resummation:

∞ ∞ reg X X d (ω, t, a0) = 2 G(2πk) . (3.260) j=0 k=1

reg Here, we have identified d (ω, t, a0) as the left–over k 6= 0 terms in the Poisson sum, eq. (3.252). (The factor of two arises because G(q) is even: g(−x) = g(x) ⇒ G(−q) = G(q).) This more “honest” approach is taken in Appendix 3.B. However, since this calculation is significantly less tiresome if we take advantage of zeta function regularization, we will present this version of the derivation herein. (The two calculations give the same result, as can be seen by comparison with the result in Appendix 3.B.) Much of the simplification arises because we can take advantage of the previous result for c(t, a0). sing Since we are guaranteed to have captured all of the divergent terms (as ω, t → 0) in d (ω, t, a0), computing the full sum in a na¨ıve way — one that does not capture these divergent terms — will give reg us d (ω, t, a0). The zeta function method is just such a na¨ıve approach! Let’s start by interchanging the j– and n–sums in eq. (3.250) and write

∞ ∞ reg X X d (ω, t, a0) = c(t, ω|n| + a0) = c(t, a0) + 2 c(t, ωn + a0) . (3.261) n=−∞ n=1

Since, the zeta function is deﬁned as ∞ X 1 ζ(s) = , (3.262) ns n=1

132 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

P∞ P∞ 0 we have that n=1 1 = n=1 1/(n ) = ζ(0) = −1/2 and so we can write

∞ reg X d (ω, t, a0) = 2 c(t, ωn + a0) − c(t, a0) . (3.263) n=1

Given the result in eqs. (3.246) and (3.247), we ﬁnd that the summand — expanded out in powers of t — is given by X r ∆c := c(t, ωn + a0) − c(t, a0) = t ∆cr (3.264) r where r ∈ {−1/2, 0, 1, 2} and where

1 1 1 ∆c = 0 , ∆c = −ωn , ∆c = − a + a2 ωn + − + a ω2n2 + ω3n3 −1/2 0 1 6 0 0 2 0 3 1 a2 a4 a 3a2 ∆c = − 0 + a3 − 0 ωn + − 0 + 0 − a3 ω2n2 (3.265) 2 60 2 0 2 2 2 0 1 1 a 1 + − + a − a2 ω3n3 + − 0 ω4n4 − ω5n5 . 6 0 0 4 2 10

Each term in ∆c proportional to ns will get a corresponding factor of ζ(−s) once the sum over n is performed. (Here, ζ(−1) = −1/12, ζ(−3) = 1/120, ζ(−5) = −1/252 and ζ(−2s) = 0 for s ≥ 1.) The end result is reg reg reg reg 2 3 d (ω, t, a0) = d0 + d1 t + d2 t + O(t ) (3.266) where

ω 1 a a2 ω3 dreg = , dreg = − + 0 − 0 ω + , (3.267) 0 6 1 36 6 6 180 1 a2 a3 a4 1 a a2 ω5 dreg = − + 0 − 0 + 0 ω + − + 0 − 0 ω3 + . 2 360 12 6 12 360 60 60 1260

As promised, the above result vanishes in the ω, t → 0 limit.

sing reg Combining the results for d (ω, t, a0) and d (ω, t, a0), we ﬁnd that

d−1 d−1/2 2 3 d(ω, t, a0) = + √ + d0 + d1 t + d2 t + O(t ) (3.268) t t where √ 1 π 1 1 1 ω2 d = , d = − a , d = − a + a2 + , −1 ω −1/2 ω 2 0 0 ω 6 0 0 6 1 1 a2 a3 a4 1 a a2 ω4 d = − 0 + 0 − 0 + − + 0 − 0 ω2 + , (3.269) 1 ω 180 6 3 6 36 6 6 180 1 1 a2 a4 a5 a6 1 a2 a3 a4 d = − 0 + 0 − 0 + 0 + − + 0 − 0 + 0 ω2 2 ω 1260 60 12 10 30 360 12 6 12 1 a a2 ω6 + − + 0 − 0 ω4 + . 360 60 60 1260

133 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

As with c(t, a0), we will often need a “tilded” version of the above sum. This is deﬁned in a similar way as before:

˜ τt d(ω, t, a0, τ) := e d(ω, t, a0) (3.270) ˜ d˜ √ d−1 −1/2 ˜ ˜ ˜ ˜ 3/2 ˜ 2 5/2 = + √ + d0 + d1/2 t + d1 t + d3/2 t + d2 t + O(t ) . t t

In terms of components,

˜ ˜ ˜ ˜ d−1 = d−1 , d−1/2 = d−1/2 , d0 = d0 + d−1 τ , d1/2 = d−1/2 τ , (3.271) d d d d d˜ = d + d τ + −1 τ 2 , d˜ = −1/2 τ 2 , d˜ = d + d τ + 0 τ 2 + −1 τ 3 . 1 1 0 2 3/2 2 2 2 1 2 6

3.C Spectra and mode sums

This appendix computes the KK spectra and mode sums that are quoted in the main text, for spins zero, half and one.

Spectrum and Mode Sum for the Scalar Field

s This appendix computes the spectrum λjn and the corresponding small-t limit of the mode sum

2 s X −tλjn X s i Ss(t) := e = si t + ··· (3.272) j,n i=−1 for a 6D charged scalar φ, with mass m and charge qg˜, on the rugby ball.

Eigenvalues

The scalar equation of motion is

MN 2 g DM DN − m φ = 0 (3.273)

where DM is the covariant derivative, and where

2 µ ν 2 2 2 2 2 ds = ηµν dx dx + r (dθ + α sin θ dϕ ) (3.274) (N − Φ) qA dxm = qA dϕ = − (cos θ − b) + b Φ dϕ . (3.275) m ϕ 2 b

In the above expression for Aϕ, the variable b is used to distinguish two patches of the gauge potential: b = +1 (−1) corresponds to the patch encompassing cos θ = +1 (−1). To verify that this gauge potential is a correct one, it is suﬃcient to check that:

dA N ϕ = F = f = (αr2 sin θ) , (3.276) dθ θφ θφ 2qr2

134 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

A A (cos θ = b) = b b . (3.277) ϕ 2π P (Recall the deﬁnitions N = ω(N − Φ), Φb = qAb/(2π) and Φ = b Φb from the text.)

Expanding out this equation of motion using the ansatz φ(x) = eik4·x Θ(θ)einϕ gives

2 2 d Θ dΘ (n − q Aφ) + cot θ + λjn − Θ = 0 (3.278) dθ2 dθ α2 sin2 θ

2 2 2 where λjn = −r (k4 +m ) are the required eigenvalues (in units of the KK scale 1/r). This is just the result for the case without a background ﬁeld, but with n → n − q Aϕ. This diﬀerence is expected given that the wave equation depends on the azimuthal covariant derivative Dϕ = ∂ϕ − iq Aϕ = inϕ i(n − q Aϕ) (since our ansatz is that φ(x) ∝ e ).

The standard approach from here is to change variables to x := cos θ, giving

d2Θ dΘ (n − q A )2 (1 − x2) − 2x + λ − φ Θ = 0 , (3.279) dx2 dx jn α2(1 − x2) and to let Θ(x) = (1 − x2)yf(x) for some convenient power y that removes the singular behaviour of (3.279) near x = ±1 (↔ θ = 0, π). However, since the gauge field is defined differently at each pole, it will help to treat the divergences at each pole separately. Therefore, we will instead take

Θ(x) = (1 − x)y(1 + x)zf(x) , which yields

d2f df ω2(n − q A )2 + k(x, y, z) (1 − x2) − 2[(1 + y + z)x + y − z] + λ − φ f(x) = 0 (3.280) dx2 dx jn (1 − x2) where ω = α−1 and

k(x, y, z) := y + z − (y − z)2 − 2(y2 − z2)x − (y + z)(y + z + 1)x2 . (3.281)

2 2 2 Requiring that the numerator [ω (n − qAϕ) + k] be proportional to (1 − x ) gives the conditions

ω ω y = |n + (N − Φ) − b Φ | , z = |n − (N − Φ) − b Φ | (3.282) 2 −b b 2 b b where + = 1 and − = 0, and the resulting regular ODE is

d2f df N 2 (1 − x2) − 2[(1 + y + z)x + y − z] + λ + − (y + z)(y + z + 1) f(x) = 0 . (3.283) dx2 dx jn 4

In passing, it should be mentioned that it is equally valid to take y and/or z to be deﬁnitely negative, rather than positive, in eliminating the singularities from eq. (3.279). However, in the absence of couplings of the scalar to the brane, we should discard these solutions since they cause φ(x) to

135 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy diverge at one or both of the branes (in conﬂict with the boundary conditions).

It is instructive to consider the special case in which q = 0 and ω = 1. In this case, we ﬁnd that y = z = |n|/2 (thereby justifying the usual choice of Θ(x) = (1 − x2)|n|/2f(x)) and

d2f df (1 − x2) − 2(1 + |n|)x + [λ − |n|(|n| + 1)] f(x) = 0 . dx2 dx jn

This is the standard differential equation that is satisfied by the |n|th derivative of a Legendre polynomial (so long as λjn = (j + |n|)(j + |n| + 1) for some j ≥ 0, allowing the corresponding power series solution to terminate). Therefore, Θ(x) can be any (linear combination) of the associated Legendre polynomials and φ(x) are (linear combinations of) spherical harmonics. It is conventional to define ` := j + |n| so that λ = `(` + 1) with degeneracy (2` + 1), representing the number of different ways of choosing combinations (j, n) that give the same `.

In the general case, the condition for termination of the power series solution to eq. (3.283) is

N 2 λ = (j + y + z)(j + y + z + 1) − . (3.284) jn 4

Mode Sum

From here, we wish to compute the small–t coeﬃcients of Ss(ω, N, Φb, t), where

∞ ∞ tN 2/4 X X Ss(ω, N, Φb, t) = e exp − t(j + y + z)(j + y + z + 1) (3.285) j=0 n=−∞ ∞ ∞ X X = eτt exp − t(j + y + z + 1/2)2 . (3.286) j=0 n=−∞

In the last line, we have introduced τ := (1 + N 2)/4.

In order to use our previously-derived results for the sums

∞ τt X 2 c˜(t, a0, τ) := e exp[−t(j + a0) ] (3.287) j=0 ∞ ∞ ˜ τt X X 2 d(ω, t, a0, τ) := e exp[−t(j + ω|n| + a0) ] , (3.288) j=0 n=−∞ we will first have to do some massaging. We will assume throughout that |Φb| < 1, since any brane– localized flux larger than this can be absorbed (one integer at a time) into the bulk flux, N. Also, we will choose to work with the north (cos θ = +1) patch of the potential, for which

ω ω y + z = |n − Φ | + |n − N + Φ | . (3.289) 2 + 2 −

136 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

To see the result for the south (cos θ = −1) patch, where

ω ω y + z = |n + N − Φ | + |n + Φ | , (3.290) 2 + 2 − we can simply make the replacements N → −N and Φb → −Φ−b in the ﬁnal result (since the spectrum only depends on the combined quantity y + z). Of course, such a transformation should be a symmetry of the result, since it is just a gauge transformation. It is helpful to ﬁrst consider the case when N > 0 and break up the n–sum in eq. (3.286) into three parts:

(+) (0) (−) Ss(ω, N, Φb, t) = Ss (ω, N, Φb, t) + Ss (ω, N, Φb, t) + Ss (ω, N, Φb, t) (3.291) where:

• for Φ− > 0, Φ+ > 0,

∞ ∞ " 2# X X ω ω 1 S(+) = eτt exp −t j + (n − Φ ) + (n − N + Φ ) + , s 2 + 2 − 2 j=0 n=N ∞ N−1 " 2# X X ω ω 1 S(0) = eτt exp −t j + (n − Φ ) + (N − Φ − n) + , s 2 + 2 − 2 j=0 n=1 ∞ −∞ " 2# X X ω ω 1 S(−) = eτt exp −t j + (Φ − n) + (N − Φ − n) + ; s 2 + 2 − 2 j=0 n=0

• for Φ− > 0, Φ+ < 0,

∞ ∞ " 2# X X ω ω 1 S(+) = eτt exp −t j + (n − Φ ) + (n − N + Φ ) + , s 2 + 2 − 2 j=0 n=N ∞ N−1 " 2# X X ω ω 1 S(0) = eτt exp −t j + (n − Φ ) + (N − Φ − n) + , s 2 + 2 − 2 j=0 n=0 ∞ −∞ " 2# X X ω ω 1 S(−) = eτt exp −t j + (Φ − n) + (N − Φ − n) + ; s 2 + 2 − 2 j=0 n=−1

• for Φ− < 0, Φ+ > 0,

∞ ∞ " 2# X X ω ω 1 S(+) = eτt exp −t j + (n − Φ ) + (n − N + Φ ) + , s 2 + 2 − 2 j=0 n=N+1 ∞ N " 2# X X ω ω 1 S(0) = eτt exp −t j + (n − Φ ) + (N − Φ − n) + , s 2 + 2 − 2 j=0 n=1 ∞ −∞ " 2# X X ω ω 1 S(−) = eτt exp −t j + (Φ − n) + (N − Φ − n) + ; s 2 + 2 − 2 j=0 n=0

137 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

• for Φ− < 0, Φ+ < 0,

∞ ∞ " 2# X X ω ω 1 S(+) = eτt exp −t j + (n − Φ ) + (n − N + Φ ) + , s 2 + 2 − 2 j=0 n=N+1 ∞ N " 2# X X ω ω 1 S(0) = eτt exp −t j + (n − Φ ) + (N − Φ − n) + , s 2 + 2 − 2 j=0 n=0 ∞ −∞ " 2# X X ω ω 1 S(−) = eτt exp −t j + (Φ − n) + (N − Φ − n) + . s 2 + 2 − 2 j=0 n=−1

If we allow ourselves to extend the notation ± to

( 1 + sgn(x) 1 , sgn(x) := x/|x| = +1 x := = (3.292) 2 0 , sgn(x) := x/|x| = −1

(recall that, before, b = (1 + b)/2 = 1 or 0), these four cases (for which N > 0) can be succintly written as

∞ ∞ " 2# X X ω∆Φ 1 + ωN S(+) = eτt exp −t j + ωn − + s 2 2 j=0 n=−Φ− ∞ " 2# X ωΦ 1 + ωN S(0) = (N − 1 + + ) × eτt exp −t j − + s −Φ− −Φ+ 2 2 j=0 ∞ −∞ " 2# X X ω∆Φ 1 + ωN S(−) = eτt exp −t j − ωn + + (3.293) s 2 2 j=0 n=−−Φ+ where ∆Φ := Φ+ − Φ−.

Performing a similar calculation for the case where N < 0, we ﬁnd something slightly diﬀerent:

∞ ∞ " 2# X X ω∆Φ 1 − ωN S(+) = eτt exp −t j + ωn − + s 2 2 j=0 n=Φ+ ∞ " 2# X ωΦ 1 − ωN S(0) = (−N − 1 + + ) × eτt exp −t j + + s Φ− Φ+ 2 2 j=0 ∞ −∞ " 2# X X ω∆Φ 1 − ωN S(−) = eτt exp −t j − ωn + + . (3.294) s 2 2 j=0 n=−Φ−

Writing η := sgn(N) = N/|N|, we ﬁnd that these expressions – for arbitrary N – take the form

∞ ∞ " 2# X X ω∆Φ 1 + ω|N| S(+) = eτt exp −t j + ωn − + s 2 2 j=0 n=−ηΦ−η

138 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

∞ " 2# X ωΦ 1 + ω|N| S(0) = (|N| − 1 + + ) × eτt exp −t j − η + s −ηΦ− −ηΦ+ 2 2 j=0 ∞ −∞ " 2# X X ω∆Φ 1 + ω|N| S(−) = eτt exp −t j − ωn + + . (3.295) s 2 2 j=0 n=−−ηΦη

Using the ﬁnal rearrangement

∞ ∞ ! X 1 X f(n) = f(|n|) ± f(0) (3.296) 2 n=∓ n=−∞

and writing aN := (1 + ω|N|)/2, we ﬁnd that the massage is complete:

1h i S(+) = d˜(ω, t, −ω∆Φ/2 + a , τ) + η sgn(Φ )c ˜(t, −ω∆Φ/2 + a , τ) s 2 N −η N (0) h i Ss = |N| − η sgn(Φ+) + sgn(Φ−))/2 c˜(t, −η ωΦ/2 + aN , τ) (3.297) 1h i S(−) = d˜(ω, t, ω∆Φ/2 + a , τ) + η sgn(Φ )c ˜(t, ω∆Φ/2 + a , τ) . s 2 N η N

This result is, by construction, invariant under N → −N,Φb → −Φ−b. Evaluating these expressions, we ﬁnd, using throughout the deﬁnitions

(n) X n (1) Fb := |Φb| (1 − |Φb|) ,F := Fb ,F := F,G(x) := (1 − x)(1 − 2x) (3.298) b

s that s−1 = 1/ω,

1 1 ω2 ss (ω, N, Φ ) = + (1 − 3F ) , (3.299) 0 b ω 6 6 " # 1 1 N 2 ω2 ω3N X ω4 ss (ω, N, Φ ) = − + (1 − 3F ) − Φ G(|Φ |) + (1 − 15F (2)) ,(3.300) 1 b ω 180 24 18 12 b b 180 b " 1 1 11 N 2 1 N 2 ω3N X ss (ω, N, Φ ) = − − + − (1 − 3F )ω2 − Φ G(|Φ |) 2 b ω 504 720 90 144 24 b b b ω4(1 − N 2) ω5N X + (1 − 15F (2)) − Φ G(|Φ |)(1 + 3F ) 360 120 b b b b ! # 1 F (2) F (3) + − − ω6 . (3.301) 1260 120 60

Spectrum and Mode Sum for the Spin-1/2 Field

f This appendix computes the spectrum λjn and the corresponding small-t limit of the mode sum

2 fξ X X −tλjn X f i Sf (t) := − e = si t (3.302) ξ j,n i=−1

139 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy for a 6D charged spin-1/2 ﬁeld ψ, with mass m and charge q, on the rugby ball. Here, the minus sign denotes fermionic degrees of freedom, and the additional sum is over spin states.

Eigenvalues

The spinor equation of motion is (D/ + m)ψ = 0 (3.303)

where DM is the covariant derivative and where the metric and background gauge ﬁeld, respectively, are

2 2 2 2 2 2 µ ν ds = r (dθ + α sin θ dϕ ) + ηµν dx dx (3.304) N − Φ qA dxM = qA dϕ = − (cos θ − b) + bΦ dϕ (3.305) M ϕ 2 b

(as before). Here, q is the charge (in units ofg ˜) of the ﬁeld ψ under the gauge ﬁeld.

M AB We make use of the frame ﬁelds eA and the spin connection ΩM to ensure that our Cartesian

Cliﬀord algebra (i.e. {ΓA, ΓB } = 2ηAB ) conforms to the spherical geometry:

i Dψ/ = ΓAe M ∂ − J ΩAB − iqA ψ . (3.306) A M 2 AB M M

(Here, JAB = −i[ΓA, ΓB ]/4 are the Lorentz generators in the spinor representation.) More precisely, these are given by two separate patches (in the same way as the gauge potential):

! 1 cos ϕ − b sin ϕ e m = α sin θ , e µ = δµ , e m = 0 (so that e M e N g = η ) , (3.307) a r cos ϕ α α α A B MN AB b sin ϕ α sin θ AB AC N K B N B A M −1 ΩM = η (eC ΓMN eK − eC ∂M eN ) (where eM := (eA ) ) . (3.308)

K (Recall that b + 1 (b = −1) for the north (south) brane.) Given that ΓMN = 0 with the exception of ϕ θ 2 Γθϕ = cot θ and Γϕϕ = −α sin θ cos θ, we ﬁnd that the spin connections mostly vanish as well, with 45 54 the exception of Ωϕ = α cos θ − b = −Ωϕ . In order to expand out the above Dirac equation, let’s use the representation

! ! ! 0 γµ 0 γ 0 −i1I Γµ = , Γ4 = 5 , Γ5 = 4 (3.309) µ γ 0 γ5 0 i1I4 0 where γµ are the usual 4D Dirac matrices:

! ! 0 σµ 1I 0 γµ = −i , γ = −iγ0γ1γ2γ3 = 2 (3.310) µ 5 −σ 0 0 −1I2

µ and σµ = (1I2, σi) = σ are the Pauli matrices.

140 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Substituting these deﬁnitions, we ﬁnd that the covariant derivative becomes

" # cos ϕ b sin ϕ cos ϕ Ω45 D/ = Γµ∂ + Γ4 ∂ − (∂ − iqA ) + ϕ µ r θ α r sin θ ϕ ϕ α r sin θ 2 " # sin ϕ cos ϕ b sin ϕ Ω45 +Γ5 ± ∂ + (∂ − iqA ) + ϕ (3.311) r θ α r sin θ ϕ ϕ α r sin θ 2 or ! 0 γµ∂ + P O − P O D/ = µ L 2 R 1 (3.312) µ γ ∂µ + PLO1 − PRO2 0 where PL(R) = (1 ± γ5)/2 and where

ibϕ 45 ! e (−i∂ϕ − qAϕ − Ω /2) O = ∂ − ϕ (3.313) 1 r θ α sin θ

−ibϕ 45 ! e (−i∂ϕ − qAϕ + Ω /2) O = ∂ + ϕ . (3.314) 2 r θ α sin θ

Since ψ is a 6D Dirac spinor, we can decompose it into two 4D Dirac spinors or, equivalently, four 4D Weyl spinors: ! ! ψ ψ + ψ ψ = 1 = 1L 1R (3.315) ψ2 ψ2L + ψ2R where the 4D Weyl spinors satisfy γ5ψiL = +ψiL and γ5ψiR = −ψiR. In terms of these, the Dirac equation gives

µ µ γ ∂µψ2R + O2ψ2L + mψ1L = 0 , γ ∂µψ2L − O1ψ2R + mψ1R = 0 (3.316) µ µ γ ∂µψ1R + O1ψ1L + mψ2L = 0 , γ ∂µψ1L − O2ψ1R + mψ2R = 0 (3.317)

2 2 (where we have applied the projection matrix identities PL = PR = 1, PLPR = 0). From these expressions, it is clear that setting m = 0 decouples ψ1 from ψ2. In other words, for massless fermions there is a halving of the number of degrees of freedom.

Decoupling (3.316) and (3.317) gives

2 2 (4 + O2O1 − m )ψ1L = 0 , (4 + O1O2 − m )ψ1R = 0 (3.318) 2 2 (4 + O1O2 − m )ψ2L = 0 , (4 + O2O1 − m )ψ2R = 0 (3.319)

µν where 4 = η ∂µ∂ν is the 4D d’Alembertian. In what follows, we will quote results only for ψ1, since a solution for ψ1L (ψ1R) will also be a solution for ψ2R (ψ2L). To obtain the mode sum for a massive spin–1/2 ﬁeld, we should remember to multiply the result for ψ1 by 2.

Using the ans¨atze

ik4·x inL(R)ϕ ψ1L(R) = e ΘL(R)(θ) e (3.320)

141 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

where nL and nR are integers satisfying nR − nL = b given eqs. (3.316), (3.317) (and because of the iϕ e factors in O1, O2), we ﬁnd that the spinor functions ΘL(R) must satisfy the following equations:

" d2Θ dΘ 1 L + cot θ L + λf − dθ2 dθ jn1/2 4

2 2 1 !# ω n − qAϕ − ω (n − qAϕ) cos θ + qFθϕ sin θ + − 1/2 1/2 4 Θ = 0 (3.321) sin2 θ L " d2Θ dΘ 1 R + cot θ R + λf − dθ2 dθ jn1/2 4

2 2 1 !# ω n − qAϕ + ω (n − qAϕ) cos θ + qFθϕ sin θ + − 1/2 1/2 4 Θ = 0 (3.322) sin2 θ R where Fθϕ = dAϕ/dθ (in our gauge, where Aθ = 0),

n1/2 := nL + b/2 = nR − b/2 , (3.323) and where λf := −(k2 + m2)r2 are the required eigenvalues (in units of the KK scale). These jn1/2 4 equations transform into one another when L ↔ R and n1/2 ↔ −n1/2, Aϕ ↔ −Aϕ. Since such a transformation is equivalent to a change of coordinates (in particular, the exchange of north and south poles), we expect that the resulting eigenvalues will be the same when solving either equation, 2 2 f as needed. The exception to this logic is the situation where k4 = −m (i.e. when λ = 0); in this case, eqs. (3.316) and (3.317) partially decouple. In this case, we will ﬁnd that only one of ψ1L, ψ1R will have zero mode solutions, given some ﬁxed value N 6= 0.

Remarkably, after: 1) performing the coordinate substitution x := cos θ; and 2) substituting

yσ zσ Θσ(x) = (1 − x) (1 + x) fσ(x) (3.324) where σ = +1 (σ = −1) denotes the left–handed (right–handed) ﬁeld, we ﬁnd that equations (3.321) and (3.322) become

2 " 2 d fσ dfσ f 1 σN (1 − x ) − 2[(yσ + zσ)x + yσ − zσ] + λ − + (3.325) dx2 dx jn1/2 4 2

2 2 1 !# ω n − qAϕ − σω(n − qAϕ)x + k(x, yσ, zσ) + − 1/2 1/2 4 f = 0 1 − x2 σ where we have borrowed the previous deﬁnition for k(x, y, z) in eq. (3.281). (We have substituted for Fθφ to obtain the term proportional to N in the equation above.) Requiring the numerator in the second line of eq. (3.325) to be proportional to (1 − x2) makes eq. (3.325) a regular ODE and

142 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

gives the following result for yσ, zσ:

1 σ 1 σ y = ω n + (N − Φ) − b Φ − , z = ω n − (N − Φ) − b Φ + . σ 2 1/2 −b b 2 σ 2 1/2 b b 2 (3.326) The resulting diﬀerential equation is

d2f df (1 − x2) σ − 2[(1 + y + z )x + y − z ] σ dx2 σ σ σ σ dx 2 f 1 N + λ − + − (yσ + zσ)(yσ + zσ + 1) fσ = 0 , (3.327) jn1/2 4 4

In passing, it should be mentioned that, if the goal is regularity of eq. (3.325), it is equally valid to take yσ and/or zσ to be deﬁnitely negative, rather than positive, in eliminating the singularities from eqs. (3.321) and (3.322). However, we should discard these solutions since they cause for ψ to diverge at one or both of the poles. In the case where the spinor does not couple to the background gauge ﬁeld (q = 0 or, equivalently,

N = Φb = 0) and the background geometry is that of a sphere (α = ω = 1), the above equations 1 1 simplify as follows. Keeping in mind that |n1/2| ≥ 1/2, we have that yL = zR = 2 (|n1/2| − 2 ) and 1 1 yR = zL = 2 (|n1/2| + 2 ) and so eq. (3.325) becomes

2 " 2# 2 d fσ dfσ f 1 (1 − x ) − 2 (1 + |n1/2|)x − 1 + λ − |n1/2| + fσ = 0 (sphere). (3.328) dx2 dx jn1/2 2

The power series solutions to these equations terminate as long as

λf = (j + |n | + 1/2)2 (sphere) (3.329) jn1/2 1/2

f for some integer j ≥ 0. If we define ` := j + |n1/2| − 1/2, the above expression for λ simplifies to the familiar (` + 1)2, as found in [21]. The degeneracy in this case is 2(` + 1), which accounts for the number of different ways one can choose (j, n1/2) to get the same `. These two formulations are equivalent. In the general case, the condition for termination of the power series solution to eqs. (3.327) (at the same order xj) is that 2 2 f 1 N λ = jσ + yσ + zσ + − (3.330) jn1/2 2 4 where jσ ∈ {0, 1,...} (and jL 6= jR generally).

Mode Sum

The spin-1/2 mode sum for ψ1 is

Sf (ω, N, Φb, t) = 2SfL(ω, N, Φb, t) + 2SfR(ω, N, Φb, t) (3.331)

143 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy where ∞ ∞ tN 2/4 X X h 2i Sfσ(ω, N, Φb, t) = −2 × e exp −t(j + yσ + zσ + 1/2) . (3.332) j=0 n=−∞ Here, σ = L = +1 (σ = R = −1) gives the result for the left-handed (right-handed) spinors. (Note that we have dropped the σ subscript on the dummy indices j, n.) Recalling that n1/2 := n + bσ/2, we can write yσ and zσ as follows:

1 h σ i σ ω fσ yσ = ω n + (N − Φ)−b − b Φb − − = n + (Nfσ − Φfσ)−b − b Φ (3.333) 2 2 2 2 b 1 h σ i σ ω fσ zσ = ω n − (N − Φ)b − b Φb − + = n − (Nfσ − Φfσ)b − b Φ (3.334) 2 2 2 2 b where, in the second expressions, we have introduced

! σ 1 X 1 N := N − σ , Φfσ := Φ − 1 − and Φ := Φfσ = Φ − σ 1 − . (3.335) fσ b b 2 ω fσ b ω b

f −1 (Later on, we will also use Φ0 := (1−ω )/2.) Given these identiﬁcations, we can write the fermionic mode sums in terms of the previously–derived scalar result:

2 2 t[N −(1+Nfσ )]/4 fσ Sfσ(ω, N, Φb, t) = −2 × e Ss(ω, Nfσ, Φb , t) (3.336) where Nfσ := ω(Nfσ − Φfσ) = N − σ. Expanding this out, we ﬁnd (using the notation introduced in fσ eq. (4.76)) that s−1 = −2/ω,

1 1 1 sfσ(ω, N , Φfσ) = − σN + + ∆F ω2 , (3.337) 0 fσ b ω 6 6 fσ " 1 7 N 2 1 (1 − 3σN ) sfσ(ω, N , Φfσ) = − + − ∆F ω2 1 fσ b ω 720 6 72 6 fσ

3 ! # ω (N − σ) X h i 7 1 (2) + ∆ Φfσ G(|Φfσ|) + + ∆[F ] ω4 , (3.338) 6 b b 720 6 fσ b " 1 31 31 N 2 7 N 2 (1 − 5N 2) sfσ(ω, N , Φfσ) = − + − − ∆F ω2 2 fσ b ω 20160 1440 2880 144 60 fσ

ω3N (1 − σN ) X 7 7 N 2 − ∆[Φfσ G(|Φfσ|)] + − 12 b b 2880 1440 b 2 ! 5 (1 − 3σN + N ) (2) ω (N − σ) X − ∆[F ] ω4 + ∆[Φfσ G(|Φfσ|)(1 + 3F fσ)] 12 fσ 60 b b b b ! # 31 ∆[F (2)] ∆[F (3)] + + fσ + fσ ω6 (3.339) 20160 60 30

144 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

(n) fσ where the Ffσ ’s are calculated using Φb , and where we have taken

∆[X] := X − X (3.340) Φb=0

fσ fσ fσ −2 to ensure that ∆[X] → 0 as Φb → 0. For example, ∆Fb = |Φb |(1 − |Φb |) − (1 − ω )/4. (Recall that ω > 1 for the rugby–ball.) It is important to remember that the derivation of these results fσ has assumed |Φb | < 1, which is distinct from the restriction |Φb| < 1 in the scalar case but of little diﬀerence near ω = 1.

Since the 6D the chiral spinor ψ1 is the sum of a left–handed and a right–handed 4D chiral spinor, f its si coefficients are given by f X fσ si = si . (3.341) σ∈{±1} (For massive 6D spinors, we must double this result.) Since we expect some cancelation when considering this sum of coefficients, let’s work out the result. We must be careful, though, since f there is some subtlety in this. For example, the spin–averaged quantity h∆Fb i depends differently f −1 f on Φb for |Φb| > Φ0 := (1 − ω )/2, as compared to when |Φb| < Φ0:

( 2 f 1 X −Φ , |Φb| ≤ Φ h∆F f i := ∆F fσ = b 0 . (3.342) b 2 b f f σ Fb − Φ0 , |Φb| ≥ Φ0

Both of these expressions have useful domains of validity: when considering small fluxes, Φb, in the f presence of some Φ0 6= 0, it is the upper function which is of interest; in the case of the larger fluxes, f or in the sphere case where ω = 1 and Φ0 = 0, it is the lower function which is relevant. We should therefore treat these two situations differently.

f In the case where both |Φb| ≤ Φ0, we ﬁnd that " ! # 1 1 1 X sf (ω, N, Φ ) = + − 2 Φ2 ω2 , (3.343) 0 b ω 3 3 b b " ! 1 7 ωN Φ N 2 1 1 X sf (ω, N, Φ ) = − − + − Φ2 ω2 1 b ω 360 2 3 36 6 b b ! # ω3N X 7 1 X − Φ (1 − 4Φ2) + − Φ2(1 − 2Φ2) ω4 , (3.344) 6 b b 360 6 b b b b " ! 1 31 ωN Φ 31 N 2 7 N 2 X 7 X sf (ω, N, Φ ) = − − + − 1 − 6 Φ2 − Φ2 ω2 2 b ω 10080 16 720 1440 72 b 240 b b b ! ω3N X 7 7 N 2 (1 − 2N 2) X − Φ (1 − 4Φ2) + − − Φ2(1 − 2Φ2) ω4 24 b b 1440 720 24 b b b b ! X 7 Φ2 Φ4 −ω5N Φ − b + b b 240 6 5 b

145 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

! # 31 X 7 Φ2 Φ4 + − Φ2 − b + b ω6 . (3.345) 10080 b 240 12 15 b

f When both |Φb| ≥ Φ0, the small–t coeﬃcients are given in terms of

Φb ˜ f X ρb := sgn(Φb) = , Φb := ω(Φb − ρbΦ0) , ρ := ρb |Φb| b X F˜b := |Φ˜ b|(1 − |Φ˜ b|) , F˜ := F˜b etc. (3.346) b as follows:

1 2 2 ω2 sf (ω, N, Φ ) = − + 2F˜ + 2ω − , (3.347) 0 b ω 3 3 " 1 1 ρN X Φ˜ 2 N 2 F˜(2) sf (ω, N, Φ ) = − + − N Φ˜ F˜ + b − + 1 b ω 45 6 b b 3 3 3 b ! # 1 F˜ N Φ˜ ρN ω4 + − + − ω2 − (3.348) 9 3 3 6 45

" ˜(2) ˜(3) ˜ 2 ˜ ˜ 4 1 1 F F ρN N X Φ Fb Φ sf (ω, N, Φ ) = − + + + − Φ˜ F˜2 + b + b 2 b ω 315 30 15 30 3 b b 2 10 b

1 F˜(2) 1 F˜(2) N X −N 2 + + − − Φ˜ G(|Φ˜ |) 45 6 90 6 6 b b b ! ! # 1 F˜ 1 F˜ ρN N Φ˜ N 2 ω6 −N 2 − ω2 + − − + + ω4 − (3.349) 18 6 90 30 30 30 90 315

(Note that we should take Φ˜ b = 0 when comparing these sets of equations when they overlap at f |Φb| = Φ0.) In the case of an uncharged, massive fermion on a sphere (i.e. when ω = 1, N = Φb = 0), these two sets of expressions both give

4 2 2 sf (1, 0, 0) = −8 , sf (1, 0, 0) = , sf (1, 0, 0) = , sf (1, 0, 0) = (3.350) −1 0 3 1 15 2 63 in agreement with the result in [23].

Spectrum and Mode Sum for the Massless Spin-1 Field

gf This appendix computes the spectrum λjn and the corresponding small-t limit of the mode sum

2 gf X −tλjn X gf i Sgf (t) := e = si t (3.351) j,n i=−1

for a 6D massless spin-1 ﬁeld AM on the rugby ball.

146 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Eigenvalues

The spin–1 equation of motion is

MN g DM FNP = 0 (3.352)

where FMN = ∂M AN − ∂N AM , DM is the covariant derivative,

ρ σ ρ Q ρ Q ρ A ρ σ (DM ) σFNP = ∂M FNP − ΓMN FQP − ΓMP FNQ − i(t ) σAM FNP , (3.353) and the metric is

2 2 2 2 2 2 µ ν ds = r (dθ + α sin θ dϕ ) + ηµν dx dx 2 2 2 2 2 i j − + = r (dθ + α sin θ dϕ ) + δijdx dx + 2dx dx . (3.354) √ In the last line above, we introduce light–cone coordinates x± := (x3 ± x0)/ 2 which are convenient for what follows. (Also, i, j run only over 1, 2.) More formally, we can introduce the (sub–)metrics gαβ and gmn over the light–cone coordinates α, β ∈ {+, −} and extra–dimensional coordinates m, n ∈ {θ, ϕ}, respectively: ! ! 0 1 r2 0 gαβ = , gmn = . (3.355) 1 0 0 α2r2 sin2 θ

The full metric gMN is then composed as follows:

gMN = gmn ⊕ δij ⊕ gαβ . (3.356)

In what follows, we assume that the generator for the background gauge ﬁeld AM is diagonal in

A σ σ the adjoint representation: (t )ρ = q δρ . Given this choice, it is convenient to suppress the group indices in what follows. We also choose the light–cone gauge, in which

A3 − A0 A− := √ = 0 . (3.357) 2

As before, the background gauge ﬁeld is taken to be

N − Φ q A dxM = q A dϕ = − (cos θ − b) + bΦ dϕ , (3.358) M ϕ 2 b where N is an integer. Expanding out the equation of motion for P = −, we ﬁnd that

ij −+ 2 mn P = − : 0 = −δ ∂i∂−Aj − g ∂−A+ − g Dm∂−An i mn = −∂− ∂iA + ∂−A+ + g DmAn . (3.359)

This can be integrated to give the constraint

i mn F−+ = ∂−A+ = −∂iA − g DmAn . (3.360)

147 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

This constraint tells us that A+ is completely determined (up to some initial conditions) once we have solved for Ak and Ap. As we shall see, this constraint also simpliﬁes the equations of motion for Ak and Ap. Evaluating the equations of motion using the above constraint for P = k ∈ {1, 2} and P = p ∈ {θ, ϕ}, respectively, we ﬁnd that

i −+ +− mn P = k : 0 = ∂ Fik + g ∂−F+k + g ∂+F−k + g DmFnk mn = (4 + g DmDn) Ak (3.361) i −+ +− mn P = p : 0 = ∂ Fip + g ∂−F+p + g ∂+F−p + g DmFnp mn mn mn = (4 + g DmDn) Ap + Dp(g DmAn) − g DmDpAn (3.362)

µν i where 4 := η ∂µ∂ν = ∂ ∂i + 2 ∂−∂+. Eqs. (3.361) are exactly the same as the one encountered in the case of a scalar field, so the corresponding spectrum for A1 and A2 will be that of a scalar. Eqs. (3.362), however, are coupled and our task is to decouple them. φ φ Given that the Christoffel symbols mostly vanish, with the exception of Γθφ = Γφθ = cot θ and θ 2 Γφφ = −α sin θ cos θ, we find that, using the ansatz

ik4·x inϕ Am(x, θ, ϕ) = e Θm(θ)e , (3.363)

Eqs. (3.362) become

2 d Θθ dΘθ Kθ + cot θ + λΘθ − = 0 (3.364) dθ2 dθ α2 sin2 θ 2 d Θϕ dΘϕ Kϕ − cot θ + λΘϕ − = 0 (3.365) dθ2 dθ α2 sin2 θ

2 2 where λ := −k4r are the required eigenvalues and where

2 2 Kθ = (n − qAϕ) + α Θθ + 2i(n − qAϕ) cot θ Θϕ (3.366) 2 2 Kϕ = (n − qAϕ) Θϕ − 2i(n − qAϕ)α cos θ sin θ Θθ . (3.367)

In order to algebraically decouple the Km terms, we should first perform a field redefinition such that the differential operators are the same for both equations. (Note that they are not currently, due to the sign difference in the cot θ dΘm/dθ terms.) With this goal in mind, we should (for convenience) switch to x := cos θ, and substitute

ym zm Θm(x) = (1 − x) (1 + x) fm(x) , (3.368)

where ym, zm are to be related in some speciﬁc way. Doing so gives

d2f df Kˆ (1 − x2) θ − 2 [(1 + z + y )x + y − z ] θ + λf − θ = 0 (3.369) dx2 θ θ θ θ dx θ 1 − x2

148 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

d2f df Kˆ (1 − x2) ϕ − 2 [(z + y )x + y − z ] ϕ + λf − ϕ = 0 (3.370) dx2 ϕ ϕ ϕ ϕ dx ϕ 1 − x2 for some Kˆm, to be specified shortly. Equating the coefficients of dfm/dx, we find that the required relations are

yθ = yϕ − 1/2 := y , zθ = zϕ − 1/2 := z . (3.371)

Given these identiﬁcations, we ﬁnd that the Kˆm are given by

2 2 2 Kˆθ = ω (n − qAϕ) + 1 + k(x, y, z) fθ + 2iω (n − qAϕ)x fϕ (3.372) 2 2 Kˆϕ = ω (n − qAϕ) + 1 + k(x, y, z) fϕ − 2i(n − qAϕ)x fθ (3.373) where k(x, y, z) is deﬁned as in eq. (3.281). From here, we can see that the complex choice

F(x) := fθ(x) + iωfϕ(x) (3.374) yields decoupled equations of motion for F (and F ∗):

d2F dF K (1 − x2) − 2 [(1 + y + z)x + y − z] + λ − F F = 0 (3.375) dx2 dx 1 − x2 where

2 2 KF F := Kˆθ + iωKˆϕ = ω (n − qAϕ) + 1 + k(x, y, z) + 2ω(n − qAϕ)x F . (3.376)

Now that we have decoupled the equations of motion, we can eliminate the singular behaviour as 2 x → ±1 by requiring that KF be proportional to (1 − x ). Doing so gives

ω 1 ω 1 y = n + (N − Φ)−b − bΦb + , z = n − (N − Φ)b − bΦb − , (3.377) 2 ω 2 ω yielding the spectrum 12 1 ω2N 2 λgf (F) = j + y + z + − − . (3.378) jn 2 4 4 The spectrum for F ∗ will be slightly diﬀerent, since complex conjugation is equivalent to taking n → −n and q → −q (or N → −N,Φb → −Φb). In this case,

ω 1 ω 1 y∗ = n + (N − Φ)−b − bΦb − , z∗ = n − (N − Φ)b − bΦb + . (3.379) 2 ω 2 ω

As is done previously, we have taken y, y∗, z, and z∗ to be positive-definite, so that the corresponding fields do not diverge near the poles at x = ±1. Before computing the mode sum, it is helpful to introduce a notion of helicity similar to that used for the spin–1/2 field. If we define a quantity υ such that ξ = +1 (ξ = −1) corresponds to the

149 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy case of F (F ∗), then we can eﬃciently write

12 1 ω2N 2 λgfξ(ω, N, Φ ) = j + y + z + − − (3.380) jn b ξ ξ 2 4 4 where

ω ξ ω ξ yξ = n + (N − Φ)−b − bΦb + , zξ = n − (N − Φ)b − bΦb − . (3.381) 2 ω 2 ω

The gauge ﬁeld is then understood to be comprised of two spin states with ξ = 0, and two spin states with each of ξ = ±1, respectively.

Mode Sum

We wish to compute the small–t limit of the sum

gfξ X ξ X X −tλjn Sgf (ω, N, Φb, t) := Sgf (ω, N, Φb, t) = e , (3.382) ξ ξ j, n

Here, the additonal sum is over spin states, ξ ∈ {0, 0, 1, −1}. Given the spectrum derived previously, we see that two of the spectra will be identical to that of a scalar, and two will be slightly different. Similar what was done for the spin–1/2 field, we can make the identifications

ξ N = N, Φgfξ := Φ − (3.383) gfξ b b ω

gfξ gf gf (or Φb := Φb − ξΦ0 where Φ0 = 1/ω) so that

N 2 N 2 λgfξ(ω, N, Φ ) + = λs (ω, N gfξ, Φgfξ) + gfξ , (3.384) jn b 4 jn b 4 where Ngfξ := ω(Ngfξ − Φgfξ) = N + 2ξ. In this case, Sgf can be written largely in terms of the scalar result:

X t(N 2−N 2 )/4 Sgf (ω, N, Φb, t) = 2 × Ss(ω, N, Φb, t) + e gfξ Ss(ω, N, Φb − ξ/ω, t) ξ∈{1,−1}

= 4 × Ss(ω, N, Φb, t) + ∆Sgf (ω, N, Φb, t) . (3.385)

P ξ In the last line above, we introduce the quantity ∆Sgf = ξ(Sgf − Ss), which is a convenient one ξ since the small–t limit of Sgf is very close to that of Ss. (In what follows, we will dispense with the level of detail as was given for the spin–1/2 field and skip to the complete result, summed over spins.) Expanding out the argument of the sum above, we find a result that behaves similarly to the one found for the spin–1/2 field, in that the result will depend on whether the fluxes Φb are either gf greater or smaller in magnitude when compared to Φ0 = 1/ω.

150 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

gf gf gf When both |Φb| ≤ Φ0 , we ﬁnd that the diﬀerences, ∆si , are given by ∆s−1 = 0, " # 1 X ∆sgf (ω, N, Φ ) = −2ω + ω2 |Φ | , (3.386) 0 b ω b b " # 1 ω2 X ω3N X ω4 X ∆sgf (ω, N, Φ ) = N 2 + ωN Φ + |Φ | − Φ |Φ | − |Φ |3 , (3.387) 1 b ω 3 b 2 b b 3 b b b b " 1 ωN 2 1 N 2 X ω3N X ∆sgf (ω, N, Φ ) = − + ω2 − |Φ | − Φ |Φ | 2 b ω 4 15 24 b 4 b b b b # ω4(1 − N 2) X ω5N X ω6 X − |Φ |3 + Φ |Φ |3 + |Φ |5 . (3.388) 6 b 4 b b 10 b b b b

s These expressions serve to cancel all the terms in si which are odd under |Φb| → −|Φb|, for those spin states with ξ = ±1. gf Since the condition |Φb| ≥ Φ0 = 1/ω does not include very much parameter space for small ω

(since we are already implicitly assuming |Φb| ≤ 1), we will not consider this case in detail here.

Spectrum and mode sum for massive gauge ﬁelds

mgf Let us now turn to the computation of the spectrum λjn and the corresponding small-t limit of the mode sum X −tλmgf Smgf (t) := e jn (3.389) j,n for a massive 6D spin–1 field AM on the rugby ball. The simplest field content for a massive gauge field includes, in addition to the metric and the gauge field, the scalar field Φ whose vev breaks the gauge group down to some subgroup H. The relevant part of the action is

Z √ 1 d6x −g − F F MN − (D Φ)† DM Φ − U(Φ) , (3.390) 4 MN M where the potential U(Φ) is assumed to have a minimum for Φ 6= 0. We would like to compute the linearized theory around a certain background solution with Φ 6= 0 (to give a bulk mass to some of the gauge fields) and the metric and gauge field set to an unwarped configuration with 4D Poincaréinvariance. So we substitute

AM → AM + VM and Φ → Φ + η ,

where VM and η are small perturbations and now gMN , AM and Φ represent the given background solution. In order to interpret Φ 6= 0 as a 6D spontaneous symmetry breaking, we require Φ to be constant and to lie at the minimum of U. Then, to solve the background scalar equation,

M DM D Φ = 0, we also demand that Φ does not break the U(1) where the background gauge ﬁeld lies:

151 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy otherwise it would not be possible to set Φ to a constant value, at least in the sphere compactiﬁcation of interest in this paper [40]. Such requirement is equivalent to demanding that the background gauge ﬁeld lives in the Lie algebra of H.

The linearized theory for the perturbations is rather complicated with mixing between VM , η and the metric ﬂuctuations. However, choosing the light-cone gauge deﬁned in the previous subsection, the bilinear action for VM (such that Tr(AM VM ) = 0) and η has a relatively simple form [37], that is

Z √ 1 1 1 d6x −g − D V DM V i − D V DM V m − RmnV V 2 M i 2 M m 2 m n e2 − V aV ib + V a V mb Φ†{T a,T b}Φ − gF˜ mnV × V (3.391) 2 i m m n 1 ∂2U 1 ∂2U ∂2U − (D η)† DM η − η η − η∗ η∗ − η∗ η M 2 ∂Φ2 2 ∂Φ∗2 ∂Φ∂Φ∗ e2 + η†T aΦ − Φ†T aη η†T aΦ − Φ†T aη , 2 where the indices are raised and lowered by the background metric and e is a collective name for the gauge couplings of the generators broken by Φ.

The simplest model of this sort having a non-trivial background ﬂux is a U(1)1 × U(1)2 gauge theory with Φ charged under the U(1)2 only, the background ﬂux embedded in the U(1)1 and the potential having a simple mexican hat form,

2 2 4 U(Φ) = −µ |Φ| + λ4|Φ| , (3.392) with µ2 > 0 (and so tachyonic) and λ a positive constant. Let us ﬁrst consider this simple option. 4 √ 2 The bilinear action for VM and η := (η1 + iη2)/ 2 is

Z √ 1 1 d6x −g − D V 2DM V i2 − D V 2 DM V m2 − e2Φ2 V 2V i2 + V 2 V m2 2 M i 2 M m i m 1 1 1 − RmnV 2 V 2 − D η DM η − D η DM η + m2 η2 − e2Φ2η2 , (3.393) 2 m n 2 M 1 1 2 M 2 2 tach 1 2

2 where we have assumed Φ real and positive without loss of generality. The ﬁelds VM and η2 form together a massive 6D vector ﬁeld with bulk mass m = 2|e|Φ, while η1 is a genuine scalar with bulk mass squared 2µ2.

In the example above the massive gauge ﬁeld is not charged under the background U(1), that is N = 0. A slightly more complicated non-abelian model can be used to illustrate the charged case. Let us consider for example an electroweak-like SU(2) × U(1)b gauge theory, where Φ is in the 2 representation. Like in the Standard Model we trigger SU(2) × U(1) → U(1) through a non 1/2 √ b q T vanishing VEV, Φ = (0, v/ 2), where U(1)q is the analogue of the electromagnetic U(1), and we assume v real and positive without loss of generality. Also we embed the background gauge ﬁeld in the U(1)q so that we can solve the scalar bulk equations for v constant and equal to the point of

152 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy minimum of U. In this case ! √1 (η + iη ) a 2 1 2 a σ η = ,VM = WM + BM b , (3.394) √1 (v + σ + iη ) 2 2 0

a where σ are the Pauli matrices, the ηi and σ are real scalars and WM and BM are the gauge field perturbations corresponding to the SU(2) and U(1) group factors. By using the bilinear action b √ √ ± 1 2 ± in the light-cone gauge, eq. (3.391), we find that W = (W ± iW )/ 2 and η = (η1 ± iη2)/ 2 1 have the same bulk mass, m = 2 v|g2|, where g2 is the gauge coupling of SU(2), and represent altogether a massive gauge field with N = ±1. Apart from this bulk mass term the light-cone-gauge bilinear action for W and η± coincide with that of a massless bulk gauge field and a massless scalar respectively. The bottom line of the examples above is that a massive gauge field leads to the 4D spectrum of a massless gauge field, given in eq. (3.128), plus that of a scalar, which we provided before. It follows that the si coefficient of a massive gauge field are

mgf s mgf s gf s−1 = 5s−1 , s0 = 5s0 + ∆s0 , mgf s gf mgf s gf s1 = 5s1 + ∆s1 , s2 = 5s2 + ∆s2 , (3.395)

s where the si are the corresponding quantities for a 6D scalar, those given in eqs. (4.247)–(4.248), and we used |N| ≤ 1, to ensure the stability (see the appendix on the massless gauge ﬁelds).

153 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

154 Bibliography

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161 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

162 Chapter 4

Accidental SUSY: Enhanced Bulk Supersymmetry from Brane Back-reaction

— C.P. Burgess, L. van Nierop, S. Parameswaran, A. Salvio, M. Williams

JHEP 1302 (2013) 120 [arXiv:hep-th/1210.5405]

4.1 Introduction

If teﬂon theories are those to which lack of experimental support does not stick, then supersymmetry is their poster child. Indeed, supersymmetry continues to play a central role in particle theory — and has done so for more than 3 decades — despite its so-far disappointing prediction: the perpetually imminent discovery of superpartners for all Standard Model particles. Its longevity in the teeth of such disappointment has many reasons, but an important one is its good ultraviolet properties. Supersymmetry is one of the few symmetries (another is scale invariance) that can suppress both scalar masses and vacuum energies when unbroken, and so potentially might be useful for the hierarchy and the cosmological-constant problems. The challenge is to enable this suppression to survive the symmetry breaking required to explain the experimental absence of superpartners. Moreover, supersymmetry arises organically in string theory, which remains our best candidate for physics at the highest energies. These observations suggest the utility of re-thinking the (apparently signature) prediction of Standard-Model superpartners, since it is the absence of evidence for these that so far provides the best evidence for absence of supersymmetry. The key assumption that underlies the prediction of superpartners (and so, more broadly, of the supersymmetric Standard Model, minimal or otherwise)

163 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy is the assumption that supersymmetry is linearly realized. After all, nonlinear realization does not require superpartners, because nonlinearly realized supersymmetry acts on a single-particle state (say, the electron) to give a two-particle state (an electron plus a goldstino) [1, 2] rather than the single-particle state (a selectron) required by linear realization. Nonlinearly realized supersymmetry also arises organically in string theory when supersymmetry is broken by the presence of branes [3]. D-branes often break half of the supersymmetries present in the bulk, and by so doing provide counter-examples [4] to previously conjectured no-go theorems [5] precluding partial supersymmetry breaking. In general, a conﬁguration of branes can break all or only part of the supersymmetries present in the bulk. This observation has spawned a variety of studies of brane-induced partial supersymmetry breaking within both string and brane-world models [6, 7].

Physically, nonlinear realization is appropriate if the symmetry-breaking scale, Ms, is larger than the UV scale, MUV , above which the theory’s UV completion intervenes [8, 9]. In this case symmetry multiplets can be split by more than MUV , and so the low-energy theory need not contain the particle content required to linearly realize the symmetry. For D-branes the UV completion is string theory itself, so the brane spectrum need never linearly realize supersymmetry in the ﬁeld theory limit below the string scale. When supersymmetry breaks on a brane it is often true that the bulk sector is more supersymmetric than the brane sector, since the bulk must pay the price of a (possibly weak) bulk-brane coupling before it learns that supersymmetry is broken. As a result, unlike for the branes, the bulk spectrum has equal numbers of bosons and fermions, whose masses could be split by as little as the Kaluza Klein scale. It therefore has the ﬁeld content to linearly realize supersymmetry, and so can have much milder UV properties than would be expected for the branes.

The gravity of SUSY

All of this suggests a somewhat unorthodox picture of how low-energy supersymmetry might be realized despite the apparent experimental absence of superpartners [10]. If Standard-Model particles were localized on a supersymmetry-breaking brane sitting within a more supersymmetric bulk, then Standard-Model superpartners would be avoided and the low-energy world would have a gravity sector that is much more supersymmetric than is the Standard-Model sector to which accelerators have access. Supersymmetric signals would be much harder to ﬁnd in such a world, and would depend somewhat on the number of degrees of freedom present in the gravity sector. Although each mode is gravitationally coupled, observable energy loss rates into the gravitational sector can be possible (such as in the speciﬁc realizations involving supersymmetric large extra dimensions [11, 10, 12]). In such scenarios the enormous phase space can compensate the small gravitational couplings, just as one obtains for gravitons in ordinary large extra dimensions [13, 14]. Can the good UV properties of supersymmetry still be useful within this kind of picture? A hint that they can comes from the observation that both the hierarchy problem — ‘Why is the weak

164 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy scale so far below the Planck scale?’ — and the cosmological constant problem — ‘Why does the vacuum energy gravitate so weakly?’ — involve gravity in their formulation. Perhaps they might be ameliorated by the same physics if the gravity sector were very supersymmetric. But there is no substitute for testing these ideas with an explicit calculation of the size of loop effects. A well-developed, fairly simple and concrete framework within which to do so is to describe the supersymmetric bulk using the field equations of 6D chiral, gauged supergravity [15], which has long been known to allow (marginally) stable compactifications to 4D on a sphere [16, 17]. All but a single modulus, ϕ, of this supergravity is stabilized by the presence of a Maxwell flux that threads the sphere in a monopole configuration. It is also known how to embed SUSY-breaking branes into this system including their back-reaction onto the extra-dimensional geometry, which (for two branes) deforms from a sphere into a rugby ball1 (with the branes located at the tips) [11] or into something even more distorted [18, 19]. (See also [20] for similar flux-stabilized rugby-ball constructions within a non-supersymmetric context.) In this paper we test the UV properties of this kind of framework by explicitly computing the contribution of bulk loops to the 1PI quantum action (as well as to the vacuum energy), including the supersymmetry-breaking influence of the branes. We do so by adapting to the supersymmetric case a general calculation of bulk loops on rugby-ball geometries [21]. For technical reasons these calculations are only for low-spin bulk fields — i.e. spins zero, half and one — but work is in progress to extend our present results to higher spins. By combining with earlier results for brane loops [22], we can piece together how the complete one-loop result depends on brane and bulk properties. In particular, because our interest is in the low-energy effects of UV modes, we track how short- wavelength bulk loops renormalize the local effective interactions both on the brane and in the bulk. In particular we ask how they depend on the single bulk modulus, ϕ, as well as on the two main brane properties relevant at low energies: their tension, Tb, and the amount of stabilizing Maxwell 2 flux, Φb, that is localized on the branes. Although Φb may seem unfamiliar, its presence is in general required in order for the full brane-bulk system to have low-energy deformations that can satisfy

flux-quantization constraints that relate Φb to Tb [23]. Both Tb and Φb correspond to the coefficients of the first two terms in a generic derivative expansion of the brane action: Z Z 4 √ 2π −φ ? Sb = − d x −γ Tb + 2 Φb e F + ··· , (4.1) W g˜ W

? where F is the 6D Hodge dual of the background 6D Maxwell ﬂux, FMN , (whose gauge coupling is

M N g˜); W denotes the 4D world-surface of the brane and γab = gMN ∂ax ∂bx is the induced metric on W. The ellipses in eq. (4.1) correspond to terms involving at least two derivatives of the bulk ﬁelds.

In general, both Tb and Φb can depend on the various bulk scalar ﬁelds — in particular on the bulk dilaton, φ, that appears in the 6D gravity supermultiplet — and generically this dependence breaks the classical scaling symmetry whose presence is responsible for the bulk geometry’s one

1North American readers should think ‘football’ here, but we use ‘rugby’ to avoid cultural disagreements about the shape of a football. 2Maxwell (or gauge) ﬂux can be localized on a 3-brane in 6 dimensions in the same way that magnetic ﬂux can be localized on a string (or vortex) in 4 dimensions.

165 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy classical modulus, ϕ. Because of this the brane-bulk backreaction combines with ﬂux quantization to ﬁx this last remaining modulus. This is why quantities like the 1PI action can depend on ϕ once branes are present, even at the classical level.

Accidental SUSY

Remarkably, we find for the simplest situation — two identical branes that do not couple at all to the 6D dilaton φ, situated at opposite ends of a rugby-ball geometry [11] — the one-loop vacuum energy precisely vanishes. On closer inspection it does so because all bulk Kaluza-Klein (KK) modes come in degenerate bose-fermi pairs. In retrospect this happens because once the bulk modulus, ϕ, adjusts to relate Φb to Tb as dictated by flux quantization, the boundary conditions at the brane allow a Killing spinor to exist in the bulk. That is, the branes unexpectedly leave unbroken the single ‘accidental’ 4D subset of 6D supersymmetry that is also left unbroken by the bulk [16]. This residual supersymmetry was not noticed earlier because its existence requires Φb to be nonzero. Consequently it is not present for the ‘pure-tension’ branes that are the usual fare of brane-world calculations. This unbroken supersymmetry is accidental in the sense that it arises automatically for two identical branes, provided these are described only up to one-derivative level (i.e. by eq. (4.1)), assuming only that Tb and Φb do not depend on φ. It is in general broken once higher-derivative effective brane-bulk interactions are also included, since these modify the boundary conditions of bulk fields in such a way as to preclude there being a Killing spinor. What is remarkable is how generic this supersymmetry is, since it depends on only to two requirements: (i) that the branes not couple to the bulk dilaton, φ; and (ii) that both branes are identical3 (such as might be enforced by a Z2 symmetry). Because of this accidental supersymmetry, the bulk contribution to the vacuum energy should vanish to all orders in the absence of brane-localized fields and of two-derivative (and higher) interactions on the brane. We explicitly verify that this is true at one-loop order, by generalizing results derived earlier for the non-supersymmetric case [21]. More generally, non-supersymmetric configurations can also be explored for which the localized flux differs on the two branes. We find that integrating out massive bulk supermultiplets at one-loop gives a low-energy vacuum energy contribution that is generically of order

C Λ ' , (4.2) (4πr2)2 where C is an order-unity constant obtained by summing the contributions of all ﬁelds in the problem (and to which bosons and fermions contribute with opposite signs). Generically C is proportional to whatever quantities break supersymmetry, for instance giving C ∝ (∆Φ)2 for branes with unequal

ﬂuxes: ∆Φ = Φ+ − Φ−. In the supersymmetric case of identical branes C = 0. Ultimately, the surprisingly small size of (4.2) has two sources. It can be partially traced to

3Even though supersymmetry breaks when (ii) is not satisfied, it is known [18, 19] that the bulk geometry obtained is still flat in the on-brane directions, provided only that (i) is satisfied.

166 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy the supersymmetry of the bulk geometry, since 6D supersymmetry strongly restricts how the bulk action is renormalized by short-wavelength UV modes. In particular, one-loop renormalizations of the bulk action (and its higher-derivative corrections) vanish once summed over a 6D supermultiplet for supersymmetric rugby balls, independent of what the brane properties are. This generalizes (for low-spin ﬁelds) to rugby balls an earlier result for Ricci-ﬂat geometries [24].

The second important ingredient underlying (4.2) is classical scale invariance, which ensures the bulk action can be written in the form Z 6 p −2φ SB = d x −gˆ e LB (ˆgMN , ∂M φ, ··· ) , (4.3)

where LB does not depend on φ undiﬀerentiated, and the Jordan-frame metric,g ˆMN is related to the φ 2φ Einstein-frame metric in 6D byg ˆMN = e gMN . This guarantees that a factor of e accompanies each loop in the Jordan frame, and so provides the bulk theory’s loop-counting parameter. eφ turns φ 2 out to be very small for large rugby balls because ﬂux stabilization dictates that e ∼ 1/(M6r) , where M6 is of order the 6D Planck scale (more about which below). Consequently each bulk loop contributes a factor proportional to 1/r4, making the one-loop vacuum energy naturally of order the KK scale.

In the 6D Einstein frame these same factors of eφ are also easily understood, since there they arise because Einstein-frame masses, m, are related to Jordan-frame masses, M, by m2 = M 2eφ.

Consequently m ' 1/r even if M ' M6. To obtain m ' M6 would require M M6, for which a proper treatment requires understanding the UV completion above M6, likely a string theory. It is here that bulk supersymmetry is likely to play an even more important role.

Finally, we use the results of the one-loop calculation to estimate the size of higher loops. In particular, we explore the size of two-loop contributions in the supersymmetric case for which the one-loop result vanishes. Here we ﬁnd the most dangerous contributions involve both a bulk and a brane loop, and in some circumstances these can contribute Λ ∝ µ2m2/(4π)4 ∝ µ2/(16π2r)2, where µ is a brane mass and m2 ' M 2eφ is the bulk mass encountered above. When present such contributions dominate, and we explore when this obtains.

The rest of this paper is organized as follows. First, §4.2 describes the bulk supergravity of interest, its rugby ball solutions and their supersymmetry properties. Then §4.3 brieﬂy recaps the results of ref. [21] for the one-loop 1PI action as computed for spins zero, half and one propagating within the rugby-ball geometry, with a focus on how short-wavelength modes renormalize the bulk and brane actions. Next, §4.4 assembles these renormalization results for individual particles into a result for several 6D supermultiplets. Then §4.5 computes how to get from the 1PI action to the 4D vacuum energy, tracking how the bulk back-reacts to the loop-changed brane energy densities, contributing an amount comparable to the direct loop-generated changes themselves. Finally, a brief summary of our conclusions, and the estimate of higher-loop bulk-brane eﬀects can be found in §4.6.

167 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

4.2 Bulk ﬁeld theory and background solution

We begin by summarizing the ﬁeld content and dynamics of the bulk ﬁeld theory of interest: six- dimensional gauged, chiral supergravity [15, 25, 16] coupled to a number of 6D gauge- and hyper- supermultiplets.

4.2.1 6D gauged, chiral supergravity

The ﬁeld content of the supergravity sector of the theory consists of the minimal supergravity multiplet plus a single chiral Kalb-Ramond tensor multiplet; that is, a metric (gMN ), antisymmetric

Kalb-Ramond field (BMN ), dilaton (φ), gravitino (ψM ) and dilatino (χ). The theory has a lagrangian formulation4 because the Kalb-Ramond field has both self-dual and anti-self-dual parts (one comes from the gravity multiplet and the other from the tensor multiplet) and this is the purpose of including the single chiral tensor multiplet. The supergravity is chiral because the fermions are all complex 6D Weyl spinors – satisfying γ7 ψM = ψM and γ7 χ = −χ. This gravity multiplet can also couple to matter supermultiplets, of which we consider two types: a a a gauge multiplets – containing a gauge potential (AM ) and a chiral gaugino (γ7 λ = λ ); or hyper- I I I multiplets — comprising two complex scalars (Φ ) and their chiral hyperini (γ7 Ψ = −Ψ ). 6D supersymmetry requires the scalars within the hypermultiplets to take values in a quaternionic manifold, and precludes them from appearing in the gauge kinetic terms or in the kinetic term for the dilaton field φ [26].

The supergravity is called ‘gauged’ because the 6D supersymmetry algebra has an abelian U(1)R symmetry that does not commute with supersymmetry and is gauged by one of the gauge multiplets. a The fermion ﬁelds ψM , χ and λ all transform under the U(1)R gauge symmetry, as do the hyperscalars, ΦI (but not the hyperini, ΨI ). For instance, the gravitino covariant derivative is

1 D ψ = ∂ − ω AB Γ − iA ψ − ΓL ψ , (4.4) M N M 4 M AB M N MN L

AB L 1 where ωM denotes the spin connection, ΓMN the metric’s Christoﬀel symbol, ΓAB := 2 [ΓA, ΓB ] is the commutator of two 6D Dirac matrices and the gauge ﬁeld AM gauges the 6D U(1)R symmetry.

Anomaly cancellation

Because the fermions are chiral there are gauge and gravitational anomalies, which must be cancelled using a version of Green-Schwarz anomaly cancellation [27, 28]. In 6D this is not possible for generic anomalous theories, but under some circumstances can be done. In particular, Green- Schwarz anomaly cancellation requires: a Kalb-Ramond field which shifts under the anomalous gauge symmetry (and so whose field strength contains a Chern-Simons term for this symmetry), and some restrictions on the gauge groups and number of chiral matter fields present [29]. In particular, the

4In general 6D supergravities need not [25], when self-dual or anti-self dual Kalb-Ramond ﬁelds are present.

168 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

number of gauge- and hyper-multiplets, nG and nH , must satisfy [29]

nH = nG + 244 . (4.5)

We see from this that anomaly-freedom ensures there are literally hundreds of matter multiplets. For the theory of interest here the Kalb-Ramond ﬁeld required by anomaly cancellation is simply

BMN of the supergravity multiplet, whose ﬁeld strength, GMNP , is required by supersymmetry to contain Chern-Simons contributions. For instance, at lowest order

κ GMNP = ∂M BNP + 2 FMN AP + (cyclic permutations) , (4.6) gR

where FMN = ∂M AN − ∂N AM is the abelian gauge ﬁeld strength for the U(1)R gauge symmetry, and gR is its coupling constant. More generally, at higher orders anomaly cancellation also requires

GMNP to contain gravitational Chern-Simons terms.

Bulk action and ﬁeld equations

The bosonic part of the classical 6D supergravity action is:5

L 1 1 √ B = e−2φ − gˆMN Rˆ + ∂ φ ∂ φ − G GMˆ Nˆ Pˆ −gˆ 2κ2 MN M N 12 MNP 2 1 a Mˆ Nˆ 1 MN I J 2gR − 2 FMN Fa − GIJ (Φ)g ˆ DM Φ DN Φ − 4 U(Φ) , (4.7) 4ga 2 κ where carets indicate curvatures, determinants or raised indices that are computed using the metric, gˆMN . Here the sum over gauge ﬁelds includes, in particular, the abelian factor that gauges the U(1)R symmetry — whose gauge coupling, gR, appears in the scalar potential on the right-hand side. GIJ (Φ) is the metric of the quaternionic coset space, M = G/H, in which the ΦI take their values. φ Eq. (4.7) can be rewritten in the 6D Einstein frame by rescalingg ˆMN = e gMN , to give

L 1 e−2φ √ B = − R + ∂ φ ∂M φ − G GMNP −g 2κ2 M 12 MNP −φ 2 e a MN 1 MN I J 2gR φ − 2 FMN Fa − GIJ (Φ) g DM Φ DN Φ − 4 e U(Φ) . (4.8) 4ga 2 κ

The potential, U(Φ), is nontrivial and depends on the gauge group and other details but in the cases for which it is known [32] it is extremized for ΦI = 0, near which

κ2 U = 1 + G (0)ΦI ΦJ + .... (4.9) 2 IJ

5Our metric is ‘mostly plus’ and we follow Weinberg’s curvature conventions [30], which differ from those of MTW [31] only by an overall sign in the definition of the Riemann tensor. To keep the same notation as [21] we adopt here a convention for gauge fields that differs in normalization by a factor of the relevant gauge coupling, ga, compared with our earlier papers on 6D supergravity.

169 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

In particular ΦI = 0 is consistent with the full equations of motion. The presence of e−2φ as an overall prefactor in eq. (4.7) reveals e2φ as the loop-counting parameter, and this action neglects higher-order corrections that are suppressed relative to the ones shown by powers of e2φ and/or higher derivatives. Among these are interactions that are related by supersymmetry to anomaly canceling terms, such as one-loop corrections to the gauge kinetic √ √ a Mˆ Nˆ φ a MN function, −gˆ FMN Fa = −g e FMN Fa [33, 34]. The equations of motion for the bosonic ﬁelds which follow from the action, eq. (4.8), after using ΦI = 0 are:

2 2 2 κ −2φ MNP κ −φ a MN 2gR φ φ + e GMNP G + 2 e FMN Fa − 2 e = 0 6 4ga κ

−2φ MNP −φ MN DM e G = 0 ,DM e Fa = 0 (4.10)

−φ MN −2φ MNP DM e F + κ e G FMP = 0

2 2 −φ κ −2φ PQ κ e a P 1 RMN + ∂M φ ∂N φ + e GMPQ GN + 2 FMP FaN + (φ) gMN = 0 , 2 ga 2

where the second-last equation is for the U(1)R gauge potential whose Chern-Simons term appears in the ﬁeld strength GMNP , as in eq. (4.6).

Massive supermultiplets

Both the gauge- and hyper- supermultiplets described above furnish representations of 6D supersymmetry for massless particles. By contrast, the particle content for a massive 6D matter multiplet consists of a massive gauge particle, a massive Dirac fermion and three scalars - a total of 8 bosonic and 8 fermionic states. Since this is also the combined field content of a gauge- plus a hyper-multiplet, one expects to be able to form a massive multiplet by having the gauge boson from a gauge multiplet ‘eat’ one of the scalars of a hypermultiplet through the Higgs mechanism. For ungauged supergravity, with vanishing scalar potential, this is indeed what happens in general as the hyperscalars can take arbitrary constant values in the vacuum. This picture is also consistent with the observation that massive states should not alter the anomaly cancellation conditions since the condition, eq. (4.5), is not modified when equal numbers of gauge and hypermultiplets are added to the system. If w denotes the v.e.v. of the field that breaks the relevant gauge symmetry, we expect the common mass of all elements of the massive supermultiplet to be of order m2 ∼ eφw2 (in the 6D Einstein frame6). This dependence of m2 on φ can be seen in several ways: for the gauge fields it arises because of the presence of e−φ in the gauge kinetic term. Alternatively, the proportionality m2 ∝ eφ can also be seen from the overall factor of eφ in the hypermultiplet scalar potential, 2 φ φ U = 2 gRe U(Φ). These factors of e play an important role in the overall size of the effects found later from loops of massive fields.

6A frame-independent way to write this is κm2 ∼ eφκw2.

170 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

4.2.2 Rugby-ball compactiﬁcations

The simplest compactiﬁed solutions [16, 11] to the ﬁeld equations (4.10) are found using the Freund- Rubin ansatz [35] in which: φ = constant and

! ! gµν (x) 0 0 0 gMN = and FMN = . (4.11) 0 gmn(y) 0 f mn(y)

Here gµν is a maximally-symmetric Lorentzian metric — i.e. de Sitter, anti-de Sitter or ﬂat space

— while gmn and mn are the metric and volume form on the two-sphere, S2. The Bianchi identity requires the quantity f appearing in the background gauge field — which could be any one of the gauge fields present in the theory — is a constant. All other fields vanish. As is easily verified, the above ansatz solves the field equations provided that the following three conditions are satisfied: Rµν = 0,

2 2 2 2 2 1 mn 2f 8 gR 2φ κ −φ p f κ −φ 2 FmnF = 2 = 4 e and Rmn = − 2 e Fmp Fn = − 2 e gmn , (4.12) gB gB κ gB gB

7 where gB is the gauge coupling, ga, for the speciﬁc gauge generator whose potential is nonzero in the background. This in general diﬀers from the gauge coupling, gR, of the abelian R-symmetry,

U(1)R (that enters through its appearance in the scalar potential). These imply the four dimensional spacetime is ﬂat, plus the two conditions

2 φ φ κ gB 2 gB gR e e = 2 2 and f = ± 2 = ± 2 . (4.13) 4gR r 2 gR r κ

Notice that these expressions determine the values of f and φ in terms of the size of the extra dimensions, implying in particular that eφ becomes very small when r is very large.

The gauge potential, Am, that gives rise to the ﬁeld strength Fmn is the potential of a magnetic monopole. As such, it is subject to the condition that the total magnetic ﬂux through the sphere is quantized:8 Z F = 4πr2f = 2πN (sphere with no branes) , (4.14) S2 with N = 0, ±1, ... This requires the normalization constant, f, to satisfy:

N f = (sphere with no branes) (4.15) 2 r2

where r is the radius of the sphere. Comparing eqs. (4.13) and eq. (4.15) then implies N = ± gB /g, which is only possible if gB is an integer multiple of g.

7 2 The coupling gB as defined here is φ-independent, and so is related to the couplingg ˜(φ) used in [21] byg ˜ (φ) = 2 φ gB e . 8This expression assumes that all charged matter fields couple to the background gauge potential with strength gB , and so differs from the corresponding one in [21] which allows the coupling strength to be qgB . Although gB can be defined so that q = 1 for any particular matter field, this cannot be done for more than one field at a time. See [21] for the expressions with general q.

171 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Of particular interest in what follows is the special case where the bulk background ﬂux lies in the U(1)R direction, in which case

gB = gR and so N = ±1 (Salam-Sezgin solution) . (4.16)

This solution turns out to preserve precisely one 4D supersymmetry [16], and in later sections we seek to identify the size of supersymmetry breaking effects due to the back-reaction of the source branes. Notice, however, that the value of r itself is not determined by the field equations, indicating the existence of a (classical) flat direction. Because of eq. (4.13) this flat direction can be parameterized either by r or φ, and its existence is a general consequence of the following rigid classical scaling symmetry of the supergravity field equations:

−c φ → φ + c and gMN → e gMN , (4.17)

(and sog ˆMN is fixed). Since this is only a symmetry of the classical bulk equations, the flat direction can be lifted, even classically, once the bulk is coupled to brane sources that break this symmetry. Alternatively, it is also generically lifted by quantum effects, with `-loop corrections to the action 2(`−1)φ proportional to e when expressed in terms of the scale-invariant metric,g ˆMN .

Brane sources

The solutions as outlined so far describe an extra-dimensional 2-sphere supported by ﬂux, with metric ds2 = r2 dθ2 + sin2 θ dϕ2 , (4.18) without the need for brane sources [16]. However brane sources can be introduced into this supergravity solution [11] simply by allowing the angular coordinate to be periodic with period ϕ ' ϕ + 2πα with α not equal to unity. Geometrically, this corresponds to removing a wedge from the sphere along two lines of longitude and identifying points on opposite sides of the wedge [20]. This introduces a conical singularity at both the north and south poles, with defect angle δ = 2π(1 − α), a geometry called the rugby ball. Physically, this geometry describes the gravitational ﬁeld of two identical brane sources, one situated at each of the two poles, with Einstein’s equations relating the defect angle to the properties of the branes. Concretely, take the action of the brane to be9 Z 4 √ Sb = − d x −γ Lb

Ab mn with Lb = Tb − 2 Fmn + ··· , (4.19) 2gB

9 ? A more covariant way of writing the term linear in Fmn is as the integral of the 6D Hodge dual, F , over the 4-dimensional brane world-sheet [23].

172 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

M N with γab := gMN ∂ax ∂bx being the brane’s induced metric and ellipses denoting terms involving two or more derivatives. In general the coefficients Tb and Ab and so on could depend on any of the 6D scalars, φ or ΦI . The back-reaction of such a brane onto the extra-dimensional geometry is governed by the near- brane boundary condition the brane induces on all bulk fields. This boundary condition relates the radial derivative of the field to the brane action, for instance implying for the hyperscalars [36]

2 h J i κ δSb lim GIJ (Φ) ρ ∂ρΦ = , (4.20) ρ→0 2π δΦI where ρ denotes proper distance from the brane. In general, a bulk ﬁeld having a nonzero derivative near a brane diverges at the brane positions, leading to curvature singularity there. But it turns out that if the coeﬃcients Tb, Ab etc. are all independent of the bulk scalars, then the singularity is fairly mild: a conical defect such as found in the above rugby-ball geometries. In this case the near-brane boundary conditions degenerate to a formula [37, 23] for the defect angle at the brane’s position: 2 δb = κ Lb . (4.21)

In the special case of a rugby-ball solution, since the defect angle is the same at both poles the same 10 must be true of Lb for the corresponding branes at each pole, with

2 2π(1 − α) = κ L± = 8πG6L± , (4.22) where b = ± labels the two poles. The presence of the brane sources complicates the flux quantization condition in two important ways. The first complication arises because the resulting defect angle changes the volume of the sphere, which appears in the flux-quantization condition when integrating over the bulk magnetic field, Z F = 4πα r2f . (4.23) S2(α) The second complication arises because the branes themselves can carry a localized flux, given by

φ 2πΦb = Ab e . (4.24)

This can be seen by asking how Ab changes the boundary conditions for the bulk gauge ﬁeld, and tracking these through the ﬂux-quantization condition, which becomes [23]

Z 2πN = 2π Φ + F = 2π Φ + 4πα r2f , (4.25) S2(α) P where Φ := b Φb. Solving this for f and comparing with the bulk ﬁeld equation, eq. (4.13), we ﬁnd that eq. (4.15)

10See [18, 19, 38] for solutions with conical singularities that can diﬀer at the two poles.

173 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy generalizes to N gB f = 2 = ± 2 , (4.26) 2 r 2 gR r where N := ω(N − Φ) = ±gB /gR and we follow [21] by deﬁning (for later convenience)

1 ω := . (4.27) α

−φ P φ Notice that if Ab ∝ e then 2πΦ = b Abe is independent of φ, and so also independent of the

ﬂat direction guaranteed by eq. (4.17) (which can be parameterized by φ). However, if Ab has any other φ-dependence (and in particular if it is φ-independent) then Φ varies with φ, and eq. (4.25) lifts the degeneracy of the ﬂat direction. It then should be regarded as an equation to be solved for Φ (and so also for φ), to give α g Φ = N − α N = N ∓ B . (4.28) gR

For instance, if gB = gR and N = ±1 (as in the Salam-Sezgin solution), then using α = 1 − δ/2π implies δ Φ = ±(1 − α) = ± (if g = g and N = ±1) . (4.29) 2π B R This can be regarded as a dynamical adjustment of Φ to track the defect angle (and so also the brane P tensions) so long as Φ depends on the ﬂat direction, φ (i.e. so long as b Ab is not proportional to e−φ).

Notice that because eq. (4.21) gives the defect angle as a function of tension and brane flux, once the brane-localized flux adjusts to track the brane tension the defect angle is completely determined by the brane tensions alone. However the presence of the flux acts to change the size of the defect angle produced by a particular tension, T , relative to its naive value. That is, for a ‘pure tension’ brane — i.e. in the absence of higher-derivative brane interactions (including brane-localized flux) — each brane’s contribution to the defect angle would be controlled by its tension

2 2π(1 − α) = κ T = 8πG6T (no brane-localized ﬂux) . (4.30)

But in the presence of brane-localizing ﬂux the brane lagrangian instead evaluates to

−φ Ab f 2πΦb e f 4π gR Φb Lb = Tb − 2 + ··· = Tb − 2 + ··· = Tb ∓ 2 + ··· , (4.31) gB gB gB κ where the ellipses represent terms like R that involve at least two derivatives (and so are down by 2 at least 1/r ). We see that for the rugby ball with equal ﬂuxes and tensions (T+ = T− = T and 1 Φ+ = Φ− = 2 Φ) combining this with eq. (4.28) gives the relation between defect angle and tension as gRN δ = 2π(1 − α) = 4πG6T + π 1 ∓ . (4.32) gB

For instance, in the Salam-Sezgin case — where gB = gR and N = ±1 — the presence of Φ makes the

174 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy defect angle δ = 2π(1 − α) half as large as it would have been – i.e. eq. (4.30) – if Φ had vanished:

κ2T δ = 2π(1 − α) = = 4π G T (with brane-localized ﬂux) . (4.33) 2 6

Having a single-derivative and no-derivative term compete in this way might raise concerns for the validity of the derivative expansion for the brane action. However the brane-localized flux can be larger than the other terms in a derivative expansion for two reasons: its dependence on the zero mode and its participation in the flux-quantization condition. On one hand the dependence on the (otherwise undetermined) zero mode makes its coefficient free to adjust to satisfy flux quantization. And on the other hand, flux quantization makes it compete with the bulk flux and so drives its coefficient out to a volume-enhanced value. The same is not true for other terms in the derivative expansion of the brane action.

4.2.3 Supersymmetry of the solutions

It was famously shown by Salam and Sezgin [16] that the spherical solution (no defect angle) using

N = ±1 unit of U(1)R flux preserves a single 4-dimensional supersymmetry. We here reproduce their argument to identify how back-reaction in the presence of branes changes this conclusion.11 We find that pure tension branes always break all of the bulk supersymmetry, but supersymmetry can be preserved if both tension and brane-localized flux are present. In particular, we find that the condition for unbroken supersymmetry is precisely the same condition as is imposed by flux quantization, as found earlier (eq. (4.29)). A background configuration does not break supersymmetry if the supersymmetry transformations all vanish when evaluated at the background solution. Since the variations of bosonic fields all vanish trivially (because all fermions vanish in the background), it suffices only to evaluate the fermionic variations. For the 6D supergravity of interest, with background U(1)R flux and vanishing hyperscalars, this requires all of √ 1 i 2 g √ −φ/2 MN R φ/2 δλ = e FMN Γ − 2 e 2 2 gR κ 1 1 δχ = √ (∂ φ)ΓM + e−φG ΓMNP κ 2 M 12 MNP √ 2 1 δψ = D + e−φ G ΓP QRΓ (4.34) M κ M 24 P QR M to vanish. First consider the variation of the dilatino, χ. Since 4D maximal symmetry and 2 internal dimensions require vanishing GMNP , the condition δχ = 0 implies the dilaton must be a constant:

∂M φ = 0. Since back-reaction relates δSb/δφ to the near-brane limit of ρ ∂ρφ, the requirement that φ be a constant implies all brane actions must be stationary with respect to dilaton variations when

11See ref. [39] for a precursor to the argument we present here.

175 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy evaluated at the background. A suﬃcient condition for this to be so is to have all of the coeﬃcient functions, Tb, Ab etc., be completely independent of the dilaton. Next, the condition δλ = 0 can be written as √ √ φ −φ/2 −φ/2 1 MN igRe 2 e 1 mn 0 = 2 e FMN Γ − 2 = 2 ± mnΓ − i , (4.35) 4gR κ 4gRr 2

2 φ 2 2 2 when evaluated with Fmn = ±mn/(2r ) and e = κ /(4gRr ). Using the following representation of 6D Gamma matrices:12 ! ! ! 0 γµ 0 γ 0 −i1I Γµ = , Γ4 = 5 , Γ5 = 4 (4.36) µ γ 0 γ5 0 i1I4 0

µ 0 1 2 3 where γ are the usual 4D Dirac matrices and γ5 = −iγ γ γ γ , we have

! mn γ5 0 mnΓ = 2i , (4.37) 0 −γ5 and so the condition δλ = 0 implies the 6D Weyl spinor satisﬁes

! ε = 4± , (4.38) 0 where ε4± is a 4D spinor that satisﬁes the 4D Weyl condition γ5ε4± = ±ε4±, with the sign correlated 2 with that of N = 2r f = ±gB /gR = ±1.

Finally the condition δψM = 0 boils down to the existence of a covariantly constant (Killing) spinor: i D = ∂ − Γ ωAB − iA = 0 , (4.39) M M 4 AB M M where the covariant derivative of depends on AM because the corresponding symmetry is an R symmetry (and so does not commute with supersymmetry). The integrability condition for such

1 PQ a spinor states [DM ,DN ] = −i 2 RMNPQΓ + FMN = 0, which for the rugby-ball background becomes i Γ − N = 0 . (4.40) 2r2 mn mn

This is automatically satisfied by eq. (4.38) together with γ5ε4± = N ε = ±ε4±. To find the Killing spinor we take two coordinate patches, centered about the North and South poles (labeled by b = ±), and use the frame fields

! 1 cos ϕ −b sin ϕ e m = α sin θ , e µ = δµ , e m = 0 , (4.41) a r cos ϕ α α α b sin ϕ α sin θ

12In what follows, we follow the conventions in Appendix C of [21].

176 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

AB to compute the following non-zero components for the spin connection ωM :

45 54 ωϕ = α cos θ − b = −ωϕ . (4.42)

The background gauge potential satisfying the near-brane boundary conditions dictated by back- reaction [23] is similarly given by

N α A = − (cos θ − b) + b Φ , (4.43) ϕ 2 b where N = ±1. The non-trivial component of the covariant derivative becomes

" ! # i γ5 0 iN α Dϕ = ∂ϕ − (α cos θ − b) + (cos θ − b) − ib Φb = 0 , (4.44) 2 0 −γ5 2 and so ε4± must satisfy 1 ∂ + ib ± (1 − α) − Φ ε = 0 . (4.45) ϕ 2 b 4± Equivalently, h i ∂ϕ − i(Φ+ − Φ−) ε4± = 0 and [±(1 − α) − Φ] ε4± = 0 , (4.46) where Φ := Φ+ + Φ−, and so solutions exist (and are constants) when the branes satisfy

Φ 1 δ Φ = Φ = = ± (1 − α) = ± . (4.47) + − 2 2 4π

We see that a single 4D supersymmetry survives when the branes are identical — i.e. have equal tensions13 and fluxes — and with localized fluxes related to their tensions by eq. (4.47). In particular, when Φb = 0 then any nonzero brane tension — α 6= 1 — breaks supersymmetry. Finally, we remark on the remarkable equivalence of the flux-quantization condition, eq. (4.29), and the supersymmetry condition, eq. (4.47), on Φ. This states that the value to which Φ is dynamically driven along the classical flat direction by flux quantization is precisely the one supersymmetric point on this flat direction. In particular, when Tb and Ab are φ-independent (which ensures com- patibility with vanishing gradients for φ) this flat direction stabilizes at the supersymmetric position for any choice (consistent with the rugby-ball condition L+ = L−) for the constant coefficients Tb and Ab.

Control of approximations

We close this section with a brief summary of the domain of validity of the previous discussions, which has two important components: weak coupling and slowly-varying fields. First, since we work within the semi-classical approximation, slowly varying fields are required to trust the effective 6D supergravity approximation for whatever theory (presumably a string theory)

13Non-rugby-ball solutions with diﬀering tensions also have nontrivial dilaton proﬁles [18, 19, 38], and so are excluded by the condition ∂M φ = 0.

177 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy provides its ultraviolet completion. In practice, without knowing the details of this UV completion, we demand ﬁelds vary slowly relative to the length scale set by κ. This is the analogue of the α0 expansion in string theory, and in the Jordan frame it requiresr ˆ2 κ wherer ˆ is the size of the extra dimensions as measured with the Jordan-frame metric,g ˆMN . In terms of the Einstein-frame radius, r, used elsewhere in the text, this condition instead is t := r2eφ/κ 1. If the classical rugby-ball solutions are to fall within this regime, eq. (4.13) shows that we must require

2 κ 4gR . (4.48)

Second, since (as remarked earlier) each bulk loop in the 6D supergravity of interest comes accompanied by a factor of e2φ, the semiclassical approximation additionally requires weak coupling: eφ 1. (This is the analogue of the condition of small string coupling for string compactiﬁcations.) This implies a semiclassical understanding of the ﬂat direction labeled by r (or φ) is possible within the regime κ eφ 1 . (4.49) r2 Next, once brane sources are included we must also demand them not to curve excessively the background geometry, and for branes with tension T this requires

2 κ Tb 1 . (4.50)

For rugby-ball geometries this ensures the defect angle satisfies δ 2π. Finally, the semiclassical approximation also restricts the properties of particles that can circulate within loops, even if these do not appear among the background fields. Most notably their masses cannot be too large if quantum effects associated with gravity are to remain under control [40]. For particles of mass m2 = M 2eφ this requires

2 2 2 2 φ 2 2 φ gR m = gR M e κm = κM e 1 . (4.51)

4.3 Mode sums and renormalization

This section summarizes the results of the companion paper [21], so readers familiar with [21] should feel free to skip to §4. Our goal is to compute the UV-sensitive part of the 1PI quantum action, Γ = S + Σ, due to bulk loops. Our starting point is the following expression

Z 1 − + X + m2 iΣ = −i d4x V = − (−)F Tr Log 6 , (4.52) 1−loop 2 µ2 for the one-loop action arising from a loop of low-spin 6D ﬁelds moving in the background rugby-ball geometry. The calculation is quite general, assuming only that the ﬁeld has statistics (−)F = ± with upper (lower) sign applying for bosons (fermions), and its kinetic operator (or, for fermions, its 2 square) can be written in the form −+X +m . We also assume the six-dimensional d’Alembertian

178 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy splits into the sum of four- and two-dimensional pieces: 6 = 4 + 2; X is some local quantity (perhaps a curvature or background ﬂux); and m is a 6D mass. This is suﬃciently general to include the spin-zero, -half and -one particles of interest in later sections.

One-loop mode sums

Specializing to rugby-ball backgrounds and Wick rotating to Euclidean signature, we have

Z d 2 2 2 ! 1 X d kE kE + m + mjn V = (−)F µ4−d ln 1−loop 2 (2π)d µ2 jn 4−d ∞ µ Z dt 2 = (−)F +1 e−t(mr) S(t) , (4.53) 2 d/2 1+d/2 2(4πr ) 0 t

2 2 where mjn = λjn/r denote the eigenvalues of −2 + X in the compactiﬁed space and d = 4 − 2 ε with regularization parameter, ε, taken to zero after all divergences in this limit are renormalized. The function S(t) is deﬁned by

F X S(t) := (−) exp [−tλjn] jn √ s−1 s−1/2 3/2 2 5/2 = + √ + s0 + s1/2 t + s1 t + s3/2t + s2 t + O(t ) , (4.54) t t and its small-t limit is of interest because this controls the UV divergences appearing in V1−loop:

C 1 µ C 1 V = + ln + Vˆ = + V , (4.55) 1−loop (4πr2)2 4 − d m f (4πr2)2 4 − d f

2 2 where Vˆf is ﬁnite and µ-independent when d → 4 and Vf := Vˆf + C ln(µ/m)/(4πr ) . The constant

C is given in terms of the si by

s s C := −1 (mr)6 − 0 (mr)4 + s (mr)2 − s . (4.56) 6 2 1 2

The coefficients si are functions of the rugby ball’s defect angle, δ = 2π(1 − α), and the background flux quantum, N, and are calculated explicitly in [21] for loops of 6D spin-zero, -half and -one bulk particles. These ultraviolet divergences also track the dominant dependence on m in the limit that m 1/r, since both UV divergences and large masses involve the short-wavelength part of a loop that can be captured as the renormalization of some local effective interaction.

Renormalization

What is perhaps unusual about the renormalizations required to absorb the UV divergences (and large-m limit) of V1−loop is that they are not done using the couplings of eﬀective interactions in the 4D theory. Because the wavelengths of interest are much shorter than the extra-dimensional

179 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy size, divergences are instead absorbed into counter-terms in both the 6D bulk and 4D brane actions.

Ref. [21] shows how to use the dependence of the si’s on α, N and r to disentangle which bulk and brane interactions absorb the divergences found in eq. (4.55), which for completeness we now brieﬂy summarize.

Bulk counterterms

The relevant bulk counterterms are identiﬁed by writing the most general derivative expansion of both the bulk lagrangian that is nonzero when evaluated at the rugby ball background:

2 LB 1 H κζAR MN ζR 2 3 √ = −U − R − 1 + R F F − R − ζ 3 R + ··· , (4.57) −g 2κ2 4 2 MN κ R

2 4 φ −φ 2 where U = (2gR/κ ) U e + δU is the bulk potential, H = e /gB + δH is the background gauge 2 3 coupling, and R (or R ) are as deﬁned in [21]: a linear combination of the most general quadratic 2 2 4 3 3 6 (cubic) gravitational terms, which together evaluate to R = Rsph = 4/r (or R = Rsph = −8/r ) on the rugby-ball background. In principle, all of the coeﬃcients in eq. (4.57) can depend on φ, but because e2φ acts as the loop-counting parameter this dependence is dictated as a series in e2φ whose order is dictated by the number of loops being computed. Keeping in mind that (4.57) is written in the Einstein frame, and the powers of eφ already present in the classical Einstein-frame action, eq. (4.8), this leads us to expect that at one loop

3φ −2 2φ φ δU ∝ e , δκ ∝ e , δH, ζR2 ∝ e , (4.58)

φ while the leading term in ζAR and ζR3 is e -independent, and so on. In practice, in the Einstein frame this φ dependence arises through the mass of the particle circulating in the loop, since a particle with a φ-independent Jordan-frame mass M has Einstein- frame mass m = Meφ/2. So the one-loop φ-dependence required by loop counting in the Einstein frame agrees with the m dependence required by dimensional analysis. For instance, in dimensional regularization one-loop corrections to U are dimensionally of order δU ∝ m6 = M 6e3φ, and this agrees with the power of eφ required by loop counting. Similarly, δκ−2 ∝ m4 = M 4e2φ and δH ∝ m2 = M 2eφ, and so on. A crucial feature of bulk counterterms is that none of the parameters like δU, δH etc. can depend on brane properties like α or Φb [21]. This is most easily seen if they are computed using Gilkey-de Witt heat-kernel techniques [41, 42] – since this calculation is explicitly boundary-condition independent (for bulk counterterms). Physically, it is because these counterterms capture the eﬀects of very short-wavelength modes, which don’t extend far enough through the extra dimensions to ‘know’ about conditions imposed at the boundaries. Ref. [21] provides a calculation of what heat kernel techniques give for generic bulk counterterms when specialized to a rugby ball geometry, and the specialization to 6D supergravity is summarized in Appendix 4.A. This means that the renormalized lagrangian evaluated on a rugby-ball background takes the

180 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy form

2 Z 1 f H κζ 4ζ 2 8ζ 3 V = − d2x L = 4πα r2 U − + 1 − AR + R − R + ··· B B κ2r2 2 r2 κ r4 r6 2 1 N H κζ 4ζ 2 8ζ 3 = 4πα r2 U − + 1 − AR + R − R + ··· , (4.59) κ2r2 8 r4 r2 κ r4 r6 where the only dependence on α arises from the overall volume integration. With this in mind it is useful to split up the quantities si in the following way:

sph si(α, N , Φb) = α si (N ) + δsi(α, N , Φb) , (4.60)

sph where si is the α-independent contribution renormalized by bulk counterterms, and the pre-factor 2 sph of α corresponds to the rugby-ball volume, 4πα r , appearing in eq. (4.59). Because si doesn’t depend on α, it can be evaluated using a Casimir energy calculation in the absence of branes — that is, on the sphere (or, equivalently, by evaluating the rugby-ball result at α = 1).

sph −2 Given si , the contributions to U, κ , H, ζAR and ζR2 can be read oﬀ by identifying the coeﬃ- cients of r2, r0, N 2/r2, N 2/r4 and 1/r2, respectively. Because the µ dependence of the renormalized quantities must cancel the explicit µ-dependence of V1−loop, they therefore satisfy [21]

∂U m6 ∂ 1 m4 µ = − ssph, 0 , µ = − ssph, 0 , (4.61) ∂µ 6(4π)3 −1 ∂µ κ2 2(4π)3 0 2 2 ∂ ζ 2 m ∂H 8 m µ R = − ssph, 0 , µ = − ssph, 2 , (4.62) ∂µ κ 4(4π)3 1 ∂µ (4π)3N 2 1

∂ζ 3 1 ∂ 8 µ R = − ssph, 0 , µ (κζ H) = − ssph, 2 . (4.63) ∂µ 8(4π)3 2 ∂µ AR (4π)3N 2 2

sph,k sph Here the quantities si denote those terms in si that are proportional to k powers of N .

Brane counterterms

The divergences contained in δsi from eq. (4.60) are absorbed in a similar way by counterterms in the brane action, whose generic derivative expansion is:

Lb Ab mn ζRb κζAb MN κζAR˜ b mn 2 √ 2 = −Tb + 2 Fmn − R − 2 FMN F + 2 R Fmn − ζR b R −γ 2gB κ 4gB 2gB 2 κ ζAR b MN − 2 RFMN F + ··· . (4.64) 8gB

Evaluating these at the background rugby-ball solution gives a contribution

2 2 2 Abf 2ζR b κζAbf 2 κζAR˜ bf 4ζR2b κ ζAR bf Vb = Tb − 2 − 2 + 2 + 2 2 + 4 − 2 2 + ··· gB κ r 2gB gB r r 2 gB r 2 2 2 AbN 2ζR b κζAbN κζAR˜ bN 4ζR2b κ ζAR bN = Tb − 2 2 − 2 + 2 4 + 2 4 + 4 − 2 6 + ··· , (4.65) 2 gB r κ r 8 gB r gB r r 8 gB r

181 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

P to the 1PI 4D eﬀective potential, with the complete result summed over all branes: Vbranes = b Vb. Again, each contribution can be disentangled by separating terms with diﬀerent powers of r and N in δsi. The running of the brane couplings that results is

4 2 ∂ Tb m 0 ∂ Ab 2 m 1 µ = 2 δs0 , µ 2 = 2 δs1 , ∂µ 2(4π) ∂µ gB (4π) N 2 ∂ ζRb m 0 ∂ κζAR˜ b 1 1 µ = 2 δs1 , µ 2 = 2 δs2 , (4.66) ∂µ κ 2(4π) ∂µ gB (4π) N ∂ζR2b 1 0 ∂ κζAb 8 2 µ = 2 δs2 , µ 2 = 2 2 δs2 , ∂µ 4(4π) ∂µ gB (4π) N

k k where, as before, δsi denotes that part of δsi proportional to N .

It remains to compute the si explicitly for various light bulk fields by performing the Kaluza- Klein mode sum. This was done in [21] for spin-zero, -half and -one fields, with results which are summarized for completeness in Appendix 4.B. In the next section we assemble the si’s using the field content of various 6D supersymmetric multiplets, to determine how these multiplets renormalize both bulk and brane counterterms.

4.4 Supermultiplets

We now use the results for the si’s for low-spin bulk ﬁelds, listed in Appendix 4.B, and combine them into the ﬁeld content of various 6D matter supermultiplets. We consider in particular two massless multiplets - the hypermultiplet and gauge multiplet, for which only s2 is relevant to renormalizations. We then combine these results to examine the renormalization due to a massive 6D multiplet.

Hypermultiplet scalars

Before combining into supermultiplets, we must first specialize the result for generic scalar fields given in Appendix 4.B so that they can apply to the hyperscalars that appear in supersymmetric hypermultiplets. The part of the action, eq. (4.8), relevant for small hyperscalar fluctuations is

Z √ 1 2g2 S = − d6x −g G (Φ) D ΦI DM ΦJ + R eφ U(Φ) , (4.67) hyp 2 IJ M κ4 where κ2 U = 1 + G ΦI ΦJ + ..., (4.68) 2 IJ

I near Φ = 0, and as before, gR is the U(1)R gauge coupling constant. Notice, in particular, that this expansion of U(Φ) near zero introduces a small universal 6D mass term for ΦI given by

2g2 1 δm2 = R eφ = , (4.69) κ2 2r2

182 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy where the last equality uses eq. (4.13) relating the background value of eφ to 1/r2. Regarding this 2 2 hs 2 mass, δm , as a shift in the hyperscalar KK spectrum, mjn = λjn/r , and using the expression for s 14 the scalar spectrum, λjn, given for minimally coupled scalars in [21], we ﬁnd that

1 λhs (ω, N, Φ ) = λs (ω, N, Φ ) + (4.70) jn b jn b 2 ω ω 12 (1 − N 2) = j + |n − Φ | + |n − N + Φ | + + . 2 − 2 + 2 4

This assumes a background flux with quantum N (where N = ±1 for the supersymmetric case of φ U(1)R flux), as well as the previously-mentioned definitions: Φb := Ab e /(2π) with b = ± denoting P the north and south branes. Finally, N := ω(N − Φ) with Φ := b Φb. The quantity Ab enters the spectrum through the boundary condition Ab := Aϕ(cos θ = b). When computing the small-t limit of the mode sum

∞ ∞ X X −tλhs Shs(ω, N, Φb, t) = e jn , (4.71) j=0 n=−∞

−t/2 we can use relation (4.70) to relate hyperscalar and scalar sums by Shs = e Ss, so using the s expressions for si from Appendix 4.B gives the following small-t coeﬃcients for the hyperscalar:

1 shs (ω, N, Φ ) = ss = , (4.72) −1 b −1 ω ss 1 1 ω2 shs(ω, N, Φ ) = ss − −1 = − + (1 − 3F ) , (4.73) 0 b 0 2 ω 3 6 " ss ss 1 17 N 2 ω2 shs(ω, N, Φ ) = ss − 0 + −1 = − − (1 − 3F ) 1 b 1 2 8 ω 360 24 36 # ω3N X ω4 − Φ G(|Φ |) + 1 − 15F (2) , (4.74) 12 b b 180 b ss ss ss shs(ω, N, Φ ) = ss − 1 + 0 − −1 2 b 2 2 8 48 " 1 1 N 2 1 N 2 ω4N 2 = − + + − (1 − 3F )ω2 − 1 − 15F (2) ω 210 180 240 144 360 # ω5N X 1 F (2) F (3) − Φ G(|Φ |)(1 + 3F ) + − − ω6 . (4.75) 120 b b b 1260 120 60 b

In the above, we adopt the following shorthand:

(n) X n (1) Fb := |Φb| (1 − |Φb|) ,F := Fb ,F := F,G(x) := (1 − x)(1 − 2x) . (4.76) b

14 The quantities N and Φb used in this paper diﬀer from those in [21] by a factor of q. (Therein, they are called N1 and Φ1b.) Unless stated otherwise, we do not to track this q-dependence, since the R-charges of interest are simply q ∈ {±1, 0}.

183 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

sph, 0 sph, 0 In the limit ω → 1, Φb → 0 these become s−1 = 1, s0 = −1/6,

1 N 2 1 N 2 ssph, 0 = , ssph, 2 = − , ssph, 0 = and ssph, 2 = − . (4.77) 1 40 1 24 2 5040 2 240

These agree with the corresponding Gilkey-de Witt calculation of Appendix 4.A. The corresponding contributions to the running of the bulk couplings are

∂U m6 ∂ 1 m4 µ = − , µ = , (4.78) ∂µ 6(4π)3 ∂µ κ2 12(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , (4.79) ∂µ κ 160(4π)3 ∂µ 40320(4π)3 −φ 2 2 2 ∂H ∂ e qhsm ∂ qhs µ = µ 2 = 3 , µ κHζAR = 3 , (4.80) ∂µ ∂µ gB 3(4π) ∂µ 30(4π) where we quote the result for the general case where the hyperscalar couples to the background ﬁeld with strength qhsgB .

2 For the brane counterterms, we similarly ﬁnd δs−1 = δs1 = 0,

1 δω δω2 ω2F δω |Φ | δs = + − b ' − b , (4.81) 0 ω 6 12 2 6 2 1 δω δω2 δω3 δω4 ω2F ω4F 2 δω |Φ | δs0 = − + + + + b − b ' − + b , (4.82) 1 ω 60 360 90 360 12 12 60 12 ω2N N Φ δs1 = − Φ G(|Φ |) ' − b , (4.83) 1 12 b b 12 1 11 δω 9 δω2 δω3 δω4 δω5 δω6 ω2F ω6F 2 ω6F 3 δs0 = + + + + + − b − b − b 2 ω 1680 1120 126 168 420 2520 80 120 60 11 δω |Φ | ' − b , (4.84) 1680 80 ω4N N Φ δs1 = − Φ G(|Φ |)(1 + 3F ) ' − b , (4.85) 2 120 b b b 120 N 2 δω 17 δω2 δω3 δω4 ω2F ω4F 2 δω |Φ | δs2 = − + + + − b − b ' −N 2 − b , (4.86) 2 ω 80 1440 180 720 48 24 80 48

and so (again generalizing to coupling qhs gB )

∂T m4 δω δω2 ω2F m4 δω µ b = + − b ' − |Φ | , (4.87) ∂µ 2(4π)2ω 6 12 2 4(4π)2 3 b 2 2 2 2 ∂ ζA˜b qhsΦb ω m qhsm Ab µ 2 = − 2 G(|Φb|) ' − 3 , (4.88) ∂µ gB 6(4π) 3(4π) ∂ ζ m2 δω δω2 δω3 δω4 ω2F ω4F 2 µ Rb = − − − − − b + b ∂µ κ 2(4π)2ω 60 360 90 360 12 12 m2 δω |Φ | ' − − b , (4.89) 2(4π)2 60 12 4 2 ∂ κζAR˜ b qhsΦb ω qhs Ab µ 2 = − 2 G(|Φb|)(1 + 3Fb) ' − 3 , (4.90) ∂µ gB 120(4π) 60(4π)

184 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 3 4 5 6 2 ∂ ζ 2 1 11 δω 9 δω δω δω δω δω ω F µ R b = − + + + + + − b ∂µ 4(4π)2ω 1680 1120 126 168 420 2520 80 ! ω6F 2 ω6F 3 1 11 δω |Φ | − b − b ' − − b , (4.91) 120 60 4(4π)2 1680 80 2 2 3 4 2 4 2 ∂ κζAb 8 qhs δω 17 δω δω δω ω Fb ω Fb µ 2 = − 2 + + + − − ∂µ gB (4π) ω 80 1440 180 720 48 24 q2 δω |Φ | ' − hs − b . (4.92) (4π)2 10 6

In these expressions δω := ω−1, and the approximate equalities give the leading terms when |δω| 1 and |Φb| 1.

4.4.1 Hypermultiplet

We are now in a position to sum the particle content of a 6D hypermultiplet, which consists of four massless hyperscalars together with a 6D Weyl fermion:

hyp hs f si := 4 si + si , (4.93)

hs f with si computed above and si given in Appendix 4.B. Although we need only really be interested hyp hyp hyp hyp in s2 for massless ﬁelds, we nonetheless keep track of s−1 , s0 and s1 as well, since these are needed when assembling a massive multiplet. sph, 0 The above combination yields s−1 = s0 = 0 (where we drop the ‘hyp’ superscript), along with

1 N 2 1 N 2 ssph, 0 = , ssph, 2 = −(q2 + 2 q2) , ssph, 0 = , ssph, 2 = −(q2 + 4q2) , (4.94) 1 6 1 hs f 6 2 60 2 hs f 60

2 where N := 2r f = ±gB /gR characterizes the bulk flux (with the second equality using the field equation, eq. (4.13)) while qhs and qf are the charges (in units of gB ) of the scalar and fermion, respectively, under the U(1) gauged by the background flux. The corresponding bulk renormalizations are

∂ζ 3 1 µ R = − (4.95) ∂µ 480(4π)3 −φ sph, 2 ! 2 2 ∂ ∂ κe ζAR 8 s2 2(qhs + 4qf ) µ κHζAR = µ 2 = − 3 2 = 3 . (4.96) ∂µ ∂µ gB (4π) N 15(4π)

From here, we can use eq. (4.59) compute the hypermultiplet contribution to the running of VB :

2 ∂VB 2 f ∂ κζAR 8 ∂ζR3 µ = 4πα r − 2 µ 2 − 6 µ ∂µ 2r ∂µ gB r ∂µ α = 1 − q2 N 2 − 4 q2N 2 . (4.97) hs f 60(4πr2)2

185 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Three choices for qhs and qf are of particular interest:

• No couplings to background ﬂuxes: qhs = qf = 0;

• Couplings to a background U(1) that commutes with supersymmetry, in which case qhs = qf = q

and N = ±gB /gR;

• Couplings to a background ﬂux that preserves supersymmetry, for which the background ﬂux

gauges the U(1)R symmetry, and so qhs = ±1, qf = 0 and N = ±1.

In the supersymmetric case a cancellation occurs, generalizing to the rugby ball a result known sph, 0 sph, 2 1 sph, 0 to apply more generally to Ricci-ﬂat geometries [43]. In this case s1 = −s1 = 6 and s2 = sph, 2 1 −s2 = 60 and so sph sph sph sph s−1 = s0 = s1 = s2 = 0 (4.98)

and µ(∂VB /∂µ) = 0 vanishes once summed over the ﬁeld content of a hypermultiplet. Notice that

MN 3 although the eﬀective couplings for both RFMN F and R do renormalize, their contributions cancel in VB . Specializing to the supersymmetric charge assignments the brane-renormalized divergences are hs f δsi = 4 δsi + δsi, and so

1 δω2 δs = δω + − 2ω2F ' δω − 2|Φ | , (4.99) 0 ω 2 b b 1 δω2 δω3 δω4 ω2F ω4F 2 |Φ | δs0 = + + + b − b ' b , (4.100) 1 ω 12 12 48 3 3 3 ω2N N Φ δs1 = − Φ G(|Φ |) ' − b , (4.101) 1 3 b b 3 1 δω 13 δω2 13 δω3 71 δω4 3 δω5 δω6 ω2F ω6F 2 ω6F 3 δs0 = + + + + + − b − b − b 2 ω 20 180 180 1440 160 320 20 30 15 δω |Φ | ' − b , (4.102) 20 20 ω4N N Φ δs1 = − Φ G(|Φ |)(1 + 3F ) ' − b , (4.103) 2 30 b b b 30 N 2 δω 17 δω2 δω3 δω4 ω2F ω4F 2 δω |Φ | δs2 = − + + + − b − b ' −N 2 − b , (4.104) 2 ω 20 360 45 180 12 6 20 12

where N = ±gB /gR = ±1.

Notice that all of the brane contributions also vanish, δsi = 0 (for all ω), in the special case that the brane fluxes are equal and the total brane flux satisfies the supersymmetric and flux-quantization conditions, eqs. (4.29) and (4.47):

Φ 1 Φ = Φ = susy = ± (1 − ω−1) . (4.105) + − 2 2 as well as N = sgn(Φ±). Similarly µ(∂Vbranes/∂µ) = 0. Once again, although δsi = 0 this is not 1 2 true for δsi and δsi separately.

186 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

In general this cancellation fails when Φ± are not equal, since this choice breaks supersymmetry.

In order to track how Vbranes deviates from zero once supersymmetry breaks, we write

1 bη Φ = Φ ± (4.106) b 2 susy ω

1 and allow η to parameterize the diﬀerence through η = ± 2 ω∆Φ where ∆Φ := Φ+ − Φ−. To ensure that η ≥ 0, we label branes such that Φ+ ≥ Φ− (Φ+ ≤ Φ−) when N = +1 (N = −1). Also, notice that the condition |Φb| ≤ 1 implies that η ≤ 1 + δω/2.

At the north (b = +1) brane, we ﬁnd that the brane contributions to the coeﬃcients δsi are given in terms of η by

2F δs = − η , (4.107) 0 ω 1 δω δω2 F F 2 ω2F δs0 = + + η − η − η , (4.108) 1 ω 6 12 2 3 6 1 δω δω2 F δs1 = − − − η (1 − 2η) , (4.109) 1 ω 6 12 3 1 δω 37 δω2 7 δω3 7 δω4 F F 2 F 3 δs0 = + + + − η + η − η 2 ω 40 720 180 720 16 20 15 F F 2 7 ω4F + η − η ω2 − η , (4.110) 24 12 240 1 δω δω2 7 δω3 7 δω4 F F 2 ω2F δs1 = − − − − + η − η − η (1 − 2η) , (4.111) 2 ω 60 15 120 480 20 10 12 1 δω 11 δω2 7 δω3 7 δω4 F 2 ω2F δs2 = − + + + + η + η , (4.112) 2 ω 120 720 360 1440 6 12 where

Fη := η(1 − η) . (4.113)

The result for the south brane can be obtained from this using η → −η if η ≤ δω/2. If instead δω/2 ≤ η ≤ 1 + δω/2 on the south brane, we ﬁnd that a more convenient quantity is

ηˆ := η − δω (4.114)

(along with Fηˆ :=η ˆ(1 − ηˆ)), in which case

2F δs = − ηˆ , (4.115) 0 ω 1 δω δω2 F F 2 ω2F δs0 = + + ηˆ − ηˆ − ηˆ , (4.116) 1 ω 6 12 2 3 6 1 δω δω2 F δs1 = + + ηˆ (1 − 2ˆη) , (4.117) 1 ω 6 12 3 1 δω 37 δω2 7 δω3 7 δω4 F F 2 F 3 δs0 = + + + − ηˆ + ηˆ − ηˆ 2 ω 40 720 180 720 16 20 15

187 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

F F 2 7 ω4F + ηˆ − ηˆ ω2 − ηˆ , (4.118) 24 12 240 1 δω δω2 7 δω3 7 δω4 F F 2 ω2F δs1 = + + + − ηˆ + ηˆ + ηˆ (1 − 2ˆη) , (4.119) 2 ω 60 15 120 480 20 10 12 1 δω 11 δω2 7 δω3 7 δω4 F 2 ω2F δs2 = − + + + + ηˆ + ηˆ . (4.120) 2 ω 120 720 360 1440 6 12

These expressions are identical to those obtained for the north brane, given the replacements η → ηˆ, 1 1 si → −si . This follows from

δω η δω (η − δω) δω ηˆ Φ = ± − = ± − − = ∓ + (4.121) − 2ω ω 2ω ω 2ω ω

1 and because si ∝ Φb rather than depending only on |Φb|, like the others.

The running of the couplings on the north — the b = +1 — brane (which we choose to be the brane whose ﬂux is larger in magnitude) are

2 3 4 2 3 ∂ζ 2 1 δω 37 δω 7 δω 7 δω F F F µ R + = + + + − η + η − η ∂µ 4(4π)2ω 40 720 180 720 16 20 15 F F 2 7 ω4F + η − η ω2 − η , (4.122) 24 12 240 2 3 4 2 2 ∂ κζAR˜ + 1 δω δω 7 δω 7 δω Fη Fη ω Fη µ 2 = 2 − − − − + − − (1 − 2η) , ∂µ gB (4π) ω 60 15 120 480 20 10 12 (4.123) 2 3 4 2 2 ∂ κζA+ 8 δω 11 δω 7 δω 7 δω Fη ω Fη µ 2 = 2 − + + + + + . (4.124) ∂µ gB (4π) ω 120 720 360 1440 6 12

As before, the corresponding expressions for the south brane when η ≤ δω/2 (η ≥ δω/2) can be found by taking η → −η (η → ηˆ = (η − δω) and ζAR˜ + → −ζAR˜ −) in the above.

For both branes — i.e. for both b = ± — and η ≤ δω/2, the hypermultiplet contribution to the total renormalized lagrangian becomes

∂Vb 4 ∂ζR2b 1 ∂ κζAR˜ b 1 ∂ κζAb µ = 4 µ + 4 µ 2 + 4 µ 2 ∂µ r ∂µ r ∂µ gB 8r ∂µ gB η2 7 η2 η4 1 η2 7 ω4 = − + + − ω2 + . (4.125) (4πr2)2ω 240 12 15 24 12 240

This expression is positive-deﬁnite on our domain of validity, 0 ≤ η ≤ 1 + δω/2, as can be checked by showing: (i) µ ∂Vb/∂µ ≥ 0 at η = 1 for any ω ≥ 1; and (ii) the smallest root of the bracketed factor, η0(ω) ≥ 1 + δω/2 for any ω ≥ 1. Also, when ω = 1, this expression vanishes as η → 0 or 1, as expected.

188 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

When δω/2 ≤ η ≤ 1 + δω/2 on the south brane, we instead ﬁnd that the renormalizations are

2 3 4 2 3 ∂ζ 2 1 δω 37 δω 7 δω 7 δω F F F µ R − = + + + − ηˆ + ηˆ − ηˆ ∂µ 4(4π)2ω 40 720 180 720 16 20 15 F F 2 7 ω4F + ηˆ − ηˆ ω2 − ηˆ , (4.126) 24 12 240 2 3 4 2 2 ∂ κζAR˜ − 1 δω δω 7 δω 7 δω Fηˆ Fηˆ ω Fηˆ µ 2 = 2 + + + − + + (1 − 2ˆη) , ∂µ gB (4π) ω 60 15 120 480 20 10 12 (4.127) 2 3 4 2 2 ∂ κζA− 8 δω 11 δω 7 δω 7 δω Fηˆ ω Fηˆ µ 2 = 2 − + + + + + . (4.128) ∂µ gB (4π) ω 120 720 360 1440 6 12

Therefore, when η ≥ δω/2 on the south brane, the beta function for the hypermultiplet contribution to the renormalized brane lagrangian is

∂V− 4 ∂ζR2− 1 ∂ κζAR˜ − 1 ∂ κζA− µ = 4 µ + 4 µ 2 + 4 µ 2 ∂µ r ∂µ r ∂µ gB 8r ∂µ gB (η − ω)2 7 (η − ω)2 (η − ω)4 1 (η − ω)2 7 ω4 = − + + − ω2 + . (4.129) (4πr2)2ω 240 12 15 24 12 240

This expression is identical to the one found for the north brane, but with the replacement η → η−ω, so its positive-deﬁniteness can be shown in the same way as before. (Recall that it is inconsistent to consider the limit η → 0 with ﬁxed ω in the above expression, because it is only valid when η ≥ δω/2.) P Finally, we sum the total contribution from both branes to compute how Vbranes = b Vb runs. When η ≤ δω/2, this is simply twice the result found in eq. (4.125). However, when η ≥ δω/2, we must sum eq. (4.125) with our last result, eq. (4.129). As expected, if we sum an even function of η with the same function evaluated at η − ω, we get an even function about η = ω/2 (which happens to be the midpoint of our domain of validity, δω/2 ≤ η ≤ 1 + δω/2):

∂V 1 7 ω2 ω4 ω6 7 ω2 ω4 ω 2 µ branes = + + + − − η − ∂µ (4πr2)2ω 480 96 160 120 6 15 2 1 ω2 ω 4 2 ω 6 − − η − + η − . (4.130) 6 3 2 15 2

This expression is positive-deﬁnite, because it is the sum of two separately positive-deﬁnite expressions.

4.4.2 Massless gauge multiplet

We next compile similar results for a massless gauge multiplet, by summing the contribution of a gauge ﬁeld and a 6D Weyl fermion: gm gf f s2 := s2 + s2 . (4.131)

189 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

sph sph Dropping the ‘gm’ superscript this combination yields s−1 = s0 = 0, and

1 q2N 2 1 q2N 2 ssph, 0 = , ssph, 2 = − f , ssph, 0 = , ssph, 2 = − f , (4.132) 1 3 1 3 2 15 2 15

where, as before, N = ±gB /gR and where qf gB is the charge of the fermion under the U(1) gauged by the background ﬂux. Since we assume the background ﬂux gauges an abelian, U(1), factor of the gauge group, we also choose qgf = 0 for the gauge boson.

The corresponding bulk beta functions are

∂ζ 3 1 µ R = − (4.133) ∂µ 120(4π)3 −φ 2 ∂ ∂ κe ζAR 8 qf µ κHζAR = µ 2 = 3 , (4.134) ∂µ ∂µ gB 15(4π)

which, when combined using eq. (4.59), give the gauge multiplet contribution to the running of VB :

2 ∂V f ∂ 8 ∂ζ 3 µ B = 4παr2 − µ κHζ − µ R ∂µ 2r2 ∂µ AR r6 ∂µ α(1 − q2N 2) = f . (4.135) 15(4πr2)2

2 This again vanishes given the supersymmetric choices qf = 1 and N = ±1 (and qgf = 0), although this comes as a cancellation between the running of ζAR and ζR3 , which both separately renormalize.

gf f We similarly compute the brane contributions, δsi = δsi + δsi, using the expressions in [21]. However, as discussed therein, there are two cases to consider because of the shift in fluxes required fσ f to calculate the fermionic mode sum. In [21], we use the quantity Φb := Φb − σΦ0 — where f −1 Φ0 := (1 − ω )/2 and σ is the spinor’s helicity — to calculate the fermionic mode sum, so this f f mode sum is dependent on whether |Φb| ≤ Φ0 or |Φb| ≥ Φ0. As it happens, we encounter the f f exact same distinction when considering unbalanced fluxes, when |Φb| > Φ0 (|Φb| < Φ0) on the north (south) brane, respectively. As for the hypermultiplet we specialize to the supersymmetric background flux, and write the brane-localized fluxes relative to the common supersymmetric value through the substitutions

Φ bη N = ±1 , Φ = susy ± , Φ = ±(1 − ω−1)(η ≥ 0) . (4.136) b 2 ω susy

˜ f At the north brane (where b = +1), we ﬁnd — using the notation [21], Φb = ω(Φb − ρbΦ0),

ρb = Φb/|Φb|, F˜b = |Φ˜ b|(1 − |Φ˜ b|), etc., as well as the shorthand Fη = η(1 − η),

1 δω δω2 1 δω δω2 2F δs = (δs ) + (δs ) = − + + − + 2F˜ = η . (4.137) 0 0 gf 0 f ω 3 3 ω 3 3 b ω

190 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Similarly, by making the identiﬁcations ρb = ±, |Φ˜ b| = η, we ﬁnd

! 1 δω δω2 F 2 ω2F δs0 = + + η − η , (4.138) 1 ω 3 6 3 3 1 δω δω2 F δs1 = − − + η (1 − 2η) , (4.139) 1 ω 3 6 3 1 δω 17 δω2 2 δω3 δω4 F 2 F 3 ω2F 2 ω4F δs0 = + + + + η + η − η − η , (4.140) 2 ω 10 180 45 90 30 15 6 30 1 δω δω2 δω3 δω4 F F 2 ω2F δs1 = − − − − + η + η − η (1 − 2η) , (4.141) 2 ω 15 10 15 60 30 10 6 ! 1 δω δω2 δω3 δω4 F 2 ω2F δs2 = − + + + − η + η (4.142) 2 ω 30 180 45 180 6 6 using the same approach. When summed, these coeﬃcients give

δs0 = δs1 = δs2 = 0 (4.143) for all ω in the supersymmetric limit of equal ﬂuxes (for which η = 0).

For nonzero η supersymmetry is broken. When η ≥ δω at the south brane, we would instead make the identifications ρb = ∓, |Φ˜ b| = η − δω, and find results similar to those above, but with 1 1 η → ηˆ := η − δω, si → −si . However, when η ≤ δω, we must instead use the fermionic result valid f when |Φb| ≤ Φ0. Substituting, we find that the coefficients at the south brane are, e.g.,

1 δω δω2 1 δω δω2 1 δs = (δs ) + (δs ) = − + + + + 2ω2Φ2 = (F + 2ωη) 0 0 gf 0 f ω 3 3 ω 3 6 b ω −η (4.144) using

F−η = −η(1 + η) , (4.145) and, similarly,

! 1 δω δω2 F 2 ωF ω2F δs0 = + + −η + −η (1 + 2η) − −η , (4.146) 1 ω 3 6 3 3 3 1 δω δω2 F δs1 = − − + −η (1 + 2η) , (4.147) 1 ω 3 6 3 1 δω 17 δω2 2 δω3 δω4 F 2 F 3 δs0 = + + + + −η + −η 2 ω 10 180 45 90 30 15 F F 2 ω2F 2 ω4F + −η + −η ω(1 + 2η) − −η − −η , (4.148) 30 10 6 30 1 δω δω2 δω3 δω4 F F 2 ω2F ωF δs1 = − − − − + −η + −η − −η (1 + 2η) − −η , (4.149) 2 ω 15 10 15 60 30 10 6 2

191 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

" # 1 δω δω2 δω3 δω4 F 2 ωF ω2F δs2 = − + + + − −η − −η (1 + 2η) + −η . (4.150) 2 ω 30 180 45 180 6 6 6

fσ The condition |Φb | ≤ 1 means that η ≤ 1, so all of these expressions lose their validity beyond this upper limit. The running of the renormalized brane couplings from a gauge supermultiplet then become, on the north brane:

2 3 4 2 3 2 2 4 ∂ζ 2 1 δω 17 δω 2 δω δω F F ω F ω F µ R + = + + + + η + η − η − η , ∂µ 4(4π)2ω 10 180 45 90 30 15 6 30 (4.151) 2 3 4 2 2 ∂ κζAR˜ + 1 δω δω δω δω Fη Fη ω Fη µ 2 = 2 − − − − + + − (1 − 2η), (4.152) ∂µ gB (4π) ω 15 10 15 60 30 10 6 2 3 4 2 2 ! ∂ κζA+ 8 δω δω δω δω Fη ω Fη µ 2 = 2 − + + + − + . (4.153) ∂µ gB (4π) ω 30 180 45 180 6 6

Therefore, the beta function for the gauge multiplet contribution to the running of Vb is ∂V+ 4 ∂ζR2+ 1 ∂ κζAR˜ + 1 ∂ κζA+ µ = 4 µ + 4 µ 2 + 4 µ 2 ∂µ r ∂µ r ∂µ gB 8r ∂µ gB η2 2 η2 η4 1 η2 ω4 = − + − + − ω2 + . (4.154) (4πr2)2ω 15 6 15 6 6 30

Several features of eq. (4.154) bear emphasis. First, it vanishes for all ω as η → 0, as appropriate to the supersymmetric limit, and when ω = 1 it also vanishes as η → 0 or 1, as also expected for the Salam-Sezgin sphere in the absence of branes. Second, for nonzero η it is positive definite throughout its domain of validity, 0 ≤ η ≤ 1, so long as ω > 1 (i.e. the branes have positive tension). This definiteness of sign can be checked by showing that: (i) µ ∂V+/∂µ ≥ 0 at, e.g., η = 1 for any ω ≥ 1; and (ii) the smallest root of the bracketed factor, η0(ω) ≥ 1 for any ω ≥ 1. For the south brane, the result can be obtained from the above in the case η ≥ δω, simply by taking η → η − δω and ζAR˜ + → −ζAR˜ −. When η ≤ δω we instead find for the south brane

2 3 4 2 3 ∂ζ 2 1 δω 17 δω 2 δω δω F F µ R − = + + + + −η + −η ∂µ 4(4π)2ω 10 180 45 90 30 15 F F 2 ω2F 2 ω4F + −η + −η ω(1 + 2η) − −η − −η , (4.155) 30 10 6 30 2 3 4 2 2 ∂ κζAR˜ − 1 δω δω δω δω F−η F−η ω F−η µ 2 = 2 − − − − + + − (1 + 2η) ∂µ gB (4π) ω 15 10 15 60 30 10 6 ωF − −η , (4.156) 2 " 2 3 4 2 2 # ∂ κζA− 8 δω δω δω δω F−η ωF−η ω F−η µ 2 = 2 − + + + − − (1 + 2η) + . ∂µ gB (4π) ω 30 180 45 180 6 6 6 (4.157)

192 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

Therefore, the gauge multiplet contribution to the running of Vb for the south brane in this case is ∂V− 4 ∂ζR2− 1 ∂ κζAR˜ − 1 ∂ κζA− µ = 4 µ + 4 µ 2 + 4 µ 2 (4.158) ∂µ r ∂µ r ∂µ gB 8r ∂µ gB 1 2 η2 η4 η6 2 η η3 η5 η2 η4 ω4η2 = − + − + − + ω + − ω2 + . (4πr2)2ω 15 6 15 15 3 5 6 6 30

When ω = 1, this expression vanishes as η → 0 or 1, again as expected. It can also be shown to be positive deﬁnite, in the same way as is done for eq. (4.154).

Finally, we sum the contributions at each brane to ﬁnd the total renormalization of Vbranes = P b Vb for a massless gauge multiplet. When η ≤ δω, this is

∂V 1 4 η2 η4 2 η6 2 η η3 η5 µ branes = − + − + − + ω ∂µ (4πr2)2ω 15 3 15 15 3 5 η2 η4 ω4η2 + − ω2 + (4.159) 3 3 15 which vanishes, as expected, when η → 0. For δω ≤ η ≤ 1, we instead have

∂V 1 ω2 5 ω4 ω6 4 5 ω2 67 ω4 ω 2 µ branes = − + − − − + η − ∂µ (4πr2)2ω 15 48 160 15 6 120 2 1 5 ω2 ω 4 2 ω 6 + − η − − η − , (4.160) 3 6 2 15 2 a function that is even about η = ω/2 (the center of its domain of validity). These expressions agree, as they should, when η = δω.

4.4.3 Massive matter multiplet

The previous sections have the drawback that they involve only massless supermultiplets, in the sense that supersymmetry forbids their 6D masses being parametrically large compared with the KK scale, 1/r. As a consequence only dimensionless couplings get renormalized when these fields are integrated out (using dimensional regularization and minimal subtraction). To get beyond this, in this section we compute the m-dependence of the Casimir coefficient for a massive 6D supermultiplet. Recall that the field content for this multiplet is a massive gauge field (mgf), two Weyl fermions (2f) and 3 real hyperscalars (3hs). Keeping in mind that the Higgs mechanism makes a massive vector equivalent to a massless vector plus a hyperscalar, the particle content of a massive supermultiplet is equivalent to the combined field content of a gauge and a hypermultiplet.15

We use this observation to compute the coeﬃcients si for the massive multiplet in terms of those

15 At first sight this is hard to reconcile with the particle U(1)R assignments. However, because hyperscalars carry nonzero U(1)R charge, we also expect the standard U(1)R symmetry to be spontaneously broken if the gauge field acquires its mass due to a nonzero hyperscalar vev. Particle states would then be labeled by the unbroken linear combination of the R symmetry and the naive generator for the heavy gauge field.

193 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy found above for gauge- and hypermultiplets, by taking

mm hm gm si = si + si . (4.161)

Our previous work in tracking all Gilkey-de Witt coefficients — even though we then only needed s2 for massless fields — pays off here, since they all contribute for a massive multiplet.

Given this prescription, a massive matter multiplet gives the following nonzero bulk coeﬃcients, sph, k si , 1 1 1 1 ssph, 0 = , ssph, 2 = − , ssph, 0 = , ssph, 2 = − , (4.162) 1 2 1 2 2 12 2 12 and so contributes the following renormalizations to the bulk couplings:

∂ U ∂ 1 µ = µ = 0 , (4.163) ∂µ ∂µ κ2

2 −φ 2 ∂ ζR2 m ∂H ∂ e 4m µ = − 3 , µ = µ 2 = 3 , (4.164) ∂µ κ 8(4π) ∂µ ∂µ gB (4π) and ∂ζR3 1 ∂ κζAR 2 µ = − 3 , µ 2 = 3 . (4.165) ∂µ 96(4π) ∂µ gB 3(4π)

Notice in particular that the nonzero renormalization of H and generation of curvature-squared terms is consistent with the known loop corrections to the gauge kinetic functions [34] required by 6D anomaly cancellation. However unbroken supersymmetry implies these contributions precisely cancel in the renormalizations of the total bulk lagrangian evaluated at the rugby ball background:

∂V µ B = 0 . (4.166) ∂µ

This is equally true when the branes break supersymmetry (i.e. η 6= 0), since bulk renormalizations do not know about brane boundary conditions.

The situation is more complicated for the brane renormalizations, however, for which µ (∂Vb/∂µ) should not vanish for unequal brane-localized ﬂuxes. We start by quoting the δsi’s for a massive multiplet, and then computing the corresponding beta functions. On the north brane, we have

δs0 = 0 , (4.167) 1 δω δω2 F ω2F δs0 = + + η − η , (4.168) 1 ω 2 4 2 2 1 δω δω2 δs1 = − − (1 − 2η) , (4.169) 1 ω 2 4 1 δω 7 δω2 δω3 δω4 F F 2 F F 2 ω4F δs0 = + + + − η + η + η − η ω2 − η , (4.170) 2 ω 8 48 12 48 16 12 24 4 16 1 δω δω2 δω3 δω4 F ω2F δs1 = − − − − + η − η (1 − 2η), (4.171) 2 ω 12 6 8 32 12 4

194 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

1 δω δω2 δω3 δω4 ω2F δs2 = − + + + + η , (4.172) 2 ω 24 48 24 96 4 which give the following renormalizations: µ ∂T+/(∂µ) = 0,

∂ ζ m2 δω δω2 F ω2F µ R+ = + + η − η (4.173) ∂µ κ 2(4π)2ω 2 4 2 2 2 2 ∂ ζA˜+ 2 m δω δω µ 2 = 2 − − (1 − 2η) (4.174) ∂µ gB (4π) ω 2 4 2 3 4 2 ∂ζ 2 1 δω 7 δω δω δω F F µ R + = + + + − η + η ∂µ 4(4π)2ω 8 48 12 48 16 12 F F 2 ω4F + η − η ω2 − η , (4.175) 24 4 16 2 3 4 2 ∂ κζAR˜ + 1 δω δω δω δω Fη ω Fη µ 2 = 2 − − − − + − (1 − 2η) , (4.176) ∂µ gB (4π) ω 12 6 8 32 12 4 2 3 4 2 ∂ κζA+ 8 δω δω δω δω ω Fη µ 2 = 2 − + + + + . (4.177) ∂µ gB (4π) ω 24 48 24 96 4

Therefore, the total contribution of a massive matter multiplet to the running of V+ is ∂V+ 1 ∂ ζA˜+ 2 ∂ ζR+ 4 ∂ζR2+ µ = − 2 µ 2 − 2 µ + 4 µ ∂µ 2 r ∂µ gB r ∂µ κ r ∂µ 1 ∂ κζAR˜ + 1 ∂ κζA+ + 4 µ 2 + 4 µ 2 r ∂µ gB 8r ∂µ gB η2 5 5 ω2 ω4 1 ω2 (ω2 − 1) = − + + + − η2 − (mr)2 . (4.178) (4πr2)2ω 48 24 16 12 4 2

Notice the appearance here of terms proportional to m2, although the entire quantity vanishes in the supersymmetric limit η → 0.

mm hm gm Following the prescription δsi = δsi + δsi on the south brane, we obtain diﬀerent results when either: 1) η ≤ δω/2; 2) δω/2 ≤ η ≤ δω; or 3) δω ≤ η ≤ 1. For the sake of brevity, we will only consider case 1 in what follows (since only in this case can the limit η → 0 be taken, with ﬁxed ω 6= 1), and refer the avid reader to Appendix 4.C for the complete results that include cases 2 and 3 as well.

When η ≤ δω/2, we ﬁnd that

δs0 = 2η , (4.179) 1 δω δω2 F ωF ω2F δs0 = + + −η + −η (1 + 2η) − −η , (4.180) 1 ω 2 4 2 3 2 1 δω δω2 δs1 = − − (1 + 2η) − ωF , (4.181) 1 ω 2 4 −η 1 δω 7 δω2 δω3 δω4 F F 2 F F 2 δs0 = + + + − −η + −η + −η + −η ω(1 + 2η) 2 ω 8 48 12 48 16 12 30 10

195 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

F F 2 ω4F + −η − −η ω2 − −η , (4.182) 24 4 16 1 δω δω2 δω3 δω4 F ω2F ωF 2 δs1 = − − − − + −η − −η (1 + 2η) − −η , (4.183) 2 ω 12 6 8 32 12 4 2 1 δω δω2 δω3 δω4 ωF ω2F δs2 = − + + + − −η (1 + 2η) + −η , (4.184) 2 ω 24 48 24 96 6 4 which give the following renormalizations:

∂T m4 η µ − = (4.185) ∂µ (4π)2 ∂ ζ m2 δω δω2 F ωF ω2F µ R− = + + −η + −η (1 + 2η) − −η (4.186) ∂µ κ 2(4π)2ω 2 4 2 3 2 2 2 ∂ ζA˜− 2 m δω δω µ 2 = 2 − − (1 + 2η) − ωF−η (4.187) ∂µ gB (4π) ω 2 4 2 3 4 2 ∂ζ 2 1 δω 7 δω δω δω F F µ R − = + + + − −η + −η ∂µ 4(4π)2ω 8 48 12 48 16 12 F F 2 F F 2 ω4F + −η + −η ω(1 + 2η) + −η − −η ω2 − −η , (4.188) 30 10 24 4 16 2 3 4 2 2 ∂ κζAR˜ − 1 δω δω δω δω F−η ω F−η ωF−η µ 2 = 2 − − − − + − (1 + 2η) − , ∂µ gB (4π) ω 12 6 8 32 12 4 2 (4.189) 2 3 4 2 ∂ κζA− 8 δω δω δω δω ωF−η ω F−η µ 2 = 2 − + + + − (1 + 2η) + . (4.190) ∂µ gB (4π) ω 24 48 24 96 6 4

Therefore, the total contribution of a massive matter multiplet to the running of V− (when η ≤ δω/2; see Appendix 4.C for the result when η ≥ δω/2) is

∂V− ∂T− 1 ∂ ζA˜− 2 ∂ ζR− 4 ∂ζR2− µ = µ − 2 µ 2 − 2 µ + 4 µ ∂µ ∂µ 2 r ∂µ gB r ∂µ κ r ∂µ 1 ∂ κζAR˜ − 1 ∂ κζA− + 4 µ 2 + 4 µ 2 r ∂µ gB 8r ∂µ gB " 1 5 5 ω2 ω4 1 ω2 (ω2 − 1) = − + + η2 + − η4 − η2(mr)2 (4πr2)2ω 48 24 16 12 4 2 # 2 η2 η4 2 2 η2 +ωη − + − − (mr)2 + (mr)4 . (4.191) 15 3 5 3 3

The ﬁrst line in eq. (4.191) is identical to the result in eq. (4.178) for the running at the north brane, and the additional piece in the second line of eq. (4.191) is odd in η.

P Lastly, we can assemble the total contribution of both branes to the running of Vbranes = b Vb: " ∂V 1 5 5 ω2 ω4 1 ω2 µ branes = − + + η2 + − η4 − (ω2 − 1)η2(mr)2 ∂µ (4πr2)2ω 24 12 8 6 2

196 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

# 2 η2 η4 2 2 η2 +ωη − + − − (mr)2 + (mr)4 (4.192) 15 3 5 3 3

Regarding the positive-definiteness of this result, we can rest assured that any m-independent contribution is positive definite, since it is simply a sum of two positive-definite contributions from the massless hyper- and gauge-multiplets. As it turns out, the m-dependent terms are also positive definite. (Checking this must be done with some care, but it can be shown that the non-trivial 2 zero, η0(ω), of the coefficient of (mr) in eq. (4.192) satisfies both η0(1) = 1 and dη0/dω > 0 for all ω ≥ 1.). However, for η larger than δω/2, the results in Appendix 4.C indicate that there is always some choice of m for which the beta function is negative. An exception to this is the special case where ω = 1, in which case η ≥ δω (= 0) for any η. As seen in Appendix 4.C, this regime is also positive definite, since the contribution of a massive matter multiplet to the renormalization of the 1PI effective potential in this case is simply the sum of the contributions of a massless hypermultiplet and a massless gauge multiplet.

4.5 The 4D vacuum energy

The previous sections give the divergent part of V1−loop obtained by integrating out low-spin bulk ﬁelds and show how these divergences are absorbed by renormalization of various bulk and brane interactions. This section computes the implication of these renormalizations for the eﬀective 4D cosmological constant, Λ, and on-brane curvature, as seen by a low-energy 4D observer. As argued more generally in [21], for codimension-2 branes this is not simply given by the sum16

V := VB + Vbranes + Vf , where Vf is the ﬁnite part of V1−loop. Instead, the changes to the branes captured by Vbranes must be combined with the contributions of bulk back-reaction – along the lines of refs. [23] – which in general need not be suppressed relative to the direct eﬀects of V1−loop, VB or

Vbranes itself [11, 44, 45]. Indeed, this back-reaction is what allows ﬂat solutions to exist at all at the classical level, despite the large classical positive tensions carried by each brane. The complete back-reacted response to V is not yet as well understood as is the response to a localized brane source. For this reason it is worth focussing exclusively on the large logarithm, ln(M/m) in Λ that our renormalization-group mechanism tracks. That is, although the µ-dependence in VB and Vbranes always cancels the explicit ln(µ/m) appearing in Vf , there is a (µ-independent) large logarithm of order ln(M/m) that survives once it has done so, where M is a typical UV scale, of generic order M6. Because part of the log always comes from the brane and bulk renormalizations, its coeﬃcient can be tracked purely using the RG calculations as given above. And the logarithm can be the dominant part of the answer when M is much greater than m.

16This result would be appropriate in the ‘probe’ approximation, but this approximation often fails for codimension-2 objects.

197 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

4.5.1 Classical bulk back-reaction

We ﬁrst recap the general results of ref. [23], to establish a common notation and to emphasize those features that are special to the supersymmetric case. Ref. [23] starts with a rugby-ball solution and R 4 √ asks how its properties respond to small changes in the brane action, δSb = − d x −γ δLb, where

γab is the metric induced on the brane from the 6D Einstein-frame metric in the bulk and 1 Ab mn δLb = δTb − δ 2 Fmn . (4.193) 2 gB

In particular, it asks how the eﬀective 4D cosmological constant is aﬀected by such a change, given that it vanishes for the unperturbed system.

The back-reaction caused by δSb is evaluated by tracking how it affects the bulk boundary conditions [36], and then solving the linearized 6D field equations to compute the change to the predicted value for the curvature, Rµν , along the brane directions. In particular, it is not assumed that the perturbed geometry has a rugby ball form. The effective 4D cosmological constant is then defined as the quantity that would give the same curvature in the low-energy 4D theory. This is a special case of a ‘matching’ calculation between the effective theory and its UV completion [46, 40]. The result found in [23] is easy to state when the bulk lagrangian density has the Einstein- Maxwell-scalar form of interest here: X N Ab X N Ab Λ = δL − δ = δT − δ , (4.194) b 2 r2 g2 b r2 g2 b B φ∗ b B φ∗

1 mn 2 where 2 Fmn = f = N /(2r ) is the background rugby-ball bulk ﬂux and the subscript φ∗ 2 indicates that δTb and δ(Ab/gB ) are to be evaluated at the classical background conﬁguration for any bulk scalar(s).

At first sight the only difference between this and the naive ‘probe-brane’ expectation, Λprobe = P b δLb, seems to be small: the additional contribution of the δAb term in the first equality of eq. (4.194). (Physically, this additional contribution to Λ arises from the energy cost imposed by adjustments to the bulk flux caused by flux quantization when the bulk volume changes in response 2 to the altered defect angle due to δLb.) Furthermore, the suppression by 1/r of the δAb terms relative to the δTb terms make it tempting to conclude that the new terms are always negligible. There are two reasons why this intuition breaks down for the 6D supergravity of interest in this paper. First, the supergravity has a classical flat direction in the absence of the branes which is stabilized by the brane-bulk couplings, implying that the brane action is itself important in determining the value for φ∗. For 6D supergravity, the position φ∗ where this stabilization occurs is related to

δLb by [23] X N Ab 1 δL − δ + L0 = 0 , (4.195) b 2 r2 g2 2 b b B φ∗ 0 where Lb := ∂Lb/∂φ where φ is the 6D dilaton. When the value of φ∗ is determined by the interplay of the brane action and the ﬂux-quantization condition the second reason for believing the δAb term

198 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

to be unsuppressed becomes operative: ﬂux quantization ensures φ∗ takes a value that makes Ab and

Tb the same order of magnitude, as found above in eqs. (4.32) and (4.33). The same is not true for higher terms in the derivative expansion of Sb because these do not enter into the ﬂux-quantization condition. Using eq. (4.195) in eq. (4.194) gives the classical supergravity result of [23]:

X 1 Λ = − L0 . (4.196) 2 b b φ∗

nφ Notice if Lb ∝ e this takes the simple form

X 1 n X Λ = − L0 = − L , (4.197) 2 b 2 b φ∗ b b φ∗ and so vanishes in particular when Lb is independent of φ (as is the case for the zeroeth-order brane P 2φ action for classical rugby ball solutions), and Λ = − b Lb if Lb ∝ e (as is true for one-loop corrections to the tension in these solutions).

4.5.2 Application to supersymmetric renormalizations

There are two complications to be checked before applying the results of [23] to the renormalized one-loop action of this paper: (i) both bulk and brane actions are renormalized; and (ii) both bulk and brane actions include corrections that are higher order in the derivative expansion. We next deal with the relevance of each of these in turn, specializing to the supersymmetric case where the bulk Maxwell ﬁeld is chosen to lie in the U(1)R direction, with unit ﬂux N = N = ±1.

Bulk counterterms

Renormalizations of bulk terms in a generic action can modify the classical ﬁeld equations, and so in general also their linearization around the bulk rugby-ball solutions. In particular they can cause the on-brane curvature to become nonzero. This is simplest to see in situations where a ﬂat on-brane metric is achieved by tuning the bulk cosmological constant, since for a generic theory this tuning need not be preserved under renormalization. Ref. [21] gives expressions for these corrections for a slightly broader class of theories than is considered in [23]. With this in mind there is much good news for the renormalizations of the 6D supergravity of sph sph interest here. First, the vanishing of s−1 and s0 (once summed over a supermultiplet) automatically ensures that neither U nor 1/κ2 get renormalized at all at one loop by the low-spin massive matter supermultiplet considered here. The bulk Maxwell action does get renormalized by a massive matter multiplet, however, as do

199 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy the curvature-squared and higher-derivative terms,

" 2 ! 3 !# √ R 1 R L δL = − −g M 2 eφ −F F MN + + −RF F MN + , (4.198) B MN 8 12 MN 8 (4π)3 where we use m2 = M 2eφ and deﬁne L := ln(M/m). We assume M/m to be independent of φ when diﬀerentiating, as is plausibly the case if the UV scale is a string theory (since then all Einstein- frame masses come with the same factor of eφ). As noted earlier, this vanishes once evaluated at a

MN 2 4 2 supersymmetric rugby ball — for which FMN F = 2f = 1/(2 r ) and R = R = −2/r . We now ask how these terms change the solutions to the background equations of motion. Differentiating eq. (4.198) with respect to φ and evaluating at the rugby ball background gives a vanishing result, and shows that δLB does not affect the background dilaton solution. Next, differentiating with respect to AM gives

Z Z √ R 4L δ d2x δL = + d2x −g M 2 eφ + F MN ∂ δA B 12 (4π)3 M N 2 2 Z 3κ M L 2 √ mn = − 1 − 2 3 4 d x −g ∂mδAn , (4.199) 2gR 3(4π) r which also vanishes at the rugby ball, for which the integrand is a total derivative.17 A similar √ argument applies when the metric is varied. In this case the variation of terms like −g gmngpq vanish once evaluated at the rugby ball background, leaving variation of the 2D curvature as the 1 only nontrivial quantity. Because in two dimensions one always has Rmnpq = 2 R(gmp gnq − gmq gnp) we may write R = R for this variation and so ﬁnd

Z Z √ 1 R2 L δ d2x δL = − d2x −g M 2 eφ R − F F mn + δR B 3 mn 8 4(4π)3 2 2 Z 3κ M L 2 √ = − 1 − 2 3 4 d x −g δR , (4.200) 2gR 3(4π) r which again involves the integral of a total derivative.

Higher-derivative brane counterterms

Since the background is unchanged by the renormalizations of the bulk action, the results of ref. [23] are almost directly applicable. The only remaining caveat is that renormalizations don’t just renormalize the first two terms of the action, eq. (4.64), but also generate the higher-derivative brane-bulk couplings. These can be neglected because (unlike for the brane-localized flux term, Ab) their effects 2 2 2 really are suppressed by powers of κ/r or gB /r . They are suppressed in this way because (unlike the Ab term) none of them are amplified by the flux-quantization condition.

17The surface terms associated with these total derivatives are not negligible, and contribute brane-localized terms once singular behaviour near the branes is excised by surrounding them with small Gaussian pillboxes. There they combine with the brane action and lead to the near-brane boundary conditions, such as the analogues of (4.20) for AM and gMN .

200 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

4.5.3 Loop-corrected 4D cosmological constant

We may now use eqs. (4.196) and (4.197) to write an expression for Λ at one loop in 6D supergravity compactified on a rugby ball stabilized by U(1)R flux, directly in terms of the renormalized quantities Vb and Vf . We do so under the assumption (necessary, but not sufficient, for unbroken supersymmetry) that the classical brane action is independent of φ: ∂Lb/∂φ = 0.

In this case the brane action acquires a dilaton dependence through its renormalization by bulk loops, with dilaton-dependence of the 1-loop corrected action arising through the powers of m, whose 4 4 2φ 2 2 2 φ appearance is dictated on dimensional grounds: δTb ∝ m = M e , δ(A/gB ) ∝ m = M e and similarly for the higher-derivative interactions. Consequently, loop-corrected brane action takes the form of eq. (4.64) with coeﬃcients

(0) (1) 4 2φ Tb(φ) = Tb + cT b M e L + ··· , (4.201)

A (φ) A(0) ζ (φ) ζ(0) b = b + c(1) M 2 eφ + ··· , Rb = Rb + c(1) M 2 eφ + ··· , (4.202) 2 2 A˜b L Rb L 2gB 2gB κ κ

κζ (φ) κζ(0) κζ (φ) κζ(0) Ab = Ab + c(1) + ··· , AR˜ b = AR˜ b + c(1) + ··· , (4.203) 2 2 Ab L 2 2 AR˜ b L 4gB 4gB 2gB 2gB and (0) (1) ζ 2 (φ) = ζ + c + ··· , (4.204) R b R2b R2b L etc., with coeﬃcients that are directly given by the brane renormalization equations, eqs. (4.66):

δs0 δs1 δs0 c(1) = − 0 , c(1) = − 1 , c(1) = − 1 T b 2(4π)2 A˜b (4π)2N Rb 2(4π)2 2 δs2 δs1 δs0 c(1) = − 2 , c(1) = − 2 , c(1) = − 2 , (4.205) Ab (4π)2N 2 AR˜ b 2(4π)2N R2b 4(4π)2 and so on, where the second equality specializes to the result computed earlier for a massive matter multiplet. All of the 1-loop terms are therefore suppressed by e2φ relative to the choices that would have been invariant under the classical scaling symmetry, eq. (4.17).

Incorporating the bulk back-reaction ﬁnally leads to a formula for the eﬀective 4D cosmological constant of the form X Λ = Λb + Λf , (4.206) b where the explicit µ-dependence that Λf inherits from Vf is canceled by the implicit µ-dependence of the renormalized couplings in Λb (just as the µ-dependence in V canceled between Vf on one hand and Vb and VB on the other). This µ-independence is most usefully exploited by choosing µ, so that all of the large-M dependence resides in Λb rather than in Λf .

Explicitly, Λb is obtained from Vb by evaluating eq. (4.196) at the rugby-ball background, as well

201 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

as VB = 0. Using eqs. (4.205) in eq. (4.197) we ﬁnd the most UV sensitive part of Λ is

(1) (1) ! (1) 4 2φ 1 2cRb N cA˜b 2 φ Λb = −c M e L − − − M e L T b 2 r2 r2 C := , (4.207) (4πr2)2 where 0 4 2 δs0 κM 1 0 1 κM C = L − δs1 + δs1 L + ··· . (4.208) 2 2 gR 2 2 gR There are several noteworthy features about this result. First, it vanishes (for all ω) in the supersymmetric limit where η → 0. Second, it is of order 1/(4πr2)2 even if κM 2 'O(1). As noted in [45], this size is a consequence of the ﬂux stabilization which, through eq. (4.13), ensures the loop-counting parameter is e2φ ∝ 1/r4. Finally, notice that massless multiplets, such as the gauge- and (hundreds of) hyper-multiplets required for anomaly cancellation, do not contribute at all, since all of the terms k proportional to δs2 are φ-independent. In the case of a massive matter multiplet with η ≤ δω/2 (i.e. the case considered in the main text), C is given by

κM 4 η (ω2 − 1) η3 κM 2 C = η L − + η2 − L + ··· (4.209) 2 gR 3 2ω 3 2 gR

P Although nothing conclusive can be said about the sign of Λbranes := b Λb for arbitrary values of 2 ω, η and M (as was done for the eﬀective potential beta functions in §4), taking (κM/2gR) > 1/3 2 guarantees positive Λbranes for a range of η’s near η = 0. Similarly, taking (κM/2gR) < 1/3 guarantees negative Λbranes near η = 0. Also, if we take the sphere limit (i.e. the case in Appendix 0 1 4.C where η ≥ δω with δω → 0), we ﬁnd that δs0 = δs1 + δs1 = 0 for any η.

4.6 Conclusions

We close with a brief summary and a sketch of the most dangerous bulk-brane higher loops.

Summary

This paper uses the recent results of [21] to compute the one-loop 1PI quantum action for 6D gauged, chiral supergravity [15], evaluated at a rugby-ball solution [11] to its classical field equations. By carefully tracking the near-brane boundary conditions as a function of the brane action [36, 23], we are able to include the effects of bulk back-reaction to these loop corrections. Our main focus is to identify the UV-sensitive part of the result, to see how it generates local effective interactions in the bulk and on the brane and how these interactions depend on the assumed properties of the branes and bulk. Because the rugby ball geometries are curved, they capture many UV-sensitive interactions that are not seen in the more familiar quantum calculations on tori [47].

202 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

We find that bulk supersymmetry strongly constrains the renormalization of bulk interactions. Renormalizations, such as corrections to the gauge coupling function and to higher-curvature terms, do occur, but with coefficients that are related to one another by supersymmetry. This is consistent with general expectations based on anomaly cancellation arguments in six dimensions [34]. The total bulk contribution to the effective vacuum energy vanishes, due to cancellations these relations permit between between gauge and curvature renormalizations. The branes are not similarly assumed to be supersymmetric a-priori, and so we use the most general brane action expanded in a derivative expansion. The first two terms of this expansion can be physically interpreted as the brane tension (no derivatives) and the amount of background flux that is localized on the brane (one derivative). In general the presence of both these terms are required to allow low-energy perturbations to exist that are consistent with flux quantization [23], and their dependence on the bulk dilaton, φ, can be used to stabilize the one modulus of the bulk geometry through a 6D analogue of the 5D Goldberger-Wise [48] mechanism. Although branes generically break all supersymmetries we find that it is possible to couple them to the bulk in a way that preserves the unbroken supersymmetry of the bulk [16], provided three conditions are satisfied.

• The branes do not couple to the bulk dilaton at all;

• The total brane-localized ﬂux and brane tension are related by eq. (4.47).

• The defect angles are the same size at both branes (as would be automatic if the two branes were identical).

What is surprising about the second condition is that (at least in the case of identical branes) it is selected automatically as the bulk modulus adjusts to satisfy the flux-quantization conditions that drive the Goldberger-Wise mechanism in 6D, leading to a supersymmetric configuration for arbitrary dilaton-independent brane tensions. This need not remain possible once two-derivative √ terms and higher — such as δLb = −γ Bb R/κ, for example — are included in the brane action, so supersymmetry is expected to break once these are included. Not too surprisingly, the entire one-loop 1PI quantum action evaluated at the rugby ball vanishes when the brane also preserves supersymmetry, at least when the brane action is only kept out to one-derivative order. This ensures that the entire one-loop vacuum energy vanishes in this case. Since higher-derivative terms need not be supersymmetric we expect the one-loop vacuum energy not to vanish generically once their influence on bulk modes is included.

Dangerous higher loops

In the absence of brane-localized particles this would be the end of the story. However when brane particles are present larger eﬀects can be possible. The simplest way to compute these is to estimate how bulk loops would renormalize the properties of brane-localized ﬁelds, and in particular what dependence on φ they introduce. Then include this φ-dependence into a brane loop (which doesn’t

203 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy itself cost an additional factor of e2φ), using a heavy and non-supersymmetric brane particle. We do so here by assuming that all φ-dependence introduced by bulk loops enters (in 6D Einstein frame) through the φ-dependent bulk mass m2(φ) = M 2eφ, with powers of m2 appearing wherever they can on dimensional grounds. To see how this goes, consider as brane lagrangian the cartoon Standard-Model form

√ 1 µ 2 2 1 µν Lb = − −γ ∂µh∂ h + M h + ψ (D/ + λh)ψ + Fµν F , (4.210) 2 0 4 where h, ψ and Aµ are brane-localized scalar, spin-half and gauge ﬁelds, with F = dA. On dimensional grounds bulk renormalizations would be expected to renormalize this action by an amount

1 √ c c δL = − −γ 1 ∂ h∂µh + c m2(φ)h2 + ψ (c D/ + c λh)ψ + 5 F F µν + ··· , (4.211) b (4π)2 2 µ 2 3 4 4 µν in addition to the renormalizations of the tension, T , and other brane-field independent coefficients considered earlier. Here the ci are dimensionless coefficients that could be calculated using techniques similar to those used in earlier sections. As usual, a power of m2 only appears for the scalar mass term, since the fermion and gauge masses are respectively protected by chiral and gauge symmetries. Proceeding now to performing a loop of brane fields [22], we focus on the scalar loop. We expect this to give contributions to the brane tension of order

4 2 2 M k 2c1 2c2M m δT ' k ' M 4 1 − + 0 + ··· , (4.212) (4π)2 (4π)2 0 (4π)2 (4π)2 where k is a calculable number and we use

2 2 2 2 2 M0 + c2m /(4π) 2 c1 c2m M ' 2 ' M0 1 − 2 + 2 + ··· . (4.213) 1 + c1/(4π) (4π) (4π)

2 2 2 2 φ What is important is the term in (4.212) proportional to M0 m = M0 M e , since this is of order 2 2 2 4 M /(16π r) (as opposed to 1/r ) when M ∼ M ∼ 1/gR ∼ M6 are much larger than 1/r. Terms independent of m2 are not dangerous since the arguments of §5 ensure that they drop out of Λ once back-reaction is included. Terms involving four powers (or more) of m are also not dangerous because they are proportional to (at least) e2φ ∝ 1/r4. Notice the dangerous term requires both of the following two ingredients:

• A brane-localized scalar, since the dangerous term only comes from scalar masses on the brane which are the only super-renormalizable interaction that is not protected by a symmetry and so can be shifted by m2(φ) (an intriguing connection to the ordinary hierarchy problem.)

• A massive bulk supermultiplet, since the dangerous terms arise from powers of m2.

The easiest way to avoid the dangerous terms is simply to postulate the absence of any massive multiplets, since these are not required by particle physics (unlike the required existence of massive Standard Model particles on the branes). This would remove the low-energy part of the cosmological

204 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy constant problem (see, e.g. [45] for a discussion), which is usually the hardest part. But it leaves open that part of the problem that asks why the UV completion (presumably a string theory) does not give a large contribution. The upshot is this: the generic size of UV eﬀects in a scenario with nonsupersymmetric branes coupled to a supersymmetric bulk is of order M 2m2/(4π)4, where M is a large brane scale and m is the KK scale. In the absence of massive brane states this leading term can vanish, leaving terms of order the KK scale alone. None of this is generic to an arbitrary extra-dimensional setup. Three additional ingredients appear to be required [11, 45]:

• The classical scale invariance of the bulk supergravity, and the associated zero mode that survives classical ﬂux stabilization of the extra-dimensional bulk;

• The systematic inclusion of brane back-reaction of the branes on the bulk geometry;

• The possibility of having codimension-2 branes with brane-localized flux, Φb, that can break this scale-invariance and so lift the flat direction through the interplay of back-reaction and flux-quantization.

The crucial role played by back-reaction in this mechanism underlines its importance, particularly for the dynamics of low-codimension objects (for which the inter-brane forces do not fall oﬀ appreciably with distance). Although this is understood reasonably well for codimension-1 objects (through the Israel junction conditions [49]), it is just beginning to be explored for higher codimension (and in particular codimension-2 [36, 23]). It is rare to ﬁnd such a vast region of unexplored territory in particle physics, and its exploration is likely to contain other surprises as well.

Acknowledgements

We would like to thank Riccardo Barbieri, Oriol Pujolàs,Fernando Quevedo, Seifallah Randjbar- Daemi and George Thompson for helpful discussions and Hyun-Min Lee for much help trying to disentangle the supergravity sector in early stages of this work. The matter supermultiplet heat- kernel calculation on spheres given in Appendix 4.A was first computed in unpublished work with Doug Hoover. Various combinations of us are grateful for the the support of, and the pleasant environs provided by, the Abdus Salam International Center for Theoretical Physics, Perimeter Institute and McMaster University. CB’s research was supported in part by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information (MRI). SLP is funded by the Deutsche Forschungsgemeinschaft (DFG) inside the “Graduiertenkolleg GRK 1463”. The work of AS was supported by the EU ITN “Unification in the LHC Era”, contract PITN-GA-2009-237920 (UNILHC) and by MIUR under contract 2006022501.

205 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

4.A Heat kernels and bulk renormalization

In this appendix we collect for convenience the explicit expressions for the heat-kernel coeﬃcients for low-spin matter supermultiplets in 6D theories compactiﬁed on a 2-sphere.

Gilkey-de Witt coeﬃcients

Heat-kernel methods provide very general results for the form of UV divergences in the presence of various background fields. Consider, for example, a collection of N fields, assembled into a column i a vector, Ψ, and coupled to a background spacetime metric, gMN , scalars, ϕ , and gauge fields, AM , with background-covariant derivative, DM , of the form

a DM Ψ = ∂M Ψ + ωM Ψ − iAM taΨ . (4.214)

Here ωM is the spin connection, and the gauge group is represented by the hermitian matrices ta.

The commutator of two such derivatives deﬁnes the matrix-valued curvature, YMN Ψ = [DM ,DN ]Ψ, which has the following form: a YMN = RMN − iFMN ta . (4.215)

Here RMN is the curvature built from the spin connection ωM , which is related to the Riemann curvature of the background spacetime in a way which is made explicit in the heat-kernel Appendix of ref. [21]. Suppose further that the one loop quantum action for such a ﬁeld is given by

1 iΣ = −(−)F Tr log − + X + m2 , (4.216) 2

F MN where (−) = + for bosons and − for fermions, and = g DM DN , X is some local quantity built from the background ﬁelds and m2 is the 6D mass matrix. In dimensional regularization the divergent part of this quantity can be written as [42]

3 1 X Z √ Σ = (−)F Γ(k − 3 + ε) d6x −g tr [m6−2k a ] (4.217) ∞ 2(4π)3 k k=0 where the divergences as ε → 0 arise from the poles of Euler’s gamma function, Γ(z), at non-positive integers. Specializing these expressions to a rugby ball and comparing to eqs. (4.55) and (4.56), clearly ak is proportional to sk−1 of the main text.

What is most useful about this expression is that in the absence of branes the coefficients, ak, are explicitly known matrix-valued local quantities constructed from X and YMN . In our conventions the first four Gilkey coefficients are [42]

a0 = I 1 a = − (R + 6X) (4.218) 1 6

206 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

1 a = 2R RABMN − 2R RMN + 5R2 − 12 R 2 360 ABMN MN 1 1 1 1 + RX + X2 − X + Y Y MN 6 2 6 12 MN 1 a = −18 2R + 17D RDM R − 2D R DLRMN − 4D R DN RML 3 7! M L MN L MN K MNLP MN M L N +9DK RMNLP D R + 28RR − 8RMN R + 24R N D D RML 35 14 14 +12R RMNLP − R3 + RR RMN − RR RABMN MNLP 9 3 MN 3 ABMN 208 64 16 + RM R RNL − RMN RKL R + RM R RNKLP 9 N ML 3 MKNL 3 N MKLP 44 80 − RAB R RMNKL − RA M R RBKNP 9 MN ABKL 9 B N AKMP 1 + 8D Y DM Y NK + 2DM Y D Y NK + 12Y MN Y (4.219) 360 M NK NM K MN M N K MNKL M NK −12Y N Y K Y M − 6R YMN YKL + 4R N YMK Y

MN 2 M 3 −5RY YMN − 6 X + 60XX + 30DM XD X − 60X

MN MN M 2 −30XY YMN + 10R X + 4R DM DN X + 12D RDM X − 30X R 2 MN ABMN + 12X R − 5XR + 2XRMN R − 2XRABMN R , where I is the unit matrix.

Specialized to the product of 4D Minkowski space with a 2-sphere, the coeﬃcients a1 through a3 simplify to

a0 = I 1 a = − R − X 1 6 1 1 1 1 a = R2 + RX + X2 + Y Y mn (4.220) 2 60 6 2 12 mn 1 1 1 1 a = − R3 − Y m Y n Y l − RY Y mn − XY Y mn 3 630 30 n l m 40 mn 12 mn 1 1 1 − X3 − X2 R − XR2 . 6 12 60

Spins zero through one

We now collect the results for X, YMN and the ultraviolet-divergent parts of the one-loop action, for the particles arising in 6D matter gauge- and hyper-multiplets. We assume also the fields in the loop do not mix appreciably with the supergravity sector, so in particular any gauge fields considered cannot be those whose background flux stabilizes the extra dimensions.

Scalars

Consider N scalars, ΦI , with action,

Z √ 1 1 1 S = − d6x −g gMN G D ΦiD Φj + V + UR + WF a F MN . (4.221) 2 ij M N 2 4 MN a

207 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

The functions U, V , W and the target-space metric, Gij, are imagined to be known functions of the Φi. The background-covariant derivative appropriate to this case is:

i i a i j DM Φ = ∂M Φ − iAM (ta) jΦ , (4.222)

i where the matrices (ta) j represent the gauge group on the scalars. The kinetic operator controlling small ﬂuctuations about a classical background is given by

i i i ∆ j = −δ j + X j , (4.223)

i with X j given by

h 1 1 i Xi = Gik V (ϕ) + RU (ϕ) + F a F MN W (ϕ) , (4.224) j kj 2 kj 4 MN a kj where subscripts on U, V and W denote differentiation with respect to the background field ϕi. i Specializing to the simple geometry and Maxwell fields of the rugby ball, these simplify to X j = ik 1 1 2 G [Vkj + 2 RUkj + 2 f Wkj] and Ymn = −igf˜ Q mn, whereg ˜ is the gauge coupling andgQ ˜ = ta is the hermitian, antisymmetric charge matrix for the background gauge field. Notice that these imply mn 2 2 2 m n l YmnY = −2˜g f Q and Y n Y l Y m = 0.

With these expressions the coeﬃcients a0 through a3 satisfy

N tr a = N , tr a = − R − tr X, (4.225) 0 1 6 and

N 1 1 1 tr a = R2 + R tr X + tr X2 − g˜2f 2 tr Q2 (4.226) 2 60 6 2 6 N 1 1 tr a = − R3 + R g˜2f 2 tr Q2 + g˜2f 2 tr (XQ2) 3 630 20 6 1 1 1 − tr X3 − R tr X2 − R2 tr X. 6 12 60

These give explicit functions of ϕ once the above expression for X is used.

Spin-half fermions

For N 6D massless spin-half Weyl fermions, ψa with a = 1, ..., N , we take the following action

Z √ 1 a S = − d6x −g G (ϕ) ψ Dψ/ b , (4.227) 2 ab

M A whereD / = eA γ DM with

1 D ψa = ∂ ψa − ωAB γ ψ − iAa t ψ , (4.228) M M 4 M AB M a

A M 1 where γ are the 6D Dirac matrices and eA the inverse sechsbein, γAB = 2 [γA, γB ], and ta de-

208 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy notes the gauge-group generator acting on the spinor ﬁelds. Since 6D Weyl spinors have 4 complex components their representation of the 6D Lorentz group has d = 8 real dimensions.

1 1 iΣ = Tr logD / = Tr log −D/ 2 1/2 2 4 1 1 1 = Tr log − − R + γAB F a t . (4.229) 4 4 4 AB a

This allows us to adopt the previous results for the ultraviolet divergences, provided we divide the result by an overall factor of 2 (and so eﬀectively d = 4 instead of 8), and use

1 1 X = − R + γAB F a t , (4.230) 4 4 AB a and i Y = − R γAB − iF a t . (4.231) MN 2 MNAB MN a The Gilkey coeﬃcients become18

N Tr [Y Y MN ] = −4 tr (t t ) F a F bMN − R RABMN 1/2 MN 1/2 a b MN 2 ABMN N = −8g ˜2f 2 tr (Q2) − R2 , (4.232) 1/2 2 where the second line specializes to rugby ball background ﬁelds. Keeping explicit the sign due to statistics, and dropping terms which vanish when traced, this leads to the following expressions for the divergent contributions of N 6D Weyl fermions:

N (−)F Tr [a ] = −4N , (−)F Tr [a ] = − R 1/2 0 1/2 1 3 N 4 (−)F Tr [a ] = R2 − g˜2f 2 tr (Q2) (4.233) 1/2 2 60 3 1/2 N 2 (−)F Tr [a ] = − R3 + g˜2f 2 R tr (Q2) . 1/2 3 504 15 1/2

Massless gauge bosons

a a For N gauge bosons, AM , with ﬁeld strength FMN and a = 1, ..., N , we use the usual Yang-Mills action Z √ 1 S = − d6x −g W (ϕ) F a F MN , (4.234) 4 MN a a a a expanded to quadratic order about the background ﬁelds: AM = AM + δAM . For an appropriate

18We adopt the convention of using Tr [...] to denote a trace which includes the Lorentz and/or spacetime indices, while reserving tr [...] for those which run only over the ‘ﬂavor’ indices which count the ﬁelds of a given spin.

209 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy choice of gauge the diﬀerential operator which governs the loop contributions becomes

aM a M aM ∆ bN = −δ b δ N + X bN , (4.235) with

aM M a a cM X bN = −R N δ b + 2i(tc) bF N , (4.236) where tc here denotes a gauge generator in the adjoint representation. Ref. [21] sums the contributions of the vector ﬁelds and ghosts to get the contribution of N physical 6D massless gauge bosons:

N (−)F Tr [a ] = 4N , (−)F Tr [a ] = R (4.237) 1 0 1 1 3 N 10 (−)F Tr [a ] = R2 + g˜2f 2 tr (Q2) 1 2 15 3 1 2N 7 (−)F Tr [a ] = − R3 + g˜2f 2 R tr (Q2) . 1 3 315 10 1

Supermultiplets

In this section, we show that the Gilkey coefficients cancel when summed over the field content of a gauge- or hypermultiplets, providing that the background flux does not break supersymmetry. Recall that unbroken supersymmetry requires the background gauge field to lie in the R-symmetry direction.

Gauge Multiplets

A gauge multiplet involves one Weyl spinor and one gauge boson. Specializing these earlier results to rugby-ball background ﬁelds, we have for Ng Weyl fermions

2N (−)F Tr [a ] = −4N , (−)F Tr [a ] = g 1/2 0 g 1/2 1 3 r2 N N 2 (−)F Tr [a ] = g − tr (Q2) (4.238) 1/2 2 15 r4 3 r4 1/2 N N 2 (−)F Tr [a ] = g − tr (Q2) , 1/2 3 63 r6 15 r6 1/2

2 2 2 where N = gB /gR is the background ﬂux quantum number.

Similarly, Ng massless spin-1 particles gives

2N (−)F Tr [a ] = 4N , (−)F Tr [a ] = − g (4.239) 1 0 g 1 1 3 r2 4N 5N 2 (−)F Tr [a ] = g + tr (Q2) 1 2 15 r4 6 r4 1 16N 7N 2 (−)F Tr [a ] = g − tr (Q2) . 1 3 315 r6 20 r6 1

210 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

These sum to give the following result for Ng massless 6D gauge supermultiplets:

F F Tr g[(−) a0] = 0 , Tr g[(−) a1] = 0 N N 2 5N 2 Tr [(−)F a ] = g − tr (Q2) + tr (Q2) (4.240) g 2 3 r4 3 r4 1/2 6 r4 1 N N 2 7N 2 Tr [(−)F a ] = g − tr (Q2) − tr (Q2) . g 3 15 r6 15 r6 1/2 20 r6 1

For the supersymmetric compactiﬁcation we must use unit background ﬂux, N 2 = 1, and the 2 2 charge assignments for U(1)R: tr 1(Q ) = 0 and tr 1/2(Q ) = Ng. In this case the above formulae simplify to F F F F Tr g[(−) a0] = Tr g[(−) a1] = Tr g[(−) a2] = Tr g[(−) a3] = 0 , (4.241) as claimed in the main text.

Hypermultiplets

Hypermultiplets contain one Weyl fermion and two complex scalars, with the scalars carrying charge

±1 under the gauge group U(1)R. The fermionic contribution to the vacuum energy is as in eq. (4.238). For the scalars we may use the results of eq. (4.226), specialized to the hypermultiplet lagrangian, for which

1 U = W = 0 and V = 2g2 eφ v(Φ) where v(Φ) = 1 + G ΦiΦj + ··· , (4.242) R 2 ij which imply i 2 φ ik 1 i X j = 2gR e G vkj = δ j . (4.243) Φ=0 2 r2 2 2 2 2 It follows that for Nh hyperscalars Tr 0I = 4Nh, Tr 0X = 2Nh/r , Tr 0(XQ ) = Tr 0(Q )/(2 r ), 2 4 3 6 Tr 0(X ) = Nh/r and Tr 0(X ) = Nh/(2 r ). Using these expressions we have the following spin-0 contribution for Nh hyperscalars:

2N Tr [(−)F a ] = 4N , Tr [(−)F a ] = − h 0 0 h 0 1 3 r2 N N 2 Tr [(−)F a ] = h − tr (Q2) (4.244) 0 2 10 r4 24r4 0 N N 2 Tr [(−)F a ] = h − tr (Q2) , 0 3 1260 r6 240r6 0

2 2 2 where, as before, N = gB /gR.

Summing this with eq. (4.238) for Nh Weyl fermions gives the result for Nh hyper-multiplets

F F Tr h[(−) a0] = 0 , Tr h[(−) a1] = 0 N N 2 N 2 Tr [(−)F a ] = h − tr (Q2) − tr (Q2) (4.245) h 2 6 r4 3 r4 1/2 24 r4 0 N N 2 N 2 Tr [(−)F a ] = h − tr (Q2) − tr (Q2) , h 3 60 r6 15 r6 1/2 240 r6 0

211 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 2 2 which, with the supersymmetric choices N = 1, tr 1/2(Q ) = 0 and tr 0(Q ) = 4Nh, gives the simple result

F F F F Tr h[(−) a0] = Tr h[(−) a1] = Tr h[(−) a2] = Tr h[(−) a3] = 0 , used in the main text.

4.B Results for spins zero, half and one

This appendix brieﬂy summarizes the results for si for spins zero, half and one, as computed in ref. [21].

Spin zero

2 Consider first the simplest case of a single minimally coupled real scalar field, satisfying (−+m )φ = 0, that is coupled to the background gauge field with monopole number N and brane–localized fluxes

Φb. In this case, using the notation ω = 1/α and

(n) X n (1) Fb := |Φb| (1 − |Φb|) ,F := Fb ,F := F,G(x) := (1 − x)(1 − 2x) , (4.246) b we ﬁnd the following for the si coeﬃcients:

1 ss = , −1 ω 1 1 ω2 ss (ω, N, Φ ) = + (1 − 3F ) , 0 b ω 6 6 " # 1 1 N 2 ω2 ω3N X ω4 ss (ω, N, Φ ) = − + (1 − 3F ) − Φ G(|Φ |) + (1 − 15F (2)) , 1 b ω 180 24 18 12 b b 180 b " 1 1 11 N 2 1 N 2 ω3N X ss (ω, N, Φ ) = − − + − (1 − 3F )ω2 − Φ G(|Φ |) 2 b ω 504 720 90 144 24 b b b ω4(1 − N 2) ω5N X + (1 − 15F (2)) − Φ G(|Φ |)(1 + 3F ) (4.247) 360 120 b b b b ! # 1 F (2) F (3) + − − ω6 . 1260 120 60

When ω = 1 and Φb = 0, these become

1 1 N 2 ssph = 1 , ssph = , ssph, 0 = , ssph, 2 = − , (4.248) −1 0 3 1 15 1 24 4 N 2 ssph, 0 = , and ssph, 2 = − 2 315 2 40

212 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy in agreement with the results in [50, 51], as well as with the result as computed using the Gilkey-de Witt coeﬃcients for a 6D scalar on a sphere using the general results found in [42]. If the scalar couples to the background ﬁeld with strength qgB , its contribution to the running of the leading bulk counterterms therefore is

∂U m6 ∂ 1 m4 µ = − , µ = − , ∂µ 6(4π)3 ∂µ κ2 6(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , (4.249) ∂µ κ 60(4π)3 ∂µ 630(4π)3 2 2 2 ∂ 1 2 q m ∂ κζAR 2 q µ 2 = 3 , µ 2 = 3 . ∂µ gB 3(4π) ∂µ gB 5(4π)

The quantities relevant to brane renormalizations are δs−1 = 0,

ω2 − 1 ωF 1 δω δω2 ω2F δω |Φ | δs = − b = + − b ' − b , 0 12 ω 2 ω 6 12 2 6 2 1 ω2 − 1 ω4 − 1 ω2F ω4F 2 δs0 = + − b − b 1 ω 36 360 6 12 1 δω 2 δω2 δω3 δω4 ω2F ω4F 2 δω |Φ | = + + + − b − b ' − b , ω 15 45 90 360 6 12 15 6 ω2N δs1 = − Φ G(|Φ |) , 1 12 b b N 2 δs2 = s2 − − = 0 , (4.250) 1 1 24 ω " # 1 ω2 − 1 ω4 − 1 ω6 − 1 F ω2F 2 ω4F 2 ω4F 3 δs0 = + + − ω2 b + b + b + b 2 ω 180 720 2520 30 24 120 60 " 1 2 δω 5 δω2 17 δω3 37 δω4 δω5 δω6 = + + + + + ω 105 252 1260 5040 420 2520 # F ω2F 2 ω4F 2 ω4F 3 −ω2 b + b + b + b , 30 24 120 60 ω2N ω4N δs1 = − Φ G(|Φ |) − Φ G(|Φ |)(1 + 3F ) , 2 24 b b 120 b b b N 2 ω2 − 1 ω4 − 1 ω2F ω4F 2 δs2 = − + − b − b 2 ω 288 720 48 24 N 2 δω 17 δω2 δω3 δω4 ω2F ω4F 2 = − + + + − b − b . ω 80 1440 180 720 48 24

The corresponding contributions to the running of the brane counterterms are

∂T m4 δω δω2 ω2F m4 δω µ b = + − b ' − |Φ | , ∂µ 2(4π)2ω 6 12 2 4(4π)2 3 b 2 2 2 2 ∂ Ab qΦb ω m q m Ab µ 2 = − 2 G(|Φb|) ' − 3 , ∂µ gB 6(4π) 3(4π)

213 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

∂ ζ m2 δω 2 δω2 δω3 δω4 ω2F ω4F 2 µ Rb = + + + − b − b ∂µ κ 2(4π)2ω 15 45 90 360 6 12 m2 δω |Φ | ' − b , 2(4π)2 15 6 2 2 2 ∂ κζAR˜ b qΦb ω ω q Ab µ 2 = − 2 G(|Φb|) + G(|Φb|)(1 + 3Fb) ' − 3 , (4.251) ∂µ gB 24(4π) 5 10(4π) " 2 3 4 5 6 ∂ ζ 2 1 2 δω 5 δω 17 δω 37 δω δω δω µ R b = + + + + + ∂µ 4(4π)2ω 105 252 1260 5040 420 2520 # F ω2F 2 ω4F 2 ω4F 3 −ω2 b + b + b + b 30 24 120 60 1 2 δω |Φ | ' − b , 4(4π)2 105 30 2 2 3 4 2 4 2 ∂ κζAb 8 q δω 17 δω δω δω ω Fb ω Fb µ 2 = − 2 + + + − − ∂µ gB (4π) ω 80 1440 180 720 48 24 q2 δω |Φ | ' − − b . (4π)2 10 6

Spin half

We next quote the results for a massive minimally coupled spin-half 6D Weyl ﬁeld minimally coupled to the background. Using the notation

1 1 δ N := N − σ , Φfσ := Φ − σΦf , Φf := 1 − ω−1 = (1 − α) = , (4.252) fσ b b 0 0 2 2 4π

19 f one finds different expressions depending on whether or not |Φb| is larger or smaller than Φ0. We f quote only the case |Φb| < Φ0, and refer the reader to [21] for the more general case. f For a 6D Weyl spinor the coefficients si become: s−1 = −4/ω, " ! # 1 1 1 X sf (ω, N, Φ ) = + − 2 Φ2 ω2 , 0 b ω 3 3 b b " ! 1 7 ωN Φ N 2 1 1 X sf (ω, N, Φ ) = − − + − Φ2 ω2 1 b ω 360 2 3 36 6 b b ! # ω3N X 7 1 X − Φ (1 − 4Φ2) + − Φ2(1 − 2Φ2) ω4 , (4.253) 6 b b 360 6 b b b b " ! 1 31 ωN Φ 31 N 2 7 N 2 X 7 X sf (ω, N, Φ ) = − − + − 1 − 6 Φ2 − Φ2 ω2 2 b ω 10080 16 720 1440 72 b 240 b b b ! ω3N X 7 7 N 2 (1 − 2N 2) X − Φ (1 − 4Φ2) + − − Φ2(1 − 2Φ2) ω4 24 b b 1440 720 24 b b b b ! X 7 Φ2 Φ4 −ω5N Φ − b + b b 240 6 5 b 19 f Both results agree when |Φb| = Φ0.

214 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

! # 31 X 7 Φ2 Φ4 + − Φ2 − b + b ω6 . 10080 b 240 12 15 b

sph In the limit ω → 1, Φb → 0 one ﬁnds s−1 = −4,

2 1 N 2 1 N 2 ssph = , ssph, 0 = , ssph, 2 = − , ssph, 0 = and ssph, 2 = − , (4.254) 0 3 1 15 1 3 2 63 2 15 in agreement with Gilkey-de Witt methods and those found in [50] for fermions on a sphere.

For a fermion with charge qgB , the corresponding contributions to the running of the bulk couplings are

∂U 2m6 ∂ 1 m4 µ = , µ = − , ∂µ 3(4π)3 ∂µ κ2 3(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , (4.255) ∂µ κ 60(4π)3 ∂µ 504(4π)3 2 2 2 ∂ 1 8 q m ∂ κζAR 8 q µ 2 = 3 , µ 2 = 3 . ∂µ gB 3(4π) ∂µ gB 15(4π)

f The quantities relevant for the brane action (when |Φb| ≤ Φ0) are δs−1 = 0 and

1 ω2 − 1 1 δω δω2 δs0 = − 2ω2Φ2 = + − 2ω2Φ2 , 0 ω 6 b ω 3 6 b 1 ω2 − 1 7(ω4 − 1) ω2Φ2 ω4Φ2(1 − 2Φ2) δs0 = + − b − b b 1 ω 72 720 6 6 1 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2(1 − 2Φ2) = + + + − b − b b , ω 15 180 180 720 6 6 N Φ ω2N Φ δs1 = − b − b (1 − 4Φ2) , δs2 = 0 , (4.256) 1 2 6 b 1 " 1 7(ω2 − 1) 7(ω4 − 1) 31(ω6 − 1) 7 ω2Φ2 ω4 δs0 = + + − b − Φ2(1 − 2Φ2) 2 ω 2880 2880 20160 240 24 b b # 7 Φ2 Φ4 −ω6Φ2 − b + b b 240 12 15 " 1 δω 101 δω2 17 δω3 257 δω4 31δω5 31 δω6 = + + + + + ω 42 2520 420 10080 3360 20160 # 7 ω2Φ2 ω4 7 Φ2 Φ4 − b − Φ2(1 − 2Φ2) − ω6Φ2 − b + b , 240 24 b b b 240 12 15 N Φ ω2N Φ 7 Φ2 Φ4 δs1 = − b − b 1 − 4Φ2 − ω4N Φ − b + b , 2 16 24 b b 240 6 5 N 2 ω2 − 1 7(ω4 − 1) ω2Φ2 ω4Φ2 δs2 = − + − b − b 1 − 2Φ2 2 ω 144 1440 12 12 b N 2 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2 = − + + + − b − b 1 − 2Φ2 . ω 30 360 360 1440 12 12 b

215 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

The brane counterterms therefore renormalize as follows:

∂T m4 δω δω2 m4 δω µ b = + − 2ω2Φ2 ' − Φ2 , ∂µ 2(4π)2ω 3 6 b (4π)2 6 b 2 2 2 2 ∂ Ab qΦb m ω 2 8 q m µ 2 = − 2 1 + (1 − 4Φb ) ' − 3 Ab , ∂µ gB (4π) 3 3(4π) ∂ ζ m2 δω 13 δω2 7 δω3 7 δω4 ω2Φ2 ω4Φ2(1 − 2Φ2) µ Rb = + + + − b − b b ∂µ κ 2(4π)2ω 15 180 180 720 6 6 m2 δω Φ2 ' − b , (4.257) 2(4π)2 15 3 2 2 4 ∂ ζAR˜ b qΦb 1 ω 2 4 7 Φb Φb µ 2 = − 2 + 1 − 4Φb + ω − + ∂µ gB (4π) 16 24 240 6 5 4 q2 ' − A , 15(4π)3 b " 2 3 4 5 6 ∂ζ 2 1 δω 101 δω 17 δω 257 δω 31δω 31 δω µ R b = + + + + + ∂µ 4(4π)2ω 42 2520 420 10080 3360 20160 # 7 ω2Φ2 ω4 7 Φ2 Φ4 − b − Φ2(1 − 2Φ2) − ω6Φ2 − b + b 240 24 b b b 240 12 15 1 δω Φ2 ' − b , 4(4π)2 42 10 2 2 3 4 2 2 4 2 ∂ ζAb 8 q δω 13 δω 7 δω 7 δω ω Φb ω Φb 2 µ 2 = − 2 + + + − − 1 − 2Φb ∂µ gB (4π) ω 30 360 360 1440 12 12 4 q2 δω ' − − Φ2 . 3(4π)2 5 b

Spin one

Massless spin one

We begin with the massless case. Deﬁning

δ N := N, Φgfξ := Φ − ξΦgf , Φgf = ω−1 = α = 1 − (4.258) gfξ b b 0 0 2π one obtains the following contributions to the bulk divergences

2 4 5 N 2 ssph = 4 , ssph = − , ssph, 0 = , ssph, 2 = , −1 0 3 1 15 1 6 16 7 N 2 ssph, 0 = , ssph, 2 = − (4.259) 2 315 2 20

216 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy and after these are subtracted the brane renormalizations are obtained from

1 (ω2 − 1) 1 δω δω2 δs = −(ω − 1) + − 2ω2F + ω2|Φ | = − + − 2ω2F + ω2|Φ | , 0 ω 3 b b ω 3 3 b b 1 (ω2 − 1) ω4 − 1 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 δs0 = + − b + b − b − b 1 ω 9 90 3 3 3 3 1 4 δω 8 δω2 2 δω3 δω4 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 = + + + − b + b − b − b , (4.260) ω 15 45 45 90 3 3 3 3 ω2N ω2N δs1 = − Φ G(|Φ |) + N Φ − Φ |Φ | , 1 3 b b b 2 b b 1 (ω2 − 1) ω4 − 1 ω6 − 1 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 δs0 = + + − b + b − b − b 2 ω 45 180 630 15 15 6 6 ω6F 2 ω6F 3 ω6|Φ |5 − b − b + b 30 15 10 1 8 δω 5 δω2 17 δω3 37 δω4 δω5 δω6 2 ω2F ω2|Φ | ω4F 2 = + + + + + − b + b − b ω 105 63 315 1260 105 630 15 15 6 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 − b − b − b + b , 6 30 15 10 ω2N ω4N ω2N Φ ω4N Φ δs1 = − Φ G(|Φ |) − Φ G(|Φ |)(1 + 3F ) − b |Φ | + b |Φ |3 , 2 6 b b 30 b b b 4 b 4 b N 2 ω − 1 ω2 − 1 ω4 − 1 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 δs2 = − + + − b + b − b − b 2 ω 8 72 180 12 24 6 6 N 2 7 δω 17 δω2 δω3 δω4 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 = − + + + − b + b − b − b ω 40 360 45 180 12 24 6 6

2 along with δs−1 = δs1 = 0 (as usual).

2 ∂ζR3 2 ∂ κζAR 14 q µ = − 3 and µ 2 = 3 , (4.261) ∂µ 315(4π) ∂µ gB 5(4π) in the bulk, and

2 3 4 5 6 ∂ζ 2 1 8 δω 5 δω 17 δω 37 δω δω δω µ R b = + + + + + ∂µ 4(4π)2ω 105 63 315 1260 105 630 2 ω2F ω2|Φ | ω4F 2 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 − b + b − b − b − b − b + b 15 15 6 6 30 15 10 1 2 δω |Φ | ' − b , (4π)2 105 60 2 4 2 4 ∂ κζAR˜ b q ω ω ω Φb ω Φb 3 µ 2 = − 2 Φb G(|Φb|) + Φb G(|Φb|)(1 + 3Fb) + |Φb| − |Φb| ∂µ gB (4π) 6 30 4 4 q Φ 2 q2 ' − b = − A , (4.262) 5(4π)2 5(4π)3 b

217 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 2 3 4 2 2 ∂ ζAb 8 q 7 δω 17 δω δω δω ω Fb ω |Φb| µ 2 = − 2 + + + − + ∂µ gB (4π) ω 40 360 45 180 12 24 ω4F 2 ω4|Φ |3 q2 7 δω |Φ | − b − b ' − − b , 6 6 (4π)2 5 3 on the brane.

Massive spin one

In this case, in the sphere limit we have

1 1 19 N 2 ssph, 0 = 5 , ssph, 0 = − , ssph, 0 = , ssph, 2 = , −1 0 3 1 3 1 24 4 3 N 2 ssph, 0 = , and ssph, 2 = − (4.263) 2 63 2 8 and so we obtain the bulk renormalizations

∂U 5 m6 ∂ 1 m4 µ = − , µ = , ∂µ 6(4π)3 ∂µ κ2 6(4π)3 2 ∂ ζ 2 m ∂ζ 3 1 µ R = − , µ R = − , (4.264) ∂µ κ 12(4π)3 ∂µ 126(4π)3 2 2 2 ∂ 1 19 q m ∂ κζAR 3 q µ 2 = − 3 , µ 2 = 3 . ∂µ gB 3(4π) ∂µ g˜ (4π)

The running of the brane couplings is similarly obtained by computing the δsi coeﬃcients:

1 δω 5 δω2 5 ω2F δs = − + − b + ω2|Φ | , 0 ω 6 12 2 b 1 δω 2 δω2 δω3 δω4 5 ω2F ω2|Φ | 5 ω4F 2 ω4|Φ |3 δs0 = + + + − b + b − b − b 1 ω 3 9 18 72 6 3 12 3 5 ω2N ω2N δs1 = − Φ G(|Φ |) + N Φ − Φ |Φ | , 1 12 b b b 2 b b 1 2 δω 25 δω2 17 δω3 37 δω4 δω5 δω6 ω2F ω2|Φ | δs0 = + + + + + − b + b 2 ω 21 252 252 1008 84 504 6 15 5 ω4F 2 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 − b − b − b − b + b , (4.265) 24 6 24 12 10 5 ω2N ω4N ω2N Φ ω4N Φ δs1 = − Φ G(|Φ |) − Φ G(|Φ |)(1 + 3F ) − b |Φ | + b |Φ |3 , 2 24 b b 24 b b b 4 b 4 b N 2 3 δω 17 δω2 δω3 δω4 5 ω2F ω2|Φ | 5 ω4F 2 ω4|Φ |3 δs2 = − + + + − b + b − b − b . 2 ω 16 288 36 144 48 24 24 6

These give

∂T m4 δω 5 δω2 5 ω2F µ b = − + − b + ω2|Φ | , ∂µ 2(4π)2ω 6 12 2 b

218 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

2 2 2 ∂ Ab 2 qm 5 ω ω µ 2 = 2 − Φb G(|Φb|) + Φb − Φb|Φb| , ∂µ gB (4π) 12 2 ∂ ζ m2 δω 2 δω2 δω3 δω4 5 ω2F ω2|Φ | µ Rb = + + + − b + b ∂µ κ 2(4π)2ω 3 9 18 72 6 3 5 ω4F 2 ω4|Φ |3 − b − b , 12 3 2 4 ∂ κζAR˜ b q 5 ω ω µ 2 = − 2 Φb G(|Φb|) + Φb G(|Φb|)(1 + 3Fb) ∂µ gB (4π) 24 24 ω2Φ ω4Φ + b |Φ | − b |Φ |3 , (4.266) 4 b 4 b 2 3 4 5 6 2 2 ∂ζ 2 1 2 δω 25 δω 17 δω 37 δω δω δω ω F ω |Φ | µ R b = + + + + + − b + b ∂µ 4(4π)2ω 21 252 252 1008 84 504 6 15 5 ω4F 2 ω4|Φ |3 ω6F 2 ω6F 3 ω6|Φ |5 − b − b − b − b + b , 24 6 24 12 10 2 2 3 4 2 2 ∂ ζAb 8 q 3 δω 17 δω δω δω 5 ω Fb ω |Φb| µ 2 = − 2 + + + − + ∂µ gB (4π) ω 16 288 36 144 48 24 5 ω4F 2 ω4|Φ |3 − b − b . 24 6

4.C Complete results for the massive multiplet

In this appendix, we compile the complete result for the renormalization of the south (b = −1) brane when η is not constrained to be η ≤ δω/2, but instead 0 ≤ η ≤ 1. At the end, we state the resulting P beta function for Vbranes := b Vb.

mm hm gm Given that δsi = δsi + δsi , and using the results from the main text for the hyper- and gauge multiplets, we ﬁnd that, on the south (b = −1) brane,

  2η , η ≤ δω/2  δs0 = −2ˆη , δω/2 ≤ η ≤ δω (4.267)   0 , η ≥ δω

 h 2 F ωF ω2F i  1 δω + δω + −η + −η (1 + 2η) − −η , η ≤ δω/2  ω 2 4 2 3 2  2 2 0 1 h δω δω Fηˆ ωFηˆ ω Fηˆ i δs1 = ω 2 + 4 + 2 + 3 (1 − 2ˆη) − 2 , δω/2 ≤ η ≤ δω (4.268)  h 2 2 i  1 δω δω Fηˆ ω Fηˆ  ω 2 + 4 + 2 − 2 , η ≥ δω

 1 h δω δω2 i  − − (1 + 2η) − ωF−η , η ≤ δω/2  ω 2 4 1  1 h δω δω2 i δs1 = − ω − 2 − 4 (1 − 2ˆη) − ωFηˆ , δω/2 ≤ η ≤ δω (4.269)  h 2 i  1 δω δω  − ω − 2 − 4 (1 − 2ˆη) , η ≥ δω

219 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

 h 2 3 4 F 2 F 2 1 δω + 7 δω + δω + δω − F−η + −η + F−η + −η ω(1 + 2η)  ω 8 48 12 48 16 12 30 10  2 4 i  F−η F−η 2 ω F−η  + − ω − , η ≤ δω/2  24 4 16  2 3 4 2 2 0 1 h δω 7 δω δω δω Fηˆ Fηˆ Fηˆ Fηˆ δs2 = ω 8 + 48 + 12 + 48 − 16 + 12 + 30 + 10 ω(1 − 2ˆη) (4.270)  F F 2 ω4F i  + ηˆ − ηˆ ω2 − ηˆ , δω/2 ≤ η ≤ δω  24 4 16  h 2 3 4 2 2 4 i  1 δω 7 δω δω δω Fηˆ Fηˆ Fηˆ Fηˆ 2 ω Fηˆ  ω 8 + 48 + 12 + 48 − 16 + 12 + 24 − 4 ω − 16 , η ≥ δω

 h 2 3 4 F ω2F ωF 2 i  1 − δω − δω − δω − δω + −η − −η (1 + 2η) − −η , η ≤ δω/2  ω 12 6 8 32 12 4 2  h 2 3 4 2  − 1 − δω − δω − δω − δω + Fηˆ − ω Fηˆ (1 − 2ˆη) 1  ω 12 6 8 32 12 4 δs2 = ωF 2 i (4.271)  − ηˆ , δω/2 ≤ η ≤ δω  2  h 2 3 4 2 i  1 δω δω δω δω Fηˆ ω Fηˆ  − ω − 12 − 6 − 8 − 32 + 12 − 4 (1 − 2ˆη) , η ≥ δω  h 2 3 4 ωF ω2F i  1 − δω + δω + δω + δω − −η (1 + 2η) + −η , η ≤ δω/2  ω 24 48 24 96 6 4  2 3 4 2 2 1 h δω δω δω δω ωFηˆ ω Fηˆ i δs2 = ω − 24 + 48 + 24 + 96 − 6 (1 − 2ˆη) + 4 , δω/2 ≤ η ≤ δω (4.272)  h 2 3 4 2 i  1 δω δω δω δω ω Fηˆ  ω − 24 + 48 + 24 + 96 + 4 , η ≥ δω which give the following renormalizations (recall thatη ˆ := η − δω):

 2η , η ≤ δω/2 4  ∂T− m  µ = 2 × −2ˆη , δω/2 ≤ η ≤ δω (4.273) ∂µ 2(4π)   0 , η ≥ δω

 2 2 δω + δω + F−η + ωF−η (1 + 2η) − ω F−η , η ≤ δω/2 2  2 4 2 3 2 ∂ ζR− m  2 2 µ = × δω δω Fηˆ ωFηˆ ω Fηˆ 2 2 + 4 + 2 + 3 (1 − 2ˆη) − 2 , δω/2 ≤ η ≤ δω ∂µ κ 2(4π) ω 2 2  δω δω Fηˆ ω Fηˆ  2 + 4 + 2 − 2 , η ≥ δω (4.274)  δω δω2  − 2 − 4 (1 + 2η) − ωF−η , η ≤ δω/2 2  ∂ ζ 2 m  2 µ A˜− = × − − δω − δω (1 − 2ˆη) + ωF , δω/2 ≤ η ≤ δω (4.275) ∂µ g2 (4π)2ω 2 4 ηˆ B  2  δω δω  − − 2 − 4 (1 − 2ˆη) , η ≥ δω

 2 3 4 F 2  δω + 7 δω + δω + δω − F−η + −η  8 48 12 48 16 12  F 2 F 2  + F−η + −η ω(1 + 2η) + F−η − −η ω2  30 10 24 4  4  − ω F−η , η ≤ δω/2  16  2 2 ∂ζ 2 1  2 3 4 F F F F µ R − = × δω + 7 δω + δω + δω − ηˆ + ηˆ + ηˆ + ηˆ ω(1 − 2ˆη) ∂µ 4(4π)2ω 8 48 12 48 16 12 30 10  2 4  Fηˆ Fηˆ ω Fηˆ  + − ω2 − , δω/2 ≤ η ≤ δω  24 4 16  2 3 4 F 2  δω 7 δω δω δω Fηˆ ηˆ  8 + 48 + 12 + 48 − 16 + 12  F 2 4  Fηˆ ηˆ 2 ω Fηˆ  + 24 − 4 ω − 16 , η ≥ δω (4.276)

220 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

 2 3 4 F ω2F − δω − δω − δω − δω + −η − −η (1 + 2η)  12 6 8 32 12 4  ωF 2  − −η , η ≤ δω/2  2 ∂ κζ 1  2 3 4 2 µ AR˜ − = × − − δω − δω − δω − δω + Fηˆ − ω Fηˆ (1 − 2ˆη) (4.277) ∂µ g2 (4π)2ω 12 6 8 32 12 4 B  ωF 2  ηˆ  + 2 , δω/2 ≤ η ≤ δω  2 3 4 2  δω δω δω δω Fηˆ ω Fηˆ  − − 12 − 6 − 8 − 32 + 12 − 4 (1 − 2ˆη) , η ≥ δω

2 3 4  δω δω δω δω ωF−η  − 24 + 48 + 24 + 96 − 6 (1 + 2η)  2  + ω F−η , η ≤ δω/2  4  2 3 4 ∂ κζA− 8 δω δω δω δω ωFηˆ µ 2 = 2 × − 24 + 48 + 24 + 96 − 6 (1 − 2ˆη) (4.278) ∂µ gB (4π) ω 2  ω Fηˆ  + 4 , δω/2 ≤ η ≤ δω  2 3 4 2  δω δω δω δω ω Fηˆ − 24 + 48 + 24 + 96 + 4 , η ≥ δω

Therefore, the total contribution of a massive matter multiplet to the running of Vb on the south brane is ∂V− ∂T− 1 ∂ ζA˜− 2 ∂ ζR− 4 ∂ζR2− µ = µ − 2 µ 2 − 2 µ + 4 µ ∂µ ∂µ 2 r ∂µ gB r ∂µ κ r ∂µ 1 ∂ κζAR˜ − 1 ∂ κζA− + 4 µ 2 + 4 µ 2 r ∂µ gB 8r ∂µ gB 2 4 2  − 5 + 5 ω + ω η2 + 1 − ω η4  48 24 16 12 4  (ω2−1) h η2 η4  − η2(mr)2 + ωη 2 − +  2 15 3 5  2 i  2 2 η 2 4  − 3 − 3 (mr) + (mr) , η ≤ δω/2     2 4 2 1  − 5 + 5 ω + ω (η − ω)2 + 1 − ω (η − ω)4 = × 48 24 16 12 4 (4.279) 2 2 2 h 2 4 (4πr ) ω  − (ω −1) η2(mr)2 + ωηˆ 1 + ηˆ − 4η ˆ +η ˆ3 − ηˆ  2 30 2 3 5  2 i  4 4η ˆ 2 4  − 3 − 2ˆη + 3 (mr) − (mr) , δω/2 ≤ η ≤ δω     2 4 2  − 5 + 5 ω + ω (η − ω)2 + 1 − ω (η − ω)4  48 24 16 12 4  2  (ω −1) 2 2 − 2 η (mr) , η ≥ δω

221 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

P Lastly, we can assemble the total contribution of both branes to Vbranes = b Vb:

 5 5 ω2 ω4 2 1 ω2 4  − 24 + 12 + 8 η + 6 − 2 η   2 2 2 h 2 η2 η4  −(ω − 1)η (mr) + ωη 15 − 3 + 5   2 2 η2 2 4i  − 3 − 3 (mr) + (mr) , η ≤ δω/2     2 4 2 2  − 5 ω + 11 ω − ω (ω −1) (mr)2  96 96 4  h 5 2 ω2 5 ω4 2 2i ω 2 ∂V 1  − − + + (ω − 1)(mr) η − µ branes = × 24 3 8 2 (4.280) 2 2 ω2 1 ω 4 h 1 ηˆ 4η ˆ2 3 ηˆ4 ∂µ (4πr ) ω  − − η − + ωηˆ + − +η ˆ −  2 6 2 30 2 3 5  2 i  − 4 − 2ˆη + 4η ˆ (mr)2 − (mr)4 , δω/2 ≤ η ≤ δω  3 3     2 4 2 2  − 5 ω + 11 ω − ω (ω −1) (mr)2  96 96 4  h 2 4 i  5 2 ω 5 ω 2 2 ω 2  − 24 − 3 + 8 + (ω − 1)(mr) η − 2   ω2 1 ω 4  − 2 − 6 η − 2 , η ≥ δω .

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E. Papantonopoulos and A. Papazoglou, “Brane-bulk matter relation for a purely conical codimension-2 brane world,” JCAP 0507 (2005) 004 [arXiv:hep-th/0501112].

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228 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

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229 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

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230 Chapter 5

Conclusion and Outlook

In this thesis we have considered two separate naturalness issues which generically come to bear on physical theories with large extra dimensions, and so a gravity scale Mg Mpl. In Chapter 2, we discussed how the violation of baryon number could be avoided by introducing an additional gauge boson. Due to the distinct nature of extra-dimensional gauge bosons—namely their unusually small gauge couplings—we enumerated constraints over broad range of masses, from the MeV to the TeV scale. We also tracked the potential impact of kinetic mixing with the hypercharge gauge boson by providing both gauge coupling constraints at various fiducial values of kinetic mixing, and also kinetic mixing constraints at various fiducial values of gauge coupling. In Chapter 3, we computed contributions to the 1-loop effective 4D potential due to bulk scalars 1 (spin 0), fermions (spin 2 ), and gauge fields (spin 1) in an extra-dimensional rugby-ball geometry. By modeling the two codimension-2 branes with the most general leading-order terms (namely, with both tensions and brane-localized fluxes), we computed the requisite running of various brane counterterms in order to obtain a final result that is independent of the renormalization scale. Since the 1-loop divergences are only sensitive to UV (i.e. short-wavelength) physics, the calculation herein demonstrates that: 1) bulk renormalization does not depend on the boundary conditions imposed at the branes; and 2) brane renormalization depends only on quantities specific to the brane. Chapter 4 showed how the results from Chapter 3 can be applied to a particular supersymmetric extra-dimensional model which was shown in [1] to have a vanishing contribution to the 4D vacuum energy at the classical level. By summing together the beta functions of various scalar, spinor, and vector fields, we computed the net renormalization effects of various supersymmetric matter multiplets. For each multiplet we also obtained its contribution to the renormalized 1-loop vacuum energy. In the case where the supergravity’s R-symmetry is gauged and where the corresponding brane- localized fluxes Φb are balanced (i.e. are equal at each brane), we found that each multiplet provides no net contribution to the vacuum energy. This is understood to be an artefact of a partial preser- vation of the 6D supersymmetry, since half of the extra-dimensional supersymmetry survives in this case. Such a configuration would lead inevitably to a supersymmetric gravity sector in 4D. This

231 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy result is unexpected since the brane lagrangian is explicitly non-supersymmetric, and since we enforce no special relationship between the brane couplings. Furthermore, the system will dynamically adjust to the supersymmetric minimum, regardless of the choice brane tensions and fluxes. With unbalanced fluxes, we find that—for small ∆Φ—the 4D vacuum energy contribution from a 6D matter multiplet with mass M is: a) positive; b) has a size set by the radius of the extra dimensions (and so serves as a demonstration of the technical naturalness of this model); and c) depends on M through the dimensionless quantity (κM/gR), where κ and gR are the TeV-scale gravity and R-symmetry gauge couplings, respectively.

New Directions

The TeV-scale phenomenology of supersymmetric large extra dimensions is likely to be very rich and intricate, and is fertile ground for original research since many aspects of the physics of codimension- 2 branes remain unrevealed. (Progress towards understanding the eﬀective scalar-scalar interactions has been made in [2].) If evidence of low-energy string excitations are to arise at the 14 TeV LHC run starting in 2015, it would certainly be desirable to have a quantitative prediction for the expected experimental signatures. Another question which remains unanswered is whether there exists a low-energy, 4-dimensional description of the large extra dimensions that would be valid at energy scales small compared to the KK scale. Such a description would necessarily contain in it a low-energy description of the dynamical cancellation mechanism which negates brane loop contributions to the 4D vacuum energy. We are at present considering a modiﬁcation of the low-energy matter lagrangian which takes the form

1 ∂L L → L˜ := L − gµν (5.1) 2 ∂gµν and which is derived from the leading-order eﬀects of back-reaction at low energies. (Substituting √ an arbitrary vacuum energy of the form L = − −g ρV demonstrates the desired cancellation.) The accompanying prescription: any physics whose energy gradients (either spatial or temporal) are slow compared to the KK scale are to be described using L˜. An immediate consequence of this modiﬁcation is that conformal systems (such as electromag- netism) remain unchanged, since the trace of stress-energy tensor,

2 ∂L T µν = √ , (5.2) −g ∂gµν vanishes. However, for massive objects—such as point particles or fluids—this modification changes both inertial and gravitational masses by a universal factor of 3/4. It appears that such a modification would have been detected, for example, in precision measurements of Avogadro’s number [3], and so a more careful dimensional reduction analysis is needed to decide whether such a description is compatible with experiment. Even with such a low-energy prescription in place, it remains necessary to reconsider physics at

232 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy energy scales between 0.01 eV and 10 TeV—keeping in mind the gradual effect of having more and more extra-dimensional KK modes available to excite—in order to pin down whether or not the precision dynamics of the standard model are significantly modified. Finally, understanding the high-energy origin of such a model (e.g. from string theory) is also topic of intense—although perhaps less pragmatic—interest.

233 Ph.D. Thesis—M. Williams—McMaster University—Dept. of Physics and Astronomy

234 Bibliography

[1] C. P. Burgess and L. van Nierop, “Technically Natural Cosmological Constant From Supersym- metric 6D Brane Backreaction,” [arXiv:hep-th/1108.0345].

[2] R. Diener and C. P. Burgess, “Bulk Stabilization, the Extra-Dimensional Higgs Portal and Missing Energy in Higgs Events,” JHEP 1305 (2013) 078, [arXiv:hep-ph/1302.6486].

[3] Andreas, B. et al., “Determination of the Avogadro Constant by Counting the Atoms in a 28Si Crystal,” Phys. Rev. Lett. 106 (2011) 030801.

235