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Accidental Supersymmetry and the Naturalness of Codimension-2 Branes ACCIDENTAL SUPERSYMMETRY AND THE NATURALNESS OF CODIMENSION-2 BRANES ACCIDENTAL SUPERSYMMETRY AND THE NATURALNESS OF CODIMENSION-2 BRANES By 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 R´emy 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.
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