Topological effects in particle physics phenomenology Joseph Enea Davighi Department of Applied Mathematics and Theoretical Physics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Jesus College October 2019 For my mum and dad Declaration This dissertation is based on original research carried out while the author was a graduate student at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Uni- versity of Cambridge, from October 2016 to July 2019. The material in Chapters 2, 3, and 5 is based on work done by the author under the supervision of Ben Gripaios, part of which is published in Refs. [1, 2]. Chapter 4 is based on work done with Ben Gripaios and Joseph Tooby-Smith [3]. In Chapter 6 we summarize work done with Benjamin Allanach [4, 5], and with Benjamin Allanach and Scott Melville [6]. Finally, Chapter 7 is based on as yet unpublished work done in collaboration with Ben Gripaios and Nakarin Lohitsiri. No part of this work has been submitted, or is being concurrently submitted, for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution. Joseph Enea Davighi October 2019 Acknowledgements I am deeply grateful to my supervisor Ben Gripaios. I could not have asked for a better teacher in the subject, one whose distinctive views on physics and mathematics have shaped my own over the course of many long and winding conversations. I am also grateful to Ben for introducing me to the beautiful subject of topology. On a more prosaic note, I thank Ben for teaching me the basics of academic writing. I am grateful to Ben Allanach for his sage advice not just about physics, but also con- cerning the early stages of a career in research. I thank Ben for putting me forward to give talks at workshops and conferences (especially Moriond 2019 where I learnt how to ski), and for fronting Rising DAMTP at two memorable Christmas parties. I also thank my other col- laborators Scott Melville, Joseph Tooby-Smith, Nakarin Lohitsiri, Marco Nardecchia, and Wolfgang Altmannshofer. It has been a special privilege to work with and learn from all of you. It’s been great fun studying for my PhD in DAMTP, and this is largely due to the com- pany of my fellow students. I would most like to thank Sam Crew for the pleasure and unpredictability of his company, for endless circular discussions about renormalization, and for his excellent services to Better Than TV, and my office mates Alex Abbott and Nakarin Lohitsiri who make every day at work enjoyable. I owe a debt to the QFT Reading Group (circa. 2016-2018), especially Alec Barns-Graham, Kai Roehrig, Sam Crew, and Nakarin Lohitsiri. Together, we were able to discover the elegance of Seiberg-Witten theory. I thank my other friends in Pavilion B for many hours of coffee and conversation, espe- cially Jonathan Rawlinson, Josh Kirklin, Alice Waterhouse, Amelia Drew, Bogdan Ganchev, Theodor Bjorkmo, Ed Walton, Felicity Eperon, Giuseppe Papallo, Alec Barns-Graham, Antoni Woss, Maeve Madigan, Ward Haddadin, Philip Boyle Smith, Dan Zhang, Shayan Iranipour, and of course Amanda Stagg for keeping Pav B going every day. I also thank Nicholas Manton, David Tong, and Pietro Genolini for some enlightening discussions, and Mihalis Dafermos for trying to convert me to GR. I thank the other members of the Cam- bridge Pheno Working Group, especially Tevong You, Bryan Webber, Christopher Lester, Alex Mitov, and Maria Ubiali. Through summer schools and conferences I have made a viii number of good friends who I have learnt a great deal from. I am most grateful to George Johnson, Jesse Liu, Mariana Carrillo González, and Darius Faroughy. This thesis is dedicated to my parents, Maria and John, who have given me so much. Most preciously, they have given me unreserved encouragement in all my interests for as long as I can remember. I also thank my wonderful siblings Katie and Laurence, and my close friends. Finally, I am lovingly indebted to Mya, who has brought me so much joy and excitement during the three years in which I worked for this PhD, and for three more years before. Abstract This thesis is devoted to the study of topological effects in quantum field theories, with a particular focus on phenomenological applications. We begin by deriving a general clas- sification of topological terms appearing in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space 퐺/퐻 (where 퐺 is an arbitrary Lie group and 퐻 ⊂ 퐺). Such models are ubiquitous in phenomenology; in three or more dimensions they cover all cases in which only some subgroup 퐻 of a dynamical symmetry group 퐺 is linearly realized in vacuo. The classification is based on the observation that, for topological terms, the maps from the worldvolume to 퐺/퐻 may be replaced by singular ho- mology cycles on 퐺/퐻. We find that such terms come in one of two types, which we refer to as ‘Aharonov-Bohm’ (AB) and ‘Wess-Zumino’ (WZ) terms. We derive a new condition for their 퐺-invariance, which we call the ‘Manton condition’, which is necessary and sufficient when the Lie group 퐺 is connected. Armed with this classification of topological terms, we then apply it to Composite Higgs models based on a variety of coset spaces 퐺/퐻 and discuss their phenomenology. For ex- ample, we point out the existence of an AB term in the minimal Composite Higgs model based on 푆푂(5)/푆푂(4), whose phenomenological effects arise only at the non-perturbative level, and lead to 푃 and 퐶푃 violation in the Higgs sector. Consideration of the Manton condition leads us to discover a rather subtle anomaly in a non-minimal model based on 푆푂(5) × 푈(1)/푆푂(4) (a model which does, however, feature an AB term not previously no- ticed in the literature). A particularly rich topological structure, with six distinct terms of various types, is uncovered for the model based on 푆푂(6)/푆푂(4), which features two Higgs doublets and one singlet. Perhaps most importantly for phenomenology, measuring the co- efficients of WZ terms that appear in any of these Composite Higgs models can allow one to probe the gauge group of the underlying microscopic theory. As a further application of our results, we analyse quantum mechanics models featur- ing such topological terms. In this context, a topological term couples the particle to a background magnetic field. The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. We show how both locality and symmetry can be x made manifest, by passing to an otherwise redundant description on a principal bundle over the original configuration space, and by promoting the original symmetry group to a central extension thereof. We then demonstrate how harmonic analysis on the extended symmetry group can be used to solve the Schrödinger equation. To conclude our study of topological terms in sigma models, we show that the classi- fication we have proposed may be rigorously justified (and generalised) using differential cohomology theory. In doing so, we introduce the notion of the ‘퐺-invariant differential characters’ of a manifold 푀. Within this language, the Manton condition follows from the homotopy formula for differential characters, and we show that it remains necessary and suf- ficient under weaker conditions than connectedness of 퐺. We prove that the abelian group of 퐺-invariant differential characters sits inside various exact sequences and commutative dia- grams, which thus provide us with some powerful algebraic tools for classifying topological terms in quantum field theories. In the remainder of the thesis we depart from the topic of sigma models and turn to gauge theories. We analyse anomalies (which may be understood as arising from topological effects) in both the Standard Model (SM) and theories Beyond the Standard Model (BSM). This analysis consists of two parts, in which we consider ‘local’ and ‘global’ anomalies in a gauge symmetry 퐺; the former depend only on the Lie algebra of 퐺, while the latter are sensitive also to its global structure, i.e. its topology. We first chart the space of anomaly-free extensions of the SM by a flavour-dependent 푈(1) gauge symmetry, using arithmetic techniques from Diophantine analysis to cancel all possible local anomalies. We then develop some of these anomaly-free theories into phe- nomenological models featuring a heavy 푍′ gauge boson, that can account for a collection of recent measurements involving 푏 → 푠휇휇 transitions which are discrepant with SM pre- dictions. We discuss how these models might also explain coarse features of the fermion mass problem, such as the heaviness of the third family. We then turn to global anomalies, which we analyse using the Dai-Freed theorem. Our principal tool here is to compute the bordism groups of the classifying spaces of various Lie groups, preserving particular spin structures, using the Atiyah-Hirzebruch spectral sequence. We show that there are no global anomalies (beyond the Witten anomaly associated with the electroweak factor) in four different ‘versions’ of the SM, in which the gauge group is taken to be 퐺SM/Γ, with 퐺SM = 푆푈(3) × 푆푈(2) × 푈(1) and Γ ∈ {0, ℤ2,ℤ3,ℤ6}. We also show that there are no new global anomalies in 푈(1)푚 extensions of the SM, which feature multiple 푍′ bosons, or in the Pati-Salam model. Table of contents List of figures xv List of tables xix Nomenclature xxi 1 Introduction 1 1.1 Topology in quantum theory .