Anomalies and the Standard Model of Particle Physics Nakarin Lohitsiri Supervisor: Professor David Tong Department of Applied Mathematics and Theoretical Physics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Trinity College August 2020 For my parents Declaration This dissertation is based on original research done by the author while he was a graduate student at the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, between October 2016 and August 2020. The material in Chapters2 and5 is based on the work done by the author under the supervision of David Tong and has been partly published in References [97, 96], while Chapters3 and4 are based on research done in collaboration with Joe Davighi, part of which is published in References [53, 52]. Except for part of Chapter3 that has been previously submitted for a degree of doctor of philosophy by Joe Davighi at the University of Cambridge, no other part of this work has been submitted, or is being concurrently submitted, for a degree or other qualification at the University of Cambridge or any other university or similar institution. Nakarin Lohitsiri August 2020 Anomalies and the Standard Model of Particle Physics Nakarin Lohitsiri This dissertation aims to study quantum anomalies and some other aspects of the Standard Model of Particle physics. In any quantum gauge field theory, anomalies place a very restrictive condition on the matter content and the dynamics. The former is due to the cancellation of gauge anomalies while ’t Hooft anomaly matching constraints produce the latter. As the Standard Model, which is our most fundamental and most accurate description of particle physics, is constructed as a gauge field theory, it is also subject to these anomalies. Here we explore subtleties in anomalies that could arise from the Standard Model and also use them to provide a consistency check as we explore its phase diagram. We start by reexamining local anomaly cancellation in the Standard Model. It has long been known that the requirement that all gauge anomalies and the mixed gauge-gravitational anomaly cancel lead to the quantisation of hypercharge and essentially give the unique hypercharge assignment to the fermion content of the theory. However, if we take the view that hypercharge must be quantised from the outset, then it is enough to prove that the fermions have the Nature-assigned hypercharges using the cancellation of gauge anomalies alone. This remarkable result is made more astounding by the fact that Fermat’s Last Theorem plays a crucial role in completing the proof. We then move on to search for subtler global anomalies in the Standard Model and beyond from the modern viewpoint of cobordism theory, where a global anomaly can be computed as a homomorphism from a bordism group of manifolds equipped with appropriate spin and gauge bundle structure to a circle group. Since the gauge interaction depends on the gauge group G only through its Lie algebra, there are many possibilities for the gauge group of a gauge theory as long as the global structure is consistent with the matter content. In the Standard Model, the options for the gauge group G are U(1) SU(2) SU(3), U(2) SU(3), × × × SU(2) U(3), or U(2) SU(3)=Z3. We compute the fifth spin-bordism group of manifolds × × Spin equipped with these G-bundle structures W5 (BG) and show that it is at most Z2. Therefore, the global anomaly that can appear in the Standard Model is a mod 2 anomaly which can be identified with the well-known Witten anomaly in the gauge group SU(2). We repeat the bordism group calculation for some beyond the Standard Model gauge groups and obtain a similar result: there is a mod 2 anomaly whenever there is an SU(2) factor in the gauge group. A curious fact from these bordism calculations is that the bordism group is trivial when U(2) appears in lieu of SU(2). Driven by this curiosity, we investigate further and find that there is an interplay between the local and the global anomaly. The condition for the gauge viii anomaly cancellation on the SU(2)-representations of the fermions coupled to a gauge theory is the same whether the gauge group is SU(2) or U(2). However, the condition comes from the cancellation of the global Witten anomaly in the former case while it arises from the mixed anomaly cancellation between the U(1) sector and SU(2) sector in the latter case. We investigate further to see whether we can give the same interpretation to the new SU(2) anomaly of Wang, Wen, and Witten when we place a U(2) gauge theory on a non-spin manifold. We find that even though the requirement that the mixed gauge and the mixed gauge-gravitational anomalies cancel automatically cancel the new SU(2) anomaly, it cannot be thought of as arising from the local anomalies. The reason is essentially because the transformation that induces the new SU(2) anomaly involves a non-trivial diffeomorphism on the underlying manifold. Mathematically, we can compute the bordism group and still see a factor of Z2 associated with this new global SU(2) anomaly. Finally, we turn our attention towards the Standard Model itself, leaving anomalies as a tool we occasionally use to provide a consistency check on the IR dynamics. We apply the philosophy that one can get information and intuition on a theory by studying a collection of theories in the parameter space to the Standard Model. In these variations of the Standard Model, we deviate the Yukawa couplings from the actual values so that they are insensitive to the generations of fermions. We then vary the relative strength between the strong nuclear force and the weak nuclear force and see what happens. The results are surprising. No phase transition seems to be present when there is only one generation of fermions. More remarkably, the leptons seem to smoothly mutate into quarks as we slowly dial the relative strength between the weak and the strong gauge group. When more than one generations of fermions are present, however, the global symmetry group on either end of the phase diagram is not a subgroup of the other, and a first-order phase transition is expected to occur. Acknowledgements First and foremost, I would like to express my enormous gratitude to my supervisor, David Tong. Through his excellent examples in teaching, writing, and his constant guidance on research, during my PhD years and when I was still an undergraduate, I have learned and enjoyed a lot of physics and gained a glimpse of understanding and experience how to progress in Academia. His delight when discussing physics is extremely contagious, and it has played a significant role to drive me forward. His supportive and caring words have pulled me out whenever I am stuck academically or emotionally. I could not wish for a better supervisor. I would like to thank my collaborators Aristomenis Donos and Joe Davighi. My first taste of theoretical physics research started with Aristos who generously accepted my request for a summer internship to work with him on the subject of holography. His encouragement made me feel a little more confident in myself to carry on along this path. Joe has been myoffice mate since my first day as a graduate student in DAMTP. His energy for physics andhis willingness to share and discuss all sorts of ideas are very stimulating and conducive to doing exciting research. I would also like to extend my gratitude to my other collaborators: Carl Turner, Nick Dorey, Alec Barnes-Graham, Mike Blake, and Ben Gripaios. I have learned a lot from them in and outside our collaboration. Of course, I cannot leave out Alex Abbott, Amanda Stagg, Sam Crew, Daniel Zhang, Josh Kirklin, Leong-Khim Wong, Amelia Louise, Jonathan Rawlinson, Ed Walton, Antoni Woss, Muntazir Abbidi, Hasan Mahmood, Wuhyun Sohn, Shayan Iranipour, Toby Crisford, Theodor Björkmo, Alice Waterhouse, Bogdan Ganchev, Philip Boyle-Smith, Roland Bittleston, Pietro Benetti Genolini, Avner Karasik, Masazumi Honda, and countless others, who have made coming to work at the CMS a pleasant experience. I can count myself very lucky to be in such a delightful work environment thanks to all of them. It is not an overstatement to say that I owe most of my success to my parents and grandparents. I am free to explore and choose my own path in life due to their openness and support. Moreover, my father and my maternal grandfather (as well as my aunts and my uncle) are bibliophiles. They have a large and diverse collection of books lying around in the houses, ranging from programming, history, sciences, mathematics, languages, waiting for x me to pick up. This spurred me to read widely from a very young age and has built a habit so essential to a researcher. Therefore, I would like to thank them all for what they have done and gone through to raise me up. Three of my Thai friends, Jiraborrirak Charoenpattarapreeda, Puthipong Worasaran, and Methawi Chomthong, have been a fixture of my life at most weekends when we take a break from research to enjoy eating out and board games. I want to thank them for interspersing my time here in Cambridge with many joyful moments. Last but not least, I am infinitely indebted to my wife Huilan, whose love and companion provide invaluable emotional support for me throughout the years. Hopefully, she will be here to support my life and research for many more years to come. Table of contents 1 Introduction1 1.1 What are anomalies? .