Non-perturbative approaches to Scattering Amplitudes by Luc´ıa G´omez C´ordova A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2019 c Luc´ıa G´omez C´ordova 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: Simon Caron-Huot Canada Research Chair in High-Energy Physics Assistant Professor of Physics, McGill University Supervisor: Pedro Vieira Faculty, Perimeter Institute for Theoretical Physics Adjunct Professor, Dept. of Physics and Astronomy, University of Waterloo Co-supervisor: Rob Myers Faculty, Perimeter Institute for Theoretical Physics Adjunct Professor, Dept. of Physics and Astronomy, University of Waterloo Internal Member: Niayesh Afshordi Associate Faculty, Perimeter Institute for Theoretical Physics Associate Professor, Dept. of Physics and Astronomy, University of Waterloo Internal Member: Freddy Cachazo Faculty, Perimeter Institute for Theoretical Physics Adjunct Professor, Dept. of Physics and Astronomy, University of Waterloo Internal-External Member: Benoit Charbonneau Associate Professor, Dept. of Pure Mathematics, University of Waterloo ii Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract This thesis is devoted to the study of scattering amplitudes using two non-perturbative approaches. In Part I we focus on a particular theory known as N = 4 Super-Yang-Mills in four spacetime dimensions. The scattering amplitudes in this theory are dual to the expectation value of null polygonal Wilson loops which can be computed non-perturbatively using integrability. The Wilson loop is decomposed into smaller polygons and computed as an evolution of the color flux tube of the theory, summing over all intermediate flux tube states. By a suitable generalization of the building blocks called pentagons we describe how this program can describe all helicity configurations of the amplitude. We also show how the contribution from all flux tube excitations can be resummed to reproduce the general kinematics result at weak coupling. In Part II we take a different approach and study the space of Quantum Field Theories (QFTs). We focus on two-dimensional theories with a mass gap and a global symmetry. By studying the consequences of unitarity, crossing symmetry and analyticity of the 2 ! 2 scattering matrix element we are able to constrain the space of allowed QFTs. At the boundary of this space we find several interesting features of the S-matrices and identify various integrable points. iv Acknowledgements I would like to express my deep gratitude to my supervisor Pedro Vieira for his enthusi- astic guidance and for encouraging me to maximize happiness when needed; it is a pleasure to learn from and work with him. I am grateful for my collaborators: Amit, Benjamin, Jo~ao C., Mart´ın and Yifei without whom this thesis would not have been possible. I want to thank all the colleagues and professors who nourished my research experience, especially: Jon, Frank, Jo~ao P., Monica, Didina and Andrea. A sincere thank you to Debbie G. for her kind help on any adminis- trative matter. I would also like to thank Rob Myers, Freddy Cachazo, Niayesh Afshordi, Benoit Charbonneau and Simon Caron-Huot for being part of my committee. I was very fortunate to spend time at ICTP-SAIFR, IPhT-Saclay and attend many schools and conferences in the past years. I would like to acknowledge the financial support given by CONACyT during my graduate studies. A big thanks to Andr´es, Nafiz, Am´erica, Rodas, Natalia, Magaly, Cl´ement, Sarah, Rachel, Juani, Naty, Pablo, Nima, Vasco, Fiona, Daniel, Tibra, Kira and all the latino crew who added much joy to my PhD experience and whose friendship kept me strong. Finally, I want to thank Emilio for his love, patience and constant support and my family for encouraging me to be a scientist. v A mis hermanos Rodrigo y Sof´ıa. vi Table of Contents 1 Introduction1 I Pentagons in N = 4 SYM4 2 Preliminary notions5 2.1 N = 4 Super-Yang-Mills............................5 2.2 Scattering Amplitudes.............................6 2.3 Wilson loop duality...............................7 2.4 Pentagon Operator Product Expansion....................8 3 POPE for all helicities I 12 3.1 Introduction................................... 12 3.2 The Charged Pentagon Program........................ 12 3.3 The Map..................................... 16 3.3.1 The Direct Map............................. 17 3.3.2 Interlude : Sanity Check........................ 21 3.3.3 The Inverse Map............................ 22 3.3.4 Easy Components and the Hexagon.................. 25 3.3.5 Parity.................................. 27 3.4 Discussion.................................... 31 vii 4 POPE for all helicities II 33 4.1 Introduction................................... 33 4.2 The abelian part................................ 36 4.2.1 The dynamical part........................... 36 4.2.2 Charged transitions and form factors................. 41 4.2.3 Consistency checks........................... 48 4.3 Comparison with data............................. 50 4.3.1 NMHV Hexagon............................ 52 4.3.2 NMHV Heptagon............................ 55 4.4 Discussion.................................... 61 5 Hexagon Resummation 65 5.1 Introduction................................... 65 5.2 Hexagon POPE and one effective particle states............... 66 5.3 Tree level resummation............................. 77 5.4 Discussion.................................... 83 II S-matrix Bootstrap 85 6 Appetizer 86 7 O(N) Bootstrap 92 7.1 Introduction................................... 92 7.2 Setup, key examples and numerics....................... 93 7.2.1 S-matrices................................ 93 7.2.2 Integrable O(N) S-matrices...................... 96 7.2.3 Numerical setup............................. 102 7.3 Numerical results................................ 103 viii 7.3.1 Reproducing integrable models.................... 103 7.3.2 Away from integrable points...................... 110 7.4 Analytic results................................. 113 7.4.1 Large N ................................. 113 7.4.2 Finite N ................................. 120 7.5 Discussion.................................... 127 8 The Monolith 131 8.1 Introduction................................... 131 8.2 The 3D Monolith................................ 132 8.3 Analytic Properties............................... 137 8.3.1 The slate................................ 137 8.3.2 General analytic properties of the Monolith............. 143 8.3.3 The σ2 = 0 line............................. 144 8.4 Some geometric aspects............................. 147 8.5 Discussion.................................... 148 9 Final Remarks 150 References 153 APPENDICES PART I 160 A More on Geometry, Pentagons and Parity 161 A.1 Variables..................................... 161 A.2 Pentagons and Weights............................. 164 A.3 Parity Map................................... 166 ix B Pentagon transitions and measures 170 B.1 Summary of transitions............................. 171 B.2 Analytic continuation to small fermions.................... 174 B.3 Measures..................................... 175 B.4 Zero momentum limit.............................. 176 C The superconformal charge Q and the flux Goldstone fermion 177 C.1 The zero momentum fermion.......................... 177 C.2 The commutator of superconformal charges.................. 179 D A second example: P1 ◦ P2 ◦ P34 181 E More on matrix part and formation of Bethe strings 184 F Measure prefactors at finite coupling 188 G Details on momentum integration 192 H Hexagon twistors 194 APPENDICES PART II 196 I Numerical Implementation 197 J Fixing parameters in analytic solution 199 x Chapter 1 Introduction Quantum Field Theory (QFT) has successfully provided a framework to describe a wide range of physical phenomena from condensed matter systems to high energy physics. How- ever, most of its success lies in describing weakly coupled systems where we can use pertur- bation theory. Taming strongly coupled theories like Quantum Chromodynamics (QCD) outside the lattice has proven much more challenging. Given the difficult situation, one can follow different strategies to understand aspects of strongly coupled QFTs. One possibility is to study first simpler theories that can serve as toy models. The hope is that by understanding how to compute observables in these simpler models one would be able to export techniques and ideas to more complicated theories. Examples of these toy models might enjoy a large number of symmetries or live in less spacetime dimensions. Another possible strategy is to constrain the space of consistent quantum field theories by imposing general principles. That is, instead of starting with the details of the theory (like a specific Lagrangian), one can get access to non-perturbative observables by asking the right type of questions. This is known as the bootstrap philosophy and has proven incredibly powerful, particularly for conformal field theories. In the present thesis we use the above two strategies to study strongly coupled QFT. The observables we focus on are scattering amplitudes.