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Bootstrapping Scattering Amplitudes in Effective Field Theories

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Bootstrapping Scattering Amplitudes in Effective Field Theories Bootstrapping Scattering Amplitudes in Effective Field Theories by Shruti Paranjape A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2021 Doctoral Committee: Professor Henriette Elvang, Chair Professor Ratindranath Akhoury Professor Finn Larsen Professor Jianming Qian Professor David Speyer Shruti Paranjape [email protected] ORCID ID: 0000-0003-2613-718X © Shruti Paranjape 2021 ACKNOWLEDGEMENTS This work would not have been possible without the mentorship, support, friendship and care I have received from many wonderful people. At the top of that list is Henriette Elvang, my PhD advisor who gave me time and room to grow and learn. I am grateful to her for her unwavering support, impeccable guidance and great textbook. I am indebted to my brilliant collaborators: Henriette Elvang, Callum Jones, Marios Hadjiantonis, Laura Johnson, Finn Larsen, Aidan Herderschee and Huan-Hang Chi. Through them, I have been introduced to new ideas and new physics. They kept my interests piqued and knowledge fresh. The Leinweber Center for Theoretical Physics has provided an amazing environment. I am grateful for the many opportunities, visitors and courses that it has hosted. I am particularly grateful to the faculty and fellow graduate students for stimulating journal clubs and discussions. I am grateful to my masters thesis advisor Gautam Mandal who was my gateway into research in quantum field theory and spent time teaching me a lot of physics. I would like to take a moment to acknowledge all the incredible women physicists I have known: Henriette Elvang, Suneeta Vardarajan, Nabamita Banerjee and my mother, Sudeshna Sinha, who were inspirational to say the least. Thanks to my friends, near and far: Kalyanee, Josh, Callum, Brian, Noah and Marina. You have been a constant source of happiness. To Ashish, my biggest cheerleader, your love and support mean more than I can express. To Ma and Pa, who taught me to do what I love, gave me everything and asked for nothing. ii TABLE OF CONTENTS Acknowledgements .................................... ii List of Figures ....................................... vii List of Tables ........................................ viii List of Appendices ..................................... x Abstract ........................................... xii Chapter 1 Introduction ...................................... 1 1.1 On-Shell Methods in Quantum Field Theory..................7 1.2 Effective Field Theory of Goldstone Modes..................9 1.3 D-Brane Effective Actions: Galileons and Born-Infeld Photons........ 12 1.4 Loop Techniques and Anomalous Symmetries................. 15 1.5 The Double Copy................................ 16 2 Soft Bootstrap and Supersymmetry ......................... 19 2.1 Introducing Exceptional EFTs......................... 19 2.2 Exceptional EFTs Studied........................... 22 2.3 Structure of the Effective Action........................ 24 2.4 Subtracted Recursion Relations......................... 26 2.4.1 Holomorphic Soft Limits and Low-Energy Theorems......... 27 2.4.2 Review of Soft Subtracted Recursion Relations............ 27 iii 2.4.3 Validity Criterion............................ 29 2.4.4 Non-Constructibility = Triviality.................... 32 2.4.5 Implementation of the Subtracted Recursion Relations........ 34 2.5 Soft Bootstrap.................................. 37 2.5.1 Pure Scalar EFTs............................ 37 2.5.2 Pure Fermion EFTs........................... 40 2.5.3 Pure Vector EFTs............................ 41 2.6 Soft Limits and Supersymmetry........................ 43 2.6.1 N = 1 Supersymmetry Ward Identities................ 43 2.6.2 Soft Limits and Supermultiplets.................... 44 2.6.3 Application to Superconformal Symmetry Breaking......... 47 2.6.4 Application to Supergravity...................... 48 2.6.5 MHV Classification and Examples of Supersymmetry Ward Identities 49 2.7 Supersymmetric Non-linear Sigma Model................... 51 1 2.7.1 N = 1 CP NLSM........................... 52 1 2.7.2 N = 2 CP NLSM........................... 55 2.8 Super Dirac-Born-Infeld and Super Born-Infeld................ 62 2.9 Galileons.................................... 62 2.9.1 Galileons and Supersymmetry..................... 63 2.9.2 Vector-Scalar Special Galileon..................... 65 2.9.3 Higher Derivative Corrections to the Special Galileon........ 67 2.9.4 Comparison with the Field Theory KLT Relations.......... 69 3 All-Multiplicity One-Loop Amplitudes in Born-Infeld Electrodynamics from Gen- eralized Unitarity ................................... 73 3.1 Special Features of Born-Infeld Theory.................... 73 3.2 Overview of Method.............................. 76 3.2.1 Generalized Unitarity and Supersymmetric Decomposition...... 77 3.2.2 Massive Scalar Extension of Born-Infeld............... 82 3.3 Calculating mDBI4 Tree Amplitudes...................... 86 3.3.1 General Structure............................ 86 3.3.2 T-Duality and Low-Energy Theorems................. 89 3.4 All Multiplicity Rational One-Loop Amplitudes................ 94 3.4.1 Diagrammatic Rules for Constructing Loop Integrands........ 94 iv 3.4.2 Self-Dual Sector............................ 95 3.4.3 Next-to-Self-Dual Sector........................ 99 3.5 Discussion.................................... 106 4 Electromagnetic Duality and D3-Brane Scattering Amplitudes Beyond Leading Order .......................................... 110 4.1 Electromagnetic Duality in Born-Infeld Theory................ 110 4.1.1 Review of D3-Brane Worldvolume EFTs............... 112 4.2 Dimensional Oxidation: the D3-brane and M2-Brane............. 115 4.2.1 World-Volume Analysis........................ 115 4.2.2 3d ! 4d Oxidation: Contact Terms.................. 118 4.2.3 3d ! 4d Oxidation: Subtracted Recursion Relations......... 120 4.2.4 Higher Derivative Corrections..................... 122 4.3 Loop Amplitudes from BCJ Double-Copy................... 125 4.3.1 Tree-level BCJ Double-Copy of BI.................. 126 4.3.2 1-loop 4-point Self-Dual Amplitude as a Double-Copy........ 127 4.4 1-loop SD and NSD Amplitudes from Unitarity................ 131 4.4.1 4-point and Counterterm........................ 131 4.4.2 n-point................................. 132 4.5 Supersymmetric (D)BI and MHV Amplitudes at 1-loop............ 134 4.5.1 Supersymmetric Born-Infeld...................... 134 4.5.2 All-Multiplicity 1-loop MHV Amplitudes in N = 4 DBI....... 135 4.5.3 1-loop MHV in BI with N -Fold SUSY and Counterterms...... 137 4.6 Rational Loop Amplitudes and Finite Counterterms.............. 139 4.7 Higher Derivative Corrections as a Double-Copy............... 145 4.7.1 KLT Double-Copy........................... 145 4.7.2 Higher-Derivative Corrections to Born-Infeld............. 146 4.7.3 Comparison with the String Theory Effective Action......... 149 4.8 Discussion.................................... 150 4.8.1 Open Questions............................. 150 4.8.2 Supergravity: Double-Copy and Symmetry Oxidation........ 153 5 Constraints on a Massive Double-Copy and Applications to Massive Gravity .. 157 5.1 A Massive BCJ Double Copy.......................... 157 5.2 Massive KLT Formula............................. 163 v 5.2.1 Equivalence of Massive BCJ and Massive KLT............ 163 5.2.2 Spurious Singularities......................... 168 5.3 Massive Gravity and (Massive Yang-Mills)2 .................. 171 5.3.1 Physical Motivation.......................... 171 5.3.2 3-point Amplitudes and Asymptotic States.............. 174 5.3.3 4-point Amplitudes and High Energy Behavior............ 176 5.3.4 5-point Amplitudes and Non-Physical Singularities.......... 180 5.4 Locality and the Spectral Condition...................... 182 5.4.1 4-point Spectral Condition....................... 183 5.4.2 5-point Spectral Conditions...................... 189 5.4.3 Non-minimal Rank........................... 192 5.5 Discussion.................................... 192 Appendices ......................................... 195 Bibliography ........................................ 272 vi LIST OF FIGURES FIGURE 3.1 Some key-properties of BI amplitudes at tree-level, in particular the double-copy construction and 4d electromagnetic duality. The idea behind the T-duality constraint [1] is that when dimensionally reduced along one direction, a linear combination of the photon polarizations become a scalar modulus of the compactified direction,. i.e. it is the Goldstone mode of the spontaneously broken translational symmetry and as such it must have enhanced O(p2) soft behavior.................. 75 3.2 Dimensional reduction commutes with the KLT product. This provides us with two alternative approaches to mDBI4........................... 86 O.1 Color-dressed tree-level 5-point amplitude organized using graphs with only cubic vertices........................................ 258 vii LIST OF TABLES TABLE 1.1 The table shows the tree-level double-copy A ⊗ B of a selection of different choices for the A and B single-color models. BAS is the cubic bi-adjoint scalar model. The other single-color models are χPT = chiral perturbation theory (NLSM), YM = Yang-Mills theory, and N = 4 super Yang Mills theory (sYM). For the results of the double-copy, sGal stands for the special Galileon, BI is Born-Infeld theory, gravity+ is Einstein gravity with a dilaton and antisymmetric 2-form, and SG stands for supergravity.................................... 18
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