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UNIVERSITY of CALIFORNIA Los Angeles Modern Applications Of UNIVERSITY OF CALIFORNIA Los Angeles Modern Applications of Scattering Amplitudes and Topological Phases A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Julio Parra Martinez 2020 © Copyright by Julio Parra Martinez 2020 ABSTRACT OF THE DISSERTATION Modern Applications of Scattering Amplitudes and Topological Phases by Julio Parra Martinez Doctor of Philosophy in Physics University of California, Los Angeles, 2020 Professor Zvi Bern, Chair In this dissertation we discuss some novel applications of the methods of scattering ampli- tudes and topological phases. First, we describe on-shell tools to calculate anomalous dimen- sions in effective field theories with higer-dimension operators. Using such tools we prove and apply a new perturbative non-renormalization theorem, and we explore the structure of the two-loop anomalous dimension matrix of dimension-six operators in the Standard Model Ef- fective Theory (SMEFT). Second, we introduce new methods to calculate the classical limit of gravitational scattering amplitudes. Using these methods, in conjunction with eikonal techniques, we calculate the classical gravitational deflection angle of massive and massles particles in a variety of theories, which reveal graviton dominance beyond 't Hooft's. Finally, we point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible topological phases, or equivalently topological superconductors, on the worldsheet. We explain how the classification of fermionic topological phases, recently achieved by the condensed matter community, provides a complete and systematic classification of ten-dimensional superstrings and gives a new perspective on the K-theoretic classification of D-branes. ii The dissertation of Julio Parra Martinez is approved. Clifford Cheung Eric D'Hoker Terence Chi-Shen Tao Zvi Bern, Committee Chair University of California, Los Angeles 2020 iii For my mother, the Mathematician, who inspired my life in science. Para mi madre, la Matem´atica, quien inspir´omi vida en la ciencia. iv Contents 1 Introduction 1 2 On-shell methods for the Standard Model Effective Theory 13 2.1 Introduction . 13 2.2 Setup and formalism . 17 2.2.1 Conventions and basic setup . 18 2.2.2 Anomalous dimensions from UV divergences . 22 2.2.3 Anomalous dimensions directly from unitarity cuts . 25 2.2.4 Simplifying strategies . 31 2.2.5 Comments on evanescent operators . 33 2.2.6 Anomalous dimensions and non-interference . 34 2.3 Non-renormalization theorem for operator mixing . 35 2.3.1 Proof of the non-renormalization theorem . 36 2.3.2 Examples . 39 2.4 One-loop amplitudes and anomalous dimensions . 43 2.4.1 One-loop amplitudes from generalized unitarity . 43 2.4.2 One-loop UV anomalous dimensions . 48 2.4.3 Structure of one-loop amplitudes and rational terms . 50 2.5 Two-loop zeros in the anomalous dimension matrix . 53 2.5.1 Zeros from length selection rules . 53 v 2.5.2 Zeros from vanishing one-loop rational terms . 54 2.5.3 General comments about scheme redefinition . 62 2.5.4 Zeros from color selection rules . 64 2.5.5 2 2 4 ............................... 64 O' F O 2.5.6 Outlook on additional zeros . 67 2.6 Implications for the SMEFT . 69 2.6.1 Mapping our theory to the SMEFT . 69 2.6.2 Verification of one-loop anomalous dimensions . 72 2.6.3 Two-loop implications . 72 2.7 Conclusions . 74 2.A Integral reduction via gauge-invariant tensors . 77 2.B Tree-level and one-loop amplitudes . 81 2.B.1 Four-vector amplitudes . 83 2.B.2 Four-fermion amplitudes . 85 2.B.3 Four-scalar amplitudes . 88 2.B.4 Two-fermion, two-vector amplitudes . 90 2.B.5 Two-scalar, two-vector amplitudes . 93 2.B.6 Two-fermion, two-scalar amplitudes . 96 3 Scattering Amplitudes and Classical Gravitational Observables 101 3.1 Introduction . 101 3.2 Universality in the classical limit of massless gravitational scattering . 106 3.2.1 The classical limit of the amplitude . 107 3.2.2 Scattering angle from eikonal phase . 110 3.2.3 Scattering angle from partial-wave expansion . 114 3.3 Extremal black hole scattering at (G3): graviton dominance, eikonal expo- O nentiation, and differential equations . 116 vi 3.3.1 Kinematics and setup . 117 3.3.2 Integrands from Kaluza-Klein reduction . 120 3.3.3 Integration via velocity differential equations . 128 3.3.4 Scattering amplitudes in the potential region . 168 3.3.5 Eikonal phase, scattering angle and graviton dominance . 171 3.3.6 Consistency check from effective field theory . 181 3.4 Conclusions . 188 3.A Dimensionally regularized integrals for the potential region . 191 3.A.1 One-loop integrals . 192 3.A.2 Two-loop integrals . 193 3.B Solution of the differential equations . 196 4 Topological Phases in String Theory 200 4.1 Introduction and summary . 200 4.1.1 Generalities . 200 4.1.2 GSO projections and K-theory classification of D-branes . 203 4.2 The (1+1)d topological superconductors . 208 4.2.1 Oriented invertible phases . 208 4.2.2 Unoriented invertible phases . 212 4.3 GSO projections . 218 4.3.1 Oriented strings . 219 4.3.2 Unoriented strings . 223 4.3.3 Branes and K-theory . 230 4.4 D-brane spectra via boundary fermions . 233 4.4.1 (0+1)d Majorana fermions and their anomalies . 233 4.4.2 D-branes and boundary fermions . 239 4.4.3 K-theory classification of branes . 242 vii 4.4.4 Vacuum manifolds of tachyons as classifying spaces . 248 4.5 D-brane spectra via boundary states . 250 4.5.1 Matching non-torsion brane spectra . 251 4.5.2 Matching torsion brane spectra . 252 4.5.3 Pin+ Type 0 theories . 255 4.6 No new Type I theories . 257 4.6.1 `Spin structure' on the Type I worldsheet . 257 d 4.6.2 The group fDPin(pt)........................... 260 4.6.3 Invertible phases for DPin structure . 262 4.A NSR formalism . 263 4.B Boundary state formalism . 267 4.B.1 Basics . 267 4.B.2 Theta functions, partition functions and boundary state amplitudes . 270 4.B.3 D-brane boundary states . 273 4.B.4 O-plane boundary states . 277 4.C Tadpole Cancellation . 281 4.D Arf and ABK from index theory . 286 4.D.1 η-invariants: generalities . 286 4.D.2 η-invariants: examples . 288 4.D.3 Quadratic forms and enhancements . 292 d 4.E fDPin(pt) via the Atiyah-Hirzebruch spectral sequence . 295 d 4.F fDPin(pt) via the Adams spectral sequence . 302 Bibliography 311 viii List of Figures 1.1 Phases of a binary merger. .5 1.2 Geodesic motion in shockwave background. .6 1.3 Pictorial representation of open and closed string worldsheets. .9 2.1 Unitarity cut relevant for the extraction of anomalous dimensions from one- loop form factors. 27 2.2 Unitarity cuts relevant for the extraction of anomalous dimensions from two- loop form factors. 28 2.3 Iterated two-particle cuts of the two-loop form factors. 31 2.4 Diagram showing the only possible two loop contribution to the renormal- ization of 2 2 by 6 , and cut of a form factor showing that 4 cannot Oφ F Oφ O renormalize 3 at two loops. 43 OF 2.5 Unitarity cut necessary for constructing a two-fermion, two-vector amplitude. 45 2.6 Unitary cuts which determine the renormalization of 4 by 2 4 or O( )1 O(D ' )1 2 4 ....................................... 56 O(D ' )2 2.7 Unitary cuts which determine the renormalization of 2 4 and 2 4 by O(D ' )1 O(D ' )2 4 or 4 ................................... 59 O( )1 O( )2 2.8 Unitary cuts which determine the renormalization of 2 2 and 2 2 by O(' F )1 O(' F )2 4 or 4 ................................... 64 O( )1 O( )2 ix 2.9 Unitary cuts which determines the renormalization of 3 by 2 2 or OF O(' F )1 2 2 ; and of 4 and 4 by 2 2 or 2 2 ............ 67 O(' F )2 O( )1 O( )2 O(' F )1 O(' F )2 3.1 The scattering configuration showing the impact parameter, b, eikonal impact parameter, be, and the scattering angle, χ.................... 111 3.2 Example of KK reduction with massless exchange. 122 3.3 One-loop topologies. 124 3.4 Massless two-loop topologies. 125 3.5 Two loop integrals that are of the \ladder" type. 127 3.6 Two loop integrals that are not of the \ladder\ type. 127 3.7 Two loop integrals that include a self interaction. 127 3.8 Kinematic setup with special variables. 131 3.9 Top level topology at one-loop. 138 3.10 Topologies for the box master integrals. 140 3.11 III topology. 146 3.12 Even- q topologies relevant for the double-box master integrals. 149 j j 3.13 Odd- q topologies relevant for the double-box master integrals. 149 j j 3.14 H topology. Indices correspond to the propagators listed in eq. (3.112). 155 3.15 Topologies relevant for the H master integrals. 156 3.16 Top-level topologies at two-loops. 161 3.17 Even q master integrals relevant for the crossed ladder topology. 163 j j 3.18 Odd q master integrals relevant for the crossed ladder topology. 163 j j 4.1 M¨obiusstrip with an orientation-reversing line. ..
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