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A Cross Section of Amplitudes By A Cross Section of Amplitudes by Timothy M. Olson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2016 Doctoral Committee: Associate Professor Henriette D. Elvang, Chair Professor Dante E. Amidei Professor Thomas Lam Professor Finn Larsen Professor James T. Liu Acknowledgements I am grateful to Henriette Elvang for guidance and collaboration on many projects. Much gratitude is due also to collaborators Nikolay Bobev, Yu-tin Huang, Cindy Keeler, Thomas Lam, Samuel Roland, and David E Speyer. Ad- ditional thanks to Ratin Akhoury, Nima Arkani-Hamed, Jake Bourjaily, Dan Freedman, Rachel Karpman, Michael Kiermaier, Finn Larsen, Jim Liu, Stefan Theisen, and Jaroslav Trnka for useful discussions. I would also like to thank all the graduate students and postdocs in the University of Michigan physics depart- ment whose interactions have been invaluable throughout my degree. Finally, I would like to thank Emily Olson for providing feedback and comments and for her constant support and love! I have been supported by a Regents Graduate Fellowship at the University of Michigan and a National Science Foundation Graduate Research Fellowship under Grant No. F031543. ii Table of Contents Acknowledgements ii List of Tablesv List of Figures vi List of Appendices ix Abstract x Chapter 1: Introduction1 1.1 Overview . .1 1.2 Background for PartI..............................3 1.3 New Results in PartI...............................9 1.4 Background for PartII.............................. 12 1.5 New Results in PartII.............................. 20 1.6 Summary of Results and Publications . 24 Part I Constraining QFTs: Dilaton Effective Action and the a-theorem 26 Chapter 2: RG Flows in d Dimensions and the Dilaton Effective Action 26 2.1 Motivation and Preview . 26 2.2 Dilaton Effective Action and Scattering in d Dimensions . 30 2.3 Example: Free Scalar in 8d . 40 2.4 Scalar Anomaly from Zeta-Function Regularization of the Free Energy . 43 2.5 Free Scalar in d Dimensions and the Dilaton Effective Action . 45 iii Chapter 3: Dilaton Effective Action with = 1 Supersymmetry 50 N 3.1 Motivation and Preview . 50 3.2 Scattering Constraints from Supersymmetry . 54 3.3 Dilaton Effective Action . 56 3.4 Matching to Superspace Calculation . 58 3.5 Dilaton and Axion Scattering in Flat Space . 60 Part II Modern Methods for Amplitudes: Grassmannians and Plabic Graphs 65 Chapter 4: Grassmannians for Amplitudes 65 4.1 Motivation and Preview . 65 4.2 = 4 SYM and the Grassmannian . 67 N 4.3 Relating the Three Grassmannian Formulations . 75 4.4 NMHV Integrals and Residues . 80 4.5 3d ABJM Grassmannian . 95 4.6 Outlook . 110 Chapter 5: Orientations of BCFW Charts on the Grassmannian 113 5.1 Motivation and Preview . 113 5.2 Background . 117 5.3 Relating Distinct Charts . 125 5.4 Applications . 137 Appendices 144 Bibliography 176 iv List of Tables 1.1 Examples of the rapidly growing number of diagrams required to compute the n-gluon color-ordered tree amplitude. 13 2.1 Results for the coefficients α, β, γ, andγ ~ of the effective action (2.49) for the case of the d-dimensional free massive scalar flow. The subscript in the table ◦ d=2 d=2 indicates that a factor of (4π)−b c was taken out, e.g. α = α (4π)−b c. The ◦ label \div" indicates that the 1-loop scalar integral diverges at and beyond this order. The boxed results are those encoding the d = 4; 6; 8 anomaly flows for the free massive scalar; see (2.76). Terms with negative mass-dimension are not needed for our study of RG flows, but we include them here to illustrate that the amplitudes match even in those higher-derivative cases . 47 D.1 Diagrams contributing to the 4-point 1-loop amplitude at (p8) in 8d..... 153 O v List of Figures 1.1 A schematic diagram of a generic scattering amplitude . Particles in the A initial state i are incoming from the left and the outgoing particles form the j i final state f on the right. .2 h j 1.2 Illustration of an RG flow interpolating between an ultraviolet fixed point at high energy and an infrared fixed point at low energy. .4 1.3 A choice of contour avoiding the simple pole at the origin. 10 1.4 A collection of puzzle pieces that make a circular disk when they are prop- erly joined together. Feynman diagrams are similar insofar as they tend to obfuscate the underlying structure. 14 1.5 A collection of rotationally symmetric puzzle pieces that also make a circular disk when they are properly assembled. Using symmetry as a guide can reveal the fundamental structure more clearly from the start. 14 1.6 The poset paths and dlog forms for a pair of distinct 2-dimensional charts on a submanifold. On the left, coordinate α corresponding to (a b) was added first, while on the right, coordinate β corresponding to (c d) was added first. 23 1.7 A suitable choice of edge weights. The product of weights around the quadri- lateral must be 1................................. 23 − 4.1 An on-shell diagram representation of the permutationσ ~ = 3; 5; 6; 7; 9; 8; 11 . f g The `left-right path' between external vertices 4 and 7 showsσ ~(4) = 7. 91 4.2 An on-shell diagram representation of the permutation σ = 4; 5; 6; 7; 9; 10; 8 . f g External legs 1 and 7 are connected by a path with all white vertices, which forces their respective momenta to be parallel. .e . 92 4.3 Two diagrams representing cells of the appropriate dimension for which the 6th and 7th minors vanish. Only the left diagram has kinematical support for generic momenta. 94 5.1 Two distinct sequences of transpositions connecting the 0-dimensional cell σ0 to the 2-dimensional cell σ............................. 115 vi 5.2 A choice of edge weights such that the product around the loop is 1. 115 − 5.3 Different types of paths through the poset of cells. 116 5.4 A GL(1) transformation with scaling factor f.................. 120 5.5 A merge operation. 120 5.6 A square move operation. 121 5.7 Adding a BCFW bridge. 123 5.8 The sequence of diagrams produced by (5.11). We have added a black vertex between the first two bridges to make the graph bipartite; it is drawn slightly smaller to distinguish it from the bridge vertices. 123 5.9 Two paths representing sequences of transpositions connecting the permuta- tions σ0 and σ. Together the paths form a closed loop. 126 5.10 Graphs representing the sequences defined in step (2i) of Algorithm1..... 130 5.11 Boundaries of a 1-dimensional cell σ1....................... 131 5.12 A sequence of equivalence moves relating the two representative graphs for a 1d cell. 131 5.13 Two sequences that are identical except for the first transposition. 132 5.14 Reference paths from Algorithm1 and interior quadrilaterals. 134 5.15 The two 0d cells at the end of the paths are connected by a sawtooth path between 1d and 0d cells. Combining this path with the input paths yields a closed loop. 136 5.16 A set of edge weights that locally match the signs obtained by direct compu- tation. 138 5.17 A factorization channel representing a physical, local pole. 140 5.18 A diagram for the permutation 4; 5; 6; 8; 9; 10; 14 . It represents an unphysi- f g cal, nonlocal singularity. 141 5.19 Grouping legs 6 and 7 in Figure 5.18 leads to a diagram of the form in Figure 5.17......................................... 142 5.20 A closeup of the left-hand subamplitude in Figure 5.19............. 142 D.1 All external momenta are outgoing. Loop momentum flows clockwise. 150 D.2 Two distinct n = 6 box diagrams with canonical labels. 151 I.1 Graph manipulations used to compare the transposition sequences in case (ii). 168 I.2 Graph manipulations used to compare the transposition sequences in case (iii).169 I.3 Simple graph manipulations used to compare the transposition sequences in case (iv). A square move is required to complete the transformation. 170 I.4 Graph manipulations used to compare the transposition sequences in case (v). 171 vii I.5 An on-shell diagram with bridges (ad) and (bc). 172 I.6 Graph manipulations used to compare the transposition sequences in case (vi), where we have defined ∆ = 1 (αj + αj+3). 173 viii List of Appendices Appendix A: Euler Density and the WZ Action 144 Appendix B: Diff Weyl Invariants in d Dimensions 145 × Appendix C: Dilaton Amplitudes in d Dimensions 148 Appendix D: Free Massive Scalar: 1-Loop Dilaton Scattering 150 Appendix E: Conformal Anomaly 154 Appendix F: Matrix Q Details 157 F.1 Derivation of det Q ................................ 157 1 F.2 Verification of Q− ................................ 159 Appendix G: Calculations of Higher-Dimensional Residues 161 Appendix H: Pole Structure of 6-Point ABJM Amplitude 165 Appendix I: Details for Algorithm 1 167 Appendix J: Assigning Edge Weights 174 J.1 Reduced Words . 174 J.2 Decorating Edges . 175 ix Abstract Scattering amplitudes provide a handle for testing and constraining quantum field theories. In this thesis, I explore two directions within the topic: 1) using amplitudes to probe how the number of degrees of freedom evolves as the energy scale changes along renormalization group flows, and 2) extending a new modern approach to computing amplitudes that reveals the intriguing underlying math- ematical structure. In the first part, I generalize a method that encodes the flow of degrees of freedom into certain scattering amplitudes from four dimensions to arbitrary dimensions and uncover new structures that become relevant in eight dimensions.
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