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SCATTERING EQUATIONS & S-MATRICES Ye Yuan SCATTERINGEQUATIONS & S-MATRICES by Ye Yuan 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, 2015 c Ye Yuan 2015 AUTHOR’SDECLARATION 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. ii ABSTRACT We introduce a new formulation for the tree-level S-matrix in theories of massless particles. Sitting at the core of this formulation are the scattering equations, which yield a map from the kinematics of a scattering process to the moduli space of punctured Riemann spheres. The formula for an amplitude is constructed by an integration of a certain rational function over this moduli space, which is localized by the scattering equations. We provide a detailed analysis of the solutions to these equations and introduce this new formulation. After presenting some illustrative examples we show how to apply this formulation to the construction of closed formulas for actual amplitudes in various theories, using only a limited set of building blocks. Examples are amplitudes in Einstein gravity, Yang–Mills, Dirac–Born–Infeld, the U(N) non-linear sigma model, and a special Galileon theory. The consistency of these formulas is checked by systematically studying locality and unitarity. In the end we discuss the implication of this formulation to the Kawai–Lewellen–Tye relations among amplitudes. iii ACKNOWLEDGEMENTS I am very grateful to Freddy Cachazo, for his supervision and mentorship, as well as his continuous encouragement and support. I am also grateful to Song He, for the many stimulating discussions and the research we collaborated. And thanks to my advisory committee members, Niayesh Afshordi, Ruxandra Moraru, and Rob Myers, for useful suggestions and feedbacks. I am pleased to thank Tim Adamo, Zvi Bern, Eduardo Casali, Yvonne Geyer, Humberto Gomez, Kurt Hinterbichler, Yu-tin Huang, Shota Komatsu, Carlos Mafra, Lionel Ma- son, Richardo Monteiro, Ruxandra Moraru, Rob Myers, Oliver Schlotterer, Dave Skin- ner, Natalia Toro, Pedro Vieira for many interesting discussions, in particular Ruxandra and Pedro for their excellent and illuminative lectures. I am indebted to Edna Cheung and Konstantin Savvidy, who introduced me to this field. I am thankful to my two officemates, Damian Galante and Pedro Ponte, as well as my classmates and friends dwelling in the Hawking Center, Tibra Ali, Lucia Cordova, Andrzej Banburski, Joao˜ Caetano, Linqing Chen, Ross Diener, Nima Doroud, Lauren Hayward, Anna Kostouki, David Kubiznak, Dalimil Mazac, Heidar Moradi, Anton van Niekerk, Sayeh Rajabi, Mohamad Shalaby, Todd Sierens, Jonathan Toledo, Tianheng Wang, and Dan Wohns. Outside physics, I would like to thank Dawn Bombay, Diana Goncalves, Debbie Guenther, and Natasha Waxman in the administrative team of Perimeter Institute, and Judy McDonnell and Brian McNamara in the University of Waterloo. Also thanks to my friends in and around Waterloo, Chen, Chenxi, Emily, Emily, Jack, Janet, Julie, Han, Hanxing, Hudson, Lois, Michael, Mingyang, Renjie, Rui, Ryan, Sophia, Tian, and many others. Finally I thank my family for their encouragement all the time. iv to my parents Xinhai Yuan &Yuqin Yao with Love and Gratitude v CONTENTS List of Tables ix 1 introduction1 1.1 The S-Matrix in Quantum Field Theories 1 1.1.1 Fields and Particles 1 1.1.2 Interactions and the S-Matrix 3 1.1.3 Computation of the Scattering Amplitudes 5 1.2 The S-Matrix Program 6 1.2.1 Particle Contents and Symmetries 7 1.2.2 Color Decomposition and Partial Amplitudes 9 1.2.3 Locality and Unitarity 10 1.3 The Scattering Equations and the CHY Representation 12 1.4 My Previous Papers Relevant to This Thesis 17 2 scattering equations 18 2.1 The Auxiliary Riemann Sphere 18 2.2 Construction 21 2.3 Counting the Number of Solutions 23 2.4 Various Limits 24 2.4.1 Soft Limits 24 2.4.2 Factorization Channels 27 3 general formulation 30 3.1 Illustrative Examples 32 3.1.1 n = 3 32 3.1.2 n = 4 32 3.1.3 n = 5 33 3.2 An Integrand-Diagram Correspondence 34 4 amplitudes with one type of boson 36 4.1 Building Blocks for Scalars 36 4.2 Scalar Amplitudes 38 4.2.1 The f3 Theory 38 4.2.2 Scalar Amplitudes in a Yang–Mills–Scalar Theory 39 vi 4.2.3 The U(N) Non-Linear Sigma Model 41 4.2.4 Scalar Amplitudes in an Einstein–Maxwell–Scalar Theory 42 4.2.5 The Scalar Sector of the Dirac–Born–Infeld Theory 43 4.2.6 A Special Galileon Theory 43 4.3 A Building Block with Polarization Vectors 44 4.4 Photon/Gluon Amplitudes 46 4.4.1 The pure Yang–Mills Theory 46 4.4.2 Photon Amplitudes in an Einstein–Maxwell Theory 46 4.4.3 The Born–Infeld Theory 47 4.5 Gravity Amplitudes 47 4.A A Brief Review of the general Galileon Theory 48 5 theories with several bosons 49 5.1 Compactification 49 5.1.1 A Yang–Mills–Scalar Theory 50 5.1.2 The Dirac–Born–Infeld Theory 52 5.1.3 An Einstein–Maxwell–(Scalar) Theory 53 5.2 The Squeezing Procedure 53 5.2.1 An Einstein–Yang–Mills Theory: First Approach 54 5.2.2 An Einstein–Yang–Mills Theory: Second Approach 57 5.2.3 Explicit Examples 60 5.2.4 Another Yang–Mills–Scalar Theory 62 5.2.5 An Extended Dirac–Born–Infeld Theory 63 5.3 A Unified Point of View 64 5.3.1 A Generalized Compactification 65 5.3.2 Summary of the Theories 66 5.A Expansion of Pf0P 68 6 locality and unitarity 71 6.1 Soft Theorems I 71 6.1.1 Scaling at the Leading Order 73 6.1.2 Single Soft Theorems at the Leading Order 75 6.2 Factorization 77 6.2.1 Setting Up the General Analysis 77 6.2.2 Behavior of the Buildng Blocks for Scalars 79 6.2.3 F3, U(N) Non-Linear Sigma Model, and the Special Galileon 81 6.2.4 Behavior of Building Blocks with Polarization Vectors 81 6.2.5 Theories with Photons/Gluons and Gravitons 85 6.3 Soft Theorems II 87 vii 6.3.1 Scaling at the Leading Order 88 6.3.2 Proof of Scalar Double Soft Theorems S(0) 90 6.3.3 More General Cases 93 7 relations among amplitudes 96 7.1 Relations among Amplitudes in Yang–Mills and Gravity 96 7.1.1 U(1) Decoupling Identities and Kleiss–Kuijf Relations 96 7.1.2 Bern–Carrasco–Johansson Relations 97 7.1.3 Kawai–Lewellen–Tye Relations 97 7.2 Kawai–Lewellen–Tye Orthogonality 98 7.2.1 Proposition 98 7.2.2 Consequence 100 7.3 The Role of Trivalent Scalar Diagrams 102 8 outlook 103 Bibliography 107 viii LISTOFTABLES Table 1 List of Theories with Their Corresponding Integrands 14 Table 2 Five-Point Examples 33 Table 3 Connections among integrands. 67 Table 4 Existing Checks for the Formulas without a Proof 68 Table 5 Leading Order Scaling of the Amplitudes 74 Table 6 Scaling of Building Blocks in Double Soft Limits 90 Table 7 Scaling of Formulas in Double Soft Limits 91 ix 1 INTRODUCTION 1.1 thes-matrix in quantum field theories One of the main purposes of quantum field theories is to provide a quantitative explanation of the nature of elementary particles. A crucial physical observable in this study is the S-matrix, which underlies the computation of both cross-sections and decay rates that particle experiments measure. Intuitively, a scattering process in particle experiments is idealized to the interaction of a set of particles prepared in the infinite past, whose product is a certain set of particles to be detected in the infinite future. An element in the S-matrix, called a scattering amplitude, is a complex- valued function of the kinematics of the scattering process, whose norm squares to the probability for this process to happen. 1.1.1 Fields and Particles In modern language what we call particles are excitations of quantum fields, ob- tained by quantizing classical fields. Classically a field is described by a function of space-time, valued in a certain target space, and its dynamics is dictated by a given Lagrangian density L (or Hamiltonian density H). This target space is normally a certain representation space of the product of the Lorentz group with possibly some additional compact group encoding internal symmetries. For instance, a scalar field f(x) is valued in R or C, while a vector field Am(x) is a real field with a Lorentz index. When non-trivial internal symmetries are present, the field may carry additional indices associated to it, which are then called the flavor indices. Typical examples of internal symmetries are U(N), SU(N), SO(N), etc. The notion of a particle is most clear in the context of free fields, where the Lagrangian density is quadratic in the field variables. In the simplest case, for a free real massless 1 scalar field we have (we choose the signature for the Minkowski metric hmn such that the component h00 = −1) 1 Lscalar := − ¶ f ¶mf .(1.1) 2 m The typical example of a gauge field, the photon field, is given by 1 Lphoton := − F Fmn , F := ¶ A − ¶ A ,(1.2) 4 mn mn m n n m which is characterized by its invariance under the gauge transformation Am(x) −! Am(x) − ¶m L(x) ,(1.3) for any function L(x).
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