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Gauge and Gravity Amplitudes from Trees to Loops

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Gauge and Gravity Amplitudes from Trees to Loops FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN PhD thesis Rijun Huang Gauge and Gravity Amplitudes from trees to loops Principal Supervisor: Poul Henrik Damgaard Submitted: November 7, 2013 University of Copenhagen Doctoral Thesis Gauge and Gravity Amplitudes from trees to loops Author: Principal Supervisor: Rijun Huang Poul Henrik Damgaard A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Niels Bohr Institute, Faculty of Science University of Copenhagen December 2013 Abstract This thesis describes two subjects that I mainly work on during my PhD study. They are both about scattering amplitudes, covering gravity and gauge theories, tree and loop level, with or without supersymmetry. The first subject is Kawai-Lewellen-Tye(KLT) relation in field theory, which mysteriously relates Yang-Mills amplitudes to gravity am- plitudes. Based on many known works about KLT and super-KLT relations, we provide a complete map between super-gravity amplitudes and super-Yang-Mills amplitudes for any number of supersymmetry that allowed in 4-dimensional theory. We also provide an explanation for vanishing identities of Yang-Mills amplitudes as violation of linear symmetry groups based on KLT relation argument. The second subject is integrand re- duction of multi-loop amplitude. The recent methods based on computational algebraic geometry make it possible to systematically study multi-loop amplitude with general- ized unitarity cut. Using Gr¨obnerbasis and primary decomposition, we thoroughly study integrand basis and solution space of equations from maximal unitarity cut for all 4-dimensional two-loop topologies. Algorithm and examples of this computation are illustrated in this thesis. We also study a special type of two-loop and three-loop dia- grams where equations of maximal unitarity cut define complex curve. Geometry genus of complex curve is a topological invariant, and characterizes the property of curve. We compute the genus of complex curve for some two-loop and three-loop diagrams from in- formation of degree and singular points of that curve using algebraic geometry method. Information of integrand basis, structure of solution space as well as geometric genus is useful for future multi-loop amplitude computation. Publication list During PhD Study 1. R. Huang and Y. Zhang, On Genera of Curves from High-loop Generalized Unitarity Cuts, JHEP 1304, 080 (2013), arXiv:1302.1023 [hep-ph]. 2. B. Feng and R. Huang, The classification of two-loop integrand basis in pure four-dimension, JHEP 1302, 117 (2013), arXiv:1209.3747 [hep-ph]. 3. P. H. Damgaard, R. Huang, T. Sondergaard and Y. Zhang, The Complete KLT- Map Between Gravity and Gauge Theories, JHEP 1208, 101 (2012), arX- iv:1206.1577 [hep-th]. Before PhD Study 1. B. Feng, Y. Jia and R. Huang, Relations of loop partial amplitudes in gauge theory by Unitarity cut method, Nucl. Phys. B 854, 243 (2012), arXiv:1105.0334 [hep-ph]. 2. J. -H. Huang, R. Huang and Y. Jia, Tree amplitudes of noncommutative U(N) Yang-Mills Theory, J. Phys. A 44, 425401 (2011), arXiv:1009.5073 [hep- th]. 3. B. Feng, S. He, R. Huang and Y. Jia, Note on New KLT relations, JHEP 1010, 109 (2010), arXiv:1008.1626 [hep-th]. 4. Y. Jia, R. Huang and C. -Y. Liu, U(1)-decoupling, KK and BCJ relations in = 4 SYM, Phys. Rev. D 82, 065001 (2010), arXiv:1005.1821 [hep-th]. N 5. B. Feng, R. Huang and Y. Jia, Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program, Phys. Lett. B 695, 350 (2011), arXiv:1004.3417 [hep-th]. 6. B. Feng, R. Huang, Y. Jia, M. Luo and H. Wang, Cross section evaluation by spinor integration: The massless case in D-4, Phys. Rev. D 81, 016003 (2010), arXiv:0905.2715 [hep-ph]. iv Acknowledgements I would like to thank my PhD supervisor Poul Henrik Damgaard and co-supervisor Emil Bjerrum-Bohr, who provide me such a wonderful opportunity studying in the Niels Bohr Institute, and also for the discussions and encouragements both in academy and personal life. The Niels Bohr Institute is the best place I have ever been, and I think it would be the best for many years before I can find a better one. I would also like to thank my master supervisor Bo Feng in Zhejiang University. His insight in life and personal style in research greatly inspire my style of doing research. I also want to thank my collaborator Yang Zhang in Niels Bohr Institute, whose hard working and knowledge of mathematics and chemistry always make me feel shame. Grateful thanks should be given to our group secretary Helle Kiilerich. She always takes care about everything, which makes the research and life in Copenhagen very comfortable. I would like to thank all my colleges at Niels Bohr Institute and Niels Bohr International Academy for so many interesting discussions about physics and life in coffee lounge. Special thanks to my family and my girl friend Yazhi Qin. I could not enjoy the academy life without their supports. v Contents Abstract iii Publication list iv Acknowledgements v 1 Introduction 1 1.1 Scattering amplitude in a nutshell ...................... 1 1.2 Outline of the thesis .............................. 3 1.3 Preliminary ................................... 5 1.3.1 Gauge group .............................. 5 1.3.2 Spinor-Helicity formalism ....................... 7 1.3.3 Three-point amplitude ......................... 9 1.3.4 BCFW recursion relation ....................... 11 1.3.5 MHV amplitude ............................ 13 1.3.6 Supersymmetry ............................. 14 2 KLT relations 17 2.1 KLT relation in field theory .......................... 18 2.1.1 General formula and properties of S-kernel ............. 18 2.1.2 Modified formulae and their applications in gravity amplitude computation .............................. 21 2.2 Super-KLT relation in field theory ...................... 25 2.2.1 Super-field representation and truncation of component fields ... 26 2.2.2 Super-KLT relations from = 8 to = 0 ............. 32 N N 2.2.3 Linear symmetry group for super-gravity theories .......... 38 3 Loop amplitude computation and Algebraic geometry method 47 3.1 Integral and integrand of loop amplitude ................... 47 3.1.1 Integral representations ........................ 47 3.1.2 The color-ordered amplitudes and non-trivial relations ....... 50 3.2 Integral reduction of loop amplitude ..................... 54 3.2.1 The traditional reduction procedure ................. 54 3.2.2 One-loop integral reduction and integral basis ............ 57 vii viii Contents 3.2.3 Discussion on multi-loop integral reduction ............. 58 3.3 Unitarity cut and generalized unitarity cut ................. 59 3.3.1 Unitarity cut in 4-dimension and d-dimension ............ 59 3.3.2 Generalized unitarity cut ....................... 61 3.3.3 One-loop amplitude computation with unitarity cut ........ 62 3.3.4 Discussion on multi-loop amplitude computation .......... 65 3.4 The algebraic geometry ............................ 66 3.4.1 Variety and ideal ............................ 66 3.4.2 The curve ................................ 70 3.4.3 Examples ................................ 72 3.5 Integrand reduction of loop amplitude .................... 75 3.5.1 Parametrization of loop momenta ................... 76 3.5.2 The integrand basis .......................... 80 3.5.3 Polynomial fitting of expansion coefficients ............. 82 4 Integrand basis for 4-dimensional two-loop amplitude 85 4.1 Topologies of two-loop amplitude ....................... 86 4.2 General discussion on equations of maximal unitarity cut ......... 90 4.3 Planar penta-triangle topology ........................ 92 4.3.1 The integrand basis .......................... 92 4.3.2 Expansion coefficients of integrand basis ............... 95 4.3.3 Irreducibility of cut equations ..................... 97 4.4 Non-planar crossed double-triangle topology ................. 99 4.4.1 The integrand basis .......................... 100 4.4.2 Irreducibility of cut equations ..................... 102 5 Genus of curve from multi-loop amplitude 111 5.1 From non-plane curve to plane curve ..................... 111 5.1.1 Birational map of non-plane curve to plane curve .......... 111 5.1.2 Compute singular points and genus .................. 113 5.2 Genus of curve from two-loop diagrams ................... 116 5.2.1 Genus of planar double-box diagram ................. 117 5.2.2 Genus of non-planar crossed-box diagram .............. 119 5.3 Genus of curve from three-loop diagrams ................... 123 5.3.1 Genus of non-planar box-crossed-pentagon diagram ......... 124 5.3.2 Genus of non-planar crossed-crossed-pentagon diagram ....... 128 5.3.3 Genus of Mercedes-logo diagram ................... 131 6 Conclusion 133 Bibliography 135 Dedicated to all the people who helped and inspired me during my growing up and then disappeared in my life. ix Chapter 1 Introduction 1.1 Scattering amplitude in a nutshell Pursuit of simplicity has never been stopped for physics research, even though we are confronting more and more complicated phenomena. In the past decade, tremendous researches have been done to uncover the simplicity of scattering amplitude[1]. Many old ideas are studied again. With a modern interpretation, their hidden power has been released for amplitude computations that can not even image to do so in old days. Traditionally, scattering amplitude is computed perturbatively by Feynman rules of Feynman diagrams. Feynman rules are derived from Lagrangian description of field theories. In the days when it is firstly proposed, it served as
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