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Studies in Field Theories: Mhv Vertices, Twistor Space, Recursion Relations and Chiral Rings

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STUDIES IN FIELD THEORIES: MHV VERTICES, TWISTOR SPACE, RECURSION RELATIONS AND CHIRAL RINGS Peter Svr·cek Advisor: Edward Witten A DISSERTATION SUBMITTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS April 2005 Copyright °c 2005 by Peter Svr·cek All rights reserved. ii Abstract In this thesis we study di®erent aspects of four dimensional ¯eld theories. In the ¯rst chapter we give introduction and overview of the thesis. In the second chapter we review the connection between perturbative Yang-Mills and twistor string the- ory. Inspired by this, we propose a new way of constructing Yang-Mills scattering amplitudes from Feynman graphs in which the vertices are o®-shell continuations of the tree level MHV amplitudes. The MHV diagrams lead to simple formulas for tree-level amplitudes. We then give a heuristic derivation of the diagrams from twistor string theory. In the third chapter, we explore the twistor structure of scattering amplitudes in theories for which a twistor string theory analogous to the one for N = 4 gauge theory has not yet been proposed. We study the di®erential equations of one-loop amplitudes of gluons in gauge theories with reduced supersymmetry and of tree level and one-loop amplitudes of gravitons in general relativity and supergravity. We ¯nd that the scattering amplitudes localize in twistor space on algebraic curves that are surprisingly similar to the N = 4 Yang-Mills case. In the next chapter we propose tree-level recursion relations for scattering am- plitudes of gravitons. We use the relations to derive simple formulas for all am- plitudes up to six gravitons. We prove the relations for MHV and next-to-MHV amplitudes and the eight graviton amplitudes. In the last two chapters, we concentrate on the nonperturbative aspects of N = 1 gauge theories. Firstly, we ¯nd the complete set of relations of the chiral op- erators of supersymmetric U(N) gauge theory with adjoint scalar. We demonstrate exact correspondence between the solutions of the chiral ring and the supersymmet- ric vacua of the gauge theory. We discuss the gaugino condensation in the vacua. Finally, we go on to study the nonperturbative corrections to the Konishi anomaly relations. We show that the Wess-Zumino consistency conditions of the chiral ro- tations of the matter ¯eld imply the absence of the corrections for a wide class of superpotentials. iii Acknowledgements First, I would like to thank my advisor Edward Witten, for his invaluable guidance and help over the last few years. I am very grateful to him for teaching me a great deal of physics, a certain style of doing research and a taste for problems. I would like to thank the members of the Princeton University High Energy Theory Group and of Physics Group of IAS School of Natural Sciences. I have enjoyed their company and have learned a great deal from them. This includes my fellow students: Chris Beasley, Chan Shih Chang, Anatoly Dymarsky, Tamar Friedmann, Davide S. Gaiotto, Joanna Karczmarek, Hai Lin, Indrajit Mitra, Sameer Murthy, John Pearson, Vasily Pestun, Mukund Rangamani, David C. Shih; post- docs; Niklas Beisert, David Berenstein, Ruth Britto, Freddy Cachazo, Sergey A. Cherkis, Michal Fabinger, Bo Feng, Simeon Hellerman, Matthew Kleban, Gra- ham Kribs, Oleg Lunin, John McGreevy, Raul Rabadan, Radu Roiban, Johannes Walcher, Martijn Wijnholt; faculty: Steven Gubser, Nissan Itzhaki, Igor Klebanov, Juan Maldacena, Chiara R. Nappi, Leonardo Rastelli, Nathan Seiberg, Paul J. Steinhardt and Herman Verlinde. Also I would like to thank for interesting discussions to Sujay Ashok, Nathan Berkovits, Zvi Bern, Lance Dixon, Louise Dolan, Sergei Gukov, Amihay Hanany, Emiliano Imeroni, Anton Kapustin, David Kosower, Valentin Khoze, Lubo·sMotl, Hitoshi Murayama, Andrew Neitzke, Marcus Spradlin and Anastasia Volovich. Finally, I would like to thank my friends and family, whose constant support was invaluable. I would especially like to thank my mother to whom I dedicate this thesis. Chapters 2 and 3 on perturbative gauge theory and twistor space are based on work performed with Freddy Cachazo and Edward Witten [1,2,3]. It is pleasure to thank Z. Bern, L. Dixon, A. De Freitas and V. Voevodsky for helpful discussions. The work reported in Chapter 4 was carried out jointly with Freddy Cachazo [4]. I would like to thank R. Britto, B. Feng, K. Intriligator and E. Witten for useful iv Acknowledgements v discussions. Chapters 5 and 6 are based on work [5,6]. It is pleasure to thank F. Cac- hazo, G. Ferretti, N. Seiberg and especially E. Witten for helpful and encouraging discussions. While this research was carried out, I was supported in part by Princeton Uni- versity Centennial fellowship and by NSF grants PHY-9802484 and PHY-0243680. Any opinions, ¯ndings and conclusions or recommendations expressed in this ma- terial are my own and do not necessarily reflect the views of the funding agencies. I also enjoyed the kind hospitality of the School of Natural Sciences at the Institute for Advanced Study during Spring of 2005. Princeton University Peter Svr·cek April 2005 vi Contents Abstract ................................ iii Acknowledgements ........................... iv 1. Introduction ............................. 1 1.0.1. Perturbative Gauge Theory and Twistor String Theory . 3 1.0.2. Twistor Structure of Scattering Amplitudes . 4 1.0.3. Tree Level Recursion Relations For Gravity Amplitudes . 5 1.0.4. Chiral Rings, Vacua of SUSY Gauge Theories . 6 1.0.5. Nonperturbative Exactness of Konishi Anomaly . 8 2. Twistor String Theory and Perturbative Yang-Mills ....... 9 2.1. Introduction . 9 2.2. Helicity Amplitudes . 11 2.2.1. Spinors . 11 2.2.2. Scattering Amplitudes . 15 2.2.3. Maximally Helicity Violating Amplitudes . 17 2.3. Twistor Space . 18 2.3.1. Conformal Invariance of Scattering Amplitudes . 18 2.3.2. Transform to Twistor Space . 19 2.3.3. Scattering Amplitudes in Twistor Space . 21 2.4. Twistor String Theory . 24 2.4.1. Brief Review of Topological Strings . 25 2.4.2. Open String B-model on a Super-Twistor Space . 27 2.4.3. D-Instantons . 30 2.5. Tree Level Amplitudes from Twistor String Theory . 31 2.5.1. Basic Setup . 31 2.5.2. Higher Degree Instantons . 35 2.5.3. MHV Diagrams . 39 2.5.4. Heuristic Derivation from Twistor String Theory . 44 2.5.5. Closed Strings . 49 3. Twistor Structure of Scattering Amplitudes ............ 51 vii Contents viii 3.1. Introduction . 51 3.2. Review of Di®erential Equations . 52 3.2.1. Higher Degree Curves . 53 3.3. Twistor Structure of One-Loop N = 1 MHV Amplitude . 57 3.3.1. Twistor Structure of N = 1 Amplitude . 58 3.3.2. Interpretation . 61 3.3.3. Holomorphic Anomaly . 63 3.4. Twistor Structure of Nonsupersymmetric One-Loop Amplitudes . 65 3.4.1. All Plus Helicity One-Loop Amplitude . 65 3.4.2. The ¡ + + + ::: + One Loop Amplitude . 67 3.4.3. Nonsupersymmetric ¡¡ + ::: + Amplitude . 68 3.4.4. Comparison of Amplitudes with Two Negative Helicity Gluons . 70 3.4.5. Cut-free Part of ¡¡ + + ::: + Amplitude . 70 3.5. Twistor Structure of Gravitational Amplitudes . 71 3.5.1. Tree Level Graviton Amplitudes . 72 3.5.2. One-Loop Graviton Amplitudes . 76 3.A. The Integral Functions . 79 3.B. KLT Relations . 81 4. Tree Level Recursion Relations For Gravity Amplitudes ..... 83 4.1. Introduction . 83 4.2. Recursion Relations . 85 4.3. Explicit Examples . 86 4.4. Derivation of the Recursion Relations . 89 4.5. Large z Behavior of Gravity Amplitudes . 91 4.5.1. Vanishing of the MHV Amplitudes . 91 4.5.2. Analysis of the Feynman Diagrams . 92 4.5.3. KLT Relations and the Vanishing of Gluon Amplitudes . 93 4.5.4. Proof of Vanishing of A(z) for NMHV Amplitudes . 94 4.A. Proof of Vanishing of A(z) up to Eight Gravitons. 97 5. Chiral Rings and Vacua of SUSY Gauge Theories ........ 100 5.1. Introduction . 100 5.2. The Chiral Ring . 101 5.2.1. Perturbative Corrections . 105 5.2.2. Nonperturbative Corrections . 107 5.2.3. U(1)free and Shift Symmetry . 114 5.3. Solution of the Chiral Ring . 116 5.3.1. U(2) Gauge Theory with Cubic Superpotential . 116 5.3.2. Classical Case . 119 5.3.3. Quantum Case . 121 5.3.4. Perturbative Chiral Ring . 123 Contents ix 5.4. Intersection of Vacua . 124 5.5. Gaugino Condensation . 126 5.5.1. Classical case . 126 5.5.2. Quantum case . 129 5.6. Examples . 131 5.6.1. Unbroken Gauge Group . 131 5.6.2. U(3) Gauge Theory with Cubic Superpotential . 133 5.A. N = 2 Chiral Ring Relations . 136 6. Nonperturbative Exactness of Konishi Anomaly ......... 140 6.1. Introduction . 140 6.2. The Algebra of Chiral Rotations . 142 6.3. Wess-Zumino Consistency Conditions for the Konishi Anomaly . 144 6.4. Nonpertubative Corrections . 146 6.5. Nonrenormalization of the Algebra of Chiral Rotations . 147 6.6. Nonperturbative Corrections to the Konishi Anomaly . 151 6.7. SO(N) and Sp(N) Gauge Theories . 153 6.8. Virasoro Constraints for the One-Matrix Model . 155 6.9. Implications for the Dijkgraaf-Vafa conjecture . 156 6.A. Second Proof of the Nonrenormalization . 158 References .............................. 162 1. Introduction Current physics theories are based on quantum mechanics and general rel- ativity. At energies far below the Planck scale, gravity is negligible. The re- maining three forces are described in terms of quantum gauge theory with group SU(3) £ SU(2) £ U(1): The fourth force, gravity, is described by a massless spin two ¯eld, the graviton. Early attempts at quantizing gravity led to divergences and ill-de¯ned results. It is believed that general relativity cannot be well de¯ned as a ¯eld theory.
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  • Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories

    Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories

    Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories by MASSIMILIANO VINCON A report presented for the examination for the transfer of status to the degree of Doctor of Philosophy of the University of London Thesis Supervisor DR ANDREAS BRANDHUBER Centre for Research in String Theory Department of Physics Queen Mary University of London Mile End Road London E1 4NS United Kingdom The strong do what they want and the weak suffer as they must.1 1Thucydides. Declaration I hereby declare that the material presented in this thesis is a representation of my own personal work unless otherwise stated and is a result of collaborations with Andreas Brandhuber and Gabriele Travaglini. Massimiliano Vincon Abstract In this thesis, we discuss novel techniques used to compute perturbative scattering am- plitudes in gauge theories. In particular, we focus on the use of complex momenta and generalised unitarity as efficient and elegant tools, as opposed to standard textbook Feynman rules, to compute one-loop amplitudes in gauge theories. We consider scatter- ing amplitudes in non-supersymmetric and supersymmetric Yang-Mills theories (SYM). After introducing some of the required mathematical machinery, we give concrete exam- ples as to how generalised unitarity works in practice and we correctly reproduce the n- gluon one-loop MHV amplitude in non-supersymmetric Yang-Mills theory with a scalar running in the loop, thus confirming an earlier result obtained using MHV diagrams: no independent check of this result had been carried out yet. Non-supersymmetric scat- tering amplitudes are the most difficult to compute and are of great importance as they are part of background processes at the LHC.