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Scattering in N = 4 Super Yang–Mills and N = 8 Supergravity

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Scattering in N = 4 Super Yang–Mills and N = 8 Supergravity Scattering in = 4 super Yang{Mills and = 8 supergravity N N Thesis by Enrico Herrmann In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2017 Defended May 15, 2017 ii c 2017 Enrico Herrmann ORCID: 0000-0002-3983-2993 All rights reserved iii Acknowledgments During my time as a graduate student at Caltech I had the pleasure to interact with a number of remarkable scientists. First and foremost I would like to thank my advisor, Mark Wise, for taking me on as a graduate student during my first year at Caltech. Even more so, I am thankful that he gave me the opportunity to follow my own research interests and for all his support along the way. Mark always had an open ear and good advice concerning practical matters like organizing travel funding or navigating the academic life in general. I am grateful that he also nominated me for a Dominic Orr Fellowship, which allowed me a one year break from all teaching duties. Along with Mark, I would like to thank my co-advisor, Jaroslav Trnka, from whom I learned an enormous amount about scattering amplitudes and physics in general. I very much appreciate all his invaluable support and encouragement throughout my graduate studies. Without him and his generous help, many of my conference visits and trips would not have been possible. In all the workshops he co-organized, Jaroslav enabled young scientists to present their research. In many respects, Jaroslav is my scientific role model. In addition to the mentorship I have received by Mark and Jaroslav, I have benefited significantly from the interaction with my collaborators Zvi Bern, Jacob Bourjaily, Sean Litsey, James Stankowicz, and Jaroslav Trnka. Without them, much of this thesis research would not have been possible. Especially Zvi has been an important academic mentor and like a third advisor to me. I enjoyed the hospitality and many great discussions during my numerous trips to UCLA, and I would like to take this opportunity to congratulate Zvi on the establishment of the Bhaumik Institute for Theoretical Physics at UCLA. Special thanks also go to Nima Arkani-Hamed, Livia Ferro, Johannes Henn, Yu-tin Huang, Henrik Johansson, Marcus Spradlin, Mathias Staudacher, and Anastasia Volovich for inviting me to various workshops and extended research visits. I have greatly ben- efited from many interesting conversations. I would also like to thank the remaining members of my thesis committee, Clifford Cheung, Ryan Patterson and Maria Spiropulu, for their time and input. iv Furthermore, I would like to thank Anders Andreassen, John Joseph Carrasco, Wei- Ming Chen, Scott Davies, Lance Dixon, Alexander Edison, Michael Enciso, David Kosower, Lionel Mason, Andrew McLeod, David Meidinger, Josh Nohle, Julio Parra- Martinez, Jan Plefka, Gregor Richter, Oliver Schlotterer, John Schwarz, Congkao Wen, Ellis Ye Yuan, and Mao Zeng for stimulating discussions around physics in general and scattering amplitudes in particular. I would also say thank you to all the people who, through their friendship, made my five years at Caltech an overwhelmingly positive experience: David Aasen, Michael Bev- erland, Thom Bohdanovicz, Aidan Chatwin-Davies, Chia-Hsien Shen, Murat Kologlu, Noah Olsman, Paul Plucinsky, Jason Pollack, Ingmar Saberi, and Christoph Uhlemann. Of course, my family deserves special credit for all their support throughout my entire life. Without them, I would not be the person who I am now. My girlfriend, Ellen Dooley, also deserves extra thanks for putting up with all my little quirks as well as my persistent lack of punctuality when coming home from work. My work was supported in part by the Walter Burke Institute for Theoretical Physics, by a Dominic Orr Fellowship and by the United States Department of Energy under Grant DE-SC0011632. v Abstract The scattering amplitudes of planar = 4 super-Yang-Mills theory (sYM) exhibit a N number of remarkable analytic structures, including dual conformal symmetry, loga- rithmic singularities of integrands, and the absence of poles at infinite loop momentum. None of these properties are apparent from our usual formulation of quantum field the- ory in terms of Lagrangians and Feynman rules. In the past years, the hidden features inspired a dual formulation for scattering amplitudes that is not built on the two pillars of locality and unitarity. Instead, a new geometric formulation in terms of Grassman- nians and the amplituhedron emerged, which is based on the key analytic properties of scattering amplitudes in the planar sector of = 4 super-Yang-Mills theory. Starting N from geometric concepts, the amplituhedron geometry derives all properties of scatter- ing amplitudes in said theory, including locality and factorization. From a practical perspective, expanding the amplitude in terms of a local diagrams, the amplituhedron construction implies that scattering amplitudes in planar = 4 super-Yang-Mills are N fully specified by a surprisingly simple subset of all unitarity cuts. Concretely, inte- grands are uniquely (up to an overall constant) fixed by demanding their vanishing on all spurious singularities. Extending an initial proposal by Arkani-Hamed, Bourjaily, Cachazo, and Trnka, we conjecture that the same analytic structures extend beyond the planar limit of = 4 N super-Yang-Mills. Furthermore we show that the dlog and no pole at infinity constraints give the key integrand level analytic information contained in dual conformal symmetry in the planar sector. While it is presently unclear how to extend either dual conformal symmetry or the amplituhedron picture beyond the planar sector, our results suggest that related concepts might exist and await discovery. In order to support our conjectures, we have analyzed several nontrivial multi-loop multi-leg amplitudes. For the nonplanar three-loop four-point and two-loop five point = 4 super-Yang-Mills amplitudes, we explicitly construct a complete basis of diagram N integrands that has only logarithmic singularities and no poles at infinity. We also give examples at three loops showing how to make the logarithmic singularity properties manifest by writing explicit dlog forms. We give additional evidence at four and five loops supporting the nonplanar logarithmic singularity conjecture. Our investigations show that the singularity structures of planar and nonplanar integrands in = 4 N super-Yang-Mills are strikingly similar. Finally, we express the complete amplitude in vi terms of our special basis diagrams, with the coefficients determined by the vanishing conditions on the amplitude. By successfully carrying out this procedure, we provide nontrivial evidence that the zero conditions also carry over into the nonplanar sector. Our analysis suggests that the concept of the amplituhedron can be extended to the nonplanar sector of = 4 super-Yang-Mills theory and one might hope to ultimately N reformulate more general quantum field theories in a geometric language. Using the marvelous squaring relation between Yang-Mills and gravity theories discov- ered by Bern, Carrasco, and Johansson (BCJ), we relate our newly gained knowledge on the Yang-Mills side to properties of gravity. We conjecture that to all loop orders, while = 8 supergravity has poles at infinity, at least at four points it has only N logarithmic singularities at finite locations. We provide nontrivial evidence for these conjectures. We describe the singularity structure of = 8 supergravity at three loops N and beyond. In order to approach a geometric formulation for scattering in gravitational theories, we retrace the initial steps taken in planar = 4 super-Yang-Mills in the gravitational N setting. In particular, we study on-shell diagrams for gravity theories with any number of supersymmetries and find a compact Grassmannian formula in terms of edge variables of the graphs. Unlike in gauge theory where the analogous form involves only dlog- factors, in gravity we find a non-trivial numerator as well as higher degree poles in the edge variables. Based on the structure of the Grassmannian formula for = 8 N supergravity we conjecture that gravity loop amplitudes also possess similar properties. In particular, we find that there are only logarithmic singularities on cuts with finite loop momentum and that poles at infinity are present. vii Published Content and Contributions Much of this thesis is adapted from articles that originally appeared in other forms. All are available as arXiv preprints as well as published journal articles. References [1,2] serve as chapters 1 and 2 in this thesis respectively; The last chapter is taken from reference [3]; Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz, and J. Trnka, Logarithmic Singularities and • Maximally Supersymmetric Amplitudes, JHEP 06 (2015) 202. doi: 10.1007/JHEP06(2015)202, [arXiv:1412.8584] All authors contributed equally. I was mostly involved in the construction of the dlog bases and constructing explicit dlog forms. Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz, and J. Trnka, Evidence for a Nonplanar • Amplituhedron JHEP 06 (2016) 098. doi: 10.1007/JHEP06(2016)098,[arXiv:1512.08591] All authors contributed equally. I was involved in the basis construction and checking the unitarity cuts. E. Herrmann, and J. Trnka, Gravity On-shell Diagrams • JHEP 11 (2016) 136. doi: 10.1007/JHEP11(2016)136,[arXiv:1604.03479] Both authors contributed equally. viii Contents Acknowledgments iii Abstractv Published Content and Contributions vii Introduction1 Logarithmic singularities and maximally
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