Multiparton webs in non-abelian gauge theories at three loops and beyond

Amplitudes in theories with a massless gauge boson su↵er from so-called infrared divergences where o↵-shell states become asymptotically close to the mass-shell in loop or phase-space momentum integrals. These singularities have been shown to cancel intricately order-by-order in the perturbative expansion. However, in order to obtain meaningful and precise predictions for physical observables, we must understand and compute such divergences to high orders. This can be accomplished by calculating webs: weighted sets of Feynman diagrams which, when exponentiated give the complete infrared singular component of the amplitude, known as the soft function. This quantity is formally equivalent to a vacuum expectation value of a product of Wilson lines. In this thesis we shall study webs correlating multiple Wilson lines, which di↵ers from the two line case due to the possibility of non-trivial colour flows. This renders the soft function matrix valued in the space of colour flows, thus making its calculation and renormalisation non-trivial. At present, the infrared singularities of non-abelian, multiparton scattering amplitudes are known only to two loops in general kinematics, and to three loops in a simplifying kinematic limit. This thesis will thus form part of a program of work aimed at calculating and understanding the three-loop singularities in general kinematics and in doing so we aim to gain all-order insights into the pertubative structure of non-abelian gauge theories. We first specialise to a subset of webs which we have called Multiple Gluon Exchange Webs (MGEWs), which contain only those diagrams with direct exchanges of soft gauge bosons directly between Wilson lines with no intervening threeor fourboson vertices. Studing their properties allows us to construct a basis of functions which describes all examples of such webs, and we conjecture will continue to do so at any order. Furthermore, we find that the basis functions can be described by a simple, one-dimensional integral over only logarithms. We iii go on to compute several examples providing evidence for the validity of our basis and demonstrate the utility of the framework we have built by computing a fourloop web and providing some all-order results for particular classes of MGEW. We then consider a step beyond MGEWs, that is, webs which contain a single three-gluon vertex sub-diagram. In particular we study the simplest web in this class correlating four lines at three loops and attempt to calculate it through the numerical fitting of a physically motivated ansatz. We show that this web cannot carry kinematic dependence through conformal invariant cross ratios, which arise when connected subdiagrams correlate at least four lines. Hence, it is subject to the same constraints as MGEWs with regards to their symbol alphabet, from the physical considerations in their lightlike limit and spacelike/timelike analytic continuation. Like all other known webs satisfying such constraints, we therefore argue that it can be written in terms of sums of products of MGEW basis functions. Symmetries inherent to our parameterisation of the cusp angles, Bose symmetry and transcendental weight further constrain this ansatz, resulting in forty parameters for which we present preliminary results of a numerical fit. iv Declaration Except where otherwise stated, the research undertaken in this thesis was the unaided work of the author. Where the work was done in collaboration with others, a significant contribution was made by the author. In particular the results of chapters 3 and 4 were produced in collaboration, and appear (or are planned to appear) in the following publications, • Chapter 3 – [1] G. Falcioni, E. Gardi, M. Harley, L. Magnea, C.D. White, Multiple Gluon Exchange Webs, JHEP 1410 (2014) 10, [arxiv:1407.3477] • Chapter 4 – [2] G. Falcioni, E. Gardi, M. Harley, L. Magnea, C.D. White, Webs containing a single three-gluon-vertex, to appear M.S. Harley October 2015 v Acknowledgements First and foremost, I would like to thank Einan Gardi, my supervisor, for his guidance, dedication, and infectious enthusiasm for physics. I will always be grateful to him for introducing me to the subject which has become my passion. Also, I would like to thank my collaborators: Giulio Falcioni, Lorenzo Magnea and Chris White for our many fascinating discussions and for the hard work we’ve shared. I look forward to continuing our work together in the near future. Many thanks go to Samuel Abreu, Øyvind Almelid, Simon Badger, Claude Duhr, Gustav Mogull, Donal O’Connell, Alexander Ochirov, Jenni Smillie, Andries Waelkens and Stephanie Yeomans for many interesting discussions. Moreover, thanks to all of the PPT group for the co↵ee break discussions, Friday whiskies, etc. I wish to express special gratitude to Sam and Jack for their conversation, the climbing and generally for their friendship. I would like to extend my sincerest thanks to Lorenzo for his hospitality during my time in Turin. I am indebted to him for the opportunity to experience such a wonderful place. I must also thank friends who made my time in Italy so special. To name a few: Giulio, Stefania, and their families; our ‘openspace’ friends; Vale, Wee Vale, Salvo, Alice, Sefora and the rest at Lungo Dora. I would also like to thank and Ruth Britto for her hospitality during the last weeks of thesis writing. I look forward to working together over the coming years. I am unable to fit on this page the words of gratitude I owe to my parents who have provided me with the love and support that have made it possible for me to reach where I am. Many thanks also to my family for their support and friendship. This thesis is dedicated to Kayleigh, my wife, without whom this work would not have been possible. Thank you, Kayleigh, for getting me through the rough times, for being there to share, and amplify, the good times and for letting me be there to share yours. vi