Amplitudes in theories with a massless gauge boson suffer from so-called infrared divergences where off-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 differs 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 three- or four- boson 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 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 four-loop 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:705357 |
| Date | January 2016 |
| Creators | Harley, Mark Stuart |
| Contributors | Gardi, Einan ; Smillie, Jennifer |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/19562 |
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