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

##### Abstract

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.