## Three-loop soft anomalous dimension of massless multi-leg scattering

##### Abstract

Infrared (IR) singularities are a salient feature of any field theory containing
massless fields. In Quantum Chromodynamics (QCD), such singularities give
rise to logarithmic corrections to physical observables. For many interesting
observables, these logarithmic corrections grow large in certain areas of phase space,
threatening the stability of perturbative expansion and requiring resummation.
It is known, however, that IR singularities are universal and exponentiate, allowing
one to study their all-order behaviour in any gauge theory by means of so-called
webs: specific linear combinations of Feynman diagrams with modified colour
factors corresponding to those of fully connected trees of gluons.
Furthermore, infrared singularities factorise from the hard cross-section into
soft and jet functions. The soft function may be calculated as a correlator of
Wilson lines, vastly simplifying the computation of IR poles and allowing analytic
computation at high loop order. Renormalisation group equations then allow the
definition of a soft anomalous dimension, which may then be directly computed
either through differential equations or by a direct, diagrammatic method.
Soft singularities are highly constrained by rescaling symmetry, factorisation, Bose
symmetry, and high energy- and collinear limits. In the case of light-like external
partons, this leads directly to a set of constraint equations for the soft anomalous
dimension, the simplest solution of which is a sum over colour dipoles. At two
loops, this so-called dipole formula is the only admissible solution, leading to the
complete cancellation of any tripole colour structure. Corrections beyond the
dipole formula may first be seen at three loops, and must take the form of weight
five polylogarithmic functions of conformal invariant cross-ratios, correlating four
hard jets through a quadrupole colour structure.
In this thesis we calculate this first correction beyond the dipole formula by
considering three-loop multiparton webs in the asymptotic limit of light-like
external partons. We do this by computing all relevant webs correlating two,
three and four lines at three loop order by means of an asymptotic expansion of
Mellin-Barnes integrals near the limit of light-like external partons.
We find a remarkably simple result, expressible entirely in terms of Brown's
single-valued harmonic polylogarithms, consistent with high-energy and forward
scattering limits.
Finally, we study the behaviour of this correction in the limit of two partons
becoming collinear, and discuss collinear factorisation properties.