Infrared divergences in scattering amplitudes from correlators of Wilson Lines

Abstract

Scattering amplitudes in theories with massless particles feature infrared (IR) divergences. In QCD, gluons are massless and when their momenta tends to zero the amplitude diverges. We call this a soft divergence. For massless external particles there are further divergences called collinear divergences, when the invariant jet mass of the external and a radiated gluon tends to zero. Even in a UV finite theory such as N = 4 super-Yang Mills there exists infrared divergences. In fact, in the planar theory there exists an all-order ansatz for the IR divergences called the BDS ansatz which amounts to the exponentiation of the one loop result with anomalous dimensions that can be computed to all orders. In this thesis we shall be considering the more complicated case of non-planar QCD with both massless and massive scattering particles. First, we shall review the IR factorisation formula for massive scattering amplitudes. Here, soft divergences are described by the soft anomalous dimension matrix. It is defined to be a vacuum expectation value of non-lightlike Wilson lines. This object is calculable in perturbation theory. It exponentiates and the exponent is a sum over webs. We will then focus on how to calculate the individual integrals that appear in webs. The technique of differential equations is explained and applied to integrals up to two loops for webs. We then discuss a basis of functions for these specific integrals with the idea of creating an ansatz for the soft anomalous dimension and other related quantities. The second half of the thesis concerns massless scattering amplitudes. By factorising not only the amplitude but also a parton distribution function we find that they share the same hard collinear behaviour. They differ in their pure soft poles which are governed by lightlike Wilson-line correlators that follow different contours dictated by the kinematics. It allows us to explain an observed relation between the subleading pole of the form factor, γG, and the coefficient of δ(1 − x) in the DGLAP splitting kernels, Bδ. We then argue that divergences of lightlike Wilson-line correlators take a general form that only depend on local features, individual line lengths, and not on the global geometry.