Few-Body Syst (2018) 59:8
https://doi.org/10.1007/s00601-017-1327-x
ChrisD.White
Wilson Lines and Webs in Higher-Order QCD
Received: 1 November 2017 / Accepted: 16 December 2017 / Published online: 1 February 2018
© The Author(s) 2018. This article is an open access publication
Abstract Wilson lines have a number of uses in non-abelian gauge theories. A topical example in QCD is the
description of radiation in the soft or collinear limit, which must often be resummed to all orders in perturbation
theory. Correlators involving a pair of Wilson lines are known to exponentiate in terms of special Feynman
diagrams called “webs”. I will show how this language can be extended to an arbitrary number of Wilson lines,
which introduces novel new combinatoric structures (web mixing matrices) of interest in their own right. I will
also summarise recent results obtained from applying this formalism at three-loop order, before concluding
with a list of open problems.
1 Introduction
It is well-known that scattering amplitudes in perturbative quantum field theory are beset by infrared (IR)
divergences. Feynman rules tell us that we must integrate over all positions of radiated fields, and the integrand
is such that a divergence occurs (in four spacetime dimensions) as we integrate out to infinity. In momentum
space, this corresponds to a low energy, or soft gluon, as distinguished from the hard particles that emit the
radiation. This hand-wavy explanation shows us that infrared divergences are not unique to QCD, but occur
in many different (non)-abelian gauge theories, including gravity.
Infrared singularities have been studied for many decades, and it is known that they formally cancel when
real and virtual diagrams are combined in suitably inclusive observables. However, large kinematic contribu-
tions remain after this cancellation, that frequently need to be summed up to all orders in perturbation theory in
order to achieve meaningful comparison of theory to data. This is called resummation, and we are always trying
to make this more precise. There are also more formal applications of IR singularities: they give us informa-
tion about the all-order structure of perturbative quantum field theory, and are thus useful in understanding the
underlying structure of both Yang–Mills theory, and its supersymmetric extensions. Furthermore, IR singular-
ities are related to Wilson lines, which occur in many contexts, including the transverse-momentum dependent
parton distributions (TMDs) discussed elsewhere at LightCone 2017. Thus, the study of IR divergences remains
a highly active subject involving various branches of high energy physics.
In order to classify the structure of IR divergences further, let us first note that the structure of an arbitrary
amplitude
A
for the production of n partons has the schematic form (see e.g. ref. [1])
A
=
H
·
S
·
n
i=1
J
i
J
i
. (1)
C. D. White (
B
)
Centre for Research in String Theory, Queen Mary University of London, 327 Mile End Road, London E1 4SN, UK
E-mail: christopher.white@qmul.ac.uk
@Phil_Robichaud
@deepthiw
@JoseServera