## Calculation of webs in non-Abelian gauge theories using unitarity cuts

##### Abstract

When calculating scattering processes in theories involving massless gauge bosons,
such as gluons in Quantum Chromodynamics (QCD), one encounters infrared
(IR), or soft, divergences. To obtain precise predictions, it is important to
have exact expressions for these IR divergences, which are present in any on-shell
scattering amplitude. Due to their long wavelength, soft gluons factorise
with respect to short-distance, or hard, interactions and can be captured by
correlators of semi-infinite Wilson lines. The latter obey a renormalisation
group equation, which gives rise to exponentiation. The exponent can be
represented diagrammatically in terms of weighted sums of Feynman diagrams,
called webs. A web with L external legs, each with ni gluon attachments, is
denoted (n1; n2; : : : ; nL). In this way all soft gluon interactions can be described
by a soft anomalous dimension. It is currently known at three loops with lightlike
kinematics, and at two loops with general kinematics. Our work is a step towards
a three-loop result in general kinematics.
In recent years, much progress has been made in understanding the general
physical properties of scattering amplitudes and in exploiting these properties
to calculate specific amplitudes. At the same time, we have discovered a lot
of structure underpinning the space of multiple polylogarithms, the functions
in terms of which most known amplitudes can be written. General properties
include analyticity, implying that scattering amplitudes are analytic functions
except on certain branch cuts, and unitarity, or conservation of probability. These
two properties are both exploited by unitarity cuts. Unitarity cuts provide a
diagrammatic way of calculating the discontinuities of a Feynman diagram across
its branch cuts, which is often simpler than calculating the diagram itself. From
this discontinuity, the original function can be reconstructed by performing a
dispersive integral.
In this work, we extend the formalism of unitarity cuts to incorporate
diagrams involving Wilson-line propagators, where the inverse propagator is linear
in the loop momenta, rather than the quadratic case which has been studied
before. To exploit this for the calculation of the soft anomalous dimension, we first
found a suitable momentum-space IR regulator and corresponding prescription,
and then derived the appropriate largest time equation (LTE). We find that, as
in the case of the scalar diagrams, most terms contributing to the LTE turn out
to be zero, albeit for different reasons. This simplifies calculations considerably.
This formalism is then applied to the calculation of webs with non-lightlike
Wilson lines. As a test, we first looked at webs that have been previously studied
using other methods. It emerges that, when using the correct variables, the
dispersive integrals one encounters here are trivial, illustrating why unitarity cuts
are a particularly useful tool for the calculation of webs. We observe that our
technique is especially efficient when looking at diagrams involving three-gluon
vertices, such as the (1; 1; 1) web and the Y diagram between two lines.
We then focus on three-loop diagrams connecting three or four external non-lightlike
lines and involving a three-gluon vertex. We calculate the previously
unknown three-loop three-leg (1; 1; 3) web in general kinematics. We obtain a
result which agrees with the recently calculated lightlike limit. We also develop
a technique to test our results numerically using the computer program SecDec,
and we find agreement with our analytical result.
The result for the (1; 1; 3) web can then be exploited to gain insight into the
more complicated three-loop four-leg (1; 1; 1; 2) web. Indeed, the (1; 1; 1; 2) web
reduces to the (1; 1; 3) web in a certain collinear limit. We propose an ansatz for
the (1; 1; 1; 2) web in general kinematics, based on a conjectured basis of multiple
polylogarithms. The result for the (1; 1; 3) web, together with the known result
for the lightlike limit of the (1; 1; 1; 2) web, imposes strong constraints on the
ansatz. Using these constraints, we manage to fix all but four coefficients in the
ansatz. We fit the remaining coefficients numerically, but find that the quality
of the fit is not good. We find possible explanations for this poor quality. This
calculation is still a work in progress.
Our results provide a major step towards the full calculation of the three-loop
soft anomalous dimension for non-lightlike Wilson lines. We calculated
new results for three-loop webs, and also deepened the understanding of webs
in general. We confirm a conjecture about the functional dependence of the soft
anomalous dimension on the cusp angles. We also confirm earlier findings about
the symbol alphabet of the relevant functions. This confirms the remarkable
simplicity found earlier in the expressions for the soft anomalous dimension.