Studying the soft anomalous dimension for massless multi-leg scattering at four loops
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Date
11/04/2023Author
Maher, Niamh
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Abstract
Infrared (IR) singularities are a salient feature of gauge theory scattering
amplitudes and their study is important from both practical and theoretical
perspectives. IR singularities exponentiate in terms of the so called soft
anomalous dimension and are the object of this research. At one loop, the
soft anomalous dimension is a sum over dipoles. Corrections to the dipole
formula start at three loops with dependence on conformally invariant cross ratios
constructed from the momentum dot products between four distinct particles.
At four loops there are additional corrections involving the quartic Casimir
invariants.
While direct computation of the soft anomalous dimension was achieved at three
loops, it is far beyond reach at four loops. In this work we develop a bootstrap
method which can potentially allow us to determine it. The starting point is
writing an ansatz for the four-loop soft anomalous dimension in terms of a suitable
basis of functions of the kinematic variables. These multiply colour structures
representing fully connected diagrams. To describe the kinematic functions in
the Euclidean region, where all Mandelstam invariants are negative and for fixed
angle scattering the amplitude is free from singularities, the soft anomalous
dimension must be expressible in terms of single-valued polylogarithms. These
can then be analytically continued to the physical region. At three loops, single-
valued harmonic polygarithms were sufficient to describe the kinematic functions.
However we find that at four loops, new higher weight single-valued multiple
polylogarithms are required. Our work employs physical limits such as collinear
limits and Regge limits, along with Bose symmetry, to constrain and eventually
determine the kinematic functions in general kinematics.
This bootstrap approach was successfully employed at three loops, reproducing
the exact result. At four loops we find new constraints on the kinematic functions
from the Regge limit for 2 → 2 scattering and from the finiteness of the momentum conserving limit. Imposing all these constraints we summarise the
state-of-the-art knowledge of the soft anomalous dimension at four loops.
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