Studying the soft anomalous dimension for massless multi-leg scattering at four loops

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.