The four-loop cusp anomalous dimension in N=4 super Yang-Mills and analytic integration techniques for Wilson line integrals

TL;DR

The paper tackles the problem of determining the velocity-dependent cusp anomalous dimension in $\mathcal{N}=4$ SYM to four loops, including planar and non-planar contributions, by developing a powerful d-log representation for Wilson line integrals and by relating the planar result to massive scattering amplitudes via Mellin-Barnes analysis. It provides an explicit planar four-loop expression in terms of harmonic polylogarithms with argument $1-x^2$ and weight seven, together with the non-planar scaling-limit result governed by a quartic Casimir invariant, and confirms the light-like cusp value. A strong-coupling extrapolation, guided by an analytic ansatz, shows good agreement with AdS/CFT predictions across a broad range of parameters, illustrating consistent behavior from weak to strong coupling. Overall, the work advances analytic control over IR structures in gauge theories, expands the class of functions describing Wilson line integrals, and strengthens the bridge between perturbative gauge theory and string-theoretic descriptions via AdS/CFT.

Abstract

Correlation functions of Wilson lines are relevant for describing the infrared structure of scattering amplitudes. We develop a new method for evaluating a wide class of such Wilson line integrals, and apply it to the calculation of the velocity-dependent cusp anomalous dimension in maximally supersymmetric Yang-Mills theory. We compute the four-loop non-planar correction in a recently introduced scaling limit. Moreover, we derive the full planar four-loop result by means of an ansatz which is based on the structure of known analytic results. We determine the coefficients in this ansatz by making use of a relationship to massive scattering amplitudes. As a corollary, our analytical result confirms the four-loop value of the light-like cusp anomalous dimension. Finally, we use the available perturbative data, as well as insight from AdS/CFT, in order to extrapolate the leading order values at strong coupling. The latter agree within two per cent with the corresponding string theory result, over a wide range of parameters.