The full four-loop cusp anomalous dimension in mathcal{N} = 4 super Yang-Mills and QCD

Abstract

We present the complete formula for the cusp anomalous dimension at four loops in QCD and in maximally supersymmetric Yang-Mills. In the latter theory it is given by {left.{Gamma}_{mathrm{cusp},mathrm{A}}right|}_{alpha_s^4}=-{left(frac{alpha_sN}{pi}right)}^4left[frac{73{pi}^6}{20160}+frac{zeta_3^2}{8}+frac{1}{N^2}left(frac{31{pi}^6}{5040}+frac{9{zeta}_3^2}{4}right)right]. Our approach is based on computing the correlation function of a rectangular light-like Wilson loop with a Lagrangian insertion, normalized by the expectation value of the Wilson loop. In maximally supersymmetric Yang-Mills theory, this ratio is a finite function of a cross-ratio and the coupling constant. We compute it to three loops, including the full colour dependence. Integrating over the position of the Lagrangian insertion gives the four-loop Wilson loop. We extract its leading divergence, which determines the four-loop cusp anomalous dimension. Finally, we employ a supersymmetric decomposition to derive the last missing ingredient in the corresponding QCD result.