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

Abstract

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 Γcusp,Aαs4=−αsNπ473π620160+ζ328+1N231π65040+9ζ324.$$ {\left.{\Gamma}_{\mathrm{cusp},\mathrm{A}}\right|}_{\alpha_s^4}=-{\left(\frac{\alpha_sN}{\pi}\right)}^4\left[\frac{73{\pi}^6}{20160}+\frac{\zeta_3^2}{8}+\frac{1}{N^2}\left(\frac{31{\pi}^6}{5040}+\frac{9{\zeta}_3^2}{4}\right)\right]. $$ 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.