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

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 begin{equation} {Gamma}^{rm}_{rm cusp}Big|_{alpha_s^4} = -left( frac{alpha_s N}{pi}right)^4 left[ frac{73 pi^6}{20160} + frac{ zeta_{3}^2}{8} + frac{1}{N^2} left( frac{31pi^6}{5040} + frac{9 zeta_3^2}{4} right) right] ,. onumber end{equation} 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.