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 \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{31\pi^6}{5040} + \frac{9 \zeta_3^2}{4} \right) \right] \,.\nonumber \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.