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

A bstract 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. {₂ₔₒ₏, ₀|}䂺䃄=- (ₛN{) }⁴73⁶20160+₃²8+1N² (31⁶5040+9₃²4). Γ cusp, A α s 4 = − α s N π 4 73 π 6 20160 + ζ 3 2 8 + 1 N 2 31 π 6 5040 + 9 ζ 3 2 4. 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.