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We present the details of the analytic calculation of the three-loopangle-dependent cusp anomalous dimension in QCD and its supersymmetricextensions, including the maximally supersymmetric $\mathcal{N}=4$ superYang-Mills theory. The three-loop result in the latter theory is new andconfirms a conjecture made in our previous paper. We study various physicallimits of the cusp anomalous dimension and discuss its relation to thequark-antiquark potential including the effects of broken conformal symmetry inQCD. We find that the cusp anomalous dimension viewed as a function of the cuspangle and the new effective coupling given by light-like cusp anomalousdimension reveals a remarkable universality property -- it takes the same formin QCD and its supersymmetric extensions, to three loops at least. We exploitthis universality property and make use of the known result for the three-loopquark-antiquark potential to predict the special class of nonplanar correctionsto the cusp anomalous dimensions at four loops. Finally, we also discuss indetail the computation of all necessary Wilson line integrals up to three loopsusing the method of leading singularities and differential equations.

The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions