Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles

Abstract: We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the corresponding conjectured expressions with full angle dependence. Taking the limit of large Minkowskian angle, we extract novel analytic results for certain terms of the light-like cusp anomalous dimension. They agree with the known numerical results. Finally, we study the anti-parallel lines limit of the cusp anomalous dimension. In a conformal theory, the latter is proportional to the static quark-antiquark potential. We use the new four-loop results to determine parts of the conformal anomaly term.