Four-Loop Anomalous Dimension of Flavor Nonsinglet Twist-Two Operator of General Lorentz Spin in QCD: ζ(3) Term

We consider the anomalous dimension of the flavor nonsinglet twist-two quark operator of arbitrary Lorentz spin N at four loops in QCD and construct its contribution proportional to ζ(3) in analytic form by applying advanced methods of number theory on the available knowledge of low-N moments. In conjunction with similar results on the ζ(5) and ζ(4) contributions, this completes our knowledge of the transcendental part of the considered anomalous dimension. This also provides important constraints on the as-yet elusive all-N form of the rational part. Via Mellin transformation, we thus obtain the exact functional form in x of the respective piece of the nonsinglet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi splitting function at four loops. This allows us to appreciably reduce the theoretical uncertainty in the approximation of that splitting function otherwise amenable from the first few low-N moments. Published by the American Physical Society 2025