Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part

Abstract

We consider the anomalous dimension ${\gamma }_{gg}^{(3)}(N)$ of the twist-two gluon operator of arbitrary Lorentz spin $N$ in the quark flavor singlet sector of a general gauge theory at four loops and construct its contribution proportional to $\zeta (3)$ in analytic form by applying the Lenstra-Lenstra-Lovász algorithm to the available low-$N$ moments. We exploit generalized Gribov-Liptov reciprocity, establish new self-tuning relations for the anomalous dimension matrix of the singlet sector, and inject information from $\mathcal{N}=1,4$ supersymmetric Yang-Mills theories. We also present the contribution to the rational part of ${\gamma }_{gg}^{(3)}(N)$ with color factor $C_F^2{n}_f^2$. Exact contributions to the four-loop splitting function $P_{gg}^{(3)}(x)$ hence resulting via inverse Mellin transformation help us to reduce theoretical uncertainties in scaling violations of parton distribution functions in QCD.

11 pages, minor corrections included, text and references added, Supplemental Material extended, accepted for publication in PRL