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

We consider the anomalous dimension γ_{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 ζ(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 N=1, 4 supersymmetric Yang-Mills theories. We also present the contribution to the rational part of γ_{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.