The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements

Future high luminosity polarized deep--inelastic scattering experiments will improve both the knowledge of the spin sub--structure of the nucleons and contribute further to the precision determination of the strong coupling constant, as well as, reveal currently yet unknown higher twist contributions in the polarized sector. For all these tasks to be performed, it is necessary to know the QCD leading twist scaling violations of the measured structure functions. Here an important ingredient consists in the polarized singlet anomalous dimensions and splitting functions in QCD. We recalculate these quantities to three--loop order in the M--scheme by using the traditional method of space--like off--shell massless operator matrix elements, being a gauge--dependent framework. Here one obtains the anomalous dimensions without referring to gravitational currents, needed when calculating them using the forward Compton amplitude. We also calculate the non--singlet splitting function $\Delta P_{\rm qq}^{(2), \rm s, NS}$ and compare the final results to the literature, also including predictions for the region of small values of Bjorken $x$.