Four-loop splitting functions in QCD -- The gluon-to-quark case

We have computed the even-$N$ moments $N\le 20$ of the gluon-to-quark splitting function $P_{\mathrm{q}\mathrm{g}}$ at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via physical cross sections. Using our new moments and all available endpoint constraints, we construct approximations for the four-loop $P_{\mathrm{q}\mathrm{g}}(x)$ that should be sufficient for a wide range of collider-physics applications. The N${}^3$LO corrections resulting from these and the corresponding quark-quark splitting functions lead to a marked improvement of the perturbative accuracy for the scale derivative of the singlet quark distribution, with effects of 1% or less at $x\gtrsim {10}^{-4}$ at a standard reference scale with ${\alpha }_s=0.2$.