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

We present the even-N moments N <= 20 of the fourth-order ((NLO)-L-3) contribution P-gq((3))(x) to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, in particular with the moments N <= 10 derived before from structure functions in deep-inelastic scattering. Using the new moments and the available endpoint constraints, we construct approximations for P-gq((3))(x) which improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions x>0.1. The resulting errors of the convolution of Pgq at (NLO)-L-3 with a typical quark distribution are small at x greater than or similar to 10(-3) and exceed 1% only at x less than or similar to 10(-4) for a strong coupling alpha(s )= 0.2. The present results for P-gq((3))(x) should thus be sufficient for most collider-physics applications.