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

We present the even-𝑁 moments 𝑁 ≤ 20 of the fourth-order (N3LO) contribution 𝑃 (3) gq (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 (3) gq (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 N3LO with a typical quark distribution are small at 𝑥 ≳ 10−3 and exceed 1% only at 𝑥 ≲ 10−4 for a strong coupling 𝛼s = 0.2. The present results for 𝑃 (3) gq (𝑥) should thus be sufficient for most collider-physics applications.