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

TL;DR

This paper computes and approximates the four-loop quark-to-gluon splitting function in QCD, significantly reducing uncertainties for high-precision collider physics at small momentum fractions.

Contribution

It provides new moments and improved approximations for the N^3LO quark-to-gluon splitting function in perturbative QCD.

Findings

Computed moments N ≤ 20 for P_{gq}^(3)(x)

Constructed accurate approximations reducing uncertainties

Errors are below 1% for x > 10^{-3} at α_s=0.2

Abstract

We present the even-N moments N =< 20 of the fourth-order (N^3LO) 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 P_{gq} at N^3LO with a typical quark distribution are small at x >~ 10^{-3} and exceed 1% only at x ~< 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.