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

TL;DR

This paper calculates four-loop gluon-to-quark splitting functions in QCD, providing new moments and approximations that improve the precision of perturbative predictions for collider physics.

Contribution

It presents the first four-loop calculations of the gluon-to-quark splitting function moments and constructs accurate approximations for practical use in collider physics.

Findings

Four-loop moments of P_qg computed analytically for N ≤ 20.

Constructed approximations for P_qg(x) applicable to collider physics.

N^3LO corrections improve perturbative accuracy for the singlet quark distribution.

Abstract

We have computed the even-$N$ moments $N \leq 20$ of the gluon-to-quark splitting function $P_{\rm qg}$ 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 $P_{\rm qg}(x)$ 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 $\alpha_s = 0.2$.