The double fermionic contribution to the four-loop quark-to-gluon splitting function

TL;DR

This paper calculates the four-loop quark-to-gluon splitting function in QCD, providing exact expressions and analyzing small-x and large-x behaviors, with new coefficients and consistency checks.

Contribution

It presents the first all-N form of the double fermionic contribution to the four-loop quark-to-gluon splitting function, derived from moments and Diophantine equations.

Findings

First all-N form of the nf^2 contribution to P_{gq}^(3)(x)

New coefficient for the 1/x ln x small-x term

Confirmed previous results for other small-x and large-x terms

Abstract

We have computed the first 30 even-N moments for the double fermionic (nf^2) part of the quark-to-gluon splitting function P_{gq} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. From these results we have determined the all-N form, and hence the exact x-space expression, using systems of Diophantine equations for its coefficients. The dominant and subdominant leading small-x nf^2 contributions to P_{gq}^(3)(x) are of the form 1/x ln x and ln^4 x, respectively; the leading large-x term is ln^4 (1-x). The coefficient of the first of these is new, the other two agree with results obtained before and thus provide checks of our results.