The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case

TL;DR

This paper computes the NNLO contributions to helicity-dependent splitting functions in QCD, extending previous calculations and providing insights into their behavior at small momentum fractions for polarized parton density evolution.

Contribution

It presents the first NNLO calculations of helicity-dependent splitting functions in QCD, including all relevant parton channels, using advanced mathematical and computational methods.

Findings

NNLO corrections are small at moderate x but large at very small x due to double-logarithmic enhancement.

Corrections are manageable down to x ≈ 10^{-4} for polarized parton evolution.

The results extend previous unpolarized calculations to the helicity-dependent case.

Abstract

We present the next-to-next-to-leading order (NNLO) contributions to the main splitting functions for the evolution of longitudinally polarized parton densities of hadrons in perturbative QCD. The quark-quark and gluon-quark splitting functions have been obtained by extending our previous all Mellin-N calculations to the structure function g_1 in electromagnetic deep-inelastic scattering (DIS). Their quark-gluon and gluon-gluon counterparts have been derived using third-order fixed-N calculations of structure functions in graviton-exchange DIS, relations to the unpolarized case and mathematical tools for systems of Diophantine equations. The NNLO corrections to the splitting functions are small outside the region of small momentum fractions x where they exhibit a large double-logarithmic enhancement, yet the corrections to the evolution of the parton densities can be unproblematic down to at least x about 10^{-4}.