The QCD Splitting Functions at Three Loops: Methods and Results

TL;DR

This work computes the complete next-to-next-to-leading order ($\mathcal{O}(\alpha_s^3)$) unpolarized QCD splitting functions that govern parton-density evolution, providing the full analytical dependence on the Mellin variable $N$ alongside the $x$-space results. The authors employ a Mellin-space framework based on the optical theorem and operator product expansion, performing an extensive three-loop reduction to master integrals with dimensional regularization and solving via recursions in $N$ using harmonic sums and harmonic polylogarithms. The results, consistent with all known fixed-$N$ moments and with resummation predictions, show NNLO corrections are typically small for $N>2$ and improve scale stability, while revealing new color structures at three loops and enriching the precision toolkit for parton-evolution and DIS coefficient functions. This methodological advance enables robust, high-precision predictions for hard processes at current and future colliders and lays groundwork for potential polarized NNLO extensions.

Abstract

We have computed the complete next-to-next-to-leading order (NNLO) contributions to the splitting functions governing the evolution of unpolarized parton densities in perturbative QCD. Our results agree with all partial results available in the literature. We illustrate the methods used for this calculation with some examples and display selected results to show the size of the NNLO corrections and their effect on the evolution.

Figures (6)

• Figure 1: The perturbative expansion of the singlet anomalous dimensions $\gamma(\alpha_s,N)$.
• Figure 2: The three-loop singlet splitting functions $P^{\,(2)}_{\rm ab}$ with the leading small-$x$ terms (dotted) and the fixed-moment estimates (dashed).
• Figure 3: The non-singlet three-loop splitting functions $P_{\rm ns}^{\, -}$ ($n_f$-independent part) and $P_{\rm ns}^{\: s}/n_f$.
• Figure 4: The perturbative expansion of the scale derivatives (\ref{['eq:evols']}) of the singlet distributions (\ref{['eq:shapesq']}),(\ref{['eq:shapesg']}).
• Figure 5: The perturbative expansion of the scale derivative (\ref{['eq:evolns']}) of the non-singlet distributions $q_{\,-}$ and $q_{\,\rm v}$ for the input (\ref{['eq:shapesns']}).