The Three-Loop Splitting Functions in QCD

TL;DR

This work tackles the problem of precisely evolving unpolarized parton distributions by computing the complete NNLO splitting functions $P^{(2)}_{ij}$ in perturbative QCD. The authors adopt a forward-DIS approach, evaluating structure functions to $\mathcal{O}(a_s^3)$ via forward Compton amplitudes and performing mass-factorization in the $\overline{\text{MS}}$ scheme to extract the $1/\varepsilon$ poles as the evolution kernels, then reconstruct the full $N$-space dependence and invert to $x$-space. They provide explicit expressions in Mellin space using shifted harmonic sums and, via inverse Mellin transforms, in Bjorken-$x$ space with harmonic polylogarithms, including the three-loop cusp anomalous dimensions that govern the end-point behaviour. The numerical analysis shows NNLO corrections are small at moderate-to-large $x$ (typically a few percent at NNLO and considerably smaller than NLO), while small-$x$ regions exhibit enhanced effects demanding further theoretical work; the results are in agreement with all known partial results and establish a solid foundation for precision collider phenomenology and future polarized NNLO studies.

Abstract

We have computed the next-to-next-to-leading-order (NNLO) contributions to the evolution of unpolarized parton distributions in perturbative QCD. In this talk, we briefly recall why this huge computation was necessary and outline how it was performed. We then illustrate the structure of the results and discuss their end-point limits which include the three-loop cusp anomalous dimensions of the Wilson lines. Finally the numerical impact of the new contributions is illustrated.

Figures (6)

• Figure 1: Sample diagrams contributing to the inclusive process $\gamma^{\ast} g \rightarrow X$ up to third order in $a_{\rm s}$.
• Figure 2: The small-$x$ behaviour of non-singlet (upper row) and singlet NNLO splitting functions for ${n^{}_{\! f}}\! =\! 4$. Also shown are the (successive) approximations (\ref{['eq:sx-ns']}) and (\ref{['eq:sx-sg']}) by the leading small-$x$ logarithms and, for the singlet cases, the errors bands vanNeerven:2000wp used in the analyses of refs. Martin:2002drAlekhin:2002fv.
• Figure 3: The perturbative expansion of the singlet anomalous dimensions $\gamma^{}_{\rm f\:\!f'}$ up to NNLO.
• Figure 4: The perturbative expansion up to NNLO of the factorization-scale derivatives $\dot{q}_{\,\rm ns}^{\, -,\rm v}$ for the initial conditions (\ref{['eq:as-ref']}), (\ref{['eq:f-ref']}) and the standard choice $\mu_r = \mu_f$ of the renormalization scale.
• Figure 5: As Fig. 4, but for the singlet quark and gluon distributions $q_{\:\!\rm s}$ and $g$. The spikes close $x = 0.1$ in the right plots reflect the zeros of $\dot{f}_k$ and do not constitute large absolute corrections.