The Three-Loop Splitting Functions in QCD: The Singlet Case

TL;DR

The authors compute the complete NNLO ($\mathcal{O}(\alpha_s^3)$) splitting functions for unpolarized flavour-singlet parton densities, providing exact results in Mellin-$N$ and Bjorken-$x$ space along with practical parametrizations. The work leverages forward-Compton amplitudes, harmonic sums, and harmonic polylogarithms to obtain the full $N$-dependent anomalous dimensions $\gamma_{ab}(N,\alpha_s)$ and their $x$-space counterparts $P^{(2)}_{ab}(x)$, including all parton channels $(qq, qg, gq, gg)$ and the pure-singlet piece. Numerical analysis shows rapid convergence for $x \gtrsim 10^{-3}$ and modest NNLO corrections with reduced renormalization-scale dependence, while small-$x$ behavior remains delicate and requires higher-order or resummed inputs. The results enable precise singlet-quark and gluon evolution and come with parametrizations and code to facilitate phenomenology, with implications for current and future collider analyses and a path toward four-loop studies.

Abstract

We compute the next-to-next-to-leading order (NNLO) contributions to the splitting functions governing the evolution of the unpolarized flavour-singlet parton densities in perturbative QCD. The exact expressions are presented in both Mellin-N and Bjorken-x space. We also provide accurate parametrizations for practical applications. Our results agree with all partial results available in the literature. As in the non-singlet case, the correct leading logarithmic predictions for small momentum fractions x do not provide good estimates of the respective complete splitting functions. We investigate the size of the corrections and the stability of the NNLO evolution under variation of the renormalization scale. The perturbative expansion appears to converge rapidly at x >~ 10^-3. Relatively large third-order corrections are found at smaller values of x.

Figures (12)

• Figure 1: The perturbative expansion of the diagonal anomalous dimensions $\gamma_{\,\rm qq}(N)$ and $\gamma_{\,\rm gg}(N)$ for four flavours at $\alpha_{\rm s} = 0.2$. The pure-singlet (ps) contribution to $\gamma_{\,\rm qq}$ is shown separately.
• Figure 2: As Fig. 1, but for the off-diagonal anomalous dimensions $\gamma_{\,\rm qg}(N)$ and $\gamma_{\,\rm gq}(N)$.
• Figure 3: The three-loop pure-singlet splitting function (\ref{['eq:Pps2']}) for four flavours, multiplied by $x$ for display purposes. Also shown is the uncertainty band derived in Ref. vanNeerven:2000wp using the lowest six even-integer moments Larin:1997wdRetey:2000nq and the leading small-$x$ term Catani:1994sq. The latter contribution is shown separately on the right-hand-side (dotted line) for $x<0.01$.
• Figure 4: As Fig. 3, but for the third-order gluon-quark splitting function specified in Eq. (\ref{['eq:Pqg2']}).
• Figure 5: As Fig. 3, but for the three-loop quark-gluon splitting function (\ref{['eq:Pgq2']}). Note that in this case the leading small-$x$ contribution was unknown before the present calculation.