Anomalous dimensions and splitting functions beyond the next-to-next-to-leading order

TL;DR

This work pushes the precision frontier of perturbative QCD for parton evolution by presenting new four-loop (N^3LO) results for flavour-singlet splitting functions and the gluon cusp anomalous dimension A_4,g, as well as first five-loop (N^4LO) results for non-singlet splitting functions. It combines Forcer-based DIS computations with OPE and large-N_c insights to determine all-N structures, including quartic color-factor contributions via d^{(4)} invariants, and derives exact and numerical values for A_{4,q} and A_{4,g}. The study reveals that quartic color terms break simple Casimir scaling at N^3LO but tend to align in the large-N_c limit, and it provides explicit α_s expansions and zeta-term structures that inform the convergence and scheme dependence of high-order evolution kernels. Overall, the results enhance the precision of PDF evolution and deepen understanding of high-order cusp anomalous dimensions in QCD, with implications for accurate predictions in hadron collider phenomenology.

Abstract

We report on recent progress on the splitting functions for the evolution of parton distributions and related quantities, the (lightlike) cusp anomalous dimensions, in perturbative QCD. New results are presented for the four-loop (next-to-next-to-next-to-leading order, N^3LO) contributions to the flavour-singlet splitting functions and the gluon cusp anomalous dimension. We present first results, the moments N=2 and N=3, for the five-loop (N^4LO) non-singlet splitting functions.

Figures (3)

• Figure 1: nameref-Fig1 fith LAB: Fig1 Moments of the singlet splitting functions at NNLO (lines) and N$^3$LO (even-$N$ points) for $\alpha_{\sf s} = 0.2$ and ${n^{}_{\! f}} = 4$, normalized to the respective NLO approximations.
• Figure 2: nameref-Fig2 fith LAB: Fig2 The dependence of the logarithmic factorization-scale derivatives of the singlet PDFs on the renormalization scale $\mu_{r}^{}$ at $N=2$ (where the very small scaling violations of $q_{\sf s}^{}$ and $g$ are related by the momentum sum rule) $N=4$ and $N=6$ for the initial distributions (\ref{['qsgInp']}).
• Figure 3: nameref-Fig3 fith LAB: Fig3 Left and middle panel: the renormalization-scale dependence of the logarithmic factorization-scale derivatives of the PDFs $q_{\rm ns}^{\,+}$ at $N=2$ and $q_{\rm ns}^{\,-}$ at $N=3$ at our standard reference point with $\alpha_{\sf s}(\mu_{f}^{\:\!2}) = 0.2$ and ${n^{}_{\! f}}=4$. Right panel: the corresponding $N=3$ results at a lower scale with $\alpha_{\sf s}(\mu_{f}^{\:\!2}) = 0.25$ and ${n^{}_{\! f}}=3$.