Towards the five-loop Beta function for a general gauge group

TL;DR

This work delivers analytical results for the five-loop QCD Beta function for a general gauge group, providing the $N_f^4$ term (in agreement with large-$N_f$ limits) and the newly computed $N_f^3$ term. Using an automated pipeline that combines diagram generation, color algebra, UV mass-regulated expansions, IBP reduction to 110 master integrals, and PSLQ for expressing results in terms of $\zeta$ values, the authors express the Beta-function coefficients in terms of $n_f$, $c_f$, and higher-order invariants $d_i$. The results are validated by reducing to SU$(3)$ and matching independent calculations, and by consistency with known large-$N_f$ expansions. The study establishes a scalable framework for completing the remaining flavor-power terms and for determining additional anomalous dimensions in high-loop QCD calculations, with potential gauge-parameter cross-checks in future work.

Abstract

We present analytical results for the $N_f^4$ and $N_f^3$ terms of the five-loop Beta function, for a general gauge group. While the former term agrees with results available from large-$N_f$ studies, the latter is new and extends the value known for SU(3) from an independent calculation.

Figures (2)

• Figure 1: Sample 5-loop diagrams that contribute to different color structures at $N_{\mathrm{f}}^4$. Straight (red) and curly (black) lines denote quarks and gluons, respectively. We did not draw arrows on the fermion lines, since all combinations occur.
• Figure 2: Sample 5-loop diagrams that contribute to different color structures at $N_{\mathrm{f}}^3$. The notation is as in figure \ref{['fig:nf4']}. Only the diagram classes whose representatives are depicted in the first line need to be considered in practice. For details, see the main text.