Five-loop beta function for gauge theories: computations, results and consequences

TL;DR

The paper reports the first complete computation of the five-loop (N$^4$LO) beta function for Yang–Mills theories with fermions and for QED, using the background-field method together with the IR-rearranged $R^*$-operation to reduce five-loop pole terms to four-loop propagator integrals. The authors developed a diagram-by-diagram implementation of $R^*$, along with tensor-projector techniques and FORM-based tooling, enabling evaluation of thousands of five-loop diagrams and extraction of the renormalization constant $Z_B$ to obtain the beta-function coefficients up to $eta_4$ for general gauge groups and representations. They provide analytical expressions for the coefficients and discuss their numerical impact on QCD running, as well as presenting results for related five-loop quantities such as Higgs decay to hadrons in the heavy-top limit and low-N non-singlet splitting-function moments; these illustrate the convergence and reliability of the N$^4$LO expansion at phenomenologically relevant scales. Overall, the work demonstrates that the five-loop expansion is perturbatively stable and yields high-precision predictions, with implications for precision collider phenomenology and for understanding the structure of perturbative gauge theories.

Abstract

At the end of 2016, we computed the five-loop (N$^4$LO) contributions to the beta function in perturbative Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Here we recall main tools used in and specifically developed for this computation and its main analytic and numerical results. The development work carried out for this project facilitated further even more involved analytic five-loop computations. We briefly summarize also their numerical QCD results for Higgs-boson decay to hadrons in the heavy-top limit and for two N$^4$LO splitting functions for the evolution of quark distributions of hadrons. The latter lead to a first realistic estimate of the five-loop contribution to another important quantity in perturbative QCD, the quark cusp anomalous dimension.

Figures (2)

• Figure 1: One of the more complicated diagrams. Single lines represent gluons, and the external double lines represent the background field. The presence of the 10 purely gluonic vertices creates a large expression after the substitution of the Feynman rules.
• Figure 2: One external line is moved to create a Feynman diagram that can be integrated, here done for the topology of fig. \ref{['fig:gluons']}. One should take into account that there can be up to 5 powers of dot products in the numerator, causing many UV subdivergences. Furthermore, the double propagator that remains on the right can introduce IR divergences. After the subdivergences are subtracted, the integral over $p$ can be performed and the remaining four-loop topology can be handled by the Forcer program.