The five-loop beta function of Yang-Mills theory with fermions

TL;DR

The paper addresses the problem of determining the five-loop beta function coefficient $\beta_4$ for Yang-Mills theory with fermions in MS-like schemes for a general simple gauge group. It employs the background-field formalism together with an enhanced $R^*$-operation and tensor-reduction boosted by the NEWRSTAR implementation, combined with the Forcer program to handle four-loop propagators, to obtain the five-loop poles of the background-field self-energy and thus $\beta_4$. The main result is an explicit expression for $\beta_4$ in terms of gauge-group invariants (such as $C_A$, $C_F$, $T_F$, $d_F^{abcd}$, $d_A^{abcd}$) and fermion content $n_f$, including SU$(N)$ QCD and QED limits, and it agrees with prior independent calculations. Numerically, the five-loop corrections are small compared to the four-loop terms for physical $n_f$, reinforcing perturbative convergence in QCD and SU$(N)$ gauge theories and supporting the reliability of running couplings computed in the $\,\overline{\text{MS}}$ scheme up to N$^4$LO; the work also provides a refined framework (NEWRSTAR) for future multi-loop renormalization tasks.

Abstract

We have computed the five-loop corrections to the scale dependence of the renormalized coupling constant for Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Our analytical result, obtained using the background field method, infrared rearrangement via a new diagram-by-diagram implementation of the R* operation and the Forcer program for massless four-loop propagators, confirms the QCD and QED results obtained by only one group before. The numerical size of the five-loop corrections is briefly discussed in the standard MSbar scheme for QCD with n_f flavours and for pure SU(N) Yang-Mills theory. Their effect in QCD is much smaller than the four-loop contributions, even at rather low scales.

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 topology that can be integrated. Here we do this for the diagram of figure \ref{['fig:gluons']}. One should take into account that there can be up to 5 powers of dot products in the numerator, causing many subdivergences. Furthermore, the double propagator that remains on the right can introduce infrared divergences. After the subdivergences have been subtracted, the integral over $p$ can be performed and the remaining four-loop topology can be handled by the Forcer program.