Running couplings in equivariantly gauge-fixed SU(N) Yang--Mills theories

TL;DR

This paper calculates the one-loop beta function for a gauge-fixed SU(N) Yang--Mills theory with partial gauge fixing, revealing that the coset interaction coupling is asymptotically free and exhibits dimensional transmutation, potentially impacting non-perturbative understanding.

Contribution

It provides the first calculation of the beta function for the coset interaction coupling in equivariantly gauge-fixed SU(N) theories, showing its asymptotic freedom and scale generation.

Findings

The coset coupling $ ilde{g}$ is asymptotically free.

Dimensional transmutation occurs for $ ilde{g}$, generating a scale $ ilde{ abla}$.

The scale $ ilde{ abla}$ can be larger than or equal to the gauge coupling scale.

Abstract

In equivariantly gauge-fixed SU(N) Yang--Mills theories, the gauge symmetry is only partially fixed, leaving a subgroup $H\subset SU(N)$ unfixed. Such theories avoid Neuberger's nogo theorem if the subgroup $H$ contains at least the Cartan subgroup $U(1)^{N-1}$, and they are thus non-perturbatively well defined if regulated on a finite lattice. We calculate the one-loop beta function for the coupling $\tilde{g}^2=\xi g^2$, where $g$ is the gauge coupling and $\xi$ is the gauge parameter, for a class of subgroups including the cases that $H=U(1)^{N-1}$ or $H=SU(M)\times SU(N-M)\times U(1)$. The coupling $\tilde{g}$ represents the strength of the interaction of the gauge degrees of freedom associated with the coset $SU(N)/H$. We find that $\tilde{g}$, like $g$, is asymptotically free. We solve the renormalization-group equations for the running of the couplings $g$ and $\tilde{g}$, and find that dimensional transmutation takes place also for the coupling $\tilde{g}$, generating a scale $\tilde{\Lambda}$ which can be larger than or equal to the scale $\Lambda$ associated with the gauge coupling $g$, but not smaller. We speculate on the possible implications of these results.