Absence of gravitational contributions to the running Yang-Mills coupling

TL;DR

This paper demonstrates through detailed calculations that quantum gravity does not modify the running of the Yang-Mills coupling, confirming the stability of asymptotic freedom against gravitational effects.

Contribution

It provides a comprehensive diagrammatic analysis showing the absence of gravitational contributions to the Yang-Mills running coupling at one loop.

Findings

Gravitational quadratic divergencies cancel in cut-off regularization.

No gravitational modification to asymptotic freedom at high energies.

Counterterms can be absorbed by field redefinitions, preserving classical results.

Abstract

The question of a modification of the running gauge coupling of (non-) abelian gauge theories by an incorporation of the quantum gravity contribution has recently attracted considerable interest. In this letter we perform an involved diagrammatical calculation in the full Einstein-Yang-Mills system both in cut-off and dimensional regularization at one loop order. It is found that all gravitational quadratic divergencies cancel in cut-off regularization and are trivially absent in dimensional regularization so that there is no alteration to asymptotic freedom at high energies. The logarithmic divergencies give rise to an extended effective Einstein-Yang-Mills Lagrangian with a counterterm of dimension six. In the pure Yang-Mills sector this counterterm can be removed by a nonlinear field redefinition of the gauge potential, reproducing a classical result of Deser, Tsao and van Nieuwenhuizen obtained in the background field method with dimensional regularization.