Absence of gravitational contributions to the running Yang–Mills coupling

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. This settles the previously open question of a potential regularization scheme dependence of the one-loop β function traditionally computed in the background field approach. Furthermore we show that the remaining logarithmic divergencies give rise to an extended effective Einstein–Yang–Mills Lagrangian with a counterterm of dimension six.