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