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Absence of gravitational contributions to the running Yang-Mills coupling. (English) Zbl 1246.81161
Summary: 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 $$\beta$$ 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.

##### MSC:
 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T13 Yang-Mills and other gauge theories in quantum field theory 81T17 Renormalization group methods applied to problems in quantum field theory 81T18 Feynman diagrams 83C45 Quantization of the gravitational field
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