Yang-Mills–gravity system by means of a [*gauge invariant*]{} running effective action. In particular we shall see that because of the semi-direct product structure of the pertinent gauge group there arises a subtlety as for the appropriate construction of the ghost action. Second, we use the resulting framework in order to find the RG flow in a simple truncation of the space of actions which, however, is general enough to allow for an approximate determination of the beta function of the Yang-Mills coupling constant in presence of quantized gravity.

The remaining sections of this paper are organized as follows. In Section 2 we describe the general setup of the effective average action for a Yang-Mills field coupled to gravity, the corresponding FRGE in particular. Then, in Section 3, we discuss the problem of background covariant ghost actions and explain its solution. The explicit computation of the running gauge coupling is performed in Section 4, and Section 5 contains a summary as well as a brief discussion of various immediate applications of our results.

The Functional RG Setup
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Schematically the dynamics of the Yang-Mills–gravity system is governed by the path integral $$\begin{aligned}
\label{PI-sketchy}
{\cal Z} = \int\:{\cal D}\gamma_{\mu\nu}{\cal D}{\cal A}^a_\mu\:{\rm e}^{-S[\gamma, 