Renormalization of gauge theories in curved space-time

We consider the renormalization of general gauge theories on curved space-time background, with the main assumption being the existence of a gauge-invariant and diffeomorphism invariant regularization. Using the Batalin-Vilkovisky formalism, one can show that the theory possesses gauge invariant and diffeomorphism invariant renormalizability at quantum level, up to an arbitrary order of the loop expansion. Starting from this point, we discuss the locality of the counterterms and the general prescription for constructing the power-counting renormalizable theories on curved background.

Figure 1

The single diagram with quadratic divergences in flat space generates an infinite set of diagrams with external lines of $h_{\mu \nu }$. Some of those diagrams have quadratic or logarithmic divergences, others are finite.

Figure 2

The single diagram with quadratic divergences in flat space generates an infinite set of diagrams with external lines of $h_{\mu \nu }$. Some of those diagrams have quadratic or logarithmic divergences, others are finite.

Figure 3

The single diagram with quartic divergences in flat space leads to the diagrams with quartic, quadratic, and logarithmic divergences due to external lines of $h_{\mu \nu }$ with the new vertices. Even though there are infinitely many new diagrams, the divergences are well controlled by covariance.