# Vacuum effective actions and mass-dependent renormalization in curved space

## TL;DR

This paper reviews the computation of non-local effective actions in semiclassical gravity, emphasizing the role of form-factors in deriving mass-dependent renormalization group functions for gravitational couplings.

## Contribution

It provides a comprehensive overview of covariant heat kernel methods for calculating non-local form-factors in curved spacetime and their application to gravitational renormalization.

## Key findings

- Non-local form-factors are crucial for understanding quantum corrections in gravity.
- Mass-dependent beta functions for gravitational couplings are constructed from these form-factors.
- The review covers results in both two and four dimensions.

## Abstract

We review past and present results on the non-local form-factors of the effective action of semiclassical gravity in two and four dimensions computed by means of a covariant expansion of the heat kernel up to the second order in the curvatures. We discuss the importance of these form-factors in the construction of mass-dependent beta functions for the Newton's constant and the other gravitational couplings.