Renormalization group equations and the recurrence pole relations in pure quantum gravity

TL;DR

This paper develops generalized renormalization group equations for perturbative quantum gravity, deriving recurrence relations for higher pole divergences and showing they can be absorbed into the bare gravitational action through renormalization.

Contribution

It introduces a generalization of RG equations in quantum gravity, providing exact forms for higher pole counter-terms and demonstrating their absorption into the bare action.

Findings

Recurrence relations relate higher pole divergences to single poles.

Higher pole counter-terms are explicitly computed at 2, 3, and 4 loops.

Divergent terms can be absorbed into the bare gravitational action.

Abstract

In the framework of dimensional regularization, we propose a generalization of the renormalization group equations in the case of the perturbative quantum gravity that involves renormalization of the metric and of the higher order Riemann curvature couplings. The case of zero cosmological constant is considered. Solving the renormalization group (RG) equations we compute the respective beta functions and derive the recurrence relations, valid at any order in the Newton constant, that relate the higher pole terms $1/(d-4)^n$ to a single pole $1/(d-4)$ in the quantum effective action. Using the recurrence relations we find the exact form for the higher pole counter-terms that appear in 2, 3 and 4 loops and we make certain statements about the general structure of the higher pole counter-terms in any loop. We show that the complete set of the UV divergent terms can be consistently (at any order in the Newton constant) hidden in the bare gravitational action, that includes the terms of higher order in the Riemann tensor, provided the metric and the higher curvature couplings are renormalized according to the RG equations.