Off-shell divergences in quantum gravity

TL;DR

This paper studies the off-shell divergences in pure quantum gravity, showing how they are constructed and renormalized, and emphasizing the role of BRST invariance and renormalisation group identities beyond one loop.

Contribution

It provides a detailed analysis of off-shell divergences in quantum gravity, demonstrating their structure and renormalization, and verifying these through explicit two-loop calculations.

Findings

Off-shell divergences depend only on the total metric at each loop order.

Background metric divergences do not appear separately and are not renormalized.

BRST invariance and renormalisation group identities ensure consistency beyond one loop.

Abstract

We investigate off-shell perturbative renormalisation of pure quantum gravity for both background metric and quantum fluctuations. We show that at each new loop order, the divergences that do not vanish on-shell are constructed from only the total metric, whilst those that vanish on-shell are renormalised by canonical transformations involving the quantum fields. Purely background metric divergences do not separately appear, and the background metric does not get renormalised. We highlight that renormalisation group identities play a crucial role ensuring consistency in the renormalisation of BRST transformations beyond one loop order. We verify these assertions by computing leading off-shell divergences to two loops, exploiting off-shell BRST invariance and the renormalisation group equations. Although some divergences can be absorbed by field redefinitions, we explain why this does not lead to finite beta-functions for the corresponding field.