# One-loop quantum gravitational corrections to the scalar two-point function at fixed geodesic distance

## TL;DR

This paper develops a gauge-invariant method for computing quantum gravitational corrections to scalar two-point functions by fixing geodesic distances, addressing divergences and gauge dependence in perturbative quantum gravity.

## Contribution

It introduces a renormalization scheme for non-local, gauge-invariant correlation functions based on geodesic distances in quantum gravity.

## Key findings

- Correlation functions are finite and gauge-independent after renormalization.
- One-loop corrections exhibit double logarithmic divergences.
- The method provides a consistent way to define gauge-invariant observables in quantum gravity.

## Abstract

We study a proposal for gauge-invariant correlation functions in perturbative quantum gravity, which are obtained by fixing the geodesic distance between points in the fluctuating geometry. These correlation functions are non-local and strongly divergent, and we show how to renormalise them by performing a "wave function renormalisation" of the geodesic embedding coordinates. The result is finite and gauge-independent, but displays unusual features such as double logarithms at one-loop order.