Gauge-invariant quantum gravitational corrections to correlation functions

TL;DR

This paper extends a gauge-invariant observable construction to general spacetimes, calculates one-loop graviton corrections to scalar correlation functions, and analyzes the impact on coupling constants, confirming expected functional forms and revealing a negative beta function.

Contribution

It generalizes gauge-invariant observable construction, computes one-loop graviton corrections in flat space, and studies their effects on scalar couplings and beta functions.

Findings

Gauge-dependent terms cancel in the corrected correlation functions.

One-loop corrections match expected functional forms.

The scalar coupling's beta function is negative for all non-minimal couplings.

Abstract

A recent proposal for gauge-invariant observables in inflation [R. Brunetti et al., JHEP 1608 (2016) 032] is examined. We give a generalisation of their construction to general background spacetimes. In flat space, we calculate one-loop graviton corrections to a scalar two-point function in a general gauge for the graviton. We explicitely show how the gauge-dependent terms cancel between the usual self-energy contributions and the additional corrections inherent in these observables. The one-loop corrections have the expected functional form, contrary to another recently studied proposal for gauge-invariant observables [M. B. Fr\"ob, Class. Quant. Grav. 35 (2018) 035005] where this is not the case. Furthermore, we determine the one-loop graviton corrections to the four-point coupling of the gauge-invariant scalar field, and the corresponding running of the coupling constant induced by graviton loops. Interestingly, the $\beta$ function is negative for all values of the non-minimal coupling of the scalar field to curvature.