Heat Kernel and Scaling of Gravitational Constants

TL;DR

This paper investigates how quantum scalar and spinor fields influence the effective gravitational constants in curved spaces, revealing that local and non-local quantum corrections depend on different heat kernel regimes, affecting the understanding of Newton's constant running.

Contribution

It clarifies the dependence of quantum corrections on early and late-time heat kernel behavior, resolving discrepancies in the running of the Newton constant.

Findings

Non-local terms are determined by early-time heat kernel behavior.

Local terms depend on asymptotic late-time heat kernel behavior.

Explains the discrepancy between RG running and quantum corrections to Newtonian potential.

Abstract

We consider the non-local energy-momentum tensor of quantum scalar and spinor fields in $2 w$-dimensional curved spaces. Working to lowest order in the curvature we show that, while the non-local terms proportional to $\Box {\cal R}$, $\Box \Box{\cal R}$, $\ldots, \Box^{w-2} {\cal R}$ are fully determined by the early-time behaviour of the heat kernel, the terms proportional to ${\cal R}$ depend on the asymptotic late-time behaviour. This fact explains a discrepancy between the running of the Newton constant dictated by the RG equations and the quantum corrections to the Newtonian potential.