A Lorentzian FRG Investigation of the Quasi-Static Weak-Field Infrared Limit of Gravity

Abstract

A common assumption in the Effective Field Theories of gravity is that their quasi-static weak-field infrared limit yields the well-known second-order Poisson operator. We examine this limit for the universality class of parity-even, symmetric, analytic gravitational theories admitting a local derivative expansion using Lorentzian FRG methods. We find that, in the curvature-squared truncation, the scalar-trace sector self-closes at $\mathcal{O}(q^4),$ allowing the projected flow to be obtained by analytic continuation of the corresponding Euclidean result. This yields a screened d'Alambertian $D_\ell \equiv (1+\ell^2 \Box)\Box$ characterised by an emergent correlation length $\ell$. We show the operator is consistent with the ADM constraint structure and thus it does not introduce propagating scalar ghosts in the scalar-trace sector. We further derive its retarded response kernel and show its static-limit Green's function in response to a point source, which reduces to Newtonian gravity for $\ell \rightarrow 0$.