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

TL;DR

The paper challenges the common assumption that the quasi-static, weak-field infrared limit of gravity reduces to a second-order Poisson operator by deriving a Lorentzian FRG endpoint within a broad local gravity universality class. Through a curvature-squared truncation and projection onto the scalar trace, it demonstrates IR closure at $O(q^4)$ and obtains a screened d'Alembertian operator $D_\ell=(1+\ell^2 \Box)\Box$, with an emergent length $\ell$ set by the IR fixed point. The Lorentzian endpoint is obtained via analytic continuation from the Euclidean flow, yielding a retarded kernel whose static limit produces an exponentially screened Newtonian potential $G(r)=(1-e^{-r/\ell})/(4\pi r)$, reducing to Newtonian gravity as $\ell\to 0$. Importantly, the analysis shows this screening does not require new propagating degrees of freedom or Ostrogradsky ghosts, being consistent with ADM constraints. The findings suggest a universal IR modification within the studied class and motivate extensions to cosmological backgrounds and matter/vacuum effects to assess broader implications for gravity EFTs.

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 $\ell \rightarrow 0$.