Classical energy momentum tensor renormalisation via effective field theory methods

TL;DR

The paper addresses how long-range fields in scalar-tensor gravity renormalize the classical energy-momentum tensor (EMT) of localized sources within the NRGR effective field theory framework. It computes EMT renormalization for point-like bodies at Newtonian and first post-Newtonian order, including a cubic self-interaction of an additional scalar, and extends the analysis to one-dimensional strings to obtain 1PN corrections to the EMT and to the string tension. The authors reconcile historical discrepancies between Dabholkar-Harvey and Buonanno-Damour by showing that certain divergences reflect source-localized contributions that depend on the computational scheme, while the EMT remains conserved. They show that in the supersymmetric (alpha^2=beta^2) case the string tension does not renormalize and that the EFT approach reproduces the Schwarzschild metric at 1PN, underscoring the consistency of this method for classical gravitational self-energy. This work demonstrates the power of EFT techniques to handle classical gravitational self-interactions and clarifies the interplay between field-mediated effects and source-localized contributions.

Abstract

We apply the Effective Field Theory approach to General Relativity, introduced by Goldberger and Rothstein, to study point-like and string-like sources in the context of scalar-tensor theories of gravity. Within this framework we compute the classical energy-momentum tensor renormalization to first Post-Newtonian order or, in the case of extra scalar fields, up to first order in the (non-derivative) trilinear interaction terms: this allows to write down the corrections to the standard (Newtonian) gravitational potential and to the extra-scalar potential. In the case of one-dimensional extended sources we give an alternative derivation of the renormalization of the string tension enabling a re-analysis of the discrepancy between the results obtained by Dabholkar and Harvey in one paper and by Buonanno and Damour in another, already discussed in the latter.

Figures (6)

• Figure 1: Contributions to the scattering amplitude of two massive objects. From left to right the diagrams represent respectively the leading Newtonian approximation, a classical contribution to the 1PN order and a negligible quantum 1-loop diagram.
• Figure 2: Feynman diagrams describing the gravitational contributions to the effective energy-momentum tensor of a particle at Newtonian level according to the parametrization (\ref{['met_nr']}) used for the metric.
• Figure 3: Feynman diagram describing the gravitational contribution to the effective energy-momentum tensor of a particle at first post-Newtonian order according to the parametrization used for the metric (\ref{['met_nr']}).
• Figure 4: Feynman diagram representing the self-interaction contribution of the massive scalar field $\psi$ to the energy-momentum tensor of a particle at next-to-leading order.
• Figure 5: Diagrams reproducing the coupling to $\phi$ (curly line) and to $\sigma_{ij}$ (long-dashed), or the effective energy-momentum tensor, of a string at next to lowest order in interaction. The diagram on the left vanishes (see discussion in the text).