Dressing the Post-Newtonian two-body problem and Classical Effective Field Theory

TL;DR

The paper develops a Classical Effective Field Theory framework (CLEFT) with Non-Relativistic Gravitation (NRG) fields to reorganize the Post-Newtonian two-body problem via dressed perturbation theory. By defining dressed charges $\rho_{dr}$ and a dressed propagator $G_{dr}$, it recasts the diagrammatic expansion into a finite set of dressed quantities governed by a classical Schwinger-Dyson–like recursion, enabling substantial reduction in computational complexity. The approach reproduces known results up to ${2PN}$ for the two-body action and provides a clear skeleton-based roadmap for ${\rm 3PN}$, while also enabling explicit calculations beyond ${2PN}$ (including ${3PN}$ and ${4PN}$ terms) through dressed inputs. This method offers a practical, gauge-compatible route to higher-order PN terms and clarifies the structure of irreducible diagrams, with potential broad applicability to classical gravitational EFT computations and beyond.

Abstract

We apply a dressed perturbation theory to better organize and economize the computation of high orders of the 2-body effective action of an inspiralling Post-Newtonian gravitating binary. We use the effective field theory approach with the non-relativistic field decomposition (NRG fields). For that purpose we develop quite generally the dressing theory of a non-linear classical field theory coupled to point-like sources. We introduce dressed charges and propagators, but unlike the quantum theory there are no dressed bulk vertices. The dressed quantities are found to obey recursive integral equations which succinctly encode parts of the diagrammatic expansion, and are the classical version of the Schwinger-Dyson equations. Actually, the classical equations are somewhat stronger since they involve only finitely many quantities, unlike the quantum theory. Classical diagrams are shown to factorize exactly when they contain non-linear world-line vertices, and we classify all the possible topologies of irreducible diagrams for low loop numbers. We apply the dressing program to our Post-Newtonian case of interest. The dressed charges consist of the dressed energy-momentum tensor after a non-relativistic decomposition, and we compute all dressed charges (in the harmonic gauge) appearing up to 2PN in the 2-body effective action (and more). We determine the irreducible skeleton diagrams up to 3PN and we employ the dressed charges to compute several terms beyond 2PN.

Figures (29)

• Figure 1: The diagram which represents the Newtonian potential interaction mediated through the Newtonian scalar field $\phi$. The notations will be fully defined later in section \ref{['dressing-PN-section']}.
• Figure 2: A class of diagrams that we interpret as the Newtonian potential interaction between dressed energy distributions of each body through a dressed propagator. The dark blobs represent any sub-diagram with an arbitrary number of vertices on the world-line and a single external leg for the Newtonian potential. There are no bulk loops, as always in classical physics. The light blob represents any sub-diagram with two external legs of the Newtonian potential, which amounts to a propagator with an arbitrary number of retardation insertions.
• Figure 3: The diagrammatic definition of the dressed energy distribution (top) and the dressed propagator (bottom). The dressed energy distribution is defined through the one point function for $\phi$ in the presence of a single source (after stripping external propagators), while the dressed propagator is defined through the full two point function for $\phi$.
• Figure 4: A schematic diagrammatic representation of the Schwinger-Dyson recursive integral equation satisfied by the dressed couplings. More details are given in the body of the paper.
• Figure 5: The Feynman rules involving $\phi$ for the static scalar field theory whose action is given by (\ref{['full-scalar-action']}). The rules describe: the propagator and the quadratic perturbation vertex -- retardation (top), the cubic bulk vertex (middle), and the world-line vertices (bottom) where the ellipsis stand for additional non-linear world-line vertices. Hereafter $k$ denotes a spatial wave-number.