We apply a dressed perturbation theory to better organize and economize thecomputation of high orders of the 2-body effective action of an inspirallingPost-Newtonian gravitating binary. We use the effective field theory approachwith the non-relativistic field decomposition (NRG fields). For that purpose wedevelop quite generally the dressing theory of a non-linear classical fieldtheory coupled to point-like sources. We introduce dressed charges andpropagators, but unlike the quantum theory there are no dressed bulk vertices.The dressed quantities are found to obey recursive integral equations whichsuccinctly encode parts of the diagrammatic expansion, and are the classicalversion of the Schwinger-Dyson equations. Actually, the classical equations aresomewhat stronger since they involve only finitely many quantities, unlike thequantum theory. Classical diagrams are shown to factorize exactly when theycontain non-linear world-line vertices, and we classify all the possibletopologies of irreducible diagrams for low loop numbers. We apply the dressingprogram to our Post-Newtonian case of interest. The dressed charges consist ofthe dressed energy-momentum tensor after a non-relativistic decomposition, andwe compute all dressed charges (in the harmonic gauge) appearing up to 2PN inthe 2-body effective action (and more). We determine the irreducible skeletondiagrams up to 3PN and we employ the dressed charges to compute several termsbeyond 2PN.
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