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The Fokker action of point-particle binaries at the fourth post-Newtonian(4PN) approximation of general relativity has been determined previously.However two ambiguity parameters associated with infra-red (IR) divergencies ofspatial integrals had to be introduced. These two parameters were fixed bycomparison with gravitational self-force (GSF) calculations of the conservedenergy and periastron advance for circular orbits in the test-mass limit. Inthe present paper together with a companion paper, we determine both theseambiguities from first principle, by means of dimensional regularization. Ourcomputation is thus entirely defined within the dimensional regularizationscheme, for treating at once the IR and ultra-violet (UV) divergencies. Inparticular, we obtain crucial contributions coming from the Einstein-Hilbertpart of the action and from the non-local tail term in arbitrary dimensions,which resolve the ambiguities.

Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order