Conservative Dynamics of Binary Systems of Compact Objects at the Fourth Post-Newtonian Order

We review our recent derivation of a Fokker action describing the conservative dynamics of a compact binary system at the fourth post-Newtonian (4PN) approximation of general relativity. The two bodies are modeled by point particles, which induces ultraviolet (UV) divergences that are cured by means of dimensional regularization combined with a renormalization of the particle's wordlines. Associated with the propagation of wave tails at infinity is the appearance of a non-local-in-time conservative tail effect at the 4PN order in the Lagrangian. In turn this implies the appearance of infrared (IR) divergent integrals which are also regularized by means of dimensional regularization. We compute the Noetherian conserved energy and periastron advance for circular orbits at 4PN order, paying special attention to the treatment of the non-local terms. One ambiguity parameter remaining in the current formalism is determined by comparing those quantities, expressed as functions of the orbital frequency, with self-force results valid in the small mass ratio limit.