  We complete the derivation of the conservative dynamics of binary systems to
fourth Post-Newtonian (4PN) order in the effective field theory (EFT) approach.
We present a self-contained (ambiguity-free) computation of the renormalized
Lagrangian, entirely within the confines of the PN expansion. While we confirm
the final results reported in the literature, we clarify several issues
regarding intermediate infrared (IR) and ultraviolet (UV) divergences, as well
as the renormalization procedure. First, we properly identify the IR and UV
singularities using (only) dimensional regularization and the method of
regions, which are the pillars of the EFT formalism. This requires a careful
study of scaleless integrals in the potential region, as well as conservative
contributions from radiation modes due to tail effects. As expected by
consistency, the UV divergences in the near region (due to the point-particle
limit) can be absorbed into two counter-terms in the worldline effective
theory. The counter-terms can then be removed by field redefinitions, such that
the renormalization scheme-dependence has no physical effect to 4PN order. The
remaining IR poles, which are spurious in nature, are unambiguously removed by
implementing the zero-bin subtraction in the EFT approach. The procedure
transforms the IR singularities into UV counter-parts. As anticipated, the
left-over UV poles explicitly cancel out against UV divergences in conservative
terms from radiation-reaction, uniquely determining the gravitational
potential. Similar artificial IR/UV poles, which are intimately linked to the
split into regions, are manifest at lower orders. Starting at 4PN, both local-
and nonlocal-in-time contributions from the radiation region enter in the
conservative dynamics. Neither additional regulators nor ambiguity-parameters
are introduced at any stage of the computations.
