Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach

We calculate the motion of binary mass systems in gravity up to the fourth post--Newtonian order. We use momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates by applying dimensional regularization. We construct the canonical transformations to ADM coordinates and to effective one body theory (EOB) to compare with other approaches. We show that intermediate poles in the dimensional regularization parameter $\varepsilon$ vanish in the observables and the classical theory is not renormalized. The results are illustrated for a series of observables for which we agree with the literature.