Analytic determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation

We complete the analytical determination, at the 4th post-Newtonian approximation, of the main radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies. The (non logarithmic) coefficient $a_5(\nu)$ measuring this 4th post-Newtonian interaction potential is found to be linear in the symmetric mass ratio $\nu$. Its $\nu$-independent part $a_5(0)$ is obtained by an analytical gravitational self-force calculation that unambiguously resolves the formal infrared divergencies which currently impede its direct post-Newtonian calculation. Its $\nu$-linear part $a_5(\nu)-a_5(0)$ is deduced from recent results of Jaranowski and Schaefer, and is found to be significantly negative.