Analytic determination of the eight-and-a-half post-Newtonian self-force contributions to the two-body gravitational interaction potential

We analytically compute, to the eight-and-a-half post-Newtonian order, and to linear order in the mass ratio, the radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies, thereby extending previous analytic results. These results are obtained by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. We emphasize the increase in "transcendentality" of the numbers entering the post-Newtonian expansion coefficients as the order increases, in particular we note the appearance of zeta(3) (as well as the square of Euler's constant gamma) starting at the seventh post-Newtonian order. We study the convergence of the post-Newtonian expansion as the expansion parameter u = GM/(c(2)r) leaves the weak-field domain u << 1 to enter the strong field domain u = O(1).