Exact static two-body gravitational potential at first self-force order

The static interaction potential of two non-spinning point masses in general relativity was previously computed through seventh post-Newtonian order using methods of effective field theory. Its probe limit, or zeroth-self-force contribution, is determined exactly by the Schwarzschild geometry. We derive a closed-form expression for the static first-self-force correction, quadratic in the light mass, and establish its weak-field expansion to all orders. Working in harmonic gauge, we construct the required Green function for the coupled linearised gravitational equations about the Schwarzschild background. We fix its homogeneous contributions and regularised coincident limit by matching to the dimensionally continued point-source theory. The result reproduces known perturbative coefficients and determines all higher static post-Newtonian coefficients in this mass sector.