Five-loop static two-body potential in massless scalar-tensor gravity

Abstract

We compute the static contribution to the conservative two-body potential in massless scalar-tensor theories through fifth post-Newtonian (5PN) order, with the strong-field scalar responses of the compact objects encoded in scalar-dependent masses. Within post-Newtonian effective field theory (EFT), evaluating the 1PN-5PN static terms requires one- through five-loop massless two-point Feynman integrals. Although the additional scalar modifies both source couplings and integrand numerators, the required scalar master topologies remain those of the corresponding static problem in general relativity. Moreover, independent reflection symmetries of the two static scalar sectors extend the odd-PN factorization mechanism to the present theory, so that no genuinely new five-loop master integrals are required. Our results reproduce the known static interaction through 2PN, provide an independent EFT determination at 3PN, and extend the explicit potential to 4PN and 5PN. We verify the general-relativity limit and body-exchange symmetry; regulator poles and residual transcendental terms cancel in the complete result, leaving rational coefficients. These high-precision results provide ingredients for increasingly precise gravitational-wave tests of gravity beyond general relativity.