Can a $ξR φ^2$ non-minimal coupling contribute to the non-relativistic gravitational potential?

Abstract: We investigate the contribution of the $\sqrt{-g}\xi R\phi^2$ non-minimal interaction for a massive scalar to the gravitational potential, found via the non-relativistic 2-2 scattering with one and two graviton exchanges up to one loop order. Such coupling can be most naturally motivated from the renormalisation of a scalar field theory with quartic self interaction in a curved spacetime. This coupling is qualitatively different from the minimal ones, like $\sim \sqrt{G} h^{\mu\nu}T_{\mu\nu}$, as the vertices corresponding to the former does not explicitly contain any scalar momenta, but instead explicitly contains the momentum of the graviton. We show that the insertion of one/two graviton-two scalar non-minimal vertices in the standard 2-2 scattering Feynman diagrams does not yield any contribution for the potential up to one loop, owing to the aforesaid explicit appearance of graviton's transfer momentum in the numerator of the Feynman amplitude. In order to complement this effect, the most natural choice seems to consider the three graviton vertex generated by the $\sim \Lambda \sqrt{-g}/G$ term in the action, where $\Lambda$ is the cosmological constant. This vertex does not contain explicitly any graviton momentum. With this, and assuming short scale scattering much small compared to the Hubble horizon, we compute the seagull, the vacuum polarisation and the fish diagrams and obtain the non-relativistic scattering amplitudes. They indeed yield a two body gravitational potential at order $\xi \Lambda G^2$. We further point out some qualitative differences of this potential with that of the standard result of $\xi =0$.