$\xi R\phi^2$ non-minimal coupling, and the long range gravitational potential for different spin fields from 2-2 scattering amplitudes

In this paper we investigate the long range gravitational effect of curvature-scalar field non-minimal coupling, in the form of $\xi R \phi^2$, in the perturbative quantum gravity framework. Such coupling is most naturally motivated from the renormalisation of a scalar field theory with a quartic self interaction in a curved spacetime background. This coupling results in two scalar-$n$ graviton vertices which contain no explicit momenta of the scalar, qualitatively different from the usual, e.g. $\kappa h^{\mu\nu}T_{\mu\nu}$-type minimal matter-graviton vertices. Assuming the dimensionless coupling parameter $\xi$ to be small, we compute the 2-2 scattering Feynman amplitudes between such scalars up to ${\cal O}(G^2 \xi)$. From the non-relativistic limit of these amplitudes, we compute the corresponding long range gravitational potential. There exists no tree level contribution $({\cal O}(\xi G))$ here, and hence the one loop ${\cal O}(G^2 \xi)$ result is leading. Recently, the effect of a cosmological constant in such non-minimal interaction and the subsequent gravitational potential was computed. In this work we take the cosmological constant to be vanishing. The resulting potential is found to have $r^{-4}$ leading behaviour. We further extend these results for scalar-massive spin-1 and massive spin-1/2 scattering. Spin and polarisation dependence of the two body potential have been explicitly demonstrated. We discuss some possible physical implications of these results.