Effect of $\xi R\phi^2$ non-minimal coupling on gravitational light bending
We investigate the bending of massless fields by a massive object in the presence of a curvature-scalar $\sqrt{-g}\xi R \phi^2$ non-minimal coupling up to one loop, using the perturbative quantum gravity computations. It is well known that without such coupling a self interacting scalar field theory cannot be renormalised in the presence of gravity. The massive object is modelled by a massive scalar $\phi$, and it is assumed to be non-relativistic, e.g., a star. We compute the 2-2 scattering of massless scalar and photons off this object via graviton exchanges. Assuming both $\xi$ and the bending angle to be small, we use the eikonal approximation to compute the angle up to ${\cal O}(\xi G^2)$. At tree level $({\cal O}(\xi G))$ we find no bending, and hence the ${\cal O}(\xi G^2)$ result happens to be leading in this case. The non-minimal vertices are qualitatively different from that of the standard minimal ones, e.g. $ \sqrt{G} h_{\mu\nu} T^{\mu\nu}$, as the former contains explicit momenta of the gravitons instead of the scalar, complementing the second. The bending angle is found to behave like $\sim b^{-7}$, where $b$ is the impact parameter. We have emphasised the qualitative differences of our results from that of the well studied minimal case.