$\xi R \phi^2$ coupling, cosmological constant and quantum gravitational correction to Newton's potential

TL;DR

This paper explores how a specific non-minimal scalar-curvature coupling and the cosmological constant influence quantum corrections to Newton's potential, revealing a potential that behaves like 1/r at leading order.

Contribution

It introduces the calculation of quantum gravitational corrections to Newton's potential from non-minimal scalar-curvature coupling and the cosmological constant, highlighting their distinct effects.

Findings

Leading correction from $\xi R ^2$ coupling behaves like $r^{-1}$.

Quantum corrections involving the cosmological constant also produce a $r^{-1}$ potential.

The $\xi R ^2$ correction is subleading compared to classical Newtonian potential.

Abstract

This letter investigates the contribution of the $\sqrt{-g}\xi R\phi^2$ interaction to the long range gravitational potential for massive scalar fields, from the non-relativistic limit of the 2-2 scattering amplitude with graviton exchanges. Such coupling is naturally motivated from the renormalisation of a scalar field theory with quartic self interaction in a curved spacetime. This is qualitatively different from the minimal ones like $ \sqrt{G} h^{\mu\nu}T_{\mu\nu}$, as the vertices corresponding to the former do not explicitly contain any scalar momenta, but instead explicitly contains the momentum carried by graviton line. For the minimal vertex, the long range gravitational potential up to one loop $({\cal O}(G), {\cal O}(G^2))$ was obtained earlier from the terms non-analytic in the transfer momentum, $q^{-2},\ q^{-1},\ \ln q^2 $, yielding potentials respectively like $r^{-1}$, $r^{-2}$, $r^{-3}$. However owing to the aforesaid explicit appearance of transfer momentum for the non-minimal vertices, the leading contribution in this case comes at ${\cal O}(\xi G^2)$, and turns out to be subleading compared to even $r^{-3}$. To complement this `screening' effect, we 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 explicitly contain any graviton momentum. With this vertex, 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 2-2 scattering amplitudes. The leading gravitational potential at ${\cal O}(\xi \Lambda G^2 )$ behaves like $ r^{-1}$, even though it is much subleading compared to Newton's potential due to the appearance of $\Lambda$. We also discuss the scenario where this potential dominates the aforesaid ${\cal O}(\xi G^2)$ one.