$ξR φ^2$ coupling, cosmological constant and quantum gravitational correction to Newton's potential

TL;DR

This work analyzes how the non-minimal $\xi R \phi^2$ coupling modifies the long-range gravitational potential between massive scalars within perturbative quantum gravity. The authors show that non-minimal vertices introduce graviton momentum in amplitudes, suppressing tree-level long-range effects and pushing leading corrections to ${\cal O}(\xi G^2)$, which are typically subleading to $r^{-3}$ terms at $\Lambda=0$. To potentially recover a stronger $1/r$ tail, they include a cosmological-constant–generated ${\cal O}(\kappa\Lambda)$ three-graviton vertex, computing seagull, vacuum polarization, and fish diagrams to obtain a total non-analytic NR amplitude and the resulting potential. The leading NR potential is found to be $V(r) = - \frac{32 G^2 \Lambda \xi m_1 m_2}{\pi r} \left[ 1 - \frac{m_1^2 + m_2^2}{m_1^2 m_2^2} \frac{1}{r^2} \right]$, highlighting a novel interplay between Planck and cosmological scales, with the $\Lambda$-driven term potentially dominating over $\xi G^2$ corrections for large masses within the subhorizon regime. The results remain subdominant in the current universe due to the tiny value of $\Lambda$, but may have implications for early-universe physics and motivate further study of higher-order corrections and RG effects involving the $\xi R \phi^2$ coupling.

Abstract

This letter investigates the contribution of the $\sqrt{-g}ξRφ^2$ 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^{μν}T_{μν}$, 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 ${\cal O}(ξG^2)$, 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 $\sim Λ\sqrt{-g}/G$ term in the action, where $Λ$ 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 ${\cal O}(ξΛG^2 )$ behaves like $ r^{-1}$, even though it is much subleading compared to Newton's potential due to the appearance of $Λ$. We also discuss the scenario where this potential dominates the aforesaid ${\cal O}(ξG^2)$ one.

Figures (4)

• Figure 1: The tree and the box diagrams for 2-2 scattering with graviton exchanges. Solid and wavy lines respectively denote scalar and graviton propagators. The thick circle denotes a non-minimal vertex of ${\cal O}(\xi \kappa)$. The external momenta are on shell, ${k^2_1}={k'}_1^2=-m_1^2$, ${k^2_2}={k'}_2^2=-m_2^2$. Both diagrams yield no non-analytic contribution in the transfer momentum, $q=k_1-k'_1= k'_2-k_2$, and hence to the long range gravitational potential. The leading contribution behaves as $r^{-4}$ at ${\cal{O}}(\xi G^2)$ from some other diagrams. See the main text for details.
• Figure 2: The seagull diagram for 2-2 scalar scattering at ${\cal O}(\xi \kappa^4 \Lambda)$, with the $\Lambda$-three graviton vertex. There are three more diagrams, found by the interchange, $(k_1, k'_1) \ \leftrightarrow \ (k_2,k'_2)$.
• Figure 3: The vacuum polarisation diagram for the 2-2 scalar scattering at ${\cal O}(\xi \kappa^4 \Lambda)$. One of the cubic graviton vertices corresponds to the $\Lambda$-three graviton vertex, \ref{["nm8'a"]}, whereas the other one is the usual, \ref{['qg25']}. Also, as earlier, the thick circle denotes a non-minimal vertex.
• Figure 4: The fish diagram for the 2-2 scalar scattering at ${\cal O}(\xi \kappa^4 \Lambda)$. The cubic vertex is the $\Lambda$-three graviton vertex.