Scattering of massive spin-2 field via graviton exchanges with different spin fields and the gravitational potential

TL;DR

This work analyzes the graviton-mediated scattering of a Fierz-Pauli massive spin-2 field off various matter fields within Einstein gravity, treating the FP field as a test quantum field. It derives the non-relativistic two-body gravitational potential from both tree-level (leading order) and loop-level (next-to-leading order) amplitudes, revealing the Newtonian potential and explicit spin/polarisation dependent corrections. At O(G^2), the authors perform a comprehensive diagrammatic calculation (ladder, cross-ladder, triangles, double seagull, vertex corrections, and vacuum polarization/ghosts) and present the resulting potential, including a quantum correction term proportional to ħ: V(r) = - (G M m)/r [1 + (G(171 m - 110 M))/(24 r) + (577 G ħ)/(80 π r^2 c^3)] (ε·ε')^2 + … . The results illustrate how higher-spin fields modify gravitational potentials in a perturbative quantum gravity framework and offer insights for the study of higher-spin theories and potential massive-gravity phenomenology, while keeping gravity itself as the massless GR interaction.

Abstract

In this work, we compute the graviton mediated scattering amplitude of a massive spin-2 Fierz-Pauli field with various other massive spin fields, and in the non-relativistic limit, find out the corresponding two-body gravitational potentials. The massive spin-2 field does not represent gravity here. Instead, the theory of gravity is taken to be the usual massless general relativity, and the massive spin-2 field is taken as a test quantum field coupled to gravity via the standard minimal prescription. We first compute the tree level 2-2 scattering of a massive spin-2 field with massive scalar, spin-1, and spin-1/2 fields with one graviton exchanges. Leading Newton potential, as well as the subleading spin or polarisation dependent terms at ${\cal O}(G)$ have been computed. We also consider the next to the leading order (${\cal O}(G^2)$) scattering of the massive spin-2 field with a massive scalar, and demonstrate the spin independent, spherically symmetric leading part of the two body gravitational potential. The present paper could be considered as an attempt to compute the gravitational potential in the context of a higher spin field theory.

Figures (11)

• Figure 1: The tree level diagram of massive spin-0-spin-0 scattering. The straight lines stand for the scalar field, whereas the wavy line represents the graviton. $q= k_1-k'_1=k_2'-k_2$ is the transfer momentum.
• Figure 2: The tree level scattering of massive spin-2-spin-0 fields. The double lines represent the massive spin-2 field.
• Figure 3: The ladder (left) and the cross-ladder diagram for massive spin-2-spin-0 fields scattering. As earlier, double and single straight lines respectively stand for the massive spin-2 and spin-0 fields, whereas a wavy line represents the graviton. $q= k_1-k'_1=k_2'-k_2$ is the transfer momentum.
• Figure 4: The triangle diagrams for massive spin-2-spin-0 fields scattering.
• Figure 5: The double seagull diagram for massive spin-2-spin-0 fields scattering.