Factorization and polarization in linearized gravity

TL;DR

This work establishes that in linearized gravity, all four-body graviton amplitudes for processes involving a graviton, a matter field X with spin < 2, and an external photon or graviton, factorize into a universal kinematic factor and a photon-like amplitude, driven by gravitational gauge invariance and Lorentz invariance. By deriving the weak-field Lagrangian and the relevant Feynman rules, the authors show that every amplitude for gX\to γX, gX\to gX, and gg\to gg can be written in a form M = (κ/2e) F times a product of lower-level photon-like amplitudes, with a common F = ((p1·k1)(p1·k2))/(k1·k2). The polarization analysis using covariant density matrices reveals that, in the massless limit, graviton polarization is mapped to photon polarization in gX\to γX and gX\to gX, while gg\to gg need not preserve helicity; these results connect gravitational interactions to familiar QED-like polarization structures. The factorization insight offers a powerful simplification for studying quantum gravity processes at low energies and suggests avenues for formal proofs, extensions to different masses, and potential links to string-theoretic approaches to gravity.

Abstract

We investigate all the four-body graviton interaction processes: $gX\rightarrow γX$, $gX\rightarrow gX$, and $gg\rightarrow gg$ with $X$ as an elementary particle of spin less than two in the context of linearized gravity except the spin-3/2 case. We show explicitly that gravitational gauge invariance and Lorentz invariance cause every four-body graviton scattering amplitude to be factorized. We explore the implications of this factorization property by investigating polarization effects through the covariant density matrix formalism in each four-body graviton scattering process.