Distinguishing Quantum Matter by Gravity with Differential Scattering Cross Section at Tree Level

TL;DR

The paper investigates whether the weak equivalence principle (WEP) holds for quantum matter by defining a quantum WEP observable based on the differential scattering cross section $\frac{d\sigma}{d\Omega}$ in perturbative quantum gravity at tree level. By computing cross sections for scalar–scalar, scalar–Dirac, and Dirac–Dirac scattering with a heavy, effectively infinite-mass target, it shows that the leading nonrelativistic Rutherford term is universal, but spin-dependent differences arise at next-to-leading order, with relativistic small-angle behavior yielding distinct $\mathcal{O}(1/\theta^2)$ or $\mathcal{O}(1/\theta^4)$ corrections depending on the target. The degree of polarization of scattered Dirac particles is analyzed, revealing that polarization observables depend on the initial polarization direction and geometry, thereby serving as a pointer to nonlocal quantum effects and WEP violation. Overall, the work demonstrates a regime in which quantum WEP can be tested and distinguished by gravity-driven differential scattering, offering a concrete observable to probe spin-dependent gravitational interactions in quantum matter.

Abstract

The definition of weak equivalence principle of quantum matter is an open problem at present. In order to reflect the probability of quantum system in the quantum version of weak equivalence principle, we proposed a quantum weak equivalence principle based on differential scattering cross section at tree level, that is, the differential scattering cross section does not depend on the mass and properties of the scattered particles when the target particles take the large mass limit. This version of the quantum equivalence principle we proposed will be broken by the spin properties of quantum matter. In the non-relativistic case, the difference of differential scattering cross sections of scattered particles with different spin properties scattered by target particles is mainly reflected in the order of $ \mathcal O (p _ {\mathrm{cm}} ^2) $. In the relativistic case , we studied the asymptotic behavior of differential scattering cross sections at small angles. When the target particles are scalar particles, the difference of light particles with different spin properties is mainly reflected in the $ \mathcal O (1/θ^2) $ order. When the target particles are Dirac particles, the difference of light particles with different spin properties is mainly reflected in the $ \mathcal O (1/θ^4) $ order. The polarization of differential scattering cross section when scattered particles are Dirac particles is investigated. The result of the degree of polarization depends on the polarization direction of the incident particles.