Bound States in Perturbative Quantum Gravity with Hydrogen-like Degeneracy

Abstract

Recent results in the study of gravitational scattering amplitudes indicate that some highly-symmetric relativistic systems may exactly conserve a version of the Laplace-Runge-Lenz (LRL) vector in two-body bound states. We make a systematic study, in the context of a generic EFT of long-range interactions due to the exchange of massless mediator particles of spins 0, $\frac{1}{2}$, 1, $\frac{3}{2}$, 2, of the conditions for the conservation of a hidden LRL vector; both classically (no orbital precession) and quantum mechanically (hydrogen-like degeneracy of bound states). The calculations require several new technical developments including the extension of relativistic post-Minkowskian potential matching to quantum corrections and the incorporation of long-range forces due to the exchange of pairs of massless fermions. We find that while classically the absence of precession is a generic property of a large class of models, including Kaluza-Klein theories, with vector and scalar exchanges, maintaining the degeneracy quantum mechanically requires a surprising cancellation between gravitons and 6 Majorana gravitinos, hinting at a special role for $\mathcal{N}=6$ supergravity.