Bethe-Salpeter equation for classical gravitational bound states

TL;DR

This paper derives and solves a classical Bethe-Salpeter equation for gravitational bound states, connecting quantum scattering amplitudes with classical bound state properties and exploring implications like Sommerfeld enhancement.

Contribution

It introduces a classical Bethe-Salpeter framework for gravitational systems, providing analytical solutions and linking bound states to scattering amplitudes and wavefunctions.

Findings

Derived the classical Bethe-Salpeter equation for gravity

Solved the equation analytically at all orders

Explored bound state poles and Sommerfeld enhancement

Abstract

The Bethe-Salpeter equation is a non-perturbative, relativistic and covariant description of two-body bound states. We derive the classical Bethe-Salpeter equation for two massive point particles (with or without spin) in a bound gravitational system. This is a recursion relation which involves two-massive-particle-irreducible diagrams in the space of classical amplitudes, defined by quotienting out by symmetrization over internal graviton exchanges. In this context, we observe that the leading eikonal approximation to two-body scattering arises directly from unitarity techniques with a coherent state of virtual gravitons. More generally, we solve the classical Bethe-Salpeter equation analytically at all orders by exponentiating the classical kernel in impact parameter space. We clarify the connection between this classical kernel and the Hamilton-Jacobi action, making manifest the analytic continuation between classical bound and scattering observables. Using explicit analytic resummations of classical (spinless and spinning) amplitudes in momentum space, we further explore the relation between poles with bound state energies and residues with bound state wavefunctions. Finally, we discuss a relativistic analogue of Sommerfeld enhancement which occurs for bound state cross sections.