Bethe-Salpeter equation for classical gravitational bound states

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.

Kinematic Hopf algebra for amplitudes from higher-derivative operators

“…An efficient method to compute the corresponding amplitudes based on Berends-Giele recursion [54] was proposed in [55]. We will be interested in scattering amplitudes of gluons coupled to two heavy scalars, which have been shown to play very important roles in the study of gravitational wave emission and black hole physics [32][33][34][35][36][37][38][39][40][41]. One way to obtain these gluon-scalar amplitudes is by performing a dimensional reduction of two gluons to two massive scalars at the level of Lagrangian, starting from (2.1).…”

“…Recently a systematic framework for the kinematic algebra was proposed in [24][25][26], where the generator is taken as the QCD current, or its heavy-mass limit. Especially in the heavy-mass effective field theory [26][27][28][29][30][31][32], which had a wide phenomenological applications in heavy quark and black hole physics [32][33][34][35][36][37][38][39][40][41], the kinematic algebra was found to be isomorphic to an infinite dimensional combinatorial algebra [42,43], i.e. a (generalised) quasi-shuffle Hopf algebra [44][45][46][47][48].…”

Self-dual black holes in celestial holography

“…Over the last few years there has been uncanny success within the perturbative amplitudes program in describing the scattering or mergers of black holes . In some circumstances exact results are obtainable, and there is even a construction of a pure 1 4D quantum state that reproduces some scattering amplitudes off of a single black hole [50,[52][53][54][55]. The exactness of some results in part stems from the existence of the Kerr-Schild form of the metric in which the linearized metric perturbation is exact [56,57].…”