Particle scattering at the Planck scale and the Aichelburg-Sexl geometry

Abstract

At energies of the order or larger than the Planck mass, the curved space-time geometry created by the particles dominates their collision process. The so-called Aichelburg-Sexl (AS) metric is relevant in this problem. We study the exact quantum particle scattering by this geometry in D space-time dimensions. The Klein-Gordon equation is ill defined and a regularization procedure is needed to give a sense to it (this does not happen for string equations). We find the troublesome result that the exact solution depends on the regularization chosen. Continuous regularization yields a scattering phase shift [ ϕ cont] that agrees with that found by 't Hooft in D=4. Discrete (lattice) regularization yields in the continuum limit a phase-shift ϕ = 2arctang[ ϕ cont]. (Actually, the same lattice results hold for the one-dimensional Dirac equation with a Dirac-δ potential and for fermionic two-dimensional field-theoretical models.) In the AS metric, we find that the scattering amplitude [ f lattice], obtained from lattice regularization, exhibits cuts in both s and t variables, and can be expressed as a coherent superposition of relativistic coulombian amplitudes [ f cont obtained from continuous regularization exhibits poles]. For small s, both f cont and f lattice= iGs( πt) −1 (i.e. the one-graviton exchange amplitude) but for small t, f lattice =i[Gsπt ln 2 (− 1 4 t)]−1 (f cont =iGs(πt) −1 for both small s and t). We compute and analyze the partial-wave amplitudes. We also compare both amplitudes in the intermediate t region and discuss their connection with the eikonal (and improved eikonal) approximation.