Universality of ultra-relativistic gravitational scattering

Paolo Di Vecchia,Carlo Heissenberg,Rodolfo Russo,Gabriele Veneziano

doi:10.1016/j.physletb.2020.135924

Abstract

We discuss the ultra-relativistic gravitational scattering of two massive particles at two-loop (3PM) level. We find that in this limit the real part of the eikonal, determining the deflection angle, is universal for gravitational theories in the two derivative approximation. This means that, regardless of the number of supersymmetries or the nature of the probes, the result connects smoothly with the massless case discussed since the late eighties by Amati, Ciafaloni and Veneziano. We analyse the problem both by using the analyticity and crossing properties of the scattering amplitudes and, in the case of the maximally supersymmetric theory, by explicit evaluation of the 4-point 2-loop amplitude using the results for the integrals in the full soft region. The first approach shows that the observable we are interested in is determined by the inelastic tree-level amplitude describing the emission of a graviton in the high-energy double-Regge limit, which is the origin of the universality property mentioned above. The second approach strongly suggests that the inclusion of the whole soft region is a necessary (and possibly sufficient) ingredient for recovering ultra relativistic finiteness and universality at the 3PM level. We conjecture that this universality persists at all orders in the PM expansion.

Highlights

At high energy and large impact parameter, gravitational scattering is dominated by the exchange of the highest-spin massless states in the theory [1, 2, 3, 4, 5]
Notice that this implies that the impact parameter b, which is related by Fourier transform to the the gravitational length stcoatlael1mRoDm−e3n∼tumGNex√csh,adnegfiendeidn the scattering, is much larger than in analogy with the Schwarzschild radius
Since the process is dominated by the exchange of gravitons, it is natural to expect a universal result for the high energy gravitational scattering

Summary

Introduction

At high energy and large impact parameter, gravitational scattering is dominated by the exchange of the highest-spin massless states in the theory [1, 2, 3, 4, 5]. We use the eikonal approach to extract the classical contribution, instead of the effective field theory comparison of [10], and we work with the full result of the integrals in the “soft region” (i.e. in the limit of small momentum transfer) Both the approach based on analyticity/crossing and the one using the explicit loop amplitudes yield the same Re(δ2) which agrees with the massless result of [6, 8, 9], suggesting that the origin of the different ultra-relativistic behaviour of [12, 14, 20, 22, 21] lies in the use of the “potential” approximation in evaluating the loop integrals. Note that both Im(δ2) and δ0(∇δ0) are IR divergent, but these divergences cancel so that physical observables derived from Re(δ2), such as the deflection angle, are finite

High energy limit of the 3-particle cut

Direct calculation of Re δ2

B Summary of the soft integrals