Bootstrapping High-Energy Observables

TL;DR

This paper develops a numerical S-matrix bootstrap using crossing symmetric dispersion relations, deriving new bounds on low-energy data and analyzing high-energy behavior of scattering amplitudes, with improved convergence over existing methods.

Contribution

It introduces a novel crossing symmetric representation of the Virasoro-Shapiro amplitude and enhances the numerical bootstrap approach for high-energy observables.

Findings

Multiple sign changes in the rho-parameter before high-energy asymptote.

New bounds on low-energy scattering data.

Improved convergence of numerical bootstrap methods.

Abstract

In this paper, we set up the numerical S-matrix bootstrap by using the crossing symmetric dispersion relation (CSDR) to write down Roy equations for the partial waves. As a motivation behind examining the local version of the CSDR, we derive a new, crossing symmetric, 3-channels-plus-contact-terms representation of the Virasoro-Shapiro amplitude in string theory that converges everywhere except at the poles. We then focus on gapped theories and give novel analytic and semi-analytic derivations of several bounds on low-energy data. We examine the high-energy behaviour of the experimentally measurable rho-parameter, introduced by Khuri and Kinoshita and defined as the ratio of the real to the imaginary part of the amplitude in the forward limit. Contrary to expectations, we find numerical evidence that there could be multiple changes in the sign of this ratio before it asymptotes at high energies. We compare our approach with other existing numerical methods and find agreement, with improvement in convergence.