An Analytical Toolkit for the S-matrix Bootstrap

TL;DR

This paper develops analytical tools for constraining the nonperturbative S-matrix in relativistic, gapped theories using analyticity, unitarity, and expansions, enhancing numerical bootstrap methods.

Contribution

It introduces two natural amplitude expansions and relates them via the Froissart-Gribov formula, providing new constraints on scattering amplitudes at finite energy and spin.

Findings

Derived threshold and large spin expansions of the amplitude.

Connected expansions through Froissart-Gribov inversion formula.

Proposed improvements to numerical S-matrix bootstrap methods.

Abstract

We revisit analytical methods for constraining the nonperturbative $S$-matrix of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering at finite energy and spin in terms of the low-energy parameters measured in the experiment. Finally, we discuss the modern numerical approach to the $S$-matrix bootstrap and how it can be improved based on the results of our analysis.