Snowmass White Paper: S-matrix Bootstrap

TL;DR

This Snowmass white paper surveys the revived S-matrix bootstrap, detailing how analyticity, crossing, and unitarity constrain scattering amplitudes and delimit the space of consistent QFTs without relying on a fixed Lagrangian. It reviews progress across dimensional settings, including 2D integrable models, higher dimensional primal and dual SDP approaches, massless amplitudes, UV input, and AdS/CFT connections, highlighting how extremal amplitudes recover known theories while saturating unitarity. The authors discuss the roles of positivity versus full unitarity, the incorporation of UV information, and the AdS flat space limit as a bridge to S-matrix data, emphasizing both limitations and opportunities. Looking forward, they propose three avenues—uncovering universal amplitude properties, constructing non perturbative UV completions, and understanding IR EFTs and gravity—to illuminate strongly coupled gauge theories and quantum gravity with non perturbative bootstrap techniques.

Abstract

The S-matrix Bootstrap originated on the idea that the S-matrix might be fully constrained by global symmetries, crossing, unitarity, and analyticity without relying on an underlying dynamical theory that may or may not be a quantum field theory. Recently this approach was revived from a somewhat different point of view. Using the same constraints, one numerically maps out the (infinite-dimensional) space of allowed S-matrices that should contain all consistent quantum field theories (and quantum theories of gravity). Moreover, in the best case scenario one finds special points in the space that can be identified with a certain quantum field theory of interest. In that case, the approach allows the numerical computation of the S-matrix without relying on the particular Lagrangian of the theory. In this white paper we summarize the state of the art and discuss the future of the topic.