[Paper Review] Snowmass White Paper: S-matrix Bootstrap

Martı́n Kruczenski, João Penedones+1 more|arXiv (Cornell University)|Mar 4, 2022

Matrix Theory and Algorithms11 citations

TL;DR

This Snowmass White Paper reviews the modern S-matrix bootstrap program, which uses analyticity, crossing symmetry, unitarity, and global symmetries to numerically map the space of consistent scattering amplitudes without relying on a Lagrangian. It identifies key theories—such as integrable 2D QFTs and potentially QCD or maximal supergravity—as special points in this space, enabling non-perturbative computation of S-matrices and uncovering universal bounds and extremal amplitudes.

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.

Motivation & Objective

• To develop a non-perturbative framework for constraining scattering amplitudes using only general principles like crossing, unitarity, and analyticity.
• To identify specific quantum field theories—such as integrable 2D models or QCD—as distinguished points in the space of allowed S-matrices.
• To explore the landscape of consistent amplitudes in higher dimensions, including strongly coupled gauge theories and quantum gravity.
• To extend the bootstrap to massless particles, spinning particles, and higher-point amplitudes, particularly in 4D and higher.
• To bridge axiomatic analyticity results with physical singularity structures like Landau thresholds, improving our understanding of scattering amplitude singularities.

Proposed method

• Numerical implementation of crossing symmetry and unitarity constraints to map the space of allowed S-matrices in a non-Lagrangian way.
• Use of basis expansions for scattering amplitudes compatible with analyticity and crossing, followed by non-linear unitarity enforcement.
• Application of semidefinite programming and optimization techniques to enforce positivity and consistency conditions on the S-matrix space.
• Adaptation of the bootstrap to include higher-point amplitudes, spinning particles, and Wilson coefficients in quantum gravity.
• Incorporation of physical principles such as soft theorems, chiral symmetry, and anyonic statistics to constrain low-energy effective theories.
• Exploration of celestial amplitudes and IR-finite observables to define physically meaningful, finite quantities in 4D scattering.

Experimental results

Research questions

• RQ1 Can the S-matrix bootstrap identify QCD as a distinguished point in the space of consistent low-energy effective field theories?
• RQ2 What are the universal bounds on scattering amplitudes in 4D, particularly for pion and nucleon scattering?
• RQ3 How can the bootstrap be extended to include massless particles and spinning particles in more than two spacetime dimensions?
• RQ4 To what extent can the analyticity domain of scattering amplitudes be extended beyond current axiomatic results, especially in relation to Landau singularities?
• RQ5 Can the bootstrap framework be used to constrain higher curvature corrections in maximal supergravity across different spacetime dimensions?

Key findings

• The S-matrix bootstrap successfully identifies integrable 2D quantum field theories as special points in the space of allowed amplitudes, confirming the method’s viability in low dimensions.
• Numerical algorithms have already uncovered universal bounds and extremal amplitudes in various settings, demonstrating the method’s power in non-perturbative analysis.
• In 2D, the bootstrap reproduces known results such as those from the LSZ formalism and the Froissart bound, validating its consistency with established physics.
• For QCD-like theories, the bootstrap has begun to constrain pion scattering amplitudes under exact isospin symmetry, suggesting a path toward identifying QCD in the landscape.
• The method reveals that higher-point amplitudes and soft theorems can be systematically incorporated, enabling constraints on chiral Lagrangians.
• Preliminary results suggest that the bootstrap can probe curvature corrections in maximal supergravity, with predictions in D=11 matching M-theory expectations.