S-Matrix Bootstrap: Constraining Scattering Amplitudes

• The S-Matrix Bootstrap Program is a framework that nonperturbatively constrains scattering amplitudes using fundamental principles like unitarity, analyticity, and crossing symmetry.
• It employs both primal and dual optimization methods—utilizing basis expansions, semidefinite programming, and neural network techniques—to systematically explore the infinite-dimensional space of allowed S-matrices.
• The program yields rigorous bounds on low-energy observables and effective field theory coefficients, while identifying extremal points that correspond to integrable models and other physical theories.

The S-Matrix Bootstrap Program is a research framework in quantum field theory aimed at nonperturbatively constraining or even determining scattering amplitudes (S-matrices) using only general physical principles—such as unitarity, analyticity, crossing symmetry, and symmetry information—without requiring an explicit underlying Lagrangian description. The program seeks to map out the infinite-dimensional space of all consistent S-matrices, identify distinguished or extremal points corresponding to physical quantum field theories, and provide rigorous bounds on low-energy observables and couplings.

1. Foundational Principles

The essential strategy of the S-matrix bootstrap is to treat the scattering amplitude as a solution to a highly constrained optimization problem, relying on the following central principles:

• Unitarity: The S-matrix must conserve probability in all allowed channels. In two-to-two scattering, this translates to strict upper bounds on the modulus of each partial wave:

$|S_\ell(s)| \leq 1 \quad \text{for %%%%0%%%% on the physical cut.}$

• Analyticity and Causality: The scattering amplitude $A(s, t)$ is analytic in the Mandelstam variables except for physical singularities (cuts for thresholds, poles for bound states). Dispersion relations, such as

$$A(s,t) = \frac{1}{\pi} \int ds' \frac{\operatorname{Im} A(s',t)}{s' - s} + (\text{subtractions}),$$

encode this analytic structure and causality.

• Crossing Symmetry: Amplitudes related by channel exchange are analytic continuations of each other. For scalars,

$A(s, t) = A(t, s)$

and, for more general situations, a web of relationships dictated by group theory and particle content.

• Global Symmetries: Both spacetime (e.g., Lorentz) and internal symmetries must be respected, restricting the allowed flavor/structure of amplitudes.
• Spectrum Assumptions: Sometimes the spectrum of stable/bound states or the absence thereof (as in the O(N) model without bound states) is imposed.

This minimal set of constraints is sufficient to carve out a convex space of all possible S-matrices compatible with general quantum field theory axioms (Kruczenski et al., 2022, Zheng, 2023).

2. Methodologies and Numerical Implementation

Modern implementations adopt two complementary perspectives: the primal bootstrap and the dual bootstrap.

• Primal Approach: The amplitude is written as a sum of basis functions (e.g., polynomials in conformal variables or via dispersion relations). Parameters of this ansatz are optimized while enforcing the full set of physical constraints at discrete points (or using a functional basis). For instance, amplitudes may be expanded as

$$A(s, t, u) = \sum_{a+b+c \leq N} \alpha_{abc} \, \rho_s^a \rho_t^b \rho_u^c$$

with $\rho_x$ conformally mapping the cut complex plane to the unit disk. Constraints are recast as semidefinite programs (SDP) or other optimization problems, and the allowed region in (for example) coupling constant space is determined by extremizing suitable functionals (Kruczenski et al., 2022, Miro et al., 2022).

• Dual Approach: Optimization duality in convex analysis is used to transform the search for physical amplitudes (primal) into the search for exclusion functionals (dual), e.g., Lagrange multipliers corresponding to unitarity and analyticity. The dual variables are “test functions” whose optimal value gives rigorous bounds from the opposite side. When both sides are saturated, one obtains a “duality gap” of zero and rigorous bounding of the true allowed region (Guerrieri et al., 2020, He et al., 2021).
• Neural and Machine Learning Approaches: Recent works incorporate neural networks as flexible ansätze for spectral densities or discontinuities, optimized using differentiable loss functions that encode physical constraints (e.g., unitarity violation as a loss). This provides an efficient way to probe or globally “scan” the space of amplitudes, especially in high dimensions or in presence of nonlinear constraints (Gumus et al., 12 Dec 2024, Dersy et al., 2023).
• Hybrid Data-Driven Bootstrap: Combining experimental/lattice data with bootstrap-constrained analytic ansatz enables one to “fit” data nonperturbatively while imposing the physical constraints above. For example, the pion scattering amplitude can be fit using a multivariate conformal ansatz, and the parameters are optimized against a $\chi^2$-type loss via methods such as Particle Swarm Optimization (Guerrieri et al., 30 Oct 2024).

These diverse numerical and analytical methods allow practitioners to explore and bound the allowed S-matrix space in both low and high dimensions, include spinning states, bound states, and accommodate various tensor structures.

3. Characterization of Allowed Spaces and Extremal Vertices

The S-matrix bootstrap finds that the set of allowed S-matrices forms a convex—potentially infinite dimensional—region. Key features include:

• Convexity and Vertices: The allowed space is convex due to unitarity and linear crossing constraints. Theories with a minimal or no continuous parameter content (e.g., the solitonic O(N) model, sine-Gordon, etc.) typically correspond to isolated vertices (corners) on this space (He et al., 2018). At these vertices, the S-matrix saturates unitarity bounds everywhere, and infinitesimal deformations violate constraints.
• Kinks and Boundaries: Amplitudes lying at the boundary may display kinks, associated with physical phenomena such as integrability, the emergence of new resonant states, or a change in the relevant partial wave (spin) dominance.
• Role of Bound/Resonant States: The boundaries can be characterized by maximizing the coupling to bound or resonant states. Analytical bounds on these couplings, such as the one derived from the analytic structure in (Doroud et al., 2018), define the maximum residue (or derivative) allowed at the location of the resonance zero.
• Special Points and Integrable Theories: In two dimensions, the boundaries frequently coincide with known integrable models (e.g., O(N) sigma model, sine-Gordon, supersymmetric sine-Gordon, Zamolodchikov's Z4 S-matrix), and distinct corners correspond to analytically known S-matrices (He et al., 2018, Bercini et al., 2019).

4. Advanced Features: EFT Bounds, Regge Asymptotics, and Symmetry

The modern bootstrap tightly connects high-energy (UV) behavior and low-energy observables:

• Low-Energy EFT Coefficient Bounds: The program constrains possible values of Wilson coefficients in effective field theory expansions (e.g., the $O(p^4)$ and $O(p^6)$ terms in chiral lagrangians for pion scattering), even in $d\geq2$ (Guerrieri et al., 2020, Miro et al., 2022). These bounds interpolate, and sometimes improve, upon earlier positivity bounds.
• Froissart–Martin and Jin-Martin Bounds: The high-energy (Regge) asymptotics of the amplitude are incorporated via (fractionally) subtracted dispersion relations, which enforce, e.g., the saturation of the Froissart or Jin-Martin bounds (Rham et al., 27 Jun 2025). By varying the subtraction order, the sensitivity of low-energy bounds to the assumed UV growth can be assessed.
• Non-invertible Symmetries and Topological Constraints: For quantum field theories with non-invertible (categorical) symmetries, the crossing symmetry constraints are modified. In such scenarios, the symmetry data (e.g., fusion categories) and topological quantum field theory structures (SymTFT) define new projector bases and Ward identities. This leads to altered bootstrap constraints and new types of extremal S-matrices (Copetti et al., 23 Aug 2024).
• Boundary and Inelastic Effects: The bootstrap has been generalized to systems with boundaries (using R-matrix/reflection matrix formalism), inelastic effects, spinning bound states, and analysis of form factors and correlation functions in deformed integrable field theories (such as $T\bar T$ perturbations) (Kruczenski et al., 2020, Castro-Alvaredo et al., 2023).

5. Applications and Phenomenological Implications

The bootstrap program has generated notable insights and concrete results:

• QCD and Pion Scattering: Analytically constrained, data-augmented S-matrix amplitudes reproduce the full pion scattering spectrum below inelastic thresholds, predict the location and widths of resonances, and tie together chiral perturbation theory with higher-energy physics. For example, analytic fits to data have confirmed not only standard spectra but provided evidence for new resonant states (e.g., candidate tetraquark excitations in the I=2 channel) (Guerrieri et al., 30 Oct 2024).
• Confining Strings and Flux Tubes: Bootstrapped S-matrices for massless Goldstones on confining flux tubes in gauge theories yield strict nonperturbative bounds on higher-derivative Wilson coefficients. These translate into constraints for lattice-measurable observables such as the spectrum and level splittings of confining strings in QCD (Miro et al., 2019).
• Dimension-6 Operators and Beyond: The S-matrix bootstrap provides nonperturbative bounds on Wilson coefficients of higher-dimension operators—including those that evade standard forward-limit positivity constraints—thus enhancing the theoretical underpinnings of EFT analyses relevant for particle phenomenology beyond the Standard Model (Miro et al., 2022).
• Emergent Integrability and Conformal/QFT Dualities: In settings such as 2d $\phi^4$ theory, nonintegrable theories have been shown to nearly saturate bootstrap bounds defined by integrable models (e.g., sinh-Gordon), suggesting emergent or approximate integrability in restricted sectors (Chen et al., 2021).

6. Extensions, Algorithms, and Open Directions

Ongoing research extends the bootstrap methodology and scope:

• Neural Optimizers for High-Dimensional Spaces: Implementation of neural network ansätze efficiently parameterizes spectral densities and amplitudes. Gradient-descent optimizers (Adam, PINNs, PSO) facilitate global or local exploration of allowed S-matrix space even in high dimensionality, overcoming the limitations of fixed-point or Newtonian iterative methods (Gumus et al., 12 Dec 2024, Dersy et al., 2023).
• Data-Driven and Nonconvex Optimization: Integration of experimental and lattice data, together with nonconvex optimization (e.g., PSO), enables precise fits of analytic amplitudes under bootstrap constraints, opening new avenues for the "bootstrapping from data" paradigm (Guerrieri et al., 30 Oct 2024).
• Fractional-Subtraction Dispersion Analysis: By employing dispersion relations with non-integer subtraction orders, the bootstrap now enables continuous interpolation of bounds as the Regge asymptotics are varied, sharpening the connection between UV growth and IR observability (Rham et al., 27 Jun 2025).
• Bootstrap with Deformed Crossing and Topology: Analysis of non-invertible symmetries and category-modified crossing extends the bootstrap to a new class of models defined by categorical/topological data (fusion categories, SymTFT), and identification of isolated integrable points within this setting (Copetti et al., 23 Aug 2024).
• Ambiguity and Phase Reconstruction: Machine learning has been successfully applied to invert the modulus-to-phase integral equations arising from unitarity. This enables reconstruction of the full amplitude from partial data, identifies regions of uniqueness vs. phase-ambiguous solutions, and refines the understanding of uniqueness within the bootstrap space (Dersy et al., 2023).
• Open Challenges: Handling higher-point amplitudes, inelastic thresholds, fully incorporating the double discontinuity (for higher partial waves), mapping nonperturbative gravity S-matrices, and isolating unique "islands" corresponding to fully realistic (e.g., Standard Model) field theories remain outstanding problems.

7. Summary Table of Key Methodological Elements

Physical Principle / Constraint	Implementation in Bootstrap	Comment
Unitarity	$|S_\ell(s)| \leq 1$ for all $s$ and $\ell$	Imposed via LMI or as a loss term in ML
Analyticity	Dispersion relations (additive subtractions)	Via basis expansions, sum rules
Crossing Symmetry	Symmetric ansatz, channel relations	Category-modified in non-invertible symmetry cases
Symmetry (global/categorical)	Tensor projection, isospin, fusion categories	Exploits group theory or category theory
Experimental/Lattice Data	$\chi^2$-like loss or direct input	Hybridizes theory with empirical spectrum
High-Energy Behavior	Regge-bound, Froissart-Martin/Jin-Martin	Controlled via subtraction order in dispersion

This table organizes the constraints and methods consistently referenced in the recent S-matrix bootstrap literature.

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The S-matrix bootstrap program is a robust, nonperturbative framework connecting physical inputs and general principles to the rigorous mathematical structure of quantum field theory amplitudes. Through analytical, numerical, and machine learning methodologies, it has established a landscape in which consistent QFTs reside, identified extremal points corresponding to known and new theories, and provided strong connections between low- and high-energy physics. The ongoing development of algorithms, inclusion of advanced symmetry and topological data, and integration with experimental/lattice information indicate a broad scope for future advances in both formal and phenomenological contexts (Kruczenski et al., 2022, Miro et al., 2022, Gumus et al., 12 Dec 2024, Guerrieri et al., 30 Oct 2024, Copetti et al., 23 Aug 2024, Rham et al., 27 Jun 2025).