Non-Perturbative S-Matrix Bootstrap

• Non-Perturbative S-Matrix Bootstrap is an approach that leverages analyticity, unitarity, and crossing symmetry to impose theory-independent constraints on scattering amplitudes without resorting to perturbation theory.
• It utilizes frameworks like conformal map parametrizations, partial wave decomposition, and semidefinite programming to rigorously bound EFT Wilson coefficients and capture resonance effects.
• These methods provide actionable insights into UV completeness, integrable RG flows, and quantitative constraints on quantum field theory spectra across various dimensions.

Non-perturbative S-matrix bootstrap techniques constitute a modern approach for extracting rigorous, theory-independent constraints on scattering amplitudes by applying the axioms of analyticity, unitarity, and crossing symmetry directly at the level of the S-matrix, without recourse to perturbation theory or a Lagrangian description. Over the past decade, these methods have been implemented across disparate domains, including relativistic effective field theories, integrable models, higher-dimensional QFTs, flux-tube worldsheet dynamics, and planar gauge theories, yielding quantitative bounds on Wilson coefficients, coupling constants, spectra of resonances, and phase shifts, as well as foundational insights into UV completeness and the landscape of quantum field theories.

1. Core Principles: Analyticity, Unitarity, and Crossing

The non-perturbative S-matrix bootstrap is grounded in three universal axioms:

• Analyticity: For a given theory (massive or massless, in $d=2$ or $d>2$), the $2 \to 2$ amplitude $A(s,t,u)$ is assumed to be analytic in the Mandelstam variables except at kinematically determined cuts and isolated (physical) poles. Mandelstam analyticity is imposed, for example, in the non-perturbative bootstrap for massless pions, demanding that $A(s,t,u)$ has only two-particle branch cuts along $s,t,u \geq 0$ and no spurious singularities (Guerrieri et al., 2020).
• Unitarity: Imposed in partial waves, it requires $|S_\ell^{(I)}(s)| \leq 1$ for all physical $s>0$ and each spin $\ell$ and isospin $I$ (or analogous discrete channel decomposition), corresponding to the conservation of probability in each partial wave (Guerrieri et al., 2020, Miro et al., 2019). In the elastic domain, saturated solutions are unitary: $|S_\ell^{(I)}(s)|=1$.
• Crossing Symmetry: Enforced either via explicit symmetrization of the amplitude ansatz (with respect to Mandelstam variable interchange) or by dual/categorical kernels in more general symmetry settings (Guerrieri et al., 2020, Copetti et al., 2024). Crossing relates amplitudes in the $s$, $t$, and $u$ channels and leads to non-trivial, often linear constraints on the set of allowed amplitudes or partial waves.

These general axioms reduce the immense space of possible amplitudes to a convex region or island whose boundaries correspond to extremal QFT data (such as maximal couplings or minimal phase shifts).

2. Mathematical Frameworks and Numerical Implementation

a. Functional and Conformal Map Parametrizations

Amplitudes are expanded using variables tailored to the analytic structure:

• Conformal Maps ($\rho$-variables): The cut $s$-plane is mapped to the unit disk via $$\rho(s) = \frac{1 - \sqrt{-s/s_*}}{1 + \sqrt{-s/s_*}}$$, where $s_*$ is an unphysical reference point. The amplitude is written as a finite sum over symmetrized monomials in $\rho(s), \rho(t), \rho(u)$, with polynomial degree $N_{\max}$ controlling the truncation (Guerrieri et al., 2020).
• Partial Wave Decomposition: For $d>2$, amplitudes are projected into isospin and angular momentum channels using $$A_{ab}^{\;cd}(s,t,u) = A(s|t,u)\delta_{ab}\delta^{cd} + \dots$$, and

$$3A(s|t,u)+A(t|s,u)+A(u|s,t) = 16\pi i \sum_{\ell=0}^\infty (2\ell+1) P_\ell(\cos\theta)[1 - S_\ell^{(0)}(s)]$$

for the $I=0$ channel, with analogous relations for $I=1,2$ (Guerrieri et al., 2020).

b. Semidefinite Programming and Unitarity Constraints

Elastic unitarity becomes a quadratic inequality on $S_\ell^{(I)}(s)$:

$|S_\ell^{(I)}(s)|^2 \leq 1$

for all relevant $s$ and channels. In practice, this is imposed on a grid of discrete $s$ points and spins up to $L_{\max}$, resulting in a large-scale semidefinite program (SDP). Additional constraints, such as analyticity (sum rules, asymptotic behaviors) and crossing relations, are incorporated either as further linear constraints or via explicit projection in the ansatz (Guerrieri et al., 2020, Miro et al., 2019, Copetti et al., 2024).

c. Convex and Dual Formulations

The bootstrap region is convex, allowing dual formulations. Here, functional variables (Lagrange multipliers) corresponding to analyticity and unitarity are introduced, yielding improved analytical control and error bounds at finite truncation (Guerrieri et al., 2020, Guerrieri et al., 2020). In high-dimensional SDPs, this duality underpins convergence diagnostics and efficient boundary determination, often enabling explicit minimization/maximization of Wilson coefficients or couplings.

3. Applications and Physical Results

a. EFT Wilson Coefficient Bounds

For massless pions in $d=4$, the $O(3)$-adjoint amplitude's low-energy expansion is

$$A(s|t,u) = \frac{s}{f_\pi^2} + \frac{1}{f_\pi^4}\left[\alpha s^2 + \beta\,(t^2 + u^2)\right] + \dots$$

Bootstrap constraints carve out a finite, "kite-shaped" allowed region in the $(\alpha, \beta)$-plane, rigorously containing QCD-like values but sharply excluding values outside the boundary. Analytically, for large positive $\alpha$:

$\beta \gtrsim -\frac{1}{48\pi^2}\log \alpha + O(1)$

and for large negative $\alpha$,

$$\alpha + 2\beta \gtrsim -\frac{1}{16\pi^2} \log |\alpha| + O(1)$$

The regional shape arises from the interplay between IR chiral symmetry breaking, crossing, and high-energy unitarity (Guerrieri et al., 2020).

b. RG Flows and Integrable Structures

In two dimensions, for massless Goldstone scattering, the non-perturbative S-matrix bootstrap extends to the flux-tube worldsheet (Miro et al., 2019) and factorized S-matrix models (Ahn, 25 Sep 2025), classifying the landscape of allowed RG flows and Y-systems. Only minimal, diagonal, and saturated S-matrices generate UV-complete flows, each with an exactly solvable TBA/Y-system and associated ADE symmetry.

c. Planar Gauge Theory Amplitudes

For planar $\mathcal N=4$ SYM, the pentagon OPE and associated S-matrix bootstrap unifies the construction of polygonal Wilson loops and planar scattering amplitudes. Bootstrap axioms (reflection, unitarity, generalized crossing) for pentagon transitions $P_{ab}(u|v)$, with the all-coupling worldsheet S-matrix $S_{ab}(u,v)$ as input, fully determine the transitions and thus the amplitude at any coupling. At strong coupling, the resulting Y-system precisely matches the AdS minimal area TBA (Basso et al., 2013).

d. Gauge Theories and Pion Amplitudes

The modern bootstrap has been extended to $SU(N_c)$ gauge theories with chiral symmetry breaking (He et al., 2023); here, the S-matrix of low-energy pions is reconstructed from analyticity, crossing, partial-wave unitarity, chiral low-energy theorems, finite-energy (SVZ) sum rules, and form-factor positivity. With only $N_c$, $N_f$, $m_q$, $\Lambda_\mathrm{QCD}$ and condensates as input, phase shifts and resonance effects (notably the $\rho$) are accurately reproduced without perturbation theory.

4. Analytic Methods and Extensions

a. Dispersion Relations and Sum Rules

Once-subtracted (or multi-subtracted) dispersion relations, rooted in analyticity and causality, link IR Wilson coefficients to integrals over the imaginary part (spectral density) of amplitudes:

$$2\delta(s) - \frac{s}{4} = \frac{2 s^3}{\pi} \int_{-\infty}^{+\infty} \frac{\Im \delta(s')}{s'^3(s'^2 - s^2)} ds'$$

Allowing for the direct translation of S-matrix bounds into operator coefficient constraints and establishing positivity of specific combinations of coefficients in the low-energy expansion (Miro et al., 2019, Correia et al., 2020).

b. Threshold Expansions and Froissart-Gribov Inversion

In $d \geq 3$, the amplitude admits threshold expansions (effective range expansion) and large-spin Froissart-Gribov inversion, which are tightly related and provide a bridge between low and high-energy regimes as well as analytic handles on the partial-wave decomposition (Correia et al., 2020).

5. Spin, Symmetry, and Advanced Implementations

a. Spinning Amplitudes and Exotic Symmetry

Generalizations to amplitudes involving spin-1/2 fermions (Hebbar et al., 2020), non-invertible fusion-category symmetries (Copetti et al., 2024), or multiple coupled species (multi-S-matrix bootstrap (Homrich et al., 2019)) employ refined representation theory and group-theoretic projectors. Modified crossing, symmetry-topological field theory (SymTFT), and categorical kernels become central, especially for kinks and vacua in non-regular module representations.

b. Inelasticity and Physical Realism

Recent advances enforce explicitly inelastic unitarity (i.e., non-saturated $|S_\ell(s)|^2 < 1$ above specific inelastic thresholds), yielding tighter, more physical bounds and enabling application to realistic hadronic amplitudes (Antunes et al., 2023). Both primal (extended ansatz) and dual functionals match and converge to the same physical exclusion regions; the same logic underpins fits to physical data for processes such as $\pi\pi$ scattering, producing amplitudes consistent with high-precision experimental and lattice phase shifts (Guerrieri et al., 2024).

6. Impact, Frontier Directions, and Outlook

The non-perturbative S-matrix bootstrap reshapes the landscape of QFT dynamics by:

• Establishing rigorous, non-perturbative constraints on low-energy Wilson coefficients, scattering lengths, and couplings, isolating physical EFTs in parameter space and excluding broad classes of non-realizable models (Guerrieri et al., 2020, Miro et al., 2022).
• Enabling global fits to data under analyticity, crossing, and full unitarity, with modern machine learning and optimization techniques practical for high-dimensional, non-convex landscapes (Gumus et al., 2024, Guerrieri et al., 2024).
• Revealing the detailed transition structure of allowed S-matrix islands across spacetime dimension—identifying threshold singularities, loss of positivity, and the breakdowns of simple dispersive arguments as key features in the high-dimensional limit (Gumus et al., 30 Dec 2025).
• Providing a template for future explorations—in higher-point, non-diagonal, massless, or gravitational scattering—by expanding the conceptual and algorithmic reach of convex/dual functionals, categorical symmetries, and data-driven (UV-complete) amplitude construction.

Open directions include analytic error bounds at finite truncation, the systematic inclusion of inelasticity and resonance/dynamical input, and the generalization to mixed correlator and higher-point (multi-particle) bootstraps, as well as the engagement with non-invertible symmetries and categorical constraints (Guerrieri et al., 2020, Copetti et al., 2024). The bootstrap establishes a robust, scalable, and highly predictive program for charting and quantifying the space of quantum field theories directly at the level of observable amplitudes.