Primal Bootstrap: Constraining Low-Energy Observables

• Primal Bootstrap Approach is a framework that builds explicit candidate solutions enforcing analyticity, crossing symmetry, and unitarity to constrain low-energy observables.
• It formulates physical and statistical consistency as semidefinite programs, transforming complex amplitude or estimator construction into tractable optimization problems.
• The method yields bounds on EFT Wilson coefficients and constrains models in QCD and holography, while extending to robust statistics and neural network optimizations.

The Primal Bootstrap Approach is a framework for deriving rigorous, nonperturbative constraints on low-energy observables in quantum field theory and statistical inference by directly constructing amplitudes or statistical measures—subject to analyticity, crossing, unitarity, and other consistency requirements—via semidefinite programming (SDP) or related convex optimization techniques. The method is "primal" in the sense that it builds explicit candidate solutions (amplitudes or estimators) that saturate or "rule-in" the allowed region, in contrast to "dual" approaches that aim to rule out forbidden regions through linear functional analysis. The primal bootstrap has found applications in S-matrix theory for effective field theories (EFTs), the study of nonperturbative bounds in quantum chromodynamics (QCD), and distributionally robust optimization in statistics.

1. Conceptual Foundations and Formal Structure

The primal bootstrap begins with an explicit parametrization (ansatz) for the objects of interest—scattering amplitudes in quantum field theory or statistical estimators in data science—that enforces key physical or structural properties at the level of construction. In the context of S-matrix theory, these are:

• Analyticity: Ensured by choosing parameterizations whose only singularities are those required by unitarity and physical branch cuts.
• Crossing Symmetry: Incorporated via symmetrized variables or crossing-symmetric basis expansions.
• Regge/High-Energy Boundedness: Built into the ansatz to reflect known asymptotic behavior.
• Infinite-Dimensional Structure: The ansatz contains an infinite or arbitrarily large collection of free parameters to span the entire desired functional space.

For instance, the effective field theory (EFT) bootstrap for large-$N$ chiral perturbation theory ($\chi$PT) utilizes an amplitude ansatz

$M(s,t) = \sum_{\substack{a,b\ge0\a+b\ge1}}^{a+b\le N_{\rm max}} \alpha_{ab} [\rho(s)]^a s^{a} t^{b} (\rho(s)+s)^{a} t^{b} + \text{sym perms},$

where $\rho(s) \equiv \sqrt{\frac{M-s}{M+s}}$ encodes the analytic structure with a branch cut for $s\ge M$ (Li, 2023). By increasing $N_{\rm max}$, one systematically introduces all higher-derivative EFT interactions.

The primal bootstrap in robust statistics similarly builds data-dependent laws (e.g., bootstrap empirical distributions) embedded within ambiguity sets (such as Wasserstein balls) and solves for robust optimizers over these sets (Summers et al., 2021).

2. Crossing Symmetry, Dispersion Relations, and Physical Constraints

In S-matrix applications, physical consistency is enforced via manifestly crossing-symmetric sum rules or fixed-t dispersion relations. For instance, the crossing-symmetric dispersive sum rules of Choudhury–Sinha–Zahed provide a powerful basis for constraint imposition without forward-limit subtractions: $$B_k(p) = \oint_{z=1,\xi,\xi^2} \frac{dz}{4\pi i} K_k(z) M(z,p) = 0,$$ with specific kernel choices ensuring the sum rules encode only valid physical relations. These contours can be deformed onto low-energy arcs plus UV branch cuts, leading to explicit relations between low-energy Wilson coefficients and the UV spectral discontinuities (Li, 2023, Rham et al., 27 Jun 2025).

For amplitudes with inelasticity, the crossing-symmetric ansatz can be augmented with additional variables (e.g., $\bar\rho$) to introduce multiple thresholds (Antunes et al., 2023), accommodating physically relevant multi-particle channels.

3. Primal Formulation as (Infinite-Dimensional) SDP

The transition to a computational problem is achieved by promoting ansatz parameters to optimization variables and expressing all physical constraints as linear or semidefinite inequalities. The primal problem typically takes the form: $$\min_x\, g_F(x) \quad \text{subject to} \quad \Im a_J(s_k;x) \ge 0,$$ for a target Wilson coefficient $g_F$, or more generally,

$\chi$0

subject to crossing, analyticity, and pointwise or integrated unitarity constraints (Li, 2023, Miro et al., 2022, Antunes et al., 2023).

For theories with large-$\chi$1 scaling, the primal formulation naturally captures the shrinking low-energy region and the dominance of even-spin, planar amplitudes (Li, 2023). Implementing an upper bound on spectral densities (i.e., $\chi$2) further produces “rule-in” bounds that are guaranteed if UV completion saturates elastic unitarity.

4. Discretization, Numerical Solution, and Primal–Dual Convergence

Solving the primal bootstrap requires discretizing the infinite set of unitarity constraints—sampling the $\chi$3 space on a sufficiently fine grid—and truncating the ansatz to a large but finite number of terms ($\chi$4, $\chi$5). The problem is then implemented as a standard SDP and solved using robust solvers such as SDPB, Mosek, or custom codes.

Algorithmic steps include:

1. Fixing normalization conditions (e.g., $\chi$6)
2. Implicit constraint evaluation and matrix construction for each grid point
3. Objective minimization (or maximization) for the target observable

Convergence is assessed by systematically raising truncations and verifying that primal and dual SDP results agree to within desired accuracy. In practice, for large-$\chi$7 $\chi$8PT, a relative primal–dual gap of $\chi$9 can be achieved at moderate truncations ($M(s,t) = \sum_{\substack{a,b\ge0\a+b\ge1}}^{a+b\le N_{\rm max}} \alpha_{ab} [\rho(s)]^a s^{a} t^{b} (\rho(s)+s)^{a} t^{b} + \text{sym perms},$0, $M(s,t) = \sum_{\substack{a,b\ge0\a+b\ge1}}^{a+b\le N_{\rm max}} \alpha_{ab} [\rho(s)]^a s^{a} t^{b} (\rho(s)+s)^{a} t^{b} + \text{sym perms},$1) (Li, 2023).

5. Extraction of Physical Bounds and Comparison with Dual Approaches

Once the extremal solution is obtained, Wilson coefficients and other observables are extracted via Taylor expansion or direct sum rule evaluation. For example, in EFT, bounds on $M(s,t) = \sum_{\substack{a,b\ge0\a+b\ge1}}^{a+b\le N_{\rm max}} \alpha_{ab} [\rho(s)]^a s^{a} t^{b} (\rho(s)+s)^{a} t^{b} + \text{sym perms},$2 are used to test the viability of candidate UV completions, such as holographic QCD models. If the predictions of a model lie outside the primal-allowed region, the model is ruled out (Li, 2023).

Comparison with dual methods (which use linear functional “rule-out”) confirms near-perfect agreement in allowed regions and convergence towards extremal amplitudes. This validates the completeness and sharpness of the primal approach for a wide variety of applications including Regge-bounded, stringy, or super-soft amplitudes (Häring et al., 2023, Rham et al., 27 Jun 2025).

6. Extensions: Inelasticity, Robust Statistics, and Neural Optimization

The primal bootstrap framework readily generalizes to incorporate explicit inelasticity profiles in S-matrix constructions. By replacing elastic unitarity bounds with user-specified profiles (e.g., step or exponential decay of $M(s,t) = \sum_{\substack{a,b\ge0\a+b\ge1}}^{a+b\le N_{\rm max}} \alpha_{ab} [\rho(s)]^a s^{a} t^{b} (\rho(s)+s)^{a} t^{b} + \text{sym perms},$3), one obtains strictly tighter or more physically relevant bounds for theories with inelastic channels (Antunes et al., 2023).

In statistical applications, the primal bootstrap underpins distributionally robust optimization (DRO) by coupling bootstrap-generated empirical distributions with ambiguity sets (e.g., Wasserstein balls), yielding optimal finite-sample guarantees and tractable reformulations as LPs or SOCPs for coherent risk measures such as CVaR (Summers et al., 2021).

Machine learning architectures, notably neural networks, have been integrated into the primal bootstrap for flexible parameterization of amplitude functions. Neural optimizers efficiently target arbitrary low-energy couplings or resonance structures and reproduce analytic SDP-generated boundaries to high precision (Gumus et al., 2024).

7. Phenomenological Applications and Future Directions

The primal bootstrap has proved particularly powerful for constraining the low-energy sector of QCD-inspired EFTs and linking nonperturbative bounds to the spectrum of candidate holographic models. For example, the Witten–Sakai–Sugimoto model’s predictions for $M(s,t) = \sum_{\substack{a,b\ge0\a+b\ge1}}^{a+b\le N_{\rm max}} \alpha_{ab} [\rho(s)]^a s^{a} t^{b} (\rho(s)+s)^{a} t^{b} + \text{sym perms},$4 land well inside the primal-allowed region for a suitable choice of resonance mass, while higher-derivative deformations are sharply bounded by the approach (Li, 2023).

A plausible implication is that further theoretical enhancements—such as incorporation of multi-particle thresholds, full double discontinuities, or advanced neural parameterization—will allow exploration of broader classes of consistent quantum field theories, systematic study of inelastic S-matrix spaces, and development of more robust decision-making frameworks in high-dimensional statistical settings.