From data to the analytic S-matrix: A Bootstrap fit of the pion scattering amplitude

Quantum Chromodynamics (QCD) governs the strong interactions of hadrons, but extracting its physical spectrum remains a significant challenge due to its non-perturbative nature. In this Letter, we introduce a novel data-driven approach that systematically enforces the fundamental principles of analyticity, crossing symmetry, and unitarity while fitting experimental data. Our Bootstrap Fit method combines S-matrix Bootstrap techniques with non-convex numerical optimization, allowing for the construction of a scattering amplitude that adheres to first-principles constraints. We apply this framework to pion-pion scattering, demonstrating that it accurately reproduces low-energy predictions from Chiral Perturbation Theory ( \chi χ PT) while also providing a non-perturbative determination of the total cross-section that is consistent with experiment. A key feature of our approach is its ability to dynamically generate physical states, yielding a spectrum of resonances consistent with QCD. Most notably, we predict the existence of a genuine doubly charged tetraquark resonance around 2 GeV, which could be observed in B-meson decays at LHCb. These results establish a robust new pathway for extracting hadronic properties directly from scattering data while enforcing fundamental physical constraints.