The near-threshold cross section of $e^+e^- \to \Omega^-\bar{\Omega}^+$: Heavy-flavor rescattering and physics-informed deep learning
Recent precision measurements of the $e^+e^- \to \Omega^-\bar{\Omega}^+$ cross section by the BESIII collaboration provide a valuable opportunity to probe complex hadronic rescattering mechanisms. In this work, we investigate a potential structure near the $D_s\bar{D}_s$ threshold using a coupled-channel framework incorporating $\Omega\bar{\Omega}$, $\Xi\bar{\Xi}$, and $D_s\bar{D}_s$ interactions. The driving potentials are derived from effective Lagrangians respecting heavy quark spin symmetry, chiral symmetry, and hidden local symmetry, and the scattering amplitude is unitarized via the on-shell factorization of the Bethe-Salpeter equation. To go beyond local fits and map theoretical uncertainties, we use a two-step machine-learning framework. First, Simulation-Based Inference with a Mixture Density Network maps the global Bayesian posterior of the effective couplings. Second, to identify the non-perturbative threshold dynamics without the instabilities of traditional root-finding across multiple Riemann sheets, we employ a Cauchy-Riemann Physics-Informed Neural Network (PINN). The network enforces mathematical analyticity, smoothly continuing the real-axis amplitude into the complex energy plane. We isolate a pole at $M = 3.847$ GeV with zero decay width, sitting $89$~MeV below the $D_s^-\bar{D}_s^+$ threshold. The corresponding S-matrix residues show an overwhelming coupling to the $D_s\bar{D}_s$ channel, indicating that the threshold dynamics are driven by a dynamically generated $D_s^-\bar{D}_s^+$ bound state.