General Hamiltonian Approach to the $\mathbf{N}$-Body Finite-Volume Formalism: Extracting the $\mathbf{\omega }$ Resonance Parameters from Lattice QCD

We present a nonperturbative Hamiltonian framework (NPHF) to address the general $N$-body problem. This framework rigorously connects finite-volume spectra from lattice QCD to scattering observables from experiment. To demonstrate its applicability, we extract the resonance parameters of the $\omega$ meson by simultaneously analyzing the isoscalar $3\pi$ and isovector $2\pi$ systems. The Hamiltonian unifies single-particle $\omega$, two-particle $\rho \pi$, and three-particle $\pi \pi \pi$ dynamics within a single unitary formalism. Using leading lattice QCD spectra from the Chinese Lattice QCD Collaboration at $m_{\pi }$ = 208 and 305 MeV, we perform a fit in the isovector and isoscalar channels, accurately describe the lattice spectra and obtain robust determinations of the $\rho$ and $\omega$ pole positions. This work establishes a foundational approach for extracting resonance dynamics from finite-volume spectra. Given the ubiquity of three-body dynamics in exotic hadrons, halo nuclei, and neutron star matter, this general formalism holds broad relevance across particle, nuclear, and astrophysical physics.