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

Abstract

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 $ω$ meson by simultaneously analyzing the isoscalar $3π$ and isovector $2π$ systems. The Hamiltonian unifies single-particle $ω$, two-particle $ρπ$, and three-particle $πππ$ dynamics within a single unitary formalism. Using leading lattice QCD spectra from the Chinese Lattice QCD Collaboration at $m_π$ = 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 $ρ$ and $ω$ 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.

Figures (2)

• Figure 1: The volume dependence of energy eigenvalues in the $\rho$ and $\omega$ channels at two different pion masses. Lattice spectra (red markers) provided in Ref. Yan:2024gwp are used to constrain the eigenvalues $E_n(L)$ (solid lines) of the finite-volume Hamiltonian with parameters determined in scheme $B_3$. The dashed lines represent the $2\pi$ or $3\pi$ non-interacting energy levels. The upper and lower rows correspond to $m_\pi=208$ and $305$ MeV, respectively. The left and right columns refer to $2\pi$ and $3\pi$ spectra, respectively.
• Figure 2: The lower half of the complex $z$-plane on the second Riemann sheet. The dashed line indicates the rotated $3\pi$ unitary cut, defined by $\text{arg}(z-3m_{\pi})=-2\theta$, when non-relativistic kinematics $\omega_\pi(p)=m_\pi+\frac{p^2}{2m_\pi}$ is adopted. For relativistic kinematics, the cut is distorted into the blue-shaded region. The orange line denotes the branch cut associated with the $\rho$-resonance, ending at $z_\rho + m_\pi$. The purple region indicates the search area for the $\omega$-resonance pole.