# The $b_1$ resonance in coupled $\pi\omega$, $\pi\phi$ scattering from lattice QCD

## TL;DR

This paper reports the first lattice QCD calculation of coupled $ ext{ } ext{ }$ scattering, revealing a narrow $b_1$-like resonance with specific coupling properties and no evidence of a $ ext{ } ext{ }$ resonance.

## Contribution

It introduces a novel lattice QCD approach to coupled three-channel scattering, including a new parameterization of the scattering matrix.

## Key findings

- Identified a narrow $b_1$-like resonance at approximately 1380 MeV.
- Found the resonance couples mainly to $S$-wave $ ext{ } ext{ }$ with suppressed $D$-wave and negligible $ ext{ } ext{ }$ coupling.
- Observed no resonant behavior in $ ext{ } ext{ }$ channel, indicating absence of a low-mass $Z_s$-like state.

## Abstract

We present the first lattice QCD calculation of coupled $\pi\omega$ and $\pi\phi$ scattering, incorporating coupled $S$ and $D$-wave $\pi\omega$ in $J^P=1^+$. Finite-volume spectra in three volumes are determined via a variational analysis of matrices of two-point correlation functions, computed using large bases of operators resembling single-meson, two-meson and three-meson structures, with the light-quark mass corresponding to a pion mass of $m_\pi \approx 391$ MeV. Utilizing the relationship between the discrete spectrum of finite-volume energies and infinite-volume scattering amplitudes, we find a narrow axial-vector resonance ($J^{PC}=1^{+-}$), the analogue of the $b_1$ meson, with mass $m_{R}\approx1380$ MeV and width $\Gamma_{R}\approx 91$ MeV. The resonance is found to couple dominantly to $S$-wave $\pi\omega$, with a much-suppressed coupling to $D$-wave $\pi\omega$, and a negligible coupling to $\pi\phi$ consistent with the `OZI rule'. No resonant behavior is observed in $\pi\phi$, indicating the absence of a putative low-mass $Z_s$ analogue of the $Z_c$ claimed in $\pi J/\psi$. In order to minimally present the contents of a unitary three-channel scattering matrix, we introduce an $n$-channel generalization of the traditional two-channel Stapp parameterization.