# Spin partners of the $Z_b(10610)$ and $Z_b(10650)$ revisited

## TL;DR

This paper explores the heavy-quark spin symmetry implications for bottomonium exotic states, modeling them as hadronic molecules with coupled-channel equations, and predicts their spin partners' properties considering experimental uncertainties.

## Contribution

It provides a dynamical coupled-channel framework including pion exchange to predict spin partner states of $Z_b$ resonances, highlighting the importance of pion tensor forces and symmetry breaking effects.

## Key findings

- Predicted masses and widths of $W_{bJ}$ states as functions of $Z_b$ pole positions.
- Demonstrated the significant impact of pion tensor forces on partner state locations.
- Showed the necessity of coupled-channel treatment for understanding hadronic molecule spin multiplets.

## Abstract

We study the implications of the heavy-quark spin symmetry for the possible spin partners of the exotic states $Z_b(10610)$ and $Z_b(10650)$ in the spectrum of bottomonium. We formulate and solve numerically the coupled-channel equations for the $Z_b$ states that allow for a dynamical generation of these states as hadronic molecules. The force includes short-range contact terms and the one-pion exchange potential, both treated fully nonperturbatively. The strength of the potential at leading order is fixed completely by the pole positions of the $Z_b$ states such that the mass and the most prominent contributions to the width of the isovector heavy-quark spin partner states $W_{bJ}$ with the quantum numbers $J^{++}$ ($J=0,1,2$) come out as predictions. Since the accuracy of the present experimental data does not allow one to fix the pole positions of the $Z_b$'s reliably enough, we also study the pole trajectories of their spin partner states as functions of the $Z_b$ binding energies. It is shown that, once the heavy-quark spin symmetry is broken by means of the physical $B$ and $B^*$ masses, especially the pion tensor force has a significant impact on the location of the partner states clearly demonstrating the need of a coupled-channel treatment of pion dynamics to understand the spin multiplet pattern of hadronic molecules.