General considerations on the nature of $Z_b(10610)$ and $Z_b(10650)$ from their pole positions

Understand

The nature of the bottomonium-like states $Z_b(10610)$ and $Z_b(10650)$ is studied by calculating the $B^{(*)}\overline B^{*}$ compositeness ($X$) in those resonances.

• We first consider uncoupled isovector $S$-wave scattering of $B^{(*)}\overline B^{*}$ within the framework of effective-range expansion (ERE).
• Expressions for the scattering length ($a$) and effective range ($r$) are derived exclusively in terms of the masses and widths of the two $Z_b$ states.
• We then develop compositeness within ERE for the resonance case and deduce the expression $X=1/\sqrt{2r/a-1}$, which is then applied to the systems of interest.