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

TL;DR

This paper investigates the internal structure of the $Z_b$ states by calculating their compositeness using effective-range expansion, providing insights into their molecular or quark-antiquark nature based on experimental data.

Contribution

It develops a novel method to compute the compositeness of near-threshold resonances directly from pole positions and branching ratios, enhancing understanding of exotic bottomonium-like states.

Findings

$Z_b(10610)$ has a compositeness of approximately 0.66.

$Z_b(10650)$ has a compositeness of approximately 0.51.

Effective-range expansion is validated as suitable for near-threshold resonance analysis.

Abstract

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. Finally, the actual compositeness parameters are calculated in terms of resonance pole positions and their experimental branching ratios into $B^{(*)}\overline{B}^*$ by using the method of Ref.[1]. We find the values $X=0.66\pm 0.11$ and $0.51\pm 0.10$ for the $Z_b(10610)$ and $Z_b(10650)$, respectively. We also compare the ERE with Breit-Wigner and Flatt\'e parameterizations to discuss the applicability of the last two ones for near-threshold resonances with explicit examples.