The line shapes of the $Z_b(10610)$ and $Z_b(10650)$ in the elastic and inelastic channels revisited

TL;DR

This paper analyzes the line shapes of the $Z_b(10610)$ and $Z_b(10650)$ states using coupled-channel equations, revealing their pole structures and the effects of long-range interactions on experimental line shapes.

Contribution

It provides a comprehensive analysis of the $Z_b$ states using unitarity-respecting solutions, clarifying their nature as virtual or shallow states near thresholds.

Findings

$Z_b(10610)$ is a virtual state below threshold.

$Z_b(10650)$ is a shallow state above threshold.

Line shapes are insensitive to higher-order interactions.

Abstract

The most recent experimental data for all measured production and decay channels of the bottomonium-like states $Z_b(10610)$ and $Z_b(10650)$ are analysed simultaneously using solutions of the Lippmann-Schwinger equations which respect constraints from unitarity and analyticity. The interaction potential in the open-bottom channels $B^{(*)}\bar{B}^{*}+\mbox{c.c.}$ contains short-range interactions as well as one-pion exchange. It is found that the long-range interaction does not affect the line shapes as long as only $S$ waves are considered. Meanwhile, the line shapes can be visibly modified once $D$ waves, mediated by the strong tensor forces from the pion exchange potentials, are included. However, in the fit they get balanced largely by a momentum dependent contact term that appears to be needed also to render the results for the line shapes independent of the cut-off. The resulting line shapes are found to be insensitive to various higher-order interactions included to verify stability of the results. Both $Z_b$ states are found to be described by the poles located on the unphysical Riemann sheets in the vicinity of the corresponding thresholds. In particular, the $Z_b(10610)$ state is associated with a virtual state residing just below the $B\bar{B}^{*}/\bar B{B}^{*}$ threshold while the $Z_b(10650)$ state most likely is a shallow state located just above the $B^*\bar{B}^{*}$ threshold.