Does the bottomonium counterpart of $X(3872)$ exist?

Abstract: A narrow line shape peak at about 10615 MeV, just above the threshold in the $B\bar B<sup>*$ channel, which can be regarded as the signal of bottomonium counterpart of $X(3872)$, $X_b$, is predicted by using the extended Friedrichs scheme. Though a virtual state is found at about 10593 MeV in this scheme, we point out that the peak is contributed mainly by the coupling form factor, which comes from the convolution of the interaction term and meson wave functions including the one from $\chi_{b1}(4P)$, but not mainly by the virtual-state pole. In this picture, the reason why $X_b$ signal is not observed in the $\Upsilon\pi<sup>+\pi<sup>-$ and $\Upsilon\pi<sup>+\pi<sup>-\pi<sup>0$ channels can also be understood. The $\chi_{b1}(4P)$ mass and width are found to be about 10771 MeV and 6 MeV, respectively and a dynamically generated broad resonance is also found with its mass and width at about 10672 MeV and 78 MeV, respectively. The line shapes of these two states are also affected by the form factor effect. Thus, this study also emphasizes the importance of the structure of the wave functions of high radial excitations in the analysis of the line shapes, and provides a caveat that some signals may be generated from the structures of the form factors rather than from poles.