\begin{document}${\rm{X}}(3872)$\end{document} in 2003 by Belle collaboration, numerous exotic candidates beyond the conventional quark model have been observed. Most of them are located in heavy quarkonium energy region. Several interpretations, e.g. compact multiquarks, hadronic molecules, hybrids, etc, are proposed to understand their internal structures. Hadronic molecules are based on the fact that most of exotic candidates have nearby thresholds, which makes them analogies of deuteron made of one proton and one neutron. Whether two or more hadrons can be form a hadronic molecule or not depends on their interactions. In this work, we study the \begin{document}${\rm{P}}$\end{document}-wave \begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document} interactions based on the \begin{document}${\rm{e^+e^-}}\to {\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document} cross sections from Belle-II experiment to study whether their interaction can form vector bottomonium-like states or not. As \begin{document}${\rm{B}}^{(*)}$\end{document} and \begin{document}$\bar{{\rm{B}}}^{(*)}$\end{document} mesons have bottom and antibottom quark, respectively, we work in the heavy quark limit, which respects both heavy quark spin symmetry and heavy quark flavor symmetry. In this framework, we construct effective contact potentials for \begin{document}$J^{{\rm{PC}}}=1^{--}$\end{document} \begin{document}${\rm{P}}$\end{document}-wave \begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document} interactions, by decomposing the \begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document} dynamic channels into heavy-light basis. That, in the heavy quark limit, heavy and light degrees of freedoms are conserved individually makes the contact potentials in a very simple form. After solving the corresponding Lippmann-Schwinger equation, one can obtain the \begin{document}${\rm{e^+e^-}}\to {\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document} scattering amplitudes. With these scattering amplitudes, we can deduce the corresponding cross sections,which can be compared with the experimental data directly. By fitting to the data, we find that the mass shifts of the considered bottomonia are small due to their small couplings to the \begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document} continuum channels. As the result, the \begin{document}$\Upsilon(4{\rm{S}})$\end{document}, \begin{document}$\Upsilon(3{\rm{D}})$\end{document}, \begin{document}$\Upsilon(5{\rm{S}})$\end{document} and \begin{document}$\Upsilon(6{\rm{S}})$\end{document} vector bottomonia express theirselves as peaks at \begin{document}$10.58\; {\rm{GeV}}$\end{document}, \begin{document}$10.87\; {\rm{GeV}}$\end{document}, \begin{document}$11.03\; {\rm{GeV}}$\end{document}. The peak at \begin{document}$10.87\; {\rm{GeV}}$\end{document} is the interference between \begin{document}$\Upsilon(3{\rm{D}})$\end{document} and \begin{document}$\Upsilon(5{\rm{S}})$\end{document}. As there are only two data points around \begin{document}$10.63\; {\rm{GeV}}$\end{document}, we cannot obtain a very clear conclusion about the peak around this energy point. To further explore its nature, both detailed scan around this energy region in experiment and improved formula in theory are needed."> Bottominium-like states in e<sup>+</sup>e<sup>–</sup> annihilation - 必威体育下载

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Ye Quan-Xing, He Guang-Zhao, Wang Qian
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  • Abstract views:1765
  • PDF Downloads:62
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  • Received Date:31 May 2023
  • Accepted Date:15 October 2023
  • Available Online:19 October 2023
  • Published Online:20 October 2023

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