Bottominium-like states in e<sup>+</sup>e<sup>–</sup> annihilation
Authors: Ye Quan-Xing , He Guang-Zhao , Wang Qian ,   Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China,   Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center, South China Normal University, Guangzhou 510006, China
Published: October 2023

In the conventional quark model, meson is made of one quark and one antiquark, and baryon is made of three quarks. Since the observation of the <inline-formula><tex-math id="M13">\begin{document}${\rm{X}}(3872)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M13.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M14">\begin{document}${\rm{P}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M14.png"/></alternatives></inline-formula>-wave <inline-formula><tex-math id="M15">\begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M15.png"/></alternatives></inline-formula> interactions based on the <inline-formula><tex-math id="M16">\begin{document}${\rm{e^+e^-}}\to {\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M16.png"/></alternatives></inline-formula> cross sections from Belle-II experiment to study whether their interaction can form vector bottomonium-like states or not. As <inline-formula><tex-math id="M17">\begin{document}${\rm{B}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M17.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M18.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M19">\begin{document}$J^{{\rm{PC}}}=1^{--}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M19.png"/></alternatives></inline-formula> <inline-formula><tex-math id="M20">\begin{document}${\rm{P}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M20.png"/></alternatives></inline-formula>-wave <inline-formula><tex-math id="M21">\begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M21.png"/></alternatives></inline-formula> interactions, by decomposing the <inline-formula><tex-math id="M22">\begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M22.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M23">\begin{document}${\rm{e^+e^-}}\to {\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M23.png"/></alternatives></inline-formula> 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 <inline-formula><tex-math id="M24">\begin{document}${\rm{B}}^{(*)}\bar{{\rm{B}}}^{(*)}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M24.png"/></alternatives></inline-formula> continuum channels. As the result, the <inline-formula><tex-math id="M25">\begin{document}$\Upsilon(4{\rm{S}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M25.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M26">\begin{document}$\Upsilon(3{\rm{D}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M26.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M26.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M27">\begin{document}$\Upsilon(5{\rm{S}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M27.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M27.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M28">\begin{document}$\Upsilon(6{\rm{S}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M28.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M28.png"/></alternatives></inline-formula> vector bottomonia express theirselves as peaks at <inline-formula><tex-math id="M29">\begin{document}$10.58\; {\rm{GeV}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M29.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M29.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M30">\begin{document}$10.87\; {\rm{GeV}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M30.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M30.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M31">\begin{document}$11.03\; {\rm{GeV}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M31.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M31.png"/></alternatives></inline-formula>. The peak at <inline-formula><tex-math id="M32">\begin{document}$10.87\; {\rm{GeV}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M32.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M32.png"/></alternatives></inline-formula> is the interference between <inline-formula><tex-math id="M33">\begin{document}$\Upsilon(3{\rm{D}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M33.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M33.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M34">\begin{document}$\Upsilon(5{\rm{S}})$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M34.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M34.png"/></alternatives></inline-formula>. As there are only two data points around <inline-formula><tex-math id="M35">\begin{document}$10.63\; {\rm{GeV}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M35.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20230908_M35.png"/></alternatives></inline-formula>, 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.