Interpretation of $\Upsilon(11020)$ as an $S$-Wave $B_1\bar{B}$--$B_1\bar{B}^*$ Molecular State

Although heavy-quark symmetry predicts a $B_1\bar{B}$ molecular partner of the $D_1\bar{D}$ molecule, no such state has been observed. We propose that the experimentally observed $\Upsilon(11020)$ may be a candidate for such a state, possibly containing a $B_1\bar{B}^{*}$ component. To test this, we interpret $\Upsilon(11020)$ as an $S$-wave $B_1\bar{B}$--$B_1\bar{B}^{*}$ molecule and compute its strong decay widths using the compositeness condition and effective Lagrangians. The couplings to $B_1$ and $\bar{B}^{(*)}$ are extracted by fitting $\Upsilon(11020)\to e^+ e^-$ and $\Upsilon(11020)\to \chi_{bJ} \pi\pi\pi$ data. Using these couplings, we evaluate partial widths into $B^{(*)}_{(s)}\bar{B}^{(*)}_{(s)}$, $\pi\pi \Upsilon(nS)$, $\pi\pi h_b(nP)$, and $\pi\pi\pi \chi_{b1}$ via hadronic loops, as well as three-body $B^{*}\pi \bar{B}^{(*)}$ decays via tree diagrams. The results indicate that $\Upsilon(11020)$ is predominantly a $B_1\bar{B}$ molecule, with its main decay channel being $B_s^{*}\bar{B}^{*}$. The $\pi\pi \Upsilon(nS)$ and $\pi\pi h_b(nP)$ widths are only a few eV, whereas $\pi\pi\pi \chi_{b1}$ reaches 0.167~MeV and the unobserved $\pi\pi\pi \chi_{b0}$ could be 0.754~keV. These distinctive decay patterns provide clear experimental signatures of the molecular nature of $\Upsilon(11020)$ and offer a test of heavy-quark symmetry.