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

TL;DR

This paper investigates whether the observed $Υ(11020)$ can be interpreted as an $S$-wave hadronic molecule formed by $B_1\bar{B}$ and $B_1\bar{B}^*$ constituents, the HQS partner of the $D_1\bar{D}$ system. The authors employ the compositeness condition and effective Lagrangians to compute strong decay widths, fitting couplings to $Υ(11020)\to e^+e^-$ and $Υ(11020)\to χ_{bJ}\pi\pi\pi$ data, with a Gaussian regulator $\Phi(p_E^2/\Lambda^2)=\exp(-p_E^2/\Lambda^2)$ determining the molecular structure. The results favor a predominantly $B_1\bar{B}$ composition ($X_{B_1\bar{B}}\approx0.75$) and predict a dominant decay to $B_s^{*}\bar{B}^{*}_s$, while $\pi\pi\Upsilon(nS)$ and $\pi\pi h_b(nP)$ widths are at the eV scale and $\pi\pi\pi\chi_{b1}$ reaches about $0.167$ MeV (with $πππχ_{b0}$ up to $0.754$ keV). These distinctive decay patterns provide clear experimental signatures to test the molecular interpretation and HQSS in the bottom sector, offering a novel perspective on the structure of $Υ(11020)$ and its relation to heavy-quark symmetry partners.

Abstract

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 $Υ(11020)$ may be a candidate for such a state, possibly containing a $B_1\bar{B}^{*}$ component. To test this, we interpret $Υ(11020)$ 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 $Υ(11020)\to e^+ e^-$ and $Υ(11020)\to χ_{bJ} πππ$ data. Using these couplings, we evaluate partial widths into $B^{(*)}_{(s)}\bar{B}^{(*)}_{(s)}$, $ππΥ(nS)$, $ππh_b(nP)$, and $πππχ_{b1}$ via hadronic loops, as well as three-body $B^{*}π\bar{B}^{(*)}$ decays via tree diagrams. The results indicate that $Υ(11020)$ is predominantly a $B_1\bar{B}$ molecule, with its main decay channel being $B_s^{*}\bar{B}^{*}$. The $ππΥ(nS)$ and $ππh_b(nP)$ widths are only a few eV, whereas $πππχ_{b1}$ reaches 0.167~MeV and the unobserved $πππχ_{b0}$ could be 0.754~keV. These distinctive decay patterns provide clear experimental signatures of the molecular nature of $Υ(11020)$ and offer a test of heavy-quark symmetry.

Figures (5)

• Figure 1: Feynman diagrams for the $\Upsilon(11020)$ decays within the $B_1\bar{B}^{(*)}$ molecular framework. $\Upsilon(nS)$ denotes the $\Upsilon(1S)$, $\Upsilon(2S)$, or $\Upsilon(3S)$ state, while $h_{b}(nP)$ represents the $h_{b}(1P)$ or $h_{b}(2P)$ state. Similarly, $\chi_{bJ}$ refers to the $\chi_{b0}$, $\chi_{b1}$, and $\chi_{b2}$ states corresponding to $J = 0, 1,$ and $2$, respectively. The definitions of the kinematic variables ($p$, $p_1$, $p_2$, $p_3$, $p_4$, $q$, $k_1$, and $k_2$) used in the calculation are also indicated.
• Figure 2: Mass operator of the $\Upsilon(11020)$, interpreted as a $B_1\bar{B}^{(*)}$ molecular state. Here, $q$ and $p$ denote the four-momenta of the $B_1$ meson and the initial $\Upsilon(11020)$, respectively.
• Figure 3: (a) Decay width of $\Upsilon(11020)\to \pi\pi\pi\chi_{bJ}(1P)$ as a function of the model parameter $\Lambda$ and the $B_1\bar{B}$ molecular component. The cyan band indicates the experimental uncertainty, while the red solid line denotes the central value. (b) Coupling constants $g_Y$ (in GeV for $\Upsilon(11020)$--$B_1\bar{B}$ and in GeV$^{-1}$ for $\Upsilon(11020)$--$B_1\bar{B}^{*}$) as a function of the molecular component, evaluated at the values of $\Lambda$ corresponding to the molecular components listed in Table \ref{['table2']}.
• Figure 4: Decay width of $\Upsilon(11020)\!\to e^{+}e^{-}$ as a function of the coupling constant $g_{B_1B^{*}\gamma}$ and the $B_1\bar{B}$ molecular component. The red solid line represents the experimental central value of $\Upsilon(11020)\!\to e^{+}e^{-}$, while the black dashed and green dash-dotted lines correspond to $g_{B_1B^{*}\gamma}=0.0218$ and $0.000173$, respectively.
• Figure 5: Decay widths of $\Upsilon(11020)$ with the red solid line representing the total width, the orange dotted line for $B^{*}\bar{B}^{*}$, the blue dash-dotted line for $B^{*}\bar{B}$, the magenta dash-dot-dotted line for $B\bar{B}$, the wine two-point-segment solid line for $B^{*}\pi\bar{B}+B^{*}\pi\bar{B}^{*}$, the dark yellow short-dashed line for $B^{*}_s\bar{B}^{*}_s$, the navy short-dotted line for $B^{*}_s\bar{B}_s$, and the purple short dash-dotted line for $B_s\bar{B}_s$. The cyan bands show the experimental total width with uncertainties, and the dashed solid line indicates the experimental central value, which matches the theoretical total width at $\Lambda=0.613$ GeV and $X_{B_1\bar{B}}=0.754$.