Scalar molecules $η_{b}B_{c}^{-}$ and $η_{c}B_{c}^{+} $ with asymmetric quark contents

TL;DR

The paper investigates scalar hadronic molecules with asymmetric heavy-quark contents, focusing on ${\mathcal{M}}_{\mathrm{b}}=\eta_{b}B_{c}^{-}$ and ${\mathcal{M}}_{\mathrm{c}}=\eta_{c}B_{c}^{+}$ using QCD sum rules. Two-point SRs yield masses $m=(15728 \pm 90)$ MeV and $\widetilde{m}=(9712 \pm 72)$ MeV with corresponding current couplings, indicating these states lie above relevant thresholds. Three-point SRs are then employed to compute strong couplings at molecule–meson–meson vertices, enabling partial widths for dominant decays ${\mathcal{M}}_{\mathrm{b}}\to\eta_{b}B_{c}^{-}$ and ${\mathcal{M}}_{\mathrm{c}}\to\eta_{c}B_{c}^{+}, J/\psi B_{c}^{\ast+}$ as well as channels arising from quark annihilation. The resulting full widths are $\Gamma[{\mathcal{M}}_{\mathrm{b}}]=(93 \pm 17)$ MeV and $\Gamma[{\mathcal{M}}_{\mathrm{c}}]=(70 \pm 10)$ MeV, with dominant decays well above thresholds, suggesting these molecules are broad, experimentally accessible states. The work connects these molecular states to related fully heavy tetraquark pictures and provides quantitative predictions to guide future searches and phenomenological analyses.

Abstract

The hadronic scalar molecules $\mathcal{M}_{b}$ and $\mathcal{M}_{c}$ with asymmetric quark contents $bb \overline{b}\overline{c}$ and $cc \overline{c} \overline{b}$ are explored by means of the QCD sum rule method. Their masses and current couplings are calculated using the two-point sum rule approach. The obtained results show that they are strong-interaction unstable particles and transform to ordinary mesons' pairs. The molecule $\mathcal{M} _{b}$ dissociates through the process $\mathcal{M}_{\mathrm{b}}\to η_{b}B_{c}^{-}$. The decays $\mathcal{M}_{\mathrm{c}}\rightarrow η_{c}B_{c}^{+}$ and $J/ψB_{c}^{\ast +}$ are dominant modes for the molecule $\mathcal{M}_{c}$. The full decay widths of the molecules $\mathcal{ M}_{b}$ and $\mathcal{M}_{c}$ are estimated using these decay channels, as well as ones generated by the annihilation of $b\overline{b}$ and $c \overline{c}$ quarks in $\mathcal{M}_{b}$ and $\mathcal{M}_{c}$, respectively. The QCD three-point sum rule method is employed to find partial widths all of these channels. This approach is required to evaluate the strong couplings at the molecule-meson-meson vertices under consideration. The mass $m=(15728 \pm 90)~\mathrm{MeV}$ and width $Γ[\mathcal{M}_b] =(93 \pm 17)~ \mathrm{MeV}$ of the molecule $\mathcal{M}_{b}$, and $ \widetilde{m}=(9712 \pm 72)~\mathrm{MeV}$ and $Γ[\mathcal{M}_c]=(70 \pm 10)~ \mathrm{MeV}$ in the case of $\mathcal{M}_{c} $ offer valuable guidance for experimental searches at existing facilities.

Figures (5)

• Figure 1: Pole contribution $\mathrm{PC}$ as a function of $M^{2}$ at some $s_{0}$. The circle labels the point $M^{2}=16.5~\mathrm{GeV}^{2}$ and $s_{0}=277.5~\mathrm{GeV}^{2}$.
• Figure 2: Dependence of the mass $m$ on the parameters $M^{2}$ (left panel), and $s_0$ (right panel).
• Figure 3: Mass $\widetilde{m}$ as a function on the parameters $M^{2}$ (left panel), and $s_0$ (right panel). The purple triangle shows point $M^{2}=9~\mathrm{GeV}^{2}$ and $s_{0}=109~\mathrm{GeV}^{2}$.
• Figure 4: The sum rule's data and extrapolating function $\mathcal{G}(Q^{2},m^2)$. The diamond is placed at $Q^{2}=-m_{B_c}^{2}$.
• Figure 5: The QCD data for the form factors $g_{1}(Q^{2})$ and $g_{3}(Q^{2})$ and fit functions $\mathcal{G}_{1}(Q^{2},m^2)$ (solid line), $\mathcal{G}_{3}(Q^{2},m^2)$ (dashed line). The red star and blue circle show positions of the points $Q^{2}=-m_{\overline{D}^{0}}^{2}$ and $Q^{2}=-m_{D^{\ast}}^{2}$, respectively.