$\varphi \varphi$ and $J/\psi \varphi$ mass spectra in decay $B_s^0\to J/\psi \varphi \varphi$

The mass spectra of the $\varphi \varphi$ and $J/\psi \varphi$ states in the decay $B_s^0\to J/\psi \varphi \varphi$ recently observed by LHCb are calculated in the model which takes into account the $J^P=0^{+},0^{-},2^{+}$ intermediate resonances $R_1$, $R_2$ in the $\varphi \varphi$ channel and the $J^P=1^{+}$ ones, $X_1$, $X_2$, in the $J/\psi \varphi$ channel. When obtaining the expressions for the effective amplitudes and mass spectra, the approximate threshold kinematics of the decay is used essentially. The $R_1-R_2$ and $X_1-X_2$ mixings arising due to the common decay modes $\varphi \varphi$ and $J/\psi \varphi$, respectively, are also taken into account. The obtained expressions for the mass spectra are applied for extracting the information about masses and coupling constants of the resonances in the $\varphi \varphi$ and $J/\psi \varphi$ final states.

Figure 1

The diagrams of the decay $B_s^0\to J/\psi \varphi \varphi$.

Figure 2

The $\varphi \varphi$ mass spectrum in the decay $B_s^0\to J/\psi \varphi \varphi$ in the model with the $R_{1,2}$ resonance quantum numbers $J^P=0^{+}$ (solid curve). The contributions of the $R_{1,2}$ and $X_{1,2}$ resonances are shown with dashed and dotted lines, respectively. LHCb data [1] are normalized to the unity in accord with Eq. (5.1).

Figure 3

The $J/\psi \varphi$ mass spectrum in the decay $B_s^0\to J/\psi \varphi \varphi$ obtained in the model with the $R_{1,2}$ resonance quantum numbers $J^P=0^{+}$. The designations of curves are the same as in Fig. 2.

Figure 4

The same as in Fig. 2, but in the model with the $R_{1,2}$ resonance quantum numbers $J^P=0^{-}$. Also shown (dot-dashed line) is the contribution of the interference term Eq. (4.4).

Figure 5

The same as in Fig. 3 but in the model with the $R_{1,2}$ resonance quantum numbers $J^P=0^{-}$.

Figure 6

The same as in Fig. 2, but in the model with the $R_{1,2}$ resonance quantum numbers $J^P=2^{+}$.

Figure 7

The same as in Fig. 3 but in the model with the $R_{1,2}$ resonance quantum numbers $J^P=2^{+}$.

Figure 8

The comparison of descriptions of the $\varphi \varphi$ mass spectrum in the models of the $R_{1,2}$ resonances with $J^P=0^{+}$ (solid line), $0^{-}$ (dashed line), and $2^{+}$ (dotted line).

Figure 9

The same as in Fig. 8 but for the $J/\psi \varphi$ mass spectrum.