Theoretical study of scalar meson $a_0(1710)$ in the ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ reaction

We investigate the process ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ by taking into account the $S$-wave $K^{*}{\overline{K}}^{*}$ and $\rho \omega$ interactions within the unitary coupled-channel approach, where the scalar meson $a_0(1710)$ is dynamically generated. In addition, the contributions from the intermediate resonances $K_0^{*}(1430{)}^{-}\to {\overline{K}}^0{\pi }^{-}$ and $K_0^{*}(1430{)}^0\to K^{+}{\pi }^{-}$ are also considered. We find a significant dip structure around 1.8 GeV, associated to the $a_0(1710)$, in the ${\overline{K}}^0{K}^{+}$ invariant mass distribution, and the clear peaks of the $K_0^{*}(1430)$ in the ${\overline{K}}^0{\pi }^{-}$ and $K^{+}{\pi }^{-}$ invariant mass distributions, consistent with the BABAR measurements. We further estimate the branching fractions $\mathcal{B}({\eta }_c\to {\overline{K}}^{*0}{K}^{*+}{\pi }^{-})=5.5\times {10}^{-3}$ and $\mathcal{B}({\eta }_c\to \omega {\rho }^{+}{\pi }^{-})=7.9\times {10}^{-3}$. Our predictions can be tested by the BESIII and Belle II experiments in the future.

Figure 1

Diagram for the process ${\eta }_c\to ({\overline{K}}^{*0}{K}^{*+}/\omega {\rho }^{+}){\pi }^{-}\to a_0(1710{)}^{+}{\pi }^{-}\to {\overline{K}}^0{K}^{+}{\pi }^{-}$.

Figure 2

Real and imaginary parts of the loop functions $G_{\omega \rho }$ and ${\tilde{G}}_{\omega \rho }$ as a function of the ${\overline{K}}^0{K}^{+}$ invariant mass.

Figure 3

Diagram for ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ via the intermediate $K_0^{*}(1430{)}^{-}$, followed by the decay $K_0^{*}(1430{)}^{-}\to {\overline{K}}^0{\pi }^{-}$.

Figure 4

Diagram for ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ via the intermediate $K_0^{*}(1430{)}^0$, followed by the decay $K_0^{*}(1430{)}^0\to K^{+}{\pi }^{-}$.

Figure 5

${\overline{K}}^0{K}^{+}$ invariant mass distribution of the process ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$. The red-solid curve stands for the total contributions, while the blue-dashed curve, the green-dot-dashed curve, and purple-dotted curve correspond to the contribution from the $a_0(1710)$ state, the intermediate $K_0^{*}(1430{)}^{-}$, and $K_0^{*}(1430{)}^0$, respectively. The BABAR data are taken from Fig. 7(a) of Ref. [33].

Figure 6

${\overline{K}}^0{K}^{+}$ invariant mass distribution of the process ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ with the parameter $V_p=0.6$, 0.8, 1.0, respectively. In addition to the BABAR measurements of the ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ [33] (labeled as BABAR 2016), we also show the BABAR measurements of the $K^{+}{K}^{-}$ invariant mass distribution of the process ${\eta }_c\to K^{+}{K}^{-}{\pi }^0$ [46] (labeled as BABAR 2014).

Figure 7

${\overline{K}}^0{K}^{+}$ invariant mass distribution of the process ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ obtained with a phase angle $\varphi =0$, $\pi /3$, $2\pi /3$, and $\pi$, respectively. See the text for details.

Figure 8

$K^{+}{\pi }^{-}$ (a) and ${\overline{K}}^0{\pi }^{-}$ (b) invariant mass distribution of the process ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$. The notations of the curves are the same as those of Fig. 5.

Figure 9

Dalitz plots for the decay ${\eta }_c\to {\overline{K}}^0{K}^{+}{\pi }^{-}$. (a) $M_{{\overline{K}}^0{K}^{+}}$ vs $M_{{\overline{K}}^0{\pi }^{-}}$; (b) $M_{{\overline{K}}^0{K}^{+}}$ vs $M_{K^{+}{\pi }^{-}}$; (c) $M_{{\overline{K}}^0{\pi }^{-}}$ vs $M_{K^{+}{\pi }^{-}}$.