Measurement of ${\chi }_{cJ}$ decaying into ${\eta }^{\prime }{K}^{+}{K}^{-}$

Using $(106.41\pm 0.86)\times 10^6\psi (3686)$ events collected with the BESIII detector at BEPCII, we study for the first time the decay ${\chi }_{cJ}\to {\eta }^{\prime }{K}^{+}{K}^{-}$ ($J=1$, 2), where ${\eta }^{\prime }\to \gamma {\rho }^0$ and ${\eta }^{\prime }\to \eta {\pi }^{+}{\pi }^{-}$. A partial wave analysis in the covariant tensor amplitude formalism is performed for the decay ${\chi }_{c1}\to {\eta }^{\prime }{K}^{+}{K}^{-}$. Intermediate processes ${\chi }_{c1}\to {\eta }^{\prime }{f}_0(980)$, ${\chi }_{c1}\to {\eta }^{\prime }{f}_0(1710)$, ${\chi }_{c1}\to {\eta }^{\prime }{f_2}^{\prime }(1525)$ and ${\chi }_{c1}\to K_0^{*}(1430{)}^{\pm }{K}^{\mp }$ ($K_0^{*}(1430{)}^{\pm }\to {\eta }^{\prime }{K}^{\pm }$) are observed with statistical significances larger than $5\sigma$, and their branching fractions are measured.

Figure 1

The invariant mass distributions of (a) ${\pi }^{+}{\pi }^{-}$ in mode I, and (b) $\gamma \gamma$ in mode II. The arrows show the $\eta$ signal region.

Figure 2

The invariant mass distributions of (a) $\gamma {\pi }^{+}{\pi }^{-}$ in the decay mode I, and (b) $\gamma \gamma {\pi }^{+}{\pi }^{-}$ in the decay mode II. The arrows show the ${\eta }^{\prime }$ signal region.

Figure 3

(a) The scatter plot of $M(\gamma {\pi }^{+}{\pi }^{-}{K}^{+}{K}^{-})$ versus $M(\gamma {\pi }^{+}{\pi }^{-})$. The two vertical lines show the ${\eta }^{\prime }$ signal region. (b) The $\gamma {\pi }^{+}{\pi }^{-}{K}^{+}{K}^{-}$ invariant mass of events with $M(\gamma {\pi }^{+}{\pi }^{-})$ outside the ${\eta }^{\prime }$ range in the ${\eta }^{\prime }$ decay mode I.

Figure 4

(a) The scatter plot of $M(\gamma \gamma {\pi }^{+}{\pi }^{-}{K}^{+}{K}^{-})$ versus $M(\gamma \gamma {\pi }^{+}{\pi }^{-})$ distribution. The two vertical lines show the ${\eta }^{\prime }$ signal region. (b) The $\gamma \gamma {\pi }^{+}{\pi }^{-}{K}^{+}{K}^{-}$ invariant mass of events with $M(\gamma \gamma {\pi }^{+}{\pi }^{-})$ outside the ${\eta }^{\prime }$ range in the ${\eta }^{\prime }$ decay mode II.

Figure 5

The $\gamma (\gamma ){\pi }^{+}{\pi }^{-}$ mass distribution within the ${\chi }_{c1}$ region for (a) ${\eta }^{\prime }$ decay mode I and (b) ${\eta }^{\prime }$ decay mode II. The band under the peak shows the ${\eta }^{\prime }$ signal region, and the other bands show ${\eta }^{\prime }$ sideband.

Figure 6

Invariant mass distribution of $\gamma {\pi }^{+}{\pi }^{-}{K}^{+}{K}^{-}$ for ${\eta }^{\prime }$ decay mode I in (a) ${\eta }^{\prime }$ signal region and (b) ${\eta }^{\prime }$ sideband region.

Figure 7

Invariant mass distribution of $\gamma \gamma {\pi }^{+}{\pi }^{-}{K}^{+}{K}^{-}$ for the ${\eta }^{\prime }$ decay mode II in (a) ${\eta }^{\prime }$ signal region and (b) ${\eta }^{\prime }$ sideband region. The fraction of nonpeaking background is very small so its line is invisible in the left plot.

Figure 8

The invariant mass distributions of $K^{+}{K}^{-}$ and ${\eta }^{\prime }{K}^{\pm }$ within the ${\chi }_{c1}$ mass range. (a),(b) for the ${\eta }^{\prime }$ decay mode I, and (c),(d) for the ${\eta }^{\prime }$ decay mode II.

Figure 9

(a) The scatter plot of $M(\gamma {\pi }^{+}{\pi }^{+}{K}^{+}{K}^{-})$ versus $M(\gamma {\pi }^{+}{\pi }^{+})$ for mode I. (b) The scatter plot of $M(\gamma \gamma {\pi }^{+}{\pi }^{+}{K}^{+}{K}^{-})$ versus $M(\gamma \gamma {\pi }^{+}{\pi }^{+})$ for mode II. The plots here are the zoom-in subregions of Fig. 3 and Fig. 4 around ${\eta }^{\prime }$ and ${\chi }_{cJ}$. The boxes defining the signal and sideband regions are described in the text.

Figure 10

Comparisons of angular distributions cos$\theta (\gamma -J/\psi )$, cos$\theta (K^{+}-K^{+}{K}^{-})$, cos$\theta (K^{+}-{\eta }^{\prime }{K}^{+})$, cos$\theta ({\eta }^{\prime }-{\eta }^{\prime }{K}^{+}{K}^{-})$, (a), (b), (c), (d) for the ${\eta }^{\prime }$ decay mode I, (e), (f), (g), (h) for the ${\eta }^{\prime }$ decay mode II. The empty histogram shows the global fit result combined with the background contribution. The filled histogram shows background.

Figure 11

Dalitz plots of $M^2({\eta }^{\prime }{K}^{+})$ versus $M^2({\eta }^{\prime }{K}^{-})$. (a) of MC projections for the ${\eta }^{\prime }$ decay mode I; (b) of data for the ${\eta }^{\prime }$ decay mode I; (c) of MC projections for the ${\eta }^{\prime }$ decay mode II; and (d) of data for the ${\eta }^{\prime }$ decay mode II.

Figure 12

The invariant mass distributions of $K^{+}{K}^{-}$ and $\gamma (\gamma ){\pi }^{+}{\pi }^{-}{K}^{\pm }$ for events within the ${\chi }_{c2}$ selection range. (a),(b) for the ${\eta }^{\prime }$ decay mode I, and (c),(d) for the ${\eta }^{\prime }$ decay mode II.