Light meson spectroscopy from Dalitz plot analyses of ${\eta }_c$ decays to ${\eta }^{\prime }{K}^{+}{K}^{-}$, ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$, and $\eta {\pi }^{+}{\pi }^{-}$ produced in two-photon interactions

We study the processes $\gamma \gamma \to {\eta }_c\to {\eta }^{\prime }{K}^{+}{K}^{-}$, ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$, and $\eta {\pi }^{+}{\pi }^{-}$ using a data sample of $519{\mathrm{fb}}^{-1}$ recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy $e^{+}{e}^{-}$ collider at center-of-mass energies at and near the $\mathrm{\Upsilon }(nS)$ ($n=2$, 3, 4) resonances. This is the first observation of the decay ${\eta }_c\to {\eta }^{\prime }{K}^{+}{K}^{-}$ and we measure the branching fraction $\mathrm{\Gamma }({\eta }_c\to {\eta }^{\prime }{K}^{+}{K}^{-})/(\mathrm{\Gamma }({\eta }_c\to {\eta }^{\prime }{\pi }^{+}{\pi }^{-})=0.644\pm 0.039_{\mathrm{stat}}\pm 0.032_{\mathrm{sys}}$. Significant interference is observed between $\gamma \gamma \to {\eta }_c\to \eta {\pi }^{+}{\pi }^{-}$ and the nonresonant two-photon process $\gamma \gamma \to \eta {\pi }^{+}{\pi }^{-}$. A Dalitz plot analysis is performed of ${\eta }_c$ decays to ${\eta }^{\prime }{K}^{+}{K}^{-}$, ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$, and $\eta {\pi }^{+}{\pi }^{-}$. Combined with our previous analysis of ${\eta }_c\to K\overline{K}\pi$, we measure the $K_0^{*}(1430)$ parameters and the ratio between its ${\eta }^{\prime }K$ and $\pi K$ couplings. The decay ${\eta }_c\to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ is dominated by the $f_0(2100)$ resonance, also observed in $J/\psi$ radiative decays. A new $a_0(1700)\to \eta \pi$ resonance is observed in the ${\eta }_c\to \eta {\pi }^{+}{\pi }^{-}$ channel. We also compare ${\eta }_c$ decays to $\eta$ and ${\eta }^{\prime }$ final states in association with scalar mesons as they relate to the identification of the scalar glueball.