Determination of the resonant parameters of excited vector strangenia with the $e^{+}{e}^{-}\to \eta \phi$ data

We determine the resonant parameters of the vector states $\phi (1680)$ and $\phi (2170)$, by doing a combined fit to the $e^{+}{e}^{-}\to \eta \phi$ cross sections from threshold to $2.85\mathrm{G}\mathrm{e}\mathrm{V}$ measured by BaBar, Belle, BESIII and CMD-3 experiments. The mass $(1678_{-3}^{+5}\pm 7)\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}{\mathrm{c}}^2$ and the width $(156\pm 5\pm 9)\mathrm{M}\mathrm{e}\mathrm{V}$ are obtained for the $\phi (1680)$, and the mass $(2169\pm 5\pm 6)\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}{\mathrm{c}}^2$ and the width $(96_{-14}^{+17}\pm 9)\mathrm{M}\mathrm{e}\mathrm{V}$ for the $\phi (2170)$. The statistical significance of $\phi (2170)$ is $7.2\sigma$. Depending on the interference between the $\phi (1680)$, $\phi (2170)$ and a non-resonant $\eta \phi$ amplitude in the nominal fit, we obtain four solutions and ${\mathrm{\Gamma }}_{\phi (1680)}^{e^{+}{e}^{-}}\cdot \mathcal{B}[\phi (1680)\to \eta \phi ]=(79\pm 4\pm 16)$, $(127\pm 5\pm 12)$, $(65_{-4}^{+5}\pm 13)$ or $(215_{-5}^{+8}\pm 11)\mathrm{e}\mathrm{V}$, and ${\mathrm{\Gamma }}_{\phi (2170)}^{e^{+}{e}^{-}}\cdot \mathcal{B}[\phi (2170)\to \eta \phi ]=(0.56_{-0.02}^{+0.03}\pm 0.07)$, $(0.36_{-0.03}^{+0.05}\pm 0.07)$, $(38\pm 1\pm 5)$ or $(41\pm 2\pm 6)\mathrm{e}\mathrm{V}$, respectively. We also search for the production of $X(1750)\to \eta \phi$ and the significance is only $2.0\sigma$, then we determine the upper limit of ${\mathrm{\Gamma }}_{X(1750)}^{e^{+}{e}^{-}}\cdot \mathcal{B}[X(1750)\to \eta \phi ]$ at $90\mathrm{\%}$ confidence level.