Determination of the resonant parameters of excited vector strangenia with the e⁺e⁻toηφ 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 $2.85~\rm GeV$ measured by BaBar, Belle, BESIII and CMD-3 experiments. The mass $(1678^{+5}_{-3} \pm 7)~\rm MeV/c^2$ and the width $(156\pm 5 \pm 9)~\rm MeV$ are obtained for the $\phi(1680)$, and the mass $(2169\pm 5 \pm 6)~\rm MeV/c^2$ and the width $(96^{+17}_{-14} \pm 9)~\rm MeV$ 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 $\Gamma^{e^{+}e^{-}}_{\phi(1680)}\cdot \mathcal{B}[\phi(1680)\to\eta\phi] = (79 \pm 4 \pm 16)$, $(127\pm 5 \pm 12)$, $(65^{+5}_{-4} \pm 13)$ or $(215 ^{+8}_{-5} \pm 11)~\rm eV$, and $\Gamma^{e^{+}e^{-}}_{\phi(2170)}\cdot \mathcal{B}[\phi(2170)\to\eta\phi] = (0.56^{+0.03}_{-0.02} \pm 0.07)$, $(0.36^{+0.05}_{-0.03} \pm 0.07)$, $(38 \pm 1 \pm 5)$ or $(41\pm 2 \pm 6)~\rm eV$, 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 $\Gamma^{e^{+}e^{-}}_{X(1750)}\cdot \mathcal{B}[X(1750)\to\eta\phi]$ at $90\%$ confidence level.