Determination of the resonant parameters of excited vector strangenia with <inline-formula><tex-math id="M1">\begin{document}${\boldsymbol e^+ \boldsymbol e^-\to\boldsymbol\eta\boldsymbol\phi }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/c51ce5ac-358d-4139-84fb-701931d64d97/CPC-2023-0194_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/c51ce5ac-358d-4139-84fb-701931d64d97/CPC-2023-0194_M1.png"/></alternatives></inline-formula> data -

\begin{document}$ \phi(1680) $\end{document} and

\begin{document}$ \phi(2170) $\end{document}

by performing a combined fit to the

\begin{document}$ e^+e^-\to \eta\phi $\end{document}

cross sections from the threshold to

\begin{document}$ 2.85\; {\rm{GeV}} $\end{document}

measured by the BaBar, Belle, BESIII, and CMD-3 experiments. The mass

\begin{document}$ (1678 ^{+5}_{-3} \pm 7)\; {{\rm{MeV}}}/c^2 $\end{document}

and width

\begin{document}$ (156 \pm 5 \pm 9)\; {\rm{MeV}} $\end{document}

are obtained for

\begin{document}$ \phi(1680) $\end{document}

, and the mass

\begin{document}$ (2169 \pm 5 \pm 6)\; {{\rm{MeV}}}/c^2 $\end{document}

and width

\begin{document}$ (96^{+17}_{-14} \pm 9)\; {\rm{MeV}} $\end{document}

are obtained for

\begin{document}$ \phi(2170) $\end{document}

. The statistical significance of

\begin{document}$ \phi(2170) $\end{document}

is

\begin{document}$ 7.2\sigma $\end{document}

. Depending on the interference between

\begin{document}$ \phi(1680) $\end{document}

,

\begin{document}$ \phi(2170) $\end{document}

, and a non-resonant

\begin{document}$ \eta\phi $\end{document}

amplitude in the nominal fit, we obtain four solutions and

\begin{document}$ \Gamma^{ e^+e^-}_{ \phi(1680)}\cdot {\cal B}[\phi(1680)\to\eta\phi] = (79 \pm 4 \pm 16) $\end{document}

,

\begin{document}$ (127\pm 5 \pm 12) $\end{document}

,

\begin{document}$ (65^{+5}_{-4} \pm 13) $\end{document}

or

\begin{document}$ (215^{+8}_{-5} \pm 11)\; {\rm{eV}} $\end{document}

, and

\begin{document}$ \Gamma^{ e^+e^-}_{ \phi(2170)}\cdot {\cal B}[\phi(2170)\to\eta\phi] = (0.56^{+0.03}_{-0.02}\pm 0.07) $\end{document}

,

\begin{document}$ (0.36^{+0.05}_{-0.03}\pm 0.07) $\end{document}

,

\begin{document}$ (38 \pm 1 \pm 5) $\end{document}

or

\begin{document}$ (41 \pm 2 \pm $\end{document}

\begin{document}$ 6)\; {\rm{eV}} $\end{document}

. We also search for the production of

\begin{document}$ X(1750)\to\eta\phi $\end{document}

, and the significance is only

\begin{document}$ 2.0\sigma $\end{document}

. We then determine the upper limit of

\begin{document}$ \Gamma^{ e^+e^-}_{X(1750)}\cdot {\cal B}[X(1750) \to \eta\phi] $\end{document}

at the 90% confidence level."> Determination of the resonant parameters of excited vector strangenia with <inline-formula><tex-math id="M1">\begin{document}${\boldsymbol e^+ \boldsymbol e^-\to\boldsymbol\eta\boldsymbol\phi }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/c51ce5ac-358d-4139-84fb-701931d64d97/CPC-2023-0194_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/c51ce5ac-358d-4139-84fb-701931d64d97/CPC-2023-0194_M1.png"/></alternatives></inline-formula> data -

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