Search for a strangeonium-like structure $Z_s$ decaying into $\phi \pi$ and a measurement of the cross section $e^+e^-\rightarrow\phi\pi\pi$.

Using a data sample of $e^+e^-$ collision data corresponding to an integrated luminosity of 108 pb$^{-1}$ collected with the BESIII detector at a center-of-mass energy of 2.125 GeV, we study the process $e^+e^-\rightarrow \phi\pi\pi$, and search for a strangeonium-like structure $Z_s$ decaying into $\phi \pi$. No signal is observed in the $\phi\pi$ mass spectrum around 1.4 GeV/$c^2$. Upper limits on the cross sections for $Z_s$ production at the 90\% confidence level are determined. In addition, the cross sections of $e^+e^-\rightarrow\phi\pi^{+}\pi^{-}$ and $e^+e^-\rightarrow \phi\pi^{0}\pi^{0}$ at 2.125 GeV are measured to be $(343.0\pm5.1\pm25.1)$ pb and $(208.3\pm7.6\pm13.5)$ pb, respectively, where the first uncertainties are statistical and the second systematic.