Study of ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}{\eta }_{c}$ and evidence for ${Z}_{c}\left(3900{\right)}^{±}$ decaying into ${\rho }^{±}{\eta }_{c}$

We study the reaction ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}{\eta }_{c}$ for the first time using data samples collected with the BESIII detector at center-of-mass energies $\sqrt{s}=4.226$, 4.258, 4.358, 4.416, and 4.600 GeV. Evidence of this process is found and the Born cross section ${\sigma }^{B}\left({e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}{\eta }_{c}\right)$, excluding ${e}^{+}{e}^{-}\to \omega {\eta }_{c}$ and $\eta {\eta }_{c}$, is measured to be $\left({46}_{-11}^{+12}±10\right)\text{}\text{}\mathrm{pb}$ at $\sqrt{s}=4.226\text{}\text{}\mathrm{GeV}$. Evidence for the decay ${Z}_{c}\left(3900{\right)}^{±}\to {\rho }^{±}{\eta }_{c}$ is reported at $\sqrt{s}=4.226\text{}\text{}\mathrm{GeV}$ with a significance of $3.9\sigma $, including systematic uncertainties, and the Born cross section times branching fraction ${\sigma }^{B}\left({e}^{+}{e}^{-}\to {\pi }^{\mp }{Z}_{c}\left(3900{\right)}^{±}\right)×B\left({Z}_{c}\left(3900{\right)}^{±}\to {\rho }^{±}{\eta }_{c}\right)$ is measured to be $\left(48±11±11\right)\text{}\text{}\mathrm{pb}$, which indicates that ${e}^{+}{e}^{-}\to {\pi }^{\mp }{Z}_{c}\left(3900{\right)}^{±}\to {\pi }^{\mp }{\rho }^{±}{\eta }_{c}$ dominates the ${e}^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}{\eta }_{c}$ process. The ${Z}_{c}\left(3900{\right)}^{±}\to {\rho }^{±}{\eta }_{c}$ signal is not significant at the other center-of-mass energies and the corresponding upper limits are determined. In addition, no significant signal is observed in a search for ${Z}_{c}\left(4020{\right)}^{±}\to {\rho }^{±}{\eta }_{c}$ with the same data samples. The ratios ${R}_{{Z}_{c}\left(3900\right)}=B\left({Z}_{c}\left(3900{\right)}^{±}\to {\rho }^{±}{\eta }_{c}\right)/B\left({Z}_{c}\left(3900{\right)}^{±}\to {\pi }^{±}J/\psi \right)$ and ${R}_{{Z}_{c}\left(4020\right)}=B\left({Z}_{c}\left(4020{\right)}^{±}\to {\rho }^{±}{\eta }_{c}\right)/B\left({Z}_{c}\left(4020{\right)}^{±}\to {\pi }^{±}{h}_{c}\right)$ are obtained and compared with different theoretical interpretations of the ${Z}_{c}\left(3900{\right)}^{±}$ and ${Z}_{c}\left(4020{\right)}^{±}$.