Improved measurement of the decays ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^{+\left(0\right)}{\pi }^{-\left(0\right)}$ and search for the rare decay ${\eta }^{\prime }\to 4{\pi }^{0}$

Using a sample of 10 billion $J/\psi $ events collected with the BESIII detector, the decays ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^{0}{\pi }^{0}$ and ${\eta }^{\prime }\to 4{\pi }^{0}$ are studied via the process $J/\psi \to \gamma {\eta }^{\prime }$. The branching fractions of ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ and ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ ${\pi }^{0}$ are measured to be $\left(8.56±0.25\left(\mathrm{stat}\right)±0.23\left(\mathrm{syst}\right)\right)×{10}^{-5}$ and $\left(2.12±0.12\left(\mathrm{stat}\right)±0.10\left(\mathrm{syst}\right)\right)×{10}^{-4}$, respectively, which are consistent with previous measurements but with improved precision. No significant ${\eta }^{\prime }\to 4{\pi }^{0}$ signal is observed, and the upper limit on the branching fraction of this decay is determined to be less than $1.24×{10}^{-5}$ at the 90% confidence level. In addition, an amplitude analysis of ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ is performed to extract the doubly virtual isovector form factor $\alpha $ for the first time. The measured value of $\alpha =1.22±0.33\left(\mathrm{stat}\right)±0.04\left(\mathrm{syst}\right)$, is in agreement with the prediction of the vector meson dominance model.