Amplitude Analysis of the Decays $\eta^\prime \rightarrow   \pi^+\pi^-\pi^0$ and $\eta^\prime \rightarrow \pi^0\pi^0\pi^0$
  Based on a sample of $1.31 \times 10^9$ $J/\psi$ events collected with the BESIII detector, an amplitude analysis of the isospin-violating decays $\eta^\prime \rightarrow \pi^+\pi^-\pi^0$ and $\eta^\prime \rightarrow \pi^0\pi^0\pi^0$ is performed. A significant $P$-wave contribution from $\eta^\prime \rightarrow \rho^{\pm} \pi^{\mp}$ is observed for the first time in $\eta^\prime \rightarrow \pi^+\pi^-\pi^0$. The branching fraction is determined to be ${\mathcal B}(\eta^\prime \rightarrow \rho^{\pm}\pi^{\mp})=(7.44\pm0.60\pm1.26\pm1.84)\times 10^{-4}$, where the first uncertainty is statistical, the second systematic, and the third model dependent. In addition to the nonresonant $S$-wave component, there is a significant $\sigma$ meson component. The branching fractions of the combined $S$-wave components are determined to be ${\mathcal B}(\eta^\prime \rightarrow \pi^+\pi^-\pi^0)_S=(37.63\pm0.77\pm2.22\pm4.48)\times 10^{-4}$ and ${\mathcal B}(\eta^\prime \rightarrow \pi^0\pi^0\pi^0)=(35.22\pm0.82\pm2.54)\times 10^{-4}$, respectively. The latter one is consistent with previous BESIII measurements. 
M. Ablikim, M. N. Achasov, X. C. Ai, O. Albayrak, M. Albrecht, D. J.   Ambrose, A. Amoroso, F. F. An, Q. An, J. Z. Bai, R. B...
hep-ex