Amplitude Analysis of the Decays ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$

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 ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$ is performed. A significant $P$-wave contribution from ${\eta }^{\prime }\to {\rho }^{\pm }{\pi }^{\mp }$ is observed for the first time in ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0$. The branching fraction is determined to be $\mathcal{B}({\eta }^{\prime }\to {\rho }^{\pm }{\pi }^{\mp })=(7.44\pm 0.60\pm 1.26\pm 1.84)\times {10}^{-4}$, 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 $\mathcal{B}({\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0{)}_S=(37.63\pm 0.77\pm 2.22\pm 4.48)\times {10}^{-4}$ and $\mathcal{B}({\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0)=(35.22\pm 0.82\pm 2.54)\times {10}^{-4}$, respectively. The latter one is consistent with previous BESIII measurements.

Figure 1

(a) ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz plot for candidate events selected from data. (b) Invariant mass distribution of ${\pi }^{+}{\pi }^{-}{\pi }^0$ candidates without the ${\eta }^{\prime }$ mass constraint applied in the kinematic fit.

Figure 2

(a) ${\eta }^{\prime }\to {\pi }^0{\pi }^0{\pi }^0$ Dalitz plot for candidate events selected from data. (b) Invariant mass of ${\pi }^0{\pi }^0{\pi }^0$ candidates without the ${\eta }^{\prime }$ mass constraint applied in the kinematic fit.

Figure 3

Comparison of the invariant mass distributions of (a) ${\pi }^{+}{\pi }^{-}$, (b) ${\pi }^{+}{\pi }^0$, (c) ${\pi }^{-}{\pi }^0$, and (d) ${\pi }^0{\pi }^0$ between data (dots with error bars) and the fit result projections (solid histograms). The dotted, dashed, dash-dotted, and dash-dot-dotted histograms show the contributions from background, $S$ wave, ${\rho }^{-}$, and ${\rho }^{+}$, respectively.