Study of Dalitz-Plot Distributions of the Decays η→π+π−π0 and η→π+π−γ

Based on 80 884 events, we obtained the matrix element squared for the decay $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}:{|M(x,y)|}^{2}=1\ensuremath{-}(1.08\ifmmode\pm\else\textpm\fi{}0.014)y+(0.03\ifmmode\pm\else\textpm\fi{}0.03){y}^{2}+(0.05\ifmmode\pm\else\textpm\fi{}0.03){x}^{2}$ where $x$ and $y$ are the usual Dalitz-plot coordinates. The $\ensuremath{\gamma}$-ray energy spectrum was measured for 18 150 $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\gamma}$ events. $\ensuremath{\rho}$ dominance of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ final state is strongly favored over a simple gauge-invariant matrix element.