Measurement of the ${K_L}^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz Plot

A Dalitz plot of 509 000 ${K_L}^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays is analyzed, using ${\mathrm{}\left|M\right|\mathrm{}}^2\propto 1+aY+b{Y}^2+cX+d{X}^2+\dots$, where $X$ and $Y$ are the Dalitz variables. We find no measurable $X$ dependence; we do find that a significant $X^2$ and $Y^2$ dependence is present. In addition, the spectrum cannot be fit with a matrix element linear in $Y$. Our best fit yields $a=-0.917\mathrm{}\pm 0.013$, $b=0.149\mathrm{}\pm 0.013$, and $\alpha =0.055\mathrm{}\pm 0.010$. Comparison with $\tau$ decay indicates a definite $\left|\Delta I\right|=\frac{1}{2}$ violation in the nonsymmetric amplitudes.