Searches for $CP$ violation and $\pi \pi$ $S$ wave in the Dalitz-plot analysis of $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$

In $e^{+}{e}^{-}$ collisions recorded using the CLEO II.V detector we have studied the Cabibbo suppressed decay of $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ with the initial flavor of the $D^0$ tagged by the decay $D^{*+}\to D^0{\pi }^{+}$. We use the Dalitz-plot analysis technique to measure the resonant substructure in this final state and observe $\rho \pi$ and nonresonant contributions by fitting for their amplitudes and relative phases. We describe the $\pi \pi$ $S$ wave with a $K$-matrix formalism and limit this contribution to the rate to be $<2.5\%@95\%$ confidence level, in contrast to the large rate observed in $D^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$ decay. Using the amplitudes and phases from this analysis, we calculate an integrated $CP$ asymmetry of ${0.01}_{-0.07}^{+0.09}\pm 0.05$.