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 ${0.01}_{-0.07}^{+0.09}\pm 0.05$.

Figure 1

Distribution of (a) $M$ within 650 keV of our measured value of $Q$ and (b) $Q$ within $44\mathrm{MeV}/c^2$ of our measured value of $M$ for the process $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$. The candidates pass all selection criteria discussed in the text. The curves show the results of the fits. The vertical lines denote the signal region. The light (dark) shaded region indicates the signal (background) contribution.

Figure 2

(a) The Dalitz-plot distribution for $D^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ candidates. (b)–(d) Projections of the results of the fit described in the text to the ${\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz plot showing both fit D (line) and the data (points). The results of fits A, C1, and C2 are indistinguishable from fit D. See text for details of the fits.

Figure 3

Projections of the $\pi \pi$ $S$-wave parametrization for (a) fit A: flat nonresonant; (b) fit C1: Breit-Wigner $\sigma (500)$; (c) fit C2: Breit-Wigner $f_0(980)$; and (d) fit D: $K$ matrix.