First determination of the $CP$ content of $D\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ and updated determination of the $CP$ contents of $D\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $D\to K^{+}{K}^{-}{\pi }^0$

Abstract

Quantum-correlated $\psi (3770)\to D\bar{D}$ decays collected by the CLEO-c experiment are used to perform a first measurement of $F_{+}^{4\pi }$, the fractional $CP$-even content of the self-conjugate decay $D\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, obtaining a value of $0.737\pm 0.028$. An important input to the measurement comes from the use of $D\to K_{\mathrm{S}}^0{\pi }^{+}{\pi }^{-}$ and $D\to K_{\mathrm{L}}^0{\pi }^{+}{\pi }^{-}$ decays to tag the signal mode. This same technique is applied to the channels $D\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and $D\to K^{+}{K}^{-}{\pi }^0$, yielding $F_{+}^{\pi \pi {\pi }^0}=1.014\pm 0.045\pm 0.022$ and $F_{+}^{KK{\pi }^0}=0.734\pm 0.106\pm 0.054$, where the first uncertainty is statistical and the second systematic. These measurements are consistent with those of an earlier analysis, based on $CP$-eigenstate tags, and can be combined to give values of $F_{+}^{\pi \pi {\pi }^0}=0.973\pm 0.017$ and $F_{+}^{KK{\pi }^0}=0.732\pm 0.055$. The results will enable the three modes to be included in a model-independent manner in measurements of the unitarity triangle angle $\gamma$ using $B^{\mp }\to D{K}^{\mp }$ decays, and in time-dependent studies of $CP$ violation and mixing in the $D\bar{D}$ system.

arXiv commentMinor revisions following journal acceptance

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