Measuring $CP$ violation and mixing in charm with inclusive self-conjugate multibody decay modes

Time-dependent studies of inclusive charm decays to multibody self-conjugate final states can be used to determine the indirect $CP$-violating observable $A_{\mathrm{\Gamma }}$ and the mixing observable $y_{CP}$, provided that the fractional $CP$-even content of the final state, $F_{+}$, is known. This approach can yield significantly improved sensitivity compared with the conventional method that relies on decays to $CP$ eigenstates. In particular, $D\to {\pi }^{+}{\pi }^{-}{\pi }^0$ appears to be an especially powerful channel, given its relatively large branching fraction and the high value of $F_{+}$ that has recently been measured at charm threshold.