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

Abstract: Quantum-correlated $\psi(3770) \to D\bar{D}$ decays collected by the CLEO-c experiment are used to perform a first measurement of $F_+<sup>{4\pi}$, the fractional $CP$-even content of the self-conjugate decay $D \to \pi<sup>+\pi<sup>-\pi<sup>+\pi<sup>-$, obtaining a value of $0.737 \pm 0.028$. An important input to the measurement comes from the use of $D \to K^0_{\rm S}\pi<sup>+\pi<sup>-$ and $D \to K^0_{\rm L}\pi<sup>+\pi<sup>-$ decays to tag the signal mode. This same technique is applied to the channels $D \to\pi<sup>+\pi<sup>-\pi<sup>0$ and $D \to K<sup>+K<sup>-\pi<sup>0$, 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 $B^\mp \to DK<sup>\mp$ decays, and in time-dependent studies of $CP$ violation and mixing in the $D\bar{D}$ system.