Improved measurement of the strong-phase difference ${\delta }_{D}^{K\pi }$ in quantum-correlated $D\overline{D}$ decays

The decay $D\to {K}^{-}{\pi }^{+}$ is studied in a sample of quantum-correlated $D\overline{D}$ pairs, based on a data set corresponding to an integrated luminosity of 2.93 fb ${}^{-1}$ collected at the $\psi \left(3770\right)$ resonance by the BESIII experiment. The asymmetry between $C\phantom{\rule{-0.166667em}{0ex}}P$ -odd and $C\phantom{\rule{-0.166667em}{0ex}}P$ -even eigenstate decays into ${K}^{-}{\pi }^{+}$ is determined to be ${A}_{K\pi }=0.132±0.011±0.007$ , where the first uncertainty is statistical and the second is systematic. This measurement is an update of an earlier study exploiting additional tagging modes, including several decay modes involving a ${K}_{L}^{0}$ meson. The branching fractions of the ${K}_{L}^{0}$ modes are determined as input to the analysis in a manner that is independent of any strong phase uncertainty. Using the predominantly $C\phantom{\rule{-0.166667em}{0ex}}P$ -even tag $D\to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ and the ensemble of $C\phantom{\rule{-0.166667em}{0ex}}P$ -odd eigenstate tags, the observable ${A}_{K\pi }^{\pi \pi {\pi }^{0}}$ is measured to be $0.130±0.012±0.008$ . The two asymmetries are sensitive to ${r}_{D}^{K\pi }cos{\delta }_{D}^{K\pi }$ , where ${r}_{D}^{K\pi }$ and ${\delta }_{D}^{K\pi }$ are the ratio of amplitudes and phase difference, respectively, between the doubly Cabibbo-suppressed and Cabibbo-favoured decays. In addition, events containing $D\to {K}^{-}{\pi }^{+}$ tagged by $D\to {K}_{S,L}^{0}{\pi }^{+}{\pi }^{-}$ are studied in bins of phase space of the three-body decays. This analysis has sensitivity to both ${r}_{D}^{K\pi }cos{\delta }_{D}^{K\pi }$ and ${r}_{D}^{K\pi }sin{\delta }_{D}^{K\pi }$ . A fit to ${A}_{K\pi }$ , ${A}_{K\pi }^{\pi \pi {\pi }^{0}}$ and the phase-space distribution of the $D\to {K}_{S,L}^{0}{\pi }^{+}{\pi }^{-}$ tags yields ${\delta }_{D}^{K\pi }={\left(187.6,{}_{-9.7}^{+8.9},{}_{-6.4}^{+5.4}\right)}^{\circ }$ , where external constraints are applied for ${r}_{D}^{K\pi }$ and other relevant parameters. This is the most precise measurement of ${\delta }_{D}^{K\pi }$ in quantum-correlated $D\overline{D}$ decays.