Improved measurement of the strong-phase difference $δ_D^{Kπ}$ in quantum-correlated $D\bar{D}$ decays

The decay $D\to K^-π^+$ is studied in a sample of quantum-correlated $D\bar{D}$ pairs, based on a data set corresponding to an integrated luminosity of 2.93\,fb$^{-1}$ collected at the $ψ(3770)$ resonance by the BESIII experiment. The asymmetry between $C\!P$-odd and $C\!P$-even eigenstate decays into $K^-π^+$ is determined to be ${\cal A}_{Kπ} = 0.132 \pm 0.011 \pm 0.007$, 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 $K^0_L$ meson. The branching fractions of the $K^0_L$ modes are determined as input to the analysis in a manner that is independent of any strong phase uncertainty. Using the predominantly $C\!P$-even tag $D\to π^+π^-π^0$ and the ensemble of $C\!P$-odd eigenstate tags, the observable ${\cal A}_{Kπ}^{πππ^0}$ is measured to be $0.130 \pm 0.012 \pm 0.008$. The two asymmetries are sensitive to $r_D^{Kπ} \cos δ_D^{Kπ}$, where $r_D^{Kπ}$ and $δ_D^{Kπ}$ are the ratio of amplitudes and phase difference, respectively, between the doubly Cabibbo-suppressed and Cabibbo-favoured decays. In addition, events containing $D \to K^-π^+$ tagged by $D \to K^0_{S,L} π^+π^-$ are studied in bins of phase space of the three-body decays. This analysis has sensitivity to both $r_D^{Kπ} \cos δ_D^{Kπ}$ and $r_D^{Kπ} \sin δ_D^{Kπ}$. A fit to ${\cal A}_{Kπ}$, ${\cal A}_{Kπ}^{πππ^0}$ and the phase-space distribution of the $D \to K^0_{S,L} π^+π^-$ tags yields $δ_D^{Kπ}= \left( 187.6 {^{+8.9}_{-9.7}}{^{+5.4}_{-6.4}} \right)$ degrees, where external constraints are applied for $r_D^{Kπ}$ and other relevant parameters. This is the most precise measurement of $δ_D^{Kπ}$ in quantum-correlated $D\bar{D}$ decays.