Improved measurement of the strong-phase difference $$\delta _D^{K\pi }$$ δ D K π in quantum-correlated $$D{\bar{D}}$$ D D ¯ decays

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