Improved measurements of the coherence factors and strong-phase differences in $D\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $D\to K^{-}{\pi }^{+}{\pi }^0$ with quantum-correlated $D\bar{D}$ decays

Improved measurements of the coherence factors and strong-phase differences in $D\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $D\to K^{-}{\pi }^{+}{\pi }^0$ decays are reported, using quantum-correlated $D\bar{D}$ pairs produced in $e^{+}{e}^{-}$ annihilation at a center-of-mass energy of $3.773\mathrm{G}\mathrm{e}\mathrm{V}$, where $D$ denotes a quantum superposition of the flavour-specific $D^0$ and ${\bar{D}}^0$ mesons. The analysis employs a dataset collected by the BESIII experiment, corresponding to an integrated luminosity of $7.93\mathrm{f}{\mathrm{b}}^{-1}$. The observables sensitive to the coherence factors and strong-phase differences are measured by reconstructing one $D$ meson in the signal mode and the other in a tag mode. These parameters provide essential inputs to the measurement of the angle $\gamma$ of the Cabibbo-Kobayashi-Maskawa Unitarity Triangle in the LHCb and Belle II experiments. The coherence factors are determined to be $R_{K3\pi }=0.51\pm 0.04$ and $R_{K\pi {\pi }^0}=0.75\pm 0.03$, and the strong-phase differences are ${\delta }_D^{K3\pi }={\left(182_{-13}^{+14}\right)}^{\circ }$ and ${\delta }_D^{K\pi {\pi }^0}={\left(209_{-8}^{+7}\right)}^{\circ }$, where the uncertainties include both statistical and systematic contributions. For $D\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$, the parameters have been further determined in four phase-space bins with improved precision compared to the previous BESIII results. The uncertainty on future $\gamma$ measurements from the knowledge of $D\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ parameters is expected to be reduced to approximately 3.5${}^{\circ }$.