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

The decays $D\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $D\to K^{-}{\pi }^{+}{\pi }^0$ are studied in a sample of quantum-correlated $D\bar{D}$ pairs produced through the process $e^{+}{e}^{-}\to \psi (3770)\to D\bar{D}$, exploiting a data set collected by the BESIII experiment that corresponds to an integrated luminosity of 2.93 fb${}^{-1}$. Here $D$ indicates a quantum superposition of a $D^0$ and a ${\bar{D}}^0$ meson. By reconstructing one neutral charm meson in a signal decay, and the other in the same or a different final state, observables are measured that contain information on the coherence factors and average strong-phase differences of each of the signal modes. These parameters are critical inputs in the measurement of the angle $\gamma$ of the Unitarity Triangle in $B^- \to DK^-$ decays at the LHCb and Belle II experiments. The coherence factors are determined to be $R_{K3\pi }=0.52_{-0.10}^{+0.12}$ and $R_{K\pi\pi^0}=0.78 \pm 0.04$,withvaluesfortheaveragestrong-phasedifferencesthatare$\delta_D^{K3\pi}=\left(167^{+31}{-19}\right)^\circ$and$\delta_D^{K\pi\pi^0}=\left(196^{+14}{-15}\right)^\circ$, where the uncertainties include both statistical and systematic contributions. The analysis is re-performed in four bins of the phase-space of the $D \to K^-\pi^+\pi^+\pi^-$ to yield results that will allow for a more sensitive measurement of $\gamma$ with this mode, to which the BESIII inputs will contribute an uncertainty of around 6${}^{\circ }$.