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

The decays $D\to K^-π^+π^+π^-$ and $D \to K^-π^+π^0$ are studied in a sample of quantum-correlated $D\bar{D}$ pairs produced through the process $e^+e^- \to ψ(3770) \to D\bar{D}$, 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 $γ$ of the Unitarity Triangle in $B^- \to DK^-$ decays at the LHCb and Belle II experiments. The coherence factors are determined to be $R_{K3π}=0.52^{+0.12}_{-0.10}$ and $R_{Kππ^0}=0.78 \pm 0.04$, with values for the average strong-phase differences that are $δ_D^{K3π}=\left(167^{+31}_{-19}\right)^\circ$ and $δ_D^{Kππ^0}=\left(196^{+14}_{-15}\right)^\circ$, where the uncertainties include both statistical and systematic contributions. The analysis is re-performed in four bins of the phase-space of the $D \to K^-π^+π^+π^-$ to yield results that will allow for a more sensitive measurement of $γ$ with this mode, to which the BESIII inputs will contribute an uncertainty of around 6$^\circ$.