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 ${Dbar{D}}$ decays

The decays $Dto K^-pi^+pi^+pi^-$ and $D to K^-pi^+pi^0$ are studied in a sample of quantum-correlated $Dbar{D}$ pairs produced through the process $e^+e^- to psi(3770) to Dbar{D}$, 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 $R_{K3pi}=0.52^{+0.12}_{-0.10}$ and $R_{Kpipi^0}=0.78 pm 0.04$, with values for the average strong-phase differences that are $delta_D^{K3pi}=left(167^{+31}_{-19}right)^circ$ and $delta_D^{Kpipi^0}=left(196^{+14}_{-15}right)^circ$, 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$^circ$.