Improved sensitivity to the CKM phase $\gamma$ through binning phase space in $B^- \to DK^-$, $D \to K^+\pi^-\pi^-\pi^+$ decays

A binning scheme is proposed for $D^0 \to K^+\pi^-\pi^-\pi^+ $ phase space that will improve the sensitivity of a $B^- \to DK^-$ analysis to the angle $\gamma$ of the Cabibbo-Kobayashi-Maskawa Unitarity Triangle. The scheme makes use of amplitude models recently reported by the LHCb collaboration. Assuming that a four-bin scheme optimised on the models retains a similar sensitivity when applied in data, it is estimated that the statistical uncertainty on $\gamma$ from the $B$-meson sample so far collected by the LHCb experiment will be as low as 5 degrees. This will be one of the most precise results available for any single decay mode in a $B^- \to D K^-$ measurement. Quantum-correlated $D\bar{D}$ data accumulated by the CLEO-c experiment are analysed to provide first constraints on the coherence factors and average strong-phase differences in the four bins, which are necessary inputs for the measurement. These constraints are compared with the predictions of the model, and consequences for the measurement of $\gamma$ are discussed.