A measurement of the CKM angle $\gamma$ from studies of $DK\pi$ Dalitz plots

Various measurements of quantities relating to $B^0_{(s)} \to D K^\pm\pi^\mp$ decays are reported from analyses building towards a measurement of the CKM angle $\gamma$. The first observation of the decay $B^0_s \to \bar{D}^0 K^-\pi^+$ is reported. Based on a data sample corresponding to an integrated luminosity of $1.0 \,{\rm fb}^{-1}$ of $pp$ collision data recorded by the LHCb detector, the branching fraction relative to that of the topologically similar decay $B^0 \to \bar{D}^0 \pi^+\pi^-$ is measured to be $$ \frac{ {\cal B}\left(B^0_s \to \bar{D}^0 K^-\pi^+\right)}{ {\cal B}\left(B^0 \to \bar{D}^0 \pi^+\pi^-\right)} = 1.18 \pm 0.05\,\text{(stat.)} \pm 0.12\,\text{(syst.)} \, . $$ In addition, the relative branching fraction of the decay $B^0 \to \bar{D}^0 K^+\pi^-$ is measured to be $$ \frac{ {\cal B}\left(B^0 \to \bar{D}^0 K^+\pi^-\right)}{ {\cal B}\left(B^0 \to \bar{D}^0 \pi^+\pi^-\right)} = 0.106 \pm 0.007\,\text{(stat.)} \pm 0.008\,\text{(syst.)} \, . $$ The resonant substructures of $B^0_s \to \bar{D}^0 K^- \pi^+$ and $B^0 \to \bar{D}^0 K^+ \pi^-$ decays are studied using a data sample corresponding to an integrated luminosity of $3.0 \,{\rm fb}^{-1}$ of $pp$ collision data recorded by the LHCb detector. In $B^0_s \to \bar{D}^0 K^-\pi^+$ decays, an excess at $m(\bar{D}^0 K^-) \approx 2.86 {\rm GeV}/c^2$ is found to be an admixture of spin-1 and spin-3 resonances. This is the first observation of a heavy flavoured spin-3 resonance, and the first time that any spin-3 particle has been seen to be produced in $B$ decays. The masses and widths of the new states and of the $D^{*}_{s2}(2573)^{-}$ meson are measured as part of this analysis, giving the most precise determinations to date, while the masses and widths of the $D^*_0(2400)^-$ and $D^{*}_{2}(2460)^{-}$ resonances are measured from the analysis of $B^0 \to \bar{D}^0 K^+ \pi^-$ decays. The complex amplitudes and fit fractions for all components included in the two amplitude models are also measured. Finally, a first determination of $\gamma$ from an amplitude analysis of $B^0 \to D K^+ \pi^-$ decays is reported based on measurements of the Cartesian parameters \begin{eqnarray*} x_{\pm} &=& r_{B} \cos \left( \delta_{B} \pm \gamma \right)\,,\\ y_{\pm} &=& r_{B} \sin \left( \delta_{B} \pm \gamma \right)\,. \end{eqnarray*}