Resonant substructure in ${\mathit{K}}^{\mathrm{-}}$${\mathrm{\pi }}^{+}$${\mathrm{\pi }}^{+}$${\mathrm{\pi }}^{\mathrm{-}}$ decays of ${\mathit{D}}^0$ mesons

We determine the resonant substructure of ${\mathit{D}}^0$→${\mathit{K}}^{\mathrm{-}}$ ${\mathrm{\pi }}^{+}$ ${\mathrm{\pi }}^{+}$ ${\mathrm{\pi }}^{\mathrm{-}}$ decays using a five-dimensional maximum-likelihood technique to extract the relative fractions and phases of the amplitudes contributing to this final state. We find that two-body decay modes account for at least 76% of all decays. We obtain branching ratios for several decay modes including B(${\mathit{D}}^0$→${\mathit{K}}^{\mathrm{-}}$ ${\mathit{a}}_1^{+}$) =(9.0±0.9±1.7)%, B(${\mathit{D}}^0$→K¯ ${}_{}{}^{\mathrm{*}0}{}_{}{}^{}\mathrm{\rho }_{}^0{}_{}{}^{}$)=(1.9±0.3±0.7)%, and B(${\mathit{D}}^0$→${\mathit{K}}_1$(1270${)}^{\mathrm{-}}$ ${\mathrm{\pi }}^{+}$) =(1.8±0.5 ±0.8)%. For the decay mode ${\mathit{D}}^0$→K¯ ${}_{}{}^{\mathrm{*}0}{}_{}{}^{}\mathrm{\rho }_{}^0{}_{}{}^{}$, the K¯ ${}_{}{}^{\mathrm{*}0}{}_{}{}^{}$ and ${\mathrm{\rho }}^0$ are found to be polarized with their spins oriented in the direction of their motion as seen from the ${\mathit{D}}^0$ frame.