The Dalitz plot distribution of $B^0 \rightarrow \bar{D}^0 K^+ \pi^$ decays is studied using a data sample corresponding to $3.0\rm{fb}^{1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. The data are described by an amplitude model that contains contributions from intermediate $K^*(892)^0$, $K^*(1410)^0$, $K^*_2(1430)^0$ and $D^*_2(2460)^$ resonances. The model also contains components to describe broad structures, including the $K^*_0(1430)^0$ and $D^*_0(2400)^$ resonances, in the $K\pi$ Swave and the $D\pi$ S and Pwaves. The masses and widths of the $D^*_0(2400)^$ and $D^*_2(2460)^$ resonances are measured, as are the complex amplitudes and fit fractions for all components included in the amplitude model. The model obtained will be an integral part of a future determination of the angle $\gamma$ of the CKM quark mixing matrix using $B^0 \rightarrow D K^+ \pi^$ decays.
Decay diagrams for the quasitwobody contributions to $ B ^0 \rightarrow D K ^+ \pi ^ $ from (a) $ B ^0 \rightarrow D K ^* (892)^0$ and (b) $ B ^0 \rightarrow D^{*}_{2}(2460)^{} K ^+ $ decays. 
Fig1a.pdf [14 KiB] HiDef png [55 KiB] Thumbnail [56 KiB] *.C file 

Fig1b.pdf [14 KiB] HiDef png [54 KiB] Thumbnail [56 KiB] *.C file 

Results of the fit to the $ B $ candidate invariant mass distribution with (a) linear and (b) logarithmic $y$axis scales. The components are as described in the legend. 
Fig2a.pdf [32 KiB] HiDef png [346 KiB] Thumbnail [267 KiB] *.C file 

Fig2b.pdf [30 KiB] HiDef png [321 KiB] Thumbnail [234 KiB] *.C file 

Distribution of $ B ^0 \rightarrow \overline{ D }{} {}^0 K ^+ \pi ^ $ candidates in the signal region over (a) the Dalitz plot and (b) the square Dalitz plot. The definition of the square Dalitz plot is given in Sec. ???. 
Fig3a.pdf [49 KiB] HiDef png [194 KiB] Thumbnail [215 KiB] *.C file 

Fig3b.pdf [49 KiB] HiDef png [219 KiB] Thumbnail [265 KiB] *.C file 

Efficiency variation as a function of SDP position for candidates triggered by (a) signal decay products and (b) by the rest of the event. The vertical white stripe is due to the $ D ^*$ veto and the curved white band is due to the $\overline{ D }{} {}^0$ veto. 
Fig4a.pdf [210 KiB] HiDef png [1 MiB] Thumbnail [667 KiB] *.C file 

Fig4b.pdf [209 KiB] HiDef png [1 MiB] Thumbnail [619 KiB] *.C file 

SDP distributions of the background contributions from (a) combinatorial background and (b) $ B ^0 \rightarrow \overline{ D }{} {}^{(*)0} \pi ^+ \pi ^ $ decays. 
Fig5a.pdf [19 KiB] HiDef png [199 KiB] Thumbnail [177 KiB] *.C file 

Fig5b.pdf [30 KiB] HiDef png [265 KiB] Thumbnail [245 KiB] *.C file 

Projections of the data and amplitude fit results onto (a) $m(\overline{ D }{} {}^0 \pi ^ )$, (c) $m( K ^+ \pi ^ )$ and (e) $m(\overline{ D }{} {}^0 K ^+ )$, with the same projections shown in (b), (d) and (f) with a logarithmic $y$axis scale. Components are described in the legend. 
Fig6a.pdf [32 KiB] HiDef png [288 KiB] Thumbnail [196 KiB] *.C file 

Fig6b.pdf [24 KiB] HiDef png [550 KiB] Thumbnail [335 KiB] *.C file 

Fig6c.pdf [31 KiB] HiDef png [257 KiB] Thumbnail [184 KiB] *.C file 

Fig6d.pdf [23 KiB] HiDef png [567 KiB] Thumbnail [335 KiB] *.C file 

Fig6e.pdf [32 KiB] HiDef png [336 KiB] Thumbnail [231 KiB] *.C file 

Fig6f.pdf [24 KiB] HiDef png [577 KiB] Thumbnail [350 KiB] *.C file 

Fig6leg.pdf [12 KiB] HiDef png [63 KiB] Thumbnail [53 KiB] *.C file 

Projections of the data and amplitude fit results onto (a) $m(\overline{ D }{} {}^0 \pi ^ )$ in the $D^*_2(2460)^$ region and (b) the low $m( K ^+ \pi ^ )$ region. Components are as shown in Fig. ???. 
Fig7a.pdf [26 KiB] HiDef png [330 KiB] Thumbnail [242 KiB] *.C file 

Fig7b.pdf [32 KiB] HiDef png [272 KiB] Thumbnail [204 KiB] *.C file 

Projections of the data and amplitude fit results onto $\cos \theta(\overline{ D }{} {}^0 \pi ^ )$ in the mass ranges (a) $2.04 < m(\overline{ D }{} {}^0 \pi ^ ) < 2.35 \mathrm{ Ge V} $ and (b) $2.35 < m(\overline{ D }{} {}^0 \pi ^ ) < 2.55 \mathrm{ Ge V} $. Components are as shown in Fig. ???. 
Fig8a.pdf [35 KiB] HiDef png [340 KiB] Thumbnail [242 KiB] *.C file 

Fig8b.pdf [33 KiB] HiDef png [328 KiB] Thumbnail [249 KiB] *.C file 

Projections of the data and amplitude fit results onto $\cos \theta( K ^+ \pi ^ )$ in the mass ranges (a) $m( K ^+ \pi ^ ) < 0.8 \mathrm{ Ge V} $, (b) $0.8 < m( K ^+ \pi ^ ) < 1.0 \mathrm{ Ge V} $, (c) $1.0 < m( K ^+ \pi ^ ) < 1.3 \mathrm{ Ge V} $ and (d) $1.4 < m( K ^+ \pi ^ ) < 1.5 \mathrm{ Ge V} $. Components are as shown in Fig. ???. 
Fig9a.pdf [31 KiB] HiDef png [288 KiB] Thumbnail [208 KiB] *.C file 

Fig9b.pdf [31 KiB] HiDef png [267 KiB] Thumbnail [209 KiB] *.C file 

Fig9c.pdf [32 KiB] HiDef png [266 KiB] Thumbnail [190 KiB] *.C file 

Fig9d.pdf [34 KiB] HiDef png [292 KiB] Thumbnail [213 KiB] *.C file 

Backgroundsubtracted and efficiencycorrected Legendre moments up to order 7 calculated as a function of $m(\overline{ D }{} {}^0 \pi ^ )$ for data (black data points) and the fit result (solid blue curve). 
Fig10a.pdf [16 KiB] HiDef png [182 KiB] Thumbnail [174 KiB] *.C file 

Fig10b.pdf [16 KiB] HiDef png [180 KiB] Thumbnail [172 KiB] *.C file 

Fig10c.pdf [16 KiB] HiDef png [164 KiB] Thumbnail [156 KiB] *.C file 

Fig10d.pdf [16 KiB] HiDef png [163 KiB] Thumbnail [149 KiB] *.C file 

Fig10e.pdf [16 KiB] HiDef png [164 KiB] Thumbnail [150 KiB] *.C file 

Fig10f.pdf [16 KiB] HiDef png [165 KiB] Thumbnail [154 KiB] *.C file 

Fig10g.pdf [16 KiB] HiDef png [162 KiB] Thumbnail [153 KiB] *.C file 

Fig10h.pdf [16 KiB] HiDef png [161 KiB] Thumbnail [154 KiB] *.C file 

Backgroundsubtracted and efficiencycorrected Legendre moments up to order 7 calculated as a function of $m( K ^+ \pi ^ )$ for data (black data points) and the fit result (solid blue curve). 
Fig11a.pdf [16 KiB] HiDef png [171 KiB] Thumbnail [154 KiB] *.C file 

Fig11b.pdf [16 KiB] HiDef png [167 KiB] Thumbnail [150 KiB] *.C file 

Fig11c.pdf [16 KiB] HiDef png [175 KiB] Thumbnail [153 KiB] *.C file 

Fig11d.pdf [16 KiB] HiDef png [169 KiB] Thumbnail [157 KiB] *.C file 

Fig11e.pdf [16 KiB] HiDef png [175 KiB] Thumbnail [163 KiB] *.C file 

Fig11f.pdf [16 KiB] HiDef png [160 KiB] Thumbnail [151 KiB] *.C file 

Fig11g.pdf [16 KiB] HiDef png [170 KiB] Thumbnail [162 KiB] *.C file 

Fig11h.pdf [17 KiB] HiDef png [165 KiB] Thumbnail [160 KiB] *.C file 

Differences between the data SDP distribution and the fit model across the SDP, in terms of the perbin pull. 
Fig12.pdf [19 KiB] HiDef png [139 KiB] Thumbnail [121 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2015017.gif Thumbnail 
Yields from the fit to the $\overline{ D }{} {}^0 K ^+ \pi ^ $ data sample. The full mass range is $5100$$5900\mathrm{ Me V} $ and the signal region is $5248.55$$5309.05\mathrm{ Me V} $. 
Table_1.pdf [54 KiB] HiDef png [111 KiB] Thumbnail [58 KiB] tex code 

Signal contributions to the fit model, where parameters and uncertainties are taken from Ref. \cite{PDG2014}. The models are described in Sec. ???. 
Table_2.pdf [58 KiB] HiDef png [98 KiB] Thumbnail [46 KiB] tex code 

Masses and widths $(\mathrm{Me V} )$ determined in the fit to data, with statistical uncertainties only. 
Table_3.pdf [43 KiB] HiDef png [60 KiB] Thumbnail [27 KiB] tex code 

Complex coefficients and fit fractions determined from the Dalitz plot fit. Uncertainties are statistical only. Note that the fit fractions, magnitudes and phases are derived quantities. 
Table_4.pdf [50 KiB] HiDef png [120 KiB] Thumbnail [59 KiB] tex code 

Experimental systematic uncertainties on the fit fractions (%) and masses and widths $(\mathrm{Me V} )$. Uncertainties given on the central values are statistical only. 
Table_5.pdf [49 KiB] HiDef png [122 KiB] Thumbnail [58 KiB] tex code 

Model uncertainties on the fit fractions (%) and masses and widths $(\mathrm{Me V} )$. Uncertainties given on the central values are statistical only. 
Table_6.pdf [49 KiB] HiDef png [117 KiB] Thumbnail [55 KiB] tex code 

Results for the complex amplitudes and their uncertainties presented (top) in terms of real and imaginary parts and (bottom) in terms and magnitudes and phases. The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. 
Table_7.pdf [48 KiB] HiDef png [92 KiB] Thumbnail [38 KiB] tex code 

Results for the fit fractions and their uncertainties (%). The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. Upper limits are given at 90 % (95 %) confidence level. 
Table_8.pdf [47 KiB] HiDef png [131 KiB] Thumbnail [69 KiB] tex code 

Results for the product branching fractions. The four quoted errors are statistical, experimental systematic, model and PDG uncertainties, respectively. Upper limits are given at 90 % (95 %) confidence level. 
Table_9.pdf [48 KiB] HiDef png [118 KiB] Thumbnail [61 KiB] tex code 

Results for the branching fractions. The four quoted errors are statistical, experimental systematic, model and PDG uncertainties, respectively. Upper limits are given at 90 % (95 %) confidence level. 
Table_10.pdf [46 KiB] HiDef png [76 KiB] Thumbnail [39 KiB] tex code 

Interference fit fractions (%) from the nominal DP fit. The amplitudes are all pairwise products involving: ($A_{0}$) $ K ^* (892)^{0}$, ($A_{1}$) $ K ^* (1410)^{0}$, ($A_{2}$) $ K ^*_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) $ K ^*_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{}$, ($A_{6}$) $D^{*}_{2}(2460)^{}$, ($A_{7}$) $D\pi$ Swave (dabba), ($A_{8}$) $D\pi$ Pwave (EFF). The diagonal elements are the same as the conventional fit fractions. 
Table_11.pdf [33 KiB] HiDef png [57 KiB] Thumbnail [29 KiB] tex code 

Statistical uncertainties on the interference fit fractions (%). The amplitudes are all pairwise products involving: ($A_{0}$) $ K ^* (892)^{0}$, ($A_{1}$) $ K ^* (1410)^{0}$, ($A_{2}$) $ K ^*_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) $ K ^*_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{}$, ($A_{6}$) $D^{*}_{2}(2460)^{}$, ($A_{7}$) $D\pi$ Swave (dabba), ($A_{8}$) $D\pi$ Pwave (EFF). The diagonal elements are the same as the conventional fit fractions. 
Table_12.pdf [27 KiB] HiDef png [86 KiB] Thumbnail [42 KiB] tex code 

Experimental systematic uncertainties on the interference fit fractions (%). The amplitudes are all pairwise products involving: ($A_{0}$) $ K ^* (892)^{0}$, ($A_{1}$) $ K ^* (1410)^{0}$, ($A_{2}$) $ K ^*_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) $ K ^*_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{}$, ($A_{6}$) $D^{*}_{2}(2460)^{}$, ($A_{7}$) $D\pi$ Swave (dabba), ($A_{8}$) $D\pi$ Pwave (EFF). The diagonal elements are the same as the conventional fit fractions. 
Table_13.pdf [27 KiB] HiDef png [85 KiB] Thumbnail [41 KiB] tex code 

Model uncertainties on the interference fit fractions (%). The amplitudes are all pairwise products involving: ($A_{0}$) $ K ^* (892)^{0}$, ($A_{1}$) $ K ^* (1410)^{0}$, ($A_{2}$) $ K ^*_{0}(1430)^{0}$, ($A_{3}$) LASS nonresonant, ($A_{4}$) $ K ^*_{2}(1430)^{0}$, ($A_{5}$) $D^{*}_{0}(2400)^{}$, ($A_{6}$) $D^{*}_{2}(2460)^{}$, ($A_{7}$) $D\pi$ Swave (dabba), ($A_{8}$) $D\pi$ Pwave (EFF). The diagonal elements are the same as the conventional fit fractions. 
Table_14.pdf [27 KiB] HiDef png [86 KiB] Thumbnail [42 KiB] tex code 
Created on 23 August 2019.Citation count from INSPIRE on 23 August 2019.