The $B^{}\to D^{+}K^{}\pi^{}$ decay is observed in a data sample corresponding to $3.0 \rm{fb}^{1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. Its branching fraction is measured to be ${\cal B}(B^{}\to D^{+}K^{}\pi^{}) = (7.31 \pm 0.19 \pm 0.22 \pm 0.39) \times 10^{5}$ where the uncertainties are statistical, systematic and from the branching fraction of the normalisation channel $B^{}\to D^{+}\pi^{}\pi^{}$, respectively. An amplitude analysis of the resonant structure of the $B^{}\to D^{+}K^{}\pi^{}$ decay is used to measure the contributions from quasitwobody $B^{}\to D_{0}^{*}(2400)^{0}K^{}$, $B^{}\to D_{2}^{*}(2460)^{0}K^{}$, and $B^{}\to D_{J}^{*}(2760)^{0}K^{}$ decays, as well as from nonresonant sources. The $D_{J}^{*}(2760)^{0}$ resonance is determined to have spin 1.
Results of the fit to the $ B ^ \rightarrow D ^+ \pi ^ \pi ^ $ candidate invariant mass distribution for the (left) TOS and (right) TISonly subsamples. Data points are shown in black, the full fitted model as solid blue lines and the components as shown in the legend. 
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Results of the fit to the $ B ^ \rightarrow D ^+ K ^ \pi ^ $ candidate invariant mass distribution for the (left) TOS and (right) TISonly subsamples. Data points are shown in black, the full fitted model as solid blue lines and the components as shown in the legend. 
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Signal efficiency across the SDP for (left) TOS and (right) TISonly $ B ^ \rightarrow D ^+ K ^ \pi ^ $ decays. The relative uncertainty at each point is typically $5 \%$. 
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The first seven Legendrepolynomial weighted moments for backgroundsubtracted and efficiencycorrected $ B ^ \rightarrow D ^+ K ^ \pi ^ $ data (black points) as a function of $m( D ^+ \pi ^ )$ in the range $2.0$$3.0\mathrm{ Ge V} $. Candidates from both TOS and TISonly subsamples are included. The blue line shows the result of the DP fit described in Sec. 7. 
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Distribution of $ B ^ \rightarrow D ^+ K ^ \pi ^ $ candidates in the signal region over (left) the DP and (right) the SDP. Candidates from both TOS and TISonly subsamples are included. 
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Square Dalitz plot distributions used in the Dalitz plot fit for (top) combinatorial background, (middle) $ B ^ \rightarrow D^{(*)+}\pi ^ \pi ^ $ decays and (bottom) $ B ^ \rightarrow D ^+_ s K ^ \pi ^ $ decays. Candidates the TOS (TISonly) subsamples are shown in the left (right) column. 
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Differences between the data SDP distribution and the fit model across the SDP, in terms of the perbin pull. 
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Projections of the data and amplitude fit onto (a) $m(D\pi)$, (c) $m(DK)$ and (e) $m(K\pi)$, with the same projections shown in (b), (d) and (f) with a logarithmic $y$axis scale. Components are described in the legend. 
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Projections of the data and amplitude fit onto $m(D\pi)$ in (a) the threshold region, (b) the $D^*_2(2460)^0$ region and (c) the $D^*_1(2760)^0$ region. Components are as shown in Fig. 8. 
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Projections of the data and amplitude fit onto the cosine of the helicity angle for the $D\pi$ system in (a) the threshold region, (b) the $D^*_2(2460)^0$ region and (c) the $D^*_1(2760)^0$ region. Components are as shown in Fig. 8. 
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Animated gif made out of all figures. 
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Measured properties of neutral $D^{**}$ states. Where more than one uncertainty is given, the first is statistical and the others systematic. 
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Yields of the various components in the fit to $ B ^ \rightarrow D ^+ \pi ^ \pi ^ $ candidate invariant mass distribution. 
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Yields of the various components in the fit to $ B ^ \rightarrow D ^+ K ^ \pi ^ $ candidate invariant mass distribution. 
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Relative systematic uncertainties on the measurement of the ratio of branching fractions for $ B ^ \rightarrow D ^+ K ^ \pi ^ $ and $ B ^ \rightarrow D ^+ \pi ^ \pi ^ $ decays. 
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Signal contributions to the fit model, where parameters and uncertainties are taken from Ref. [9]. States labelled with subscript $v$ are virtual contributions. 
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Masses and widths determined in the fit to data, with statistical uncertainties only. 
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Complex coefficients and fit fractions determined from the Dalitz plot fit. Uncertainties are statistical only. 
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Experimental systematic uncertainties on the fit fractions and complex amplitudes. 
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Model uncertainties on the fit fractions and complex amplitudes. 
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Breakdown of experimental systematic uncertainties on the fit fractions (%) and masses $(\mathrm{Me V} )$ and widths $(\mathrm{Me V} )$. 
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Breakdown of model uncertainties on the fit fractions (%) and masses $(\mathrm{Me V} )$ and widths $(\mathrm{Me V} )$. 
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Results for the complex amplitudes and their uncertainties. The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. 
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Results for the complex amplitudes and their uncertainties. The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. 
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Results for the fit fractions and their uncertainties (%). The three quoted errors are statistical, experimental systematic and model uncertainties, respectively. 
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Results for the product branching fractions ${\cal B}( B ^ \rightarrow R K ^ ) \times {\cal B}(R \rightarrow D ^+ \pi ^ )$ ($10^{6}$). The four quoted errors are statistical, experimental systematic, model and inclusive branching fraction uncertainties, respectively. 
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Results for the fit fractions and complex coefficients for the secondary minima with $2{\rm NLL}$ values 2.8 and 3.3 units greater than that of the global minimum of the NLL function. 
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Interference fit fractions (%) and statistical uncertainties. The amplitudes are: ($A_0$) $D^*_v(2007)^0$, ($A_1$) $D^*_0(2400)^0$, ($A_2$) $D^*_2(2460)^0$, ($A_3$) $D^*_1(2760)^0$, ($A_4$) $B^*_v$, ($A_5$) nonresonant Swave, ($A_6$) nonresonant Pwave. The diagonal elements are the same as the conventional fit fractions. 
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Experimental systematic uncertainies on the interference fit fractions (%). The amplitudes are: ($A_0$) $D^*_v(2007)^0$, ($A_1$) $D^*_0(2400)^0$, ($A_2$) $D^*_2(2460)^0$, ($A_3$) $D^*_1(2760)^0$, ($A_4$) $B^*_v$, ($A_5$) nonresonant Swave, ($A_6$) nonresonant Pwave. The diagonal elements are the same as the conventional fit fractions. 
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Model systematic uncertainies on the interference fit fractions (%). The amplitudes are: ($A_0$) $D^*_v(2007)^0$, ($A_1$) $D^*_0(2400)^0$, ($A_2$) $D^*_2(2460)^0$, ($A_3$) $D^*_1(2760)^0$, ($A_4$) $B^*_v$, ($A_5$) nonresonant Swave, ($A_6$) nonresonant Pwave. The diagonal elements are the same as the conventional fit fractions. 
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Created on 10 July 2020.