The Dalitz plot analysis technique is used to study the resonant substructures of $B^{} \to D^{+} \pi^{} \pi^{}$ decays in a data sample corresponding to 3.0 ${\rm fb}^{1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. A modelindependent analysis of the angular moments demonstrates the presence of resonances with spins 1, 2 and 3 at high $D^{+}\pi^{}$ mass. The data are fitted with an amplitude model composed of a quasimodelindependent function to describe the $D^{+}\pi^{}$ Swave together with virtual contributions from the $D^{*}(2007)^{0}$ and $B^{*0}$ states, and components corresponding to the $D^{*}_{2}(2460)^{0}$, $D^{*}_{1}(2680)^{0}$, $D^{*}_{3}(2760)^{0}$ and $D^{*}_{2}(3000)^{0}$ resonances. The masses and widths of these resonances are determined together with the branching fractions for their production in $B^{} \to D^{+} \pi^{} \pi^{}$ decays. The $D^{+}\pi^{}$ Swave has phase motion consistent with that expected due to the presence of the $D^{*}_{0}(2400)^{0}$ state. These results constitute the first observations of the $D^{*}_{3}(2760)^{0}$ and $D^{*}_{2}(3000)^{0}$ resonances.
Results of the fit to the $B$ candidate invariant mass distribution shown with (left) linear and (right) logarithmic $y$axis scales. Contributions are as described in the legend. 
Fig1a.pdf [31 KiB] HiDef png [272 KiB] Thumbnail [86 KiB] *.C file 

Fig1b.pdf [31 KiB] HiDef png [347 KiB] Thumbnail [99 KiB] *.C file 

Distribution of $ B ^ \rightarrow D ^+ \pi ^ \pi ^ $ candidates in the signal region over (left) the DP and (right) the SDP. 
Fig2a.pdf [626 KiB] HiDef png [422 KiB] Thumbnail [119 KiB] *.C file 

Fig2b.pdf [342 KiB] HiDef png [643 KiB] Thumbnail [171 KiB] *.C file 

The first seven unnormalised angular moments for backgroundsubtracted and efficiencycorrected data (black points) as a function of $m( D ^+ \pi ^ )$ in the range $2.0$$4.0\mathrm{ Ge V} $. The blue line shows the result of the DP fit described in Sec. 7. 
Fig3a.pdf [19 KiB] HiDef png [167 KiB] Thumbnail [59 KiB] *.C file 

Fig3b.pdf [20 KiB] HiDef png [195 KiB] Thumbnail [66 KiB] *.C file 

Fig3c.pdf [19 KiB] HiDef png [170 KiB] Thumbnail [59 KiB] *.C file 

Fig3d.pdf [21 KiB] HiDef png [192 KiB] Thumbnail [67 KiB] *.C file 

Fig3e.pdf [20 KiB] HiDef png [180 KiB] Thumbnail [63 KiB] *.C file 

Fig3f.pdf [22 KiB] HiDef png [184 KiB] Thumbnail [70 KiB] *.C file 

Fig3g.pdf [23 KiB] HiDef png [179 KiB] Thumbnail [69 KiB] *.C file 

Unnormalised angular moments $\left\langle P_7 \right\rangle$ and $\left\langle P_8 \right\rangle$ for backgroundsubtracted and efficiencycorrected data (black points) as a function of $m( D ^+ \pi ^ )$ in the range $2.0$$4.0\mathrm{ Ge V} $. The blue line shows the result of the DP fit described in Sec. 7. 
Fig4a.pdf [24 KiB] HiDef png [186 KiB] Thumbnail [72 KiB] *.C file 

Fig4b.pdf [25 KiB] HiDef png [182 KiB] Thumbnail [71 KiB] *.C file 

Zoomed views of the fourth and sixth unnormalised angular moments for backgroundsubtracted and efficiencycorrected data (black points) as a function of $m( D ^+ \pi ^ )$. The blue line shows the result of the DP fit described in Sec. 7. 
Fig5a.pdf [17 KiB] HiDef png [163 KiB] Thumbnail [57 KiB] *.C file 

Fig5b.pdf [19 KiB] HiDef png [169 KiB] Thumbnail [65 KiB] *.C file 

Signal efficiency across the SDP for $ B ^ \rightarrow D ^+ \pi ^ \pi ^ $ decays. The relative uncertainty at each point is typically $5 \%$. 
Fig6.pdf [59 KiB] HiDef png [383 KiB] Thumbnail [146 KiB] *.C file 

Square Dalitz plot distributions for (left) combinatorial background and (right) $ B ^ \rightarrow D^{(*)+} K ^ \pi ^ $ decays. 
Fig7a.pdf [48 KiB] HiDef png [151 KiB] Thumbnail [57 KiB] *.C file 

Fig7b.pdf [50 KiB] HiDef png [157 KiB] Thumbnail [61 KiB] *.C file 

Real and imaginary parts of the Swave amplitude, shown in an Argand diagram. The knots are shown with statistical uncertainties only, connected by the cubic spline interpolation used in the fit. The leftmost point is that at the lowest value of $m( D ^+ \pi ^ )$, with mass increasing along the connected points. Each point, labelled 113, corresponds to the position of a knot in the spline, at values of $m( D ^+ \pi ^ ) = \{ 2.01, 2.10, 2.20, 2.30, 2.40, 2.50, 2.60, 2.70, 2.80, 2.90, 3.10, 4.10, 5.14 \} \mathrm{ Ge V} $. The points at $(0.5,0.0)$ and $(0.0,0.0)$ are fixed. The anticlockwise rotation of the phase at low $m( D ^+ \pi ^ )$ is as expected due to the presence of the $ D ^{*}_{0}(2400)^{0}$ resonance. 
Fig8.pdf [55 KiB] HiDef png [144 KiB] Thumbnail [63 KiB] *.C file 

Differences between the SDP distribution of the data and fit model, in terms of the normalised residual in each bin. No bin lies outside the $z$axis limits. 
Fig9.pdf [53 KiB] HiDef png [168 KiB] Thumbnail [66 KiB] *.C file 

Projections of the data and amplitude fit onto (top) $m( D ^+ \pi ^ )_{\rm min}$, (middle) $m( D ^+ \pi ^ )_{\rm max}$ and (bottom) $m(\pi ^ \pi ^ )$, with the same projections shown (right) with a logarithmic $y$axis scale. Components are described in the legend. 
Fig10a.pdf [57 KiB] HiDef png [239 KiB] Thumbnail [69 KiB] *.C file 

Fig10b.pdf [57 KiB] HiDef png [388 KiB] Thumbnail [98 KiB] *.C file 

Fig10c.pdf [59 KiB] HiDef png [327 KiB] Thumbnail [82 KiB] *.C file 

Fig10d.pdf [58 KiB] HiDef png [449 KiB] Thumbnail [108 KiB] *.C file 

Fig10e.pdf [56 KiB] HiDef png [292 KiB] Thumbnail [75 KiB] *.C file 

Fig10f.pdf [54 KiB] HiDef png [371 KiB] Thumbnail [91 KiB] *.C file 

Fig10g.pdf [12 KiB] HiDef png [83 KiB] Thumbnail [25 KiB] *.C file 

Projections of the data and amplitude fit onto (left) $m( D ^+ \pi ^ )$ and (right) the cosine of the helicity angle for the $ D ^+ \pi ^ $ system in (top to bottom) the low mass threshold region, the $ D ^{*}_{2}(2460)^{0}$ region, the $ D ^{*}_{1}(2680)^{0}$  $ D ^{*}_{3}(2760)^{0}$ region and the $ D ^{*}_{2}(3000)^{0}$ region. Components are as shown in Fig. 10. 
Fig11a.pdf [52 KiB] HiDef png [213 KiB] Thumbnail [69 KiB] *.C file 

Fig11b.pdf [66 KiB] HiDef png [222 KiB] Thumbnail [71 KiB] *.C file 

Fig11c.pdf [55 KiB] HiDef png [245 KiB] Thumbnail [74 KiB] *.C file 

Fig11d.pdf [69 KiB] HiDef png [260 KiB] Thumbnail [74 KiB] *.C file 

Fig11e.pdf [56 KiB] HiDef png [264 KiB] Thumbnail [86 KiB] *.C file 

Fig11f.pdf [71 KiB] HiDef png [350 KiB] Thumbnail [94 KiB] *.C file 

Fig11g.pdf [55 KiB] HiDef png [281 KiB] Thumbnail [80 KiB] *.C file 

Fig11h.pdf [70 KiB] HiDef png [328 KiB] Thumbnail [85 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2016026.gif Thumbnail 
Measured properties of neutral excited charm states. World averages are given for the 1P resonances (top part), while all measurements are listed for the heavier states (bottom part). Where two uncertainties are given, the first is statistical and second systematic; where a third is given, it is due to model uncertainty. The uncertainties on the averages for the $ D ^{*}_{0}(2400)^{0}$ mass and the $D_1(2420)^0$ and $ D ^{*}_{2}(2460)^{0}$ masses and widths are inflated by scale factors to account for inconsistencies between measurements. The quoted $ D ^{*}_{2}(2460)^{0}$ averages do not include the recent result from Ref. [12]. 
Table_1.pdf [53 KiB] HiDef png [120 KiB] Thumbnail [20 KiB] tex code 

Yields of the various components in the fit to $ B ^ \rightarrow D ^+ \pi ^ \pi ^ $ candidate invariant mass distribution. Note that the yields in the signal region are scaled from the full mass range. 
Table_2.pdf [46 KiB] HiDef png [59 KiB] Thumbnail [10 KiB] tex code 

Signal contributions to the fit model, where parameters and uncertainties are taken from Ref. [19]. States labelled with subscript $v$ are virtual contributions. The model "MIPW" refers to the quasimodelindependent partial wave approach. 
Table_3.pdf [55 KiB] HiDef png [82 KiB] Thumbnail [13 KiB] tex code 

Masses and widths determined in the fit to data, with statistical uncertainties only. 
Table_4.pdf [44 KiB] HiDef png [97 KiB] Thumbnail [15 KiB] tex code 

Complex coefficients and fit fractions determined from the Dalitz plot fit. Uncertainties are statistical only. 
Table_5.pdf [56 KiB] HiDef png [92 KiB] Thumbnail [14 KiB] tex code 

Breakdown of experimental systematic uncertainties on the fit fractions (%) and masses and widths $(\mathrm{Me V} )$. 
Table_6.pdf [62 KiB] HiDef png [164 KiB] Thumbnail [27 KiB] tex code 

Breakdown of model uncertainties on the fit fractions (%) and masses and widths $(\mathrm{Me V} )$. 
Table_7.pdf [61 KiB] HiDef png [147 KiB] Thumbnail [22 KiB] tex code 

Results for the complex amplitudes. The three quoted errors are statistical, experimental systematic and model uncertainties. 
Table_8.pdf [55 KiB] HiDef png [203 KiB] Thumbnail [34 KiB] tex code 

Results for the \DpiSwave amplitude at the spline knots. The three quoted errors are statistical, experimental systematic and model uncertainties. 
Table_9.pdf [47 KiB] HiDef png [275 KiB] Thumbnail [54 KiB] tex code 

Results for the fit fractions. The three quoted errors are statistical, experimental systematic and model uncertainties. 
Table_10.pdf [52 KiB] HiDef png [159 KiB] Thumbnail [26 KiB] tex code 

Results for the product branching fractions ${\cal B}( B ^ \rightarrow R\pi ^ ) \times {\cal B}(R \rightarrow D ^+ \pi ^ )$. The four quoted errors are statistical, experimental systematic, model and inclusive branching fraction uncertainties. 
Table_11.pdf [52 KiB] HiDef png [146 KiB] Thumbnail [25 KiB] tex code 

Interference fit fractions (%) and statistical uncertainties. The amplitudes are: ($A_0$) $ D ^{*}_{v}(2007)^{0}$ , ($A_1$) \DpiSwave , ($A_2$) $ D ^{*}_{2}(2460)^{0}$ , ($A_3$) $ D ^{*}_{1}(2680)^{0}$ , ($A_4$) $B^{*0}_v$, ($A_5$) $ D ^{*}_{3}(2760)^{0}$ , ($A_6$) $ D ^{*}_{2}(3000)^{0}$ . The diagonal elements are the same as the conventional fit fractions. 
Table_12.pdf [33 KiB] HiDef png [46 KiB] Thumbnail [8 KiB] tex code 

(Top) Experimental and (bottom) model systematic uncertainties on the interference fit fractions (%). The amplitudes are: ($A_0$) $ D ^{*}_{v}(2007)^{0}$ , ($A_1$) \DpiSwave , ($A_2$) $ D ^{*}_{2}(2460)^{0}$ , ($A_3$) $ D ^{*}_{1}(2680)^{0}$ , ($A_4$) $B^{*0}_{v}$, ($A_5$) $ D ^{*}_{3}(2760)^{0}$ , ($A_6$) $ D ^{*}_{2}(3000)^{0}$ . The diagonal elements are the same as the conventional fit fractions. 
Table_13.pdf [27 KiB] HiDef png [56 KiB] Thumbnail [8 KiB] tex code 
Created on 14 June 2019.Citation count from INSPIRE on 14 June 2019.