The resonant substructures of $B^0 \to \overline{D}^0 \pi^+\pi^$ decays are studied with the Dalitz plot technique. In this study a data sample corresponding to an integrated luminosity of 3.0 fb$^{1}$ of $pp$ collisions collected by the LHCb detector is used. The branching fraction of the $B^0 \to \overline{D}^0 \pi^+\pi^$ decay in the region $m(\overline{D}^0\pi^{\pm})>2.1$ GeV$/c^2$ is measured to be $(8.46 \pm 0.14 \pm 0.29 \pm 0.40) \times 10^{4}$, where the first uncertainty is statistical, the second is systematic and the last arises from the normalisation channel $B^0 \to D^*(2010)^\pi^+$. The $\pi^+\pi^$ Swave components are modelled with the Isobar and Kmatrix formalisms. Results of the Dalitz plot analyses using both models are presented. A resonant structure at $m(\overline{D}^0\pi^) \approx 2.8$ GeV$/c^{2}$ is confirmed and its spinparity is determined for the first time as $J^P = 3^$. The branching fraction, mass and width of this structure are determined together with those of the $D^*_0(2400)^$ and $D^*_2(2460)^$ resonances. The branching fractions of other $B^0 \to \overline{D}^0 h^0$ decay components with $h^0 \to \pi^+\pi^$ are also reported. Many of these branching fraction measurements are the most precise to date. The first observation of the decays $B^0 \to \overline{D}^0 f_0(500)$, $B^0 \to \overline{D}^0 f_0(980)$, $B^0 \to \overline{D}^0 \rho(1450)$, $B^0 \to D_3^*(2760)^ \pi^+$ and the first evidence of $B^0 \to \overline{D}^0 f_0(2020)$ are presented.
Examples of tree diagrams via $\bar{b} \rightarrow \bar{c}u\bar{d}$ transition to produce (a) $\pi ^+ \pi ^ $ resonances, (b) nonresonant threebody decay and (c) $\overline{ D }{} {}^0 \pi ^ $ resonances. 
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Invariant mass distribution of $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^$ candidates. Data points are shown in black. The fit is shown as a solid (red) line with the background component displayed as dashed (green) line. 
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Density profile of the combinatorial background events in the Dalitz plane obtained from the upper $m(\overline{ D }{} {}^0 \pi^+\pi^)$ sideband with a looser selection applied on the Fisher discriminant. 
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Efficiency function for the Dalitz variables obtained in a fit to the LHCb simulated samples. 
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Dalitz plot distribution of candidates in the signal region, including background contributions. The red line shows the Dalitz plot kinematic boundary. 
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Projections of the data and Isobar fit onto (a) $m^2(\pi^+\pi^)$ and (c) $m^2(\overline{ D }{} {}^0 \pi^)$ with a linear scale. Same projections shown in (b) and (d) with a logarithmic scale. Components are described in the legend. The lines denoted $\overline{ D }{} {}^0 \pi^$ and $\pi^+\pi^$ include the coherent sums of all $\overline{ D }{} {}^0 \pi^$ resonances, $\pi^+\pi^$ resonances, and $\pi^+\pi^$ Swave resonances. The various contributions do not add linearly due to interference effects. 
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Projections of the data and Kmatrix fit onto (a) $m^2(\pi^+\pi^)$ and (c) $m^2(\overline{ D }{} {}^0 \pi^)$ with a linear scale. Same projections shown in (b) and (d) with a logarithmic scale. Components are described in the legend. The lines denoted $\overline{ D }{} {}^0 \pi^$ and $\pi^+\pi^$ include the coherent sums of all $\overline{ D }{} {}^0 \pi^$ resonances, $\pi^+\pi^$ resonances, and $\pi^+\pi^$ Swave resonances. The various contributions do not add linearly due to interference effects. 
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Comparison of the $\pi^+\pi^$ Swave obtained from the Isobar model and the Kmatrix model, for (a) amplitudes and (b) phases. The Kmatrix model is shown by the red solid line, two scenarios for the Isobar model with (black long dashed line) and without (blue dashed line) $f_0(1370)$ and $f_0(1500)$ are shown. 
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Distributions of $m^2(\pi^+\pi^)$ in the $\rho(770)$ mass region. The different fit components are described in the legend. Results from (a) the Isobar model and (b) the Kmatrix model are shown. 
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Distributions of $m^2(\overline{ D }{} {}^0 \pi ^ )$ in the $D_J^*(2760)^$ mass region. The different fit components are described in the legend. Both results from (a) the Isobar model and (b) the Kmatrix model are shown. 
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Invariant mass distributions of (a) $m(\overline{ D }{} {}^0 \pi^+\pi^)$ and (b) $m(\overline{ D }{} {}^0 \pi^)$ for $B^0 \rightarrow D^*(2010)^\pi^+$ candidates. The data is shown as black points with the fit superimposed as red solid lines. 
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Cosine of the helicity angle distributions in the $m^2(\overline{ D }{} {}^0 \pi^)$ range [7.4, 8.2] GeV$^2/c^4$ for (a) the Isobar model and (b) the Kmatrix model. The data are shown as black points. The helicity angle distributions of the Dalitz plot fit results, without the $D^*_J(2760)^$ and with the different spin hypotheses of $D^*_J(2760)^$, are superimposed. 
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Mixing angle as a function of form factor ratio for the (a) $q\bar{q}$ model and (b) $[qq'][\bar{q}\bar{q'}]$ tetraquark model. Green band gives 1$\sigma$ interval around central values (black solid line). 
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The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for backgroundsubtracted and efficiencycorrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^$ data and the Dalitz plot fit results as a function of $m^2(\overline{ D }{} {}^0 \pi^)$. 
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The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for backgroundsubtracted and efficiencycorrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^$ data and the Dalitz plot fit results as a function of $m^2(\pi^+\pi^)$. 
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The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for backgroundsubtracted and efficiencycorrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^$ data and the Dalitz plot fit results as a function of $m^2(\overline{ D }{} {}^0 \pi^)$. Only results in the region $m^2(\overline{ D }{} {}^0 \pi^)< 10$ GeV$^2$/$c^4$ are shown. 
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The first eight unnormalised Legendre polynomial weighted moments (0 to 7 correspond to (a) to (h)) for backgroundsubtracted and efficiencycorrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^$ data and the Dalitz plot fit results as a function of $m^2(\pi^+\pi^)$. Only results in the region $m^2(\pi^+\pi^)< 3$ GeV$^2$/$c^4$ are shown. 
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Animated gif made out of all figures. 
PAPER2014070.gif Thumbnail 
Results of the fit to the invariant mass distribution of $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^$ candidates. Uncertainties are statistical only. 
Table_1.pdf [59 KiB] HiDef png [87 KiB] Thumbnail [41 KiB] tex code 

The Kmatrix parameters used in this paper are taken from a global analysis of $\pi^+\pi^$ scattering data [22]. Masses and coupling constants are in units of $ {\mathrm{ Ge V /}c^2}$ . 
Table_2.pdf [58 KiB] HiDef png [121 KiB] Thumbnail [57 KiB] tex code 

Resonant contributions to the nominal fit models and their properties. Parameters and uncertainties of $\rho(770)$, $\omega(782)$, $\rho(1450)$ and $\rho(1700)$ come from Ref. [92], and those of $f_2(1270)$ and $f_0(2020)$ come from Ref. [32]. Parameters of $f_0(500)$, $f_0(980)$ and Kmatrix formalism are described in Sec. 4. 
Table_3.pdf [57 KiB] HiDef png [151 KiB] Thumbnail [75 KiB] tex code 

Systematic uncertainties on $\cal B ( B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ )$. 
Table_4.pdf [45 KiB] HiDef png [75 KiB] Thumbnail [36 KiB] tex code 

Statistical significance ($\sigma$) of $\pi^+\pi^$ resonances in the Dalitz plot analysis. For the statistically significant resonances, the effect of adding dominant systematic uncertainties is shown (see text). 
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Measured masses ($m$ in $ {\mathrm{ Me V /}c^2}$ ) and widths ($\Gamma$ in $\mathrm{ Me V}$ ) of the $D_0^*(2400)^$, $D_2^*(2460)^$ and $D_3^*(2760)^$ resonances, where the first uncertainty is statistical, the second and the third are experimental and modeldependent systematic uncertainties, respectively. 
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The moduli of the complex coefficients of the resonant contributions for the Isobar model and the Kmatrix model. The first uncertainty is statistical, the second and the third are experimental and modeldependent systematic uncertainties, respectively. 
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The phase of the complex coefficients of the resonant contributions for the Isobar model and the Kmatrix model. The first uncertainty is statistical, the second and the third are experimental and modeldependent systematic uncertainties, respectively. 
Table_8.pdf [54 KiB] HiDef png [153 KiB] Thumbnail [81 KiB] tex code 

The fit fractions of the resonant contributions for the Isobar and Kmatrix models with $m(\overline{ D }{} {}^0 \pi^{\pm})> 2.1$ $ {\mathrm{ Ge V /}c^2}$ . The first uncertainty is statistical, the second and the third are experimental and modeldependent systematic uncertainties, respectively. 
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Correction factors due to the $D^*(2010)^$ veto. 
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Measured branching fractions of $\cal B ( B ^0 \rightarrow r h_3) \times \cal B (r \rightarrow h_1 h_2)$ for the Isobar and Kmatrix models. The first uncertainty is statistical, the second the experimental systematic, the third the modeldependent systematic, and the fourth the uncertainty from the normalisation $B^0 \rightarrow D^*(2010)^ \pi^+$ channel. 
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Systematic uncertainties on $r^f$. The sum in quadrature of the uncertainties is also reported. 
Table_12.pdf [47 KiB] HiDef png [279 KiB] Thumbnail [111 KiB] tex code 

Results of $R_{D\rho}$ and $\cos\delta_{D\rho}$. 
Table_13.pdf [51 KiB] HiDef png [67 KiB] Thumbnail [29 KiB] tex code 

Systematic uncertainties on the $\overline{ D }{} {}^0 \pi ^ $ resonant masses (MeV/$c^2$) and widths (MeV) for the Isobar model. 
Table_14.pdf [40 KiB] HiDef png [180 KiB] Thumbnail [90 KiB] tex code 

Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the Isobar model. The moduli are normalised to that of $\rho(770)$. 
Table_15.pdf [40 KiB] HiDef png [228 KiB] Thumbnail [104 KiB] tex code 

Systematic uncertainties on the phases ($^{\circ}$) of the complex coefficients of the resonant contributions for the Isobar model. The phase of $\rho(700)$ is set to $0^{\circ}$ as the reference. 
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Systematic uncertainties on the fit fractions (%) of the resonant contributions for the Isobar model. 
Table_17.pdf [40 KiB] HiDef png [196 KiB] Thumbnail [87 KiB] tex code 

Systematic uncertainties on the $\overline{ D }{} {}^0 \pi ^ $ resonant masses (MeV/$c^2$) and widths (MeV) for the Kmatrix model. 
Table_18.pdf [40 KiB] HiDef png [156 KiB] Thumbnail [78 KiB] tex code 

Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the Kmatrix model. The moduli are normalised to that of $\rho(770)$. 
Table_19.pdf [40 KiB] HiDef png [219 KiB] Thumbnail [105 KiB] tex code 

Systematic uncertainties on the phases ($^{\circ}$) of the complex coefficients of the resonant contributions for the Kmatrix model. The phase of $\rho(700)$ is set to $0^{\circ}$ as reference. 
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Systematic uncertainties on the fit fractions (%) of the resonant contributions for the Kmatrix model. 
Table_21.pdf [40 KiB] HiDef png [201 KiB] Thumbnail [95 KiB] tex code 

Interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant Swave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{10})$ $D_0^*(2400)^$, $(A_{11})$ $D_2^*(2460)^$, $(A_{12})$ $D_3^*(2760)^$. The diagonal elements correspond to the fit fractions given in Table 9. 
Table_22.pdf [33 KiB] HiDef png [58 KiB] Thumbnail [27 KiB] tex code 

Interference fit fractions (%) of the resonant contributions for the Kmatrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) Kmatrix Swave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{7})$ $D_0^*(2400)^$, $(A_{8})$ $D_2^*(2460)^$, $(A_{9})$ $D_3^*(2760)^$ The diagonal elements correspond to the fit fractions given in Table 9. 
Table_23.pdf [33 KiB] HiDef png [55 KiB] Thumbnail [28 KiB] tex code 

Statistical uncertainties on the interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant Swave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{10})$ $D_0^*(2400)^$, $(A_{11})$ $D_2^*(2460)^$, $(A_{12})$ $D_3^*(2760)^$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9. 
Table_24.pdf [47 KiB] HiDef png [88 KiB] Thumbnail [44 KiB] tex code 

Experimental systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant Swave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{10})$ $D_0^*(2400)^$, $(A_{11})$ $D_2^*(2460)^$, $(A_{12})$ $D_3^*(2760)^$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9. 
Table_25.pdf [33 KiB] HiDef png [88 KiB] Thumbnail [44 KiB] tex code 

Modeldependent systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) nonresonant Swave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{10})$ $D_0^*(2400)^$, $(A_{11})$ $D_2^*(2460)^$, $(A_{12})$ $D_3^*(2760)^$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9. 
Table_26.pdf [33 KiB] HiDef png [92 KiB] Thumbnail [45 KiB] tex code 

Statistical uncertainties on the interference fit fractions (%) of the resonant contributions for the Kmatrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$)Kmatrix Swave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{7})$ $D_0^*(2400)^$, $(A_{8})$ $D_2^*(2460)^$, $(A_{9})$ $D_3^*(2760)^$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9. 
Table_27.pdf [33 KiB] HiDef png [79 KiB] Thumbnail [41 KiB] tex code 

Experimental systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the Kmatrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) Kmatrix Swave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{7})$ $D_0^*(2400)^$, $(A_{8})$ $D_2^*(2460)^$, $(A_{9})$ $D_3^*(2760)^$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9. 
Table_28.pdf [33 KiB] HiDef png [80 KiB] Thumbnail [41 KiB] tex code 

Modeldependent systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the Kmatrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{ Ge V /}c^2}$ . The resonances are: ($A_0$) Kmatrix Swave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^$ Pwave, $(A_{7})$ $D_0^*(2400)^$, $(A_{8})$ $D_2^*(2460)^$, $(A_{9})$ $D_3^*(2760)^$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9. 
Table_29.pdf [33 KiB] HiDef png [78 KiB] Thumbnail [41 KiB] tex code 

The moduli and phases of the Kmatrix parameters. The first uncertainty is statistical, the second the experimental systematic, and the third the modeldependent systematic. The moduli are normalised to that of the $\rho(770)$ contribution and the phase of $\rho(770)$ is set to 0$^{\circ}$. 
Table_30.pdf [43 KiB] HiDef png [117 KiB] Thumbnail [67 KiB] tex code 

Systematic uncertainties on the moduli of the Kmatrix parameters. The moduli are normalised to that of $\rho(770)$. 
Table_31.pdf [40 KiB] HiDef png [115 KiB] Thumbnail [58 KiB] tex code 

Systematic uncertainties on the phases ($^{\circ}$) of the Kmatrix parameters. The phase of $\rho(700)$ is set to 0$^{\circ}$ as the reference. 
Table_32.pdf [40 KiB] HiDef png [111 KiB] Thumbnail [58 KiB] tex code 
Created on 18 October 2019.