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Dalitz plot analysis of $B^0 \to \overline{D}^0 \pi^+\pi^-$ decays

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Abstract

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 $\pi^+\pi^-$ S-wave components are modelled with the Isobar and K-matrix 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 spin-parity 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.

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Examples of tree diagrams via $\bar{b} \rightarrow \bar{c}u\bar{d}$ transition to produce (a) $\pi ^+ \pi ^- $ resonances, (b) nonresonant three-body 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 $\pi^+\pi^-$ S-wave resonances. The various contributions do not add linearly due to interference effects.

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Projections of the data and K-matrix 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 $\pi^+\pi^-$ S-wave resonances. The various contributions do not add linearly due to interference effects.

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Comparison of the $\pi^+\pi^-$ S-wave obtained from the Isobar model and the K-matrix model, for (a) amplitudes and (b) phases. The K-matrix 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 K-matrix 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 K-matrix 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 K-matrix 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 background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ 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 background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ 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 background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ 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 background-subtracted and efficiency-corrected $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ 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.

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Tables and captions

Results of the fit to the invariant mass distribution of $B^0 \rightarrow \overline{ D }{} {}^0 \pi^+\pi^-$ candidates. Uncertainties are statistical only.

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The K-matrix 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{\,GeV\!/}c^2}$ .

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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 K-matrix formalism are described in Sec. 4.

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Systematic uncertainties on $\cal B ( B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^- )$.

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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{\,MeV\!/}c^2}$ ) and widths ($\Gamma$ in $\mathrm{\,MeV}$ ) 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 model-dependent systematic uncertainties, respectively.

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The moduli of the complex coefficients of the resonant contributions for the Isobar model and the K-matrix model. The first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively.

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The phase of the complex coefficients of the resonant contributions for the Isobar model and the K-matrix model. The first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively.

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The fit fractions of the resonant contributions for the Isobar and K-matrix models with $m(\overline{ D }{} {}^0 \pi^{\pm})> 2.1$ $ {\mathrm{\,GeV\!/}c^2}$ . The first uncertainty is statistical, the second and the third are experimental and model-dependent 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 K-matrix models. The first uncertainty is statistical, the second the experimental systematic, the third the model-dependent 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.

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Results of $R_{D\rho}$ and $\cos\delta_{D\rho}$.

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Systematic uncertainties on the $\overline{ D }{} {}^0 \pi ^- $ resonant masses (MeV/$c^2$) and widths (MeV) for the Isobar model.

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Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the Isobar model. The moduli are normalised to that of $\rho(770)$.

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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.

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Systematic uncertainties on the $\overline{ D }{} {}^0 \pi ^- $ resonant masses (MeV/$c^2$) and widths (MeV) for the K-matrix model.

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Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the K-matrix model. The moduli are normalised to that of $\rho(770)$.

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Systematic uncertainties on the phases ($^{\circ}$) of the complex coefficients of the resonant contributions for the K-matrix 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 K-matrix model.

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Interference fit fractions (%) of the resonant contributions for the Isobar model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the fit fractions given in Table 9.

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Interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the fit fractions given in Table 9.

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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{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9.

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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{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9.

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Model-dependent 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{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) nonresonant S-wave, ($A_1$) $f_0(500)$, ($A_2$) $f_0(980)$, ($A_3$) $f_0(2020)$, ($A_4$) $\rho(770)$, ($A_5$) $\omega(782)$, ($A_6$) $\rho(1450)$, $(A_7)$ $\rho(1700)$, $(A_8)$ $f_2(1270)$, $(A_9)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{10})$ $D_0^*(2400)^-$, $(A_{11})$ $D_2^*(2460)^-$, $(A_{12})$ $D_3^*(2760)^-$. The diagonal elements correspond to the statistical uncertainties on the fit fractions given in Table 9.

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Statistical uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{\,GeV\!/}c^2}$ . The resonances are: ($A_0$)K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9.

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Experimental systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9.

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Model-dependent systematic uncertainties on the interference fit fractions (%) of the resonant contributions for the K-matrix model with $m(\overline{ D }{} {}^0 \pi^{\pm})>2.1$ $ {\mathrm{\,GeV\!/}c^2}$ . The resonances are: ($A_0$) K-matrix S-wave, ($A_1$) $\rho(770)$, ($A_2$) $\omega(782)$, ($A_3$) $\rho(1450)$, $(A_4)$ $\rho(1700)$, $(A_5)$ $f_2(1270)$, $(A_6)$ $\overline{ D }{} {}^0 \pi^-$ P-wave, $(A_{7})$ $D_0^*(2400)^-$, $(A_{8})$ $D_2^*(2460)^-$, $(A_{9})$ $D_3^*(2760)^-$ The diagonal elements correspond to the statistical uncertainties on the fit fractions shown in Table 9.

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The moduli and phases of the K-matrix parameters. The first uncertainty is statistical, the second the experimental systematic, and the third the model-dependent systematic. The moduli are normalised to that of the $\rho(770)$ contribution and the phase of $\rho(770)$ is set to 0$^{\circ}$.

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Systematic uncertainties on the moduli of the K-matrix parameters. The moduli are normalised to that of $\rho(770)$.

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Systematic uncertainties on the phases ($^{\circ}$) of the K-matrix parameters. The phase of $\rho(700)$ is set to 0$^{\circ}$ as the reference.

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Created on 09 December 2018.Citation count from INSPIRE on 09 December 2018.