Dalitz plot analysis of B 0 → D̄ 0 π þ π − decays

The resonant substructures of B0→D¯0π+π− 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 B0→D¯0π+π− decay in the region m(D¯0π±)2.1  GeV/c2 is measured to be (8.46±0.14±0.29±0.40)×10−4, where the first uncertainty is statistical, the second is systematic and the last arises from the normalization channel B0→D∗(2010)−π+. The π+π− S-wave components are modeled with the isobar and K-matrix formalisms. Results of the Dalitz plot analyses using both models are presented. A resonant structure at m(D¯0π−)≈2.8  GeV/c2 is confirmed and its spin-parity is determined for the first time as JP=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 B0→D¯0h0 decay components with h0→π+π− are also reported. Many of these branching fraction measurements are the most precise to date. The first observation of the decays B0→D¯0f0(500), B0→D¯0f0(980), B0→D¯0ρ(1450), B0→D∗3(2760)−π+ and the first evidence of B0→D¯0f0(2020) are presented.


I. INTRODUCTION
The study of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism [1,2] is a central topic in flavor physics.Accurate measurements of the various CKM matrix parameters through different processes provide sensitivity to new physics effects, by testing the global consistency of the Standard Model.Among them, the CKM angle β is expressed in terms of the CKM matrix elements as argð−V cd V Ã cb =V td V Ã tb Þ.The most precise measurements have been obtained with the B 0 → ðccÞK ðÃÞ0 decays by BABAR [3], Belle [4] and more recently by LHCb [5].The decay 1 B 0 → D0 π þ π − through the b → c ūd transition has sensitivity to the CKM angle β [6][7][8][9][10] and to new physics effects [11][12][13][14].
The Dalitz plot analysis [15] of B 0 → D0 π þ π − decays, with the D0 → K þ π − mode, is presented as the first step towards an alternative method to measure the CKM angle β.Two sets of results are given, where the π þ π − S-wave components are modeled with the isobar [16][17][18] and K-matrix [19] formalisms.Dalitz plot analyses of the decay B 0 → D0 π þ π − have already been performed by Belle [20,21] and BABAR [22].Similar studies for the charged B decays B − → D ðÃÞþ π − π − have been published by the B-factories [23,24].The LHCb data set offers a larger and almost pure signal sample.Feynman diagrams of the dominant tree level amplitudes contributing to the decay B 0 → D0 π þ π − are shown in Fig. 1.
In addition to the interest for the CKM parameter measurements, the analysis of the Dalitz plot of the B 0 → D0 π þ π − decay is motivated by its rich resonant structure.
The decay B 0 → D0 π þ π − contains information about excited D mesons decaying to Dπ, with natural spin and parity J P ¼ 0 þ ; 1 − ; 2 þ ; … A complementary Dalitz plot analysis of the decay B 0 s → D0 K − π þ was recently published by LHCb [25,26], and constrains the phenomenology of the D0 K − (D − sJ ) and K − π þ states.The spectrum of excited D mesons is predicted by theory [27,28] and contains the known states D Ã ð2010Þ; D Ã 0 ð2400Þ; D Ã 2 ð2460Þ, as well as other unknown states not yet fully explored.An extensive discussion on theory predictions for the c ū, c d and cs mass spectra is provided in Refs.[26,29].More recent measurements performed in inclusive decays by BABAR [30] and LHCb [29] have led to the observation of several new states: D Ã ð2650Þ; D Ã ð2760Þ, and D Ã ð3000Þ.However, their spin and parity are difficult to determine from inclusive studies.Orbitally excited D mesons have also been studied in semileptonic B decays (see a review in Ref. [31]) with limited precision.These are of prime interest both in the extraction of the CKM parameter jV cb j, where longstanding differences remain between exclusive and inclusive methods (see review in Ref. [32]), and in recent studies of B → D ðÃÞ τν τ [33] which have generated much theoretical discussion (see, e.g., Refs.[34,35]).
A measurement of the branching fraction of the decay B 0 → D0 ρ 0 is also presented.This study helps in understanding the effects of color suppression in B decays, which is due to the requirement that the color quantum numbers of the quarks produced from the virtual W boson must match those of the spectator quark to form a ρ 0 meson [36][37][38][39][40].Moreover, using isospin symmetry to relate the decay amplitudes of B 0 → D0 ρ 0 , B 0 → D − ρ þ and B þ → D0 ρ þ , effects of final-state interactions (FSI) can be studied in those decays (see a review in Refs.[37,41]).The previous measurement for the branching fraction of B 0 → D0 ρ 0 has limited precision, ð3.2 AE 0.5Þ × 10 −4 [21], and is in agreement with theoretical predictions that range from 1.7 to 3.4 × 10 −4 [38,42].
The analysis of the decay B 0 → D0 π þ π − presented in this paper is based on a data sample corresponding to an integrated luminosity of 3.0 fb −1 of pp collision data collected with the LHCb detector.Approximately one third of the data was obtained during 2011 when the collision center-of-mass energy was ffiffi ffi s p ¼ 7 TeV and the rest during 2012 with ffiffi ffi s p ¼ 8 TeV.
The paper is organized as follows.A brief description of the LHCb detector as well as the reconstruction and simulation software is given in Sec.II.The selection of signal candidates and the fit to the B 0 candidate invariant mass distribution used to separate and to measure signal and background yields are described in Sec.III.An overview of the Dalitz plot analysis formalism is given in Sec.IV.Details and results of the amplitude analysis fits are presented in Sec.V.In Sec.VI the measurement of the B 0 → D0 π þ π − branching fraction is documented.The evaluation of systematic uncertainties is described in Sec.VII.The results are given in Sec.VIII, and a summary concludes the paper in Sec.IX.

II. THE LHCB DETECTOR
The LHCb detector [59] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks.The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region [60], a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes [61] placed downstream of the magnet.The tracking system provides a measurement of momentum, p, with a relative uncertainty that varies from 0.4% at low momentum to 0.6% at 100 GeV=c.The minimum distance of a track to a primary vertex, the impact parameter (IP), is measured with a resolution of ð15 þ 29=p T Þ μm, where p T is the component of p transverse to the beam, in GeV=c.Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [62].Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter.Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [63].
The online event selection is performed by a trigger which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction.At the hardware trigger stage, events are required to have a muon with high p T or a hadron, photon or electron with high transverse energy in the calorimeters.For hadrons, the transverse energy threshold is 3.5 GeV.The software trigger requires a two-, three-or four-track secondary vertex with a significant displacement from the primary pp interaction vertices (PVs).At least one charged particle must have a transverse momentum p T > 1.7 GeV=c and be inconsistent with originating from a PV.A multivariate algorithm [64] is used for the identification of secondary vertices consistent with the decay of a b hadron.The p T of the photon from D Ã− s decay is too low to contribute to the trigger decision.
Simulated events are used to characterize the detector response to signal and certain types of background events.In the simulation, pp collisions are generated using PYTHIA [65] with a specific LHCb configuration [66].Decays of hadronic particles are described by EVTGEN [67], in which final-state radiation is generated using PHOTOS [68].The interaction of the generated particles with the detector and its response are implemented using the GEANT4 toolkit [69] as described in Ref. [70].

III. EVENT SELECTION
Signal B 0 candidates are formed by combining D0 candidates, reconstructed in the decay channel K þ π − , with two additional pion candidates of opposite charge.Reconstructed tracks are required to be of good quality and to be inconsistent with originating from a PV.They are also required to have sufficiently high p and p T and to be within kinematic regions where reasonable particle identification (PID) performance is achieved, as determined by calibration samples of D Ãþ → D 0 ðK − π þ Þπ þ decays.The four final-state tracks are required to be positively identified by the PID system.The D0 daughters are required to form a good quality vertex and to have an invariant mass within 100 MeV=c 2 of the known D0 mass [32].The D0 candidates and the two charged pion candidates are required to form a good vertex.The reconstructed D0 and B 0 vertices are required to be significantly displaced from the PV.To improve the B 0 candidate invariant mass resolution, a kinematic fit [71] is used, constraining the D0 candidate to its known mass [32].
By requiring the reconstructed D0 vertex to be displaced downstream from the reconstructed B 0 vertex, backgrounds from both charmless B decays and direct prompt charm production coming from the PV are reduced to a negligible level.Background from D Ã ð2010Þ − decays is removed by requiring mð D0 are also removed by this requirement.
To further distinguish signal from combinatorial background, a multivariate analysis based on a Fisher discriminant [72] is applied.The sPlot technique [73] is used to statistically separate signal and background events with the B 0 candidate mass used as the discriminating variable.Weights obtained from this procedure are applied to the candidates to obtain signal and background distributions that are used to train the discriminant.The Fisher discriminant uses information about the event kinematic properties, vertex quality, IP and p T of the tracks and flight distance from the PV.It is optimized by maximizing the purity of the signal events.
Signal candidates are retained for the Dalitz plot analysis if the invariant mass of the B 0 meson lies in the range [5250, 5310] MeV=c 2 and that of the D0 meson in the range [1840, 1890] MeV=c 2 (called the signal region).Once all selection requirements are applied, less than 1% of the events contain multiple candidates, and in those cases one candidate is chosen randomly.
Background contributions from decays with the same topology, but having one or two misidentified particles, are estimated to be less than 1% and are not considered in the Dalitz analysis.These background contributions include decays like Partially reconstructed decays of the type B 0 → D0 π þ π − X, where one or more particles are not reconstructed, have similar efficiencies to the signal channel decays.They are distributed in the region below the B 0 mass.By requiring the invariant mass of B 0 candidates to be larger than 5250 MeV=c 2 , these backgrounds are reduced to a negligible level, as determined by simulated samples of B 0 → DÃ0 π þ π − and B 0 → DÃ0 ρð770Þ with DÃ0 decaying into D0 γ or D0 π 0 under different hypotheses for the DÃ0 helicity.
The signal and combinatorial background yields are determined using an unbinned extended maximum likelihood fit to the invariant mass distribution of B 0 candidates.The invariant mass distribution is shown in Fig. 2, with the fit result superimposed.The fit uses a Crystal Ball (CB) function [76] convoluted with a Gaussian function for the signal distribution and a linear function for the combinatorial background distribution in the mass range of [5250, 5500] MeV=c.Simulated studies validate this choice of signal shape and the tail parameters of the CB function are fixed to those determined from simulation.Table I summarizes the fit results on the free parameters, where μ B 0 is the mean peak position and σ G is the width of the Gaussian function.The parameter σ CB is the width of the Gaussian core of the CB function.The parameters f CB and p 1 give the fit fraction of the CB function and the slope of the linear function that describes the background distribution.The yields of signal (ν 0 s ) and background (ν 0 b ) events given in Table I are calculated within the signal region.The purity is ð97.8AE 0.2Þ%.

IV. DALITZ PLOT ANALYSIS FORMALISM
The analysis of the distribution of decays across the Dalitz plot [15] allows a determination of the amplitudes contributing to the three-body B 0 → D0 π þ π − decay.Two of the three possible two-body invariant mass-squared combinations, which are connected by are sufficient to describe the kinematics of the system.The two observables m 2 ð D0 π − Þ and m 2 ðπ þ π − Þ, where resonances are expected to appear, are chosen in this paper.These observables are calculated with the masses of the B 0 and D0 mesons constrained to their known values [32].The invariant mass resolution has negligible effect and therefore it is not modeled in the Dalitz plot analysis.
The total decay amplitude is described by a coherent sum of amplitudes from resonant or nonresonant intermediate processes as The complex coefficient c i and amplitude A i ðxÞ describe the relative contribution and dynamics of the ith intermediate state, where x represents the ðm 2 ð D0 π − Þ; m 2 ðπ þ π − ÞÞ coordinates in the Dalitz plot.The Dalitz plot analysis determines the coefficients c i .In addition, fit fractions and interference fit fractions are also calculated to give a convention-independent representation of the population of the Dalitz plot.The fit fractions are defined as and the interference fit fractions between the resonances i and j (i < j) are defined as where the integration is performed over the full Dalitz plot with mð D0 π AE Þ > 2.1 GeV=c.Due to these interferences  between different contributions, the sum of the fit fractions is not necessarily equal to unity.
The amplitude A i ðxÞ for a specific resonance r with spin L is written as The functions F ðLÞ B ðq; q 0 Þ and F ðLÞ r ðp; p 0 Þ are the Blatt-Weisskopf barrier factors [77] for the production, B 0 → rh 3 , and the decay, r → h 1 h 2 , of the resonance, respectively.The parameters p and q are the momenta of one of the resonance daughters (h 1 or h 2 ) and of the bachelor particle (h 3 ), respectively, both evaluated in the rest frame of the resonance.The value p 0 (q 0 ) represents the value of p (q) when the invariant mass of the resonance is equal to its pole mass.The spin-dependent F B and F r functions are defined as where z ð0Þ is equal to ðr BW × q ð0Þ Þ 2 or ðr BW × p ð0Þ Þ 2 .The value for the radius of the resonance, r BW , is taken to be 1.6 GeV −1 × ℏc (¼ 0.3 fm) [78].The function T L ðxÞ represents the angular distribution for the decay of a spin L resonance.It is defined as The helicity angle, θ, of the resonance is defined as the angle between the direction of the momenta p and q.The y dependence accounts for relativistic transformations between the B 0 and the resonance rest frames [79,80], where Finally, RðxÞ is the resonant line shape and is described by the relativistic Breit-Wigner (RBW) function unless specified otherwise, where s ¼ m 2 ðh 1 h 2 Þ and m r is the pole mass of the resonance; Γ ðLÞ ðsÞ, the mass-dependent width, is defined as where Γ 0 is the partial width of the resonance, i.e., the width at the peak mass s ¼ m r .
The line shapes of ρð770Þ, ρð1450Þ and ρð1700Þ are described by the Gounaris-Sakurai (GS) function [81], where The ρ − ω interference is taken into account by where Γ 0 is used, instead of the mass-dependent width Γ ðLÞ ðsÞ, for ωð782Þ [82].The D Ã ð2010Þ − contribution is vetoed as described in Sec.III.Possible remaining contributions from the D Ã ð2010Þ − RBW tail or general D0 π − P-waves are modeled as where β 1 and β 2 are free parameters.
The π þ π − S-wave contribution is modeled using two alternative approaches, the isobar model [16][17][18] or the K-matrix model [19].Contributions from the f 0 ð500Þ, f 0 ð980Þ, f 0 ð2020Þ resonances and a nonresonant component are parametrized separately in the isobar model and globally by one amplitude in the K-matrix model.
In the isobar model, the f 0 ð2020Þ resonance is modeled by a RBW function and the modeling of the f 0 ð500Þ, f 0 ð980Þ resonances and the nonresonant contribution are described as follows.The Bugg resonant line shape [83] is employed for the f 0 ð500Þ contribution, where The parameters are fixed to m r ¼ 0.953 426 GeV 2 =c 4 and g 4π ¼ 0.011 GeV=c 2 [83].The phase-space factors of the decay channels ππ, KK and ηη correspond to ρ 1;2;3 ðsÞ, respectively, and are defined as 1;2;3 s s ; 1; 2; and 3 ¼ π; K and η: The Flatté formula [84] is used to describe the f 0 ð980Þ line shape, where The parameters g 1ð2Þ and m r [46] Its modulus equals unity, and a slowly varying phase over m 2 ðπ þ π − Þ accounts for rescattering effects of the π þ π − final state and α is a free parameter of the model.The K-matrix formalism [19] describes the production, rescattering and decay of the π þ π − S-wave in a coherent way.The scattering matrix S, from an initial state to a final state, is where I is the identity matrix, ρ is a diagonal phase-space matrix and T is the transition matrix.The unitarity requirement SS † ¼ I gives The K-matrix is a Lorentz-invariant Hermitian matrix, defined as K −1 ¼ T −1 þ iρ.The amplitude for a decay process, is computed by combining the K-matrix obtained from a scattering experiment with a production vector to describe process-dependent contributions.The K-matrix is modeled as a five-pole structure, where the indexes i; j ¼ 1; 2; 3; 4; 5 correspond to five decay channels: ππ, K K, ηη, ηη 0 and multimeson (mainly 4π states) respectively.The coupling constant of the bare state α to the decay channel i, g α i , is obtained from a global fit of scattering data and is listed in Table II.The mass m α is the bare pole mass and is in general different from the resonant mass of the RBW function.The parameters f scatt ij and s scatt 0 are used to describe smooth scattering processes.The last factor of the K-matrix, 1−s A0 s−s A0 ðs − s A m 2 π 2 Þ, regulates the singularities near the π þ π − threshold, the so-called "Adler zero" [85,86].The Hermitian property of the K-matrix imposes the relation f scatt ij ¼ f scatt ji , and since only π þ π − decays are considered, if i ≠ 1 and j ≠ 1, f scatt ij is set to 0. The production vector is modeled with where f prod 1j and β α are free parameters.The singularities in the K-matrix and the production vector cancel when calculating the amplitude matrix element.

V. DALITZ PLOT FIT
An unbinned extended maximum likelihood fit is performed to the Dalitz plot distribution.The likelihood function is defined by where The background probability density function (PDF) is given by f b ðx; θ b Þ and is described in Sec.VA.The signal PDF, f s ðx i ; θ s Þ, is described by where the decay amplitude, Mðx; θ s Þ, is described in Sec.IV and the efficiency variation over the Dalitz plot, εðxÞ, is described in Sec.V B. The fit parameters, θ s and θ b , include complex coefficients and resonant parameters like masses and widths.The value N is the total number of reconstructed candidates in the signal region.The number of signal and background events, ν s and ν b , are floated and constrained by the yields, ν 0 s and ν 0 b , determined by the D0 π þ π − mass fit and shown with their statistical uncertainties in Table I.

A. Background modeling
The only significant source of candidates in the signal region, other than with a looser requirement on the Fisher discriminant, and is shown in Fig. 3.The looser requirement gives a similar distribution in the Dalitz plane but with lower statistical fluctuations.The Dalitz plot distribution of the combinatorial background events lying in the uppermass sideband is considered to provide a reliable description of that in the signal region, as no dependence on mð D0 π þ π − Þ is found by studying the Dalitz distribution in a different upper-mass sideband region.The combinatorial background is modeled with an interpolated nonparametric PDF [87,88] using an adaptive kernel-estimation algorithm [89].TABLE II.The K-matrix parameters used in this paper are taken from a global analysis of π þ π − scattering data [22].Masses and coupling constants are in units of MeV=c 2 .

B. Efficiency modeling
The efficiency function εðxÞ accounts for effects of reconstruction, triggering and selection of the B 0 → D0 π þ π − signal events, and varies across the Dalitz plane.Two simulated samples are generated to describe its variation with several data-driven corrections.One is uniformly distributed over the phase space of the Dalitz plot and the other is uniformly distributed over the square Dalitz plot, which models efficiencies more precisely at the kinematic boundaries.The square Dalitz plot is parametrized by two variables m 0 and θ 0 that each varies between 0 and 1 and are defined as The two samples are fitted simultaneously with common fit parameters.A fourth-order polynomial function is used to describe the efficiency variation over the Dalitz plot.As the efficiency in the simulation is approximately symmetric over . The fitted efficiency distribution over the Dalitz plane is shown in Fig. 4.
The efficiency is corrected using dedicated control samples with data-driven methods.The corrections applied to the simulated samples include known differences between simulation and data that originate from the trigger, PID and tracking.

C. Results of the Dalitz plot fit
The Dalitz plot distribution from data in the signal region is shown in Fig. 5.The analysis is performed using the isobar model and the K-matrix model.The nominal fit model in each case is defined by considering many possible resonances and removing those that do not significantly contribute to the Dalitz plot analysis.The resulting resonant contributions are given in Table III while the projections of the fit results are shown in Fig. 6 (Fig. 7) for the isobar (K-matrix) model.
The comparisons of the S-wave results for the isobar model and the K-matrix model are shown in Fig. 8.The results from the two models agree reasonably well for the amplitudes and phases.In the π þ π − mass-squared region of ½1.5; 4.0 GeV 2 =c 4 , small structures are seen in the K-matrix model, indicating possible contributions from f 0 ð1370Þ and f 0 ð1500Þ states.These contributions are not significant in the isobar model and are thus not included in the nominal fit: adding them results in marginal changes and shows similar qualitative behavior to the K-matrix model as displayed in Fig. 8.The measured S-waves from both models qualitatively agree with predictions given in Ref. [91].
To see more clearly the resonant contributions in the region of the ρð770Þ resonance, the data are plotted in the π þ π − invariant mass-squared region ½0.0;In the π þ π − S-wave distributions of both the isobar model and the K-matrix model, a peaking structure is seen in the region ½0.9; 1.0 GeV 2 =c 4 , which corresponds to the f 0 ð980Þ resonance.The structure in the region ½1.3; 1.8 GeV 2 =c 4 corresponds to the spin-2 f 2 ð1270Þ resonance.
Distributions in the invariant mass-squared region ½6.4; 10.4 GeV 2 =c 4 of m 2 ð D0 π − Þ are shown in Fig. 10.There is a significant contribution from the D Ã J ð2760Þ − resonance observed in Ref. [29] and a spin-3 assignment gives the best description.A detailed discussion on the determination of the spin of D Ã J ð2760Þ is provided in Sec.VIII B.
The fit quality is evaluated by determining a χ 2 value by comparing the data and the fit model in N bins ¼ 256 bins that are defined adaptively to ensure approximately equal population with a minimum bin content of 37 entries.A value of 287 ( 296) is found for the isobar (K-Matrix) model based on statistical uncertainties only.TABLE III.Resonant contributions to the nominal fit models and their properties.Parameters and uncertainties of ρð770Þ, ωð782Þ, ρð1450Þ and ρð1700Þ come from Ref. [90], and those of f 2 ð1270Þ and f 0 ð2020Þ come from Ref. [32].Parameters of f 0 ð500Þ, f 0 ð980Þ and K-matrix formalism are described in Sec.IV.

Resonance
Spin Model m r (MeV=c 2 ) Γ 0 (MeV) D0 π − P-wave 1 Eq.( 14) Further checks of the consistency between the fitted models and the data are performed with the unnormalized Legendre polynomial weighted moments as a function of

VI. MEASUREMENT OF THE B
Measuring the branching fractions of the different resonant contributions requires knowledge of the B 0 → D0 π þ π − branching fraction.This branching fraction is normalized relative to the B 0 → D Ã ð2010Þ − π þ decay that has the same final state, so systematic uncertainties are reduced.Identical selections are applied to select 11 and are fitted simultaneously to determine the signal and background contributions.The D Ã ð2010Þ − signal distribution is modeled by three Gaussian functions to account for resolution effects while its background is modeled by a phase-space factor.The modeling of the signal and background shapes in the mð The efficiencies for selecting B 0 → D Ã ð2010Þ − π þ and B 0 → D0 π þ π − decays are obtained from simulated samples.To take into account the resonant distributions in the Dalitz plot, the B 0 → D0 π þ π − simulated sample is weighted using the model described in the previous sections.The average efficiencies are ð1.72 AE 0.05Þ × 10 −4 and ð4.96 AE 0.05Þ × 10 −4 for the Using the branching fractions of where the first uncertainty is statistical and the second uncertainty comes from the branching fraction of the normalization channel.

VII. SYSTEMATIC UNCERTAINTIES A. Common systematic uncertainties and checks
Two categories of systematic uncertainties are considered, each of which is quoted separately.They originate from the imperfect knowledge of the experimental conditions and from the assumptions made in the Dalitz plot fit model.The Dalitz model-dependent uncertainties also account for the precision on the external parameters.The various sources are assumed to be independent and summed in quadrature to give the total.
Experimental systematic uncertainties arise from the efficiency and background modeling and from the veto on the D Ã ð2010Þ − resonance.Those corresponding to the signal efficiency are due to imperfect estimations of PID, trigger, tracking reconstruction effects, and to the finite size of the simulated samples.Each of these effects is evaluated by the differences between the results using efficiencies computed from the simulation and from the data-driven methods.The systematic uncertainties corresponding to the modeling of the small residual background are estimated by using different subsamples of backgrounds.The systematic uncertainty due to the veto on the D Ã ð2010Þ − resonance is assigned by changing the selection requirement from The systematic uncertainties related to the Dalitz models considered (see Sec. IV) include effects from other possible resonant contributions that are not included in the nominal fit, from the modeling of resonant line shapes and from imperfect knowledge of the parameters of the modeling, i.e., the masses and widths of the π þ π − resonances considered, and the resonant radius.
The nonsignificant resonances added to the model for systematic studies are the f 0 ð1300Þ, f 0 ð1500Þ, f 0 2 ð1525Þ, and D Ã ð2650Þ − [f 0 2 ð1525Þ and D Ã ð2650Þ − ] mesons for the isobar (K-matrix) model [29,32,48,49].The spin of the D Ã ð2650Þ − resonance is set to 1.The differences between each alternative model and the nominal model are conservatively assigned as systematic uncertainties.
The radius of the resonances ðr BW Þ is set to a unique value of 1.6 GeV −1 × ℏc in the nominal fit.In the systematic studies, it is floated as a free parameter and its best fit value is 1.84 AE 0.05 GeV −1 × ℏc (1.92 AE 0.31 GeV −1 × ℏc) for the isobar (K-matrix) model.The value 1.85 GeV −1 × ℏc is chosen to estimate the systematic uncertainties due to the imperfect knowledge of this parameter.
The masses and widths of the π þ π − resonances considered are treated as free parameters with Gaussian constraints according to the inputs listed in Table III.The differences between the results from those fits and those of the nominal fits are assigned as systematic uncertainties.
For the isobar model, additional systematic uncertainties due to the modeling of the f 0 ð500Þ and f 0 ð980Þ resonances are considered.The Bugg model [83] for the f 0 ð500Þ resonance and the Flatté model [84] for the f 0 ð980Þ resonance, used in the nominal fit, are replaced by more conventional RBW functions.The masses and widths, left as free parameters, give 553 AE 15 MeV=c 2 and 562 AE 39 MeV for the f 0 ð500Þ meson and 981 AE 13 MeV=c 2 and 191 AE 39 MeV for the f 0 ð980Þ meson.The resulting differences to the nominal fit are assigned as systematic uncertainties.
The kinematic variables are calculated with the masses of the D0 and B 0 mesons constrained to their known values [32].These kinematic constraints affect the extraction of the masses and widths of the D0 π − resonances.The current world average value for the B 0 meson mass is 5279.58AE 0.17 MeV=c 2 and for the D0 meson is 1864.84AE 0.07 MeV=c 2 [32].A conservative and direct estimation of the systematic uncertainties on the masses and widths of the D0 π − resonances is provided by the sum in quadrature of the B 0 and D0 mass uncertainties.The effects of mass constraints are found to be negligible for the fit fractions, moduli and phases of the complex coefficients.
The systematic uncertainties are summarized for the isobar (K-matrix) model Dalitz analysis in Appendix B. Systematic uncertainties related to the measurements performed with the isobar formalism are listed in Tables XIV-XVII, while those for the K-matrix formalism are given in Tables XVIII-XXI.In most cases, the dominant systematic uncertainties are due to the D Ã ð2010Þ − veto and the model uncertainties related to other resonances not considered in the nominal fit.In the isobar model, the modeling of the f 0 ð500Þ and f 0 ð980Þ resonances also has non-negligible systematic effects.
Several cross-checks have been performed to study the stability of the results.The analysis was repeated for different Fisher discriminant selection criteria, different trigger requirements and different subsamples, corresponding to the two data-taking periods and to the two half-parts of the D0 π þ π − invariant mass signal region, above and below the B 0 mass [32].Results from those checks demonstrate good consistency with respect to the nominal fit results.No bias is seen; therefore no correction is applied, nor is any related uncertainty assigned.

B. Systematic uncertainties on the
The systematic uncertainties related to the measurement of the B 0 → D0 π þ π − branching fraction are listed in Table IV.The systematic uncertainties on the PID, trigger, reconstruction and statistics of the simulated samples are calculated in a similar way to those of the Dalitz plot analysis.Other systematic uncertainties are discussed below.
The systematic uncertainty on the modeling of the D0 π − and D Ã ð2010Þ − π þ invariant mass distributions is estimated by counting the number of signal events in the B 0 signal region assuming a flat background contribution.The D Ã ð2010Þ − mass region is restricted to the range [2007, 2013] MeV=c 2 for this estimate.The calculated branching fraction is nearly identical to that from the mass fit and thus has a negligible contribution to the systematic uncertainty.The signal purity of B 0 → D Ã ð2010Þ − π þ is more than 99%.
To account for the effect of resonant structures on the signal efficiency, the data sample is divided using an adaptive binning scheme.The average efficiency is calculated in a model-independent way as where N i is the number of events in bin i and ε i is the average efficiency in bin i calculated from the efficiency model.The difference between this model-independent method and the nominal is assigned as a systematic uncertainty.

A. Significance of resonances
The isobar and K-matrix models employed to describe the Dalitz plot of the B 0 → D0 π þ π − decay include all of the resonances listed in Table III.The statistical significances of well-established π þ π − resonances are calculated directly with their masses and widths fixed to the world averages.They are computed as the relative change of the minimum of the negative logarithm of the likelihood (NLL) function with and without a given resonance.Besides the π þ π − resonances listed in Table III, the significances of the f 0 ð1370Þ, f 0 ð1500Þ and f 0 2 ð1525Þ are also given.The results, expressed as multiples of Gaussian standard deviations (σ), are summarized in Table V.All of the other π þ π − resonances not listed in this Table have large statistical significances, well above five standard deviations.
To test the significance of the D Ã J ð2760Þ − state, where J ¼ 3 (see Sec. VIII B), an ensemble of pseudoexperiments is generated with the same number of events as in the data sample, using parameters obtained from the fit with the D Ã J ð2760Þ − resonance excluded.The difference of the minima of the NLL when fitting with and without   this analysis.Finally, an extra D0 π − resonance, with different spin hypotheses (J ¼ 0; 1; 2; 3; 4) and with its mass and width allowed to vary, is added to the nominal fit model and no significant contribution is found.
The significance of each of the significant ωð782Þ, f 0 ð980Þ, ρð1450Þ, f 0 ð2020Þ and D Ã J ð2760Þ − states is checked while including the dominant systematic uncertainties (see Sec. VII A), namely, the modeling of the f 0 ð500Þ and f 0 ð980Þ resonances, the addition of other resonant contributions and the modification of the D Ã ð2010Þ − veto criteria.In all configurations, the significances of the ωð782Þ, f 0 ð980Þ, ρð1450Þ and D Ã J ð2760Þ − resonances are greater than 7.7σ, 7.0σ, 8.7σ, and 10.8σ, respectively.The significance of the f 0 ð2020Þ drops to 4.3σ when using a RBW line shape for the f 0 ð500Þ resonance.The abundant f 0 ð500Þ contribution is highly significant under all of the applied changes.

B. Spin of resonances
As described in Sec.V C, a spin-3 D Ã J ð2760Þ − contribution gives the best description of the data.To obtain the significance of the spin-3 hypothesis with respect to other spin hypotheses (J ¼ 0; 1; 2; 4), test statistics are built.Their computations are based on the shift of the minimum of the NLL with respect to the nominal fit model when using different spin hypotheses.The mass and width of the D Ã J ð2760Þ − resonance are floated in all the cases.Pseudoexperiments are generated using the fit parameters obtained using the other spin hypotheses.Significances are calculated according to the distributions obtained from the pseudoexperiments of the test statistic and its values from data.These studies indicate that data are inconsistent with other spin hypotheses by more than 10σ.Following the discovery of the D Ã sJ ð2860Þ − meson, which is interpreted as the superposition of two particles with spin 1 and spin 3 [25,26], a similar configuration for the D Ã J ð2760Þ − has been tested and is found to give no significant improvement in the description of the data.To illustrate the preference of the spin-3 hypothesis, the cosine of the helicity angle distributions in the masssquared region of ½7.4; 8.2 GeV 2 =c 4 for m 2 ð D0 π − Þ are shown in Fig. 12 under the various scenarios.Based on our result, D Ã J ð2760Þ − is interpreted as the D Ã 3 ð2760Þ − meson.Recently, LHCb observed a neutral spin-1 D Ã ð2760Þ 0 state [92].The current analysis does not preclude a charged spin-1 D Ã state at around the same mass, but it is not sensitive to it with the current data sample size.
Studies have also been performed to validate the spin-0 hypothesis of the D Ã 0 ð2400Þ − resonance, as the spin of this state has never previously been confirmed in experiment   [32].When moving to other spin hypotheses, the minimum of the NLL increases by more than 250 units in all cases, which confirms the expectation of spin 0 unambiguously.

C. Results of the Dalitz plot analysis
The shape parameters of the π þ π − resonances are fixed from previous measurements except for the nonresonant contribution in the isobar model.The fitted value of the parameter α defined in Eq. ( 20) is −0.363AE 0.027, which corresponds to a 10σ statistical significance compared to the case where there is no varying phase.An expansion of the model by including a varying phase in the D0 π − axis is also investigated but no significantly varying phase in that system is seen.The results indicate a weak, but nonnegligible, rescattering effect in the π þ π − states, while the rescattering in the D0 π − states is not significant.The masses, widths and other shape parameters of the D0 π − contributions are allowed to vary in the analysis.The values of the shape parameters of the D0 π − P-wave component, defined in Eq. ( 14), are β 1 ¼ 0.95 AE 0.05 (0.90 AE 0.04) and β 2 ¼ 0.51 AE 0.06 (0.43 AE 0.05) for the isobar (K-matrix) model.
The measurements of the masses and widths of the three resonances D Ã 0 ð2400Þ − , D Ã 2 ð2460Þ − and D Ã 3 ð2760Þ − are listed in Table VI.The present precision on the mass and width of the D Ã 0 ð2400Þ − resonance is improved with respect to Refs.[29,32].The result for the width of the D Ã 2 ð2460Þ − meson is consistent with previous measurements, whereas the result for the mass is above the world average which is dominated by the measurement using inclusive production by LHCb [29].In the previous LHCb inclusive analysis, the broad D Ã 0 ð2400Þ − component was excluded from the fit model due to a high correlation with the background line shape parameters, while here it is included.The present result supersedes the former measurement.The Dalitz plot analysis used in this paper ensures that the background under the D Ã 2 ð2460Þ − peak and the effect on the efficiency are under control, resulting in much lower systematic uncertainties compared to the inclusive approach.
The moduli and the phases of the complex coefficients of the resonant contributions, defined in Eq. ( 2), are displayed in Tables VII and VIII.Compatible results are obtained using both the isobar and K-matrix models.The results for the fit fractions are given in Table IX, while results for the interference fit fractions are given in Appendix C. Pseudoexperiments are used to validate the fitting procedure and no biases are found in the determination of parameter values.

D. Branching fractions
The measured branching fraction of the  [22], obtained in a slightly larger phase-space region.A multiplicative factor of 94.5% (96.2%) is required to scale the Belle (BABAR) results to the same phase-space region as in this analysis.
The branching fraction of each quasi-two-body decay, B 0 → r i h 3 , with r i → h 1 h 2 , is given by where the resonant states ðh The fit fractions F i , defined in Eq. ( 3), are obtained from the Dalitz plot analysis and are listed in Table IX.The correction factors, ε corr i , account for the cutoff due to the D Ã ð2010Þ − veto.They are obtained by generating pseudoexperiment samples for each resonance over the Dalitz plot and applying the same requirement [mð D0 π AE Þ > 2.1 GeV=c 2 ].They are summarized in Table X.The correction factors are the same for the isobar model and the K-matrix model.The effects due to the uncertainties of the masses and widths of the resonances are included in the uncertainties given in the table.
Using the overall B 0 → D0 π þ π − decay branching fraction, the fit fractions (F i ) and the correction factors (ε corr i ), , and the first evidence of B 0 → D0 f 0 ð2020Þ are reported.The present world averages [32] of the branching fractions When accounting for the branching fractions of the ωð782Þ and f 2 ð1270Þ to π þ π − , one obtains the following results for the isobar model: and For the K-matrix model, one obtains and In both models, the fifth uncertainty is due to knowledge of the π þ π − decay rates [32].The results are consistent with the measurement of the decay B 0 → D0 ωð782Þ, using the dominant ωð782Þ → π þ π − π 0 decay [32,40].
E. Structure of the f 0 ð980Þ and f 0 ð500Þ resonances In the isobar model, significant contributions from both B 0 → D0 f 0 ð500Þ and B 0 → D0 f 0 ð980Þ decays are observed.The related branching fraction measurements can be used to obtain information on the substructure of the f 0 ð980Þ and f 0 ð500Þ resonances within the factorization approximation.As discussed in Sec.I, two models for the quark structure of those states are considered: q q or ½qq 0 ½ q q0 (tetraquarks).In both models, mixing angles between different quark states are determined using our measurements.In the q q model, the mixing between ss and u ū or d d can be written as where jn ni ≡ ðju ūi þ jd diÞ= ffiffi ffi 2 p and φ mix is the mixing angle.In the ½qq 0 ½ q q0 model, the mixing angle, ω mix , is introduced and the mixing becomes In both cases, the following variable is defined: where Φð500Þ and Φð980Þ are the integrals of the phasespace factors computed over the resonant line shapes and the phase-space factors are proportional to the momentum computed in the B 0 rest frame.The value of their ratio is Φð500Þ=Φð980Þ ¼ 1.02 AE 0.05.The value of the branching fraction Bðf 0 ð500Þ → π þ π − Þ ¼ 2=3 is obtained from the isospin Clebsch-Gordan coefficients and assumes that there are only contributions from ππ final states.The ratio Bðf 0 ð980Þ 15  −0.14 , obtained from an average of the measurements by the BABAR [93] and BES [94] collaborations, is used to estimate the branching fraction Bðf 0 ð980Þ → π þ π − Þ.Assuming that the ππ and KK decays are dominant in the f 0 ð980Þ decays, taking into account the systematic uncertainties, as listed in Table XII.The parameter r f is related to the mixing angle by the equation in the q q model and by in the ½qq 0 ½ q q0 tetraquark model [57,58].The form factors FðB 0 → f 0 ð980ÞÞ and FðB 0 → f 0 ð500ÞÞ are evaluated at the four-momentum transfer squared equal to the square of the D0 mass.Finally, values of the mixing angles as a function of form factor ratio are obtained in Fig. 13 for the q q model and the ½qq 0 ½ q q0 tetraquark model.Such angles have also been computed by LHCb for the decays The expectation is that the ratio of form factors should be close to unity.However, LHCb has recently performed a search for the decay B 0 s → D0 f 0 ð980Þ [95].The limit set on this decay is below the value expected in a simple model based on our measured value of BðB 0 → D0 f 0 ð500ÞÞ and assuming equal form factors.More complicated models may be needed in order to explain all results.
The above discussion is one possible interpretation of the results.Another possible mechanism [91,96] involves the generation of pseudoscalar resonances through the interactions of π þ π − mesons.

F. Isospin analysis of the B → Dρ system
The measured branching fraction of the B 0 → D0 ρð770Þ 0 decay, presented in Table XI, can be used to perform an isospin analysis of the B → Dρ system.Isospin symmetry relates the amplitudes of the decays B þ → D0 ρð770Þ þ , B 0 → D − ρð770Þ þ , and B 0 → D0 ρð770Þ 0 , which can be written as linear combinations of the isospin eigenstates A I with I ¼ 1=2 and 3=2 [37,41] The strong phase difference between the amplitudes A 1=2 and A 3=2 is denoted by δ Dρ .Final-state interactions between the states D0 ρ 0 and D − ρ þ may lead to a value of δ Dρ different from zero and through constructive interference, to a larger value of BðB 0 → D0 ρ 0 Þ than the prediction obtained within the factorization approximation.In the heavy-quark limit, the factorization model predicts [97,98]  Using our measurement of BðB 0 → D0 ρ 0 Þ together with the world average values of and With a frequentist statistical approach [99], R Dρ and cos δ Dρ are calculated for the isobar and K-matrix models in Table XIII.These results are not significantly different from the predictions of factorization models.As opposed to the theoretical expectations [37,41] and in contrast to the B → D ðÃÞ π system [40], nonfactorizable final-state interaction effects do not introduce a sizable phase difference between the isospin amplitudes in the B → Dρ system.The precision on R Dρ and cos δ Dρ is dominated by that of the branching fractions of the decays [32].The precision of the branching fraction of the B 0 → D0 ρð770Þ 0 decay is 7.3% (9.2%) for the isobar (K-matrix) model (see Table XI).

IX. CONCLUSION
A Dalitz plot analysis of the B 0 → D0 π þ π − decay is presented.The decay model contains four components from D0 π − resonances, four P-wave π þ π − resonances and one D-wave π þ π − resonance.Two models are used to describe the S-wave π þ π − resonances.The isobar model uses four components, including the f 0 ð500Þ, f 0 ð980Þ, f 0 ð2020Þ resonances and a nonresonant contribution.The K-matrix approach describes the π þ π − S-wave using a 5 × 5 scattering matrix with a production vector.The overall branching fraction of B 0 → D0 π þ π − and quasitwo-body decays are measured.Significant contributions from the f 0 ð500Þ, f 0 ð980Þ, ρð1450Þ and D Ã 3 ð2760Þ − mesons are observed for the first time.For the latter, this is a confirmation of the observation from previous inclusive measurements, and the spin-parity of this resonance is determined for the first time to be J P ¼ 3 − .This suggests a spectroscopic assignment of 3 D 3 , and shows that the 1D family of charm resonances can be explored in the Dalitz plot analysis of B-meson decays in the same way as recently seen for the charm-strange resonances [25,26].Evidence for the f 0 ð2020Þ meson is also seen for the first time.The measured branching fractions of two-body decays are more precise than the existing world averages and there is good agreement between values from the isobar and K-matrix models.
The masses and widths of the D0 π − resonances are also determined.The measured masses and widths of the D Ã 0 ð2400Þ − and D Ã 3 ð2760Þ − states are consistent with the previous measurements.The precision on the D Ã 0 ð2400Þ − meson is much improved.For the measurement on the mass and width of the D Ã 2 ð2460Þ − meson, the broad D Ã 0 ð2400Þ − component was excluded from the fit model in the former LHCb inclusive analysis [29], due to a high correlation with the background line shape parameters, while here it is included.The present result therefore supersedes the former measurement.
The significant contributions found for both the f 0 ð500Þ and f 0 ð980Þ allow us to constrain the mixing angle between the f 0 ð500Þ and f 0 ð980Þ resonances.An isospin analysis in the B → Dρ decays using our improved measurement of the branching fraction of the decay B 0 → D0 ρ 0 is performed, indicating that nonfactorizable effects from final-state interactions are limited in the Dρ system.
(France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom).We are indebted to the communities behind the multiple open source software packages on which we depend.We are also thankful for the computing resources and the access to software R&D tools provided by Yandex LLC (Russia).Individual groups or members have received support from EPLANET, Marie Skłodowska-Curie Actions            Van Swinderen Institute, University of Groningen, Groningen, The Netherlands (associated with Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands) † Deceased.a Also at Università di Firenze, Firenze, Italy.b Also at Università di Ferrara, Ferrara, Italy.c Also at Università della Basilicata, Potenza, Italy.d Also at Università di Modena e Reggio Emilia, Modena, Italy.e Also at Università di Milano Bicocca, Milano, Italy.f Also at LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain.g Also at Università di Bologna, Bologna, Italy.h Also at Università di Roma Tor Vergata, Roma, Italy.i Also at Università di Genova, Genova, Italy.j Also at Scuola Normale Superiore, Pisa, Italy.k Also at Università di Cagliari, Cagliari, Italy.l Also at Politecnico di Milano, Milano, Italy.m Also at Universidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil.n Also at AGH-University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland.o Also at Università di Padova, Padova, Italy.p Also at Hanoi University of Science, Hanoi, Viet Nam.q Also at Università di Bari, Bari, Italy.r Also at Università degli Studi di Milano, Milano, Italy.s Also at Università di Roma La Sapienza, Roma, Italy.t Also at Università di Pisa, Pisa, Italy.u Also at Università di Urbino, Urbino, Italy.v Also at P.N.Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia.

FIG. 2 (
FIG. 2 (color online).Invariant mass distribution of B 0 → D0 π þ π − 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.

FIG. 3 (
FIG. 3 (color online).Density profile of the combinatorial background events in the Dalitz plane obtained from the upper mð D0 π þ π − Þ sideband with a looser selection applied on the Fisher discriminant.

2 . 1 FIG. 5 (
FIG. 4 (color online).Efficiency function for the Dalitz variables obtained in a fit to the LHCb simulated samples.

Fig. 9 .
Fig.9.In the region around 0.6 GeV 2 =c 4 , interference between the ρð770Þ and ωð782Þ resonances is evident.In the π þ π − S-wave distributions of both the isobar model and the K-matrix model, a peaking structure is seen in the region ½0.9; 1.0 GeV 2 =c 4 , which corresponds to the f 0 ð980Þ resonance.The structure in the region ½1.3; 1.8 GeV 2 =c 4 corresponds to the spin-2 f 2 ð1270Þ resonance.Distributions in the invariant mass-squared region ½6.4; 10.4 GeV 2 =c 4 of m 2 ð D0 π − Þ are shown in Fig.10.There is a significant contribution from the D Ã J ð2760Þ − resonance observed in Ref.[29] and a spin-3 assignment gives the best description.A detailed discussion on the determination of the spin of D Ã J ð2760Þ is provided in Sec.VIII B.The fit quality is evaluated by determining a χ 2 value by comparing the data and the fit model in N bins ¼ 256 bins that are defined adaptively to ensure approximately equal population with a minimum bin content of 37 entries.A value of 287 (296) is found for the isobar (K-Matrix) model based on statistical uncertainties only.

FIG. 6 (
FIG. 6 (color online).Projections of the data and isobar fit onto (a) m 2 ðπ þ π − Þ and (c) m 2 ð D0 π − Þ with a linear scale.Same projections shown in (b) and (d) with a logarithmic scale.Components are described in the legend.The lines denoted D0 π − and π þ π − include the coherent sums of all D0 π − resonances, π þ π − resonances, and π þ π − S-wave resonances.The various contributions do not add linearly due to interference effects.
The corresponding distributions are shown in Appendix A.

FIG. 8 (
FIG. 8 (color online).Comparison of the π þ π − 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 ð1370Þ and f 0 ð1500Þ are shown.

FIG. 7 (
FIG. 7 (color online).Projections of the data and K-matrix fit onto (a) m 2 ðπ þ π − Þ and (c) m 2 ð D0 π − Þwith a linear scale.Same projections shown in (b) and (d) with a logarithmic scale.Components are described in the legend.The lines denoted D0 π − and π þ π − include the coherent sums of all D0 π − resonances, π þ π − resonances, and π þ π − S-wave resonances.The various contributions do not add linearly due to interference effects.

FIG. 9 (FIG. 10 (
FIG. 9 (color online).Distributions of m 2 ðπ þ π − Þ in the ρð770Þ 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.
− is used as a test statistic.It corresponds to 11.4σ (11.5σ) for the isobar (K-matrix) model and confirms the observation of D Ã J ð2760Þ − reported in Ref. [29].The two other orbitally excited D resonances, D Ã J ð2650Þ − and D Ã J ð3000Þ − , whose observations are presented in the same paper, are added into the nominal fit model with different spin hypotheses and tiny improvements are found.They also do not describe the data in the absence of the D Ã J ð2760Þ − .Those resonances are thus not confirmed by

FIG. 12 (
FIG. 12 (color online).Cosine of the helicity angle distributions in the m 2 ð D0 π − Þ range ½7.4; 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 Ã J ð2760Þ − and with the different spin hypotheses of D Ã J ð2760Þ − , are superimposed.

FIG. 13 (
FIG. 13 (color online).Mixing angle as a function of form factor ratio for the (a) q q model and (b) ½qq 0 ½ q q0 tetraquark model.The green band gives the 1σ interval around central values (black solid line).
where m b represents the b quark mass and Λ QCD the QCD scale.

TABLE I .
Results of the fit to the invariant mass distribution of B 0 → D0 π þ π − candidates.Uncertainties are statistical only.

TABLE V .
Statistical significance (σ) of π þ π − resonances in the Dalitz plot analysis.For the statistically significant resonances, the effect of adding dominant systematic uncertainties is shown (see text).

TABLE VII .
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.

TABLE VIII .
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.

TABLE X .
Correction factors due to the D Ã ð2010Þ − veto.

TABLE IX .
The fit fractions of the resonant contributions for the isobar and K-matrix models with mð D0 π AE Þ > 2.1 GeV=c 2 .The first uncertainty is statistical, the second and the third are experimental and model-dependent systematic uncertainties, respectively.
the branching fractions of quasi-two-body decays are calculated in TableXI.The first observation of the decays

TABLE XI .
Measured branching fractions of BðB 0 → rh 3 Þ × Bðr → h 1 h 2 Þ 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 normalization B 0 → D Ã ð2010Þ − π þ channel.

TABLE XII .
Systematic uncertainties on r f .The sum in quadrature of the uncertainties is also reported.

TABLE XIV .
Systematic uncertainties on the D0 π − resonant masses (MeV=c 2 ) and widths (MeV) for the isobar model.

TABLE XV .
Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the isobar model.The moduli are normalized to that of ρð770Þ.

TABLE XVI .
Systematic uncertainties on the phases (°) of the complex coefficients of the resonant contributions for the isobar model.The phase of ρð700Þ is set to 0°as the reference.

TABLE XVII .
Systematic uncertainties on the fit fractions (%) of the resonant contributions for the isobar model.

TABLE XVIII .
Systematic uncertainties on the D0 π − resonant masses (MeV=c 2 ) and widths (MeV) for the K-matrix model.

TABLE XIX .
Systematic uncertainties on the moduli of the complex coefficients of the resonant contributions for the K-matrix model.The moduli are normalized to that of ρð770Þ.

TABLE XXI .
Systematic uncertainties on the fit fractions (%) of the resonant contributions for the K-matrix model.

TABLE XX .
Systematic uncertainties on the phases ( °) of the complex coefficients of the resonant contributions for the K-matrix model.The phase of ρð700Þ is set to 0°as reference.

TABLE XXXI .
Systematic uncertainties on the moduli of the K-matrix parameters.The moduli are normalized to that of ρð770Þ.

TABLE XXXII .
Systematic uncertainties on the phases (°) of the K-matrix parameters.The phase of ρð700Þ is set to 0°as the reference.Yandex School of Data Analysis, Moscow, Russia (associated with Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia) 66 Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC, Valencia, Spain (associated with Universitat de Barcelona, Barcelona, Spain)