Measurement of CP-Violating Asymmetries in B0-->(rhopi)0 Using a Time-Dependent Dalitz Plot Analysis

We report a measurement of CP-violating asymmetries in B0-->(rhopi)0-->pi+pi-pi0 decays using a time-dependent Dalitz plot analysis. The results are obtained from a data sample of 375 million Y(4S) -->BBbar decays, collected by the BaBar detector at the PEP-II asymmetric-energy B Factory at SLAC. We measure 26 coefficients of the bilinear form-factor terms occurring in the time-dependent decay rate of the B0 meson. We derive the physically relevant quantities from these coefficients. In particular, we measure a constraint on the angle alpha of the Unitarity Triangle.


I. INTRODUCTION
Measurements of the parameter sin2β [1,2] have established CP violation in the B 0 meson system. These measurements provide strong support for the Kobayashi and Maskawa model of this phenomenon as arising from a single phase in the three-generation CKM quark-mixing matrix [3]. We present in this paper results from a timedependent analysis of the B 0 → π + π − π 0 [4] Dalitz plot which is dominated by intermediate vector resonances (ρ). The goal of this analysis is the simultaneous extraction of the strong transition amplitudes and the weak interaction phase α ≡ arg [−V td V * tb /V ud V * ub ] of the Unitarity Triangle. In the Standard Model, a non-zero value for α is responsible for the occurrence of mixing-induced CP violation in this decay. The BABAR and Belle experiments have obtained constraints on α from the measurement of effective quantities sin2α eff in B decays to π + π − [5,6] and from ρ + ρ − [7,8], using an isospin analysis [9].
Unlike π + π − , ρ ± π ∓ is not a CP eigenstate and four flavor-charge configurations (B 0 (B 0 ) → ρ ± π ∓ ) must be considered. The corresponding isospin analysis [10] is unfruitful with the present data sample since two pentagonal amplitude relations with 12 unknowns have to be solved (compared to 6 unknowns for the π + π − and ρ + ρ − systems). However, it has been pointed out by Snyder and Quinn [11] that one can obtain the necessary degrees of freedom to constrain α without ambiguity by explicitly including in the analysis the variation of the strong phases of the interfering ρ resonances in the Dalitz plot.

A. DECAY AMPLITUDES
We consider the decay of a spin-zero B 0 meson with four-momentum p B into the three daughters π + , π − , π 0 , with p + , p − , and p 0 their corresponding four-momenta. We take as the independent (Mandelstam) variables the invariant squared masses of the charged and neutral pions The invariant squared mass of the positive and negative pion, s 0 = (p + + p − ) 2 , is obtained, from energy and momentum conservation, The differential B 0 decay rate distribution as a function of the variables defined in Eq. (1) (i.e., the Dalitz plot) ‡ Also with Università della Basilicata, Potenza, Italy § Also with IPPP, Physics Department, Durham University, Durham DH1 3LE, United Kingdom reads dΓ(B 0 → π + π − π 0 ) = 1 (2π) 3 |A 3π | 2 32m 3 where A 3π is the Lorentz-invariant amplitude of the three-body decay [12]. We assume in the following that the amplitude A 3π and its CP conjugate A 3π , corresponding to the transitions B 0 → π + π − π 0 and B 0 → π + π − π 0 , respectively, are dominated by the three resonances ρ + , ρ − and ρ 0 . The ρ resonances are assumed to be the sum of the ground state ρ(770) and the radial excitations ρ(1450) and ρ(1700), with masses and widths determined by a combined fit to τ + → ν τ π + π 0 and e + e − → π + π − data [13]. Since the hadronic environment is different in B decays, we do not rely on this result for the relative ρ(1450) and ρ(1700) amplitudes but instead simultaneously measure them with the CP parameters from the fit. Variations of the other parameters and possible contributions to the B 0 → π + π − π 0 decay other than the ρ resonances are studied as part of the systematic uncertainties (Section V).
We write the A 3π and A 3π amplitudes [11,14] A where the f κ (with κ = {+, −, 0} denoting the charge of the ρ from the decay of the B 0 meson) are functions of the Dalitz variables s + and s − that incorporate the kinematic and dynamical properties of the B 0 decay into a vector ρ resonance and a pseudoscalar pion. The A κ are complex amplitudes that include weak and strong transition phases and that are independent of the Dalitz variables. Following Ref. [13], the ρ resonances are parameterized in f κ (where κ is the charge) as a sum of the ρ(770), ρ(1450), and ρ(1700) resonances: f κ (s) ∝ F ρ(770) (s) + a ρ ′ e iφ ρ ′ F ρ(1450) (s) + (6) a ρ ′′ e iφ ρ ′′ F ρ(1700) (s) , where the F ρ are modified relativistic Breit-Wigner functions introduced by Gounaris and Sakurai (GS) [15] and the a ρ (φ ρ ) are the magnitudes (phases) of the higher mass ρ resonances, relative to the ρ(770). In this analysis, we assume that the a and φ for f + and f − are the same while for f 0 , corresponding to the much smaller ρ 0 component, we fix a ′ ρ and a ′′ ρ to zero. Note that the definitions (4) and (5) are based on the assumption that the relative phases between the ρ(770) and its radial excitations are CP -conserving.
Due to angular momentum conservation, the spin-one ρ resonance is restricted to a helicity-zero state. For a ρ κ resonance with charge κ, the GS function is multiplied by the kinematic function −4|p κ ||p τ | cos θ κ , where p κ is the momentum of either of the daughters of the ρ resonance, p τ is the momentum of the particle not from the ρ decay, and cos θ κ is the cosine of the helicity angle of the ρ κ all defined, in the ρ-resonance rest frame. For the ρ + (ρ − ), θ + (θ − ) is defined by the angle between the π 0 (π − ) momentum in the ρ + (ρ − ) rest frame and the ρ + (ρ − ) flight direction in the B 0 rest frame. For the ρ 0 , θ 0 is defined by the angle between the π + momentum in the ρ 0 rest frame and the ρ 0 flight direction in the B 0 rest frame. With these definitions, each pair of GS functions interferes destructively at equal masses.
The factor of cos θ κ in the kinematic functions leads to an increased population in the interference regions between the different ρ bands in the Dalitz plot, and thus increases the sensitivity of this analysis [11].

B. TIME DEPENDENCE
With ∆t ≡ t 3π −t tag defined as the proper time interval between the decay of the fully reconstructed B 0 3π and that of the other meson B 0 tag from the Υ (4S), the time- where τ B 0 is the mean neutral B lifetime and ∆m d is the B 0 B 0 mass difference. Here, we have assumed that CP violation in B 0 B 0 mixing is absent (|q/p| = 1) and the lifetime difference between B H and B L is ∆Γ B d = 0. Inserting the amplitudes (4) and (5), one obtains for the terms in Eq. (7) with The 27 coefficients (9)- (14) are real-valued parameters that multiply the f κ f * σ bilinears (where κ and σ denote the charge of the ρ resonances) [16]. These coefficients are the observables that are determined by the fit. Each of the coefficients is related in a unique way to physically more intuitive quantities, such as tree-level and penguintype amplitudes, the angle α, or the quasi-two-body CP and dilution parameters [17] (cf. Section VI). The parameterization (8) is general: the information on the mirror solutions (e.g., on the angle α) that are present in the transition amplitudes A κ , A κ is conserved.
The decay rate (7) is used as a probability density function (PDF) in a maximum-likelihood fit and must therefore be normalized: where where ... denotes the expectation value over the Dalitz plot. The complex expectation values f κ f * σ are obtained from Monte Carlo integration of the Dalitz plot (3), taking into account acceptance and resolution effects. In this paper, we determine the relative values of U and I coefficients to U + + leaving 26 free coefficients. The choice to fit for the U and I coefficients rather than fitting for the complex transition amplitudes and the weak phase α directly is motivated by the following technical simplifications: (i) in contrast to the amplitudes, there is a unique solution for the U and I coefficients requiring only a single fit to the selected data sample; (ii) in the presence of background, we find that the errors on the U and I coefficients are approximately Gaussian, which in general is not the case for the amplitudes; and (iii) the propagation of systematic uncertainties and the averaging between different measurements are straightforward for the U and I coefficients.
The U + κ coefficients are related to resonance branching fractions and charge asymmetries; the U − κ coefficients determine the relative abundance of the B 0 decay into ρ + π − and ρ − π + and the time-dependent direct CP asymmetries. The I κ measure mixing-induced CP violation and are sensitive to strong phase shifts. Finally, the U ±,Re(Im) κσ and I Re(Im) κσ coefficients describe the interference pattern in the Dalitz plot, and their presence distinguishes this analysis from the quasi-two-body analysis previously reported in Ref. [17]. They represent the additional degrees of freedom that allow one to determine the unknown penguin contribution and the relative strong phases. However, because the overlap regions of the resonances are small and because the events reconstructed in these regions suffer from large misreconstruction rates and background, a substantial data sample is needed to perform a fit that constrains all amplitude parameters.
We determine the physically relevant quantities in a subsequent least-squares fit to the measured U and I coefficients.

C. THE SQUARE DALITZ PLOT
Both the signal events and the combinatorial e + e − → qq (q = u, d, s, c) continuum background events populate the kinematic boundaries of the Dalitz plot due to the low final state masses compared with the B 0 mass. We find the representation (3) is inconvenient when one wants to use empirical reference shapes in a maximum-likelihood fit. Large variations occurring in small areas of the Dalitz plot are very difficult to describe in detail.These regions are particularly important since it is here that the interference between ρ-resonances, and hence our ability to determine the strong phases, occurs. We therefore apply the transformation which defines the Square Dalitz plot (SDP). The new coordinates are Jacobian is given by where |p * + | = E * + − m 2 π + and |p * 0 | = E * 0 − m 2 π 0 , and where the energies E * + and E * 0 are defined in the π + π − rest frame. Figure 1 shows the determinant of the Jacobian as a function of the SDP parameters m ′ and θ ′ . If the events in the nominal Dalitz plot were distributed according to a uniform (non-resonant) three-body phases space, their distribution in the SDP would match the plot of | det J|.
The effect of the transformation (17) is illustrated in Fig. 2, which displays the nominal and square Dalitz plots for simulated signal events generated with Monte Carlo. As is shown, the transformation is benificial because: (i) it expands the regions of interference so that equal size bins cover this region in more detail; and (ii) it avoids the curved edge of bins on the boundary. This simulation does not take into account any detector effects and corresponds to a particular choice of the decay amplitudes for which destructive interferences occur where the ρ resonances overlap. To simplify the comparison, hatched areas showing the interference regions between ρ bands and dashed isocontours √ s +,−,0 = 1.5 GeV/c 2 have been superimposed on both Dalitz plots.

II. THE BABAR DETECTOR AND DATASET
The data used in this analysis were collected with the BABAR detector at the PEP-II asymmetric-energy e + e − storage ring at SLAC between October 1999 and August 2006. The sample consists of about 346 fb −1 , corresponding to (375 ± 4) × 10 6 BB pairs collected at the Υ (4S) resonance ("on-resonance"), and an integrated luminosity of 21.6 fb −1 collected about 40 MeV below the Υ (4S) ("off-resonance"). In addition, we use GEANT4 [18] simulated Monte Carlo (MC) events to study detector efficiency and backgrounds.
A detailed description of the BABAR detector is presented in Ref. [19]. The tracking system used for charged particle and vertex reconstruction has two main components: a silicon vertex tracker (SVT) and a drift chamber (DCH), both operating within a 1.5-T magnetic field generated by a superconducting solenoid. Photons are identified in an electromagnetic calorimeter (EMC) surrounding a detector of internally reflected Cherenkov light (DIRC), which associates Cherenkov photons with tracks for particle identification (PID). Muon candidates are identified with the use of the instrumented flux return (IFR) of the solenoid.

III. ANALYSIS METHOD
The U and I coefficients and the B 0 → π + π − π 0 event yield are determined by a maximum-likelihood fit of the signal and background model to the selected candidate events. Kinematic and event shape variables exploiting the characteristic properties of the events are used in the fit to discriminate signal from background.

A. EVENT SELECTION AND BACKGROUND SUPPRESSION
We reconstruct B 0 → π + π − π 0 candidates from pairs of oppositely-charged tracks and a π 0 → γγ candidate. In order to ensure that all events are within the Dalitz plot boundary, we constrain the three-pion invariant mass to the B mass after final selections have been made. The largest source of background is from continuum e + e − → qq production. We use information from the tracking system, EMC, and DIRC to remove tracks for which the PID is consistent with the electron, kaon, or proton hypotheses. In addition, we require that at least one track has a signature in the IFR that is inconsistent with the muon hypothesis. This selection retains 92% of signal events while rejecting 42% of continuum background events. The π 0 candidate mass m(γγ) must satisfy 0.11 < m(γγ) < 0.16 GeV/c 2 , where each photon, γ, is required to have an energy greater than 50 MeV in the laboratory frame (LAB) and to exhibit a lateral profile of energy deposition in the EMC consistent with an electromagnetic shower.
A B-meson candidate is characterized kinemati-cally by the beam-energy substituted mass and (E 0 , p 0 ) are the four-vectors of the B-candidate and the initial electron-positron systems respectively. The asterisk denotes the center-of-mass (CM) frame and s is the square of the CM energy. We require 5.272 < m ES < 5.288 GeV/c 2 , which retains 81% of the signal and 8% of the continuum background events. The ∆E resolution exhibits a dependence on the π 0 energy and therefore varies across the Dalitz plot. To avoid bias in the Dalitz plot, we introduce the transformed Backgrounds arise primarily from random combinations of π ± and π 0 candidates in continuum events. Continuum events tend to have a more "jet-like" structure than B decays which are produced nearly at rest in the CM system. To enhance discrimination between signal and continuum, we use a neural network (NN) [20] to combine four discriminating variables: the angles with respect to the beam axis of the B momentum and B thrust axis in the Υ (4S) frame, and the zeroth and second order polynomials L 0,2 of the energy flow about the B thrust axis. The polynomials are defined by L n = i p i ·|cos θ i | n , where θ i is the angle with respect to the B thrust axis of any track or neutral cluster i, p i is its momentum, and the sum excludes the B candidate. The NN is trained with off-peak data and simulated signal events. The final sample of signal candidates is selected with a requirement on the NN output that retains 77% (8%) of the signal (continuum) events. A total of 35444 on-peak data events pass the selection.
The time difference ∆t is obtained from the measured distance between the z positions (along the beam direction) of the B 0 3π and B 0 tag decay vertices, and the boost βγ = 0.56 of the e + e − system: ∆t = ∆z/βγc. The B 0 tag vertex is determined from the charged particles in the event not included in the signal B. To determine the flavor of the B 0 tag we use the B flavor-tagging algorithm of Ref. [1]. This produces six mutually exclusive tagging categories. We improve the efficiency of the signal selection by retaining untagged events in a seventh category which contribute to the measurement of direct CP violation.
Multiple B candidates passing the full selection occur in 16% (ρ ± π ∓ ) and 9% (ρ 0 π 0 ) of ρ(770) MC events. If the multiple candidates have different π 0 candidates, we choose the B candidate with the reconstructed π 0 mass closest to the nominal π 0 mass; in the case that more than one candidate have the same π 0 , we arbitrarily chose a reconstructed B candidates passing the selection (this occurs in 4% of events).
The signal efficiency determined from MC simulation is 24% for B 0 → ρ ± π ∓ and B 0 → ρ 0 π 0 events, and 11% for non-resonant B 0 → π + π − π 0 events. The signal efficiency distribution on the SPD is shown in Figure 3. The signal events passing the event selection are a combination of correctly reconstructed ("truth-matched", TM) events and mis-reconstructed ("self-cross-feed", SCF) events. Of the selected signal events, 22% of B 0 → ρ ± π ∓ , 13% of B 0 → ρ 0 π 0 , and 6% of nonresonant events are mis-reconstructed, according to MC. Mis-reconstructed events occur when a track or neutral cluster from the tagging B is assigned to the reconstructed signal candidate. This occurs most often for low-momentum particles and photons; hence the misreconstructed events are concentrated in the corners of the standard Dalitz plot. Since these are also the areas where the ρ resonances overlap strongly, it is important to model the mis-reconstructed events correctly. The details of the model for the distributions of misreconstructed events in the Dalitz plot are described in Section III C 1.

B. BACKGROUND FROM OTHER B DECAYS
We use MC simulated events to study the background from other B decays. More than one-hundred channels were considered in these studies, of which 29 are included in the final likelihood model. These exclusive B-background modes are grouped into eighteen different classes according to their kinematic and topological properties: six for charmless B + decays, eight for charmless B 0 decays and four for exclusive charmed B 0 decays. Two additional classes account for inclusive B 0 and B + charmed decays. Table I summarizes the twenty background classes that are used in the fit. For each mode, the expected number of selected events is computed by multiplying the selection efficiency (estimated using MC simulated decays) by the branching fraction, scaled to the dataset luminosity (346 fb −1 ). The world average branching ratios have been used for the experimentally known decay modes [12,21]. When only upper limits are given, they have been translated into branching ratios including additional conservative hypotheses (e.g., 100% longitudinal polarization for B → ρρ decay) if needed.

C. THE MAXIMUM-LIKELIHOOD FIT
We perform an unbinned extended maximumlikelihood fit to extract the total B 0 → π + π − π 0 event yield, and the U and I coefficients defined in Eqs. (9)- (14). The fit uses the variables ∆t, m ′ , θ ′ , m ES , ∆E ′ , and NN output to discriminate signal from background. The ∆t distribution is sensitive to mixing-induced CP violation but also provides additional continuum-background rejection.
The selected on-resonance data sample is assumed to consist of signal, continuum-background, and Bbackground components, separated by the flavor and tagging category of the tag side B decay. The probability density function P c i for event i in tagging category c is the sum of the probability densities of all components, 19.1 ± 3.5 57 ± 11 where N 3π is the total number of π + π − π 0 signal events in the data sample; f c 3π is the fraction of signal events that are in tagging category c; f c SCF is the fraction of SCF events in tagging category c, averaged over the Dalitz plot; P c 3π−TM,i and P c 3π−SCF,i are the products of PDFs of the discriminating variables used in tagging category c for TM and SCF events, respectively; N c qq is the number of continuum events that are in tagging category c; q tag,i is the tag flavor of the event, defined to be +1 for a B 0 tag and −1 for a B 0 tag ; A qq, tag parameterizes possible flavor tag asymmetry in continuum events; P c qq,i is the continuum PDF for tagging category c; N B + class (N B 0 class ) is the number of charged (neutral) B-related background classes considered in the fit; where i is the event index and j is a B-background class.
The extended likelihood over all tagging categories is given by where N c is the total number of events expected in category c.
A total of 68 parameters, including the inclusive signal yield N 3π and the 26 U and I coefficients from Eq. (7), are varied in the fit. Most of the parameters describing the continuum distributions are also free in the fit. The parameterizations of the PDFs are described below and are summarized in Tab. II.

THE ∆t AND DALITZ PLOT PDFS
The Dalitz plot PDFs require as input the Dalitz plotdependent relative selection efficiency ε = ε(m ′ , θ ′ ), and the SCF fraction, f SCF = f SCF (m ′ , θ ′ ). Both quantities are taken from MC simulation. Away from the Dalitz plot corners the efficiency is uniform, while it decreases when approaching the corners where one of the three particles in the final state is almost at rest in the LAB frame so that the acceptance requirements on the particle reconstruction become restrictive. Combinatorial backgrounds, and hence SCF fractions, are large in the corners of the Dalitz plot due to the presence of soft neutral clusters and tracks.
For an event i, we define the time-dependent Dalitz plot PDFs where P 3π−TM,i and P 3π−SCF, i are normalized. The normalization involves the expectation values and similarly for f SCF ε | det J| f κ f σ * , where all quantities in the integrands are Dalitz-plot dependent.
Equation (20) invokes the phase space-averaged SCF fraction f SCF ≡ f SCF | det J| f κ f σ * . The PDF normalization is decay-dynamics-dependent and is computed iteratively. We determine the average SCF fractions separately for each tagging category from MC simulation.
The width of the dominant ρ(770) resonance is large compared to the mass resolution for TM events (about 8 MeV/c 2 Gaussian resolution). We therefore neglect resolution effects in the TM model. Mis-reconstructed events have a poor mass resolution that strongly varies across the Dalitz plot. These events are described in the fit by a two-dimensional resolution function which represents the probability to reconstruct at the coordinate (m ′ r , θ ′ r ) an event that has the true coordinate (m ′ t , θ ′ t ). This function obeys the unitary condition and is convolved with the signal model. The R SCF function is obtained from MC simulation. The dynamical information in the signal model is described in Section I A and is connected with ∆t via the matrix element in Eq. (7), which serves as the PDF. The PDF is modified by the effects of mistagging and the limited vertex resolution [17]. The ∆t resolution function for signal and B-background events is a sum of three Gaussian distributions, with parameters determined by a fit to fully reconstructed B 0 decays [1]. Since the majority of SCF events arise from mis-reconsructed π 0 decays which do not affect the vertex resolution, we use the same resolution function for TM and SCF events.
The Dalitz plot-and ∆t-dependent PDFs factorize for the charged-B background modes, but not necessarily for the B 0 background due to B 0 B 0 mixing.
The charged B-background contribution to the likelihood (20) involves the parameter A B + , tag , multiplied by the tag flavor q tag of the event. In the presence of significant "tag-'charge" correlation (represented by an effective flavor tag versus Dalitz coordinate correlation), it parameterizes possible fake direct CP violation or asymmetries due to detector effects in these events. We also use separate square Dalitz plot PDFs for B 0 and B 0 flavor tags, and a flavor-tag-averaged PDF for untagged events. The PDFs are obtained from MC simulation and are described with the use of non-parametric functions.
The ∆t resolution parameters are determined by a fit to fully reconstructed B + decays. For each B + -background class we obtain effective lifetimes from MC to account for the mis-reconstruction of the event that modifies the nominal ∆t resolution function. The neutral-B background is parameterized with PDFs that depend on the flavor tag of the event. In the case of CP eigenstates, correlations between the flavor tag and the Dalitz coordinate are expected to be small. However, non-CP eigenstates, such as a ± 1 π ∓ , may exhibit such correlations. Both types of decays can have direct and mixing-induced CP violation. A third type of decay involves charged kaons (e.g. ρ ± K ∓ ) and does not exhibit mixing-induced CP violation, but usually has a strong correlation between the flavor tag and the Dalitz plot coordinate, because these decays correspond to Bflavor eigenstates. The Dalitz plot PDFs are obtained from MC simulation and are described with the use of non-parametric functions. For neutral-B background, the signal ∆t resolution model is assumed.
The Dalitz plot treatment of the continuum events is similar to that used for charged-B background. The square Dalitz plot PDF for continuum background is obtained from on-resonance events selected in the m ES sidebands (defined as 5.225 < m ES < 5.265) and corrected for a 5% feed-through from B decays. A large number of cross checks have been performed to ensure the high fidelity of the empirical shape parameterization. The continuum ∆t distribution is parameterized as the sum of three Gaussian distributions with common mean and three distinct widths. The widths scale with the estimated ∆t uncertainty for each event. This yields six shape parameters that are determined by the fit. The model is motivated by the observation that the ∆t average is independent of its error, and that the ∆t RMS depends linearly on the ∆t error.

PARAMETERIZATION OF THE OTHER VARIABLES
The m ES distribution of TM signal events is parameterized by a bifurcated Crystal Ball function [22], which is a combination of a one-sided Gaussian and a Crystal Ball function, given as: The peak position of this function, m, is determined by the fit to on-peak data while the other parameters are taken from signal MC. A non-parametric function [23] is used to describe the SCF signal component. The ∆E ′ distribution of TM events is parameterized by a double Gaussian function, where all five parameters de-pend linearly on m 2 0 . The parameters of the narrow Gaussian are determined by the fit to data while the others are obtained from signal MC. Mis-reconstructed events are parameterized by a broad single Gaussian function whose parameters are taken from signal MC.
Both m ES and ∆E ′ PDFs are parameterized by nonparametric functions for all B-background classes. Continuum events are parameterized with an Argus shape function [24] f and a second-order polynomial in ∆E ′ , with parameters determined by the fit. The value of m max ES is 5.2886 GeV/c 2 .
We use non-parametric functions to empirically describe the distributions of the NN outputs found in the MC simulation for TM and SCF signal events, and for B-background events. We distinguish tagging categories for TM signal events to account for differences observed in the shapes.
The continuum NN distribution is parameterized by a third-order polynomial. The coefficients of the polynomial are determined by the fit. Continuum events exhibit a correlation between the Dalitz plot coordinate and the inputs to the NN. To account for this correlation, we introduce a linear dependence of the polynomial coefficients on the distance of the Dalitz plot coordinate from kinematic boundaries of the Dalitz plot. The parameters describing this dependence are determined by the fit.

IV. FIT RESULTS
The maximum-likelihood fit results in a B 0 → π + π − π 0 event yield of N 3π = 2067 ± 86, where the error is statistical only. The results for the U and I coefficients are given together with their statistical and systematic errors in Table III. The corresponding correlation matrix is given in Table IV. We have generated a sample of Monte Carlo experiments to determine the probability density distributions of the fit parameters. Within the statistical uncertainties of this sample we find Gaussian distributions for the fitted U and I coefficients. This allows us to use the least-squares method to derive other quantities from these (Section VI).    The signal is dominated by B 0 → ρ ± π ∓ decays. We observe an excess of ρ 0 π 0 events (see, mainly, U + 0 ), which is in agreement with our previous upper limit [25] and the latest measurement from the Belle collaboration [26]. We find the ratio of ρ(1450)/ρ(770) ( ρ(1700)/ρ(770) ) rates to be 0.13 ± 0.04 (0.07 ± 0.04) where the errors are statistically only. For the relative strong phase be-tween the ρ(770) and ρ(1450) (ρ(1700)) amplitudes we find (163±22) • ((5±36) • (statistical errors only), which is compatible with the result from τ and e + e − data. These results for the ρ amplitudes are compatible with the findings in τ and e + e − decays [13]. on ∆t, as well as the Dalitz plot variables m ′ and θ ′ . All distributions are enhanced in signal content by selecting on the ratio of the probability the event is signal to the total, P s ig/ P , excluding the variable plotted. Figure 5 shows the distribution of the minimum of the three dipion invariant masses, again enhanced in signal content. This plot shows clearly that ρ(770) dominates the signal component.

V. SYSTEMATIC STUDIES
The contributions to the systematic error on the signal parameters are summarized in Table V. Table VI summarizes the correlation coefficients extracted from the systematic covariance matrix. For a given systematic effect, we vary a parameter in the fit (e.g. the ρ(770) mass), refit the data, and construct the systematic covariance matrix for that source based on the deviations of the U and I coefficients from the nominal values. The (i, j) matrix element is given as where δ i is the difference between the two two fits for variable i. The total systematic covariance matrix is obtained by adding together the covariance matrices in quadrature from the different systematic sources.
To estimate the contribution to B 0 → π + π − π 0 decay from other resonances and non-resonant decays, we fit the on-peak data including these other possible decays in the fit model. For simplicity, we assume a uniform Dalitz distribution for the non-resonant events and consider possible non-ρ resonances including f 0 (980), f 2 (1270), and a low mass S-wave σ whose mass and width we take to be 478 MeV/c 2 and 324 MeV/c 2 , respectively [12]. The fit does not find a significant signal for any of those decays. However, the inclusion of the broad, low mass π + π − S-wave component significantly degrades our ability to identify ρ 0 π 0 events. The systematic effect (contained in the "Dalitz plot model" rows in Table V) is estimated by generating Monte Carlo samples including the other B 0 → π + π − π 0 modes and fitting with the nominal setup, where only ρ(770) is taken into account. We vary the mass and width of the ρ(770), ρ(1450), and ρ(1700) resonances within ranges that exceed twice the errors found for these parameters extracted from τ decays and e + e − annihilations [13], and assign the observed shifts in the measured U and I coefficients as systematic uncertainties ("ρ, ρ ′ , ρ ′′ lineshape" in Table V). Since some of the U and I coefficients exhibit significant dependence on the ρ(1450) and ρ(1700) contributions, we leave their amplitudes (phases and fractions) free to vary in all fits.
To validate the fitting tool, we perform fits on large MC samples with the measured proportions of signal, continuum, and B-background events. No significant biases are observed in these fits. The statistical uncertainties on their fit parameters are taken as systematic uncertainties ("Fit bias" in Table V).
Another potentially large source of systematic uncertainty is the B-background model. The expected event yields from the background modes are varied according to the uncertainties in the measured or estimated branching fractions ("N Background " in Table V). Since B-background modes may exhibit CP violation, the corresponding parameters are varied either within their measured ranges (if available) or within ±0.5 (if unmeasured) ("B background CP " in Table V).
Other systematic effects are much less important to the measurements of U and I coefficients and are combined in the "Others" field in Table V. Details are given below.
The parameters for the continuum events are determined by the fit. No additional systematic uncertainties are assigned to them. An exception to this is the Dalitz plot PDF; to estimate the systematic uncertainty from the m ES sideband extrapolation, we select large samples of off-resonance data by loosening the requirements on ∆E and the NN. We compare the distributions of m ′ and θ ′ between the m ES sideband and the signal region. No significant differences are found. We assign as systematic error the effect seen when weighting the continuum Dalitz plot PDF by the ratio of the 2-dimensioal histograms taken from the signal region and sideband data sets. This effect is mostly statistical in origin.
The uncertainties associated with ∆m d and τ are estimated by varying these parameters within the uncertain- I Im  ties on the world averages [12].
The systematic effects due to the signal PDFs comprise uncertainties in the PDF parameterization, the treatment of misreconstructed decays, the tagging performance, and the modeling of the signal contributions.
When the signal PDFs are determined from fits to a control sample of fully reconstructed B decays to exclusive final states with charm, the uncertainties are obtained by varying the parameters within the statistical uncertainties. In other cases, the dominant parameters have been left free to vary in the fit, and the differences observed in these fits are taken as systematic errors.
The average fraction of misreconstructed signal events predicted by the MC simulation has been verified with fully reconstructed B → Dρ events [17]. No significant differences between data and the simulation are found. We vary f SCF for all tagging categories relatively by 25% to estimate the systematic uncertainty.  As is done for the signal PDFs, we vary the ∆t resolution parameters and the flavor-tagging parameters within their uncertainties and assign the differences observed in the data fit with respect to the nominal fit as systematic errors.
The systematic errors for the parameters that measure interference effects are dominated by the uncertainty in the signal model, mainly the description of the ρ resonance tails. For the other parameters, the uncertainty on the fit bias and the B-background contamination are important.
As a validation of our treatment of the time dependence we allow τ B 0 to vary in the fit. We find τ B 0 = (1.513 ± 0.066) ps, while the remaining free parameters are consistent with the nominal fit. To validate the SCF modeling, we leave the average SCF fractions per tagging category free to vary in the fit and find results that are consistent with the MC prediction.

VI. INTERPRETATION OF THE RESULTS
We can use the results of this time-dependent Dalitz analysis to extract the B 0 (B 0 ) → ρ ± π ∓ parameters defined in Ref. [17]: where Q tag = 1(−1) when the tagging meson B 0 tag is a B 0 (B 0 ). The time-and flavor-integrated charge asymmetry A ρπ measures direct CP violation and the quantities S and C parameterize mixing-induced CP violation related to the angle α, and flavor-dependent direct CP violation, respectively. The parameters ∆C ρπ and ∆S ρπ are insensitive to CP violation.
The U and I coefficients are related to the parameters as follows: where C = (C + + C − )/2, ∆C = (C + − C − )/2, S = (S + +S − )/2, and ∆S = (S + −S − )/2 . The definitions of Eq. (33) explicitly account for the presence of interference effects, and are thus exact even for a ρ with finite width, as long as the U and I coefficients are obtained with a Dalitz plot analysis. This treatment leads to slightly increased statistical uncertainties compared to the results obtained neglecting the interference effects. Using a least-squares method including statistical and systematic correlations for the U and I coefficients, we obtain: where the first errors are statistical and the second are the systematic uncertainties. For the other parameters in the description of the B 0 (B 0 ) → ρπ decay-time dependence, we measure ∆C = 0.39 ± 0.09 ± 0.09 , ∆S = −0.01 ± 0.14 ± 0.06 . In addition, we measure the B 0 → ρ 0 π 0 CP -violation parameters and decay fraction to be The systematic errors are dominated by the uncertainty on the CP content of the B-related backgrounds. Other contributions are the signal description in the likelihood model (including the limit on non-resonant B 0 → π + π − π 0 events), and the fit bias uncertainty. The large systematic error on C 00 is due to the possible π + π − Swave contribution. The correlation matrix, including statistical and systematic uncertainties, of the eight quasitwo-body parameters is given in Table VII. One can transform the experimentally convenient (uncorrelated) direct-CP violation parameters C and A ρπ into A +− ρπ , A −+ ρπ , defined by where κ +− = (q/p)A − /A + and κ −+ = (q/p)A + /A − , so that A +− ρπ (A −+ ρπ ) involves only diagrams where the ρ (π) meson is formed from the W boson. We find A +− ρπ = 0.03 ± 0.07 ± 0.04 , A −+ ρπ = −0.37 ± 0.16 +0.09 −0.10 , with a correlation coefficient of 0.62 between A +− ρπ and A −+ ρπ . The confidence level contours including systematic uncertainties are shown in Fig. 6. The significance, including systematic uncertainties and calculated by using a minimum χ 2 method, of direct CP violation is less than 3.0σ.
The measurement of the resonance interference terms allows us to determine the relative phase between the amplitudes of the decays B 0 → ρ − π + and B 0 → ρ + π − . Through the definitions in Eqs. (9)-(14), we can derive a constraint on δ +− from the measured U and I coefficients by performing a least-squares minimization with the six complex amplitudes as free parameters. The constraint can be improved with the use of strong isospin symmetry. The amplitudes A κ represent the sum of tree-level and penguin-type amplitudes, which have different CKM factors: the tree-level (T κ ) B 0 → ρ κ π κ transition amplitude is proportional to V ud V * ub , while the corresponding penguin-type amplitude (P κ ) involves V qd V * qb , where q = u, c, t. Here we denote by κ the charge conjugate of κ, where κ = 0 when κ = 0. Using the unitarity of the CKM matrix one can reorganize the amplitudes and obtain [14] A κ = T κ e −iα + P κ , (q/p)A κ = T κ e +iα + P κ , where the magnitudes of the CKM factors have been absorbed in T κ , P κ , T κ and P κ . The Eqs. (46) represent 13 unknowns of which two can be fixed due to an arbitrary global phase and the normalization condition U + + = 1. Using strong isospin symmetry one can identify P 0 = −(P + + P − )/2, which reduces the number of unknowns to be determined by the fit to nine. This set of parameters provides the constraint on δ +− , shown in the left plot of Fig. 7. We find for the solution that is favored by the fit where the errors include both statistical and systematic effects. There is only a marginal constraint on δ +− obtained at 95% confidence level (C.L.). Finally, following the same procedure, we can also derive a constraint on α from the measured U and I coefficients. The resulting C.L. function versus α is given in the right-hand plot of Fig. 7, including systematic uncertainties. Ignoring the mirror solution at α + 180 • , we find at 68% C.L.
Almost no constraint on α is achieved at two sigma and beyond.

VII. SUMMARY
We have presented a measurement of CP -violating asymmetries in B 0 → π + π − π 0 decays dominated by the ρ resonance. The results are obtained from a data sample of 375 million Υ (4S) → BB decays. We perform a timedependent Dalitz plot analysis. From the measurement of the coefficients of 26 form-factor bilinears we determine the three CP -violating and two CP -conserving quasi-twobody parameters, and find no evidence of direct CP violation. Taking advantage of the interference between the ρ resonances in the Dalitz plot, we derive constraints on the relative strong phase between B 0 decays to ρ + π − and ρ − π + , and on the angle α of the Unitarity Triangle. These measurements are consistent with the results obtained by Belle [27] as well as with the expectation of a SM fit to all constraints on the CKM matrix [28,29].

VIII. ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US