Dalitz Plot Analysis of the Decay B0(B0bar) -->K+/- pi-/+ pi0

We report a Dalitz-plot analysis of the charmless hadronic decays of neutral B mesons to K^+/- pi^-/+ pi^0. With a sample of (231.8+/-2.6) 10^6 Upsilon(4S) to B Bbar decays collected by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC, we measure the magnitudes and phases of the intermediate resonant and nonresonant amplitudes for B0 and B0bar decays and determine the corresponding CP-averaged branching fractions and charge asymmetries. The inclusive branching fraction and CP-violating charge asymmetry are measured to be BF(B0 -->K+ pi^- pi0)=(35.7-1.5+2.6 +/-2.2)10^(-6), and A_CP=-0.030+0.045-0.051 +/-0.055 where the first errors are statistical and the second systematic. We observe the decay B0 -->K^{*0}(892) pi^0 with the branching fraction BF(B0 -->K^{*0}(892) pi^0)=(3.6_{-0.8}^{+0.7} +/-0.4) 10^{-6}. This measurement differs from zero by 5.6 standard deviations (including the systematic uncertainties). The selected sample also contains B0 -->D^0bar pi^0 decays where D^0bar -->K+ pi-, and we measure BF(B0 -->D0bar pi^0)=(2.93 +/- 0.17 +/-0.18) 10^{-4}.


I. INTRODUCTION
Amplitude analyses of three-body decays of B mesons with no charm particle in the final state are well suited to study the Cabibbo-Kobayashi-Maskawa (CKM) framework [1] for charged current weak interactions. In the analysis of a Dalitz plot the strong phase motions along the lineshapes of interfering resonances are measured and can be used to constrain the weak phases related to the CKM parameters which, in the Standard Model govern CP -violation. Following the path [2,3,4] of the 3-pion B meson decays which give constraints on the CKM angle α CKM ≡ arg(−V td V * tb /V ud V * ub ), it has been shown in [5,6] that B decays into a kaon and two pions are sensitive to the angle γ CKM ≡ arg(−V ud V * ub /V cd V * cb ). In this paper we present a time independent amplitude analysis of the flavor-specific B 0 → K + π − π 0 decay [7] which finalizes preliminary studies [8,9]. The analysis compares the Dalitz plots of the B 0 and B 0 decays where low-mass Kπ and ππ resonances interfere. Previous measurements of the three-body final state [10,11] and subdecays [12,13] to a vector and a pseudoscalar meson have been published. Other B → Kππ decays have been studied in [14,15,16,17]. A phenomenological study of three-body B meson decays without charm in the final state is presented in [18]. This paper is organized as follows. We first present in Section II the decay model based on an isobar expansion of the three-body decay amplitude. The complex coefficients of the expansion are the unknowns we seek to determine by means of an unbinned extended maximum likelihood fit. We describe the detector and dataset in Section III, the procedure used to select the data sample in Section IV, and the fit method in Section V. The results are then described in Section VI together with the accounting of the systematic uncertainties (in Section VII). Finally in Section VIII, we summarize our findings and present an interpretation of our results.

II. DECAY AMPLITUDES
The B 0 → K + π − π 0 decay amplitude is a function of two independent kinematic variables commonly chosen to be the invariant masses squared 1 , x = m 2 K ± π ∓ and y = m 2 K ± π 0 . The Dalitz plot (DP) is the x, y bidimensional distribution. It is customary to express the decay USA ‡ Now at Tel Aviv University, Tel Aviv, 69978, Israel § Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy ¶ Also with Università della Basilicata, Potenza, Italy * * Also with Universita' di Sassari, Sassari, Italy † † Now at Cadence Design Systems Inc., San Jose, CA 95134, USA 1 We use natural units whereh = c = 1 in our algebraic equations. amplitude as a sum over intermediate (isobar) states: A(x, y) = j t j e iφj f j (x, y), (1) and similarly for the B 0 → K − π + π 0 Dalitz plot, The isobar coefficients t j e iφj are constant over the Dalitz plot. For each decay channel, the isobar phase φ j is the sum of a strong phase, the same for B 0 and B 0 decays, and a weak phase which changes sign. The decay dynamics of an intermediate state are described by the f j (x, y) function which structures the Dalitz plot. For instance a resonance formed in the K + π − system gives a contribution which factorizes as: where R j (x) is the resonance mass distribution or lineshape and T j (x, y) models the angular dependence. W B j (x) = B B (Rp * (x)) B j (Rq(x)), the product of Blatt-Weisskopf damping factors [19], slightly deviates from unity as a function of x through the breakup momenta 2 of the (quasi) two body B and resonance decays multiplied by a range parameter R. The f j are normalized, DP |f j (x, y)| 2 dx dy = 1.
We use the Zemach tensor formalism [20,21] for the angular distribution T (J) j (x, y) of a process by which a pseudoscalar B meson produces a spin-J resonance in association with a bachelor pseudoscalar meson. For J = 0, 1, 2, we have: where 3 p(x, y) [ q(x)] is the momentum vector of the bachelor particle (the resonance decay product Q defined below) measured in the resonance rest frame. For a neutral (charged) Kπ resonance, Q is the pion (kaon), and for a dipion resonance, Q is the π 0 . Notice that these choices define for each two-body system the helicity angle θ j = ( p j , q j ) between 0 and π. I: The nominal model for the decay B 0 → K + π − π 0 comprises a nonresonant part and five intermediate states.
The three types of lineshape are described in the text. The resonances masses and widths are from [22], except for the LASS shape [23]. We use the same LASS parameters for both neutral and charged Kπ systems. Additional resonances that may contribute are included in extended models which we study to estimate the systematic uncertainties.
Intermediate state Lineshape Parameters Nominal model  (Table I) for the decay B 0 → K + π − π 0 includes a nonresonant contribution which is uniformly distributed over the Dalitz plot, and five resonant intermediate states: ρ − (770), K * (892) +,0 and (Kπ) * +,0 0 . The latter notation introduced by the BABAR experiment [14] denotes phenomenological amplitudes describing the neutral and charged (Kπ) S-waves each by a coherent superposition of an elastic effective range term and a term for the K * 0 (1430) scalar resonance. It describes current knowledge on low energy Kπ systems with a small number of parameters. Variations to the nominal model are used to estimate the model-dependent systematic uncertainty in the results. Three lineshape parameterizations R j (x) are used. Parameters are taken from [22] unless stated otherwise.

The relativistic Breit-Wigner distribution
The relativistic Breit-Wigner (RBW) parameterization is used for K * (892) +,0 , K * 2 (1430) +,0 , and K * (1680) +,0 : The mass-dependence of the total width Γ (J) j can be ignored for high-mass states. For the low-mass states which decay only elastically, it is defined by where m j is the mass of the resonance j, Γ 0 j = Γ j (m 2 j ) its width, and the barrier factors (squares of the Blatt-Weisskopf damping factors [19]) are: All range parameters were set to R = 0 in the analysis but we checked that the systematic deviations associated with more realistic values taken or interpolated from [24] are below the numerical accuracy we use to quote our results.

The Gounaris-Sakurai distribution
The Gounaris-Sakurai (GS) parameterization [25] [Eq. (9)] is used for ρ − (770), ρ − (1450) and ρ − (1700): with the same x-dependence of the width as for the RBW. The expressions of the constant d j and the function f j (x) in terms of m j and Γ 0 j are given in Reference [25].

The LASS distribution
For the Kπ S-wave resonances, (Kπ) * +,0 0 , which dominate for m Kπ below m max j = 2 GeV/c 2 , an effectiverange parameterization was suggested [26] to describe the slowly increasing phase as a function of the Kπ mass. We use the parameterization as in the LASS experiment [23], tuned for B decays: a is the scattering length, and r the effective range (Table I). A LASS isobar coherently comprises the actual K * 0 (1430) (82%), the effective range term (57%) and the (destructive) interference term (39%).
In this analysis we use a maximum likelihood fit to measure the inclusive branching fraction and CPviolation asymmetry, as well as the isobar fractions F F j , (CP -averaged over B 0 and B 0 ) and the CP -violation charge asymmetries: and the isobar phases relative to those of the K * ± π ∓ channel. In those expressions, t j and t j are the fitted magnitudes for the intermediate state j; φ j and φ j are the fitted relative phases. Note that, due to interference, the fractions F F k in general do not add up to unity. Nevertheless, we define the quasi-two-body branching fraction for an intermediate state j as follows:

B. THE SQUARE DALITZ PLOT
The accessible phase space of charmless three-body B decays is unusually large. Most contributing resonances have masses much lower than the B mass. Hence signal events cluster along the Dalitz plot boundaries. This is also true for background events. Past experience has shown that another set of variables, defining the Square Dalitz Plot (SDP) [27] is well suited to such configurations. It is defined by the mapping: where m = √ x and θ are respectively the invariant mass and helicity angle of the K ± π ∓ system. m max = m B − m π 0 and m min = m K + + m π − are the kinematic limits of m. The new variables both range between 0 and 1.

III. THE BABAR DETECTOR AND DATASET
The data used in this analysis were collected with the BABAR detector at the PEP-II asymmetric e + e − storage rings between October 1999 and July 2004. The sample consists of about 210.6 fb −1 corresponding to N BB = 231.8 ± 2.6 million BB pairs taken on the peak of the Υ (4S) resonance (on resonance) and 21.6 fb −1 recorded at a center-of-mass (CM) energy 40 MeV below (off resonance).
A detailed description of the BABAR detector is given in [28]. Charged-particle trajectories are measured by a five-layer, double-sided silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH) immersed in a 1.5 T magnetic field. Charged-particle identification is achieved by combining the information from a ringimaging Cherenkov device with the ionization energy loss (dE/dx) measurements from the DCH and SVT. Photons are detected in a CsI(Tl) electromagnetic calorimeter (EMC) inside the coil. Muon candidates are identified in the instrumented flux return of the solenoid.We use GEANT4-based [29] software to simulate the detector response and account for the varying beam and environmental conditions.

A. SIGNAL SELECTION AND BACKGROUND REJECTION
To reconstruct B 0 → K + π − π 0 decays, we select two charged particles and two photons. The charged particle candidates are required to have transverse momenta above 100 MeV/c and at least 12 hits in the DCH. They must not be identified as electrons or muons or protons. A K + candidate must be identified as a kaon (with a misidentification probability smaller than 4%) and a π − candidate must not be identified as a kaon (with a misidentification probability smaller than 7%). The misidentification probabilities are momentum dependent and therefore vary across the Dalitz plot. A π 0 candidate is built from a pair of photon candidates, each with an energy greater than 50 MeV in the laboratory frame (LAB) and a lateral energy deposition profile in the EMC consistent with an electromagnetic shower. The invariant mass of a π 0 candidate must satisfy 0.11 < m γγ < 0.16 GeV/c 2 , a wide enough window to accommodate the variation of the resolution across the Dalitz Plot from 4.5 to 8 MeV/c 2 .
At the Υ (4S) resonance, B mesons are characterized by two nearly independent kinematic variables, the beam energy substituted mass and the energy difference: where E and p are energy and momentum, the subscripts 0 and B refer to the e + e − -beam system and the B candidate respectively; s is the square of the center-of-mass energy and the asterisk labels the CM frame. We require that m ES > 5.27 GeV/c 2 . To avoid a bias in the Dalitz plot from the dependence on the π 0 energy of the reso-lution in ∆E, we introduce the dimensionless quantity: where the bounds obtained from simulation, ∆E max = 0.08−0.0014·m K + π − , ∆E min = −0.14+0.005·m K + π − , all in units of GeV follow the variation of the ∆E resolution. We require |∆E ′ | ≤ 1. Continuum e + e − → qq (q = u, d, s, c) events are the dominant background. To enhance discrimination between signal and continuum, we select events by using a neural network [30] with an output N N which combines three discriminating variables: the angles of the B momentum and the B thrust axis with respect to the e + beam direction in the CM frame and the difference 2L 2 − L 0 between two variables characterizing the energy flow about the B thrust axis. We have L n ≡ i p i ·| cos θ i | n , where the sum runs over all charged and neutral particles in the event (except for those in the B candidate) whose momenta p i make angles θ i with the B thrust axis. The neural network was trained on off resonance data and correctly reconstructed signal Monte Carlo events. A data sample with about 4000 fully reconstructed B 0 → D * − π + decays with D * − → D 0 π − and D 0 → K + π − is used to validate the shapes of the distributions on which the selection procedure is based.
Between 3% and 17% of the signal events have multiple reconstructed B candidates (usually two) depending on their location in the Dalitz plot. When distinct π 0 candidates make acceptable B combinations, we choose that with the reconstructed π 0 mass closest to the nominal value of 0.1349 GeV/c 2 . When several acceptable B combinations can be made of distinct charged particle sets, we arbitrarily choose one (in a reproducible fashion by using the date and time at which the event was recorded as a random number).
There are 8014 events in the data sample after the selection. The B meson candidate in each event is mass constrained to ensure that the measurement falls within the Dalitz plot boundary. The resulting standard and square Dalitz plots are shown in Fig. 1. We now describe the composition of the selected data sample.

B. TRUTH-MATCHED AND SELF-CROSS-FEED SIGNAL EVENTS
Using the Monte Carlo simulation as in [3], we distinguish between the correctly reconstructed and the misreconstructed signal events. A correctly reconstructed event where the three particles of the B candidate match the generated ones, is called a Truth-Matched (TM) event. A misreconstructed signal event contains a B meson which decays to the signal mode, but one or more reconstructed particles in the B candidate are not actually from the decay of that B. Misreconstructed signal is called Self-Cross-Feed (SCF). Misreconstruction is primarily due to the presence of low momentum pions. Con-sequently the SCF fraction varies across the Dalitz plot as shown in Fig. 2. For each point in the Dalitz plot there is an efficiency ε(m ′ , θ ′ ) to reconstruct an event either correctly or incorrectly. The quality of the reconstruction is poor where the SCF fraction f SCF (m ′ , θ ′ ), is high. This occurs in the corners of the Dalitz plot where one of the final-state particles has a low momentum in the LAB frame. These variations can be seen in Fig. 2 and in Table II which compares efficiencies and f SCF values averaged over the Dalitz plot for different B 0 → K + π − π 0 subdecays 4 computed using high statistics Monte Carlo samples (4.9 × 10 6 B 0 → K + π − π 0 events). It is important to keep a high efficiency in the Dalitz plot corners where the low-mass vector resonances interfere. Overall the total efficiency is close to 15%. The main sources of inefficiency are the ∆E ′ selection (ε ≈ 70 − 80%), the kaon identification (ε ≈ 69%) and the neural network selection (ε ≈ 60%). C. BACKGROUND

Continuum background
Although the neural network selection rejects 96% of the continuum events, this background is the dominant II: Signal selection efficiencies (overall and truthmatched) and fraction of misreconstructed events (SCF) for different signal modes after full selection. As explained in the text, ε is the overall efficiency for (TM and SCF) signal events. Hence εTM = ε(1 − fSCF). The relative statistical precision on these numbers is a few tenths of a percent.

Background from other B decays
Since there is no restriction on any two-body invariant mass of the final state particles, large backgrounds from other B decays occur. We use high statistics Monte Carlo samples to study these backgrounds. Conservative assumptions about unknown branching fractions are made. For instance when only an upper limit U is known for the branching fraction of a decay channel, we use U/2 ± U/2 as an input to the simulation.
Inclusive and exclusive B decays with or without charm are grouped into ten classes to be used in the fit. Rates, and topological and kinematical similarities are studied to define the classes listed in Table III. There are two classes with inclusive b → c decays which are distinguished according to whether a genuine D 0 is part of the B candidate. We keep the exclusive decays B 0 → D 0 π 0 with D 0 → K + π − in the data sample because the copious yield of approximately 400 well identified events helps to control the fit algorithm.

V. THE MAXIMUM LIKELIHOOD FIT
We perform an unbinned extended maximum likelihood fit to determine the total B 0 → K + π − π 0 event yield, the magnitudes t j , t j and the phases φ j , φ j of the complex isobar coefficients of the decay amplitude defined in Eq. (1) and (2). The fit uses the variables m ′ , θ ′ , m ES , ∆E ′ and NN, to discriminate signal from background.  V). For each channel, we give (anticipating Section VI) either the fitted number of events in the data sample if its yield is allowed to vary in the fit procedure or the expected number otherwise.

Class Mode
Events

A. THE LIKELIHOOD FUNCTION
The selected on resonance data sample is assumed to consist of signal, continuum-background and background from other B decays. Accordingly the likelihood function of event i is written: where the sum runs over all the B-background classes listed in Table III. All background likelihood functions have the same expression: where q K i is the kaon charge in event i, and A back is the charge asymmetry.
We consider B 0 and B 0 separately to build the signal likelihood function.
Each part has two terms, one for the TM and one for the SCF events: and similarly for L − sig,i . f SCF , the fraction of SCF-events averaged over the Dalitz plot, assumed to be the same for both flavors, is discussed below. We denote by N comp the expected number of events for species comp. The fit maximizes the extended likelihood function 5 : where N = 8014 events is the size of the data sample and N tot = N sig + N qq + c N Bbg,c , is the expected number of events. The five-dimensional probability density functions (PDF) P are the products of the four PDFs of the measured discriminating variables The correlations among the measurements are handled by building conditional PDFs where appropriate (between NN and the Dalitz variables for the continuum, and between ∆E ′ and the Dalitz variables for TM signal events). Systematic uncertainties account for the correlations we neglect. A total of 37 parameters are varied in the fit (see Section V.D). A summary of the PDF parametizations is given in Table IV.

B. THE DALITZ PROBABILITY DENSITY FUNCTIONS
Since the decay B 0 → K + π − π 0 is flavor-specific (the charge of the kaon identifies the b flavor), the B 0 and B 0 Dalitz plots are independent. However, because the backgrounds are essentially flavor blind, we get a more robust procedure by fitting them simultaneously. It is enough to describe only the B 0 Dalitz plot PDF 6 . A change from A to A (Eq. (1) and (2)) gives the B 0 PDF.

Signal
The signal Dalitz model has been described in Section II. The free parameters are the magnitudes and phases defined in Eq. (1) and (2) for all the intermediate states of the signal model given in Table I. Since the measurement is done relative to the K * + (892)π − final state, the phases of this and the charge conjugate channels are fixed to zero. The amplitude of B 0 → K * + (892)π − is also fixed but not that of B 0 → K * − (892)π + in order to be sensitive to direct CP -violation. IV: Summary of the PDF parameterizations. G=Gaussian, P1=1st order polynomial, NP=non-parametric, and CB=Crystal Ball. The notation GG(DP) designates a double Gaussian PDF with parameters which vary over the Dalitz plot. The Dalitz plot signal model is described in Section II. The numbers associated with the B background in the last row are the class indices in Table III. (3 and 9) signal-like signal-like one NP for both see text The normalization of the component signal PDFs: is model dependent. J is the Jacobian matrix of the mapping to the square Dalitz plot. The symbol ⊗ stands for a convolution and the R matrix is described below [Eq. (33)]. The normalization requires the computation of the integrals where the notations of Eq. (1) are used. The integrations over the square Dalitz plot are performed numerically. The weight in Eq. (22) ensures that the total signal PDF is normalized. The PDF normalization depends on the decay dynamics and is computed iteratively. In practice the computation of f SCF rapidly converges to a value which we fix after a few exploratory fits. Studies in simulation have shown that the experimental resolutions of m ′ and θ ′ need not be introduced in the TM signal PDF. On the contrary, misreconstructed events often incur large migrations, when the reconstructed m ′ r , θ ′ r are far from the true values m ′ t , θ ′ t . We use the Monte Carlo simulation to compute a normalized two-dimensional resolution function R SCF is convolved with the signal model in the expression of P SCF [Eq. (28)].

Background
Except for events coming from exclusive B → D decays, all background Dalitz PDF are non-parametric smoothed histograms. The continuum distributions are extracted from a combination of off resonance data and a sideband (5.20 < m ES < 5.25 GeV/c 2 ) of the on resonance data from which the B-background has been subtracted. The square Dalitz plot is divided into three regions where different smoothing parameters are applied in order to optimally reproduce the observed wide and narrow structures by using a two-dimensional kernel estimation technique [32]. For m ′ > 0.8 and all θ ′ , a fine grained model is needed to follow the peak from D 0 continuum production. For m ′ < 0.8 there are two different wide structures corresponding to the continuum production of the ρ's for θ ′ below 0.8 and the charged K * 's above.
The B-background (Table III) Dalitz PDFs are obtained from the Monte Carlo simulation. For the components which model b → c decays with real D 0 mesons, a fine grained binning around the D mass is used to construct the smoothed histograms. The exclusive signallike components B 0 → D 0 π 0 , D 0 → K + π − and B 0 → D − K + , D − → π − π 0 are modeled with TM-like and SCF-like PDFs. The former are products of a Gaussian distribution in m Kπ (then transformed into m ′ ) and a fifth order polynomial in θ ′ . The D 0 Gaussian mass and widths are free parameters in the fit, the TM-like D 0 helicity polynomial coefficients were obtained via an ancillary fit to data, all other parameters are fixed to their value in the Monte Carlo. For the B 0 → D − K + where D − → π − π 0 , the Dalitz Plot PDF is a two-dimensional smoothed histogram obtained from the Monte Carlo.

Signal
The m ES PDF for TM-signal events is a Crystal Ball function [31], the mean and the width of which are free parameters in the fit. For SCF-signal events we use a non-parametric shape taken from the Monte Carlo simulation. ∆E ′ is correlated with the Dalitz plot variables for TM-signal events. To account for the correlation, we choose a double Gaussian PDF the mean and standard deviation of which vary linearly with m 2 K + π − . These parameters (intercept and slope) are taken from the Monte Carlo except for a global mean of the double Gaussian distribution. A wide single Gaussian distribution with mean and standard deviation taken from the Monte Carlo is used for the SCF-signal ∆E ′ PDF.
The NN PDFs for TM and SCF events are nonparametric distributions taken from the Monte Carlo.

Background
We use the Argus function [33] f as the continuum m ES PDF. The endpoint m max ES and the shape parameter ξ are fixed to 5.2897 GeV/c 2 and −22 respectively (ξ = −22 ± 7 from a fit to off resonance data). The ∆E ′ PDF is a linear polynomial whose slope is free to vary in the fit. The shape of the NN distribution for continuum is correlated with the event location in the Dalitz plot. To account for that effect we use for the NN PDF a function that varies with the closest distance ∆ dalitz between the point representing the event and the boundary of the standard Dalitz plot, The A and B coefficients are linear functions of ∆ dalitz . Their expressions in terms of the a i parameters that are varied in the likelihood fit are: We use two-dimensional non-parametric distributions to describe the B-background classes in the (m ES , ∆E ′ ) plane, except for the exclusive B decays to D mesons (classes 3 and 9) in Table III for which we use the same shapes as for the signal. For each B-background class the NN PDF is a non-parametric distribution taken from the Monte Carlo. Classes 3 and 9 share the same NN PDF.

D. THE FIT PARAMETERS
The following parameters are varied in the fit: • Five yields for signal (N sig ), continuum (N qq ) and three B background classes (c=3, 7 and 8 defined in Table III).
• One CP -asymmetry for the continuum events.
• Four parameters related to narrow particle masses: the mass and mass resolution for the B 0 and the D 0 .
• Two parameters, the global mean (slope), of the ∆E ′ distribution for the TM-signal (continuum) events.
• Four parameters which account for the correlation between the NN output and the event location in the Dalitz plot [Eq. (36)].
• Twenty-one isobar amplitudes and phases. There are 6 intermediate states (5 resonances and a nonresonant term) and two Dalitz plots. We fix one reference amplitude, that of B 0 → K * + (892)π − and two phases for the latter and its conjugate. Therefore we end up with 11 magnitudes and 10 phases to be determined by the fit.

VI. RESULTS
The maximum likelihood fit results in a B 0 (B 0 ) → K ± π ∓ π 0 event yield of N sig = 1377 ± 70 events, where the uncertainty is statistical only. There are 5395 ± 104 continuum events and 424 ± 25 exclusive B → D 0 π 0 decays where D 0 → K + π − , (not included in N sig ). The remaining 833 B background events are detailed in Table III. When the fit is repeated starting from input parameter values randomly chosen within wide ranges of one units. There are four degenerate solutions. The event yields we quote are the averages of the four solutions for which the relative spreads are less than 1%. Monte Carlo (m ′ , θ ′ ) scatter plots generated according to the fitted signal model in the four solutions are barely distinguishable. We checked with simulated datasets that multiple quasi degenerate solutions are to be expected when applying our fitting procedure to samples containing as many events as the experiment.
In the Appendix, the fitted parameters are given for the four solutions in Table IX, together with their correlation matrix (for solution-I) in Tables X and XI. The isobar fractions, which are not required to sum to one in order to accomodate interference, are not significantly different from unity. We observe that the fit fractions and the CP asymmetries are consistent within less than three standard deviations among the solutions, and indeed agree much better for the subdecays to a pseudoscalar meson and a vector meson. The phases differ substantially. The four solutions are actually two solutions for the B 0 phases and two solutions for the B 0 phases. The uncertainties and correlation coefficients given in these tables are statistical only. They are underestimated because the fitting program assumes a parabolic shape for the NLL in the vicinity of its minimum. This assumption overlooks the fit degeneracy. Before we explain how we derive consistent statistical and systematic uncertainties, we proceed to describe the goodness of the fit. We postpone the discussion of the physical meaning of the fitted signal model to Section VIII.
To check the validity of the fits and to study the results, we have generated 1000 pseudo experiments with as many events as in the data sample using PDFs with the fitted parameter values. We check that the NLL of the experimental fit falls well within the NLL distribution in the pseudo experiments. The goodness of the fits and their ability to reproduce the real data are similar for all four solutions. We show the results of solution-I in the following. We compare the likelihood ratio r lik ≡ LT M L (see Eq. (20) and (24)) in the data and in the pseudo experiments and see good agreement (Fig. 3). The distributions of the discriminating variables (m ES , ∆E ′ and NN) are shown in Fig. 4 for samples that have been enriched in signal events by restricting r lik (computed without the plotted variable) to large enough values in order to maximize the signal significance. Fig. 5 shows the Dalitz plot mass spectra over their full range with no restriction on r lik . A zoom of the lowmass resonance region (below 1.75 GeV/c 2 ) is shown for signal-enriched events in Fig. 6. More details are shown in Fig. 7 (Fig. 8) which display m K + π − , (m K + π 0 ) for different intervals of the helicity angles θ K + π − , (θ K + π 0 ). The interference between the scalar and vector K * is evident from the opposite sign of the forward-backward helicity asymmetries below and above the K * (892). This effect is seen with sufficient statistics in these plots which include both B 0 and B 0 decays, because the measured phase differences are similar in both cases (Table IX).  The signal-enriched spectra for masses between threshold and 1.75 GeV/c 2 : m K + π − (a), m K + π 0 (b) and m π − π 0 (c). The fractions of accepted signal and rejected continuum events are respectively 51% and 89%. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and grey shaded areas represent the continuum and B background, respectively. The light grey region shows the SCF contribution. Signal-enriched spectra of m K + π − in the low-mass resonance region and different ranges of the helicity angle, θ K + π − . (a) 0 < θ K + π − < 90 degrees, (b) 90 < θ K + π − < 180 degrees. The data sample is enriched in signal events as in Fig. 6. An interference between the vector and scalar K * 0 is apparent through a positive forward-backward asymmetry below the K * (892) and a negative one above. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and grey shaded areas represent the continuum and B background, respectively. The light grey region shows the SCF contribution. Signal-enriched spectra of m K + π 0 in the low-mass resonance region and different ranges of the helicity angle, θ K + π 0 . (a) 0 < θ K + π 0 < 90 degrees, (b) 90 < θ K + π 0 < 180 degrees. The data sample is enriched in signal events as in Fig. 6. An interference between the vector and scalar K * is apparent through a negative forward-backward asymmetry below the K * (892) and a positive one above. The data are shown as points with error bars. The solid histogram shows the projection of the fit result. The dark and grey shaded areas represent the continuum and B background, respectively. The light grey region shows the SCF contribution. We take into account all four solutions of the fit to quote the final results of the analysis. To determine the statistical uncertainty on a physical parameter p, we fix it at successive values spanning its range of interest and repeat the fit, maximizing the projected likelihood function (Eq. (25) where p is frozen). NLL(p), the minimum where the fit converges given p, is not always associated with the same solution. Therefore the NLL(p) envelope curve is far from a parabola. The flatness of its shape around the overall minimum reflects the fit degeneracy. The parameter values at which ∆χ 2 ≡ 2 (N LL − N LL bestf it ) reaches unity bound the one-standard deviation confidence interval. The scan of the K * 0 (892)π 0 isobar fraction is shown in Fig. 9. The graph also shows the envelope curve obtained when the systematic uncertainty is included. The null value of the branching fraction is excluded with a statistical and systematic significance of 5.6 standard deviations. Thus, this is the first observation of the decay B 0 → K * 0 (892)π 0 . Other scans are presented in the Appendix.

VII. SYSTEMATIC UNCERTAINTIES
Variations around the nominal fit are tried to study the dominant systematic effects which are summarized in Table V. For each parameter of interest p (F F , A CP , φ), the positive (negative) deviations from each effect are summed in quadrature to obtain total upward (downward) systematic errors δp + (δp − ). The total projected likelihood function L tot (p) is computed as the convolution of the fit projected likelihood function defined at the end of the previous section, by a bifurcated Gaussian distribution with a lower (upper) standard deviation δp − (δp + ). The scan of L tot drawn as a solid line on Fig. 9 (and subsequent ones) is used to determine the total confidence interval which accounts for both statistical and systematic uncertainties. Finally, the upward (downward) systematic uncertainty on p (Table VI) is the quadratic difference between the upward (downward) limits of the total and statistical confidence intervals. Note that when p central values significantly differ between the four fit solutions, the lowest p min and highest p max are used in the determination of the downward (upward) uncertainties and an additional systematic uncertainty ±(p max − p min )/2 is quoted in Table VI. While simple methods can be used to estimate the systematic effects on the isobar fractions and CPasymmetries, it is often necessary to estimate upper limits for phases. In such cases (the rows labeled Other in Table V) we conservatively use the upper limits at the 90% confidence level as the systematic uncertainties.
To estimate the contribution of other resonances, we fit the on resonance data with extended signal models including one extra-resonance on top of those in the nominal signal model. The ρ − (1450)K + , ρ − (1700)K + , K * 0 2 (1430)π 0 , K * + 2 (1430)π − , K * 0 (1680)π 0 and K * + (1680)π − final states have been studied. These fits are not significantly better than the nominal fit. The isobar fractions of all extra resonances are below 5%. None is significant. The isobar fractions of the nominal resonances change by non significant amounts which we record as (Dalitz Plot Model) systematic uncertainties in Table V. Changes in A CP are also recorded. As we explain in Section VIII our fitting procedure is sensitive to some phase differences but not others. When there is sensitivity the systematic Dalitz Plot Model effects are insignificant. When there is no sensitivity dramatic deviations occur. However since these phase differences are essentially undetermined by the fit we do not record these changes as systematic effects. Note that we have not included the low mass κ resonance which is not established as it is an alternative to the (Kπ) * 0 0 isobar from the model. It has been verified however that adding it to the model results into a destructive interference with the latter state, an insignificant κ amplitude and no change in the K * 0 (892)π 0 numeric results.
There are fixed parameters in the nominal fit model. We estimate the associated (Shape parameters, and B Background) systematic uncertainties by repeating the fit giving the studied parameter values at ±1 standard deviation from its fixed value or at the limits of a conservative range. The fixed parameters to which the isobar fractions are sensitive are f SCF [Eq. (32)] and the ∆E ′ signal PDF parameters. f SCF was varied between 10 and 16%, a range that is inferred from the comparison of the data and simulation of B → Dρ events as explained in References [3,34]. The A CP mainly depend on the fixed CP -asymmetries of B background classes 1 and 7 (see Table III). We vary them by ±0.5 and ±0.2 to determine the (B background ) systematic uncertainties. These intervals were chosen after inspection of the latest V: Summary of the systematic uncertainties. The systematic uncertainties are equal for the phases φ and φ. All phases φ are referenced to φ K * + (892)π − and φ to φ K * − (892)π + . The systematic uncertainties associated with the Dalitz plot model and the lineshapes are negligible for the phases of the Kπ S-waves which are the only phase angles that are measured with some statistical accuracy. For the others which are essentially undetermined by the fit we do not quote these undefined systematic uncertainties. available measurements [35].
The variations of the physical parameters of the resonances in the nominal signal model are recorded as (Lineshape) systematic uncertainties. In particular, variations of the barrier coefficients R in [Eq. (8)] cause no significant effects.
The method used to determine the continuum square Dalitz plot PDF (Section V.B) has been extensively tested in the Monte Carlo. The spread of the yields across those tests is used to estimate the associated (continuum DP PDF) systematic errors. Among the dominant effects is the variation of the distortion of the Dalitz plot distribution as a function of m ES due to the B meson mass constraint.
To estimate the Fit Bias uncertainties stemming from the imperfection of the fit model (most importantly the too simple description of the SCF but also the neglected correlations between m ES and ∆E ′ for TM signal and B-background events, and the neglected resolution effect smearing the K * ± (892) peak), we record the fitted biases and spreads in fits performed on large Monte Carlo samples with GEANT4 signal events and the actual amount of background generated along with their PDFs. Since the SCF prevails in the Dalitz plot corners where the resonances interfere, this kind of systematic uncertainty is the dominant one for the phase measurements.
For the B meson branching fraction measurements, we have to include additionally the effect of imperfections in the event reconstruction on the efficiency. Adding in quadrature the uncertainties associated with tracking (1.6%), charged particle identification (2%), π 0 reconstruction (3%), the efficiency of the selection requirements (0.3% for m ES , 1.2% for ∆E, 2% for the NN), and the integrated luminosity (1.1%), we obtain a global systematic percent error of 4.7%.
The subdecay branching fraction for B 0 → K * + (892)π − agrees with previous measurements from Belle [11,17] and BABAR [16] in both K * + → K + π 0 and K 0 S π + decay channels. Averaging the BABAR measurements one obtains B(B 0 → K * + (892)π − ) = (11.7 +1.3 −1.2 ) × 10 −6 and A CP (B 0 → K * + (892)π − ) = −0.14 ± 0.12. The branching fraction for B 0 → ρ − K + is lower than those measured by Belle [11] and CLEO [12] but in agreement within errors. If we assume that the (Kπ) * +,0 0 isobars are pure isospin-1/2 and neglect possible non Kπ final states, we determine the following effective branching fractions: +2.7 −1.9 ± 3.6) × 10 −6 . A consistency check of our analysis is provided by our measurement of the branching ratio, B(B 0 → D 0 π 0 ) = (2.93 ± 0.17 ± 0.18) × 10 −4 in agreement with its world average and that of the branching fraction of the decay There is no evidence of direct CP -violation in any resonant subdecay. In Tables VI and VIII we give upper limits at the 90% statistical confidence level on the branching fractions of resonances that might contribute to K + π − π 0 but are not part of the nominal signal model.
We also search for direct CP -violation in the interference between pairs of isobars (R i , R j ) by comparing the interference patterns in the B 0 and B 0 Dalitz plots. In the figures described in what follows, we display three NLL scans for in turn ∆φ ij ≡ φ j −φ i , the phase difference between the resonances in B 0 decays, ∆φ ij ≡ φ j − φ i , the same for B 0 decays and δφ ij ≡ ∆φ ij − ∆φ ij . A marked minimum in a ∆φ (∆φ) scan indicates a sizable interference in B 0 (B 0 ) decays. Evidence for direct CP -violation would be seen if δφ min ≡ ∆φ min − ∆φ min were significantly away from zero. One standard deviation confidence intervals (1σ c.i.) bounds are graphically seen as the intersection points between the solid scan curves (which incorporate both statistical and systematic uncertainties) and the ∆χ 2 = 1 horizontal line. The results are collected in Table VII. All isobar pairs for which the 95% confidence intervals in δφ are non trivial (smaller than ±180 degrees) exhibit sizeable interference patterns in B 0 and B 0 decays. The ∆χ 2 value for δφ = 0 corresponds to the square of the direct CP -violation significance in standard deviation units. A scan of the R i and R j lineshapes with more statistics could enable one to disentangle the strong phase motions and determine the weak phase. For the scans where interference is elusive, we quote in Table VII the maximum ∆χ 2 over the ±180 degrees scanned range. For the Kπ systems, we see no significant interference pattern between charged and neutral P-waves (Fig. 10) at the 95% confidence level; similarly for the S-waves (Fig. 11). Only a small range (∆φ between -131 and -75 degrees) is excluded at the two standard deviation level for the B 0 P-waves. These observations are not unexpected for the K * (892)π final states since the K * resonances are quite narrow and therefore have a small overlap in the Dalitz plot. Furthermore TABLE VI: Final results for rates and CP -asymmetries. The quasi-two-body branching fractions Bj have been computed from the isobar fractions F Fj using Eq. (15). The statistical uncertainties are given first. The statistical and systematic uncertainties are calculated by scanning the NLL close to its minimum taking into account the four fit solutions and recording the values where the NLL increases by one unit above its minimum. For the final states with Kπ S-waves, a second systematic uncertainty covers the spread between the best fit values from the four solutions. For resonances that are considered in extended fit models, we quote upper limits at the 90% confidence level (based on statistical uncertainties only) for the isobar fractions and the quasi-two-body branching fractions.  When there is sensitivity, the results are the one standard deviation confidence interval (1 σ c.i.) for the phase angle measurements (in degrees). The statistical and systematic uncertainties, determined by the NLL scan method described in Fig. 9 are included. The interval bounds can be seen on the graphs referenced in the second column, as the intersections of the solid scan curves with the (∆χ 2 = 1) horizontal dashed lines. The ∆χ 2 evaluated for zero δφ measures the significance (squared) for direct CP -violation. When there is no sensitivity we give the maximum ∆χ 2 reached over the scanned region.
Interference pattern Graph Results ∆φ for B 0 ∆φ for B 0 δφ ≡ ∆φ − ∆φ ∆χ 2 (δφ = 0) K π neutral−charged P-waves Fig. 10 Fig. 14   the coherent sum of amplitudes that interfere might behave like the model in [5,6] with a single weak phase (equal to γ CKM in the absence of electroweak penguin diagrams). In such a scheme, one weak phase would be missing to enable direct CP -violation to take place. In contrast, Fig. 12 and Fig. 13 show that we measure, with uncertainties smaller than ±36 degrees, the phase dif- ferences between the Kπ S-and P-waves of the same charge for the B 0 and B 0 decays. Moreover, the associated CP -observables δφ are measured to 35 degrees with negligible systematic uncertainties (less than 10 degrees). An interval of order 220 degrees is excluded at the 95% confidence level. The charged and neutral S-and P-wave interference patterns thus provide sensitivity to two weak phases. The scans of the phase differences between ρK and K * (892)π show no evidence for interference at the three standard deviation level as shown in Fig. 14. Here again the overlap in phase space between the interfering resonances is small. This contrasts with what we observe in the scans of the phase differences between the nonreso- FIG. 14: The phase difference between the ρK and the charged Kπ P-wave. The three diagrams are the NLL scans for the B 0 (a) and B 0 (b) decays as well as their differences (c). The lines are drawn as for Fig. 10. The vertical scale cuts off ∆χ 2 = 4, however it has been checked that all phase differences are consistent with the data at the three standard deviation level.
nant K ± π ∓ π 0 and the S-waves in Fig. 15 and Fig. 16. We see that they are somewhat constrained. These observations are in agreement with the fact that the fit finds a sizable nonresonant B 0 → K + π − π 0 component that populates the Dalitz plot far from the boundary.  [24] have been used. The upper limits at 90% confidence level, UL are based on statistical uncertainties only.

IX. SUMMARY
We have measured the branching fraction and CPasymmetry of the B ± → K ± π ∓ π 0 decay and compared the Dalitz plots of B 0 → K + π − π 0 and B 0 → K − π + π 0 using an isobar model. We have extracted the CP -averaged isobar branching fractions and CPasymmetries assuming no interference. We observe the B 0 → K * 0 (892)π 0 with 5.6 standard deviation significance. We have looked at the interference patterns in the Dalitz plots and put significant constraints on phase differences between wide intermediate states which have a sizable overlap in phase space. The phase shifts between S and P-waves in the charged and neutral Kπ and Kπ are constrained to within ±70 degrees or less at the 95% confidence level. Weaker constraints are observed for the phase shifts between the K(K)π and nonresonant components which extend widely over the Dalitz plots. The phase shift differences between Kπ and Kπ S and P-waves are measured and found to be consistent with no direct CP -violation within 2 standard deviations. Additionally we determine the branching fraction for the decay B 0 → D 0 π 0 with an accuracy comparable to that of the world average value of this quantity. The phase difference between the nonresonant K + π − π 0 and the neutral Kπ S-wave. The three diagrams are the NLL scans for the B 0 (a) and B 0 (b) decays as well as their differences (c). The lines are drawn as for Fig. 10. Ranges with widths of 90 (a) and 110 (b) degrees are excluded at the 95% confidence level for the B 0 and B 0 decays. No significant difference between the B 0 and B 0 interference patterns is seen.

X. 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 Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariatà l'Energie Atomique and Institut National

APPENDIX
The four solutions of the fit are displayed in Table IX. The correlation coefficients of solution-I are given in Tables X and XI. As explained in Section VI the statistical uncertainty of each solution does not reflect the actual experimental uncertainty and should not be used. The procedure we have devised blends all four solutions and determines reliable statistical and systematic uncertainties. For illustration, we display the four NLL around their minima in Fig. 17 for the isobar fractions and Fig. 18 for the CP -asymmetries. For the neutral S-wave final states, the spread due to the degeneracy of the fitted fractions and asymmetries is quite large.
When the NLL are far from being parabolic at their minima, actual scans as described at the end of Section V are performed to derive the results. Fig. 19 shows two examples of such scans for the sum of the isobar fit fractions (or the total fit fraction) and for the global CPasymmetry A CP [Eq. (13)].    (14)]. The phases φ for the B 0 decays and φ for the B 0 decays are measured relative to B 0 (B 0 ) → K * ± π ∓ . The uncertainties are statistical only. They are underestimated because a parabolic approximation is made for the shape of the NLL close to minimum. 104.5 +4.5