Time-dependent amplitude analysis of B0 -->K0S pi+ pi-

We perform a time-dependent amplitude analysis of $B^0 \to K^0_S\pi^+\pi^-$ decays to extract the CP violation parameters of $f_0(980) K^0_S$ and $\rho^0(770) K^0_S$ and the direct CP asymmetry of $K^{*+}(892) \pi^-$. The results are obtained from a data sample of $(383 \pm 3) \times 10^6$ $B\bar{B}$ decays, collected with the BABAR detector at the PEP-II asymmetric-energy $B$ factory at SLAC. We find two solutions, with an equivalent goodness-of-fit. Including systematic and Dalitz plot model uncertainties, the combined confidence interval for values of the CP parameter $\beta_{eff}$ in $B^0$ decays to $f_0(980) K^0_S$ is $18^o<\beta_{eff}<76^o$ at 95% confidence level (C.L). CP conservation in $B^0$ decays to $f_0(980) K^0_S$ is excluded at $3.5\sigma$ including systematic uncertainties. For $B^0$ decays to $\rho^0(770) K^0_S$, the combined confidence interval is $-9^o<\beta_{eff}<57^o$ at 95% C.L. In decays to $K^{*+}(892) \pi^-$ we measure the direct CP asymmetry to be $A_{CP}=-0.20 \pm 0.10 \pm 0.01 \pm 0.02$. The measured phase difference (including $B^0\bar{B}^0$ mixing) between decay amplitudes of $B^0 \to K^{*+}(892) \pi^-$ and $\bar{B}^0 \to K^{*-}(892) \pi^+$, excludes the interval $-137^o<\Delta\Phi(K^{*+}(892) \pi^-)<-5^o$ at 95% C.L.


I. INTRODUCTION
The Cabibbo-Kobayashi-Maskawa (CKM) mechanism [1,2] for quark mixing describes all transitions between quarks in terms of only four parameters: three rotation angles and one irreducible phase. Consequently, the flavor sector of the Standard Model (SM) is highly predictive. One particularly interesting prediction is that mixing-induced CP asymmetries in decays governed by b → qqs (q = u, d, s) transitions are, to a good approximation, the same as those found in b → ccs transitions. Since flavor changing neutral currents are forbidden at tree-level in the Standard Model, the b → s transition proceeds via loop diagrams (penguins), which are affected by new particles in many extensions of the SM.
Various b → s dominated charmless hadronic B decays have been studied in order to probe this prediction. The values of the mixing-induced CP asymmetry measured for each (quasi-)two-body mode can be compared to that measured in b → ccs transitions (typically using B 0 → J/ψK 0 S ). A recent compilation [3] of results shows that they tend to have central values below that for b → ccs. Recent theoretical evaluations [4,5,6,7,8,9,10,11,12] suggest that SM corrections to the b → qqs mixing-induced CP violation parameters should be small, in particular for the modes φK 0 , η ′ K 0 , and K 0 S K 0 S K 0 S , and tend to increase the values, i.e. the opposite trend to that seen in the data. However, there is currently no convincing evidence for new physics effects in these transitions. Clearly, more precise experimental results are required.
The compilation given in [3] includes several threebody modes, which may be used either by virtue of being CP eigenstates (K 0 S K 0 S K 0 S , K 0 S π 0 π 0 ) [13] or because their CP content can be determined experimentally (K + K − K 0 ) [14,15]. It also includes quasi-twobody (Q2B) modes, such as f 0 (980)K 0 S and ρ 0 (770)K 0 S , which are reconstructed via their three-body final states (K 0 S π + π − for these modes). The precision of the Q2B approach is limited as other structures in the phase space may cause interference with the resonances considered as signal. Therefore, more precise results can be obtained using a time-dependent amplitude analysis covering the complete phase space, or Dalitz plot (DP), of B 0 → K 0 S π + π − . Furthermore the interference terms allow the cosine of the effective weak phase difference in mixing and decay to be determined, helping to resolve ambiguities which arise from the Q2B analysis. This ap- * * Now at University of South Alabama, Mobile, Alabama 36688, USA † † Also with Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France ‡ ‡ Also with Università di Sassari, Sassari, Italy proach has been successfully used in a time-dependent DP analysis of B 0 → K + K − K 0 [15].
The discussion above assumes that the b → s penguin amplitude dominates the decay. However, for each mode contributing to the K 0 S π + π − final state, there is also the possibility of a b → u tree diagram. These are doubly CKM suppressed compared to the b → s penguin diagram (the tree is O(λ 4 ) whereas the penguin is O(λ 2 ), where λ is the usual Wolfenstein parameter [16,17]). However, hadronic factors may enhance the tree amplitudes, resulting in a significant "tree pollution." These hadronic factors may be different for each Q2B state, thus the relative magnitudes of each tree and penguin amplitudes, |T /P |, and the strong phase difference may be different as well. Nontheless, the relative weak phase between these two amplitudes is the same -and in the Standard Model is equal to the CKM unitarity triangle angle γ. An amplitude analysis, in contrast to a Q2B analysis, yields sufficient information to extract relative phases and magnitudes. Measurements of decay amplitudes in the DP analysis of B 0 → K 0 S π + π − (and similar modes) can therefore be used to set constraints on the CKM parameters (ρ, η) [18,19,20,21].
Recently published results on time-dependent DP analysis of B 0 → K 0 S π + π − are available [22]. Previous studies of the B 0 → K 0 S π + π − decay were either based on a Q2B approach [23], or were amplitude analyses that did not take into account either time-dependence or flavor-tag dependence [24]. The available results for B 0 → K 0 S π + π − are consistent with studies obtained from other B → Kππ decay modes: K + π − π 0 [25,26] and K + π + π − [27,28]. The latter results indicate evidence for direct CP violation in the B + → ρ 0 (770)K + channel. If confirmed, this will be the first observation of CP violation in the decay of any charged particle. The relevance of B → Kππ is further highlighted by recent theoretical calculations [29] suggesting that large CP violation effects are expected in several B → K * π and B → Kρ resonant modes.
In this paper we present results from a time-dependent amplitude analysis of the B 0 → K 0 S π + π − decay. In Sec. II we describe the time-dependent DP formalism, and introduce the signal parameters that are extracted in the fit to data. In Sec. III we briefly describe the BABAR detector and the data set. In Sec. IV, we explain the selection requirements used to obtain the signal candidates and suppress backgrounds. In Sec. V we describe the fit method and the approach used to control experimental effects such as resolution. In Sec. VI we present the results of the fit, and extract parameters relevant to the contributing intermediate resonant states. In Sec. VII we discuss systematic uncertainties in the results, and finally we summarize the results in Sec. VIII.

II. ANALYSIS OVERVIEW
Taking advantage of the interference pattern in the DP, we measure relative magnitudes and phases for the different resonant decay modes using a maximum-likelihood fit. Below, we detail the formalism used in the present analysis.

A. Decay amplitudes
We consider the decay of a spin-zero B 0 with fourmomentum p B into the three daughters π + , π − , and K 0 S with p + , p − , and p 0 their corresponding four-momenta. Using as independent (Mandelstam) variables the invariant squared masses can be obtained from energy and momentum conservation: The differential B 0 decay width with respect to the variables defined in Eq. (1) (i.e. the Dalitz plot) reads where A is the Lorentz-invariant amplitude of the threebody decay. In the following, the amplitudes A and A correspond to the transitions B 0 → K 0 S π + π − and B 0 → K 0 S π + π − , respectively. We describe the distribution of signal events in the DP using an isobar approximation, which models the total amplitude as resulting from a coherent sum of amplitudes from the N individual decay channels where F j are DP-dependent dynamical amplitudes described below, and c j complex coefficients describing the relative magnitude and phase of the different decay channels. All the weak phase dependence is contained in c j , and F j contains strong dynamics only; therefore, The resonance dynamics are contained within the F j terms, which are represented by the product of the invariant mass and angular distribution probabilities, i.e., where • m is the invariant mass of the decay products of the resonance, • R j (m) is the resonance mass term or "lineshape" (e.g. Breit-Wigner), • L is the orbital angular momentum between the resonance and the bachelor particle, • p ⋆ is the momentum of the bachelor particle evaluated in the rest frame of the B, • p and q are the momenta of the bachelor particle and one of the resonance daughters, respectively, both evaluated in the rest frame of the resonance (for K 0 S π − , K 0 S π + , and π + π − resonances, q is assigned to the momentum of the K 0 S , π + , and π − , respectively), • X L are Blatt-Weisskopf barrier factors [30] with parameters r ′ (taken to be 2 ( GeV/c) −1 ) and r (given in Table I), and • T j (L, p, q) is the angular distribution: The helicity angle of a resonance is defined as the angle between p and q. Explicitly, for K 0 S π − , K 0 S π + , and π + π − resonances the helicity angle is defined between the momenta of the bachelor particle and of the K 0 S , π + , and π − , respectively, in the resonance rest frame.
For most resonances in this analysis the R j are taken to be relativistic Breit-Wigner (RBW) [31] lineshapes: where m 0 is the nominal mass of the resonance and Γ(m) is the mass-dependent width. In the general case of a spin-J resonance, the latter can be expressed as The symbol Γ 0 denotes the nominal width of the resonance. The values of m 0 and Γ 0 are listed in Table I. The symbol q 0 denotes the value of q when m = m 0 . For the f 0 (980) lineshape the Flatté form [32] is used. In this case the mass-dependent width is given by the sum of the widths in the ππ and KK systems: where Γ ππ (m) = g π 1 3 1 − 4m 2 π 0 /m 2 + (14) The fractional coefficients arise from isospin conservation and g π and g K are coupling constants for which the values are given in Table I. For the ρ 0 (770) we use the Gounaris-Sakurai (GS) parameterization [33], that describes the P -wave scattering amplitude for a broad resonance, decaying to two pions: where , and the function h(m) is defined as with The normalization condition at R j (0) fixes the parameter d = f (0)/(Γ 0 m 0 ). It is found to be: The 0 + component of the Kπ spectrum is not well understood [34,35]; we dub this component (Kπ) * ± 0 and use the LASS parameterization [34] which consists of the K * (1430) resonance together with an effective range nonresonant (NR) component: where cot δ B = 1 aq + 1 2 rq. The values we have used for the scattering length (a) and effective range (r) parameters of this distribution are given in Table I. The effective range part of the amplitude is cut off at m cutoff Kπ = 1800 MeV/c 2 . Integrating separately the resonant part, the effective range part, and the coherent sum we find that the K * (1430) resonance accounts for 81.7%, the effective range term 44.1%, and destructive interference between the two terms is responsible for the excess 25.8%.
A flat phase space term has been included in the signal model to account for NR B 0 → K 0 S π + π − decays. We determine a nominal signal Dalitz-plot model using information from previous studies [23,24] and the change in the fit likelihood value observed when omitting or adding resonances. The components of the nominal signal model are summarized in Table I. Other components, taken into account only to estimate the DP model uncertainty, are discussed in Sec. VII.

B. Time dependence
With ∆t ≡ t sig − t tag defined as the proper time interval between the decay of the fully reconstructed B 0 → K 0 S π + π − (B 0 sig ) and that of the other meson (B 0 tag ) from the Υ (4S), the time-dependent decay rate |A + sig (∆t)| 2 (|A − sig (∆t)| 2 ) when the B 0 tag is a B 0 (B 0 ) is given by where τ B 0 is the neutral B meson lifetime and ∆m d is the B 0 B 0 mass difference. In the last formula and in the following, the DP dependence of the amplitudes is implicit.
Here, we have assumed that there is no CP violation in mixing, and have used a convention whereby the phase from B 0 B 0 mixing is absorbed into the B 0 decay amplitude (i.e. into thec j terms). In other words, we assume that the B 0 B 0 mixing parameters satisfy |q/p| = 1 and absorb q/p intoc j . Lifetime differences in the neutral B meson system are assumed to be negligible.

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 DP due to the low final state masses compared with the B 0 mass. The representation in Eq. (3) is inconvenient when empirical reference shapes are to be used. Large variations occurring in small areas of the DP are very difficult to describe in detail. We therefore apply the transformation which defines the square Dalitz plot (SDP). The new coordinates are where m 0 = √ s 0 is the π + π − invariant mass, m max 0 = m B 0 − m K 0 S and m min 0 = 2m π + are the kinematic limits of m 0 , θ 0 is the π + π − resonance helicity angle and J is the Jacobian of the transformation. Both variables range between 0 and 1. The determinant of the Jacobian is given by where |p * and where the π + (K 0 S ) energy E * + (E * 0 ), is defined in the π + π − rest frame. This transformation was introduced in Ref. [37], and has been used in several B decay DP analyses.

III. 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 an integrated luminosity of 347.3 fb −1 , corresponding to (383 ± 3) × 10 6 BB pairs collected at the Υ (4S) resonance ("on-resonance"), and 36.6 fb −1 collected about 40 MeV below the Υ (4S) ("offresonance").
A detailed description of the BABAR detector is presented in Ref. [38]. The tracking system used for track and vertex reconstruction has two components: a silicon vertex tracker (SVT) and a drift chamber (DCH), both operating within a 1.5 T magnetic field generated by a superconducting solenoidal magnet. Photons are identified in an electromagnetic calorimeter (EMC). It surrounds a detector of internally reflected Cherenkov light (DIRC), which associates Cherenkov photons with tracks for particle identification. Muon candidates are identified with the use of the instrumented flux return (IFR) of the solenoid.

IV. EVENT SELECTION AND BACKGROUNDS
We reconstruct B 0 → K 0 S π + π − candidates from pairs of oppositely-charged tracks and a K 0 S → π + π − candidate, which are required to form a good quality vertex. In order to ensure that all events are within the DP boundaries, we constrain the invariant mass of the final state to the B mass. For the π + π − pair from the B, we use information from the tracking system, EMC, and DIRC to remove tracks consistent with electron, kaon, and proton hypotheses. In addition we require at least one track to be inconsistent with the muon hypothesis based on information from the IFR. The K 0 S candidate is required to have a mass within 15 MeV/c 2 of the nominal K 0 mass [31], and a lifetime significance of at least five standard deviations. The last requirement ensures that the decay vertices of the B 0 and the K 0 S are well separated. In addition, combinatorial background is suppressed by requiring the cosine of the angle between the K 0 S flight direction and the vector connecting the B-daughter pions and the K 0 S vertices to be greater than 0.999. A B-meson candidate is characterized kinematically by the energy-substituted mass and (E i , p i ) are the four-vectors of the B-candidate and the initial electron-positron system, respectively. The asterisk denotes the Υ (4S) frame, and s is the square of the invariant mass of the electron-positron system. We require 5.272 < m ES < 5.286 GeV/c 2 and |∆E| < 0.065 GeV. Following the calculation of these kinematic variables, each of the B candidates is refitted with its mass constrained to the world average value of the B-meson mass [31] in order to improve the DP position resolution, and ensure that Eq. (2) holds.
Backgrounds arise primarily from random combinations in continuum events. To enhance discrimination between signal and continuum, we use a neural network (NN) [39] 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 monomials L 0,2 of the energy flow about the B thrust axis. The monomials 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 track or neutral cluster i and p i is the magnitude of its momentum. The sum excludes the B candidate and all quantities are calculated in the Υ (4S) frame. The NN is trained using off-resonance data as well as simulated signal events, all of which passed the selection criteria. The final sample of signal candidates is selected with a requirement on the NN output that retains 90% of the signal and rejects 71% of the continuum.
The time difference ∆t is obtained from the measured distance between the positions of the B 0 sig and B 0 tag decay vertices, using the boost βγ = 0.56 of the e + e − system. B 0 candidates with |∆t| > 20 ps are rejected, as are candidates for which the error on ∆t is higher than 2.5 ps. To determine the flavor of B 0 tag we use the B flavor tagging algorithm of Ref. [40]. This algorithm combines several different signatures, such as charges, momenta, and decay angles of charged particles in the event to achieve optimal separation between the two B flavors. This produces six mutually exclusive tagging categories: lepton, two different kaon categories, slow pion, kaon-slow pion, and a category that uses a combination of other signatures. We also retain untagged events in a seventh category since although these events do not contribute to the measurement of the time-dependent CP asymmetries they do provide additional statistics for the measurements of direct CP violation and CP -conserving quantities such as the branching fractions [41]. Multiple B candidates passing the full selection occur between ∼ 1% of the time for NR signal events and ∼ 8% of the time for B 0 → f 0 (980)K 0 S signal events. If an event has more than one candidate, we select one using a reproducible pseudo-random procedure based on the event timestamp.
With the above selection criteria, we obtain a signal efficiency determined from Monte Carlo (MC) simulation of 21 − 25%, depending on the position in the DP.
Of the selected signal events, 8% of events are misreconstructed. Misreconstructed events occur when a track from the tagging B is assigned to the reconstructed signal candidate. This occurs most often for low-momentum tracks and hence the misreconstructed events are concentrated in the corners of the DP. Since these are also where the low-mass resonances overlap strongly with other resonances, it is important to model the misreconstructed events correctly. The model used to account for misreconstructed events is detailed in Sec. V A.
We use MC events to study the background from other B decays (B background). More than fifty channels were considered in preliminary studies, of which twenty are included in the final likelihood model -those with at least two events expected after selection. These exclusive B background modes are grouped into ten different classes that gather decays with similar kinematic and topological properties: nine for neutral B decays, one of which accounts for inclusive decays, and one for inclusive charged B decays. Table II summarizes the ten B background classes that are used in the fit. The yields of those classes that have a clear signature in the DP are allowed to float in the maximum-likelihood fit, the remainder are fixed. When the yield of a class is varied in the maximum-likelihood fit the quoted number of events corresponds to the fit results. For the other modes, the expected numbers of selected events are computed by multiplying the selection efficiencies (estimated using MC simulated decays) by the world average branching fractions [3,31], scaled to the data set luminosity (347 fb −1 ).

V. THE MAXIMUM-LIKELIHOOD FIT
We perform an unbinned extended maximumlikelihood fit to extract the inclusive B 0 → K 0 S π + π − event yield and the resonant amplitudes. The fit uses the variables m ES , ∆E, the NN output, and the SDP to discriminate signal from background. The ∆t measurement allows the determination of mixing-induced CP violation and provides additional continuum background rejection.
The selected on-resonance data sample is assumed to consist of signal, continuum background, and B background components. The signal likelihood consists of the sum of a correctly reconstructed ("truth-matched," TM) term and a misreconstructed ("self-cross-feed," SCF) term. Generally, the components in the fit are separated by the flavor and tagging category of the tag side B decay.
The probability density function (PDF) P c i for an event i in tagging category c is the sum of the probability densities of all components, namely The variables are defined in Table III. The PDFs P c X (X = {sig − TM, sig − SCF, qq, B + , B 0 }) are the product of the four PDFs of the discriminating variables, x 1 = m ES , x 2 = ∆E, x 3 = NN output, and the triplet   where i is the event index and j is a B background class. Not all the PDFs depend on the tagging category; the general notations P c X,i(j) and P c X,i(j) are used for simplicity. Correlations between the tag and the position in the DP are absorbed in tag-flavor-dependent SDP PDFs that are used for continuum and charged B backgrounds. The parameters A B + and A qq parametrize any potential asymmetry between these PDFs. The extended likelihood over all tagging categories is given by whereN c is the total number of events expected in category c.
A total of 75 parameters are varied in the fit. They include the 12 inclusive yields (signal, four B background classes, and seven continuum yields, one per tagging category), 30 parameters for the complex amplitudes from Eq. (22), and 33 parameters of the different PDFs. The latter include most of the parameters describing the continuum distributions.

A. The ∆t and Dalitz plot PDFs
The SDP PDFs require as input the DP-dependent selection efficiency, ε = ε(m ′ , θ ′ ), and SCF fraction, f SCF = f SCF (m ′ , θ ′ ). Both quantities are taken from MC simulation. Away from the DP corners the efficiency is uniform. It decreases when approaching the corners, where one of the three particles in the final state is nearly at rest so that the acceptance requirements on the particle reconstruction become restrictive. Combinatorial backgrounds and hence SCF fractions are large in the corners of the DP due to the presence of soft tracks.
For an event i we define the time-dependent SDP PDFs and similarly for ε f SCF | det J| F k F * k ′ , where all quantities in the integrands are DP-dependent.
Equation (26) 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 resonances are large compared to the mass resolution for TM events (about 8 MeV/c 2 core Gaussian resolution). We therefore neglect resolution effects in the TM model. Misreconstructed events have a poor mass resolution that strongly varies across the DP. It is described in the fit by a 2 × 2dimensional resolution function which represents the probability to reconstruct at the coordinates (m ′ r , θ ′ r ) an event that has the true coordinates (m ′ t , θ ′ t ). It obeys the unitarity condition and is convolved with the signal model. The R SCF function is obtained from MC simulation.
We use the signal model described in Sec. II A. It contains the dynamical information and is connected with ∆t via the matrix element in Eq. (22), which intervenes in the signal PDFs defined in Eq. (29) and (30). The PDFs are diluted by the effects of mistagging and the limited vertex resolution [42]. The ∆t resolution function for signal (both TM and SCF) and B background events is a sum of three Gaussian distributions. The parameters of the signal resolution function are determined by a fit to fully reconstructed B 0 decays [40].
The charged B background contribution to the likelihood, given in Eq. (26), uses distinct SDP PDFs for each reconstructed B flavor tag, and a flavor-tag-averaged PDF for untagged events. The PDFs are obtained from MC simulation and are described by histograms. The ∆t resolution parameters are determined by a fit to fully reconstructed B + decays. For the B + background class we adjust the effective lifetime to account for the misreconstruction 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 coordinates 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 D mesons and does not exhibit mixing-induced CP violation, but usually has a strong correlation between the flavor tag and the DP coordinates because it consists of B-flavor eigenstates. Direct CP violation is also possible in these decays, though it is set to zero in the nominal model. The DP PDFs are obtained from MC simulation and are described by histograms. For neutral B background, the signal ∆t resolution model is assumed. Note that the SDP-and ∆tdependent PDFs factorize for the charged B background modes, but not necessarily for the neutral B background due to B 0 B 0 mixing.
The DP treatment of the continuum events is similar to that used for charged B background. The SDP PDF for continuum background is obtained from on-resonance events selected in the m ES sidebands and corrected for feed-through from B decays. A large number of cross checks have been performed to validate the empirical shape used. The continuum ∆t distribution is parameterized as the sum of three Gaussian distributions with common mean and three distinct widths that scale with the ∆t per-event error. This introduces six shape parameters that are determined by the fit. The model is motivated by the observation that the mean of the ∆t distribution is independent of the per-event error, and that the width depends linearly on this error.

B. Description of the other variables
The m ES distribution of TM signal events is parameterized by a bifurcated Crystal Ball function [43,44,45], which is a combination of a one-sided Gaussian and a Crystal Ball function. The mean and the two widths of this function are determined by the fit. The ∆E distribution of TM signal events is parameterized by a double Gaussian function. The five parameters of this function are determined by the fit. Both m ES and ∆E PDFs are described by histograms, taken from the distributions found in appropriate MC samples, for SCF signal events and all B background classes. Exceptions to this are the m ES PDFs for the B 0 → D − π + and B 0 → J/ψ K 0 S components, and the ∆E PDF for B 0 → D − π + , which are the same as the corresponding distributions of TM signal events. The m ES and ∆E PDFs for continuum events are parameterized by an ARGUS shape function [46] and a first-order polynomial, respectively, with parameters determined by the fit.
We use histograms to empirically describe the distributions of the NN output found in the MC simulation for TM and SCF signal events and for all B background classes. 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 that is constrained to take positive values in the range populated by the data. The coefficients of the polynomial are determined by the fit. Continuum events exhibit a correlation between the DP coordinates and the shape of the event that is exploited in the NN. To correct for this effect, we introduce a linear dependence of the polynomial coefficients on the variable ∆ DP , defined as the smallest of the three invariant masses, and is thus a measure of the distance of the DP coordinates from the kinematic boundaries of the DP. The parameters describing this dependence are determined by the fit.

VI. FIT RESULTS
The standard and square Dalitz plots of the selected data sample are shown in Fig. 1.
The maximum-likelihood fit of 22525 candidates results in a B 0 → K 0 S π + π − event yield of 2182 ± 64 and a continuum yield of 14240 ± 126, where the uncertainties are statistical only. The remaining number of events is covered by the yields of backgrounds from charged and neutral B decays, where the dominant contributions are 3361 ± 60 B 0 → D − π + and 1804 ± 44 B 0 → J/ψ K 0 S events. When the fit is repeated starting from input parameter values randomly chosen within wide ranges above and below the nominal values for the magnitudes and within the [0 − 360 • ] interval for the phases, we observe convergence toward two solutions with minimum values of the negative log likelihood function −2 log L that are equal within 0.32 units. In the following, we refer to them as solution I (the global minimum) and solution II (a local minimum). Between the two solutions, the fit values for most free parameters are very similar. Exceptions occur among isobar parameters, and most particularly isobar phases, some of which can differ significantly.
For a given event i, we define the likelihood ratio as R ≡ P sig−TM,i /P i (see Eq. (26) and explanations below). Figure 2 shows distributions of log R for all the events entering the fit, and for the signal-like region. We obtain signal enriched samples that are used in some of the figures below, by removing events with small values of R; in each case R is computed excluding the variable being plotted. Figure 3 shows distributions of ∆E, m ES , and the NN output which are enhanced in signal content by requirements on R. Figures 4 to 7 show similar distributions for m(π + π − ), m(K 0 s π), and ∆ DP . These distributions illustrate the good quality of the fit in the signalenhanced regions. Signal enriched distributions of ∆t and ∆t asymmetry for events in the regions of f 0 (980)K 0 S and ρ 0 (770)K 0 S are shown in Fig. 8. In the fit, we measure directly the relative magnitudes and phases of the different components of the signal model. The magnitude and phase of the B 0 → f 0 (980)K 0 S amplitude are fixed to 4 and 0, respectively, as a reference. The results corresponding to the two solutions are given together with their statistical uncertainties in Table IV. The full (statistical, systematic and model dependent) correlation matrices between the magnitudes and the phases for the two solutions are given    in the Appendix. The measured relative amplitudes c k , where the index represents an intermediate resonance, are used to extract the Q2B parameters defined below. For a resonant decay mode k which is a CP eigenstate, the following Q2B parameters are extracted: the angle β eff defined as and the direct and mixing-induced CP asymmetries, de-fined as: For a flavor-specific resonant decay mode k such as B 0 → K * + (892)π − , it is customary to define the direct S π + π − signal with a requirement on the likelihood ratio R computed without the variable being plotted. In each case the applied cut rejects 99% of continuum background, while retaining 28% of signal for ∆E and mES, and 16% for N N . A veto in the D − π + and J/ψ K 0 S bands has been applied.     FIG. 5: Distribution of m π + π − for a sample enhanced in B 0 → K 0 S π + π − signal, showing the f0(980)K 0 S and ρ 0 (770)K 0 S signal region for positive (left) and negative (right) π + π − helicity. The contribution from fX (1300)K 0 S and f2(1270)K 0 S are also visible. A veto in the Dπ band has been applied. The ∆t and DP PDFs have been excluded from the likelihood ratio R used to enhance the sample in signal events. The cut on R retains 21% of signal, while rejecting 99% of continuum. S π for a sample enhanced in B 0 → K 0 S π + π − signal, showing the K * (892)π and K * (1430)π signal region for positive (left) and negative (right) K 0 S π helicity. A veto in the J/ψK 0 S and ψ(2S)K 0 S bands has been applied. The ∆t and DP PDFs have been excluded from the definition of the likelihood ratio used to enhance the sample in signal events. The cut on R retains 18% of signal while rejecting 94% of continuum. An interference between the vector and scalar K * + is apparent through a positive (negative) forward-backward asymmetry below (above) the K * (892).
The ∆DP variable is defined as min(m K 0 S π + , m K 0 S π − , m π + π − ). Small (large) values of ∆DP correspond to the edges (center) of the DP. On the left (right) side of the figure, for ∆DP < 1.9 GeV/c 2 (> 1.9 GeV/c 2 ), the dominant contribution to the signal is from the light resonances (the NR) component of the signal model. A veto in the Dπ, J/ψK 0 S , and ψ(2S)K 0 S bands has been applied. The ∆t and DP PDFs have been excluded from the likelihood ratio R used to enhance the sample in signal events. The cut on R retains 37% of signal while rejecting 88% of continuum. These distributions correspond to samples where the D − π + and J/ψ K 0 S bands are removed from the DP, and the ∆t and DP PDFs have been excluded from the likelihood ratio R used to enhance the sample in signal events. The cut on R retains 24% of signal while rejecting 98% of continuum.
CP asymmetry parameter A CP as: For a pair of resonances k and k ′ , the phase φ(k, k ′ ) relating their amplitudes c k and c k ′ , defined as can be accessed by exploiting the interference pattern in the DP areas where k and k ′ overlap; correspondingly, the phaseφ(k, k ′ ) for the CP -conjugated amplitudesc k andc k ′ isφ (k, k ′ ) = arg(c kc * k ′ ).
From these two phases, the difference ∆φ(k, k ′ ) = φ(k, k ′ ) − φ(k, k ′ ), can be extracted. This parameter is a direct CP violation observable, and can only be accessed in an amplitude analysis. For a resonant decay mode k, the phase relating its amplitude c k to its charge conjugatec k is defined as here it is worth recalling that we use a convention in which the B 0 decay amplitudes have absorbed the phase from B 0 B 0 mixing, and so the phase of q/p is implicit in the ∆Φ(k) parameter. Although the definition of this parameter is technically similar to the β eff phase defined in Eq. (34), they differ in their physical interpretation. The parameter β eff quantifies the time-dependent mixing-induced CP asymmetry, and therefore is most relevant for the CP eigenstate modes, such as ρ 0 (770)K 0 S and f 0 (980)K 0 S . On the other hand the ∆Φ(k) parameter concerns mostly flavor-specific modes, such as B 0 → K * + (892)π − , for which there is no interference be-tween decays with and without mixing. For such modes, sensitivity to ∆Φ(k) is provided indirectly by the interference pattern of the resonance k with other modes that are accessible both to B 0 and B 0 decays.
We also extract the relative fit fraction F F of a Q2B channel k, which is calculated as: where the terms are obtained by integration over the complete Dalitz plot. The total fit fraction is defined as the algebraic sum of all fit fractions. This quantity is not necessarily unity due to the potential presence of net constructive or destructive interference. Using the relative fit fractions, we calculate the branching fraction B for the intermediate mode k as where B(B 0 → K 0 S π + π − ) is the total inclusive branching fraction We compute the average efficiency,ε, by weighting MC events with the measured intensity distribution of signal events, (|A(DP)| 2 + |Ā(DP)| 2 )/2. The term N BB is the total number of BB pairs in the sample. Finally, we use the following integrals of amplitudes over the complete Dalitz plot to measure the inclusive direct CP -asymmetry: The Q2B parameters and fit fractions are given in Table V, together with their statistical and systematic errors. The branching fractions are shown in Table VI.
To extract the statistical uncertainties on the Q2B parameters we perform likelihood scans, not relying on any assumption about the shape of the likelihood function. Since the Q2B parameters are not directly used in the fit, we instead must perform the scan fixing one or two parameters among the signal model magnitudes and phases. These are chosen in such a way that the resulting likelihood curve can be trivially interpreted in terms of the Q2B parameter of interest. In each case the chosen parameters are fixed at several consecutive values, for each of which the fit to the data is repeated. The error on the Q2B parameter is determined by the points, or the contour, where the −2 log L function changes by one unit with respect to its minimum value. Systematic uncertainties are discussed in Sec. VII. Results of the likelihood scans in terms of −2∆ log L are shown in Fig. 9 to 16. The measurements of time-dependent CP -violation in the f 0 (980)K 0 S and ρ 0 (770)K 0 S modes are presented as two-dimensional likelihood scans in the (β eff , C) plane, shown in Fig. 9. The scans are displayed as confidence level contours after two-dimensional convolution with the covariance matrix of systematic uncertainties. On the same figure are also displayed the one-dimensional likelihood scans of β eff . For f 0 (980)K 0 S the two solutions lie below and above 45 degrees and correspond very closely to the trigonometric ambiguity between a given value of β eff and 90 • − β eff (mirror solutions). On the other hand, for ρ 0 (770)K 0 S both solutions are below 45 degrees. In this case the local solutions corresponding to the trigonometric ambiguities of the two observed solutions are suppressed at 3.6 and 2.0 standard deviations, respectively.
The (β eff , C) plane can be transformed to the more familiar (S, C) plane using Eq. (34) to (36). The corresponding two-dimensional contours are shown in Fig. 10. While a part of the information on the phases is lost, this representation has nonetheless the advantage of allowing direct comparison with the measurement of sin 2β and C in b → ccs modes. For f 0 (980)K 0 S , the results agree with the expectation based on b → ccs to 1.1σ; for ρ 0 (770)K 0 S the agreement is better than 1σ. For the measured values of (β eff , C) for f 0 (980)K 0 S , CP conservation is excluded at 3.5σ. For ρ 0 (770)K 0 S , the measurement of (β eff , C) is consistent with CP conservation within 1σ.
The measurement of the phase ∆Φ(K * + (892)π − ) is presented as a one-dimensional likelihood scan in Fig. 11. For this flavor-specific mode, there is virtually no region in phase space that is accessible both to B 0 and B 0 ; thus, sensitivity to this phase difference is limited. Simulation shows that interference of the K * + (892)π − with the f 0 (980)K 0 S and ρ 0 (770)K 0 S modes (for which B 0 and B 0 amplitudes interfere via mixing) provides most of the sensitivity to ∆Φ(K * + (892)π − ); unfortunately, the overlap in phase space of these resonances is small. As a consequence, only the (−137, −5) • interval is excluded at 95% confidence level. Figure 11 also shows the measurement of the similar phase difference for the (Kπ) * 0 component. As for K * (892), the measurement sets no strong constraint on this phase. Only the interval [−132, +25] • is excluded at 95% confidence level.
In contrast, due to the sizable overlap in phase space between the Kπ S-and P-waves of the same charge, the relative phases φ((Kπ) * ± 0 , K * ± (892)) are measured to ±13 • including systematics. The onedimensional scans are shown in Fig. 12. The associated observable ∆φ((Kπ) * ± 0 , K * ± (892)) is compatible with CP conservation. Figure 12 also shows the scans for φ(f 0 (980)K 0 S , ρ 0 (770)K 0 S ), φ(ρ 0 (770)K 0 S , K * + (892)π − ), and their corresponding CP -conjugates. It is clear from this figure and from Table V that the phases for the former are measured to a better accuracy. This is due to the larger overlap in phase space between the f 0 (980) and the ρ 0 (770). In both cases the associated observables ∆φ are compatible with CP conservation.
Mode model: f X (1300)K 0 S , f 2 (1270)K 0 S , NR, and χ c0 (1P )K 0 S , we scan the likelihood as a function of the corresponding fit fractions. These scans are shown in Fig. 13. We obtain a total (statistical and systematic) significance of 4.8 and 3.8 standard deviations for the NR and χ c0 (1P )K 0 S components, respectively. The significance for the sum of fit fractions of the f 2 (1270)K 0 S and f X (1300)K 0 S components is 4.8 standard deviations while their individual significances are 2.9σ and 2.4σ, respectively.
The (Kπ) * 0 component is modeled in our analysis by the LASS parametrization [34], which consists of a NR effective range term plus a relativistic Breit-Wigner term for the K * (1430) resonance. We separate from the corresponding branching fraction, quoted in Table VI, the contribution of the K * (1430) resonance and find it to be (29.9 +2.3 −1.7 ± 1.6 ± 0.6 ± 3.2) × 10 −6 . This value is corrected for the secondary branching fraction using B(K * (1430) → Kπ) from Ref. [31] and the isospin relation B(K * + (1430) → K 0 π + )/B(K * + (1430) → K + π 0 ) = 2. The first uncertainty is statistical, the second is systematic, the third represents the DP signal model dependence, and the fourth is due to the uncertainty on the secondary branching fraction. In addition we calculate the total NR contribution by combining coherently the effective range part of the LASS parametrization and the flat phase-space NR component. We find this total NR fit fraction to be 22.1 +2.8 −2.0 ± 2.1 ± 0.7 %. Note that this number accounts for the destructive interference between the two NR terms. The corresponding branching fraction is (11.07 +2.51 −0.99 ± 0.81 ± 0.40) × 10 −6 .
As a validation of our treatment of the timedependence, we allow τ B 0 and ∆m d to vary in the fit. We find τ B 0 = 1.579±0.061 ps and ∆m d = 0.497±0.035 ps −1 while the remaining free parameters are consistent with the nominal fit. The numbers for τ B 0 and ∆m d are in agreement with current world averages [3]. In addition we perform a fit floating the S parameters for B 0 → J/ψK 0 S and B 0 → ψ(2S)K 0 S events. We find S = sin(2β) = 0.690 ± 0.077 and 0.73 ± 0.27 for J/ψK 0 S and ψ(2S)K 0 S respectively. These numbers are in agreement with the current world average [3]. Signal enhanced distributions of ∆t and the ∆t asymmetry for events in the J/ψK 0 S region are shown in Fig. 17. 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 estimation.
As a further cross-check of the results, we performed an independent analysis and obtained compatible results [47]. The main differences between this cross-check analysis and the one presented here were the use of a Fisher discriminant instead of a NN, the removal of bands in invariant mass to cut away the B 0 → D − π + , B 0 → J/ψ K 0 S and B 0 → ψ(2S)K 0 S contributions, and the use of Cartesian isobar parameters.

VII. SYSTEMATIC STUDIES
To estimate the contribution to B 0 → K 0 S π + π − decay via other resonances, we first fit the data including these other decays in the fit model. We consider possible resonances, including ω(782), ρ 0 (1450), ρ 0 (1700), f 0 (1710), f 2 (1810), K * ± (1680), K * ± 2 (1430), χ c2 (1P ), and a low mass σ. A relativistic Breit-Wigner lineshape is used to parameterize these additional resonances, with masses and widths from Ref. [31]. As a second step we simulate high statistic samples of events, using a model based on the previous fits, including the additional resonances. Finally, we fit these simulated samples using the nominal signal model. The systematic effect (contained in the "DP Model" field in Table VII) is taken from the difference observed between the generated and fitted values. We quote this DP model uncertainty separately from other systematics.
We vary the mass, width, and any other parameters of all isobar fit components within their errors, as quoted in Table I, and assign the observed differences in the measured amplitudes as systematic uncertainties ("Lineshape" in Table VII).
To validate the fitting tool, we perform fits on large MC samples of fully-reconstructed events with the measured proportions of signal, continuum, and B background events. No significant biases are observed in these fits and therefore no corrections are applied. The statistical uncertainties on the fit parameters are taken as systematic uncertainties ("Fit Bias" in Table VII).
Another major 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. Since B background modes may exhibit CP violation, the corresponding parameters are varied within their uncertainties, or, if unknown, within the physical range. 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 these fits with respect to the nominal fit as systematic errors. These errors are listed as "B Background " in Table VII.
Other systematic effects are much less important for the measurements of the amplitudes and are combined in the "Other" field in Table VII. Details are given below.
The parameters of the continuum PDFs are determined by the fit. No additional systematic uncertainties are assigned to them. An exception to this is the DP PDF: to estimate the systematic uncertainty from the m ES sideband extrapolation, we use large samples of e + e − → qq MC data (q = u, d, s, c). We compare the distributions of m ′ and θ ′ between sidebands at different ranges in m ES and find the two such sidebands that show the maximum discrepancy. We assign as systematic uncertainty the effect seen when weighting the continuum DP PDF by the ratio of these two data sets.
The uncertainties associated with ∆m d and τ are estimated by varying these parameters within the uncertain-   ties on the world average [31].
The signal PDFs for the ∆t resolution and tagging fractions are determined from fits to a control sample of fully reconstructed B decays to exclusive final states with charm, and the uncertainties are obtained by varying the parameters within the statistical uncertainties.
Finally, the uncertainties due to particle identification, tracking efficiency corrections, K 0 S reconstruction, and the calculation of N BB are 2.0%, 1.6%, 0.9%, and 1.1%, respectively. These contribute only to the branching fraction systematic uncertainties.
The average fraction of misreconstructed signal events (f SCF ) predicted by the MC simulation has been verified with fully reconstructed B → Dρ events [42]. No significant differences between data and the simulation were found. To estimate a systematic uncertainty from f SCF , we vary these fractions, for all tagging categories. Tagging efficiencies, dilutions, and biases for signal events  are varied within their experimental uncertainties.

VIII. SUMMARY
We have presented results from a time-dependent Dalitz plot analysis of B 0 → K 0 S π + π − decays obtained from a data sample of 383 million Υ (4S) → BB decays.
Using an amplitude analysis technique, we measure 15 pairs of relative magnitudes and phases for the different resonances, taking advantage of the interference between them in the Dalitz plot. From the measured decay amplitudes, we derive the Q2B parameters of the resonant decay modes. Two solutions, with equivalent goodnessof-fit, were found.
Including systematic and Dalitz plot model uncertainties, the combined confidence interval for the measured values of β eff in B 0 decays to f 0 (980)K 0 S is 18 • < β eff < 76 • at 95% C.L. CP conservation in B 0 decays to f 0 (980)K 0 S is excluded at 3.5σ, including systematics. For B 0 decays to ρ 0 (770)K 0 S , the combined confidence interval is −9 • < β eff < 57 • at 95% C.L. These results are both consistent with the measurements in b → ccs modes.