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Time-dependent Dalitz Plot Measurement of CP Parameters in B0 -->Ks Pi+ Pi- Decays

We present a time-dependent Dalitz plot measurement of CP violation parameters in B0 -->Ks Pi+ Pi- decays. These results are obtained from a large data sample that contains 657 million BBbar pairs collected at the Y(4S) resonance with the Belle detector at the KEKB asymmetric-energy e+e- collider. For the CP violation parameters, we obtain two consistent solutions that describe the data well. The first of these solutions may be preferred by external information from other measurements. There is no evidence for direct CP violation in B0 -->rho0(770) Ks, B0 -->f0(980) Ks and B0 -->K*+(892) Pi-, while measurements of mixing-induced CP violation in B0 -->rho0(770) Ks, B0 -->f0(980) Ks decays are consistent with that of b -->c cbar s decays. We also measured the phase difference between B0 -->K*+(892) Pi- and B0B -->K*-(892) Pi+ which may be used to extract phi3.


Abstract
We present a time-dependent Dalitz plot measurement of CP violation parameters in B 0 → K 0 S π + π − decays. These results are obtained from a large data sample that contains 657 × 10 6 BB pairs collected at the Υ(4S) resonance with the Belle detector at the KEKB asymmetric-energy e + e − collider. For the CP violation parameters, we obtain two consistent solutions that describe the data well. The first of these solutions may be preferred by external information from other measurements. There is no evidence for direct CP violation in B 0 → ρ 0 (770)K 0 S , B 0 → f 0 (980)K 0 S and B 0 → K * + (892)π − , while measurements of mixing-induced CP violation in B 0 → ρ 0 (770)K 0 S and B 0 → f 0 (980)K 0 S decays are consistent with that of b → ccs decays. We also measured the phase difference between B 0 → K * + (892)π − andB 0 → K * − (892)π + , which may be used to extract φ 3 .

INTRODUCTION
CP violation in the Standard Model (SM) is due to a complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [1,2]. At present, mixing-induced CP violation has been clearly observed by the BaBar [3] and Belle [4] Collaborations in the b → ccs induced decay, B 0 → J/ψ K 0 S , while many other modes provide additional information on CP violating parameters. Of recent interest is CP violation in b → qqs transitions, which proceeds by loop diagrams that may be affected by new particles in various extensions of the SM. Furthermore, the CP asymmetries in b → qqs transitions are expected in the SM to be slightly higher than those observed in b → ccs transitions [5,6,7,8,9,10,11,12,13]. However, current experimental measurements [14] tend to be lower than those for b → ccs transitions motivating more precise experimental determinations.
This analysis is concerned with time-dependent CP violation in B 0 → K 0 S π + π − decays including the quasi-two-body modes B 0 → ρ 0 (770)K 0 S and B 0 → f 0 (980)K 0 S . Quasi-twobody time-dependent analyses have been performed on B 0 → ρ 0 (770)K 0 S by the BaBar Collaboration [15] and B 0 → f 0 (980)K 0 S by both the BaBar and Belle Collaborations [16,17]. However, a two-body approach to these modes is not ideal due to interference with other resonances as well as non-resonant decays into the same final state. These effects can be taken into account with a time-dependent Dalitz plot fit. In addition, the interference terms in some cases may be sensitive to the cosine of the effective weak phase difference (φ eff 1 ) in mixing, potentially resolving the two-fold ambiguity in φ eff 1 characteristic of quasi-twobody analyses. Current direct measurements of cos 2φ 1 from b → ccs prefer the solution of φ 1 = 21.5 • over φ 1 = 68.5 • [18,19,20,21,22,23] and similarly, it is important to attempt to resolve this quadratic ambiguity in b → qqs modes as well.
Another mode of interest included in B 0 → K 0 S π + π − decays is B 0 → K * + (892)π − . Since direct CP violation has been observed in B 0 → K + π − [24,25], CP violation could also be present in B 0 → K * + (892)π − decays, though existing measurements are consistent with zero [26]. This measurement may also be related to the A CP (Kπ) puzzle; the asymmetry in B 0 → K + π − is found to be significantly different from that in B ± → K ± π 0 . This unexpected result may indicate the presence of new physics (NP) or poor understanding of strong interaction effects in B decays. A model-independent test for NP is possible via an isospin sum rule with high statistics [27]. Furthermore, it has been proposed that the phase difference between B 0 → K * + (892)π − andB 0 → K * − (892)π + can be used to extract φ 3 [28,29,30,31]. The BaBar collaboration has also performed a time-dependent Dalitz plot analysis on B 0 → K 0 S π + π − decays and released a preliminary result [32].

DECAY AMPLITUDE
The Dalitz plot variables are defined as the invariant squared masses, where p ± (p 0 ) are the 4-momenta of the π ± (K 0 S ) respectively. The final combination, s 0 ≡ (p + + p − ) 2 can be obtained from 4-vector conservation, The differential B 0 decay width with respect to the Dalitz plot variables is where A is the Lorentz-invariant amplitude of the decay. In the isobar approximation, the total amplitude of B 0 (B 0 ) → K 0 S π + π − is written as the sum of intermediate decay channel amplitudes with the same final state, where a ′ i ≡ |a ′ i |e ib ′ i are complex coefficients describing the relative magnitudes and phases between the decay channels, which also carry the weak phase dependence. The amplitudes, F i (s + , s − ) contain only strong dynamics and thus, F i (s + , s − ) =F i (s − , s + ). They can be expanded in terms of invariant mass and angular distribution probabilities, where p * is the momentum of the bachelor particle in the B 0 rest frame, p and q are the momenta of the bachelor particle and one of the resonance daughters in the resonance's rest frame respectively, L is the orbital angular momentum between the resonance and the bachelor particle while X L i are the Blatt-Weisskopf barrier factors [33]. As these factors are unknown in general and the Blatt-Weisskopf forms do not improve this model, they are taken to be unity with the uncertainty in this choice being treated in the systematic errors. The angular distribution, The mass shapes are denoted as R i (s + , s − ), which differ depending on the decay channel. We utilize the Relativistic Breit-Wigner (RBW) [34], Gounaris-Sakurai (GS) [35] and Flatté [36] line shapes. Table I summarizes the components considered in the signal model, which was motivated by the previous Belle Dalitz plot analysis of B 0 → K 0 S π + π − [37]. The f X (1300) resonance of unknown spin that appears in the table was first described in Ref. [37] and is assumed to be a scalar.

TIME-DEPENDENCE
The decay of the Υ(4S) produces a BB pair of which one (B 0 Rec ) may be reconstructed as B 0 → K 0 S π + π − while the other (B 0 Tag ) may reveal its flavor. The proper time interval between the B 0 Rec and B 0 Tag mesons that decay at times, t Rec and t Tag , respectively, is defined as ∆t ≡ t Rec − t Tag . For coherent BB production in a Υ(4S) decay, the time-dependent decay rate when B 0 Tag possesses flavor, q (B 0 : q = +1,B 0 : q = −1), is given by  [34] with the exception of those for the f 0 (980) and f X (1300), which are taken from Refs. [38] and [37], respectively.
where τ B 0 is the B 0 lifetime and ∆m d is the mass difference between the two mass eigenstates of the neutral B meson. This assumes no CP violation in mixing, |q/p| = 1, and that the total decay rate difference between the two mass eigenstates is negligible. The amplitudes, A, were defined previously; we choose a convention where the B 0 -B 0 mixing phase of q/p is absorbed into theB 0 decay amplitude,ā ′ i . These complex coefficients can be redefined in a way that depends on the decay amplitude, forĀ, and thus a resonance, i, has a direct CP violation asymmetry given by For a CP eigenstate, the CKM angle, φ eff 1 (i), is reduced to a fit parameter, and its effective mixing-induced CP violation asymmetry is calculated as where η i is the CP eigenvalue of the final state. Note that A CP (i) and S eff CP (i) are restricted by these definitions to lie in the physical region. For flavor-specific states the phase difference is calculated as

DATA SET AND BELLE DETECTOR
This time-dependent Dalitz plot measurement of CP violating parameters in B 0 → K 0 S π + π − is based on a data sample that contains 657 × 10 6 BB pairs collected with the Belle detector at the KEKB asymmetric-energy e + e − (3.5 on 8 GeV) collider [39]. Operating with a peak luminosity that exceeds 1.7 × 10 34 cm −2 s −1 , the collider produces the Υ(4S) resonance ( √ s = 10.58 GeV) with a Lorentz boost of βγ = 0.425, opposite to the positron beam direction, z, which usually decays into a BB pair. The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) comprised of CsI (Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron fluxreturn located outside of the coil is instrumented to detect K 0 L mesons and to identify muons (KLM). The detector is described in detail elsewhere [40]. Two inner detector configurations were used. A 2.0 cm beampipe and a 3-layer silicon vertex detector (SVD1) was used for the first sample of 152 ×10 6 BB pairs, while a 1.5 cm beampipe, a 4-layer silicon detector (SVD2) and a small-cell inner drift chamber was used to record the remaining 505 ×10 6 BB pairs [41]. We use a GEANT-based Monte Carlo (MC) simulation to model the response of the detector and determine its acceptance [42].

EVENT SELECTION
We reconstruct B candidates from a K 0 S candidate and a pair of oppositely-charged tracks. Charged tracks satisfy loose criteria on their impact parameters relative to the interaction point (IP), dr < 0.4 cm and |dz| < 5.0 cm, where r is the radial coordinate of the Belle detector. With information obtained from the CDC, ACC and TOF, particle identification (PID) is determined with the likelihood ratio, L i /(L i + L j ). Here, L i (L j ) is the likelihood that the particle is of type i (j). To suppress background from particle misidentification, vetoes are applied on particles consistent with kaon, electron or proton hypotheses. The transverse momenta of charged tracks are required to be greater than 100 MeV and the additional SVD requirements of two z hits and one r − φ hit [43] are imposed so that a good quality vertex of the reconstructed B candidate can be determined. We only consider K 0 S → π + π − candidates with vertices that are displaced from the IP and that lie in the mass window, |m(π + π − ) − m(K 0 S )| < 15 MeV/c 2 . Reconstructed Rec is from continuum (e + e − → qq) events. Since their topology tends to be jet-like in contrast to the spherical BB decay, continuum can be suppressed with a Fisher discriminant based on modified Fox-Wolfram moments [44]. This discriminant is combined with the polar angle of the B candidate in the CMS, cos θ B , which follows a 1 − cos 2 θ B distribution for BB events while being flat for continuum. A requirement that rejects 91% of the continuum background while retaining 73% of the signal events is applied. The second largest background comes from charm decays of the B meson and peaks in the signal region. We apply the charm vetoes summarized in Table II. The fraction of events having more than one reconstructed candidate in regions dominanted by particular quasi-two-body modes is 28% for B 0 → ρ 0 (770)K 0 S , 25% for B 0 → f 0 (980)K 0 S and 21% for B 0 → K * + (892)π − . Selecting the B candidate having an M bc value closest to the nominal B meson mass, the fraction of misreconstructed events is 3% for This criteria introduces negligible bias into the ∆E distribution. From this best candidate, the Dalitz plot coordinates in the signal region, s ± , are calculated after a mass-constrained fit of the B meson to ensure all the events are within the Dalitz plot boundaries.

Region vetoed
Mode vetoed Rec and B 0 Tag mesons are approximately at rest in the Υ(4S) CMS, the difference in decay time between the two B mesons, ∆t, can be determined from the displacement in z between the final state decay vertices, The vertex of reconstructed B candidates is determined from the charged daughters using the known IP. The IP profile is smeared in the plane perpendicular to z to account for the finite flight length of the B meson in this plane. To obtain the ∆t distribution, we reconstruct the tag-side vertex from the tracks not used to reconstruct B 0 Rec [43] and employ the flavor tagging routine described in Ref. [45]. The tagging information is represented by two parameters, the B 0 Tag flavor, q and r. The parameter r is an event-by-event, MC determined flavor-tagging dilution factor that ranges from r = 0 for no flavor discrimination to r = 1 for unambiguous flavor assignment. The total effective tagging efficiency is determined to be 0.29 ± 0.01.

SIGNAL YIELD
The signal yield is extracted with an extended one-dimensional unbinned maximum likelihood fit to ∆E with the requirement, 5.27 GeV/c 2 < M bc < 5.29 GeV/c 2 . The signal probability density function (PDF), P Sig , is modeled with a sum of two Gaussians, where the fraction and shape parameters of the tail Gaussian are fixed from MC relative to the main Gaussian. The qq component, P qq , is modeled with a first-order polynomial whose slope is allowed to float in the fit while the BB component, P BB is described by a histogram determined from MC. The total likelihood for N events becomes where the qq fraction, f qq , is a free parameter. A fit to data yields N Sig = 1944 ± 98 events with the fit result shown in Fig. 1. In the signal region, there are a total of 4547 B 0 → K 0 S π + π − candidates that are used to perform the time-dependent Dalitz plot analysis. The average signal purity in this region is 0.41 ± 0.02 and the contribution of continuum to the total background is found to be 0.89 ± 0.02.

SQUARE DALITZ PLOT
In B 0 → K 0 S π + π − decays, both signal and continuum events tend to concentrate near the Dalitz plot kinematic boundaries making large variations in these small areas difficult to describe if non-parametric shapes such as histograms are used to describe the continuum background. In order to alleviate this problem, a transformation that expands these regions, is introduced to define the Square Dalitz Plot [46]. Its coordinates are where m 0 is the π + π − invariant mass, m min S are the kinematic limits and θ 0 is the angle between the π − and K 0 S in the π + π − frame. The determinant of the Jacobian of this transformation is

SIGNAL PDF
The signal PDF is given by which accounts for CP dilution from the probability of incorrect flavor tagging, w and the wrong tag difference between B 0 andB 0 , ∆w, both of which are determined from flavor specific control samples [45]. This PDF is convolved with the ∆t resolution function, R B 0B0 , described in Ref. [43]. On the other hand, detector resolution is ignored in the Dalitz plot because the widths of the dominant resonances are large compared to the mass resolution. For example, the detector resolution in the region of the narrowest resonance, the f 0 (980), is typically 4 MeV, while most estimates of the f 0 (980) width tend to be at least an order of magnitude larger [34]. The relative signal detection efficiency variation across the Dalitz plane due to detector acceptance, ǫ(m ′ , θ ′ ), is also considered and is taken from MC. Misreconstructed events usually occur around the corners of the Dalitz plot with a slow pion from the tag side. The fraction of misreconstruction can be as high as 10% in these regions. As misreconstruction is not accounted for in the signal model, its effect will be included in the systematic errors.

CONTINUUM PDF
The PDF for the time-dependent Dalitz plot for the qq continuum is determined from a narrow sideband, 5.250 GeV/c 2 < M bc < 5.265 GeV/c 2 and −0.1 GeV < ∆E < 0.1 GeV, which ensures that the resonances present in continuum are not shifted significantly as the Dalitz variables are scaled to originate from B decays. There is a flavor tag asymmetry in θ ′ due to the jet-like topology of continuum; as a high momentum π + (π − ) in B 0 Rec is correlated with a high momentum π − (π + ) on the tag side. To account for this, we fit the Dalitz plot asymmetry with an empirical two-dimensional PDF, The Dalitz plot component of the qq PDF will be modeled with a two-dimensional histogram, H qq (m ′ , θ ′ ), including a term accounting for the flavor tag asymmetry with parameters fixed from the two-dimensional fit to sideband. To reduce the effects of statistical fluctuations with limited data, the symmetry condition, The ∆t distribution is modeled with a lifetime and prompt component, convolved with a double Gaussian, which uses the event-dependent ∆t error constructed from the vertex resolution, σ ≡ ( σ 2 Rec + σ 2 Tag )/βγc as a scale factor. The total qq PDF is given by As a check of this PDF, we perform fits in various Dalitz plot regions to extract the ∆t shape, however no correlation between ∆t and the Dalitz plot coordinates was found.

CHARGED B PDF
Like continuum, B + B − events also exhibit a flavor tag asymmetry. Using a generic MC sample that contains charm and rare B decays, a similar procedure is performed to model the asymmetry in the Dalitz plane. The Dalitz component of the B + B − PDF is modeled with a two-dimensional histogram symmetrized as Since charged B events are misreconstructed by borrowing a particle from the tag side, the average ∆t lifetime tends to be smaller and should be taken into account. The ∆t distribution is modeled with an exponential PDF with effective lifetimes determined from MC, convolved with the ∆t resolution function for B + , R B + B − , The B + B − time-dependent Dalitz plot PDF becomes NEUTRAL B PDF As in the charged B case, this PDF is also fixed from MC. However, no correlation between flavor and the Dalitz coordinate is found. A two-dimensional histogram is chosen to represent the Dalitz distribution that again incorporates the symmetry condition, Lifetime fits to the ∆t distribution are also consistent with the nominal lifetime value, therefore the current world-average B 0 lifetime will be used as the default value in all fits.
As neutral BB pairs mix, a time-dependent CP violating PDF is used to model ∆t and q, with A B 0B0 = S B 0B0 = 0. The total B 0B0 PDF is, Significant B 0B0 backgrounds with measured CP parameters are treated separately using Eq. 26. There are ∼ 20 B 0 → η ′ K 0 S and ∼ 4 B 0 → a 1 (1260) + π − events expected in the data sample. In these decay channels no flavor tag asymmetry in MC is found and lifetime measurements are consistent with the nominal value. The CP parameters for B 0 → η ′ K 0 S and B 0 → a 1 (1260) + π − are taken from Ref. [34].

OUTLIER PDF
To account for events with ∆t values not yet described by either signal or background PDFs, an outlier PDF is introduced, where the width of the Gaussian, σ Out , is determined in Ref. [43] and H(m ′ , θ ′ ) is the Dalitz plot histogram of data.

TIME-DEPENDENT DALITZ PLOT PDF
The full time-dependent Dalitz plot PDF is given by where f Sig (∆E, r) is the event-dependent signal probability, .
The r-bin dependent purity, p Sig (r), is calculated in the signal region and is included to increase signal sensitivity and p qq ≡ N qq /(N qq + N BB ) is the qq contribution to the total background in the signal region. The event-dependent qq probability is calculated as and S are constants defined in the signal region as where N i are the expected number of events in background category i.
As there is only sensitivity to the relative amplitudes and phases between decay modes, we fix a K * + (892) = 1 and b K * + (892) = 0. In addition, the f 2 (1270), f X (1300) and non-resonant components share common CP parameters giving a total of 27 free parameters.

FIT RESULT
We perform a time-dependent Dalitz plot fit to data and find four solutions with consistent CP parameters given in Table III. These were obtained by performing numerous fits with randomly generated initial values for the free parameters. For each resonance, i, a relative fraction can be calculated as where the sum of fractions over all decay channels may not be 100% due to interference. Table IV summarizes the relative fractions for all solutions. Solutions 1 and 2 were found in the previous Belle analysis [37] and correspond to the two solutions found in the higher statistics Belle Dalitz plot analysis of the isospin partner mode, B + → K + π + π − [47]. The signal models used in this analysis and that of the isospin partner are almost identical. These two solutions are due to an interplay between the two broad S-wave amplitudes, K * + 0 (1430)π − and the non-resonant component, and are characterized by their different relative fractions. However, they give almost indistinguishable Dalitz plot distributions; studies with high statistics pseudo-experiments confirm the existence of these two solutions. Solution 3 has a high f 0 (980)K 0 S fraction compared with Solution 2 that may indicate a similar type of interference between f 0 (980) and the non-resonant component. Unlike the previous case, Solution 3 cannot be found in a high statistics toy MC sample generated with Solution 2, neither can Solution 2 be reproduced from Solution 3 indicating that one of these solutions cannot survive with higher experimental statistics. Considering that a solution with a high f 0 (980)K 0 S fraction was not found in the previous analysis or the higher statistics B + → K + π + π − analysis [47], we therefore conclude that Solution 3 is due to statistical fluctuations in the small B 0 → K 0 S π + π − sample and is unlikely to be a physical solution. Solution 4 appears to be statistically consistent with Solution 2 for all parameters, while Solution 4 cannot be reproduced from a high statistics toy MC sample generated with Solution 2 and vice versa. This indicates that these two solutions are likely due to a statistical fluctuation of a single solution and we select Solution 2 as the nominal solution of the two because of its slightly better likelihood. Consequently, Solutions 1 and 2 are treated as the solutions of this analysis. The high K * + 0 (1430)π − fraction of Solution 1 is in agreement with some phenomenological estimates [48] and may also be qualitatively favored by the total K − π S-wave phase shift as a function of m(Kπ) when compared with that measured by the LASS collaboration [49]. As the likelihood difference is not found to be significant using an ensemble of pseudoexperiments corresponding to the luminosity of the data sample, we retain both solutions.
To assess how well the fit to the Dalitz plot represents data as shown in Fig. 2, the time and flavor-integrated Dalitz plot is divided into variable size bins so that each bin contains at least 25 events. A goodness-of-fit statistic for the multinomial distribution is then calculated as the sum over N bins, where n i is the number of events observed in the i-th bin and p i is the number of events in the i-th bin as given by the fit [50]. The distribution of this statistic is bounded by a χ 2 distribution with N − 1 degrees of freedom and one with N − k − 1 degrees of freedom where k = 19 is the number of Dalitz plot fit parameters. The χ 2 /N value for the best fit is 201.9/(189 − 19 − 1). We measure the time-dependent CP parameters of B 0 → K 0 S π + π − , Solution 1: −2 log L = 18472.5 S π + π − . When plotting a two-particle mass projection, we require the invariant mass of the other two two-particle combinations to be greater than 1.5 GeV/c 2 . The solid curve shows the fit result while the shaded and dashed curves show the qq and total background components, respectively.
where the first error is statistical, the second is systematic and the third is the Dalitz plot signal model uncertainty. The ∆t and raw asymmetry fit projections for B 0 → ρ 0 (770)K 0 S and B 0 → f 0 (980)K 0 S are shown in Fig. 3, and the statistical correlation coefficients between the CP parameters of both solutions are given in Tables V and VI. The full correlation  matrices are given in Tables IX-XIV. Likelihood scans of φ eff 1 for both solutions are obtained by fixing φ eff 1 and redoing the fit. The statistical error for S eff CP is also determined from a likelihood scan that fixes A CP and φ eff 1 . Similarly, likelihood scans of ∆φ are also produced. In addition, we also perform scans that include the systematic and model errors by convolving the likelihood with a Gaussian t (ps) ∆

SYSTEMATIC UNCERTAINTIES
The systematic errors in the vertex reconstruction include uncertainties in the IP profile, charged track selection based on track helix errors, helix parameter corrections, ∆t and vertex goodness-of-fit selection, ∆z bias and SVD misalignment. The parameters for w and ∆w, resolution function, physics parameters, background shape and signal probability are varied by ±1σ. Where histograms were used, all bins are simultaneously but independently varied by their respective errors. The effect of CP asymmetry in the background is estimated by varying the CP parameters of the entire neutral B component within the physical range except for B 0 → η ′ K 0 S , which is varied within its known uncertainties. Toy MC samples showed some small fitting bias for CP parameters due to low statistics in each sample. We take this bias as a systematic uncertainty. The effect of misreconstructed signal events is also investigated using signal MC. This is achieved by comparing the fit result of the signal MC sample with another fit on the same sample, which required that the events were reconstructed correctly. The efficiency histogram also includes systematic uncertainties in    correction factors due to tracking, K 0 S selection and particle identification. The systematics from these data to MC efficiency ratios are calculated from independent studies at Belle. Tag-side interference, which only affects CP eigenstates, comes from CP violation on the tag side [51], and is estimated with B → D * lν. We generate MC pseudo-experiments and perform an ensemble test to obtain systematic biases from tag-side interference [52]. In the signal model, the Dalitz plot mass, widths and other parameters are varied by their errors and the Blatt-Weisskopf barrier factors are included. The Dalitz plot model is the dominant systematic source and is quoted separately. We introduce additional resonances, K * 2 (1430), the non-resonant component is empirically chosen, so different parametrizations are tested. They include modeling the non-resonant part with the tail of a Breit-Wigner, R NR (s; α) = iα/(s + iα) and a power law whose exponent is a fit parameter, R NR (s; α) = s −α . The fit differences from these alternate Dalitz plot parametrizations were summed in quadrature. The systematic errors for both solutions are summarized in Table VII and Table VIII. As the systematic uncertainty from the Dalitz plot model is the largest, it is quoted separately.

CONCLUSION
In summary, we perform a time-dependent Dalitz plot measurement of CP parameters in B 0 → K 0 S π + π − decays and find two solutions that describe the data well. The first of these solutions may be preferred by external information from other measurements; however, we retain both solutions. This is Belle's first measurement of CP violation parameters in the B 0 → ρ 0 (770)K 0 S channel and the first measurement of the CP parameters in B 0 → f 0 (980)K 0 S decays using a time-dependent Dalitz plot technique. There is currently no evidence for direct CP violation in B 0 → ρ 0 (770)K 0 S , B 0 → f 0 (980)K 0 S and B 0 → K * + (892)π − , while mixing-induced CP violation in B 0 → ρ 0 (770)K 0 S and B 0 → f 0 (980)K 0 S decays deviates from zero by roughly 2σ and is consistent with measurements in b → ccs transitions. We also measured the phase difference between B 0 → K * + (892)π − andB 0 → K * − (892)π + , which may be used to extract φ 3 .