First model-independent Dalitz analysis of $B^0 \to DK^{*0}$, $D\to K_S^0\pi^+\pi^-$ decay

We report a measurement of the amplitude ratio $r_S$ of $B^0 \to D^0K^{*0}$ and $B^0 \to \bar{D^0}K^{*0}$ decays with a Dalitz analysis of $D\to K_S^0\pi^+\pi^-$ decays, for the first time using a model-independent method. We set an upper limit $r_S<0.87$ at the 68\% confidence level, using the full data sample of $772\times10^6$ $B\bar{B}$ pairs collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB $e^+e^-$ collider. This result is obtained from observables $x_- = +0.4 ^{+1.0 +0.0}_{-0.6 -0.1} \pm0.0$, $y_- = -0.6 ^{+0.8 +0.1}_{-1.0 -0.0} \pm0.1$, $x_+ = +0.1 ^{+0.7 +0.0}_{-0.4 -0.1} \pm0.1$ and $y_+ = +0.3 ^{+0.5 +0.0}_{-0.8 -0.1} \pm0.1$, where $x_\pm = r_S \cos(\delta_S \pm \phi_3)$, $y_\pm = r_S \sin(\delta_S \pm \phi_3)$ and $\phi_3~(\delta_S)$ is the weak (strong) phase difference between $B^0 \to D^0K^{*0}$ and $B^0 \to \bar{D^0}K^{*0}$.

We report a measurement of the amplitude ratio rS of B 0 → D 0 K * 0 and B 0 →D 0 K * 0 decays with a Dalitz analysis of D → K 0 S π + π − decays, for the first time using a model-independent method. We set an upper limit rS < 0.87 at the 68% confidence level, using the full data sample of 711 fb −1 corresponding to 772 × 10 6 BB pairs collected at the Υ(4S) resonance with the Belle detector at the KEKB e + e − collider. This result is obtained from observables x− = +0.

INTRODUCTION
Determination of parameters of the standard model (SM) plays an important role in the search for new physics. In the SM, the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1] gives a successful description of current all measurements of CP violation. The CP -violating parameters φ 1 , φ 2 and φ 3 are the three angles of the most equilateral of the CKM unitarity triangles, of which φ 3 ≡ arg (−V ud V ub * /V cd V cb * ) is the least accurately determined. In the usual quarkphase convention, where the complex phase is negligible in the CKM matrix elements other than V ub and V td [2], the measurement of φ 3 is equivalent to the extraction of the phase of V ub . To date, φ 3 measurements have been performed mainly with B meson decays into D ( * ) K ( * ) final states [3][4][5][6][7][8][9][10][11][12], all of which exploit the interference between theD ( * )0 and D ( * )0 decaying into a common final state. In particular, Dalitz analyses of B ± → D ( * ) K ( * )± , D → K 0 S π − + π − provide the most precise determination of φ 3 . The Dalitz analysis technique for the measurement of φ 3 was proposed in Ref. [13]. Belle reported the first φ 3 measurement with the model-independent Dalitz analysis technique in Ref. [14], which exploits a set of measured strong phases instead of relying on a D decay model into a three-body final state.
In this paper, we present the first measurement of the amplitude ratio of B 0 → D 0 K * 0 and B 0 →D 0 K * 0 decays with a model-independent Dalitz analysis. We reconstruct B 0 → DK * 0 , with K * 0 → K + π − (Throughout the paper, charge-conjugate processes are implied; K * 0 refers to K * (892) 0 and D refers to either D 0 orD 0 when the D 0 flavor is untagged). Here, the flavor of the B meson is identified by the kaon charge. Neutral D mesons are reconstructed in the K 0 S π + π − decay mode. The reconstructed final states are accessible through b → c and b → u processes via the diagrams shown in Fig. 1. In this analysis, we use the variables r S , k, and δ S to parameterize the strong dynamics of the decay. These parameters are defined as [15] where the integration is over the B 0 → DK + π − Dalitz distribution region corresponding to the K * 0 resonance. Here, A b→c (A b→u )(p) is the magnitude of the amplitude for the b → c (u) transition and δ(p) is the relative strong phase, where the variable p indicates the position within the DK + π − Dalitz distribution. If the B 0 decay can be considered as a DK * 0 two-body decay, r S becomes the ratio of the amplitudes for b → u and b → c and k becomes 1. According to a simulation study using a Dalitz model based on the measurements in Ref. [16], the value of k is 0.95 ± 0.03 within the phase space of the DK * 0 resonance. The value of r S is expected to be around 0.4, which corresponds naïvely to | V ub V * cs | / | V cb V * us | but also depends on strong interaction effects. For r S , the best experimental value is reported by LHCb [17] as r S = 0.240 +0.055 −0.048 (different from zero by 2.7 σ) from B 0 → DK * 0 , D → K + K − , π + π − , K ± π ∓ decay.
In the model-dependent analysis, one deals directly with the Dalitz distribution density and the functions C and S are obtained from a model based upon a fit to the D 0 → K 0 S π + π − amplitude. On the other hand, in the modelindependent approach [18], where the assumption of a model for D 0 → K 0 S π + π − decay is not necessary, the Dalitz plot is divided into 2N bins symmetric under the exchange m 2 − ↔ m 2 + . The bin index i ranges from −N to N (excluding 0); the exchange m 2 − ↔ m 2 + corresponds to the exchange i ↔ −i. The expected number of signal events in bin i of the Dalitz distribution of the D mesons from B 0 → DK * 0 is where N +(−) stands for the number of B 0 (B 0 ) meson decays, h is the normalization constant and K +i is the number of events in the i th bin of a flavor-tagged D 0 → K S π + π − decays measured with a sample of inclusively reconstructed D * + → D 0 π + decays. Equation (7) can be obtained by integrating Eq. (4) over the i th bin region. Here, K i ∝ Di |A| 2 dD, and D represents the Dalitz plane and D i is the bin over which the integration is performed. The values of K i are measured from a sample of flavor-tagged D 0 mesons obtained by reconstructing D * ± → Dπ ± decays. The terms c i and s i are the amplitude-weighted averages of the functions C and S over the bin: The terms s i are defined similarly with C substituted by S. The absence of CP violation in the D decay implies c i = c −i and s i = −s −i . The values of c i and s i can be measured using quantum-correlated D pairs produced at charm-factory experiments operating at the threshold of DD pair production. The CLEO Collaboration has reported c i and s i values from CP tagged and flavor tagged DD events data, and this analysis is performed with the optimal binning in Refs. [19,20], as shown in Fig. 2. Given that c i and s i are measured and K i and k are known, Eq. (7) has only three free parameters (x, y, and h B ) for each of B 0 andB 0 , and can be solved. We use the values of (c i , s i ) for the "optimal D 0 → K 0 S π + π − binning" reported in Table XVI of Ref. [20], "the optimal binning" K i values reported in Table II of Ref. [14], and k = 0.95 ± 0.03 [16]. We have neglected charm-mixing effects in D decays from both the B 0 → DK * 0 process and in the quantum-correlated DD production [21].

EVENT RECONSTRUCTION AND SELECTION
This analysis is based on a data sample that contains 711 fb −1 corresponding to 772 × 10 6 BB pairs, collected with the Belle detector at the KEKB asymmetric-energy e + e − (3.5 on 8 GeV) collider [22] operating at the Υ(4S) resonance. The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of FIG. 2: Binning of the optimal binning in Ref. [19,20].
time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons. The detector is described in detail elsewhere [23]. We reconstruct B 0 → DK * 0 events with K * 0 → K + π − and D → K 0 S π + π − . The event selection described below is developed from studies of continuum data taken at center-of-mass energies just below the Υ(4S) resonance and Monte Carlo (MC) simulated events.
The K 0 S candidates are identified using the output of a neural network. Inputs to the network for a pair of oppositelycharged pions are the invariant mass, 20 kinematic parameters and particle identification (PID) information from the ACC, TOF and the ionization energy loss in the CDC. The K 0 S selection has a simulated purity of 92.2% and an efficiency of 75.1%. Charged kaon and pion candidates are identified using PID information. The efficiency is 80-90% and the probability of misidentification is 6-10%, depending upon the momentum of hadrons and obtained using dedicated data control samples. We reconstruct neutral D mesons by combining a K 0 S candidate with a pair of oppositely-charged pion candidates. We require that the invariant mass be within ±15 MeV/c 2 (±3σ) of the nominal D 0 mass. K * 0 candidates are reconstructed from K + π − pairs. We require that the invariant mass be within ±50 MeV/c 2 of the nominal K * 0 mass. We combine D and K * 0 candidates to form B 0 mesons. Candidate events are identified by the energy difference ∆E where E b is the beam energy and p i and E i are the momenta and energies, respectively, of the B 0 meson decay products in the e + e − center-of-mass (CM) frame. We select events with 5.21 GeV/c 2 < M bc < 5.29 GeV/c 2 and −0.10 GeV < ∆E < 0.15 GeV.
Among other B decays, the most serious background is fromB 0 decaying to the same final state as B 0 → DK * 0 . To suppress this background, we exclude candidates for which the invariant mass of the K * 0 π + system is within ±4 MeV/c 2 of the nominal D + mass. This criterion leads to a negligible contamination from this mode and a relative loss of 0.6% in the signal efficiency.
The large combinatorial background of true D 0 and random K + and π − combinations from the e + e − → cc process and other BB decays is reduced if D 0 candidates that are a decay product of D * + → D 0 π + are eliminated. We use the mass difference ∆M between the [K 0 S π + π − ] D π + and [K 0 S π + π − ] D systems for this purpose: if ∆M > 0.15 GeV/c 2 for any additional π + candidate not used in the B candidate reconstruction, the event is retained. This requirement 1 Fisher discriminants based on modified Fox-Wolfram moments [24]. 2 The angle in the CM frame between the thrust axes of the B decay and that of remaining particles. 3 The signed difference of the vertices between the B candidate and the remaining charged tracks. 4 The distance of closest approach between the trajectories of the K * and D candidates. 5 The expected flavor dilution factor described in Ref. [25]. 6 The angle θ between the B meson momentum direction and the beam axis in the CM frame. 7 The angle between the D and Υ(4S) directions in the rest frame of the B candidate. 8 The projection of the sphericity vector with the largest eigenvalue onto the e + e − beam direction.

9
The angle of the sphericity vector with the largest eigenvalue with respect to that of the remaining particles. 10 The angle of the sphericity vector with the second largest eigenvalue. 11 The angle of the sphericity vector with the smallest eigenvalue. 12 The magnitude of the thrust of the particles not used to reconstruct the signal. removes 19% of cc background and 11% of BB background according to MC simulation. The relative loss in signal efficiency is 5.5%.
In the rare case where there are multiple candidates in an event, the candidate with M bc closest to the nominal value is chosen. The relative loss in signal efficiency is 0.8%.
To discriminate signal events from the large combinatorial background dominated by the two-jet-like e + e − → qq continuum process, where q indicates u, d, s or c, a multivariate analysis is performed using the 12 variables introduced in Table I. To effectively combine these 12 variables, we employ the NeuroBayes neural network package [26]. The NeuroBayes output is denoted as C NB and lies within the range [−1, 1]; events with C NB ∼ 1 are signal-like and events with C NB ∼ −1 are qq-like. Training of the neural network is performed using signal and qq MC samples. The C NB distribution of signal events peaks at C NB ∼ 1 and is therefore difficult to represent with a simple analytic function. However, the transformed variable where C NB,low = −0.6 and C NB,high = 0.9992, has a distribution that can be modeled by a Gaussian for signal as well as background. The events with C NB < C NB,low are rejected; the relative loss in signal efficiency is 7.4%.

ANALYSIS PROCEDURE
In this section we describe the fit to determine the physics parameters. In Sec. we describe the signal and background shape parametrization. In Sec. we describe how we correct for the effect of migration and acceptance variations between bins. In Sec. the fit to extract the values of (x, y) is described.

Signal and background parametrization
The number of signal events is obtained by fitting the three-dimensional distribution of variables M bc , ∆E, and C NB using the extended maximum likelihood method. We form three-dimensional PDFs for each component as the product of one-dimensional PDFs for ∆E, M bc and C NB , since the correlations among the variables are found to be small. The fit region is defined as ∆E ∈ [−0.1, 0.15] GeV and M bc > 5.21 GeV/c 2 .
Backgrounds are divided into the following components: • Continuum background from qq events.
• BB background, in which the tracks forming the B 0 → DK * 0 candidate come from decays of both B mesons in the event. The number of possible B decay combinations that contribute to this background is large; therefore, both the Dalitz distribution and distribution of the fit parameters are quite smooth. BB backgrounds are further subdivided into two components: events reconstructed with a true D → K 0 S π + π − decay, referred to as D true BB background, and those reconstructed with a combinatorial D candidate, referred to as D fake BB background.
• Peaking BB background, in which all tracks forming the B 0 → DK * 0 candidate arise from the same B meson.
This background has two types: events with one pion misidentified as a kaon, such as D 0 [π + π − ] ρ 0 , and one pion misidentified as a kaon and one pion not reconstructed, such as D 0 [π + π + π − ] a + 1 . The backgrounds come from individual B decays and are well separated from the signal.
The ∆E PDFs are parameterized by a double Gaussian for the signal, an exponential function for the D true BB background, an exponential function for the D fake BB background, a linear function for the qq background, a double Gaussian for theD 0 ρ 0 background, and a Gaussian for theD 0 a + 1 background. The M bc PDFs are a Gaussian for signal, a Crystal Ball function [27] for D true BB background, an ARGUS function [28] for D fake BB background, an ARGUS function for qq background, a sum of a Gaussian and ARGUS functions forD 0 ρ 0 background and a Gaussian forD 0 a + 1 background. For each component, the C NB PDF is the sum of a Gaussian and bifurcated Gaussian. The shape parameters of the PDFs are fixed from MC samples.
The numbers of events in each bin are free parameters in the fit. This procedure has been justified for background that is either well separated from the signal (such as peaking BB background) or is constrained by a much larger number of events than the signal (such as qq background). The results of the fit to the full Dalitz plot are shown in Fig. 3

Corrections to the bin-by-bin yields
There are further effects that must be accounted for before the values of (x, y) can be determined from a binned fit. Equation 7 only holds if there is no migration between bins and the Dalitz acceptance is uniform. Here, we consider the crossfeed for bin-by-bin yields as the migration and the acceptance as the event reconstruction efficiency.
First, we discuss migration which is due to momentum resolution and flavor misidentification. Momentum resolution leads to migration of events among the bins. In the binned approach, this effect can be corrected in a non-parametric way. The migration can be described by a linear transformation of the number of events in each bin where N i is the the number of events that bin i would contain without the migration with acceptance and N obs,i is the reconstructed number of events in bin i. The migration matrix α ik is nearly the unit matrix; it is obtained from a signal MC simulation generated with the amplitude model reported in Ref. [5]. Most of the off-diagonal elements are null; only a few have values |α ik | ≤ 0.04. In the case of a D → K 0 S π + π − decay from a B, the migration depends on the parameters x and y. However, this is a minor correction to an already small effect and so is neglected.
The second migration effect to be considered is due to misidentification of the B flavor. Double misidentification in K * 0 reconstruction from K + π − where K − is misidentified as π − and π + is misidentified as K + at the same time leads to migration of events between N + i ↔ N − −i due to assignment of the wrong flavor to the B candidate. If the fraction of doubly-misidentified events is β, the number of events in each bin can be written as The value of β is obtained from MC simulation and is found to be (0.12 ± 0.01)%. Therefore, the effect of flavor misidentification is neglected. The final state radiation also causes migration between bins. The measured values of c i and s i by CLEO are not corrected for the radiation and the effect upon our analysis is found to be negligible [14].
Second, we consider the effect of the variation of the efficiency profile over the Dalitz plane. We note that Eq. (4) does not change under the transformation P → P when the efficiency profile (m 2 + , m 2 − ) is symmetric: (m 2 + , m 2 − ) = (m 2 − , m 2 + ). The effect of non-uniform efficiency over the Dalitz plane cancels when using a flavor-tagged D sample with kinematic properties that are similar to the sample from the signal B decay. This approach allows for the removal of the systematic uncertainty associated with the possible inaccuracy of the detector acceptance description in the MC simulation. With the efficiency taken into account (that is, in general non-uniform across the bin region), the number of events reconstructed is Here, p is the probability density on the Dalitz plane and D is the position on Dalitz plane. Clearly, the efficiency does not factorize. One can use an efficiency averaged over the bin, then correct for it in the analysis: Here, N i are the number of events corrected for variations in acceptance and migration, which should be used for (x ± , y ± ) extraction. The averaged efficiency¯ i can be determined from MC. The assumption that the efficiency profile depends only on the D momentum is tested using MC simulation and the residual difference is treated as a systematic uncertainty. The correction for c i and s i due to efficiency variation within a bin cannot be calculated in a completely model-independent way, since the correction terms include the amplitude variation inside the bin. Calculations using the Belle D → K 0 S π + π − model [5] show that this correction is negligible even for very large non-uniformity of the efficiency profile.

Fit to determine (x, y)
If N i in each bin is measured, x ± and y ± can be obtained according to Eq. (7) by minimizing where N i are the expected number of signal events in the bin i obtained from Eq. (7). Here, N i and σ Ni are the observed number of events in data and the uncertainty on N i , respectively. The procedure described above does not make any assumptions about the Dalitz distribution of the background events, since the fits in each bin are independent. Thus, there is no uncertainty related to the Dalitz model. However, in our case, where there are a small number of events and many background components, such independent fits are not feasible. Therefore, we obtain (x ± , y ± ) from a combined fit with a common likelihood for all bins. The relative numbers of background events in each bin are constrained to the numbers found in the MC. The amount of the D true BB background in bins from the ratio of D 0 (K i ) andD 0 (K −i ) from MC and the amount of the D fake , BB, qq and the background from individual B decays from the MC. The yields integrated over the Dalitz plot of the background components are additional free parameters. Thus, the variables (x ± , y ± ) become free parameters of the combined likelihood fit and the assumption that the signal yield obeys a Gaussian distribution is not needed. While the normalization parameter h B is also a free parameter of the fit, we do not mention it in the following as it is not a quantity of interest.  Table II. In this study, these (x, y) values from the likelihood distribution of the combined fit are corrected using the frequentist approach with Feldman-Cousins ordering [29], which is described in Sec. .

SYSTEMATIC UNCERTAINTIES
The systematic uncertainties of (x, y) are obtained by taking deviation from the default procedure under various assumptions. The systematic uncertainties are summarized in Table III; most are negligible compared to the statistical uncertainty. There is an uncertainty due to the Dalitz efficiency variation because of the difference in average efficiency over each bin for the flavor-tagged D and B 0 → DK * 0 samples. A maximum difference of 1.5% is obtained in a MC study. The uncertainty is taken as the maximum of two quantities: • the root mean square of x and y from smearing the numbers of events in the flavor-tagged sample K i by 1.5%,

or
• the bias in x and y between the fits with and without efficiency correction for K i obtained from signal MC.
The uncertainty due to migration of events between bins is estimated by taking the bias between the fits with and without the migration correction. The uncertainties due to the fixed parameterization of the signal and background PDFs are estimated by varying them by ±1σ. The uncertainty due to the C NB PDF distributions for BB is estimated by replacing them with the signal C NB PDF. The uncertainty due to the D true and D fake BB fractions is estimated by varying them between 0 and 1. The uncertainty arising from the finite sample of flavor-tagged D → K 0 S ππ decays is evaluated by varying the values of K i within their statistical uncertainties. The uncertainty due to the limited precision of c i and s i parameters is obtained by smearing the c i and s i values within their total errors and repeating the fits for the same experimental data. The uncertainty due to k in Eq. (2) is evaluated by varying the value of k(= 0.95 ± 0.03) within its error [16]. Total systematic uncertainties in the (x, y) are obtained by summing all uncertainties in quadrature and listed in Table III. RESULT We use the frequentist approach with Feldman-Cousins ordering [29] to obtain the physical parameters µ = (φ 3 , r S , δ S ) (or true parameters µ = z true = (x − , y − , x + , y + )) from the measured parameters z = z meas = (x − , y − , x + , y + ) taken from the likelihood distribution. In essence, the confidence level α for a set of physical parameters µ is calculated as where p(z | µ) is the probability density to obtain the measurement z given by the set of true parameters µ. The integration domain D(µ) is given by the likelihood ratio (Feldman-Cousins) ordering: p(z | µ) p(z | µ best (z)) > p(z 0 | µ) p(z 0 | µ best (z 0 )) , where µ best (z) is µ that maximizes p(z | µ) for the given z, and z 0 is the result of the data fit. This PDF is taken from MC pseudo-experiments. Systematic uncertainties in µ are obtained by varying the measured parameters z within their systematic uncertainties assuming a nominal distribution. In this calculation, we ignore the correlations of uncertainties between the B 0 andB 0 as the two samples are independent.
As a result of this procedure, we obtain the confidence levels (C.L.) for (x, y) and the physical parameter r S . The C.L. contours on (x, y) are shown in Fig. 6. The 1 − C.L. as a function of r S is shown in Fig. 7. The final results are: x − = +0.4 +1.0+0.0 −0.6−0.1 ± 0.0, y − = −0.6 +0.8+0.1 −1.0−0.0 ± 0.1, x + = +0.1 +0.7+0.0 −0.4−0.1 ± 0.1, r S < 0.87 at 68% C.L., where the first error is statistical, the second is systematic without uncertainties in (c i , s i ), and the third is from the (c i , s i ) precision from CLEO. CONCLUSION We report the first measurement of the amplitude ratio r S using a model-independent Dalitz analysis of D → K S π + π − decays in the process B 0 → DK * 0 with the full data sample of 711 fb −1 corresponding to 772 × 10 6 BB pairs collected by the Belle detector at the Υ(4S) resonance. Model independence is achieved by binning the Dalitz plot of the D → K 0 S π + π − decay and using the strong-phase coefficients with binning as in the CLEO experiment [20]. We obtain the value r S < 0.87 at 68% C.L. This measurement results in lower statistical precision than the modeldependent measurement from BaBar with the B 0 → DK 0 mode [8] despite the larger data sample due to the smaller B 0 → DK * 0 signal observed. The result is consistent with the most precise r S measurement reported by the LHCb