Measurement of Branching Fraction and Dalitz Distribution for B0->D(*)+/- K0 pi-/+ Decays

We present measurements of the branching fractions for the three-body decays B0 ->D(*)-/+ K0 pi^+/-$ and their resonant submodes $B0 ->D(*)-/+ K*+/- using a sample of approximately 88 million BBbar pairs collected by the BABAR detector at the PEP-II asymmetric energy storage ring. We measure: B(B0->D-/+ K0 pi+/-)=(4.9 +/- 0.7(stat) +/- 0.5 (syst)) 10^{-4} B(B0->D*-/+ K0 pi+/-)=(3.0 +/- 0.7(stat) +/- 0.3 (syst)) 10^{-4} B(B0->D-/+ K*+/-)=(4.6 +/- 0.6(stat) +/- 0.5 (syst)) 10^{-4} B(B0->D*-/+ K*+/-)=(3.2 +/- 0.6(stat) +/- 0.3 (syst)) 10^{-4} From these measurements we determine the fractions of resonant events to be : f(B0->D-/+ K*+/-) = 0.63 +/- 0.08(stat) +/- 0.04(syst) f(B0->D*-/+ K*+/-) = 0.72 +/- 0.14(stat) +/- 0.05(syst)

From these measurements we determine the fractions of resonant events to be f (B 0 → D ∓ K * ± ) = 0.63 ± 0.08 stat ± 0.04 syst and f (B 0 → D * ∓ K * ± ) = 0.72 ± 0.14 stat ± 0.05 syst . PACS numbers: 13.25.Hw,14.40.Nd Several independent measurements are needed to test the Standard Model description of CP violation. The angle γ can be determined using decays of the type B → D ( * ) K ( * ) [1]. The experimental challenges are color suppression of the b → u transition, reconstruction of D 0 CP eigenstates, and interfering doubly-Cabibbosuppressed decays (DCSD) [2]. Also, two-body mode analyses are complicated because there are eight degenerate solutions for γ in the interval [0, 2π].
In recent papers [3,4] three-body decays have been suggested for measuring γ, since these do not suffer from the color suppression penalty. Furthermore, the channels B 0 → D ( * )∓ K 0 π ± do not have the above problems with CP states and DCSD interference, and can resolve most of the ambiguities [3]. The angle γ can be extracted from a time-dependent Dalitz analysis of these decay modes.
The analysis presented here is based on 81.8 fb −1 of data taken at the Υ (4S) resonance, corresponding to approximately 88 million BB pairs, with the BABAR detector [5] at the PEP-II storage ring. We measure the branching fractions of the B 0 → D ( * )∓ K 0 π ± decays and consider their distribution in the Dalitz plot.
We reconstruct D + mesons in the decay mode K − π + π + and D * + mesons in the mode D 0 π + , with the D 0 decaying to K − π + , K − π + π 0 , and K − π + π − π + . Here and throughout the paper charge conjugate states are implied. Tracks from the D decay are required to originate from a common vertex. Positive kaon identification is enforced on kaons from D meson decays, except for the D 0 → K − π + mode.
The D + candidates are required to have a mass within 12 MeV/c 2 (2σ) of the D + mass, while the mass of D 0 candidates decaying to charged daughters only is required to lie within 15 MeV/c 2 (2.5σ) of the D 0 mass, where σ is the experimental resolution. The D 0 → K − π + π 0 candidates are required to have a mass within 30 MeV/c 2 (2.5σ) of the D 0 mass and to be located at a point in the D 0 Dalitz plot where the density of events is larger than 1.4% of the maximum density.
The D * + candidates are accepted if the mass differ-ence m D * + −m D 0 is within 2 MeV/c 2 (3σ) of the nominal value, except for the D 0 → K − π + π 0 candidates where we use 1.5 MeV/c 2 to reduce this mode's larger combinatoric background. We combine oppositely-charged tracks from a common vertex into K 0 S candidates. The K 0 S candidates are required to have a mass within 7 MeV/c 2 (3σ) of the K 0 S mass and a transverse flight length that is significantly (4σ) greater than zero.
To form B 0 candidates, the D ( * )+ candidates are combined with a K 0 S candidate and a π − , for which the particle identification (PID) is inconsistent with being a kaon or an electron. The probability of a common vertex is required to be above 0.1%. Using the beam energy, two almost-independent kinematic variables are constructed: the beam-energy substituted mass , and the difference between the B 0 candidate's measured energy and the beam energy, ∆E ≡ E * B − √ s/2. The asterisk denotes evaluation in the Υ (4S) CM frame. B 0 candidates are required to have ∆E in the range [−0.1, 0.1] GeV, and m ES in the range [5.24, 5.29] ([5.20, 5.288 To suppress the dominant continuum background events, which have a more jet-like shape than BB events, we use a linear combination, F , of four variables: , and the absolute values of the cosine of the polar angles of the B momentum and of the B thrust direction [7]. Here, p i is the momentum and θ i is the angle with respect to the thrust axis of the signal B candidate of the tracks and clusters not used to reconstruct the B. All of these variables are calculated in the CM frame. The coefficients are chosen to maximize the separation between signal Monte Carlo and 9.6 fb −1 of continuum events from data taken 40 MeV below the Υ (4S) resonance (off-resonance data). F has negligible correlations with m ES and ∆E.
After the event selection, approximately 5% of the events have more than one B 0 candidate. We choose the one with m D closest to the expected value and correct for differences between data and simulation. In simulated signal events, the final selection is 19.3% efficient for B 0 → D ∓ K 0 π ± and 15.5%, 3.9% and 8.2% efficient for B 0 → D * ∓ K 0 π ± in the three D 0 decay modes K − π + , K − π + π 0 and K − π + π − π + , respectively.
We perform an unbinned extended maximum likelihood fit with the variables m ES , ∆E, and F on the selected candidates, using the logarithm of the likelihood: (1) where P ij is the product of probability density functions (PDFs) for event i of m ES , ∆E, and F, and N j is the number of events of each sample component j: signal, continuum, combinatoric BB decays, and BB events that peak in m ES but not in ∆E signal region (denoted peaking BB background).
The signal is described by a Gaussian distribution in m ES , two Gaussian distributions with common mean in ∆E, and a Gaussian distribution with different widths on each side of the mean ("bifurcated Gaussian distribution") in F . Their shape is obtained from the highstatistics data control samples B 0 → D ( * )∓ a ± 1 (similar topology of the final state as the signal) for m ES and ∆E, and B 0 → D * ∓ π ± for F , and all nine parameters are fixed in the fit.
The continuum and combinatoric BB backgrounds are described by empirical endpoint functions [8] in m ES , linear functions in ∆E, and bifurcated Gaussian distributions in F . The F distribution of continuum is obtained from off-resonance data, while the F distribution of the BB backgrounds is obtained from Monte Carlo simulation, and compared with data in high-statistics samples to ensure that there is no significant difference. The two F distributions and the common endpoint in m ES are fixed in the fit, while the m ES shape and ∆E distributions are left free to float, leaving four out of eleven parameters free in the fit.
The peaking BB background is parametrized by a Gaussian distribution in m ES , an exponential distribution in ∆E, and shares the PDF in F with the nonpeaking BB background. The mean and width in m ES of the peaking BB background are fixed to values obtained from Monte Carlo simulation, which are consistent with values measured in ∆E sideband of data, thus adding one free and two fixed parameters.
The likelihood function is determined by the 27 parameters described above, of which all four yields and five background shape parameters are fitted. Subsequent to the fit, possible residual backgrounds from combinatoric D and K 0 S candidates are estimated using the sidebands of m D and m K 0 S , and subtracted. The three-body and quasi-two-body (that is B 0 → D ( * )∓ K * ± ) branching fractions are obtained by fitting first without regard to event positions in the Dalitz plot, Events appropriately weighted by Wsig (see text) to exhibit the signal distribution [9] are shown as solid points over which the fitted signal PDF is superimposed. For comparison, the mES distribution obtained with |∆E| < 25 MeV (2σ) is included (dotted points).
and then with the requirement that the K 0 S π + invariant mass lies within 100 MeV/c 2 of the K * + (892) mass. Due to the relatively small number of background events in the second fit, all BB shape parameters are kept fixed to the value obtained in the first fit.
The results are shown in Fig. 1, while yields and puri-ties (defined as N sig /σ 2 (N sig )) are listed in Table I, with the K * + resonant part included in the three-body state.
To determine the three-body branching fractions opti- mally, a mapping of the efficiency across the Dalitz plot is needed. This is obtained from simulated signal events. Incorporating the efficiency variations (∼ ±30%) across the Dalitz plot requires a measure of the (apriori unknown) event distribution in the Dalitz plot. We obtain the number of signal events from the likelihood fit using weights defined as: where N j and P ij are defined as in Eq. (1), and V sig,j is the signal row of the covariance matrix of the component yields obtained from the likelihood fit. These weights W i sig , which in the absence of correlations are signal probabilities P sig /P total , contain the signal distribution and its uncertainty for any quantity uncorrelated with the variables in the likelihood fit [9].
The efficiency-corrected Dalitz distributions, weighted by W sig , are shown in Fig. 2. The K * (892) + resonance is dominant in both the B 0 → D ∓ K 0 π ± and B 0 → D * ∓ K 0 π ± modes, while no other resonant structures are significant. In the B 0 → D ∓ K 0 π ± channel, the spin-1 K * ± meson has the helicity distribution dN/d cos θ ∝ cos 2 θ, where θ is the angle between the K * ± and the K 0 in the K * ± center of mass frame. This can be seen in Fig. 3.
The systematic errors are summarized in Table II. Most systematic errors are due to possible differences between data and Monte Carlo. The tracking efficiency residuals and associated systematic error are obtained from a large sample of τ decays. The efficiency correction as a function of the position in the Dalitz plot obtained from simulated signal events comes with systematic uncertainties due to resolution effects and binning, which are mostly of statistical origin. A ±1σ variation of all fixed variables in the fit, including relevant correlations, is used to obtain the systematic from the uncertainty in the PDFs.
Our final branching fraction results, weighting the three D 0 modes according to their combined statistical  To summarize, a clear signal is seen in both the B 0 → D ∓ K 0 π ± and B 0 → D * ∓ K 0 π ± channels, and in both modes the K * (892) + resonance is dominant.
, we obtain the fractions f (B 0 → D ∓ K * ± ) = 0.63±0.08 stat ± 0.04 syst and f (B 0 → D * ∓ K * ± ) = 0.72 ± 0.14 stat ± 0.05 syst , respectively, where the systematic errors are mainly from correcting for any possible non-resonant contributions. Both the method of this analysis and the resulting threebody branching fraction measurements are the first of their kind, while the resonant decay modes have been measured before [10]. To determine the sensitivity to γ of these modes, a time-dependent Dalitz fit is required, for which the data sample is inadequate. However, the branching fractions and Dalitz distributions suggest that these modes will be useful for measuring γ at the B-

factories.
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), IHEP (China), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), and PPARC (United Kingdom). Individuals have received support from the A. P. Sloan Foundation, Research Corporation, and Alexander von Humboldt Foundation.