Search for B+ ->D+ K0 and B+->D+ K*0 decays

We report a search for the rare decays $B^+ \to D^+ K^0$ and $B^+ \to D^+ K^{*0}$ in an event sample of approximately 465 million $B\bar B$ pairs collected with the BaBar detector at the PEP-II asymmetric-energy $e^+e^-$ collider at SLAC. We find no significant evidence for either mode and we set 90\% probability upper limits on the branching fractions of $B(B^+ \to D^+ K^0)<2.9\times 10^{-6}$ and $B(B^+ \to D^+ K^{*0})<3.0\times 10^{-6}$.


INTRODUCTION
Charged B meson decays in which neither constituent quark appears in the final state, such as B + → D + K ( * )0 , are expected to be dominated by weak annihilation diagrams with the bu pair annihilating into a W + boson. Such processes therefore can provide insight into the internal dynamics of B mesons, in particular the overlap between the b and the u quark wave functions. Annihilation amplitudes cannot be evaluated with the commonlyused factorization approach [1]. As a consequence, there are no reliable estimates for the corresponding decay rates. Annihilation amplitudes are expected to be proportional to f B /m B where m B is the mass of the B meson and f B is the pseudoscalar B meson decay constant. The quantity f B represents the probability amplitude for the two quark wave functions to overlap. Numerically, f B /m B is approximately equal to λ 2 , where λ is the sine of the Cabibbo angle [1,2]. In addition, these amplitudes are also suppressed by the CKM factor |V ub | ∼ λ 3 . So far, there has been no observation of a hadronic B meson decay that proceeds purely through weak annihilation diagrams, although evidence for the leptonic decay B → τ ν has been found [3]. In theoretical calculations of nonleptonic decays, the assumption is often made that these amplitudes may be neglected.
Some studies indicate that the branching fractions of weak-annihilation processes could be enhanced by socalled rescattering effects, in which long-range strong interactions between B decay products, rather than the decay amplitudes, lead to the final state of interest [2]. Figure 1 shows the Feynman diagram for the decays B + →D + K ( * )0 and B + →D + s π 0 [4], and the hadron-level diagram for the rescattering of D + s π 0 into D + K ( * )0 . Significant rescattering could thus mimic a large weak annihilation amplitude. It has been argued [2] that rescattering effects might be suppressed by only λ 4 , compared to λ 5 for the weak annihilation amplitudes, rendering the B + → D + K ( * )0 decay rate due to rescattering comparable to the isospin-related color-suppressed B 0 → D 0 K ( * )0 decay rate of approximately 5 × 10 −6 . B + → D + K ( * )0 decays are also of interest because their decay rates can be used to constrain the annihilation amplitudes in phenomenological fits [1,5]. This allows the translation of the measurements of the B + →D 0 K ( * )+ amplitudes into estimations of the |V ub | suppressed amplitudes B 0 →D 0 K ( * )0 [5,6]. None of the modes studied here has been observed so far, and a 90% confidence level upper limit on the branching fraction B(B + →D + K 0 ) < 5 × 10 −6 has been established by BABAR [7]. No study of B + → D + K * 0 has previously been published.
The results presented here are obtained with 426 fb −1 of data collected at the Υ (4S) resonance with the BABAR detector at the PEP-II asymmetric e + e − collider [9] corresponding to 465 × 10 6 BB pairs (N BB ). An additional 44.4 fb −1 of data ("off-resonance") collected at a center-of-mass (CM) energy 40 MeV below the Υ (4S) resonance is used to study backgrounds from e + e − → qq (q = u, d, s, or c) processes, which we refer to as continuum events.
The BABAR detector is described in detail elsewhere [10]. Charged-particle tracking is provided by a five layer silicon vertex tracker (SVT) and a 40 layer drift chamber (DCH). In addition to providing precise position information for tracking, the SVT and DCH measure the specific ionization, which is used for particle identification of low-momentum charged particles. At higher momenta (p > 0.7 GeV/c) pions and kaons are identified by Cherenkov radiation detected in a ring-imaging device (DIRC). The position and energy of photons are measured with an electromagnetic calorimeter (EMC) consisting of 6580 thallium-doped CsI crystals. These systems are mounted inside a 1.5 T solenoidal superconducting magnet. Muons are identified by the instrumented magnetic-flux return, which is located outside the magnet.

EVENT RECONSTRUCTION AND SELECTION
The event selection criteria are determined using Monte Carlo (MC) simulations of e + e − → Υ (4S) → BB ("BB" in the following) and continuum events, and the off-resonance data. The selection criteria are optimized by maximizing the quantity S/ √ S + B, where S and B are the expected numbers of signal and background events, respectively. We assume the signal branching fraction to be 5 × 10 −6 in the optimization procedure.
The charged particle candidates are required to have transverse momenta above 100 MeV/c and at least twelve hits in the DCH.
Candidate D + mesons are reconstructed in the D + → K − π + π + (Kππ in the following), D + → K 0 S π + (K 0 S π), D + → K − π + π + π 0 (Kπππ 0 ) and D + → K 0 S π + π 0 (K 0 S ππ 0 ) modes for the B + → D + K 0 decay channel (DK). Only the first two modes are used for the B + → D + K * 0 decay channel (DK * in the following) since we find that including the Kπππ 0 and K 0 S ππ 0 modes in this channel does not appreciably improve the sensitivity of the analysis.
The D + candidates are reconstructed by combining kaons (either charged or neutral depending on the channel) and the appropriate number of pions. The charged kaons used to reconstruct the D + and K * 0 candidates are required to satisfy kaon identification criteria obtained using a likelihood technique based on the opening angle of the Cherenkov light measured in the DIRC and the ionization energy loss measured in the SVT and DCH. These criteria are typically 85% efficient, depending on the momentum and polar angle, with misidentification rates at the 2% level. Kaons and pions from D decays are required to have momenta in the laboratory frame greater than 200 MeV/c and 150 MeV/c, respectively.
The reconstructed D + candidates are required to satisfy the invariant mass (M D ) selection criteria given in Table I.
The K 0 S candidates are reconstructed from pairs of oppositely-charged pions with invariant mass within 5-7 MeV/c 2 of the nominal K 0 S mass [11]. This mass cut corresponds to 2-2.8 standard deviations of the experimental resolution and varies slightly among channels due to the different amounts of background per channel. For the prompt K 0 S candidates from the B + → D + K 0 S decay, we require 1 − cos is the angle between the momentum vector of the K 0 S candidate and the vector connecting the B + and K 0 S decay vertices. For K 0 S daughters of a D + decay, we require The π 0 candidates are reconstructed from pairs of photon candidates each with an energy greater than 70 MeV, and a lateral shower profile in the EMC consistent with a single electromagnetic deposit. These pairs must have a total energy greater than 200 MeV, a CM momentum greater than 400 MeV/c, and an invariant mass within 10 MeV/c 2 (for the Kπππ 0 mode) or 12 MeV/c 2 (for the K 0 S ππ 0 mode) of the nominal π 0 mass [11]. The K * 0 candidates are reconstructed in the decay channel K * 0 → K + π − . These charged tracks are constrained to originate from a common vertex. The reconstructed invariant mass, whose width is dominated by the K * 0 natural width, is required to lie within 40 MeV/c 2 of the nominal K * 0 mass [11]. We define θ H as the angle between the direction of flight of the charged K and the direction of flight of the B in the K * 0 rest frame. The probability distribution of cos θ H is proportional to cos 2 θ H for longitudinally polarized K * 0 mesons from B → DK * 0 decays, due to angular momentum conservation, and is approximately flat for fake (random combinations of tracks) or unpolarized background K * 0 candidates. To suppress fake and background K * 0 candidates we require | cos θ H | > 0.5.
The B + candidates are reconstructed by combining one D + and one K 0 S or K * 0 candidate, constraining them to originate from a common vertex. The probability distribution of the cosine of the B polar angle with respect to the beam axis in the CM frame, cos θ B , is expected to be proportional to 1 − cos 2 θ B . Selection criteria on | cos θ B | are channel dependent and are summarized in Table I.
We measure two almost independent kinematic variables: the beam-energy substituted mass m ES ≡ where E and p are energy and momentum, the subscripts B and 0 refer to the candidate B and to the e + e − system, respectively, and the asterisk denotes a calculation made in the CM frame. Signal events are expected to peak at the B meson mass for m ES and at zero for ∆E. Channel-dependent selection crite-ria on |∆E| are given in Table I. We retain candidates with m ES in the range [5.20, 5.29] GeV/c 2 for subsequent analysis.
In less than 1% of the cases, multiple B + candidates are present in the same event, and in those cases we choose the one with the reconstructed D + mass closest to the nominal mass value [11]. If more than one B + candidate shares the same D + candidate, then we choose the B + candidate with ∆E closest to zero.

BACKGROUND CHARACTERIZATION
After applying the selection criteria described above, the remaining background is composed of non-signal BB events and continuum events, the latter being the dominant contribution. Continuum background events, in contrast to BB events, are characterized by a jet-like shape, which can be used in a Fisher discriminant F [12] to reduce this background component. The discriminant F is a linear combination of four variables trained to peak at 1 for signal and at −1 for continuum background. The first variable is the cosine of the angle between the B thrust axis and the thrust axis of all the other reconstructed charged tracks and neutral energy deposits (rest of the event), where the thrust axis is defined as the direction that maximizes the sum of the longitudinal momenta of all the particles. The second and third variables are the event shape moments L 0 = i p i , and L 2 = i p i | cos θ i | 2 , where the index i runs over all tracks and energy deposits in the rest of the event; p i is the momentum and θ i is the angle with respect to the thrust axis of the B candidate. These three variables are calculated in the CM. Finally we use |∆t|, the absolute value of the measured proper time interval between the two B decays [8]. It is calculated using the measured separation along the beam direction ∆z between the decay points of the reconstructed B and the other B and the Lorentz boost between the laboratory and CM frames. The other B decay point is obtained from the tracks that do not belong to the reconstructed B, with constraints from the reconstructed B momentum and the beam-spot location. The coefficients of F , chosen to maximize the separation between signal and continuum background, are determined with samples of simulated signal and continuum events, and validated using off-resonance data. We denote two regions: the fit region, defined as 5.20 < m ES < 5.29 GeV/c 2 and −5 < F < 5, and the signal region, defined as 5.27 < m ES < 5.29 GeV/c 2 and 0 < F < 5.
To reduce the importance of the continuum background in the final sample we divide the events according to their flavor-tagging category [8]. We define the following exclusive tagging categories: • lepton category, events contain at least one lepton I: Main selection criteria used to distinguish between signal and background events. MD, PDG is the nominal mass of the D + meson [11].  in the decay of the other B meson; • kaon category, events contain at least one kaon in the decay of the other B meson, which do not belong to the first category; • other category contains all the events not included in the two previous categories.
The first two categories are expected to be less contaminated by continuum background. We fit all three categories simultaneously. Studies of simulated events show that using the tagging categories reduces the statistical uncertainty on the measured branching fraction for the Kππ mode by 5%, but leads to little gain for the other modes (which are less statistically significant themselves). Hence, we use tagging information only for the Kππ channel. The BB background is divided into two components: non-peaking (combinatorial) and peaking. The latter can occur when one or several particles of a background channel are replaced by a low momentum charged π + and the resulting candidate still contributes to the signal region. The largest contributions to the BB peaking background for the B + → D + K 0 channel arise from the following decays: B 0 → D + ρ − with D + decaying into signal channels, B 0 → D 0 K 0 and B 0 → D * 0 K 0 . To further reduce the contribution from the B 0 → D + ρ − background, the variable | cos θ K 0 S | has been introduced, where θ K 0 S is the K 0 S helicity angle, i.e., the angle between one of the two pions from the K 0 S and the D + in the K 0 S rest frame. We reject events with | cos θ K 0 S | greater than 0.8 for the Kππ mode and 0.9 for all other modes. Based on MC studies, we expect no more than 1 BB peaking background event per mode in the signal region, after applying all selection criteria (see Table II). A similar study is performed for the B + → D + K * 0 decay modes. The main peaking backgrounds arise from B 0 → D + ρ − , B 0 → D + K * − , and B 0 → D + a − 1 . In all cases, the D + decays into the signal decay modes. The number of BB peaking background events expected in the signal region for the DK * mode are shown in Table II. Charmless B decays may also contribute to the peaking background. These decays can produce π and K mesons with characteristics similar to those of signal events without forming a real D meson. The charmless background is evaluated from data using the D + sidebands: events are required to satisfy the criteria 1.774 < M D < 1.840 GeV/c 2 or 1.900 < M D < 1.954 GeV/c 2 . We obtain −1.7 ± 1.0 events for DK decays and −0.7 ± 2.1 events for DK * decays. We estimate the charmless peaking background contribution to be negligible and assign a systematic uncertainty based on this assumption.
The overall reconstruction and selection efficiencies for signal events, as well as the numbers of expected events for each background category, are given in Table II.

FIT PROCEDURE
The signal and background yields are extracted by maximizing the unbinned extended likelihood Here x j = {m ES ; F }, θ is a set of parameters, N is the number of events in the selected sample, N ′ is the expectation value for the total number of events, and with f sig (x|θ) and f Bi (x|θ) the probability density functions (PDFs) for the hypothesis that the event is a signal or a background event, respectively. The B i are the different background categories used in the fit: continuum background, combinatorial BB background, and peaking BB background. N sig is the number of signal events, and N Bi is the number of events for each background species B i . The individual probability density functions are defined by the product of the one-dimensional distributions of m ES and F . Absence of the correlations between these distributions is checked using the MC samples. The signal m ES distribution is modeled with a Gaussian function. The continuum and non-peaking BB background m ES distributions are modeled with two different threshold ARGUS functions defined [13] as follows: where x 0 represents the maximum allowed value for the variable x and c accounts for the shape of the distribution. The m ES distribution of the peaking BB background is modeled with a Crystal Ball (CB) function [14]. The CB function is a Gaussian modified to include a power-law tail on the low side of the peak. The F distributions are modeled as the sum of two asymmetric Gaussians for signal and continuum background events, and with a Gaussian for the combinatorial BB background. For the peaking BB background we use a Gaussian distribution for the DK mode. For the DK * mode, an asymmetric Gaussian is used for the Kππ mode and a sum of two asymmetric Gaussians for the K 0 S π mode. The shape parameters of the threshold function for continuum background are determined from data. All other PDF parameters are derived from the simulated events.
In the fits we fix the numbers of peaking BB background events, which are estimated from the PDG branching fractions [11] and MC efficiency evaluations.
The number of signal events determined by the fit (N sig ) is used to calculate the branching fraction as where N B + is the total number of charged B mesons in the data sample (equal to the total number of all BB pairs produced, since we assume equal production of B + B − and B 0 B 0 ), B D and B K 0 S are the branching fraction for each D meson decay channel and for K 0 S → π + π − respectively [11], and ǫ sig is the reconstruction efficiency for each D decay channel evaluated from MC events. The expression for B(B + → D + K * 0 ) is obtained replacing B K 0 S /2 with the branching fraction of K * 0 → K + π − , B K * 0 . The likelihoods for individual channels are combined to derive average branching fractions for B + → D + K 0 and B + → D + K * 0 .
The fit procedure is validated using an ensemble of simulated experiments (toy MC studies) with all yields generated according to Poisson distributions. The nonfloating parameters of the fits as well as the shapes of the background threshold functions are fixed to the values obtained from the MC samples. We define the pull for a variable x as the difference between the fitted x fit and the mean generated value x gen , divided by the error σ err , x pull = (x fit − x gen )/σ err . We use the negative errors for fitted values that are smaller than the generated ones and the positive errors in the opposite case. The procedure gives Gaussian-like pull distributions for each channel and thus no biases of the fit model were found. In Table III we show resulting expectations of asymmetric errors for each channel. The 95% probability ranges for these errors obtained from toy MC studies are also shown. Tests of the fit procedure performed on the full MC samples give values for the yields compatible with the generated ones.
The main results of the fit to the data are reported in Table IV, which gives the values of the fitted parameters for each D channel and for the combination of fits. The background yields are close to the expectations and the errors obtained on the branching fractions are in good agreement with the values reported in Table III. The leading contribution (as expected) is obtained from the Kππ mode. Likelihood fit projections of the m ES and F distributions are shown in Fig. 2. In Fig. 3 we also show for illustrative purposes the fit projection for m ES , after requiring F > 0, to visually enhance any possible signal.

SYSTEMATIC ERRORS
We consider various sources of systematic error. One of the largest contributions comes from the uncertainties on the PDF parameterizations. To evaluate the contributions related to the m ES and F PDFs, we repeat the fit varying the MC-obtained PDF parameters within their statistical errors, taking into account correlations among the parameters (labeled as "PDF -MC" in the final list of systematic error sources). Differences between the data and MC (labeled as "Data -MC PDF shapes" in the final list of systematic error sources) for the shapes of m ES and F distributions are studied for signal components using data control samples. B 0 → D + π − and B 0 → D + ρ − selected events are used to obtain the m ES and F parameters for the DK and DK * modes, respectively. The analysis strategy is the same as for the signal events except for specific criteria to select K 0 S or K * 0 . For the continuum background, we estimate this uncertainty by repeating the fit using the PDF parameters obtained from off-resonance data instead of those from continuum MC. Finally, for the BB background, we estimate this uncertainty by leaving the parameters that describe the BB combinatorial background as free variables in the fit (separately for m ES and F ). The systematic uncertainty is defined as the difference in the branching fraction results from the nominal and alternative fits summed in quadrature.
The systematic errors on the signal reconstruction efficiency include the uncertainty due to limited MC statistics, uncertainties on possible differences between data and MC in tracking efficiency, K 0 S and π 0 reconstruction, and charged-kaon identification. In addition, there are additional contributions to these uncertainties originating from the disagreement between data and MC distributions for all the variables used in the selection. These are estimated by comparing the data and simulation performance in control samples. To evaluate the uncertainties arising from peaking background contributions, we repeat the fit by varying the numbers of these events within their statistical errors. The uncertainties on the branching fractions of the sub-decay modes are also taken into account. The uncertainty on N BB (1.1%) has a negligible effect on the total error.
The systematic uncertainties on the branching fractions are summarized in Table V. All the uncertainties are considered to be uncorrelated and are treated sepa-  Kππ and (f) K 0 S π for the B + → D + K * 0 mode. The data are indicated with black dots and error bars and the different fit components are shown: signal (black solid curve), combinatorial BB (green dotted), continuum (magenta dot-dashed) and BB peaking background (red dotted) and the blue solid curve is the projection of the fit. We require F > 0 to visually enhance the signal component. Such a cut has an approximate efficiency of 70% for signal, while it rejects more than 80% of the continuum background. rately for each channel.

RESULTS FOR BRANCHING FRACTIONS
The final likelihood for each decay mode is obtained by convolving the likelihoods for the measured branching fractions with Gaussian functions of width equal to the systematic uncertainty.
Since the measurements for the branching fractions are not statistically significant, following a Bayesian approach and assuming a flat prior distribution for the branching fractions, we integrate over the positive portion of the likelihood function to obtain the following upper limits at 90% probability: B(B + → D + K 0 ) < 2.9 × 10 −6 , B(B + → D + K * 0 ) < 3.0 × 10 −6 .
The B + → D + K 0 result represents an improvement over, and supersedes, our previous result [7], while the B + → D + K * 0 result is the first for this channel.

CONCLUSIONS
In summary, we have presented a search for the rare decays B + → D + K 0 and B + → D + K * 0 , which are predicted to proceed through annihilation or rescattering amplitudes. We do not observe any significant signal and we set 90% probability upper limits on their branching fractions. V: Systematic errors on branching fractions for B + → D + K 0 and B + → D + K * 0 decay channels. All quantities are given in units of 10 −6 .