Measurement of the Branching Fractions of the Rare Decays B0->Ds(*)+pi-, B0->Ds(*)+rho-, and B0->Ds(*)-K(*)+

We report the measurement of the branching fractions of the rare decays B0->Ds(*)+pi-, B0->Ds(*)+rho-, and B0->Ds(*)-K(*)+ in a sample of 381 million Y(4S) decays into BBbar pairs collected with the BABAR detector at the PEP-II asymmetric-energy e+e- storage ring. We present evidence for the decay B0->Ds-K*+ and the vector-vector decays B0->Ds*+rho- and B0->Ds*-K*+, as well as the first measurement of the vector meson polarization in these decays. We also determine the ratios of the CKM-suppressed to CKM-favored amplitudes r(D(*)pi) and r(D(*)rho) in decays B0->D(*)pi and B0->D(*)rho, and comment on the prospects for measuring the CP observable sin(2beta+gamma).


I. INTRODUCTION
The Cabibbo-Kobayashi-Maskawa (CKM) quark flavor-mixing matrix [1] provides an elegant explanation of the origin of CP violation within the Standard Model. CP violation manifests itself as a non-zero area of the unitarity triangle [2]. While it is sufficient to measure one of the angles to demonstrate the existence of CP violation, the unitarity triangle needs to be over-constrained by experimental measurements in order to demonstrate that the CKM mechanism is the correct explanation of this phenomenon. Precision measurements of the sides and angles of the unitarity triangle are the focus of the physics program at the B Factories. While several theoretically clean measurements of the angle β exist [3], constraining the other two angles α and γ is significantly more challenging. A theoretically clean measurement of sin(2β + γ) can be obtained from the study of the time evolution for B 0 → D ( * )− π + [4] and B 0 → D ( * )− ρ + decays, which are available in large samples at the B factories, and for the corresponding CKM-suppressed modes B 0 →D ( * )+ π − and B 0 →D ( * )+ ρ − [5]. Measurements of CP asymmetries in decays B 0 →D ( * )∓ π ± and B 0 →D ∓ ρ ± decays have recently been published [6,7].
Such long-distance effects could also affect the vector meson polarization in B 0 → D * − s K * + decays. The angular distribution in vector-vector decays B 0 → D * s V (V = ρ, K * ) is given by where θ D * s and θ V are the helicity angles of D * + s and the vector meson V , respectively, f L = |A 0 | 2 /(Σ|A λ | 2 ) is the longitudinal polarization fraction, and A λ=−1,0,+1 are the helicity amplitudes. These distributions are integrated over the angle between the decay planes of D * + s and V . For amplitudes dominated by the short-range (electroweak) currents, f L is predicted to be near unity [13], with corrections of order O(m 2 V /m 2 B ), where m V is the mass of the vector meson produced by the weak current, and m B is the mass of the B meson. Thus, the measurement of f L can constrain the size of the long-distance contributions in B 0 → D * − s K * + decays [12]. The branching fractions B(B 0 → D ( * )+ s π − ) and B(B 0 → D ( * )− s K + ) have been measured previously by the BABAR Collaboration [14]. In this paper we present the first evidence for the decays B 0 → D * + s ρ − and B 0 → D ( * )− s K * + , and a limit on the rate of B 0 → D + s ρ − . We also update the measurements of the branching frac- improved precision, using a 65% larger dataset.

II. DATA SAMPLE AND THE DETECTOR
We use a sample of 381 × 10 6 Υ (4S) decays into BB pairs collected with the BABAR detector at the PEP-II asymmetric-energy e + e − collider [15]. A detailed description of the BABAR detector is available elsewhere [16]. The components of the detector crucial to this analysis are summarized below.
Charged particle tracking is provided by a five-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH). For charged-particle identification, ionization energy loss (dE/dx) in the DCH and SVT, and Cherenkov radiation detected in a ring-imaging device (DIRC) are used. Photons and neutral pions are identified and measured using an electromagnetic calorimeter (EMC), which comprises 6580 thallium-doped CsI crystals. These systems are mounted inside a 1.5-Tesla solenoidal superconducting magnet. We use the GEANT4 [17] software to simulate interactions of particles traversing the BABAR detector, taking into account the varying detector conditions and beam backgrounds.

III. EVENT SELECTION AND ANALYSIS
The selection of events of interest proceeds in two steps. First, we preselect events with at least three reconstructed charged-particle tracks and a total measured energy greater than 4.5 GeV, as determined using all charged particles and neutral particles with energy above 30 MeV. In order to reject e + e − → qq(q = u, d, s, c) continuum background, the ratio of the second to zeroth order Fox-Wolfram moments [18] must be less than 0.5.
Candidates for D + s mesons are reconstructed in the D + s → φπ + , K 0 S K + and K * 0 K + final states, with φ→K + K − , K 0 S →π + π − , and K * 0 →K − π + . The K 0 S candidates are reconstructed from two oppositely-charged tracks, and their momentum is required to make an angle |θ flight | < 11 • with the line connecting their vertex and the e + e − interaction point. All other tracks are required to originate from the e + e − interaction region, loosely defined by |d 0 | < 1.5 cm and |z 0 | < 10 cm, where d 0 and z 0 are the distances of closest approach to the primary e + e − vertex in the directions perpendicular and parallel to the beams, respectively. In order to reject background from D + →K 0 S π + or K * 0 π + , the K + candidate in the reconstruction of D + s →K 0 S K + or K * 0 K + is required to satisfy positive kaon identification criteria, which have an efficiency of 85% and a 5% pion misidentification probability. The same selection is used to identify kaon daughters of the B 0 and K * + mesons in decays The selection is based on the ratios of likelihoods for kaon, pion, and proton identification in the SVT, DCH, and DIRC. The detector likelihoods are calibrated over a wide range of momenta using particles identified kinematically in clean decay chains, such as D * + → D 0 π + , D 0 → K − π + , and Λ → pπ − . In all other cases, kaons are not positively identified, but instead candidates passing a likelihood-based pion selection are rejected. The selection efficiency of this "pion veto" is 95% for the kaons and 20% for the pions. Pion daughters of B 0 and ρ − mesons in the decays B 0 → D ( * )+ s π − and B 0 → D ( * )+ s ρ − are required to be positively identified. Decay products of φ, K * 0 , D + s , and B 0 candidates are constrained to originate from a single vertex.
We reconstruct ρ + → π + π 0 candidates by combining a well-identified charged pion with a π 0 → γγ candidate. The K * + candidates are reconstructed through the decays K * + → K + π 0 and K * + → K 0 S π + . The neutral pion candidates are reconstructed from a pair of photons each with a minimum energy of 30 MeV. The invariant mass of the photon pair is required to be within ±25 MeV/c 2 of the nominal value [20]. The selected candidates are constrained to the nominal π 0 mass before forming the ρ + or K * + candidates. We require that the invariant mass of the two pions forming the ρ − candidate be within ±320 MeV/c 2 of the nominal value [20], and the invariant mass of the K + π 0 and K 0 S π + pairs be within ±75 MeV/c 2 of the nominal K * + mass [20]. K 0 S π + pairs are constrained to a common geometric vertex.
We reconstruct D * + s candidates in the mode D * + s →D + s γ by combining D + s and photon candidates. Photon candidates are required to be consistent with an electromagnetic shower in the EMC, and to have an energy greater than 100 MeV in the laboratory frame. When forming a D * + s , the D + s candidate is required to have an invariant mass within 10 MeV/c 2 of the nominal value [20]. For B 0 → D * + s ρ − and B 0 → D * − s K * + modes, we apply a "π 0 veto" by rejecting photons that in combination with any other photon in the event form an invariant mass that falls within 125 < m γγ < 145 MeV/c 2 .
The efficiency of the initial preselection discussed above varies between 14% (B 0 → D * + s ρ − , D + s → K * 0 K + ) and 48% (B 0 → D + s π − , D + s → φπ + ). After the preselection, we identify signal B decay candidates using a likelihood ratio R L = L sig /(L sig + L bkg ), where L sig = j P sig (x k ) is the multivariate likelihood for the signal hypothesis and L bkg = i P bkg (x k ) is the likelihood for the background hypothesis. Here x k represents one of the discriminating variables described below, which are computed for each event. The likelihoods for the signal and background hypotheses are computed as a product of the probability density functions (PDFs) P sig (x k ) and P bkg (x k ), respectively, for the following selection variables: invariant masses of the φ, K * 0 , ρ + , K * + , and K 0 S candidates; χ 2 confidence level of the vertex fit for the B 0 and D + s mesons; the helicity angles of the φ, K * 0 , ρ + , K * + , and D * + s meson decays; the mass difference ∆m(D * + s ) = m(D * + s ) − m(D + s ); the polar angle θ B of the B candidate momentum vector with respect to the beam axis in the e + e − center-of-mass (c.m.) frame; the angle θ T between the thrust axis of the B candidate and the thrust axis of all other particles in the event in the c.m. frame; the event topology variable F , and the kinematic variable ∆E, described below. Correlations among these variables are small.
The helicity angle θ H is defined as the angle between one of the decay products of a vector meson and the flight direction of its parent particle in the meson's rest frame. Polarization of the vector mesons in the signal decays causes the cosines of their helicity angles to be distributed as cos 2 θ H (φ, K * 0 , ρ + , and K * + ) or 1 − cos 2 θ H (D * + s ), while the random background combinations tend to produce a more uniform distribution in cos θ H , with a peak in the forward direction (which corresponds to a low-energy π 0 ) for ρ + and K * + candidates. We do not include the helicity angles for D * + s , ρ + , and K * + mesons in the likelihood ratio R L for the vector-vector B 0 → D * + s ρ − and B 0 → D * − s K * + modes, since the polarizations of the vector mesons in these decays are not known a priori. Instead, the helicity angles are used in the multi-dimensional likelihood fit to determine the polarizations, as discussed below.
The variables cos θ B , cos θ T , and F discriminate between spherically-symmetric BB events and jet-like continuum background using event topology. The polar angle θ B is distributed as sin 2 θ B for real B decays, while being nearly flat in cos θ B for the continuum. BB pairs form a nearly uniform | cos θ T | distribution, while the | cos θ T | distribution for continuum events peaks at 1. A linear (Fisher) discriminant F is derived from the values of sphericity and thrust for the event, and the two Legendre moments L 0 and L 2 of the energy flow around the B-candidate thrust axis [19].
The ratio R L has a maximum at R L = 1 for signal events, and at R L = 0 for background originating from continuum events. It also discriminates well against B decays without a real D icantly smaller than the energy resolution of the reconstructed B mesons, and at the same time larger than the momentum resolution. The momentum of the signal candidates is included in the beam-energy-substituted mass is the four-momentum of the initial e + e − system, and p B is the B 0 candidate momentum, both measured in the laboratory frame. The second For signal events, the m ES distribution is nearly Gaussian and centered at the B meson mass with a resolution of about (2.5-2.8) MeV/c 2 , and the ∆E distribution has a maximum near zero with a resolution of (17-25) MeV. We include ∆E in the definition of the likelihood ratio R L ; m ES is used as a discriminating variable in the maximum likelihood fit described below.
We parameterize the signal and background PDFs using large samples of simulated events. We select in which the kinematics is similar to those of our signal events, and find that they are consistent with Monte Carlo esti- modes. The fraction of continuum background events passing the selection varies between 2% and 15%, depending on the mode.
Less than 30% of the selected events in the contain two or more candidates that satisfy the criteria listed above. In such events we select a single B 0 candidate based on an event χ 2 formed from ∆E, m(D s ) and (where appropriate) ∆m(D * + s ), m ρ , m K * , m π 0 and m Ks , and their uncertainties.

IV. EXTRACTION OF SIGNAL YIELDS
After the R L requirement is applied, we define the region of interest using the beam-energy-substituted mass m ES and the mass of the D + s candidate m(D s ). We require 5.2 < m ES < 5.3 GeV/c 2 and |m( [20]. The invariant mass m(D s ) has a resolution of (5-6) MeV/c 2 , depending on the D + s decay mode. The selection is significantly broader than the region populated by the signal events, and allows us to constrain backgrounds in the signal region. For Five classes of background events contribute to the fit region. First is the combinatorial background , in which a true or fake D ( * ) s candidate is combined with a randomly-selected light meson. Second, B meson decays such as B 0 →D ( * )+ π − or B 0 →D ( * )+ ρ − with D + →K 0 S π + or K * 0 π + can constitute a background for the B 0 → D  For each B decay, we simultaneously fit distributions in the three D + s decay modes, constraining the signal branching fractions to a common value. The likelihood function contains the contributions of the signal and the five background components discussed above. The function to be maximized is where n jm is the number of events for each event type j (signal and all background modes) in each D s decay mode m, and P jm ( ζ i ) is the probability density function of the variables ζ i = (m ES , m(D s ), cos θ D * s , cos θ V ) for the ith event. The likelihood product is computed over all candidates N cand in the region of interest. We parameterize the event yields as where m stands for D + s → φπ + , D + s → K * 0 K + , or D + s → K 0 S K + , N BB = 381 × 10 6 , B j is the B decay branching fraction, B Ds m is the branching fraction of the m-th D + s mode, and ε m is the reconstruction efficiency. The branching fractions of the channels contributing to the reflection background are fixed in the fit to the current world average values [20] and the branching fractions of the crossfeed backgrounds are determined by iterating the fits over each B decay mode. The branching fractions of the non-resonant backgrounds are fixed to the values recently measured by BABAR [21]. In the case of B 0 → D ( * )− s K + π 0 , which can contribute to B 0 → D ( * )− s K * + (K * + → K + π 0 ), we estimate the branching fraction by This scaling assumes that the dominant mechanism for producing both D ( * )− s K + π 0 and D ( * )− s K + π + final states is a sub-threshold production of a charmed D * * 0 meson, which subsequently decays into D The expected yields of the dominant B-decay backgrounds are listed in Table I. The PDFs and efficiencies for the signal, reflection, and crossfeed backgrounds are determined independently for each D + s decay mode using Monte Carlo samples. The signal contribution is modeled as a Gaussian (B 0 → D + s π − and B 0 → D − s K + ) or a "Crystal Ball" function [22] in m ES and a double Gaussian in m(D s ). The combinatorial background is described in m ES by a threshold function [23], dN/dx ∝ x 1 − 2x 2 /s exp −ξ 1 − 2x 2 /s , characterized by the shape parameter ξ. This shape parameter, common to all D + s modes, is allowed to vary in the fit. In m(D s ), the combinatorial background is well described by a combination of a first-order polynomial (fake D + s candidates) and a Gaussian with (5-6) MeV/c 2 resolution (true D + s candidates). The charmless background is parameterized by the signal Gaussian shape in m ES and a first order polynomial in m(D s ).
Ideally, the distribution of the helicity angles in the vector-vector decays is given by Eq. (3). The helicity angle θ D * s is defined as the angle between the direction of the photon in D * s → D s γ and the direction of the B in the rest frame of the D * s candidate. The helicity angle θ V is similarly defined by the direction of the charged daughter particle in the decays ρ + → π + π 0 , K * + → K + π 0 , and K * + → K 0 S π + . Since the momenta of the decay products in the laboratory frame depend on the helicity angles, acceptance and efficiency effects modify the ideal angular distribution. We determine the PDFs of the signal events using the Monte Carlo simulation, and measure the angular distribution of the combinatorial background in the data region m ES < 5.27 GeV/c 2 .  Table II.

V. SYSTEMATIC UNCERTAINTIES
For the branching fractions, the systematic errors are dominated by the 13% relative uncertainty for B(D + s → φπ + ) [20]. The uncertainty in the branching fraction ratio B(D + s →K * 0 K + )/B(D + s →φπ + ) contributes (2-4)%, depending on the decay channel. For B(D + s →K 0 S K + ), we use the most recent measurement from the CLEO Collaboration [24], which differs from the previously reported central value [20] by about 50%. We estimate uncertainties due to modeling of the resonance (K * 0 , φ, ρ, and K * + ) lineshapes by measuring the effect of the lineshape variation on signal selection efficiency.
The uncertainties in the signal selection efficiency are determined by the accuracy with which the detector effects are modeled in the Monte Carlo simulations. Tracking, particle identification (PID), photon, π 0 and K 0 S reconstruction efficiencies are measured across the wide range of particle momenta in the dedicated data control samples. The tracking efficiency and resolution are adequately reproduced by the simulations. The simulated distributions are corrected for the efficiency and resolution of the π 0 reconstruction. The efficiency of the R L cut is also measured in the data control samples, as discussed in Section III.
The uncertainties due to the knowledge of the signal and background PDFs in the ML fit are estimated by measuring the variation of the fitted values of the branching fractions when PDF parameters are varied within their uncertainties. The correlations between parameters are taken into account. The uncertainties in the signal PDF parameters for the key discriminants ∆E, m ES , m(D s ), ∆m(D * + s ), and cos θ D * + s are determined by comparing data and Monte Carlo simulations for the samples of decays B 0 → D − π + , D − ρ + (D − → K + π − π − , K 0 S π − ) and B + → D * 0 π + , D * 0 ρ + (D * 0 → D 0 γ, D 0 → K − π + ). The uncertainties in the signal PDFs for cos θ ρ,K * and the PDFs for the peaking backgrounds are determined by Monte Carlo simulations. These distributions depend on the modeling of the charged track and π 0 reconstruction, discussed above. The helicity angle PDFs for the continuum background are determined in the data sideband m ES < 5.27 GeV/c 2 , and their uncertainties are statistical in nature.
Uncertainties due to reflection and crossfeed backgrounds include the uncertainties in the branching fractions of the relevant modes, and also account for the contributions of the sub-dominant modes that are not explicitly included in the ML fit. These contributions dominate the systematic uncertainty for the B 0 → D + s ρ − mode, which has a small absolute branching fraction.
As ML estimators may be biased in small samples, we measure the bias using a large ensemble of simulated experiments. In each of these experiments, generated according to the sample composition observed in data, the signal and B-decay background events are fully simulated, and the combinatorial background events are generated from their PDFs. The bias is found to be negligible for the 1-and 2-dimensional ML fits (B 0 → D On the other hand, we find that in the vectorvector modes (B 0 → D * + s ρ − and B 0 → D * − s K * + decays), the 3-dimensional ML fits underestimate the true values of the signal branching fraction and the fraction of the longitudinal polarization. We correct for the biases of ∆B = (−0.37 ± 0.03) × 10 −5 and ∆f L = (−5.3 ± 0.6)% (B 0 → D * + s ρ − ) and ∆B = (−0.14 ± 0.04) × 10 −5 and ∆f L = (−5.5 ± 0.8)% (B 0 → D * − s K * + ). We assign a conservative uncertainty of 50% of the bias to this correction.
For the longitudinal polarization fractions f L in the vector-vector modes, the systematic errors are dominated by the uncertainties in the shapes of the signal and background PDFs and the fit bias. The systematic uncertainties for each mode are summarized in Tables III-V.

VI. RESULTS
We estimate the significance of a non-zero signal yield by computing S = −2 log(L 0 /L max ), where L max is the maximum likelihood value, and L 0 is the likelihood for a fit in which the signal contribution is set to zero. Including systematic uncertainties and assuming Gaussiandistributed errors, we obtain signal observation significances of 3.9 (B 0 → D * + s ρ − ), 4.6 (B 0 → D − s K * + ), and 3.1 (B 0 → D * − s K * + ) standard deviations, providing the first evidence for these decays. We test that S measures the probability for the background events to fluctuate to the observed number of signal events with a large ensemble of simulated experiments. For each such experiment, we generate a set of pure background events according to the PDFs and sample composition observed in our dataset. We then fit each simulated experiment and measure the signal and background yields and, for the vectorvector modes, the polarization fraction f L . By counting the fraction of such pseudo-experiments in which the signal yields are at least as large as the yield observed in the real dataset, we confirm that S 2 follows closely the χ 2 distribution with one degree of freedom.
The branching fraction and polarization results are collected in Table II. Since we do not observe a significant event yield in B 0 → D + s ρ − , we set a 90% confidencelevel Bayesian upper limit assuming a constant prior for

VII. CONCLUSIONS
We report the following improved measurements of the branching fractions for the rare decays where the first quoted uncertainty is statistical, and the second is systematic. where the first error is statistical, the second includes experimental systematics, and the last accounts for the uncertainty in the theoretical value of f D ( * ) s /f D ( * ) [9]. These amplitude ratios are below 2%, which implies small CP asymmetries in B 0 →D ( * )∓ π ± and B 0 →D ( * )∓ ρ ± decays, making it difficult to measure sin(2β + γ) precisely in these decays. The results presented here supersede our previously published measurements [14].

VIII. ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariatà l'Energie Atom-   II: The number of reconstructed candidates (Nraw), the signal yield (Nsig), computed from the fitted branching fractions, the combinatorial background (N comb ), and the sum of charmless, reflection, non-resonant, and crossfeed contributions (N peak ), extracted from the likelihood fit. Also given are the reconstruction efficiency (ε), the signal significance S, the measured branching fraction B, and the fraction of longitudinal polarization fL (where appropriate). The first uncertainty is statistical, and the second is systematic.