Measurement of Decay Amplitudes of B -->(c cbar) Kstar with an Angular Analysis, for (c cbar)=J/psi, psi(2S) and chi_c1

We perform the first three-dimensional measurement of the amplitudes of $B\to \psi(2S) K^*$ and $B\to \chi_{c1} K^*$ decays and update our previous measurement for $B\to J/\psi K^*$. We use a data sample collected with the BaBar detector at the PEP2 storage ring, corresponding to 232 million $B\bar B$ pairs. The longitudinal polarization of decays involving a $J^{PC}=1^{++}$ $\chi_{c1}$ meson is found to be larger than that with a $1^{--}$ $J/\psi$ or $\psi(2S)$ meson. No direct {\it CP}-violating charge asymmetry is observed.

We perform the first three-dimensional measurement of the amplitudes of B → ψ(2S)K * and B → χc1K * decays and update our previous measurement for B → J/ψ K * . We use a data sample collected with the BABAR detector at the PEP-II storage ring, corresponding to 232 million BB pairs. The longitudinal polarization of decays involving a J P C = 1 ++ χc1 meson is found to be larger than that with a 1 −− J/ψ or ψ(2S) meson. No direct CP -violating charge asymmetry is observed.
PACS numbers: 13.25.Hw, 12.15.Hh,11.30.Er In the context of measuring the parameters of the Unitarity Triangle of the CKM matrix, B 0 decays to charmonium-containing final states (J/ψ , ψ(2S), χ c1 )K * , defined collectively here as B 0 → (cc)K * , are of interest for the precise measurement of sin 2β, where , in a similar way as for B 0 → J/ψ K 0 . Furthermore, the J/ψ K * channel allows the measurement of cos 2β [1].
For the modes considered in this paper, the final state consists of two spin-1 mesons, leading to three possible values of the total angular momentum with different CP eigenvalues (L = 1 is odd, while L = 0, 2 are even). The different contributions must be taken into account in the measurement of sin 2β. The amplitude for longitudinal polarization of the two spin-1 mesons is A 0 . There are two amplitudes for polarizations of the mesons transverse to the decay axis, here expressed in the transversity basis [2]: A for parallel polarization and A ⊥ for their perpendicular polarization. Only the relative amplitudes are measured, so that |A 0 | 2 + |A | 2 + |A ⊥ | 2 = 1. Previous measurements by the CLEO [3], CDF [4], BABAR [1] and Belle [5] collaborations for the B → J/ψ K * channels are all compatible with each other, and with a CP -odd intensity fraction |A ⊥ | 2 close to 0.2.
Factorization predicts that the phases of the transversity decay amplitudes are the same. BABAR has observed [1,6] a significant departure from this prediction.
B → (cc)K ( * ) decays provide a clean environment for the measurement of the CKM angle β because one tree amplitude dominates the decay. Very small direct CP -violating charge asymmetries are expected in these decays, and no such signal has been found [7]. While more than one amplitude with different strong and weak phases are needed to create a charge asymmetry in a simple branching fraction measurement, London et al. have suggested [11] that an angular analysis of vector-vector decays can detect charge asymmetries even in the case of vanishing strong phase difference. Belle has looked for, and not found, such a signal [5].
In this paper we present the amplitude measurement of charged and neutral B → (cc)K * using a selection similar to that of Ref. [7], and a fitting method similar to that of Ref. [1]. We use the notation ψ for the 1 −− states J/ψ and ψ(2S). ψ (χ c1 ) candidates are reconstructed in their decays to ℓ + ℓ − (J/ψ γ), where ℓ represents an electron or a muon. Decays to the flavor eigenstates K * 0 → K ± π ∓ , K * ± → K 0 S π ± and K * ± → K ± π 0 are used. The relative strong phases are known to have a two-fold ambiguity when measured in an angular analysis alone. In contrast to earlier publications [3,4,6] we use here the set of phases predicted in Ref. [12], with arguments based on the conservation of the s-quark helicity in the decay of the b quark. We have confirmed experimentally this prediction through the study of the variation with Kπ invariant mass of the phase difference between the K * (892) amplitude and a non-resonant Kπ S-wave amplitude [1].
The data were collected with the BABAR detector at the PEP-II asymmetric e + e − storage ring, and correspond to an integrated luminosity of about 209 fb −1 at the centerof-mass energy near the Υ (4S) mass. The BABAR detector is described in detail elsewhere [13]. Charged-particle tracking is provided by a five-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH). For chargedparticle identification (PID), ionization energy loss in the DCH and SVT, and Cherenkov radiation detected in a ring-imaging device (DIRC) are used. Photons are identified by the electromagnetic calorimeter (EMC), which comprises 6580 thallium-doped CsI crystals. These systems are mounted inside a 1.5-T solenoidal superconducting magnet. Muons are identified in the instrumented flux return (IFR), composed of resistive plate chambers and layers of iron that return the magnetic flux of the solenoid. We use the GEANT4 [14] software to simulate interactions of particles traversing the detector, taking into account the varying accelerator and detector conditions.
J/ψ → e + e − (µ + µ − ) candidates must have a mass between 2.95 − 3.14 (3.06 − 3.14) GeV/c 2 . ψ(2S) candidates are required to have invariant masses 3.44 < m e + e − < 3.74 GeV/c 2 or 3.64 < m µ + µ − < 3.74 GeV/c 2 . Electron candidates are combined with photon candidates in order to recover some of the energy lost through Bremsstrahlung. J/ψ candidates and γ candidates with an energy larger than 150 MeV, are combined to form χ c1 candidates, which must satisfy 350 < m ℓ + ℓ − γ − m ℓ + ℓ − < 450 MeV/c 2 . π 0 → γγ candidates must satisfy 113 < m γγ < 153 MeV/c 2 . The energy of each photon has to be greater than 50 MeV. K 0 S → π + π − candidates are required to satisfy 489 < m π + π − < 507 MeV/c 2 . In addition, the K 0 S flight distance from the ψ vertex must be larger than three times its uncertainty. K * 0 and K * + candidates are required to satisfy 796 < m Kπ < 996 MeV/c 2 and 792 < m Kπ < 992 MeV/c 2 , respectively. In addition, due to the presence of a large background of low-energy non-genuine π 0 's, the cosine of the angle θ K * between the K momentum and the B momentum in the K * rest frame has to be less than 0.8 for K * → K ± π 0 . In events where two B's reconstruct to modes with the same cc and K candidate, one with a π ± and the other with a π 0 , the B candidate with a π 0 is discarded due to the high background induced by fake π 0 's. B candidates, reconstructed by combining cc and K * candidates, are characterized by two kinematic variables: the difference between the reconstructed energy of the B candidate and the beam energy in the center-of-mass frame ∆E = E * B − √ s/2, and the beam-energy substi- where subscript 0 and B correspond to Υ (4S) and the B candidate in the laboratory frame. For a correctly reconstructed B meson, ∆E is expected to peak near zero and m ES near the B-meson mass 5.279 GeV/c 2 . The analysis is performed in a region of the m ES vs ∆E plane defined by 5.2 < m ES < 5.3 GeV/c 2 and −120 < ∆E < 120 MeV. The signal region is defined as m ES > 5.27 GeV/c 2 and |∆E| smaller than 40 (30) MeV for channels with (without) a π 0 . For events that have multiple candidates, the candidate having the smallest |∆E| is chosen. m ES distributions are available in Ref. [18].
The B decay amplitudes are measured from the differential decay distribution, expressed in the transversity basis [1,6], Fig. 1, with conventions detailed in Ref. [15]. θ K * is the helicity angle of the K * decay. It is defined in the rest frame of the K * meson, and is the angle between the kaon and the opposite direction of the B meson in this frame. θ tr and φ tr are defined in the ψ (χ c1 ) rest frame and are the polar and azimutal angle of the positive lepton (J/ψ daughter of χ c1 ) , with respect the axis defined by: • x tr : opposite direction of the B meson; • y tr : perpendicular to x tr , in the (x tr , p K * ) plane, with a direction such that p K * · y tr > 0; • z tr : to complete the frame, ie: z tr = x tr × y tr .
In terms of the transversity angular variables ω ≡ (cos θ K * , cos θ tr , φ tr ), the time-integrated differential decay rate for the decay of the B meson is where the amplitude coefficients A i and the angular functions f k (ω), k = 1 · · · 6 are listed in Table I. The ψ decays to two spin-1/2 particles, while the χ c1 decays to two vector particles. The angular dependencies are therefore different [15]. The symbol A ≡ (A 0 , A , A ⊥ ) denotes the transversity amplitudes for the decay of the B meson, and A for the B meson decay. In the absence of direct CP violation, we can choose a phase convention in which these amplitudes are related by A 0 = +A 0 , A = +A , A ⊥ = −A ⊥ , so that A ⊥ is CP -odd and A 0 and A are CP -even. The phases δ j of the amplitudes, where j = 0, , ⊥, are defined by A j = |A j |e iδj . Phases are defined relative to δ 0 = 0. We perform an unbinned likelihood fit of the threedimensional angle probability density function (PDF). The acceptance of the detector and the efficiency of the event reconstruction may vary as a function of the transversity angles, in particular as the angle θ K * is strongly correlated with the momentum of the final kaon and pion. We use the acceptance correction method developped in Ref. [1]. The PDF of the observed events, g obs , is : where ε(ω) is the angle-dependent acceptance and is the average acceptance. We take into account the presence of cross-feed from channels with the same cc candidate and a different K * candidate that has (due to isospin symmetry) the same A dependence as the signal. The observed PDF for channel b (b = K ± π ∓ , K 0 S π ± , K ± π 0 ) is then I: Amplitude coefficients A k and angular functions f k (ω) that contribute to the differential decay rate. An overall normalization factor 9/32π (for ψ) and 9/64π (for χc1) has been omitted. In the case of a B decay, the ℑm terms change sign.
The coefficients Φ b k are computed with exclusive signal MC samples obtained using a full simulation of the experiment [14,16]. PID efficiencies measured with data control samples are used to adjust the MC simulation to the observed performance of the detector. Separate coefficients are used for different charges of the final state mesons, in particular to take into account the charge dependence of the interaction of charged kaons with matter, and a possible charge asymmetry of the detector. Writing the expression for the log-likelihood L b (A) for the PDF g b obs (ω i ; A) for a pure signal sample of N S events, the relevant contribution is since the remaining term NS i=1 ln ε b (ω i ) does not depend on the amplitudes.
We use a background correction method [1] in which background events from a pure background sample of N B events are added with a negative weight to the loglikelihood that is maximized where L(ω; A) = ln(g b obs (ω; A)). The fit is performed within the m ES signal region. Background events used here for subtraction are from generic (BB, qq) MC samples.ñ B is an estimate of the unknown number n B of background events that are present in the signal region in the data sample.
As L ′b is not a log-likelihood, the uncertainties yielded by the minimization program Minuit [17] are biased estimates of the actual uncertainties. An unbiased estimation of the uncertainties is described and validated in Appendix A of Ref. [1]. With this pseudo-log-likelihood technique, we avoid parametrizing the acceptance as well as the background angular distributions.
The measurement is affected by several systematic uncertainties. The branching fractions used in the crossfeed part of the acceptance cross section are varied by ±1σ, and the largest variation is retained. The uncertainty induced by the finite size of the MC sample used to compute the coefficients Φ b k is estimated by the statistical uncertainty of the angular fit on that MC sample [6]. The uncertainty due to our limited understanding of the PID efficiency is estimated by using two different methods to correct for the MC-vs-data differences. The background uncertainty is obtained by comparing MC and data shapes of the m ES distributions for the combinatorial component and by using the corresponding branching errors for the peaking component. The uncertainty due to the presence of a Kπ S wave under the K * (892) peak is estimated by a fit including it. The differential decay rate is described by Eqs. (6-9) of Ref. [1].
The results are summarized in Table II. The values of |A 0 | 2 , |A | 2 , |A ⊥ | 2 are negatively correlated due to the constraint |A 0 | 2 + |A | 2 + |A ⊥ | 2 = 1. In particular, |A | 2 ,   δA4 δA6 (K + π − ) 0.002 ± 0.025 ± 0.005 −0.011 ± 0.043 ± 0.016 (K + π 0 ) −0.017 ± 0.047 ± 0.023 −0.051 ± 0.098 ± 0.064 (K 0 S π + ) −0.008 ± 0.049 ± 0.011 0.075 ± 0.089 ± 0.009 which would be the least precisely measured parameter in separate one-dimensional fits, is strongly anti-correlated with |A 0 | 2 , which would be the best measured. The one-dimensional (1D) distributions, acceptance-corrected with an 1D Ansatz and background-subtracted, are overlaid with the fit results and shown on Figure 2. In contrast with the dedicated method used in the fit, for the plots, we simply computed the 1D efficiency maps from the distributions of the accepted events divided by the 1D PDF. As in lower statistics studies, the cos θ K * forward backward asymmetry due to the interference with the S wave is clearly visible. Our measurement of the amplitudes of B decays to J/ψ are compatible with, and of better precision than, previous measurements. A comparison of neutral and charged B decays (not shown) yields results consistent with isospin symmetry. The strong phase difference δ − δ ⊥ is obtained from a fit in which the phase origin is δ ⊥ ≡ 0. We confirm our previous observation that the strong phase differences are significantly different from zero, in contrast with what is predicted by factorization. For B → J/ψ K * , it amounts to δ − δ ⊥ = 0.45 ± 0.05 ± 0.02. The presence of direct CP -violating triple-products in the amplitude would produce a B to B difference in the interference terms A 4 and A 6 : δA 4 and δA 6 . Our results (see Table III), with improved precision relative to Ref. [19], are consistent with no CP violation.
In summary, we have performed the first threedimensional analysis of the decays to ψ(2S) and χ c1 . The longitudinal polarization of the decay to ψ(2S) is lower than that to J/ψ , while the CP -odd intensity fraction is higher (by 1.4 and 1.0 standard deviations, respectively). This is compatible with the prediction of models of meson decays in the framework of factorization. The longitudinal polarization of the decay to χ c1 is found to be larger than that to J/ψ , in contrast with the predictions of Ref. [8], which include non-factorizable contributions. The CP -odd intensity fraction of this decay is compatible with zero. The parallel and longitudinal amplitudes for χ c1 seem to be aligned (|δ − δ 0 | ∼ 0) while for ψ they are anti-aligned (|δ − δ 0 | ∼ π).
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 hospi-