Measurement of the polarisation amplitudes in B0 ->J/psi K*(892)0 decays

An analysis of the decay B0 ->J/psi K*(892)0 is presented using data, corresponding to an integrated luminosity of 1.0fb^-1, collected in pp collisions at a centre-of-mass energy of 7TeV with the LHCb detector. The polarisation amplitudes and the corresponding phases are measured to be |A_{\parallel}|^2 = 0.227 +- 0.004 (stat.) +- 0.011 (syst.), |A_{\perp}|^2 = 0.201 +- 0.004 (stat.) +- 0.008 (syst.), \delta_{\parallel} [rad] = -2.94 +- 0.02 (stat.) +- 0.03 (syst.), \delta_{\perp} [rad] = 2.94 +- 0.02 (stat.) +- 0.02 (syst.). Comparing B0 ->J/psi K*(892)0 and \bar{B}0 ->J/psi \bar{K}*(892)0 decays, no evidence for direct CP violation is found.


Introduction
The measurement of the polarisation content of the decay B 0 → J/ψ(µ + µ − )K * 0 (K + π − ) and its charge-conjugate B 0 → J/ψ(µ + µ − )K * 0 (K − π + ) is presented in this paper, where the notation K * 0 is used to refer to the K * (892) 0 meson. Recent measurements have been performed by BaBar (2007, [1]), Belle (2005, [2]) and CDF (2005, [3]). A detailed comparison can be found in Sec. 7. The decay can be decomposed in terms of three transversity states, corresponding to the relative orientation of the linear polarisation vectors of the two vector mesons. The amplitudes are referred to as P-wave amplitudes since the Kπ system is in a P-wave state, and are denoted by A 0 (longitudinal), A (transverse-parallel) and A ⊥ (transverse-perpendicular), where the relative orientations are shown in parentheses. An additional S-wave amplitude corresponding to a non-resonant Kπ system is denoted by A S . The strong phases of the four amplitudes are δ 0 , δ , δ ⊥ and δ S , respectively, and by convention δ 0 is set to zero. The parity of the final states is even for A 0 and A , and odd for A ⊥ and A S . The Standard Model (SM) predicts that the B 0 → J/ψ(µ + µ − )K * 0 (K + π − ) decay is dominated by a colour-suppressed tree diagram (Fig. 1a), with highly-suppressed contributions from gluonic and electroweak loop (penguin) diagrams (Fig. 1b). Neglecting the penguin contributions and using naïve factorisation for the tree diagram leads to predictions for the P-wave amplitudes |A 0 | 2 ≈ 0.5, and A ≈ A ⊥ [4]. In the absence of final state interactions, the phases δ and δ ⊥ are both predicted to be 0 or π rad. Corrections of order 5% to these predictions from QCD have been incorporated in more recent calculations [5,6]. The signal decay is flavour specific, with K * 0 → K + π − or K * 0 → K − π + indicating a B 0 or B 0 decay, respectively. In the SM, the amplitudes for the decay and its charge-conjugate are equal, but in the presence of physics beyond the SM (BSM) the loop contributions could be enhanced and introduce CP -violating differences between the B 0 and B 0 decay amplitudes [7,8,9]. An analysis of the angular distributions of the decay products gives increased sensitivity to BSM physics through differences in the individual amplitudes [10]. A further motivation for studying B 0 → J/ψK * 0 decays is that the magnitudes and phases of the amplitudes should be approximately equal to those in B 0 s → J/ψφ decays [11]. Both decay modes are dominated by colour-suppressed tree diagrams and have similar branching fractions, B(B 0 → J/ψK * 0 ) = (1.29 ± 0.14) × 10 −3 [12] (S-wave subtracted) and B(B 0 s → J/ψφ) = (1.05 ± 0.11) × 10 −3 [13]. Any BSM effects observed in B 0 → J/ψK * 0 may also be present in B 0 s → J/ψφ, where they would modify the time-dependent CP violation and the CP -violating phase φ s [14].

Angular analysis
To measure the individual polarisation amplitudes (A 0 , A , A ⊥ , A S ) the decay is analysed in terms of three angular variables, denoted as Ω = {cos θ, cos ψ, ϕ} in the transversity basis (Fig. 2). For a B 0 decay, the angle between the µ + momentum direction and the z axis in the J/ψ rest frame is denoted θ and ϕ is the azimuthal angle of the µ + momentum direction in the same frame. ψ is the angle between the momentum direction of the K + meson and the negative momentum direction of the J/ψ meson in the K * 0 → K + π − rest frame. For B 0 decays, the angles are defined with respect to the µ − and the K − meson.
In this analysis the flavour of the B meson at production is not measured. Therefore, the observed B 0 → J/ψK * 0 decays arise from both initial B 0 or B 0 mesons as a result of oscillations. Summing over both contributions, the differential decay rate can be written as [15,16] where t is the decay time and Γ d is the total decay width of the B 0 meson; h k are combinations of the polarisation amplitudes and the f k are functions of the three transversity angles. These factors can be found in Table 1. The h k combinations are invariant under the phase transformation (δ , δ ⊥ , δ S ) ←→ (−δ , π − δ ⊥ , −δ S ). This two-fold ambiguity can be resolved by measuring the phase difference between the S-and P-wave amplitudes as a function of m(K + π − ) (see Sec. 7). The difference in decay width between the heavy and light eigenstates, ∆Γ d , has been neglected. The differential decay rate for B 0 → J/ψK * 0 is obtained from Eq. 1 by defining the angles using the charge conjugate final state particles, and multiplying the interference terms f 4 , f 6 and f 9 in Table 1 by −1. To allow for possible direct CP violation, the amplitudes are changed from A i to A i (i = 0, , ⊥, S).   [15,16]. The f k are functions defined such that their integrals over Ω are unity.

LHCb detector
The LHCb detector [17] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c, and impact parameter resolution of 20 µm for tracks with high transverse momentum (p T ). Charged hadrons are identified using two ring-imaging Cherenkov detectors [18]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. In the simulation, pp collisions are generated using Pythia 6.4 [19] with a specific LHCb configuration [20]. Decays of hadronic particles are described by EvtGen [21], in which final state radiation is generated using Photos [22]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [23] as described in Ref. [24].

Data samples and candidate selection
In the following B 0 → J/ψK * 0 refers to both charge-conjugate decays unless otherwise stated. The selection of B 0 → J/ψK * 0 candidates is based upon the decays of the J/ψ → µ + µ − and the K * 0 → K + π − final states. Candidates must satisfy the hardware trigger [25], which selects events containing muon candidates that have high transverse momentum with respect to the beam direction. The subsequent software trigger [25] is composed of two stages. The first stage performs a partial event reconstruction and requires events to have two well-identified oppositely-charged muons with invariant mass larger than 2.7 GeV/c 2 . The second stage of the software trigger performs a full event reconstruction and only retains events containing a µ + µ − pair that has invariant mass within 120 MeV/c 2 of the known J/ψ mass [26] and forms a vertex that is significantly displaced from the nearest primary pp interaction vertex (PV). The J/ψ candidates are formed from two oppositely-charged tracks, being identified as muons, having p T > 500 MeV/c and originating from a common vertex. The invariant mass of this pair of muons must be in the range 3030 − 3150 MeV/c 2 .
The K * 0 candidates are formed from two oppositely-charged tracks, one identified as a kaon and one as a pion which originate from the same vertex. It is required that the K * 0 candidate has p T > 2 GeV/c and invariant mass in the range 826 − 966 MeV/c 2 .
The B 0 candidates are reconstructed from the J/ψ and K * 0 candidates, with the invariant mass of the µ + µ − pair constrained to the known J/ψ mass. The resulting B 0 candidates are required to have an invariant mass m(J/ψK + π − ) in the range 5150 − 5400 MeV/c 2 . The decay time of the B 0 candidate is calculated from a vertex and kinematic fit that constrains the B 0 candidate to originate from its associated PV [27]. The χ 2 per degree of freedom of the fit is required to be less than 5. For events with multiple B 0 candidates, the candidate with the smallest fit χ 2 per degree of freedom is chosen. Only  In the data sample, corresponding to an integrated luminosity of 1.0 fb −1 , collected in pp collisions at a centre-of-mass energy of 7 TeV with the LHCb detector, a total of 77 282 candidates are selected. The invariant mass distribution is shown in Fig. 3. From a fit the number of signal decays is found to be 61 244 ± 132. The uncertainties on the signal yields quoted here and in Sec. 7 come from propagating the uncertainty on the signal fraction evaluated by the fit.

Maximum likelihood fit
The parameters used in this analysis are |A | 2 , |A ⊥ | 2 , F S , δ , δ ⊥ and δ S , where we introduce the parameter F S = |A S | 2 /(1 + |A S | 2 ) to denote the fractional S-wave component. The parameter |A 0 | 2 is determined by the constraint |A 0 | 2 + |A | 2 + |A ⊥ | 2 = 1. The best fit values of these parameters are determined with an unbinned maximum log-likelihood fit to the decay time and angular distributions of the selected B 0 candidates. In order to subtract the background component, each event is given a signal weight, W i , using the sP lot [28] method with m(J/ψK + π − ) as the discriminating variable. The invariant mass distribution of the signal is modelled as the sum of two Gaussian functions with a common mean. The mean and widths of both Gaussian functions, as well as the fraction of the first Gaussian are parameters determined by the fit. The effective resolution of the mass peak is determined to be 9.3 ± 0.8 MeV/c 2 . The invariant mass distribution of the background is described by an exponential function. The signal fraction in a ±30 MeV/c window around the known B 0 mass [26] is approximately 93%.
A maximum likelihood fit is then performed with each candidate weighted by W i . The fit uses a signal-only probability density function (PDF) which is denoted S. It is a function of the decay time t and angles Ω, and is obtained from Eq. 1. The exponential decay time function is convolved with a Gaussian function to take into account the decay time resolution of 45 fs [14]. The effect of the time and angular resolution on this analysis has been studied and found to be negligible [16]. The fit minimises the negative log likelihood summed over the selected candidates where α = i W i / i W 2 i is a normalisation factor accounting for the effect of the weights in the determination of the uncertainties [29].
The selection applied to the data is almost unbiased with respect to the decay time. The measurements of amplitudes and phases are insensitive to the decay time acceptance since ∆Γ d ∼ 0 and the time dependence of the PDF factorises out from the angular part. Nevertheless, the small deviation of the decay time acceptance from uniformity is determined from data using decay time unbiased triggers as a reference, and is included in the fitting procedure. The acceptance as a function of the decay angles is not uniform because of the forward geometry of the detector and the momentum selection requirements applied to the final state particles. A three-dimensional acceptance function, A(Ω), is determined using simulated events subject to the same selection criteria as the data, and is included in the fit. Figure 4 shows the acceptance as a function of each decay angle, integrated over the two other angles. The variation in acceptance is asymmetric for cos ψ, due to the selection requirements on the π − and the K * 0 mesons.
The phase of the P-wave amplitude increases rapidly as a function of the K + π − invariant mass, whereas the S-wave phase increases relatively slowly [30]. As a result the phase difference between the S-and P-wave amplitudes falls with increasing K + π − invariant mass. A fit which determines the phase difference in bins of m(K + π − ) can therefore be used to select the physical solution and hence resolve the ambiguity described in Sec. 2. This method has previously been used to measure the sign of ∆Γ s in the B 0 s system [31]. In the analysis the data are divided into four bins of m(K + π − ), shown in Fig. 5 and defined in Table 2. A simultaneous fit to all four bins is performed in which the P-wave parameters are common, but F S and δ S are independent parameters in each bin. Consistent results are obtained with the use of two or six bins.
To correct for the variation of the S-wave relative to the P-wave over the m(K + π − ) range of each bin, a correction factor is introduced in each of the three interference terms f 8 , f 9 and f 10 in Eq. 1. The S-wave lineshape is assumed to be uniform across the m(K + π − ) range and the P-wave shape is described by a relativistic Breit-Wigner function. The correction factor is calculated by integrating the product ps * where p and s are the P-and S-wave lineshapes normalised to unity in the range of integration, * is the complex conjugation operator, m L K + π − and m H K + π − denote the boundaries Table 2: Bins of m(K + π − ) and the corresponding C SP correction factor for the S-wave interference terms, assuming a uniform distribution for the non-resonant K + π − contribution and a relativistic Breit-Wigner shape for decays via the K * 0 resonance. of the m(K + π − ) bin, C SP is the correction factor and θ SP is absorbed in the measurements of δ S − δ 0 . The C SP factors tend to unity (i.e. no correction) as the bin width tends to zero. The C SP factors calculated for this analysis are given in Table 2. The factors are close to unity, and hence the analysis is largely insensitive to this correction.

Systematic uncertainties
To estimate the systematic uncertainties arising from the choice of the model for the B 0 invariant mass, the signal mass PDF is changed from a double Gaussian function to either a single Gaussian or a Crystal Ball function. The largest differences observed in the fitted values of the parameters are assigned as systematic uncertainties.
To account for uncertainties in the treatment of the combinatorial background, an alternative fit to the data is performed without using signal weights. An explicit background model, B, is constructed, with the time distribution being described by two exponential functions, and the angular distribution by a three-dimensional histogram derived from the sidebands of the B 0 invariant mass distribution. A fit is then made to the unweighted data sample with the sum of S and B. The results of this fit are consistent with those from the fit using signal weights and the small differences are included as systematic uncertainties.
A very small contribution from the decay B 0 s → J/ψK * 0 [32] in the high-mass sideband of the B 0 invariant mass distribution of Fig. 3 has a negligible effect on the fit results. The only significant background that peaks in the B 0 mass region arises from candidates where one or more of the tracks are misreconstructed, in most of the cases the pion track. From simulation studies we find that this corresponds to 3.5% of the signal yield and has a similar B 0 mass distribution to the signal but a significantly different angular distribution. The yield and shape of the background are taken from simulated events, and are used to explicitly model this background in the data fit. The effect on the fit results is taken as a systematic uncertainty. Other background contributions are found to be insignificant. The angular acceptance function is determined from simulated events, and a systematic uncertainty is included to take into account the limited size of the simulated event sample. An observed difference in the kinematic distributions of the final state particles between data and simulation is largely attributed to the S-wave component, which is not included in the simulation. To account for the S-wave, the simulated events are reweighted to match the signal distributions expected from the best estimate of the physics parameters from data (including the S-wave). After this procedure, small differences remain in the pion and kaon momentum distributions. The simulated events are further reweighted to remove these differences, and the change in the fit results is taken as the systematic uncertainty due to the modelling of the acceptance. The C SP factors do not affect the P-wave amplitudes and only have a small effect on the S-wave amplitudes. The fit is performed with each C SP factor set to unity, and the differences in the S-wave parameters are taken as a systematic uncertainty.
This analysis assumes only P-and S-wave contributions to the K + π − system, but makes no assumption about the m(K + π − ) mass model itself (except in the determination of the C SP factors). The S-wave fractions reported in Table 5 correspond to a shape that does not exhibit an approximately linear S-wave (as might be naïvely expected). A separate study of the m(K + π − ) mass spectrum and angular distribution has been performed over a wider m(K + π − ) mass range. This study indicates that there may be contributions from additional resonances, e.g. κ(800), K * (1410), K * 2 (1430) and K * (1680) states. Of particular interest is the K * 2 (1430) contribution, which is a D-wave state and can interfere with the P-wave. Using simulated pseudo experiments such interferences are observed to change the shape of the observed m(K + π − ) spectrum from that corresponding to a simple linear S-wave, and that by ignoring such possible additional resonances the P-and S-wave parameters may be biased. These biases are estimated using simulated experiments containing these additional resonances and they are assigned as systematic uncertainties. The systematic uncertainties are summarised in Table 3.

Results
The values of the P-wave parameters obtained from the fit to the combined B 0 → J/ψK * 0 and B 0 → J/ψK * 0 samples, assuming no direct CP violation, are shown in Table 4 with their statistical and systematic uncertainties. The projections of the decay time and the transversity angles are shown in Fig. 6. Although we have included the decay time distribution in the fit, we do not report a lifetime measurement here, which will instead be included in a forthcoming publication. Figure 7 shows the values for F S and δ S − δ 0 as a function of the K + π − mass. The phase δ 0 = 0 is inserted explicitly to emphasise that this is the phase difference between the S-and P-waves. The error bars include both the statistical and systematic uncertainties. The solid points of Fig. 7(b) correspond to the physical solution with a decreasing phase difference. Table 5 presents the values of F S and δ S − δ 0 for the physical solution. The correlation matrix for the P-and S-wave parameters is shown in Table 6. Integrating the S-wave fraction over all four m(K + π − ) bins gives an average value of F S = (6.4 ± 0.3 ± 1.0)% in the full window of ±70 MeV/c 2 around the known K * 0 mass [26]. The BaBar collaboration [1] measured an S-wave component of (7.3 ± 1.8)% in B 0 → J/ψK + π − in a K + π − mass range from 0.8 to 1.0 GeV/c 2 . 2.94 ± 0.02 ± 0.02 1.53 ± 0.03 ± 0.11 Table 6: Correlation matrix for the four-bin fit.  The results of separate fits to 30 896 ± 95 B 0 → J/ψK * 0 and 30 442 ± 92 B 0 → J/ψK * 0 background subtracted candidates are shown in Table 7, along with the direct CP asymmeties. Only the P-wave amplitudes are allowed to vary in the fit; the S-wave parameters in each m(K + π − ) bin are fixed to the values determined with the combined fit. The fit allows for a difference between the angular acceptance due to charge asymmetries in the detector. The systematic uncertainties are calculated similarly as described in Sec. 6; the uncertainty due to the angular acceptance partially cancels in the direct CP asymmetry calculation. The B 0 and B 0 fit results are consistent within uncertainties, with the largest difference being approximately 2 standard deviations in |A ⊥ | 2 . There is no evidence for BSM contributions to direct CP violation at the current level of precision.
In previous analyses of the B 0 → J/ψK * 0 polarisation amplitudes and phases fits have been performed using a single bin in m(K + π − ) and no S-wave component has been included. To allow comparison with recent results, the fit is repeated in a single m(K + π − ) bin with the S-wave component set to zero. The results are summarised in Table 8 and   are consistent with the previous results, and are more accurate by a factor of 2 to 3. BaBar has also resolved the two-fold ambiguity in the strong phases [30,33] but has not reported S-wave fractions in separate bins.

Conclusion
A full angular analysis of the decay B 0 → J/ψK * 0 has been performed. The polarisation amplitudes and their strong phases are measured using data, corresponding to an integrated luminosity of 1.0 fb −1 , collected in pp collisions at a centre-of-mass energy of 7 TeV with the LHCb detector. The results are consistent with previous measurements and confirm the theoretical predictions mentioned in Sec. 1. The ambiguity in the strong phases is resolved by measuring the relative S-and P-wave phases in bins of the K + π − invariant mass. No significant direct CP asymmetry is observed.