Study of $\eta-\eta^{\prime}$ mixing from measurement of $B^0_{(s)} \rightarrow J/\psi \eta^{(\prime)}$~decay rates

A study of $B^0$~and $B^0_s$ meson decays into $J/\psi\eta$~and $J/\psi\eta^{\prime}$ final states is performed using a data set of proton-proton collisions at centre-of-mass energies of 7 and 8TeV, collected by the LHCb experiment and corresponding to 3.0fb$^{-1}$ of integrated luminosity. The decay $B^0 \rightarrow J/\psi \eta^{\prime}$ is observed for thefirst time. The following ratios of branching fractions are measured: $$ \frac{\mathcal{B}(B^0 \rightarrow J/\psi \eta^{\prime})}{\mathcal{B}(B^0_s \rightarrow J/\psi \eta^{\prime})} = (2.28\pm0.65\,(stat)\,\pm0.10\,(syst)\,\pm0.13\,(f_{s}/f_{d}))\times10^{-2},$$ $$ \frac{\mathcal{B}(B^0 \rightarrow J/\psi \eta)}{\mathcal{B}(B^0_s \rightarrow J/\psi \eta)} = (1.85\pm0.61\,(stat)\,\pm0.09\,(syst)\,\pm0.11\,(f_{s}/f_{d}))\times10^{-2},$$ where the third uncertainty is related to the present knowledge of $f_{s}/f_{d}$, the ratio between the probabilities for a $b$ quark to form a $B^0_s$ or $B^0$ meson. The branching fraction ratios are used to determine the parameters of $\eta-\eta^{\prime}$ meson mixing. In addition, the first evidence for the decay $B^0_s \rightarrow \psi(2S) \eta^{\prime}$ is reported, and the relative branching fraction is measured, $$\frac{\mathcal{B}(B^0_s \rightarrow \psi(2S) \eta^{\prime})}{\mathcal{B}(B^0_s \rightarrow J/\psi \eta^{\prime})} = (38.7\pm9.0\,(stat)\,\pm1.3\,(syst)\,\pm0.9(\mathcal{B}))\times10^{-2},$$ where the third uncertainty is due to the limited knowledge of the branching fractions of $J/\psi$ and $\psi(2S)$~mesons.


Introduction
Decays of beauty mesons to two-body final states containing a charmonium resonance (J/ψ , ψ(2S), χ c , η c , ...) allow the study of electroweak transitions, of which those sensitive to charge-parity violation are especially interesting. In addition, a study of these decays provides insight into strong interactions at low-energy scales. The hypothesis that η and η mesons contain gluonic and intrinsic cc components has long been used to explain experimental results, including the recent observations of large branching fractions for some decay processes of J/ψ and B mesons into pseudoscalar mesons [1,2].
In this paper, the measurement of the ratios of branching fractions for B 0 (s) → ψη ( ) decays is presented, where ψ represents either the J/ψ or ψ(2S) meson, and charge-conjugate decays are implicitly included. The study uses a sample corresponding to 3.0 fb −1 of pp collision data, collected with the LHCb detector [35] at centre-of-mass energies of 7 TeV in 2011 and 8 TeV in 2012. The results are reported as .
Due to the similar kinematic properties, decay topology and selection requirements applied, many systematic uncertainties cancel in the ratios.

LHCb detector and simulation
The LHCb detector [35] 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 [36], 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 [37] placed downstream of the magnet.
The tracking system provides a measurement of momentum, p, with a relative uncertainty that varies from 0.4% at low momentum to 0.6% at 100 GeV/c. The minimum distance of a track to a primary vertex (PV), the impact parameter, is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors [38]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of a scintillating-pad detector (SPD), preshower detectors (PS), an electromagnetic calorimeter and a hadronic calorimeter.
Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [39]. This analysis uses events collected by triggers that select the µ + µ − pair from the ψ decay with high efficiency. At the hardware stage a muon with p T > 1.5 GeV/c or a pair of muons is required to trigger the event. For dimuon candidates, the product of the p T of muon candidates is required to satisfy √ p T 1 p T 2 > 1.3 GeV/c and √ p T 1 p T 2 > 1.6 GeV/c for data collected at √ s = 7 and 8 TeV, respectively. At the subsequent software trigger stage, two muons are selected with a mass in excess of 2.97 GeV/c 2 and consistent with originating from a common vertex. The common vertex is required to be significantly displaced from the pp collision vertices.
In the simulation, pp collisions are generated using Pythia [40] with a specific LHCb configuration [41]. Decays of hadronic particles are described by EvtGen [42], in which final-state radiation is generated using Photos [43]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [44] as described in Ref. [45].

Event selection
Signal decays are reconstructed using the ψ → µ + µ − decay. For the B 0 (s) → ψη channels, η candidates are reconstructed using the η → ρ 0 γ and η → ηπ + π − decays, followed by ρ 0 → π + π − and η → γγ decays. For the B 0 (s) → J/ψ η channels, η candidates are reconstructed using the η → π + π − π 0 decay, followed by the π 0 → γγ decays. The η → γγ decay, which has a larger branching fraction and reconstruction efficiency, is not used for the reconstruction of B 0 (s) → J/ψ η candidates due to a worse mass resolution, which does not allow to resolve the B 0 s and B 0 peaks [16,46]. The selection criteria, which follow Refs. [16,46], are common to all decay channels, except for the requirements directly related to the photon kinematic properties.
The muons and pions must be positively identified using the combined information from RICH, calorimeter, and muon detectors [47,48]. Pairs of oppositely charged particles, identified as muons, each having p T > 550 MeV/c and originating from a common vertex, are combined to form ψ → µ + µ − candidates. The resulting dimuon candidate is required to form a good-quality vertex and to have mass between −5σ and +3σ around the known J/ψ or ψ(2S) masses, where the mass resolution σ is around 13 MeV/c 2 . The asymmetric mass intervals include the low-mass tail due to final-state radiation.
The charged pions are required to have p T > 250 MeV/c and to be inconsistent with being produced in any primary vertex. Photons are selected from neutral energy clusters in the electromagnetic calorimeter, i.e. clusters that do not match the geometrical extrapolation of any track [48]. The photon quality criteria are further refined by exploiting information from the PS and SPD detectors. The photon candidate's transverse momentum inferred from the energy deposit is required to be greater than 500 MeV/c for η → ρ 0 γ and η → γγ candidates, and 250 MeV/c for π 0 → γγ candidates. In order to suppress the large combinatorial background from π 0 → γγ decays, photons that, when combined with another photon in the event, form a π 0 → γγ candidate with mass within 25 MeV/c 2 of the π 0 mass (corresponding to about ±3σ around the known mass) are not used in the reconstruction of η → ρ 0 γ candidates. The π + π − mass for the η → ρ 0 γ channel is required to be between 570 and 920 MeV/c 2 . Finally, the masses of π 0 , η and η candidates are required to be within Finally, the measured proper decay time of the B 0 (s) candidate, calculated with respect to the associated primary vertex, is required to be between 0.1 mm/c and 2.0 mm/c . The upper limit is used to remove poorly reconstructed candidates.
4 Study of B 0 (s) → J/ψ η and B 0 (s) → J/ψ η decays with η → ηπ + π − and η → π + π − π 0 The mass distributions of the selected B 0 (s) → J/ψ η and B 0 (s) → J/ψ η candidates are shown in Fig. 2, where the η and η states are reconstructed in the ηπ + π − and π 0 π + π − decay modes, respectively. The B 0 (s) → J/ψ η ( ) signal yields are estimated by unbinned extended maximum-likelihood fits. The B 0 s and B 0 signals are modelled by a modified Gaussian function with power-law tails on both sides [51], referred to as "F function" throughout the paper. The mass resolutions of the B 0 s and B 0 peaks are the same; the difference of the peak positions is fixed to the known difference between the B 0 s and the B 0 meson masses [49] and the tail parameters are fixed to simulation predictions. The background contribution is modelled by an exponential function. The fit results are presented in Table 2. The decays η → ηπ + π − and η → π + π − π 0 are used in the reconstruction of J/ψ η and J/ψ η candidates, respectively. The total fit function (solid blue) and the combinatorial background contribution (dashed black) are shown. The long-dashed red line represents the signal B 0 s contribution and the yellow shaded area shows the B 0 contribution. ), B 0 s signal peak position (m 0 ) and mass resolution (σ) in B 0 (s) → J/ψ η and B 0 (s) → J/ψ η decays, followed by η → ηπ + π − and η → π + π − π 0 decays, respectively. The quoted uncertainties are statistical only. Mode To verify that the signal originates from B 0 (s) → J/ψ η ( ) decays, the sPlot technique is used to disentangle signal and the background components [52]. Using the µ + µ − π + π − γγ mass distribution as the discriminating variable, the distributions of the masses of the intermediate resonances are obtained. For each resonance in turn the mass window is released and the mass constraint is removed, keeping other selection criteria as in the baseline analysis. The background-subtracted mass distributions for η → ηπ + π − , η → γγ and J/ψ → µ + µ − combinations from B 0 (s) → J/ψ η signal candidates are shown in Fig. 3 and the mass distributions for η → π + π − π 0 , π 0 → γγ and J/ψ → µ + µ − from B 0    Table 2, the mass resolutions are consistent with expectations from simulation, and peak positions agree with the known meson masses [49]. The sizes of the non-resonant components are consistent with zero for all cases, supporting the hypothesis of a fully resonant structure for the decays B 0 (s) → J/ψ η ( ) .
5 Study of B 0 (s) → ψη decays with η → ρ 0 γ The mass distributions of the selected ψη candidates, where the η state is reconstructed using the η → ρ 0 γ decay, are shown in Fig. 5. The B 0 (s) → ψη signal yields are estimated by unbinned extended maximum-likelihood fits, using the model described in Sect. 4. Studies of the simulation indicate the presence of an additional background due to feeddown from the decay B 0 → ψK * 0 , followed by the K * 0 → K + π − decay. The charged kaon is misidentified as a pion and combined with another charged pion and a random photon to form an η candidate. This background contribution is modelled in the fit using a probability density function obtained from simulation. The fit results are summarised in Table 3. For both final states, the positions of the signal peaks are consistent with the known B 0 s mass [49] and the mass resolutions agrees with those of the simulation. The statistical significances of the B 0 s → ψ(2S)η and B 0 → J/ψ η signals are determined by a simplified simulation study, as described in Sect. 4. The significances are found to be 4.3σ and 3.5σ for B 0 s → ψ(2S)η and B 0 → J/ψ η , respectively. By combining the latter ), B 0 s signal peak position (m 0 ) and mass resolution (σ) in B 0 (s) → ψη decays, followed by the η → ρ 0 γ decay. The quoted uncertainties are statistical only. Mode

Efficiencies and systematic uncertainties
The ratios of branching fractions are measured using the formulae where N represents the observed yield, ε is the total efficiency and f s /f d is the ratio between the probabilities for a b quark to form a B 0 s and a B 0 meson. Equal values of f s /f d = 0.259 ± 0.015 [53][54][55][56] at centre-of-mass energies of 7 TeV and 8 TeV are assumed. The branching fractions for η, η and π 0 decays are taken from Ref. [49]. For the ratio of the J/ψ → µ + µ − and ψ(2S) → µ + µ − branching fractions, the ratio of dielectron branching fractions, 7.57 ± 0.17 [49], is used.
The total efficiency is the product of the geometric acceptance, and the detection, reconstruction, selection and trigger efficiencies. The ratios of efficiencies are determined using simulation. For R (s) , the efficiency ratios are further corrected for the small energydependent difference in photon reconstruction efficiency between data and simulation. The photon reconstruction efficiency has been studied using a large sample of B + → J/ψ K * + decays, followed by K * + → K + π 0 and π 0 → γγ decays [16,46,57,58]. The correction for the ratios ε B 0 (s) →J/ψ η /ε B 0 (s) →J/ψ η is estimated to be (94.9 ± 2.0)%. For the R η ( ) and R ψ(2S) cases no such corrections are required because photon kinematic properties are similar. The ratios of efficiencies are presented in Table 4. The ratio of efficiencies for the ratio R ψ(2S) exceeds the others due to the p T (η ) > 2.5 GeV/c requirement and the difference in p T (η ) spectra between the two channels.
Since the decay products in each of the pairs of channels involved in the ratios have similar kinematic properties, most uncertainties cancel in the ratios, in particular those related to the muon and ψ reconstruction and identification. The remaining systematic uncertainties, except for the one related to the photon reconstruction, are summarised in Table 5 and discussed below.
Systematic uncertainties related to the fit model are estimated using alternative models for the description of the mass distributions. The tested alternatives are first-or second- 1.352 ± 0.016 degree polynomial functions for the background description, a model with floating mass difference between B 0 and B 0 s peaks, and a model with Student's t-distributions for the signal shapes. For the B 0 (s) → J/ψ η followed by η → ηπ + π − decays, and B 0 (s) → J/ψ η decays, an additional model with signal widths fixed to those obtained in simulation is tested. For each alternative fit model, the ratio of event yields is calculated and the systematic uncertainty is determined as the maximum deviation from the ratio obtained with the baseline model. The resulting uncertainties range between 0.8% and 2.9%.
Another important source of systematic uncertainty arises from the potential disagreement between data and simulation in the estimation of efficiencies, apart from those related to π 0 and γ reconstruction. This source is studied by varying the selection criteria, listed in Sect. 3, in ranges that lead to as much as 20% change in the measured signal yields. The agreement is estimated by comparing the efficiency-corrected yields within these variations. The largest deviations range between 2.9% and 3.7% and these values are taken as systematic uncertainties.
To estimate a possible systematic uncertainty related to the knowledge of the B 0 s production properties, the ratio of efficiencies determined without correcting the B 0 s transverse momentum and rapidity spectra is compared to the default ratio of efficiencies determined after the corrections. The resulting relative difference is less than 0.2% and is therefore neglected. The trigger is highly efficient in selecting B 0 (s) meson decays with two muons in the final state. For this analysis the dimuon pair is required to be compatible with triggering the event. The trigger efficiency for events with ψ → µ + µ − produced in beauty hadron decays is studied in data. A systematic uncertainty of 1.1% is assigned based on the comparison of the ratio of trigger efficiencies for samples of B + → J/ψ K + and B + → ψ(2S)K + decays in data and simulation [59]. The final systematic uncertainty originates from the dependence of the geometric acceptance on the beam crossing angle and the position of the luminosity region. The observed channel-dependent 0.8% − 1.5% differences are taken as systematic uncertainties. The effect of the exclusion of photons that potentially originate from π 0 → γγ candidates is studied by comparing the efficiencies between data and simulation. The difference is found to be negligible. The total uncertainties in Table 5 are obtained by adding the individual independent uncertainties in quadrature.

Results and conclusions
The ratios of branching fractions involving B 0 (s) → J/ψ η ( ) decays, R η ( ) and R (s) , are determined using Eqs. (5) and (6) with the results from Sects. 4, 5 and 6, where the third uncertainty is associated with the uncertainty of f s /f d for the ratios R η ( ) and the uncertainties of the branching fractions for η ( ) decays for the ratios R (s) . The R s determination is in good agreement with previous results [14,16] and has better precision. The ratios R η and R η allow a determination of the mixing angle ϕ P using the expressions where θ C is the Cabibbo angle. These relations are similar to those discussed in Ref. [4].
In comparison with Eq. (2) these expressions are not sensitive to gluonic contributions and have significantly reduced theory uncertainties related to the B (s) → J/ψ form-factors. The values for the mixing angle ϕ P determined from the ratios R η and R η are 43.8 +3.9 The measured ratios R and R s , together with Eqs. (2) and (3), give tan 4 ϕ P = 1.26 ± 0.55, cos 4 ϕ G = 1.58 ± 0.70.
The contours of the two-dimensional likelihood function L (ϕ P , |ϕ G |), constructed from Eqs. (2) and (3) are presented in Fig. 7. The estimates for each angle are obtained by treating the other angle as a nuisance parameter and profiling the likelihood with respect to it, ϕ P | R (s) = (43.5 +1.4 −2.8 ) • , ϕ G | R (s) = (0 ± 24.6) • , where the uncertainties correspond to ∆ ln L = 1/2 for the profile likelihood. This result does not support a large gluonic contribution in the η meson. Neglecting the gluonic component, the angle ϕ P is determined using Eq. (2) separately from the ratios R and R s to be (49.9 +6.1 −11.5 ) • and (43.4 +1.4 −1.3 ) • , respectively. The combination yields 3 ) • , which is consistent with the result from R η ( ) . The measured η-η mixing parameters are in agreement with earlier measurements and have comparable precisions.
The reported branching-fraction ratios correspond to the decay-time-integrated rates, while theory predictions usually refer to the branching fractions at the decay time t = 0. Due to a sizeable decay width difference in the B 0 s system [66], the difference can be as large as 10% for B 0 s → ψη ( ) decays, depending on the decay dynamics [67]. The corresponding change in the angle ϕ P can be up to 3 • .
In summary, a study of B 0 and B 0 s meson decays into J/ψ η and J/ψ η final states is performed in a data set of proton-proton collisions at centre-of-mass energies of 7 and 8 TeV, collected by the LHCb experiment and corresponding to 3.0 fb −1 of integrated luminosity. All four B 0 (s) → J/ψ η ( ) decay rates are measured in a single experiment for the first time. The first observation of the decay B 0 → J/ψ η and the first evidence for the decay B 0 s → ψ(2S)η are reported. All these results are among the most precise available from a single experiment and contribute to understanding the role of the strong interactions in the internal composition of mesons. [49] Particle Data Group, K. A. Olive et al., Review of particle physics, Chin. Phys. C38 (2014) 090001. [