Observation of the decay $\overline{B_s^0} \rightarrow \chi_{c2} K^+ K^- $ in the $\varphi$ mass region

The $\overline{B_s^0} \rightarrow \chi_{c2} K^+ K^- $ decay mode is observed and its branching fraction relative to the corresponding $\chi_{c1}$ decay mode, in a $\pm 15 \textrm{MeV}/c^2$ window around the $\phi$ mass, is found to be $\frac{\mathcal{B}(\overline{B_s^0} \rightarrow \chi_{c2} K^+ K^-) }{ \mathcal{B}(\overline{B_s^0} \rightarrow \chi_{c1} K^+ K^-)} = (17.1 \pm 3.1 \pm 0.4 \pm 0.9)\%,$ where the first uncertainty is statistical, the second systematic and the third due to the knowledge of the branching fractions of radiative $\chi_c$ decays. The decay mode $\overline{B_s^0} \rightarrow \chi_{c1} K^+ K^- $ allows the $ B_s^0$ mass to be measured as $m(B_s^0) = 5366.83 \pm 0.25 \pm 0.27 \, \textrm{MeV}/c^2,$ where the first uncertainty is statistical and the second systematic. A combination of this result with other LHCb determinations of the $B_s^0$ mass is made.

In this paper, the decay B 0 s → χ c2 K + K − with χ c2 → J/ψ γ and J/ψ → µ + µ − is observed using the LHCb data set collected in pp collisions up to the end of 2016. The data corresponds to an integrated luminosity of 3.0 fb −1 collected at centre-of-mass energies of 7 and 8 TeV during 2011 and 2012, together with 1.9 fb −1 collected at a centre-of-mass energy of 13 TeV during 2015 and 2016. This analysis focuses on the low K + K − mass region, where the B 0 s → χ cJ K + K − decay is expected to be dominated by the decay of an intermediate φ meson, as shown in Fig. 1. The same data set allows a measurement of the B 0 s mass with high precision due to the relatively small energy release. These studies build on the previous observation of the B 0 s → χ c1 φ mode [4].

Detector and simulation
The LHCb detector [7,8] 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 [9], 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 [10] placed downstream of the magnet.
The tracking system provides a measurement of the momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. The momentum scale is calibrated using samples of J/ψ → µ + µ − and B + → J/ψ K + decays collected concurrently with the data sample used for this analysis [11,12]. The relative accuracy of this procedure is estimated to be 3 × 10 −4 using samples of other fully reconstructed b-hadron, narrow-Υ , and K 0 S decays. The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ringimaging Cherenkov (RICH) detectors. Photons, electrons and hadrons 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 [13].
The online event selection is performed by a trigger [14], which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, where a full event reconstruction is made. Candidate events are required to pass the hardware trigger, which selects muon and dimuon candidates with high p T based upon muon-system information. The subsequent software trigger is composed of two stages. The first performs a partial event reconstruction and requires events to have two well-identified oppositely charged muons with an invariant mass larger than 2.7 GeV/c 2 . The second stage performs a full event reconstruction. Events are retained for further processing if they contain a J/ψ → µ + µ − candidate. The distance between the decay vertex of the J/ψ and each PV, divided by its uncertainty, is required to be larger than three.
To study the properties of the signal and the most important backgrounds, simulated pp collisions are generated using Pythia [15] with a specific LHCb configuration [16]. Decays of hadronic particles are described by EvtGen [17], in which final-state radiation is generated using Photos [18]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [19] as described in Ref. [20]. Other sources of background, such as those from b → ψ(2S) transitions, where the ψ(2S) decays radiatively to a χ cJ meson, are studied using the RapidSim fast simulation package [21].

Selection
A two-step procedure is used to optimize the selection of B 0 s → χ c1,c2 K + K − candidates. These studies use simulation samples together with the high-mass sideband of the data, 5550 < m(χ c2 K + K − ) < 6150 MeV/c 2 , which is not used for subsequent analysis. In a first step, loose selection criteria are applied to reduce the background significantly whilst retaining high signal efficiency. Subsequently, a multivariate selection is used to reduce further the combinatorial background.
The selection starts from a pair of oppositely charged particles, identified as muons, that form a common decay vertex. Combinatorial background is suppressed by requiring that the χ 2 IP of the muon candidates, defined as the difference between the χ 2 of the PV reconstructed with and without the considered particle, be larger than four for all reconstructed PVs. The invariant mass of the dimuon candidate must be within 50 MeV/c 2 of the known J/ψ mass [22].
Photons are selected from well-identified neutral clusters, reconstructed in the electromagnetic calorimeter [8], that have a transverse energy in excess of 700 MeV/c. Selected J/ψ and photon candidates are combined to form χ c1,c2 candidates. The invariant mass of the combination, obtained from a kinematic fit [23] with a J/ψ mass constraint [22], is required to be within the range 3400-3700 MeV/c 2 .
Pairs of oppositely charged kaons with p T > 200 MeV/c and displaced from all PVs (χ 2 IP > 4) are selected. Good kaon identification is achieved by using information from the RICH detectors. This is combined with kinematic and track quality information using neural networks which provide a response that varies between 0 and 1 for each of the different mass hypotheses: kaon (P K ), pion (P π ), and proton (P p ). The closer to one this value is, the higher the likelihood that the particular mass hypothesis is correct. The chosen requirements on these variabless have an efficiency of (86.8 ± 0.2)% and (86.4 ± 0.2)% for the B 0 s → χ c1 K + K − and B 0 s → χ c2 K + K − modes, respectively. The invariant mass of the selected kaon pair is required to be within 15 MeV/c 2 of the known value of the φ mass [22]. These criteria substantially reduce background from K * (892) 0 decays where a pion is misidentified as a kaon. To reduce background from Λ 0 b decays to excited Λ states a loose proton veto is applied to both kaon candidates.
The χ c1,c2 candidate is combined with the pair of kaons to make a candidate B 0 s meson, which is associated to the PV giving the minimum χ 2 IP . A kinematic fit is performed in which the candidate is constrained to point to this PV and the dimuon mass is constrained to the known value of the J/ψ mass [22]. The reduced χ 2 of this fit is required to be less than five. Combinatorial background is further reduced by requiring the decay time of the B 0 s candidate to be larger than 0.3 ps and its χ 2 IP to be less than 20. Several vetoes are applied to remove background from fully reconstructed b-hadron decay modes. By combining kinematic and particle-identification information it is possible to impose requirements that are almost fully efficient for signal decays. The upper-mass sideband is found to be polluted by fully reconstructed b-hadron decays where a random photon is added. The most important of these is the B 0 s → J/ψ φ decay mode. This is removed by rejecting candidates in which the reconstructed J/ψ K + K − invariant mass, calculated with a J/ψ mass constraint, is within 18 MeV/c 2 (±3σ) of the known B 0 s mass [22]. A similar background is possible from the B 0 → J/ψ K + π − decay mode where the pion is misidentified as a kaon. The candidate is rejected if either of the two possible J/ψ K + π − masses is within 18 MeV/c 2 of the known B 0 mass. These two requirements reject a negligible number of signal decays. Finally, candidates in which either of the kaons is consistent with being a proton (P p > P K ) are rejected if the reconstructed J/ψ pK − mass is within 18 MeV/c 2 of the known Λ 0 b mass. The efficiency of this veto is 99.3% for signal decays. Background from the Λ 0 b → χ c1,c2 pK − decay mode peaks in the signal regions. Therefore, a veto is applied to each kaon candidate in turn. The candidate is rejected if the χ c1,c2 pK − mass is within 10 MeV/c 2 of the Λ 0 b mass (a ±2σ window) and the proton well identified. After these requirements a broad signal for the B 0 s → J/ψ γK + K − decay mode is observed (Fig. 2) above a large combinatorial background. The second step of the selection process is based on a multilayer perceptron (MLP) classifier [24], trained using the simulated signal sample and the high-mass sideband of the data. As input, the classifier uses ten variables, related to the displacement of the candidate from the associated PV and kinematics, that show good agreement between data and simulation. Figure 3 shows the output of the MLP for the training samples and the B 0 s → χ c1 K + K − signal in data where the background is subtracted using the sPlot technique [25]. The MLP gives excellent separation between signal and background and shows good agreement between data and simulation.
The requirement on the MLP output is chosen to maximize the figure of merit /(a/2 + √ N B ) [26], where is the signal efficiency for the χ c2 mode obtained from the simulation, a = 5 is the target signal significance, and N B is the background yield in a ±25 MeV/c 2 window centred on the known B 0 s mass [22] estimated from the sideband. The chosen threshold of 0.85 has an efficiency of (65.

Mass fit
The energy resolution of the LHCb calorimeter results in an invariant-mass resolution for the χ c1 and χ c2 states of about 50 MeV/c 2 . This makes it difficult to separate the two states based on the J/ψ γ invariant mass alone. To improve the mass resolution, the approach used in previous LHCb analyses [4,6] is followed. A kinematic fit is made in which constraints are applied to ensure the pointing of the candidate to the associated primary vertex, on the J/ψ mass and either on the χ c2 or χ c1 mass. Owing to the small radiative branching fraction any contribution from the B 0 s → χ c0 K + K − decay mode can be ignored. As can be seen in Fig. 4 the two components are then separable from the B 0 s invariant mass calculated from this fit. A mass model for the B 0 s → χ c1,c2 K + K − signal is developed using the simulation. This factorizes the observed width of the mass distribution into a component related to the constraints and a component related to the detector resolution.
The effect of applying the χ c2 mass constraint can be seen as follows. 2 To satisfy the constraint, the kinematic fit adjusts the photon momentum, which is the most poorly measured quantity, by a factor, 1 − α, where J/ψ γ and m χ c2 and m J/ψ are the known values of the χ c2 and J/ψ masses [22], respectively. For each event in the simulation the value of α can be calculated using the generated four-momenta. Then the generated four-momentum of the photon is scaled by 1 − α and the four-momentum of the B 0 s meson recalculated. In this way the effect of the constraint is emulated. For genuine B 0 s → χ c2 K + K − decays, applying a χ c2 mass constraint transforms the true B 0 s invariant-mass distribution from a δ-function to a Breit-Wigner distribution whose width is equal to the natural width of the χ c2 state. In the case of genuine B 0 s → χ c1 K + K − decays the distribution is shifted upwards in mass by an amount equal to the mass splitting between the χ c2 and χ c1 states and is broadened. The RMS of the resulting distribution is 9.5 MeV/c 2 , which allows the separation of the χ c1 and χ c2 components.
To obtain the mass models for the χ c1 and χ c2 components, the distributions described above are convolved with a resolution function that accounts for the uncertainty in the measurement of the kaon four-momenta by the tracking system. Using the simulation, the resolution model is found to be well described by a Student's t-distribution which has two resolution parameters: s which describes the core and n which controls the tail of the distribution. As part of the systematic studies, the following alternative resolution models are also considered: Gaussian, sum of two Gaussians, double-sided Crystal Ball [27,28] and Bukin [29] functions. The advantage of factorizing the mass distribution in this way is that it leads to a model where all parameters can be fixed from physics considerations apart from an overall resolution scale factor, s f , that accounts for differences between data and simulation. The simulation is tuned to match the mass resolution seen in data for the B + → J/ψ K + , B 0 → J/ψ K + π − and B 0 s → J/ψ φ decay modes with a precision of 5%. The validity of this tuning for B 0 s → χ c1,c2 K + K − decays is cross-checked using Λ 0 b → χ c1,2 pK − candidates, which have a similar topology, selected using the criteria described in Ref. [6]. Similar agreement between data and simulation is found and consequently in this analysis a Gaussian constraint is applied, s f = 1.00 ± 0.05.
After the selection described in Sec. 3 three sources of background remain and are included in the mass fit. By default, combinatorial background is modelled by a first-order polynomial. Both a power law and an exponential function are considered as systematic variations. Partially reconstructed background from B 0 s → ψ(2S)K + K − decays, with the subsequent decay ψ(2S) → χ cJ γ, is studied using RapidSim and the resulting template is added to the fit. The residual background from B 0 → χ c1,c2 K * (892) 0 and partially reconstructed B 0 → ψ(2S)K * (892) 0 decays is estimated to be 7 ± 2 candidates and is included as a fixed component in the fit with the shape modelled using the simulation.
Extended unbinned maximum likelihood fits are applied separately to the χ c1 K + K − and χ c2 K + K − invariant-mass distributions. The former fit is used to make further crosschecks of the mass resolution and to determine the B 0 s mass. The latter fit is used to determine the yield of the B 0 s → χ c1,c2 K + K − components. The χ c1 K + K − fit has six free parameters: the B 0 s → χ c1 K + K − decay yield, N χ c1 , the B 0 s → χ c2 K + K − decay yield relative to that of the χ c1 mode, f , the B 0 s mass, m(B 0 s ), the yield of the partially reconstructed background, N part , the combinatorial background yield, N comb , and the slope of the combinatorial background. In addition, s f is allowed to vary within the Gaussian constraint of 1.00 ± 0.05. The B 0 s → χ c2 K + K − fit has the same free parameters apart from m(B 0 s ), which is fixed to its known value [22]. The fit procedure is validated using both the full simulation and pseudoexperiments which are fits to simulated distributions generated according to the density functions described above and using the yields from the fit to the data. No significant bias is found and the uncertainties estimated by the fit agree with the results of the pseudoexperiments.
The results of the fits to the data are shown in Fig. 4 and the relevant parameters listed in Table 1. The quality of the fit is judged to be good from the residuals and by a binned χ 2 test. The value of N part is consistent with the expectation based on the relevant branching fractions [22]. The significance of the B 0 s → χ c2 K + K − component, including systematic uncertainties due to the choice of fit model and evaluated using the fit with χ c2 γ mass constraint, is evaluated to be 6.7σ using Wilks' theorem [30]. The values of f determined from the two fits are consistent. That from the χ c2 K + K − fit is more precise since, as can be seen from Fig. 4, the width of the B 0 s → χ c2 K + K − component is narrower than in the B 0 s → χ c1 K + K − case. Hence, this value is used in the determination of the ratio of branching fractions.
5 Determination of the B 0 s → χ c2 K + K − branching fraction The ratio of branching fractions is calculated as using the values given in Ref. [22]. The ratio of reconstruction and selection efficiencies between the two modes, r , is evaluated using the simulation to be (92.0 ± 1.6)% where the uncertainty is statistical. Thus, the ratio of branching fractions is where the uncertainty is statistical.
Since the signal and normalization modes are identical in topology, systematic uncertainties largely cancel in the ratio of branching fractions. The assigned systematic

Source of systematic uncertainty
Relative uncertainty (%) Simulation sample size 1.8 Fit model 3.6 Sum in quadrature of external uncertainties 5.0 uncertainties are listed in Table 2. The limited size of the available simulation samples leads to a relative uncertainty of 1.8%. The uncertainty from the choice of the fit model is evaluated to be 1.5% using the discrete profiling method described in Ref. [31]. Propagating the uncertainty on the yield of the K * (892) 0 background leads to an additional 0.3% uncertainty. The effect of possible differences in the B 0 s kinematics between the data and simulation is studied by weighting the simulation such that p T spectra in data and simulation agree for the B 0 s → χ c1 K + K − decay mode. Based on this study, a 0.4% uncertainty is assigned. Summing in quadrature, the total systematic uncertainty amounts to 2.4%. No systematic uncertainty is included for the admixture of CP -odd and CP -even B 0 s eigenstates in the decays, which is assumed to be the same for both channels [32]. In the extreme case that one decay is only from the short-lifetime eigenstate and the other only from the long-lifetime eigenstate, the ratio would change by 2.8%.
Both decay modes are expected to be dominated by contributions from an intermediate φ resonance that decays to a K + K − pair. Additional S-wave contributions may also be present. To check if this is the case, the resonance structure of the m(K + K − ) invariantmass distribution is studied using the sPlot technique [25], with weights determined from the B 0 s fit described in Sec. 4. To increase the sensitivity to an S-wave contribution, the K + K − mass window 1000-1050 MeV/c 2 is considered. The resulting K + K − invariant-mass distribution is shown in Fig. 5 for the two decay modes.
The observed K + K − invariant-mass distribution is modelled with two components. The first is a relativistic P-wave Breit-Wigner function with Blatt-Weisskopf form factors [33]. The natural width is fixed to the known value of the φ meson [22] and a meson radius parameter of 3 c GeV −1 is used. The detector resolution of 0.9 MeV/c 2 is accounted for by convolving the resonance lineshape with a Gaussian distribution. The second contribution to the K + K − invariant-mass distribution is the S-wave. This is assumed to be nonresonant in nature and is modelled by a phase-space function. The fit model has two free parameters: the φ mass and the nonresonant S-wave fraction, f s . Applying an unbinned maximum likelihood fit of this model to the B 0 s → χ c1 K + K − sample gives f s = (13.9 ± 2.3)%, where the statistical uncertainty is evaluated using pseudoexperiments. This value is consistent at the 2σ-level with that found in the previous LHCb study [4] of this mode, f s = (3.3 ± 5.1)%. This corresponds to an S-wave fraction of (9.2 ± 1.5)% in a ±15 MeV/c 2 window around the φ mass.
The same procedure is used for the B 0 s → χ c2 K + K − sample. In this case the central value of f s returned by the fit is zero, that is at the physical boundary. Pseudoexperiments are used to set a limit f s < 0.30 at 90% confidence level in the 50 MeV/c 2 wide K + K − mass window. This corresponds to an S-wave fraction of less than 21% in a ±15 MeV/c 2 window around the φ mass.
6 Measurement of the B 0 s mass The fit to the χ c1 K + K − invariant-mass distribution in Fig. 4 gives m(B 0 s ) = 5366.83 ± 0.25 MeV/c 2 , where the uncertainty is statistical. The dominant source of systematic uncertainty on the B 0 s mass comes from the knowledge of the momentum scale. This is found to be 0.26 MeV/c 2 by adjusting the momentum scale by the 3 × 10 −4 uncertainty on the calibration procedure and rerunning the mass fit. A further uncertainty arises from the knowledge of the amount of material in the spectrometer. This is known to 10% accuracy [8] and results in a 0.02 MeV/c 2 uncertainty on the B 0 s mass. The uncertainty from the choice of the fit model is evaluated to be 0.01 MeV/c 2 using the discrete profiling method described in Ref. [31]. Finally, uncertainties of 0.08 MeV/c 2 and 0.02 MeV/c 2 arise from the current knowledge [22] of the χ c1 and K + masses, respectively.
These uncertainties are summarized in Table 3. Adding them in quadrature results in a systematic uncertainty of 0.27 MeV/c 2 .

Summary
The B 0 s → χ c2 K + K − decay mode is observed for the first time with a significance of 6.7σ. The branching fraction of this decay relative to that of the B 0 s → χ c1 K + K − mode within a ±15 MeV/c 2 window around the φ mass is measured to be This ratio agrees with the value measured for the corresponding B 0 decay by LHCb [4] B In the ±15 MeV/c 2 window around the φ mass, the nonresonant S-wave fraction for the B 0 s → χ c1 K + K − mode is measured to be (9.2 ± 1.5)% whilst for the B 0 s → χ c2 K + K − mode it is limited to < 21% at 90% confidence level.
The This result is in good agreement with and has similar precision to previous LHCb measurements of the B 0 s mass made using the B 0 s → J/ψ φ [34] and B 0 s → J/ψ φφ [35] decay modes. The LHCb results are combined, taking the statistical uncertainties and those related to the fit procedure to be uncorrelated and those due to the detector material budget and K + mass to be fully correlated. The uncertainty due to the momentum scale in Ref. [35] is also taken to be fully correlated, whereas in Ref. [34] a different procedure was used and so the corresponding uncertainty is considered to be uncorrelated with the other measurements. The result of this combination is This value is in good agreement with the value published by the CDF collaboration, m(B 0 s ) = 5366.01 ± 0.73 (stat) ± 0.33 (syst) MeV/c 2 [36], and is the most precise value to date.