$\chi_{c1}$ and $\chi_{c2}$ resonance parameters with the decays $\chi_{c1,c2}\to J/\psi\mu^+\mu^-$

The decays $\chi_{c1} \rightarrow J/\psi \mu^+ \mu^-$ and $\chi_{c2} \rightarrow J/\psi \mu^+ \mu^-$ are observed and used to study the resonance parameters of the $\chi_{c1}$ and $\chi_{c2}$ mesons. The masses of these states are measured to be m(\chi_{c1}) = 3510.71 \pm 0.04(stat) \pm 0.09(syst)MeV\,, m(\chi_{c2}) = 3556.10 \pm 0.06(stat) \pm 0.11(syst)MeV\,, where the knowledge of the momentum scale for charged particles dominates the systematic uncertainty. The momentum-scale uncertainties largely cancel in the mass difference m(\chi_{c2}) - m(\chi_{c1}) = 45.39 \pm 0.07(stat) \pm 0.03(syst)MeV\,. The natural width of the $\chi_{c2}$ meson is measured to be $$\Gamma(\chi_{c2}) = 2.10 \pm 0.20(stat) \pm 0.02(syst)MeV\,.$$ These results are in good agreement with and have comparable precision to the current world averages.

Offline, J/ψ candidates are combined with a pair of oppositely charged muons to form χ c1,c2 → J/ψ µ + µ − candidates. Several criteria are applied to reduce the background and maximize the sensitivity for the mass measurement. Selected muon candidates are required to be within the range 2 < η < 4.9. Misreconstructed tracks are suppressed by the use of a neural network trained to discriminate between these and real particles. Muon candidates are selected with a neural network trained using simulated samples to discriminate muons from hadrons and electrons. Finally, to improve the mass resolution, a kinematic fit is performed [25]. In this fit the mass of the J/ψ candidate is constrained to the known mass of the J/ψ meson [20] and the position of the χ c1,c2 candidate decay vertex is constrained to be the same as that of the primary vertex. The χ 2 per degree of freedom of this fit is required to be less than four, which substantially reduces the background while retaining almost all the signal events.
In the simulation, pp collisions are generated using Pythia [26] with a specific LHCb configuration [27]. For this study, signal decays are generated using EvtGen [28] with decay amplitudes that depend on the invariant dimuon mass, m(µ + µ − ), using the model described in Ref. [29]. This model assumes that the decay proceeds via the emission of a virtual photon from a pointlike meson and is known to provide a good description of the corresponding dielectron mode [9]. Final-state radiation is accounted for using Photos [30]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [31] as described in Ref. [32].
The signal yields and parameters of the χ c1,c2 resonances are determined with an extended unbinned maximum likelihood fit performed to the J/ψ µ + µ − invariant mass distribution. In this fit, the χ c1 and χ c2 signals are modelled by relativistic Breit-Wigner functions with Blatt-Weisskopf form factors [33] with a meson radius parameter of 3 GeV −1 . Jackson form factors [34] are considered as an alternative to estimate the uncertainty associated with this choice. The orbital angular momentum between the J/ψ meson and the µ + µ − pair is assumed to be 0 (1) for the χ c1 (χ c2 ) cases.
The relativistic Breit-Wigner functions are convolved with the detector resolution. Three resolution models are found to describe the simulated data well: a double-Gaussian function, a double-sided Crystal Ball function [35,36] and a symmetric variant of the Apollonios function [37]. The double-Gaussian function is used by the default model and the other functions are considered to estimate the systematic uncertainty. The parameters of the resolution model are determined by a simultaneous fit to the χ c1 and χ c2 simulated samples. All the parameters apart from the core resolution parameter, σ, are common between the two decay modes. For all the models in the simulation it is found that α ≡ σ χ c2 /σ χ c1 = 1.13 ± 0.01. This is close to the value expected, α = 1.11, from the assumption that the resolution scales with the square root of the energy release.
Combinatorial background is modelled by a second-order polynomial function. The total fit function consists of the sum of the background and the χ c1 and χ c2 signals. The free parameters are the yields of the two signal components, the yield of the background component, the two background shape parameters, the χ c1 and χ c2 masses, σ χ c1 and the natural width of the χ c2 resonance, Γ(χ c2 ). The other resolution parameters are fixed to the simulation values. Since the natural width of the χ c1 state Γ(χ c1 ) = 0.84 ± 0.04 MeV [20] is less than the detector resolution (σ χ c1 = 1.41 ± 0.01 MeV), this study has limited sensitivity to its value. By applying Gaussian constraints on the natural width of the χ c1 state (to the value from Ref. [20]) and α (to the value found in the simulation) the χ c2 width is determined in a data-driven way using the observed resolution for the χ c1 state. propagated to the systematic uncertainty.

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Other uncertainties arise from the fit modeling. To minimize statistical effects these 198 are studied using a large toy sample of events generated using the default fit model. The

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The maximal difference is found to be 8 keV/c 2 , the same for χ c1 and χ c2 states. The fit of this model to the full data sample is shown in Fig. 1 and the resulting parameters of interest are summarized in Table 1. The fitted value of σ χ c1 is 1.51±0.04 MeV, which agrees at the level of 5% with the value found in the simulation. Figure 2 shows the m(µ + µ − ) mass distribution for selected candidates where the background has been subtracted using the sPlot technique [38]. The data agree well with the model described in Ref. [29].
The dominant source of systematic uncertainty on the mass measurements comes from the knowledge of the momentum scale. This is evaluated by adjusting the momentum scale by the 3 × 10 −4 uncertainty on the calibration procedure and rerunning the mass fit. Uncertainties of 88 keV and 102 keV are assigned to the χ c1 and χ c2 mass measurements, respectively. A further uncertainty arises from the knowledge of the correction for energy loss in the spectrometer, which is known with 10 % accuracy [12]. Based on the studies in Table 1: Signal yields and resonance parameters from the nominal fit. No correction for final-state radiation is applied to the mass measurements at this stage.
The distortion of the lineshape due to final-state radiation introduces a bias on the mass. This bias is evaluated using the simulation to be 47 ± 7 keV (29 ± 10 keV) for the χ c1 (χ c2 ) where the uncertainty is statistical. The central values of the mass measurements are corrected accordingly and the uncertainties are propagated.
Other uncertainties arise from the fit modelling and are studied using a simplified simulation. Several variations of the relativistic Breit-Wigner distribution are considered. Using Jackson form factors, modifying the meson radius parameter and varying the orbital angular momentum, the observed χ c1 (χ c2 ) mass changes by at most 15 (24) keV, which is assigned as a systematic uncertainty. Similarly, fitting with a double-sided Crystal Ball or Apollonios model, variations of 7 keV and 2 keV are seen for the χ c1 and χ c2 masses and assigned as systematic uncertainties. Finally, varying the order of the polynomial background function results in a further uncertainty of 2 keV. The uncertainties due to the momentum scale and energy loss correction largely cancel in the mass difference. The assigned systematic uncertainties on the mass measurements are summarized in Table 2.
The main uncertainty on the determination of the natural width of the χ c2 is due to the knowledge of the detector resolution. This is accounted for in the statistical uncertainty since the resolution scale is determined using the χ c1 signal in data. Similarly, the uncertainty on the knowledge of the χ c1 width is propagated via the Gaussian constraint in the mass fit. By running fits with and without the constraint the latter is evaluated to be 40 keV. Further uncertainties of 10 keV and 20 keV arise from the assumed Breit-Wigner parameters and resolution model, respectively. Other systematic uncertainties, e.g. due to the background model, are negligible. The stability of the results is studied by dividing the data into different running periods and also into kinematic bins and repeating the fit. None of these tests shows evidence of a systematic bias.
In summary, the decays χ c1 → J/ψ µ + µ − and χ c2 → J/ψ µ + µ − are observed and the mass of the χ c1 meson together with the mass and natural width of the χ c2 are measured. The results for the mass measurements are m(χ c1 ) = 3510.71 ± 0.04 ± 0.09 MeV, m(χ c2 ) = 3556.10 ± 0.06 ± 0.11 MeV, where the first uncertainty is statistical and the second is systematic. The dominant systematic uncertainty is due to the knowledge of the momentum scale and largely cancels in the mass difference. It can be seen in Table 3 that the measurements are in good agreement with and have comparable precision to the best previous ones, made using pp annihilation at threshold by the E760 [39] and E835 experiments [40] at Fermilab. They are considerably more precise than the best measurement based on the final-state reconstruction [41]. It should be noted that the world average for the χ c1 mass has a scale factor of 1.5 to account for the poor agreement between the results [20]. The result for It has similar precision to and is in good agreement with previous measurements [20]. The observations presented here open up a new avenue for hadron spectroscopy at the LHC. These decay modes can be used to measure the production of χ c1 and χ c2 states with a similar precision to the converted photon study presented in Ref. [6]. Importantly, it will be possible to extend measurements down to very low p T (χ c1,c2 ) probing further QCD predictions [42][43][44]. In addition, measurements of the transition form factors [45] will provide inputs on the interaction between charmonium states and the electromagnetic field. With larger data samples, studies of the Dalitz decays of other heavy-flavour states will become possible. For example, measurement of the transition form factor of the X(3872) via its Dalitz decay may help elucidate the nature of this enigmatic state [9].   [