Determination of the number of $J/\psi$ events with inclusive $J/\psi$ decays

A measurement of the number of $J/\psi$ events collected with the BESIII detector in 2009 and 2012 is performed using inclusive decays of the $J/\psi$ . The number of $J/\psi$ events taken in 2009 is recalculated to be $(223.7\pm1.4)\times 10^6$, which is in good agreement with the previous measurement, but with significantly improved precision due to improvements in the BESIII software. The number of $J/\psi$ events taken in 2012 is determined to be $(1086.9\pm 6.0)\times 10^6$. In total, the number of $J/\psi$ events collected with the BESIII detector is measured to be $(1310.6\pm 7.0)\times 10^6$, where the uncertainty is dominated by systematic effects and the statistical uncertainty is negligible.


Introduction
Studies of J/ψ decays have provided a wealth of information since the discovery of the J/ψ in 1974 [1] [2]. Decays of the J/ψ offer a clean laboratory for light hadron spectroscopy, provide an insight into decay mechanisms and help in distinguishing between conventional hadronic states and exotic states.
A lot of important progress in light hadron spectroscopy has been achieved based on a sample of (225.3±2.8)×10 6 J/ψ events collected by the BESIII experiment [3] in 2009. To further comprehensively study the J/ψ decay mechanism, investigate the light hadron spectrum, and search for exotic states, e.g. glueballs, hybrids and multi-quark states, an additional, larger J/ψ sample was collected in 2012. A precise determination of the number of J/ψ events is essential for analyses based on these data samples. With improvements in the BESIII software, particularly in Monte Carlo (MC) simulations and the reconstruction of tracks in the main drift chamber (MDC), it is possible to perform a more precise measurement of the number of J/ψ events taken in 2009 and 2012. The relevant data samples used in this analysis are listed in Table 1.
We implement the same method as that used in the previous study [4] to determine the number of J/ψ events. The advantage of this approach is that the detection efficiency of inclusive J/ψ decays can be extracted directly from the data sample taken at the peak of the ψ(3686). This is useful because the correction factor of the detection efficiency is less dependent on the MC model for the inclusive J/ψ decay and therefore the systematic uncertainty can be reduced significantly. The number of J/ψ events, N J/ψ , is calculated as where N sel is the number of inclusive J/ψ events selected from the J/ψ data; N bg is the number of background events estimated with continuum data taken at √ s = 3.08 GeV; ǫ trig is the trigger efficiency; ǫ ψ(3686) data is the inclusive J/ψ detection efficiency determined experimentally using the J/ψ sample from the reaction ψ(3686) → π + π − J/ψ. f cor is a correction factor that accounts for the difference in the detection efficiency between the J/ψ events produced at rest and those produced in ψ(3686) → π + π − J/ψ. f cor is expected to be unity approximately, and is determined by the MC simulation sample with where ǫ J/ψ MC is the detection efficiency of inclusive J/ψ events determined from the MC sample of J/ψ events produced directly in the electron-positron collision, and ǫ ψ(3686) MC is that from the MC sample of ψ(3686) → π + π − J/ψ (J/ψ → inclusive) events. In MC simulation, the J/ψ and ψ(3686) resonances are simulated with KKMC [5]. The known decay modes of the J/ψ and ψ(3686) are generated by EVTGEN [6,7] with branching fractions taken from the Review of Particle Physics [8], while the remaining decays are generated according to the LUNDCHARM model [9,10]. All of the MC events are fed into a GEANT4-based [11] simulation package, which takes into account the detector geometry and response. 2 Inclusive J/ψ selection criteria To distinguish the inclusive J/ψ decays from Quantum Electro-Dynamics (QED) processes (i.e. Bhabha and dimuon events) and background events from cosmic rays and beam-gas interactions, a series of selection criteria are applied to the candidate events. The charged tracks are required to be detected in the MDC within a polar angle range of | cos θ| < 0.93, and to have a momentum of p < 2.0 GeV/c. Each track is required to originate from the interaction region by restricting the distance of closest approach to the run-dependent interaction point in the radial direction, V r < 1 cm, and in the beam direction, |V z | < 15 cm. For photon clusters in the electromagnetic calorimeter (EMC), the deposited energy is required be greater than 25 (50) MeV for the barrel (endcap) region of | cos θ| < 0.83 (0.86 < | cosθ| < 0.93). In addition, the EMC cluster timing T must satisfy 0 < T ≤ 700 ns, which is used to suppress electronics noise and energy deposits unrelated to the event.
The candidate event must contain at least two charged tracks. The visible energy E vis , defined as the sum of charged particle energies computed from the track momenta by assuming a pion mass and from the neutral shower energies deposited in the EMC, is required to be greater than 1.0 GeV. A comparison of the E vis distribution between the J/ψ data, the data taken at √ s = 3.08 GeV, and the inclusive J/ψ MC sample is illustrated in Fig. 1. The requirement E vis > 1.0 GeV removes one third of the background events while retaining 99.4% of the signal events. Since Bhabha (e + e − → e + e − ) and dimuon (e + e − → µ + µ − ) events are two-body decays, each charged track carries a unique energy, close to half of the center-of-mass energy. Therefore, for events with only two charged tracks, we require that the momentum of each charged track is less than 1.5 GeV/c in order to remove Bhabha and dimuon events. This requirement is depicted by the solid lines in the scatter plot of the momenta of the two charged tracks (Fig. 2). The Bhabha events are characterized by a significant peak around 1.5 GeV in the distribution of energy deposited in the EMC, shown in Fig. 3. Hence an additional requirement that the energy deposited in the EMC for each charged track is less than 1 GeV is applied to further reject the Bhabha events. After the above requirements, N sel = (854.60 ± 0.03)×10 6 candidate events are selected from the J/ψ data taken in 2012. The distributions of the track parameters of closest approach in the beam line and radial directions (V z and V r ), the polar angle (cos θ), and the total energy deposited in the EMC (E EMC ) after subtracting background events estimated with the continuum data taken at √ s = 3.08 GeV (see Sec. 3 for details) are shown in Fig. 4. Reasonable agreement between the data and MC samples is observed. The multiplicity of charged tracks (N good ) is shown in Fig. 5, where the MC sample generated according to the LUNDCHARM model agrees very well with the data while the MC sample generated without the LUNDCHARM model deviates from the data. The effect of this discrepancy on the determination of the number of J/ψ events is small, as described in Sec. 6.

Background analysis
In this analysis, the data samples taken at √ s = 3.08 GeV and in close chronological order to the J/ψ sample are used to estimate the background due to QED processes, beam-gas interactions and cosmic rays. To normalize the selected background events to the J/ψ data, the integrated luminosity for the data samples taken at the J/ψ peak and at √ s = 3.08 GeV are determined using the precess e + e − → γγ, respectively.
To determine the integrated luminosity, the candidate events e + e − → γγ are selected by requiring at least two showers in the EMC. It is further required that the energy of the second most energetic shower is between 1.2 and 1.6 GeV and that the polar angles of the two showers are in the range | cos θ| < 0.8. The number of signal events is determined from the number of events in the signal region |∆φ| < 2.5 • and the background is estimated from those in the sideband region 2.5 < |∆φ| < 5 • , where ∆φ = |φ γ1 −φ γ2 |−180 • and φ γ1/2 is the azimuthal angle of the photon. Taking into account the detector efficiency obtained from the MC simulation and the cross section of the QED process e + e − → γγ, the integrated luminosities of the J/ψ data sample and the sample taken at √ s = 3.08 GeV taken in 2012 are determined to be 315.02±0.14 pb −1 and 30.84±0.04 pb −1 , respectively, where the errors are statistical only.
After applying the same selection criteria as for the J/ψ data, N 3.08 = 1, 440, 376 ± 1, 200 events are selected from the continuum data taken at √ s = 3.08 GeV. Assuming the same detection efficiency at √ s = 3.08 GeV as for the J/ψ peak and taking into account the energy-dependent cross section of the QED processes, the number of background events for the J/ψ sample, N bg , is estimated to be where L J/ψ and L 3.08 are the integrated luminosities for the J/ψ data sample and the data sample taken at √ s = 3.08 GeV, respectively, and s J/ψ and s 3.08 are the corresponding squares of the center-of-mass energies. The background is calculated to be 1.7% of the selected inclusive J/ψ events taken in 2012.
According to the studies of the MC sample and the V z distribution, the QED background fraction is found to be about 1.5% of the total data. It is known that the beam status for the data taken in 2009 was worse and the background is much higher than for the 2012 sample. With the same method, the total background (including the QED contribution) for the 2009 sample is estimated to be 3.7%.

Determination of the detection efficiency and correction factor
In the previous study, the detection efficiency was determined using a MC simulation of the reaction J/ψ → inclusive, assuming that both the physics process of the inclusive J/ψ decay and the detector response were simulated well. In this analysis, to reduce the uncertainty related to the discrepancy between the MC simulation and the data, the detection efficiency is determined experimentally using a sample of J/ψ events from the reaction ψ(3686) → π + π − J/ψ. To ensure that the beam conditions and detector status are similar to those of the sample collected at the J/ψ peak, a dedicated ψ(3686) sample taken on May 26, 2012 is used for this study.
To select ψ(3686) → π + π − J/ψ events, there must be at least two soft pions with opposite charge in the MDC within the polar angle range | cos θ| < 0.93, having V r < 1 cm and |V z | < 15 cm, and momenta less than 0.4 GeV/c. No further selection criteria on the remaining charged tracks or showers are required. The distribution of the invariant mass recoiling against all possible soft π + π − pairs is shown in Fig. 6 (a). A prominent peak around 3.1 GeV/c 2 , corresponding to the decay of ψ(3686) → π + π − J/ψ, J/ψ → inclusive, is observed over a smooth background. The total number of inclusive J/ψ events, N inc = (1147.8 ± 1.9) × 10 3 , is obtained by fitting a double-Gaussian function for the J/ψ signal plus a second order Chebychev polynomial for the background to the π + π − recoil mass spectrum.
To measure the detection efficiency of inclusive J/ψ events, the same selection criteria as described in Sec. 2 are applied to the remaining charged tracks and showers at the event level. The distribution of the invariant mass recoiling against π + π − for the remaining events is shown in Fig. 6 (b); it is fitted with the same function described above. The number of selected inclusive J/ψ events, N sel inc , is determined to be (877.6±1.7)×10 3 . The detection efficiency of inclusive J/ψ events, ǫ ψ(3686) data = (76.46±0.07)%, is calculated by the ratio of the number of inclusive J/ψ events with and without the inclusive J/ψ event selection criteria applied.
Since the J/ψ particle in the decay ψ(3686) → π + π − J/ψ is not at rest, a correction factor, defined in Eq. (2), is used to take into account the kinematical effect on the detection efficiency of the inclusive J/ψ event selection. Two large statistics, inclusive ψ(3686) and J/ψ MC samples are produced and are subjected to the same selection criteria as the data samples. The detection efficiency of inclusive J/ψ events are determined to be ǫ ψ(3686) MC = (75.76±0.06)%, and ǫ J/ψ MC = (76.58 ± 0.04)% for the two inclusive MC samples, respectively. The correction factor f cor for the detection efficiency is therefore taken as  Table 2. The trigger efficiency of the BESIII detector is 100%, based on the study of various reactions [12].
With the same procedure, the number of J/ψ events taken in 2009 is determined to be (223.72±0.01)×10 6 . Here, the statistical uncertainty is from the number of J/ψ events only, while the statistical fluctuation of N bg is taken into account as part of the systematic uncertainty (see Sec. 6.4). The systematic uncertainties from different sources are discussed in detail in Sec. 6.

Systematic uncertainty
The sources of systematic uncertainty and their corresponding contributions are summarized in Table 3, and are discussed in detail below.

MC model uncertainty
In the measurement of the number of J/ψ events, only the efficiency correction factor, f cor , is dependent on the MC simulation. To evaluate the uncertainty due to the MC model, we generate a set of MC samples without the LUNDCHARM model and compare the correction factor determined using these samples to its nominal value. According to the distributions of the charged track multiplicity shown in Fig. 5, the MC simulation without the LUNDCHARM model poorly describes the data, which means this method will overestimate the systematic uncertainty. The studies show that the correction factor has a slight dependence on the MC mode of inclusive J/ψ decays. To be conservative, the change in the correction factor, 0.42% (0.36%), is taken as the systematic uncertainty due to the MC model on the number of J/ψ events taken in 2012 (2009).

Track reconstruction efficiency
According to studies of the track reconstruction efficiency, the difference in track reconstruction efficiencies between the MC and data samples of J/ψ decays is less than 1% for each charged track.
In the analysis, the ψ(3686) data sample used to determine the detection efficiency is taken in close chronological order to the J/ψ sample. The consistency of track reconstruction efficiency between the MC and data samples in ψ(3686) decays is assumed to be exactly the same as that in J/ψ decays. Therefore the track reconstruction efficiencies in both J/ψ and ψ(3686) MC samples are varied by −1% to evaluate the uncertainty due to the MDC tracking. As expected, the change in the correction factor is very small, 0.03%, and this is taken as a systematic uncertainty.
In the determination of the number of J/ψ events taken in 2009, the J/ψ and ψ(3686) data samples were collected at different times, which may lead to slight differences in the tracking efficiency between the two data sets due to the imperfect description of detector performance and response in the MC simulation. To estimate the corresponding systematic uncertainty, we adjust the track reconstruction efficiency by −0.5% in the J/ψ MC sample, keeping it unchanged for the ψ(3686) MC sample. The resulting change in the correction factor, 0.30%, is taken as a systematic uncertainty on the number of J/ψ events in 2009.

Fit to the J/ψ peak
In this measurement, the selection efficiency of inclusive J/ψ events is estimated experimentally with the ψ(3686) data sample (ψ(3686) → π + π − J/ψ), and the yield of J/ψ events used in the efficiency calculation is determined by a fit to the invariant mass spectra recoiling against π + π − . The following systematic uncertainties of the fit are considered: (a) the fit : we propagate the statistical uncertainties of the J/ψ signal yield from the fit to the selection efficiency, and the resulting uncertainties, 0.09% and 0.08% for ǫ ψ(3686) data and ǫ ψ(3686) M C respectively, are considered to be the uncertainty from the fit itself. (b) the fit range: we change the fit range on the π + π − recoiling mass from [3.07, 3.13] GeV/c 2 to [3.08, 3.12] GeV/c 2 , and the resulting difference, 0.08% is taken as a systematic uncertainty. (c) the signal shape: we perform an alternative fit by describing the J/ψ signal with a histogram obtained from the recoil mass spectrum of π + π − in ψ(3686) → π + π − J/ψ, J/ψ → µ + µ − , and the resulting change, 0.12%, is considered to be the associated systematic uncertainty. (d) the background shape: the uncertainty due to the background shape, 0.02%, is estimated by replacing the second order Chebychev polynomial with a first order Chebychev polynomial. By assuming all of the sources of systematic uncertainty are independent, the fit uncertainty for the 2012 J/ψ sample, 0.19%, is obtained by adding all of the above effects in quadrature.
The same sources of systematic uncertainty are considered for the J/ψ sample taken in 2009. The fit has an uncertainty of 0.02% for ǫ ψ(3686) data and 0.07% for ǫ ψ(3686)

MC
. The uncertainty from the fit range, signal function and background shape are 0.02%, 0.15% and 0.02%, respectively. The total uncertainty from the fit for the 2009 data is 0.17%.

Background uncertainty
In the measurement of the number of J/ψ events, the background events from QED processes, cosmic rays, and beam-gas interactions are estimated by normalizing the number of events in the continuum data sample taken at √ s = 3.08 GeV according to Eq. (3). Therefore the background uncertainty mainly comes from the normalization method, the statistics of the sample taken at √ s = 3.08 GeV, the statistical uncertainty of the integrated luminosity and the uncertainty due to beam associated backgrounds.
In practice, Eq. (3) is improper for the normalization of the background of cosmic rays and beamgas. The number of cosmic rays is proportional to the time of data taking, while beam-gas interaction backgrounds are related to the vacuum status and beam current during data taking in addition to the time of data taking. Assuming a stable beam and vacuum status, the backgrounds of cosmic rays and beam-gas interactions are proportional to the integrated luminosity. Therefore, the difference in the estimated number of backgrounds between that with and without the energy-dependent factor in Eq. (3) is considered to be the associated systematic uncertainty.
In 2012, two data samples with √ s = 3.08 GeV were taken at the beginning and end of the J/ψ data taking. To estimate the uncertainty of the background related with the stability of the beam and vacuum status, we estimated the background with Eq. (3) for the two continuum data samples, individually. The maximum change in the nominal results, 0.05%, is taken as the associated systematic uncertainty. In the background estimation for data taken in 2009, only one continuum data sample was taken. The corresponding uncertainty is estimated by comparing the selected background events from the continuum sample to that from the J/ψ data, which is described in detail in [4].
After considering the above effects, the uncertainties on the number of J/ψ events related to the back-ground are 0.06% and 0.13% for the data taken in 2012 and 2009, respectively. The uncertainties are determined from the quadratic sum of the above individual uncertainties, assuming all of them to be independent.

Noise mixing
In the BESIII simulation software, the detector noise and machine background are included in the MC simulation by mixing the simulated events with events recorded by a random trigger. To determine the systematic uncertainty associated with the noise realization in the MC simulation, the ψ(3686) MC sample is reconstructed by mixing the noise sample accompanying the J/ψ data taking. The change of the correction factor for the detection efficiency, 0.09%, is taken as the systematic uncertainty due to noise mixing for the number of J/ψ events taken in 2012.
In the determination of the number of J/ψ events collected in 2009, 106 million of ψ(3686) events taken in 2009 are used to determine the detection efficiency, and the corresponding uncertainty related to the noise realization is estimated to be 0.10% with the same method. However, the noise level was not entirely stable during the time of the ψ(3686) data taking. To check the effect on the detection efficiency related to the different noise levels, the ψ(3686) data and the MC samples are divided into three subsamples, and the detection efficiency and the correction factor are determined for the three sub-samples individually. The resulting maximum change in the number of J/ψ events, 0.06%, is taken as an additional systematic uncertainty associated with the noise realization. The total systematic uncertainty due to the noise is estimated to be 0.12% for the J/ψ events taken in 2009.

Selection efficiency uncertainty of two soft pions
According to the MC study, the selection efficiency of soft pions, ǫ π + π − , recoiling against the J/ψ in ψ(3686) → π + π − J/ψ is found to depend on the multiplicity of the J/ψ decays. Differences between the data and MC samples may lead to a change in the number of J/ψ events. We compare the multiplicity distribution of J/ψ decays in the ψ(3686) → π + π − J/ψ data sample to that of the J/ψ data at rest to obtain the dependence of ǫ π + π − in the data. The efficiency determined from the ψ(3686) → π + π − J/ψ(J/ψ → inclusive) MC sample, ǫ ψ(3686) MC in Eq. (2), is reweighted with the dependence of ǫ π + π − from the data sample. The resulting change in the number of J/ψ events, 0.28% (0.34%) is taken as the uncertainty for the data taken in 2012 (2009).
The systematic uncertainties from the different sources studied above are summarized in Table 3. The total systematic uncertainty for the number of J/ψ in 2012 (2009), 0.55% (0.63%), is the quadratic sum of the individual uncertainties.

Summary
Using inclusive J/ψ events, the number of J/ψ events collected with the BESIII detector in 2012 is determined to be N J/ψ2012 = (1086.9 ± 6.0) × 10 6 , where the uncertainty is systematic only and the statistical uncertainty is negligible. The number of J/ψ events taken in 2009 is recalculated to be N J/ψ2009 = (223.7±1.4)×10 6 , which is consistent with the previous measurement [4], but with improved precision.
In summary, the total number of J/ψ events taken with BESIII detector is determined to be N J/ψ = (1310.6±7.0)×10 6 . Here, the total uncertainty is determined by adding the common uncertainties directly and the independent ones in quadrature.