Number of J/ψ events at BESIII ^*

As a charmonium ground state, the $J/\psi$ offers a unique laboratory for studying light hadron spectroscopy. In particular, $J/\psi$ decays can be used to search for exotic hadrons composed of light quarks and gluons, which are key to a more comprehensive understanding of the nature of the strong interaction.

Many important results in light hadron spectroscopy [1] have been reported based on $(1310.6 \pm 7.0)\times 10^6$ $J/\psi$ events collected by the BESIII experiment [2] in 2009 and 2012. An additional large sample of $J/\psi$ events was collected by BESIII during 2017–2019 to improve the precision of the measurements and search for new processes. The three data samples of $J/\psi$ events collected at BESIII are summarized in Table 1.

Table 1. Data samples used in the determination of the number of $J/\psi$ events. $\psi(3686)$ and QED samples (continuum data samples at $\sqrt{s} = 3.08$ GeV) are taken in close chronological order to each $J/\psi$ sample with only different c.m. energies to make sure the efficiency and background estimation reliable.

Data set	$\sqrt{s}$	${\mathcal{L}}_\text{online}$	Date (duration) (YYYY-MM-DD)
$J/\psi$	3.097 GeV	$2678$ pb $^{-1}$	2017-08-12 – 2019-06-02
QED1	3.08 GeV	$48$ pb $^{-1}$	2018-04-12 – 2018-04-14
QED2	3.08 GeV	$88$ pb $^{-1}$	2019-02-07 – 2019-02-11
$\psi(3686)$	3.686 GeV	$25$ pb $^{-1}$	2018-05-20
$J/\psi$	3.097 GeV	$323$ pb $^{-1}$	2012-04-10 – 2012-05-22
QED1	3.08 GeV	$13$ pb $^{-1}$	2012-04-08
QED2	3.08 GeV	$17$ pb $^{-1}$	2012-05-23 – 2012-05-24
$\psi(3686)$	3.686 GeV	$7.5$ pb $^{-1}$	2012-05-26
J/ψ	3.097 GeV	$82$ pb $^{-1}$	2009-06-12 – 2009-07-28
QED	3.08 GeV	$0.3$ pb $^{-1}$	2009-06-19
$\psi(3686)$	3.686 GeV	$150$ pb $^{-1}$	2009-03-07 – 2009-04-14

This paper reports a precise determination of the total number of $J/\psi$ events. The number of $J/\psi$ events for the new samples collected in 2017–2019 was determined with an inclusive method as the one used in the previous measurements [3, 4]. In addition, in this analysis we also recalculate the number of $J/\psi$ events for the two data samples taken in 2009 and 2012, reconstructed using the latest BESIII software [5, 6]. The number of $J/\psi$ events, $N_{J/\psi}$, was calculated as

$$\begin{eqnarray} &&N_{ J/\psi}=\frac{N_{\rm{sel}}-N_{\rm{bg}}}{\epsilon_{\rm{trig}} \times \epsilon^{\psi(3686)}_{\rm{data}} \times f_{\rm{cor}}}, \end{eqnarray} \tag{ 1 }$$

where $N_{\rm{sel}}$ is the number of inclusive $J/\psi$ decays selected from the $J/\psi$ data; $N_{\rm{bg}}$ is the number of background events estimated with continuum data taken at $\sqrt{s} =$ 3.08 GeV; $\epsilon_{\rm{trig}}$ is the trigger efficiency; $\epsilon^{\psi(3686)}_{\rm{data}}$ is the inclusive $J/\psi$ detection efficiency determined experimentally using the $J/\psi$ sample from the reaction $\psi(3686) \rightarrow \pi^+ \pi^- J/\psi$. $f_{\rm{cor}}$ is a correction factor that accounts for the difference in the detection efficiency between the $J/\psi$ events produced at rest and those produced in $\psi(3686)\rightarrow \pi^+ \pi^- J/\psi$. $f_{\rm{cor}}$ is expected to be approximately unity and is determined by the Monte Carlo (MC) simulation sample with

$$\begin{eqnarray} &&f_{\rm{cor}} = \frac {\epsilon^{J/\psi}_{\rm MC}} {\epsilon^{\psi(3686)}_{\rm{MC}}}, \end{eqnarray} \tag{ 2 }$$

where $\epsilon^{J/\psi}_{\rm{MC}}$ is the detection efficiency of inclusive $J/\psi$ events determined from the MC sample of $J/\psi$ events produced directly in electron-positron collisions, and $\epsilon^{\psi(3686)}_{\rm{MC}}$ is that from the MC sample of $\psi(3686)\to \pi^+\pi^- J/\psi$. For the number of $J/\psi$ events determined with Eq. (1), only $f_{\rm{cor}}$ depends on MC simulation. According to Eq. (2), the uncertainties related to MC simulation (including generator, detector response, etc.) almost cancel out because they impact both the numerator and denominator, which improves the precision of the number of $J/\psi$ events. In the MC simulation, the production of $J/\psi$ and $\psi(3686)$ resonances was simulated with a GEANT4-based [7] MC software, which includes the geometric description of the BESIII detector and the detector response. The simulation models the beam energy spread and initial state radiation (ISR) in the $e^+e^-$ annihilations with the generator KKMC [8]. The known decay modes were modeled with EVTGEN [9, 10] using branching fractions taken from the Particle Data Group [11], and the remaining unknown charmonium decays were modeled with LUNDCHARM [12, 13].

Candidate events must contain two or more charged tracks that must have a momentum less than 2.0 GeV/c and to be within a polar angle (θ) range of $|\cos \theta| \lt 0.93$, where θ is defined with respect to the axis of the Main Drift Chamber (MDC). The distance of closest approach to the interaction point (IP) must be less than 15 cm along the z-axis, $|V_z|$, and less than 1 cm in the transverse plane, $V_r$. Photon candidates were identified using isolated showers in the electromagnetic calorimeter (EMC). The deposited energy of each shower must be more than 25 MeV in the barrel region ( $|\cos\theta| \lt 0.83$) and more than 50 MeV in the end cap region ( $0.86 \lt |\cos\theta| \lt 0.93$). To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within [0, 700] ns.

To suppress events from Quantum Electrodynamics (QED) processes (i.e., Bhabha and dimuon events), from cosmic rays, beam-induced backgrounds, and electronic noise, a series of selection criteria were applied to the candidate events.

The sum of charged particle energies computed from the track momenta assuming a pion mass and the neutral shower energies deposited in the EMC, $E_{\rm{vis}}$, is required to be greater than 1.0 GeV. Fig. 1 shows a comparison of the $E_{\rm{vis}}$ distribution between the $J/\psi$ data, the data taken at $\sqrt{s} = 3.08$ GeV, and the inclusive $J/\psi$ MC sample. The requirement $E_{\rm{vis}} \gt 1.0\; {\rm{GeV}}$ removes one third of the background events while retaining $99.5\%$ of the signal events.

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For events with only two charged tracks, the momentum of each track is required to be less than 1.5 GeV/c to exclude Bhabha and dimuon events. Fig. 2 shows a scatter plot of the momenta of the two charged tracks, and the solid lines depict the momentum requirement. Fig. 3 displays the distribution of energy deposited by the charged particles in the EMC; a significant peak around 1.5 GeV is from Bhabha events, which can be rejected by requiring the energy deposited in the EMC to be less than 1 GeV for each charged track.

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After the above requirements, $N_{\rm{sel}} = (6912.03 \pm 0.08)\times 10^{6}$ candidate events were selected from the $J/\psi$ data taken in 2017–2019. The distributions of the track parameters of closest approach along the beam line and in radial direction ( $V_z$ and $V_r$), the polar angle ( $\cos\theta$), and the total energy deposited in the EMC ( $E_{\rm{EMC}}$) after subtracting background events estimated with the continuum data taken at $\sqrt{s} = 3.08$ GeV (see Sec. III for details) are illustrated in Fig. 4. The multiplicity of charged tracks ( $N_{\rm{good}}$) is shown in Fig. 5, where the MC sample generated according to the standard MC model agrees very well with the data while the MC sample generated with an 'incomplete' MC model deviates from the data. The standard MC model includes the known decay processes listed in the PDG and the unknown ones modeled with LUNDCHARM [12, 13], while the incomplete MC model only includes the known decay modes listed in the PDG. The MC sample generated with the incomplete MC model was used to test the effect of this discrepancy on the determination of the number of $J/\psi$ events, which is small as described in Sec. VI.

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In this analysis, the potential background sources include QED processes, beam-induced background, cosmic rays, and electronic noise. The continuum data samples at $\sqrt{s}=3.08$ GeV were taken in close chronological order to each $J/\psi$ sample to estimate these backgrounds.

The integrated luminosity was determined using the process $e^+e^-\rightarrow \gamma\gamma$. The candidate events were selected by requiring at least two showers in the EMC within $|\cos\theta | \lt 0.8$ and with the energy of the second most energetic shower between 1.2 and 1.6 GeV. The number of signal events was determined from the number of events in the signal region $|\Delta \phi| \lt 2.5^\circ$, and the background was estimated from those in the side-band region $2.5^\circ \lt |\Delta \phi| \lt 5^\circ$, where $\Delta \phi = |\phi_{\gamma1} - \phi_{\gamma2}| - 180^\circ$ and $\phi_{\gamma1/2}$ are the azimuthal angles of the two photon candidates. Taking into account the detector efficiency obtained from the MC simulation and the cross section of the QED process $e^+e^- \rightarrow \gamma\gamma$, the integrated luminosities of the $J/\psi$ data sample and the sample taken at $\sqrt{s}=3.08$ GeV in 2017–2019 were determined to be $(2568.07 \pm 0.40) \;{\rm{pb}}^{-1}$ and $(136.22 \pm 0.09)\;{\rm{pb}}^{-1}$, respectively, where the errors are statistical only.

After applying the same selection criteria as for the $J/\psi$ data, $N_{3.08}=6,363,941 \pm 2,523$ events were selected from the continuum data taken at $\sqrt{s} = 3.08$ GeV. Assuming the same detection efficiency at $\sqrt{s} = 3.08$ GeV as for the $J/\psi$ peak and taking into account the energy-dependent cross section of the QED processes, the number of background events for the $J/\psi$ sample, $N_{\rm{bg}}$, was estimated to be

$$\begin{eqnarray} &&N_{\rm{bg}}=N_{3.08}\times \frac{ {\mathcal{L}}_{J/\psi}}{ {\mathcal{L}}_{3.08}} \times \frac{s_{3.08}}{s_{J/\psi}}=(118.66 \pm 0.05)\times 10^{6} , \end{eqnarray} \tag{ 3 }$$

where ${\mathcal{L}}_{ J/\psi}$ and ${\mathcal{L}}_{3.08}$ are the integrated luminosities for the $J/\psi$ data sample and the data sample taken at $\sqrt{s}=3.08$ GeV, and $s_{ J/\psi}$ and $s_{3.08}$ are the corresponding squares of the center-of-mass energies. The background was calculated to be $(1.717 \pm 0.002)\%$ of the number of selected inclusive $J/\psi$ events taken in 2017–2019.

In this analysis, the detection efficiency was determined experimentally using a sample of $J/\psi$ events from the reaction $\psi(3686) \rightarrow \pi^+ \pi^- J/\psi$ to reduce the uncertainty related to any discrepancies between the MC simulation and the data. To ensure that the beam conditions and detector status are similar to those of the sample collected at the $J/\psi$ peak, a dedicated $\psi(3686)$ sample taken on May 20, 2018 was used for this study.

For a candidate $\psi(3686) \rightarrow \pi^+\pi^- J/\psi$ event, there must be at least two soft pions with opposite charge detected in the MDC with $|\cos\theta| \lt 0.93$. Each candidate pion is required to have a momentum less than $0.4\;{\rm{GeV}}/c$, and the distance of closest approach to the IP must satisfy $|V_z| \lt 15$ cm and $V_r \lt 1$ cm. No further selection criteria on the remaining charged tracks or showers are required. The distribution of the invariant mass recoiling against all possible soft $\pi^+\pi^-$ pairs is shown in Fig. 6. A prominent peak around $3.1\;{\rm{GeV}}/c^2$, corresponding to the decay of $\psi(3686) \rightarrow \pi^+\pi^- J/\psi$ is observed over a smooth background. The number of inclusive $J/\psi$ events, $N_{\rm{inc}} = (3538.5 \pm 3.6)\times 10^{3}$, was obtained by fitting a double-Gaussian function for the $J/\psi$ signal plus a second-order Chebychev polynomial for the background to the $\pi^+$ $\pi^-$ recoil mass spectrum.

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To measure the detection efficiency of inclusive $J/\psi$ events, the same selection criteria as described in Sec. II were applied to the remaining charged tracks and showers. The number of selected inclusive $J/\psi$ events, $N_{\rm{inc}}^{\rm{sel}}$, was determined to be $(2717.6 \pm 3.4)\times 10^{3}$ using a fit to the recoil mass distribution of the selected events with the same function as described above. The detection efficiency of inclusive $J/\psi$ events, $\epsilon^{ \psi(3686)}_{\rm{data}} = (76.80 \pm 0.05)\%$, was calculated from the ratio of the number of inclusive $J/\psi$ events with and without the inclusive $J/\psi$ event selection criteria applied.

To account for the efficiency difference between the $J/\psi$ produced at rest and the $J/\psi$ from the decay $\psi(3686) \rightarrow \pi^+\pi^- J/\psi$, a correction factor, defined in Eq. (2), was used. Two large statistics MC samples, inclusive $\psi(3686)$ and inclusive $J/\psi$ events, were produced and subjected to the same selection criteria as the data samples. The detection efficiencies of inclusive $J/\psi$ events were determined to be $\epsilon^{\psi(3686)}_{\rm{MC}} = (76.93 \pm 0.02)$% and $\epsilon^{J/\psi}_{\rm{MC}}=(77.56 \pm 0.01)$% for the two inclusive MC samples, respectively. The correction factor $f_{\rm{cor}}$ for the detection efficiency was therefore taken as

$$\begin{eqnarray} &&f_{\rm{cor}} = \frac {\epsilon^{J/\psi}_{\rm{MC}}} {\epsilon^{\psi(3686)}_{\rm{MC}}}=1.0082 \pm 0.0007, \end{eqnarray} \tag{ 4 }$$

where the error is statistical only.

With Eq. (1) and the corresponding parameter values summarized in Table 2, the number of $J/\psi$ events collected in 2017-2019 was determined to be $(8774.0 \pm 0.2) \times 10^6$. The trigger efficiency of the BESIII detector was taken to be 100%, based on a study of various reactions [14]. With the same procedure, the numbers of $J/\psi$ events taken in 2012 and 2009 were recalculated to be $(1088.5 \pm 0.1) \times 10^6$ and $(224.0 \pm 0.1) \times 10^6$, respectively, where the uncertainties are statistical only. The statistical uncertainties of $N_{\rm{bg}}$ were taken into account as part of the systematic uncertainty (see Sec. VI.D). The systematic uncertainties from different sources are discussed in detail in Sec. VI.

Table 2. The values used in the calculation, and the resulting number of $J/\psi$ events, where the uncertainties are statistical.

Item	2017–2019	2012	2009
$N_\text{sel}(\times 10^6)$	$6912.03 \pm 0.08$	$860.59 \pm 0.03$	$180.84 \pm 0.01$
$\epsilon_\text{trig}$	1.00	1.00	1.00
$N_\text{QED} (\times 10^3)$	$6363.9 \pm 2.5$	$1616.6 \pm 1.3$	$24.6 \pm 0.2$
$N_\text{bg}(\times 10^6)$	$118.66 \pm 0.05$	$15.32 \pm 0.02$	$6.89 \pm 0.04$
$\epsilon^{\psi(3686)}_\text{data}$	$\dfrac{(2717.6 \pm 3.4) \times 10^3 }{(3538.5 \pm 3.6) \times 10^3} =0.7680 \pm 0.0005$	$\dfrac{(981.6 \pm 2.0) \times 10^3 }{(1274.9 \pm 2.1) \times 10^3} =0.7699 \pm 0.0005$	$\dfrac{(17048.6 \pm 8.9) \times 10^3}{(22120.0 \pm 9.4) \times 10^3} =0.7707 \pm 0.0001$
$\epsilon^{\psi(3686)}_\text{MC}$	$\dfrac{(4450.2 \pm 3.2) \times 10^3 }{(5784.7 \pm 3.5) \times 10^3}=0.7693 \pm 0.0002$	$\dfrac{(4668.8 \pm 3.1) \times 10^3}{(6056.3 \pm 3.4) \times 10^3} =0.7709 \pm 0.0002$	$\dfrac{(4626.8 \pm 3.3) \times 10^3}{(5991.4 \pm 3.4) \times 10^3}=0.7723 \pm 0.0002$
$\epsilon^{J/\psi}_\text{MC}$	$\dfrac{(6747.59 \pm 0.04) \times 10^6}{8700 \times 10^6}=0.7756 \pm 0.0001$	$\dfrac{(7775.5 \pm 1.4) \times 10^3}{10^7} =0.7776 \pm 0.0001$	$\dfrac{(174.26 \pm 0.02) \times 10^6}{ 224 \times 10^6}=0.7780 \pm 0.0001$
$f_\text{cor}$	$1.0082 \pm 0.0007$	$1.0086 \pm 0.0008$	$1.0074 \pm 0.0003$
$N_{J/\psi}(\times 10^6)$	$8774.0 \pm 0.2$	$1088.5 \pm 0.1$	$224.0 \pm 0.1$

The sources of systematic uncertainties, including the MC model, track reconstruction efficiency, fit to the $J/\psi$ peak, background estimation, random trigger mixing and the efficiency of selecting the two soft pions recoiling against $J/\psi$, are investigated in detail below, and the corresponding contributions are summarized in Table 3.

Table 3. Sources of systematic uncertainties and the corresponding contributions to the number of $J/\psi$ events, where the superscript * indicates that the error is common for the same item in different data samples.

Sources	2017–2019(%)	2012 (%)	2009(%)
MC model uncertainty	0.18 $^*$	0.18 $^*$	0.27 $^*$
Tracking efficiency	0.02 $^*$	0.03 $^*$	0.31
Fit to $J/\psi$ peak	0.10	0.21	0.09
Background uncertainty	0.04	0.10	0.14
Noise mixing	0.06	0.02	0.12
$\epsilon_{\pi^+\pi^-}$ uncertainty	0.40 $^*$	0.27 $^*$	0.32 $^*$
Total	0.45	0.40	0.56

A. MC model uncertainty

B. Track reconstruction efficiency

C. Fit to the $\boldsymbol{J/\psi}$ peak

D. Background uncertainty

E. Random Trigger mixing

F. Uncertainty of selection efficiency of two soft pions

G. Summary of systematic uncertainties

Using the inclusive $J/\psi$ decays, the number of $J/\psi$ events collected with the BESIII detector in 2017–2019 was determined to be $(8774.0 \pm 39.4)\times10^{6},$ where the uncertainty is completely dominated by systematics, and the statistical uncertainty is negligible. The numbers of $J/\psi$ events taken in 2009 and 2012 were recalculated to be $(224.0 \pm 1.3)\times10^{6}$ and $(1088.5 \pm 4.4)\times10^{6},$ which are consistent with the previous measurements [4] but with improved precision.

The total number of $J/\psi$ events taken with BESIII detector was determined to be $N_{J/\psi}= (10087 \pm 44)\times10^{6}$. Here, the total uncertainty was determined by adding the common uncertainties linearly and the independent ones in quadrature.

The BESIII collaboration thanks the staff of BEPCII and the IHEP Computing Center for their strong support.