Measurements of the center-of-mass energies at BESIII via the di-muon process^*

The BESIII detector operating at the BEPCII accelerator is designed to study physics in the τ-charm energy region (2–4.6 GeV) [1]. From 2011 to 2014, the BESIII experiment accumulated 5 fb⁻¹ of e⁺ e⁻ collision data at center-of-mass energies between 3.810 and 4.600 GeV to study the charmonium-like and higher excited charmonium states [2]. In the past, BESIII has taken large data samples at the J/ψ, ψ(3686) and ψ(3770) peaks. The corresponding beam energy was fine tuned by a J/ψ or ψ(3686) mass scan before the data-taking. However, around 4 GeV, there is no narrow resonance in e⁺e⁻ annihilation, and the ψ(3686) peak is too far away to be used to calibrate the beam energy. The Beam Energy Measurement System (BEMS), which was installed in 2008, is designed to measure the beam energy with a relative systematic uncertainty of 2×10⁻⁵ [3] based on the energies of Compton back-scattered photons. The performance of BEMS is verified through measurement of the ψ(3686) mass, but 4 GeV is beyond the working range of BEMS. To precisely measure the masses of the newly observed Z_c [4, 5] particles, especially for those which are observed by a partial reconstruction method [6, 7], a precise knowledge of the center-of-mass energy (E_cms) is crucial.

In this paper, we develop a method to measure the E_cms using the di-muon process

where γ_ISR/FSR represents possible initial state radiative (ISR) or final state radiative (FSR) photons. The E_cms can be written as

where M(μ⁺μ⁻) is the invariant mass of μ⁺μ⁻, and ΔM_ISR/FSR is the mass shift due to ISR/FSR radiation. In the analysis, ΔM_ISR/FSR is estimated from a Monte Carlo (MC) simulation of the di-muon process by turning the ISR/FSR on or off, where the ISR/FSR is simulated by babayaga3.5 [8]. To make sure the measured invariant mass M(μ⁺μ⁻) is unbiased, we validate the reconstructed momentum of μ⁺/μ⁻ with the J/ψ signal from the process e⁺e⁻→ γ_ISRJ/ψ with J/ψ → μ⁺μ⁻(γ_FSR) in the same data samples.

The BESIII detector is described in detail in Ref. [9]. The detector is cylindrically symmetric and covers 93% of the solid angle around the collision point. The detector consists of four main components: (a) A 43-layer main drift chamber (MDC) provides momentum measurement for charged tracks with a momentum resolution of 0.5% at 1 GeV/c in a 1 T magnetic field. (b) A time-of-flight system (TOF) composed of plastic scintillators has a time resolution of 80 ps (110 ps) in the barrel (endcaps). (c) An electromagnetic calorimeter (EMC) made of 6240 CsI(Tl) crystals provides an energy resolution for photons of 2.5% (5%) at 1 GeV in the barrel (endcaps). (d) A muon counter (MUC), consisting of 9 (8) layers of resistive plate chambers in the barrel (endcaps) within the return yoke of the magnet, provides 2 cm position resolution. The electron and positron beams collide with an angle of 22 mrad at the interaction point (IP) in order to separate the e⁺ and e⁻ beams after the collision. A geant4 [10] based detector simulation package is developed to model the detector response for MC events.

In total, there are 25 data samples taken at different center-of-mass energies or during different periods, as listed in Table 1. The data sets are listed chronologically, and the ID number is the requested E_cms. The offline luminosity is measured through large-angle Bhabha scattering events with a precision of 1% [11]. In this paper, we measure E_cms for all the 25 data samples and examine its stability during each data taking period.

The measurement of high momentum muons is validated with J/ψ → μ⁺μ⁻ candidates selected via the process e⁺e⁻ → γ_ISRJ/ψ. Events must have only two good oppositely charged tracks. Each good charged track must be consistent with originating from the IP, by requiring the impact parameter to be within 1 cm in the radial direction (V_xy < 1 cm) and 10 cm in the z direction (|V_z| < 10 cm) from the run-dependent IP, and within the polar angle region |cosθ| < 0.8 (i.e. accepting only tracks in the barrel region). The energy deposition in the EMC (E) for each charged track is required to be less than 0.4 GeV to suppress background from radiative Bhabha events. A further requirement on the opening angle between the two tracks, cosθ_μ⁺μ⁻ > − 0.98, is used to remove cosmic rays. The background remaining after the above selection comes from the radiative di-muon process, which has exactly the same final state and cannot be completely removed. The radiative di-muon events show a smooth distribution in M(μ⁺μ⁻). With the above selection criteria imposed, the distribution of M(μ⁺μ⁻) of each sample, having a tail on the low mass side due to final state radiation (FSR) effects, is fitted with an asymmetric function of a crystal-ball function [12] for the J/ψ signal and a linear function to model the background. Figure 1 shows the fit result for data sample 4600 as an example. In order to reduce the fluctuation of M(μ⁺μ⁻), adjacent data samples with small statistics are combined. Due to FSR, J/ψ → μ⁺μ⁻γ_FSR, the measured M^obs(μ⁺μ⁻) is slightly lower than the nominal J/ψ mass [13]. The mass shift due to the FSR photon(s) ΔM_FSR is estimated by simulated samples of the process e⁺e⁻→ γ_ISRJ/ψ with 50,000 events each, generated at different energies using the generator photos [15] with FSR turned on and off. The mass shift ΔM_FSR at each E_cms is obtained as the difference in M^obs(μ⁺μ⁻) between the MC samples with FSR turned on and off. These simulation studies validate that ΔM_FSR is independent of E_cms. A weighted average, ΔM_FSR = (0.59±0.04) MeV/c², is obtained by fitting the ΔM_FSR versus E_cms. The measured mass corrected by ΔM_FSR, M^cor(μ⁺μ⁻), is plotted in Fig. 2 and listed in Table 1 (column 4). The values of M^cor(μ⁺μ⁻) for the different data samples are consistent within errors. By fitting the M^cor(μ⁺μ⁻) of all data samples with a first-order polynomial, the average M^cor(μ⁺μ⁻) is obtained to be M^cor(μ⁺μ⁻) = 3096.79±0.08 MeV/c², which agrees with the nominal J/ψ mass within errors. The goodness of the fit is χ²/ndf = 11.68/11 = 1.06. The small difference is considered as a systematic uncertainty in Section 7.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The E_cms of the initial e⁺e⁻ pair is measured via the di-muon process e⁺e⁻ → γ_ISR/FSRμ⁺μ⁻. However, due to the emission of radiative photons, the invariant mass of the μ⁺μ⁻ pair is less than the E_cms of the initial e⁺e⁻ pair by ΔM_ISR/FSR. In general, the mass shift due to FSR is small, about 0.6 MeV/c² at 3.097 GeV, and the mass shift due to ISR is 2–3 MeV/c², which has been well studied theoretically [14]. In the analysis, the ΔM_ISR/FSR is estimated with MC simulation using babayaga3.5 [8]. We generate 50000 di-muon MC events for each sample with ISR/FSR turned on and off, and take the difference in M(μ⁺μ⁻) as the mass shift ΔM_ISR/FSR caused by ISR and FSR. In order to avoid possible bias, the same event selection criteria for the di-muon process applied for data (as described in Section 5) are imposed to the MC samples.

The distributions of M(μ⁺μ⁻) with ISR/FSR on and off are fitted with a Gaussian function in a range around the peak (same method with data in Section 5). The difference in peak positions (the mass shift ΔM_ISR/FSR) versus E_cms is seen to increase with E_cms, as shown in Fig. 3. The ΔM_ISR/FSR is fitted with a linear function. The fit result is ΔM_ISR/FSR = (−3.53±1.11) + (1.67 ± 0.28) × 10⁻³ × E_cms/MeV with statistical uncertainty only. The goodness of the fit is χ²/n.d.f = 6.3/13. The resulting E_cms-dependent ΔM_ISR/FSR will be used to correct the measured M^obs(μ⁺μ⁻) for the effects of ISR and FSR.

Zoom In Zoom Out Reset image size

The mass shift due to FSR only, ΔM_FSR, is estimated by comparing MC samples of di-muon production with FSR turned on and off. We find that ΔM_FSR increases with E_cms and we parameterize the E_cms dependence with a first-order polynomial as ΔM_FSR = (−1.34 ± 0.84) + (0.56 ± 0.21)×10⁻³×E_cms/MeV, where the corresponding correlation coefficient between the parameters is − 0.964. So the corresponding ΔM_FSR at 3.81 GeV (4.6 GeV) is 0.79 ± 0.09 MeV/c² (1.24 ± 0.14 MeV/c²).

To select the di-muon process e⁺e⁻ → (γ_ISR/FSR) μ⁺μ⁻, the requirement for charged tracks is the same as the γ_ISRJ/ψ selection. To achieve the best precision, only events with both tracks in the barrel region (i.e., in the polar angle region |cosθ|<0.80) are accepted. A requirement on the opening angle between the two tracks of 178.60° < θ_μμ < 179.64° is applied to suppress cosmic ray and di-muon events with high-energy radiative photons. To further remove cosmic ray events, the TOF timing difference between the two tracks is required to be |Δt| < 4 ns. The background contribution following the above selection criteria is less than 0.001% compared to signal and is therefore neglected in the following.

After applying the above selection, the distribution of M^obs(μ⁺μ⁻) for selected di-muon events has a tail in the low mass side which cannot be described by a single Gaussian. Since only the peak position of the distribution will be used, we estimate it by fitting with a Gaussian function in the range of (−1σ,2σ) around the peak, where σ is the standard deviation of the Gaussian. To examine the stability of E_cms over time for each data sample, the fit procedure is performed for each run of the data samples, where a run normally corresponds to one hour of data taking. The fit result for one run of the 4600 data sample is shown in Fig. 4. The measured μ⁺μ⁻ masses versus the run number for the samples 4009^1,2, 4260^1,2, 4360, 4230^2,3, 4600, and 4420^2,3 are plotted in Fig. 5. For the sample 4260¹ (4230²), the measured M^obs(μ⁺μ⁻) changes slowly and is fitted with a linear function. The fit gives (4367.37±53.53) + (−3.75±1.80) × 10⁻³ × N_run ((4316.81±7.84) + (−2.87±0.24) × 10⁻³ × N_run) in unit of MeV/c², where N_run is the run number. Since the uncertainty is N_run dependent, we take the largest value from error propagation as the corresponding statistical uncertainty. For other data samples, M^obs(μ⁺μ⁻) remains stable, and the average value is used to calculate E_cms. The samples 4009¹ (4420²) and 4009² (4420³) are separated because they show a sudden drop in the average energies. Table 1 (column 5) summarizes the measured M^obs(μ⁺μ⁻) for each sample.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

E_cms is finally obtained by adding the energy-dependent mass shift ΔM_ISR/FSR due to ISR/FSR obtained in Section 4 to the measured M^obs(μ⁺μ⁻). The measured E_cms is listed in Table 1 (column 6); the systematic uncertainty will be discussed in Section 7.

Each of the data sets 4009, 4230, 4260, and 4420 is split into several sub-samples. We calculate the luminosity-weighted average E_cms for each, and the largest systematic uncertainty of the sub-samples is taken as the systematic uncertainty. In Table 2, we summarize the weighted average E_cms for all data samples.

The processes of e⁺e⁻ → π⁺π⁻K⁺K⁻ and e⁺e⁻ → π⁺π⁻pp̅ are used to check the measurement of E_cms. Similar to the di-muon process e⁺e⁻→ γ_ISR/FSRμ⁺μ⁻, the E_cms of the initial e⁺e⁻ system is estimated by the corrected invariant masses of the final state particles M^cor(π⁺π⁻K⁺K⁻) and M^cor(π⁺π⁻pp̅). The measurement of the low momentum charged tracks is validated using the decay channels D⁰ → K⁻π⁺ and D̅⁰ → K⁺π⁻. The measured mass, M^obs(K⁻π⁺/K⁺π⁻) = 1864.00 ± 0.70 MeV/c² (statistical uncertainty only) is consistent with the nominal D⁰/D̅⁰ mass [13] with a deviation of 0.84 ± 0.71 MeV/c². Both the corrected M^cor(π⁺π⁻K⁺K⁻) and M^cor(π⁺π⁻pp̅) are found to be consistent with E_cms obtained using the di-muon process, with the largest deviation of 0.53 ± 0.75 MeV found in sample 4420.

The systematic uncertainty in E_cms in this analysis is estimated by considering the uncertainties from the momentum measurement of the μ^±, the estimation of the mass shift ΔM_ISR/FSR due to ISR/FSR, the generator, and the corresponding fit procedure.

We use the J/ψ invariant mass via the process J/ψ → μ⁺μ⁻ to check the momentum reconstruction. The measured J/ψ mass corrected for FSR effects at each energy, M^cor(J/ψ), is close to the nominal J/ψ mass. To be conservative, we use a first-order polynomial to fit the M^cor(J/ψ) versus E_cms distribution, and find the largest difference in the J/ψ mass between the fit result and the nominal value to be 0.34 MeV/c². We take as the systematic uncertainty due to the momentum measurement.

The mass shift ΔM_ISR/FSR due to ISR/FSR is E_cms dependent, and is obtained from MC samples with 50,000 generated events each. The standard deviation of the distribution of ΔM_ISR/FSR versus E_cms is given by

where ΔM_ISR/FSR is the value from the fit (Fig. 3), and N is the number of points in Fig. 3. A value of 0.37 MeV/c² is taken as systematic uncertainty due to the ISR/FSR correction.

Additionally, we use two different generators (babay- aga3.5 and babayaga@nlo) to estimate the mass shift ΔM_ISR/FSR. The averaged difference in ΔM_ISR/FSR from the two generators is 0.036 ± 0.067 MeV/c², which reflects the contribution to the systematic uncertainty of the ISR/FSR correction from the generator; it is negligibly small.

M^obs(μ⁺μ⁻) is measured run-by-run and is found to be stable during data-taking for most samples. For the runs in each sample (except for samples 4230² and 4260¹, which are described by a first-order polynomial), the average E_cms is provided to reduce the statistical fluctuation. If the energy shifts gradually during the data-taking, the simple average value will cause a systematic uncertainty. To estimate this systematic error for each sample, we fit the distribution of M^obs(μ⁺μ⁻) versus run-number by a first-order polynomial and take the largest difference between the fitting result and the average value, less than 0.25 MeV/c² on average, as the systematic uncertainty.

The uncertainties from other sources, such as background and event selection, are negligible. Assuming all the sources of systematic uncertainty are independent, the total systematic uncertainty is obtained by adding all items in quadrature, giving the values listed in Table 1 (column 6). The uncertainty is smaller than 0.8 MeV for all the data samples.

The center-of-mass energies of the data taken from 2011 to 2014 for the studies of the charmonium-like and higher excited charmonium states are measured with the di-muon process e⁺e⁻→(γ_ISR/FSR)μ⁺μ⁻. The corresponding statistical uncertainty is very small, and the systematic uncertainty is found to be less than 0.8 MeV. The measured E_cms is validated by the processes e⁺e⁻ → π⁺π⁻K⁺K⁻ and e⁺e⁻ → π⁺π⁻pp̅. The stability of E_cms over time for the data samples is examined. For samples 4009, 4230, 4260, 4420, we also give the luminosity-weighted average E_cms. The results are essential for the discovery of new states and investigation of the transition of charmonium and charmonium-like states [4–7].

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.