Measurement of the center-of-mass energies at BESIII via the di-muon process

From 2011 to 2014, the BESIII experiment collected about 5 fb$^{-1}$ data at center-of-mass energies around 4 GeV for the studies of the charmonium-like and higher excited charmonium states. By analyzing the di-muon process $e^{+}e^{-}\rightarrow\gamma_{\rm ISR/FSR}\mu^{+}\mu^{-}$, the center-of-mass energies of the data samples are measured with a precision of 0.8 MeV. The center-of-mass energy is found to be stable for most of time during the data taking.


I. INTRODUCTION
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 centerof-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 the BEMS is verified through the 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 − , ∆M ISR/FSR is the mass shift due to ISR/FSR radiation, which equals to the difference between the invariant mass of the µ + µ − pair and the E cms of the initial e + e − system. In the analysis, ∆M ISR/FSR is estimated from a Monte Carlo (MC) simulation of the di-muon process by turning on or off the ISR/FSR, where the ISR/FSR is simulated by BABAYAGA3.5 [9]. 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 ISR J/ψ with J/ψ → µ + µ − (γ FSR ) in the same data samples.

II. THE BESIII DETECTOR AND DATA SETS
The BESIII detector is described in detail in Ref. [10]. 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 [11] based detector simulation package is developed to model the detector response for MC events.
In total, there are 25 data samples taken at different centerof-mass energies or during different periods, as listed in Table I. 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% [12]. In this paper, we measure E cms for all the 25 data samples and examine its stability during each data taking period.

III. MUON MOMENTUM VALIDATION WITH J/ψ SIGNAL
The high momentum measurement of muons is validated with J/ψ → µ + µ − candidates selected via the process e + e − → γ ISR J/ψ. Events must have only two good oppositely charged tracks. Each good charged track is required to be consistent with originating from the IP within 1 cm in radial direction (V xy < 1 cm) and 10 cm in z direction (|V z | < 10 cm) to the run-dependent IP, and within the polar angle region | cos θ| < 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, cos θ 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 can not 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 is fitted with a crystal-ball function [13] for the J/ψ signal and a linear function to model the background. Figure 1 shows the fit result for the 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 final state radiation, J/ψ → µ + µ − γ FSR , the measured M obs (µ + µ − ) is slightly lower than the nominal J/ψ mass [14]. The mass shift due to the FSR photon(s) ∆M FSR is estimated by simulated samples of the process e + e − → γ ISR J/ψ with 50,000 events each, generated at different energies using the generator PHOTOS [15] with FSR turned on and off. The mass shift ∆M 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 ∆M FSR is independent of E cms . A weighted average, ∆M FSR = (0.59 ± 0.04) MeV/c 2 , is obtained by fitting the ∆M FSR versus E cms . The measured mass corrected by ∆M FSR , M cor (µ + µ − ), is plotted in Fig. 2 Table I (column 4). The values of M cor (µ + µ − ) for the different data samples are consistent within errors, and the average is M cor (µ + µ − ) = 3096.79±0.08 MeV/c 2 , which agrees with the nominal J/ψ mass within errors. The small difference is considered as systematic uncertainty in Section VII.

IV. THE MASS SHIFT ∆M ISR/FSR
The E cms of the initial e + e − pair is measured via the dimuon process e + e − → γ ISR/FSR µ + 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 ∆M ISR/FSR . In general, the mass shift due to the FSR is small, about 0.6 MeV/c 2 at 3.097 GeV, and the mass shift due TABLE I. Summary of the data sets, including ID, run number, offline luminosity, the measured M cor (J/ψ), M obs (µ + µ − ), and Ecms. The first uncertainty is statistical, and the second is systematic. Superscripts indicate separate samples acquired at the same Ecms. The "-" indicates samples which are combined with the previous one(s) to measure M cor (µ + µ − ).

ID
Run number Offline lum.  to the ISR is 2∼3 MeV, which has been well studied theoretically [8]. In the analysis, the ∆M ISR/FSR is estimated with MC simulation using BABAYAGA3.5 [9]. We generate 50,000 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 ∆M 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 V) are im- ness of the fit is χ 2 /n.d.f = 6.3/13. The resulting E cmsdependent ∆M ISR/FSR will be used to correct the measured M obs (µ + µ − ) for the effects of ISR and FSR. The mass shift due to FSR only, ∆M FSR , is estimated by comparing MC samples of di-muon production with FSR turned on and off. We find that ∆M FSR increases with E cms and we parameterize the E cms dependence with a first-order polynomial as ∆M FSR = (−1.34 ± 0.84) + (0.56 ± 0.21) × 10 −3 × E cms , where E cms is in unit of MeV and the error matrix of the fit parameters is (0.693, −0.170 × 10 −3 , −0.170 × 10 −3 , 0.042 × 10 −6 ). So the corresponding ∆M FSR at 3.81 GeV (4.6 GeV) is 0.79±0.09 MeV (1.24±0.14 MeV).

V. THE MEASUREMENT OF Ecms
To select the di-muon process e + e − → γ ISR/F SR µ + µ − , the requirement for charged tracks is the same as the γ ISR J/ψ selection. To achieve best precision, only events with both tracks in the barrel region (i.e., in the polar angle region | cos θ| < 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 |∆t| < 4 ns. The background contribution following above selection criteria is less than 0.001% compared to signal and is therefore neglected in the following.
We estimate the peak position of the distribution of M obs (µ + µ − ) for selected di-muon events 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 the 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 where N run is the run number, and the largest value from error propagation is taken 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 4009 1 (4420 2 ) and 4009 2 (4420 3 ) are separated because they show a sudden drop in the average energies. Table I (  The E cms is finally obtained by adding the energydependent mass shift ∆M ISR/FSR due to ISR/FSR obtained in Section IV to the measured M obs (µ + µ − ). The measured E cms is listed in Table I (column 6); the systematic uncertainty will be discussed in Section VII.
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 samples is taken as the systematic uncertainty. In Table II, we summarize the weighted average E cms for all data samples.

VI. CROSS CHECK
The processes of e + e − → π + π − K + K − and e + e − → π + π − pp are used to check the measurement of the E cms . Similar to the di-muon process e + e − → γ ISR/FSR µ + 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 − pp). The measurement of the low momentum charged tracks is validated using the decay channels D 0 → K − π + andD 0 → K + π − . The measured mass, M obs (K − π + /K + π − ) = 1864.00 ± 0.7 MeV (statistical uncertainty only) is consistent with the nominal D 0 /D 0 mass [14] with a deviation of 0.84 ± 0.71 MeV. Both the corrected M cor (π + π − K + K − ) and M cor (π + π − pp) are found to be consistent with E cms obtained using the dimuon process, with the largest deviation of 0.53 ± 0.75 MeV found in sample 4420.

VII. SYSTEMATIC UNCERTAINTIES
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 ∆M 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/c 2 . We take 0.34 3096.92 = 0.011% as the systematic uncertainty due to the momentum measurement.
The mass shift ∆M 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 ∆M ISR/FSR versus E cms is given by where ∆M 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/c 2 is taken as systematic uncertainty due to the ISR/FSR correction.
Additionally, we use different generator versions (BABAYAGA3.5 and BABAYAGA@NLO) to estimate the mass shift ∆M ISR/FSR . The averaged difference in ∆M ISR/FSR from the two generators is 0.036 ± 0.067 MeV/c 2 , which reflects the contribution to the systematic uncertainty of the ISR/FSR correction from the generator; it is negligibly small.
The 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 the samples of 4230 2 and 4260 1 , 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 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, which is listed in Table I (column 6). The uncertainty is smaller than 0.8 MeV for all the data samples.

VIII. SUMMARY
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 − pp. The stability of E cms over time for the data samples is also examined. For the 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][5][6][7].