Observation of $\eta_c(2S) \to 3(\pi^+\pi^-)$ and measurements of $\chi_{cJ} \to 3(\pi^+\pi^-)$ in $\psi(3686)$ radiative transitions

The hadronic decay $\eta_c(2S) \to 3(\pi^+\pi^-)$ is observed with a statistical significance of 9.3 standard deviations using $(448.1\pm2.9)\times10^6$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII collider. The measured mass and width of $\eta_c(2S)$ are $(3643.4 \pm 2.3 (\rm stat.) \pm 4.4 (\rm syst.))$ MeV/$c^2$ and $(19.8 \pm 3.9 (\rm stat.) \pm 3.1 (\rm syst.))$ MeV, respectively, which are consistent with the world average values within two standard deviations. The product branching fraction $\mathcal{B}[\psi(3686)\to \gamma \eta_c(2S)]\times\mathcal{B}[\eta_c(2S)\to3(\pi^+\pi^-)]$ is measured to be $(9.2 \pm 1.0 (\rm stat.) \pm 0.9 (\rm syst.))\times10^{-6}$. Using $\mathcal{B}[\psi(3686)\to \gamma \eta_c(2S)]=(7.0^{+3.4}_{-2.5})\times10^{-4}$, we obtain $\mathcal{B}[\eta_c(2S) \to 3(\pi^+\pi^-)] = (1.31 \pm 0.15 (\rm stat.) \pm 0.13 (\rm syst.)(^{+0.64}_{-0.47}) (\rm extr))\times10^{-2}$, where the third uncertainty is from $\mathcal{B}[\psi(3686) \to \gamma \eta_c(2S)]$. We also measure the $\chi_{cJ} \to 3(\pi^+\pi^-)$ ($J=0, 1, 2$) decays via $\psi(3686) \to \gamma \chi_{cJ}$ transitions. The branching fractions are $\mathcal{B}[\chi_{c0} \to 3(\pi^+\pi^-)] = (2.080\pm0.006 (\rm stat.)\pm0.068 (\rm syst.))\times10^{-2}$, $\mathcal{B}[\chi_{c1} \to 3(\pi^+\pi^-)] = (1.092\pm0.004 (\rm stat.)\pm0.035 (\rm syst.))\times10^{-2}$, and $\mathcal{B}[\chi_{c2} \to 3(\pi^+\pi^-)] = (1.565\pm0.005 (\rm stat.)\pm0.048 (\rm syst.))\times10^{-2}$.


I. INTRODUCTION
In recent years, remarkable experimental and theoretical progress has been made in charmonium studies. All predicted charmonium states below the open-charm threshold have been observed experimentally, and to a large extent, the measured spectra agree with the theoretical predictions based on Quantum Chromodynamics (QCD) [1][2][3] and QCD-inspired potential models [4][5][6]. However, there are still problems or puzzles that need to be understood.
It is predicted that the ratio of branching fractions of ψ(3686) and J/ψ decaying into the same light hadron final states is around 12%, which was first proposed by Appelquist and Politzer using perturbative QCD [7]. This is valid in most of the measured hadronic channels [8], except for the decays into pseudoscalar vector pairs and vector tensor pairs, which are suppressed by at least an order of magnitude.
The η c (2S) and η c (1S) are the spin-singlet partners of ψ ′ (ψ ′ ≡ ψ(3686)) and J/ψ, but two different ratios are calculated by the authors of Refs. [10,11], suggesting two possibilities: respectively. Using information on light hadronic final states [12], the authors of Ref. [13] recently examined this branching fraction ratio in several decay modes and found that the experimental data are significantly different from both of the two theoretical predictions, e.g., B[ηc(1S)→KKπ] = 0.27 +0.10 −0.07 [13]. The η c (2S) is the first excited state of the pseudoscalar ground state η c (1S), lying just below the mass of its vector counterpart, ψ ′ . It was first observed by the Belle Collaboration in B meson decay, 14]. This state was hereafter confirmed by BABAR [15], CLEO [16], and Belle [17] in the two-photon fusion process γγ → η c (2S) →KKπ, and by BABAR [18] and Belle [19] in the double charmonium production process e + e − → J/ψ + cc. The magnetic dipole (M1) transition between ψ ′ and η c (2S) was first observed by BESIII, where η c (2S) was reconstructed with the KKπ mode [20]. Currently, only seven decay modes of η c (2S) have been observed experimentally, of which the branching fractions of four have been measured with uncertainties larger than 50% [12]. The sum of the four decay modes is around 5% of the to-tal width of η c (2S). Among the discovered decay modes of η c (1S), the decay rate of η c (1S) → 3(π + π − ) is relatively large, while the decay of η c (2S) → 3(π + π − ) has not yet been seen. CLEO searched for this mode in ψ ′ radiative decay but no signal was observed [21].
We present herein a study of the η c (2S) → 3(π + π − ) and χ cJ → 3(π + π − ) in ψ ′ radiative transitions. The measurement is based on a data sample corresponding to an integrated luminosity of 668.55 pb −1 ((448.1 ± 2.9) × 10 6 ψ ′ events [24]) produced at the peak of the ψ ′ resonance and collected by the BESIII detector at the BEPCII collider. Additional datasets recorded at the c.m. energies of 3.581 and 3.670 GeV with integrated luminosities of 85.7 and 84.7 pb −1 , respectively, are used to determine the nonresonant continuum background contributions.

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector [25] records e + e − collisions provided by the BEPCII storage ring [26], which operates with a peak luminosity of 1 × 10 33 cm −2 s −1 in the centerof-mass (c.m.) energy range from 2.00 to 4.95 GeV. The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel. The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for the electrons from Bhabha scattering at 1 GeV. The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end cap) region. The time resolution of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps.
Simulated data samples produced with geant4based [28] Monte Carlo (MC) software, which includes the geometric description of the BESIII detector and the detector response, are used to determine detection efficiencies and to estimate background contributions. The simulation models the beam energy spread and initial state radiation (ISR) in the e + e − annihilations with the generator kkmc [29]. The generic MC sample includes the production of the ψ ′ resonance, the ISR production of the J/ψ, and the continuum processes incorporated in kkmc [29]. The known decay modes are generated with evtgen [30] using branching fractions taken from the Particle Data Group [12], and the remaining unknown charmonium decays are modeled with lundcharm [31]. Final state radiation (FSR) from charged final state particles is incorporated using photos [32]. The exclusive decays of ψ ′ → γX are generated following the angular distribution of (1 + λ cos 2 θ), where X refers to η c (2S) or χ cJ , θ is the polar angle of the radiative photon in the rest frame of ψ ′ , and the value of λ is set to 1 for η c (2S) and to 1, −1/3, 1/13 for χ cJ (J = 0, 1, 2) [33], respectively. The X → 3(π + π − ) decay is generated uniformly in phase space (PHSP). Additionally, two exclusive MC samples, ψ ′ → π 0 3(π + π − ) and ψ ′ → (γ FSR )3(π + π − ), are generated according to PHSP to estimate background contamination.

III. EVENT SELECTION
We search for η c (2S) in the exclusive decay ψ ′ → γη c (2S) with η c (2S) → 3(π + π − ) in events containing at least one radiative photon and six charged tracks. Charged tracks detected in the MDC are required to be within a polar angle (θ) range of | cos θ| < 0.93, where θ is defined with respect to the symmetry axis of the MDC (z-axis). For charged tracks, the distance of closest approach to the interaction point must be less than 10 cm along the z-axis, and less than 1 cm in the transverse plane. Charged-particle identification (PID) is based on the combined information from the energy deposited in the MDC (dE/dx) and the flight time measured by the TOF, which are used to determine a variable χ 2 PID (h) for each track, where h denotes the pion, kaon, or proton hypothesis.
Photon candidates are identified using isolated showers in the EMC. The deposited energy of each shower must be larger than 25 MeV in both the barrel region (| cos θ| < 0.80) and end cap region (0.86 < | cos θ| < 0.92). 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.
Candidate events having exactly six charged tracks with net charge zero and at least one candidate photon are retained. To improve the mass resolution and suppress background, the total four-momentum of the charged tracks and the photon candidate is constrained to the initial ψ ′ four-momentum by a kinematic fit (4C fit). If there is more than one photon candidate, the one resulting in the minimum χ 2 from the 4C fit (χ 2 4C ) is selected as the radiative photon. The background due to incorrect PID is suppressed by using χ 2 tot = χ 2 4C + i χ 2 PID (π), where i runs over the six charged tracks. The candidate events satisfying χ 2 tot < 200 and χ 2 4C < 44 are kept as the γ3(π + π − ) candidate events. Background events from the ψ ′ → π + π − J/ψ pro-cess are removed by a J/ψ veto, which requires the recoil masses of all π + π − pairs be below the J/ψ mass (M rec π + π − < 3.05 GeV/c 2 ). Background events from ψ ′ → ηX with η → γπ + π − are greatly suppressed by the η veto, which requires the invariant mass of γπ + π − be outside the η signal region, M γπ + π − > 0.554 or M γπ + π − < 0.538 GeV/c 2 .
The χ 2 4C and η mass window requirements are optimized by maximizing S/ √ S + B, where S and B are the numbers of expected η c (2S) signal and background events in the η c (2S) signal region in data, respectively. The η c (2S) signal region is defined as (3.60, [24] and ǫ is the detection efficiency.
] are taken from the world average values [12], and B is estimated with the generic MC sample.
The background from ψ ′ → 3(π + π − ) with a fake photon satisfying the 4C fit contributes to a peak near the η c (2S) mass and decreases rapidly with higher mass in the 3(π + π − ) mass spectrum due to the threshold of 25 MeV for a photon. The inclusion of a fake photon in the 4C kinematic fit shifts the 3(π + π − ) invariant mass peak lower compared to the true mass. This shift can be corrected by performing a modified kinematic fit in which the energy of the measured photon is allowed to vary in the fit (3C fit). The η c (2S) mass resolution from the 3C fit is similar to that from the 4C fit, while the former can separate the background events from the η c (2S) signal significantly (see Fig. 1). Therefore, the invariant mass spectrum from the 3C fit, M 3C 3(π + π − ) , is used to determine the signal yield.
Comparison between 3C and 4C kinematic fits (unnormalized). Shown in the plot are the invariant mass distributions of 3(π + π − ) of signal events from the 3C fit (blue solid line) and 4C fit (red dashed line), those of background events of ψ ′ → 3(π + π − ) from the 3C fit (black solid line) and 4C fit (black dashed line).
The background events from the ψ ′ → 3(π + π − ) process with an FSR photon have the same final state as the signal process and can contaminate the η c (2S) signal due to a long tail from 3.60 to 3.66 GeV/c 2 in the M 3C distribution. The size of the tail depends on the FSR fraction, which is defined as R FSR = N FSR /N nonFSR , where N FSR (N nonFSR ) is the number of events with (without) an FSR photon surviving the selection. The difference in the FSR fraction between data and MC simulation is studied by using a control sample ψ ′ → γχ c0 , χ c0 → (γ FSR )3(π + π − ) and subsequently corrected for by the FSR correction factor.
The FSR photon in the control sample is required to be reconstructed. The event selection criteria of the control sample are similar to those of the signal sample, except that the final states contain two photons. The softer photon is treated as the FSR photon, and in the 3C fit, the energy of the FSR photon is free. The process ψ ′ → π 0 3(π + π − ) is a dominant background contribution to the control sample, since it has the same final state particles as the control sample. The candidate events satisfying 0.115 < M γγ < 0.150 GeV/c 2 are rejected to remove the background events from ψ ′ → π 0 3(π + π − ), where M γγ is the invariant mass of γγ. The requirement of E hard < 0.2 GeV is placed to remove the background events from ψ ′ → γχ c1,2 , χ c1,2 → (γ FSR )3(π + π − ), where E hard is the energy of the harder photon. For this sample, R Data FSR is determined by fitting the M 3C spectrum. The FSR and nonFSR events are described by the corresponding MC shapes determined from the ψ ′ → (γ FSR )3(π + π − ) MC simulation and convolved with a Gaussian function to account for the resolution difference between data and MC simulation. The parameters of the Gaussian function are free in the fit. The background events from ψ ′ → π 0 3(π + π − ) and ψ ′ → γχ c1,2 , χ c1,2 → (γ FSR )3(π + π − ) are dominant, and their shapes are determined directly from the MC simulated events surviving the γγ3(π + π − ) selection. The distribution of the remaining background events is smooth and thus modeled by an ARGUS function [35]. The number of events of each component is a free parameter, and the fit result is shown in Fig. 2 Though emitting an FSR photon changes the charge conjugate parity of the 3(π + π − ) system, the fact that the f FSR only accounts for the difference on FSR fraction between data and MC simulation and the FSR fraction in MC simulation depends on the masses of the mother and daughter particles [32] proves the independence of the f FSR on the charge conjugate parity of the system. Thus, the f FSR obtained from χ c0 decay is applied for ψ ′ decay in this work. In the fit to determine the numbers of η c (2S) and χ cJ signal events, the background shape from ψ ′ → (γ FSR )3(π + π − ) is described by the sum of MC simulated shapes ψ ′ → 3(π + π − ) and ψ ′ → γ FSR 3(π + π − ) with the FSR fraction corrected by f FSR . 3(π + π − ) from ψ ′ → γχc0, χc0 → (γFSR)3(π + π − ) events. The black dots with error bars are the data, the red solid line is the best fit result, the blue dotted line represents the nonFSR component, the brown dashed line represents the FSR component, the green long-dashed line shows the contribution from known background components, and the pink dash-dotted line represents the smooth shape used to describe the remaining background events.

C. Continuum contribution
The background contribution from the continuum processes (including the initial state radiation) is estimated using the datasets taken at the c.m. energies (E c.m. ) of 3.581 and 3.670 GeV. The momenta and energies of final state particles are scaled to account for the difference in c.m. energy. The mass spectrum is normalized based on the differences in the integrated luminosity and cross section. The energy dependence of the cross section is measured by the BABAR experiment [36], and we determine it to be proportional to 1/s 2.21±0.67 (s = E 2 c.m. ) by a fit to the measured cross sections of e + e − → 3(π + π − ), where the uncertainty includes the statistical and systematic uncertainties.

V. SIGNAL DETERMINATION
The signal yields are determined by a fit to the M 3C 3(π + π − ) spectrum using an unbinned maximum likelihood method, as shown in Fig. 3. The fit range is from 3.325 to 3.700 GeV/c 2 , which includes the χ cJ signals. The line shape of η c (2S) is described by where m is the mass of 3(π + π − ), E γ is the energy of the transition photon in the rest frame of ψ ′ , BW (m) is the Breit-Wigner function, f d (E γ ) is a function to damp the diverging tail from E 3 γ , ǫ(m) is the efficiency curve as a function of m, and G(δm, σ) is a Gaussian function describing the detector resolution determined from MC simulation. The f d (E γ ) form proposed by the KEDR Collaboration [37], , is used in the nominal fit, where E 0 is the most probable energy of the transition photon. The efficiency curve is parametrized by (m/GeV/c 2 )(1 − (m/3.6747 GeV/c 2 ) 2 ) 0.303 × e −3.75(1−(m/3.6747 GeV/c 2 ) 2 ) , obtained by fitting the efficiencies determined at each m using an ARGUS function [35]. The χ cJ line shapes are obtained directly from the MC simulations. The η c (2S) and χ cJ line shapes are convolved with additional Gaussian functions to account for the mass resolution difference between data and MC simulations. For χ cJ , the parameters of the Gaussian function are determined in the fit, while for η c (2S) they are fixed to the values extrapolated from the χ cJ results assuming a linear relationship. The background shapes have been estimated as described above. For ψ ′ → π 0 3(π + π − ) and the continuum contributions, the shapes and the numbers of events are fixed. For ψ ′ → (γ FSR )3(π + π − ), the shape is from MC simulation convolved with a Gaussian function with floating parameters and the number of events is left free. In MC simulation, the fraction of FSR events is corrected by f FSR . The remaining background distribution is smooth, thus is described by an ARGUS function with the number of events a free parameter in the fit. It is assumed there is no interference between the signal and the continuum events. With toy MC samples, we validate that the output values of the mass and width of the η c (2S) signal, as well as the numbers of η c (2S) and χ cJ signal events, are consistent with the inputs within one standard deviation, suggesting that the fit procedure has no bias.
The signal yields (N sig data ) obtained from the fit are summarized in Table I. The χ 2 /ndf value of the fit is 557.2/359 = 1.51, where ndf is the number of degrees of freedom. The statistical significance of η c (2S) is calculated to be 10.8σ from the difference of the logarithmic likelihoods [38], −2 ln(L 0 /L max ), where L max and L 0 are the maximized likelihoods with and without the signal component, respectively. The difference in the number of degrees of freedom (∆ndf=3) has been taken into account. The largest systematic uncertainty, as described in Sec. VI, is from the fit to the mass spectrum. Alternative fits to the M 3C 3(π + π − ) spectrum under different fit conditions are performed, and the η c (2S) signal significance is larger than 9.3σ in all cases. We measure the mass and width of η c (2S), which are M = (3643.4±2.3±4.4) MeV/c 2 and Γ = (19.8±3.9±3.1) MeV, respectively. The branching fraction is calculated using where X refers to η c (2S) or χ cJ , B(ψ ′ → γX) is the branching fraction of ψ ′ → γX [12], and ǫ corr = . Dots with error bars are data, the red solid curve is the best fit result, the blue long-dashed lines show the ηc(2S) and χcJ signal shapes, the cyan dotted line represents the contribution from ψ ′ → (γFSR)3(π + π − ), the green dashed line shows the contribution from ψ ′ → π 0 3(π + π − ), the pink dash-dotted line is the continuum contribution, and the orange dash-dot-dotted line represents the smooth background.
ǫ MC * f corr is the corrected signal detection efficiency. The correction factor of the efficiency takes into account small differences between data and simulation in the single-track reconstruction efficiency; it is f corr = [ N sel i 6 j w ij (p t , cos θ)]/N sel , where N sel is the number of signal MC simulated PHSP events surviving the event selection, i and j run over the surviving events and the six charged tracks, respectively, w ij is the charged track reconstruction weight factor in bins of (p t , cos θ), and p t is the transverse momentum of the track. The values of w ij are obtained using the control sample J/ψ → π + π − π 0 . f corr is calculated using a sampling method, where w ij is sampled according to G(w ij , ∆w ij ). Here ∆w ij is the uncertainty of w ij , G is a Gaussian function. Using 10000 samples, the resulting f corr follows a Gaussian distribution. The mean value of the Gaussian distribution is taken as the nominal efficiency correction factor, and the standard deviation is taken as the uncertainty, labeled as ∆f corr . Table I lists the corrected efficiencies and calculated branching fractions, where the first uncertainty is statistical, the second systematic, and the third from the uncertainty of B[ψ ′ → γη c (2S)].
Our measurement indicates that the branching fractions for χ cJ → 3(π + π − ) are about twice as large as the current world average. To validate our results, we perform a number of cross-checks. We divide the dataset into two sub-datasets, collected in 2009 and 2012, respectively, and compare the branching fractions of χ cJ → 3(π + π − ) obtained from the two sub-datasets. The results are consistent with each other and agree with the nominal results. Additionally, two individual studies of measurements of branching fractions of χ cJ → 3(π + π − ) are performed and the results from two cross-checks agree well with the reported results. Table II summarizes the sources of systematic uncertainties in measuring the branching fractions and the parameters of η c (2S). They are described in the following.

VI. SYSTEMATIC UNCERTAINTIES
The systematic uncertainty from the efficiency correction factor is given by ∆f corr /f corr .
Based on the study of the photon detection efficiency using the control samples of J/ψ → ρ 0 π 0 and e + e − → γγ [39], the systematic uncertainty due to photon reconstruction is 1% per photon.
In the kinematic fit, the helix parameters of charged tracks in MC simulation have been corrected to improve the consistency between data and simulation [40]. The systematic uncertainty from the kinematic fit is taken as half of the difference between the efficiencies with and without track helix parameter correction.
To estimate the uncertainty from the J/ψ veto, we vary the maximum requirement on M rec π + π − from 3.00 to 3.08 GeV/c 2 in steps of 0.01 GeV/c 2 . The maximum difference of the results with respect to the nominal one is taken as the systematic uncertainty from the J/ψ veto. To estimate the uncertainty from the η veto, the mass window of the η veto is changed to be (0.538, 0.552), (0.536, 0.554), (0.534, 0.556), (0.532, 0.558), (0.530, 0.560), (0.528, 0.562), or (0.526, 0.564) GeV/c 2 . The results under the different windows are obtained, and the maximum difference with the nominal one is taken as the systematic uncertainty from the η veto.
An alternative damping function used by the CLEO Collaboration [41], f d (E γ ) = exp(−E 2 γ /8β 2 ), is cho-sen to estimate the uncertainty from the damping function, where β is a free parameter. The difference between the two damping functions is taken as the systematic uncertainty. We model the efficiency curve using a threshold function with the parametrized form of 4 3.675 − (m/GeV/c 2 ) 1.5 − 1.5(3.675 − (m/(GeV/c 2 )) , and take the difference between the two functions as the systematic uncertainty. The resolutions of two Gaussian functions for the line shape of the η c (2S) signal are varied by ±1σ, and the largest difference is taken as the systematic uncertainty.
The systematic uncertainties related to the background contributions are from the FSR process, continuum, and ψ ′ → π 0 3(π + π − ), of which the first two may influence our fit results in the η c (2S) mass region. We vary the FSR correction factor by ±1σ and take the largest difference as the uncertainty from the shape of FSR process. The number of continuum events is determined by the normalization factor calculated using the integrated luminosity and cross section, where the latter is proportional to 1/s 2.21±0.67 . Taking into account a 1.0% uncertainty of the integrated luminosity, a new normalization factor is calculated, and the mass spectrum is fitted with the new fixed number of continuum events. The differences of the measured results with respect to the nominal ones are taken as the uncertainty from the luminosity. Additionally, we change the cross section by varying the exponent by ±0.67 and refit the mass spectrum. The uncertainty from the cross section is estimated to be the difference of the measured result. Finally, the systematic uncertainty from the number of continuum events is determined by adding the uncertainties from the luminosity and the cross section in quadrature. We modify the number of the background events of ψ ′ → π 0 3(π + π − ) by ±1σ to estimate the uncertainty from the number of ψ ′ → π 0 3(π + π − ) events. The line shapes of continuum and ψ ′ → π 0 3(π + π − ) background are smoothed by RooKeysPDF [42] in the nominal fit. We use RooHistPDF [43] and take the difference under the two methods as the systematic uncertainty.
There are obvious ρ and f 0 (980) intermediate states in the M π + π − distribution from the data, as shown in Fig. 4, while these states are not taken into account in the PHSP MC simulation. We correct the M π + π − distribution from PHSP MC simulation to agree with data, as shown by the red line in Fig. 4. Additionally, the distributions of track momenta disagree between data and MC simulation, and we correct the MC sample similarly. The difference in detection efficiency with and without these corrections is taken as the systematic uncertainty. The uncertainty of assuming ψ ′ → γχ cJ as a pure electric dipole transition is studied by considering the contribution from higher-order multipole amplitudes [44? ? ] in the MC simulation, and the differences of the efficiency, 1.0% for χ c1 and 0.2% for χ c2 , are taken as the systematic uncertainties. As a conservative estimate, we assign a 1.0% uncertainty for assuming ψ ′ → γη c (2S) I: Signal yields, corrected signal efficiencies, and branching fractions for ηc(2S) and χcJ decays. The branching fractions of χcJ → 3(π + π − ) from the world average values [12] are also shown.
568. 8  , where f = 3(π + π − ), and absolute systematic uncertainties in the ηc(2S) mass (MeV/c 2 ) and width (MeV) measurements. For the ηc(2S) → 3(π + π − ) mode, the value in parentheses is the total systematic uncertainty without including the uncertainty from B(ψ ′ → γηc(2S)) [13]. as a pure magnetic dipole transition due to the absence of experimental measurements of higher-order multipole amplitudes. The systematic uncertainty from the MC simulation is calculated by adding above uncertainties in quadrature.
There could be interference between the signal process and the e + e − → γ3(π + π − ) process when the quantum numbers of the two systems are the same. To estimate the uncertainty of the assumption of no interference in the nominal fit, we include the contribution from possible interferences into the mass spectrum fit and take the maximum differences of the measured branching fractions and the measured mass and width of η c (2S) relative to the nominal ones as the uncertainty. Using the difference of the maximum likelihoods and the number of degrees of freedom with and without the interference effect, the significance of the interference contribution is calculated to be 0.54σ.
The number of ψ ′ events is determined to be (448.1 ± 2.9)× 10 6 [24]; therefore 0.6% is taken as the uncertainty. The systematic uncertainties from branching fractions of ψ ′ → γX are taken from the world average values [12], where X is η c (2S) or χ cJ . The uncertainty from the statistics of the signal MC sample is considered as well.
Using B[η c (1S) → 3(π + π − )] = (1.7 ± 0.4)% [12], we calculate the ratio of branching fractions where the uncertainty is obtained by assuming the systematic uncertainties of both branching fractions are uncorrelated. This central value seems to lean slightly towards the prediction from Ref. [11] over the prediction from Ref. [10], but our result is compatible with both predictions.