Measurement of the phase between strong and electromagnetic amplitudes of $J/\psi$ decays

Using 16 energy points of $e^{+}e^{-}$ annihilation data collected in the vicinity of the $J/\psi$ resonance with the BESIII detector and with a total integrated luminosity of around 100 pb$^{-1}$, we study the relative phase between the strong and electromagnetic amplitudes of $J/\psi$ decays. The relative phase between $J\psi$ electromagnetic decay and the continuum process ($e^{+}e^{-}$ annihilation without the $J/\psi$ resonance) is confirmed to be zero by studying the cross section lineshape of $\mu^{+}\mu^{-}$ production. The relative phase between $J/\psi$ strong and electromagnetic decays is then measured to be $(84.9\pm3.6)^\circ$ or $(-84.7\pm3.1)^\circ$ for the $2(\pi^{+}\pi^{-})\pi^{0}$ final state by investigating the interference pattern between the $J/\psi$ decay and the continuum process. This is the first measurement of the relative phase between $J/\psi$ strong and electromagnetic decays into a multihadron final state using the lineshape of the production cross section. We also study the production lineshape of the multihadron final state $\eta\pi^{+}\pi^{-}$ with $\eta\to\pi^{+}\pi^{-}\pi^{0}$, which provides additional information about the phase between the $J/\psi$ electromagnetic decay amplitude and the continuum process. Additionally, the branching fraction of $J/\psi\to 2(\pi^{+}\pi^{-})\pi^{0}$ is measured to be $(4.73\pm0.44)\%$ or $(4.85\pm0.45)\%$, and the branching fraction of $J/\psi\to\eta\pi^{+}\pi^{-}$ is measured to be $(3.78\pm0.68)\times10^{-4}$. Both of them are consistent with the world average values. The quoted uncertainties include both statistical and systematic uncertainties, which are mainly caused by the low statistics.


INTRODUCTION
The relative phase between the strong and electromagnetic (EM) amplitudes of quarkonium decays is a basic parameter that provides insight into the dynamics of quarkonium decays. As shown in Fig. 1, in the vicinity of the J/ψ, the annihilation of e + e − into a hadronic final state proceeds through three processes: strong decay of the J/ψ (mediated by gluons), EM decay of the J/ψ (mediated by a virtual photon), and the continuum process (without a J/ψ intermediate state and mediated by a virtual photon). For leptonic final states, on the other hand, the strong decay is absent. In perturbative quantum chromodynamics, the relative phase (Φ g,γ ) between the charmonium strong decay amplitude (A g ) and the EM amplitude (A γ ) is predicted to be 0 • or 180 • [1,2] at lowest order. In contrast to this prediction, model-dependent analyses using SU(3) flavor symmetry suggest that Φ g,γ is 90 • for J/ψ two-body decays into meson pairs with quantum numbers (J P ) of 1 − 0 − [3,4], 0 − 0 − [5,6,7], 1 − 1 − [7], and 1 + 0 − [8], and for J/ψ decays into NN baryon pairs [9,10]. Similar analyses suggest ψ(2S) decays to pairs of pseudoscalar mesons also have a phase Φ g,γ around 90 • , but ψ(2S) decays to pairs of mesons with 1 − 0 − and 1 + 0 − have a different value of Φ g,γ [8,11].
Several theoretical ideas regarding the origin and implications of Φ g,γ have been proposed. Based on unsubtracted dispersion relations and asymptotic freedom, the Okubo-Zweig-Iizuka-rule-violating amplitude with respect to the virtual photon contribution is predominately imaginary [12]. An orthogonal phase in J/ψ decays is also expected if any vector quarkonium is assumed to be coupled to a vector glueball [13,14,15]. Furthermore, it has been advocated [8,15] that different phases for the J/ψ and ψ(2S) decay, namely ∼ 90 • and ∼ 180 • , respectively, can explain the long-standing ρπ puzzle of charmonium physics. However, there is no simple explanation that the phase should be 90 • .
An independent approach for measuring the relative phases of the diagrams in Fig. 1 consists of extracting the interference pattern of the e + e − reaction cross section as a function of the center-of-mass (CM) energy (W ) in the vicinity of a resonance. The Born cross section of a pure EM process can be written as σ 0 (W )∝|A γ (W )e iΦγ,cont + A cont (W )| 2 .
The relative phase (Φ γ,cont ) between the J/ψ EM amplitude (A γ ) and the continuum amplitude (A cont ) has previously been assumed to be zero degrees and this assumption has been shown to be consistent with the observed interference pattern in J/ψ decays to lepton pairs [16,17,18,19]. The full Born cross section for processes including the strong and EM amplitudes can be written as If we take the phase Φ γ,cont to be zero, as measurements suggest, the Born cross section is simplified to be: where Φ g,EM is the phase between the strong and the full EM amplitudes.
It is argued that the relative phases Φ γ,cont and Φ g,EM are universal in all exclusive decay modes [20]. In this Letter, we first analyze the process e + e − → µ + µ − and confirm the phase Φ γ,cont is consistent with zero. We also use this process to extract the CM energy spread and the overall energy scale, which are essential accelerator parameters that are used as input for the other analyses. Then, we measure the phase Φ g,EM by analyzing the process e + e − → 2(π + π − )π 0 (abbreviated as 5π). We chose this process because it both has a large branching fraction in J/ψ decays and has a sizable cross section of the continuum decay. We also study the process e + e − → ηπ + π − with η decaying into π + π − π 0 . Since it proceeds largely through ηρ 0 , which is an EM process due to G-parity conservation, this process is used to gain further information about Φ γ,cont . This is the first measurement of the phases Φ g,EM and Φ γ,cont in the interference pattern of the cross section lineshape in the vicinity of the J/ψ and the first time using multihadron final states.
This letter is organized as follows: in Section 2, the BESIII detector and the data sets being used are described. In Section 3, the event selection, the efficiency, the observed cross section and the systematic uncertainties of e + e − → µ + µ − , 5π and ηπ + π − are described. In Section 4, the fit to the cross section lineshapes of e + e − → µ + µ − , 5π and ηπ + π − as well as the results are reported. The results are summarized in Section 5.

BESIII EXPERIMENT AND DATA SETS
The BEPCII is a double-ring e + e − collider running at CM energies between 2.0 − 4.6 GeV and it has reached its design luminosity of 1.0 × 10 33 cm −2 s −1 at a CM energy of 3770 MeV. The cylindrical BESIII detector has an effective geometrical acceptance of 93% of 4π solid angle and it is divided into a barrel section and two endcaps. It consists of a small-cell, helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), a CsI(TI) (Thallium doped Cesium Iodide) crystal electromagnetic calorimeter (EMC) and a muon system containing resistive plate chambers in the iron return yoke of a 1 Tesla (0.9 Tesla for data sets used in this letter) superconducting solenoid. The momentum resolution for charged tracks is 0.5% for 1 GeV/c momentum tracks. The time resolution in the barrel (endcaps) is 80 ps (110 ps). The photon energy resolution at 1 GeV is 2.5% (5%) in the barrel (endcaps) of the EMC. Further details about the BESIII detector are described in Ref. [21].
This analysis uses data samples collected in 2012 at 16 different CM energies with a total integrated luminosity of about 100 pb −1 [22]. The CM energies, W i , and the integrated luminosities, L i , of each data sample are summarized in Table 1. The CM energies are measured by the Beam Energy Measurement System (BEMS), in which photons from a CO 2 laser are Compton back-scattered off the electron beam and detected by a high-purity Germanium detector [23]. The integrated luminosities are determined using two-gamma events [22].
A geant4-based [24] simulation software package including a description of the geometry and material and the detector response is used to generate Monte Carlo (MC) samples. The babayaga [25] generator which includes interference between A cont and A γ is used to simulate the e + e − → µ + µ − and the e + e − → e + e − events. The samples e + e − → 5π and intermediate processes e + e − → ρ 0 ρ ± π ∓ , e + e − → ρ 0 f 2 (1275)π 0 , e + e − → ωπ + π − and e + e − → ηπ + π − are generated assuming a uniform phase space distribution. The intermediate decays e + e − → ηρ 0 and ηω and the subsequent decay of all intermediate states are generated with evtgen [26,27]. For the 5π system, the polar angular distributions for each of the pions in the e + e − CM frame are tuned to be the same as those in data. The mcgpj [28] generator is used to incorporate radiation effects in the e + e − → 5π process. The possible interference between A g and A cont (or A γ ) is included in the mcgpj generator. The output cross section from the mcgpj generator is tuned to be the same as the observed cross section of e + e − → 5π. A MC sample of J/ψ inclusive decays is used to explore possible hadronic background. In this sample, the known decay modes are generated with evtgen incorporating the branching fractions from the Particle Data Group (PDG) [29] and the remaining unknown decays are generated according to the lundcharm [30] model. The CM energy spread is incorporated in all MC samples.

ANALYSIS
3.1. Event selection for the e + e − → µ + µ − process Events of e + e − → µ + µ − are required to have only two charged tracks with opposite charge. The charged tracks are required to originate from the interaction region which is defined as a cylinder with a radius of Table 1: The CM energy (Wi) and the integrated luminosity (Li) for each data point. The uncertainty of Wi is from the BEMS measurement, and the uncertainty of Li is the statistical and systematic uncertainties added in quadrature [22].

No.
W 1 cm and an axial distance from the interaction point of ±10 cm. The polar angle θ of each track with respect to the positron beam is required to be within the barrel region (| cos θ| < 0.8). Each charged track must have hit information in the EMC, and its measured energy deposit divided by its momentum obtained from the MDC (E/p) is required to be less than 0.3 to suppress e + e − → e + e − and hadronic final state events. Cosmic rays are rejected by requiring ∆T ≡ |T trk1 − T trk2 | < 4 ns, where T trk1 and T trk2 are the measured flight times in the TOF detector for the two tracks. The improved track parameters obtained from the vertex fit, which constrains the two tracks to a common vertex, are used in further analysis. The momenta of muon candidates must satisfy (p the − 4σ p ) < p µ ± < (p the + 3σ p ), where p the and σ p are the nominal value and experimental resolution of the momentum of µ ± , respectively. Figure 2 (a) shows the momentum distributions of data and the babayaga MC sample at W = 3092.32 MeV. Throughout this letter, all the performance plots are from the same energy point. The data appear to be consistent with the MC simulations. Potential two-body decay backgrounds are estimated by investigating the exclusive MC samples of e + e − → pp, K + K − , π + π − , and e + e − . Only the process e + e − → π + π − is found to be a potential background. According to Ref. [31], the cross section of π + π − is about 10 −2 nb at 3000 MeV, which is negligible compared to that of µ + µ − of about 10 nb. At the J/ψ peak, the ratio between the branching fraction of J/ψ → π + π − and that of J/ψ → µ + µ − is about 0.2%. Taking into account the selection efficiency, where that of e + e − → π + π − is about one third of that of e + e − → µ + µ − , the background from the π + π − final state can safely be ignored. From a study of the J/ψ inclusive MC sample, the contribution from the remaining multihadron events is about 0.2% of the surviving events and is also negligible.

3.2.
Event selection for the e + e − → 5π and ηπ + π − processes The events are required to have four charged tracks with a net charge of zero and at least two photons. The charged tracks are required to originate from the interaction region, while their polar angles are required to be within a range of | cos θ| < 0.93. Charged particle identification is performed by combining the ionization energy loss (dE/dx) in the MDC and the flight times in the TOF. For each track, the probability for the pion particle hypothesis is required to be larger than that for the kaon particle hypothesis. The photons are required to have a deposited energy greater than 50 MeV in the endcap (0.86 < | cos θ| < 0.92) or 25 MeV in the barrel (| cos θ| < 0.8) of the EMC. To suppress electronic noise and energy deposits unrelated to the event, the time of the cluster signal given by the EMC must be within 700 ns after the reconstructed event start time. To exclude clusters originating from charged tracks, the angle between the photon candidate and the nearest charged track is required to be greater than 10 • .
After constraining the four charged tracks to a common vertex using a vertex fit, a four-constraint (4C) kinematic fit imposing energy and momentum conservation is performed to the e + e − → 2(π + π − )γγ hypothesis. Events with χ 2 4C < 200 are retained, and at least 80% of the background is rejected and about 95% signal is retained. If there are more than two photons, all combinations of photon pairs are tried and that with the least χ 2 4C value is retained. Figure 2 (b) shows the distribution of χ 2 4C for the data and MC simulation. The invariant mass of the photon pair M γγ is required to be within the range (0.0, 0.3) GeV/c 2 . The decay angle (θ decay ) of a photon is defined as the polar angle measured in the π 0 rest frame with respect to the π 0 direction in the e + e − CM frame. The cosine of the decay angle (cos θ decay ) is required to be lower than 0.9 to remove wrong photon combinations.
Only the e + e − → γ5π makes a peaking background of less than 1% of the π 0 events on the spectrum of M γγ .
The surviving candidate events include events from the process with an η intermediate state, i.e. e + e − → ηπ + π − with η decays to π + π − π 0 . Due to G-parity conservation, the dominant process e + e − → ηρ 0 → ηπ + π − is allowed only via EM decay, and will affect the measurement of Φ g,EM for the process e + e − → 5π. Thus, the process of e + e − → ηπ + π − will be separated from the inclusive e + e − → 5π, and measured alone. In the inclusive e + e − → 5π candidate events, we reconstructed the η signal with the π + π − γγ combination whose invariant mass M η π + π − γγ is closest to the η nominal mass. The signal candidate of e + e − → 5π is then selected by imposing a further requirement of M η π + π − γγ < 0.517 MeV/c 2 or M η π + π − γγ > 0.577 MeV/c 2 . The corresponding yield is determined by fitting the distribution of γγ invariant mass, M γγ , with a double Gaussian function for the signal and a second-order polynomial function for background, as shown in Fig. 2 (c). The yield of e + e − → ηπ + π − is determined by fitting the M η π + π − γγ distribution, where the η signal is modeled by a Gaussian function and the background is described by a third-or lower-order polynomial function, as presented in Fig. 2 (d). To better describe the data, the parameters of the η and π 0 signal lineshapes are fixed to values obtained from fits to distributions summed over all CM energies.

3.3.
Cross sections of e + e − → µ + µ − , 5π and ηπ + π − The observed cross section is calculated with where N i is the number of observed signal events, i is the efficiency given by the MC simulations, and L i is the luminosity listed in Table 1. In the equation, B denotes the branching fractions of intermediate decays, and is B(π 0 → γγ) for e + e − → 5π and B(η → π + π − π 0 ) × B(π 0 → γγ) for e + e − → ηπ + π − . For e + e − → µ + µ − , the efficiency from the babayaga simulation includes the radiative effects [25].
For the process e + e − → 5π, to take into account kinematic effects of the intermediate states, the weighted-average efficiency com i obtained according to the relative production rates between the processes with different intermediate states is used. The interference among different intermediate processes is assumed to be independent of the phase measurement and not taken into account. To take into account the radiation effect, an additional CM energy-dependent correction factor, f EC i is used, which is the ratio of the detection efficiencies of e + e − → 5π at the i-th CM energy point estimated with the generator mcgpj to that at the J/ψ peak. The generator mcgpj models radiation effect for the process e + e − → 5π properly by adjusting the output cross section to be the same as the calculated σ obs From the PDG, we know the decays J/ψ → ηρ, ηω, and ηπ + π − also exist, even though the measured branching ratios are very old and have large uncertainties. According to MC simulations, the efficiencies for these processes are nearly the same. Thus, the efficiency of the MC sample for e + e − → ηπ + π − , without intermediate states, is used in the cross section calculation. The efficiency correction factor f EC i is not implemented due to the large statistical uncertainty of its cross section and the small effect of f EC i on the phase measurement (see the results of the 5π in Section 4). The calculated cross sections for e + e − → µ + µ − , 5π and ηπ + π − , together with the efficiencies and the number of events, are listed in Table 2. The number of events, efficiency and the observed cross section for e + e − → µ + µ − , 5π and ηπ + π − at each energy point. Statistical uncertainties are quoted for the number of events and the efficiencies, while both statistical and systematic uncertainties are quoted for the cross section.

Systematic uncertainties
Systematic uncertainties are divided into two categories. Those that are universal among the different energy points include those related to the event selection efficiencies, intermediate states in e + e − → ηπ + π − , and the branching fractions of intermediate state decays. Those that are not universal are treated separately for all energy points, which include the uncertainties related to the fits to the spectra, com i of e + e − → 5π, and the luminosities.
The systematic uncertainty of the tracking of muons is studied with a control sample of J/ψ → µ + µ − selected with more stringent criteria on one tagged charged track. The efficiency is the rate to detect another charged track on the recoil side of the tagged track. The difference on the efficiency is 1% between data and MC simulation, which is treated as the systematic uncertainty. The systematic uncertainties associated with the tracking and the particle identification for pion candidates are investigated using a control sample of J/ψ → ppπ + π − , and are found to be 1% individually [33]. Dedicated studies on e + e − → γµ + µ − [34] and J/ψ → π + π − π 0 [35] conclude that the systematic uncertainty due to photon identification is 1% per photon. The systematic uncertainty related to the 4C kinematic fit is determined by changing the χ 2 4C requirement, 9 and found to be 1%. The uncertainties of the branching fractions for the intermediate-state decays π 0 → γγ and η → π + π − π 0 from the PDG [29] are considered in the systematic uncertainty.
The requirements of cos θ, E/p, |∆T | and p µ ± in the selection of e + e − → µ + µ − , and M η π + π − γγ and cos θ decay in the selection of e + e − → 5π are varied at all energy points. The largest difference of the cross section with respect to the nominal result at each energy point is taken as the deviation of each requirement. The weighted-average deviation (with weights of statistics of each energy point) of each item is taken as the uncertainties. The uncertainties of cos θ, E/p, |∆T |, p µ ± , M η π + π − γγ and cos θ decay are determined as 0.16%, 0.09%, 0.05%, 0.26%, 0.04%, and 0.40%, respectively. The uncertainties of the requirement of cos θ decay are the same for the processes of e + e − → 5π and e + e − → ηπ + π − .
The uncertainties associated with the fit procedure on the M γγ and M η π + π − γγ distributions are estimated by changing the signal shapes to the Crystal Ball function and MC simulated histograms, respectively, extending or shrinking the fit ranges, changing the background shapes to a higher or lower order of the polynomial functions, and changing the interval width of each spectrum. The largest deviations of results for the different fit scenarios with respect to the nominal values are regarded as the individual systematic uncertainties and are added in quadrature to be the systematic uncertainty associated with the fit procedure. Due to the low statistics in the process of e + e − → ηπ + π − , ensembles of simulated data samples (toy MC samples) at each energy point are generated according to the nominal fit result with the same statistics as data, then fitted by the alternative fitting scenario. These trials are performed 1000 times, and the average signal yields are taken as the results. For the data with the CM energy being 3101.92, 3106.14, 3112.62, and 3120.44 MeV, the statistics are extremely low and the uncertainties of the fit procedure are assigned to be the same as that for data at CM energy of 3099.61 MeV. Totally, the fit procedure introduces systematic uncertainties of about 1 − 2% and 11% for the channels e + e − → 5π and ηπ + π − , respectively.
The systematic uncertainty due to the intermediate states in e + e − → ηπ + π − is about 3.0%, estimated as the difference between the weighted-average efficiency which takes into account the efficiencies and the relative branching fractions of J/ψ → ηπ + π − and J/ψ → ηρ 0 and the efficiency of J/ψ → ηπ + π − .
The uncertainty associated with com i in the decay e + e − → 5π mainly comes from the statistical uncertainty of the relative ratios among different processes. Besides, the measured angular distributions of the pions in the MC samples are corrected to be the same as those measured in data. The systematic uncertainty due to the correction is estimated to be 0.1%-5.8% depending on the statistics of each dataset. The uncertainty of the luminosity determination is determined to be 1.1-1.3%, as listed in Table 1.
All the systematic uncertainties discussed above are combined in quadrature to obtain the overall systematic uncertainties.

RESULTS
Due to the effects of radiation and CM energy spread, the observed cross section cannot be directly compared with the Born cross section. In this section, the fit formulas for the observed cross section and fit results are presented.
The Born cross section of e + e − → µ + µ − , consisting of only the pure EM contributions, A γ and A cont , is conventionally expressed as [16,17,18,19]: where α is the fine structure constant, M and Γ are the mass and width of the J/ψ, and Γ ee and Γ µµ are the partial widths of J/ψ → e + e − and µ + µ − , respectively. Incorporating the radiative correction F (x, W ), the cross section reads: where W min is the minimum invariant mass of the µ + µ − system, x = 2Eγ √ s , E γ is the energy of the radiation photon, and F (x, W ) is approximated as [36]: with δ = 3 4 β + α π ( π 2 3 − 1 2 ) + β 2 ( 9 32 − π 2 12 ) and β = 2α π (2 ln W me − 1). The CM energy spread (S E ) is included by convolving with a Gaussian function with a width of S E , and the expected cross section σ (W i ) at W i is obtained as Finally, the minimizing function is built with the factored minimization method separating the correlated and uncorrelated systematic uncertainties, and the effective variance-weighted least squares method [37] including the uncertainty ∆W along the X-axis by projecting it along the Y -axis. The resulting χ 2 reads where f is the normalization factor, and ∆f is its uncertainty and set as the total correlated systematic uncertainty. The term ∆σ obs i is the combined statistical and uncorrelated systematic uncertainties. Thus, the obtained uncertainties of results in this section include statistical and systematic ones. To be more efficient in the fitting procedure, instead of two integrations, an approximation [38,39,40] is used for σ (W ), which is decomposed into a resonance and interference term σ R+I (W ) as well as a continuum term σ C (W ). The formulas can be found in the Appendix. The vacuum polarization is included by quoting the value of Γ ee , Γ µµ and Γ from the world averaged values [29,41].
We perform the minimized χ 2 fit to the measured cross section of e + e − → µ + µ − with the free parameters S E , M , and Φ γ,cont and the fit curve is presented in Fig. 3 (a). The minimization gives Φ γ,cont = (3.0±5.7) • , which is consistent with zero as expected. A scan of the parameter Φ γ,cont in the full range (−180 • , 180 • ) confirms that only one solution exists. By further fixing the parameter Φ γ,cont to be zero, an alternative fit is carried out, resulting in values of S E and M consistent with the previous fit. The resultant M is higher than its world average value M J/ψ [29], and indicates a deviation of the absolute energy calibration for the BEMS. Therefore, to obtain the proper detection efficiencies, the CM energies of MC samples are corrected by shifting the absolute values with ∆M = M − M J/ψ . To estimate the effect of fixed Γ ee , Γ µµ and Γ in the fit, the alternative fits are performed by letting these variables free in the fit. We also perform two additional fits, one with an alternative analytical formula in Ref. [18] which takes a different approximation, and the other is without the factored minimization method, in which the ∆σ obs i is the statistical and systematic uncertainties added in quadrature. All of the results from various fits turn out to be consistent with each other. Taking into account the differences among different approaches and parameterizations, we obtain Φ γ,cont = (3.0 ± 10.0) • , S E = (0.90 ± 0.03) MeV and ∆M = (0.57 ± 0.05) MeV/c 2 , which will be used in the fit for the hadronic final state.
By assuming the Φ γ,cont to be zero, the Born cross section for e + e − → 5π is written as where C is the ratio of |Ag| |Aγ | , A W 2 is the form factor, and A is a free parameter in the fits. The decay width of J/ψ → 5π can be calculated as Γ 5π = ( A W 2 ) 2 Γ µµ |Ce iΦg,EM + 1| 2 , and the corresponding branching fraction is then B(J/ψ → 5π) = Γ 5π /Γ. The analytical form σ (W ) for the e + e − → 5π is similar to that for e + e − → µ + µ − (see Appendix). Similar to the fit for e + e − → µ + µ − , we fix the parameters Γ ee , Γ µµ and Γ to the world average values [29] in the fit. By further fixing the values of M and S E obtained in e + e − → µ + µ − and constraining Φ g,EM to be within (0, 180) • , the minimization fit yields Φ g,EM = (84.9 ± 2.6) • and B(J/ψ → 5π) = (4.73 ± 0.41)%. An alternative fit with M and S E being free parameters is performed to estimate the systematic uncertainty associated with the fixed M and S E . The uncertainties associated with those of Γ ee , Γ µµ and Γ are evaluated by shifting the corresponding values within one standard deviation. The impact of f EC i is estimated by fitting the cross sections without incorporating it. The minimization function without the factor f is also applied to consider its influence. All results of Φ g,EM and B(J/ψ → 5π) are consistent within the uncertainty and the differences are included into the final uncertainties which listed in Table 3.  Constraining Φ g,EM in the interval (−180, 0) • , the second solution is found. The similar alternative fits described above are also carried out to estimate the corresponding systematic uncertainties. The results are listed in Table 3, too. As a check, the branching ratio of J/ψ → 5π via only A γ can be calculated by B(J/ψ → 5π) 1 |1+C| 2 , and it is consistent with the result calculated from B(J/ψ → µ + µ − ) σ(e + e − →5π)
The process e + e − → ηπ + π − is dominated by e + e − → ηρ 0 (where the ρ 0 mixes with the ω), while the non-resonant e + e − → ηπ + π − decay is much smaller. Due to the limited statistics, the non-resonant decay is not considered in the fitting formula. The process with a ρ 0 intermediate state violates G parity and is therefore a pure EM process, while that with an ω intermediate state can also proceed through the strong interaction. Thus, the Born cross section for ηπ + π − can be written as where C 1 represents the contribution from e + e − → ηρ 0 , and C 2 and Φ are the ratio and the relative phase between the processes J/ψ → ηω and J/ψ → ηρ 0 , respectively. The ratio between the branching fractions B(J/ψ→ηω,ω→π + π − ) B(J/ψ→ηρ 0 ,ρ 0 →π + π − ) is (0.138 +0.025 −0.026 ) according to PDG [29]. Thus, C 2 is fixed to be √ 0.138 in the fit. The branching fraction of J/ψ → ηπ + π − is determined by ( A W 2 ) 2 |C 1 e iΦγ,cont (1 + C 2 e iΦ )| 2 B(J/ψ → µ + µ − ).
Analogously, fixing the Γ ee , Γ µ + µ − and Γ as well as M and S E , and assuming the relative phase Φ to be 0 • or 90 • for two extreme cases since it cannot be extracted from the fit, the fit yields Φ γ,cont to be (−2 ± 36) • or (−22 ± 36) • with the same goodness of fit. The fit curve with Φ = 0 • is shown in Fig. 3 (c), where the lineshape of e + e − → ηπ + π − is very similar to that of the process e + e − → µ + µ − , but different from that of e + e − → 5π. The branching fractions in the two cases are both calculated to be (3.78 ± 0.66) × 10 −4 . In alternative fits with floating M and S E , changing Γ ee , Γ µ + µ − , Γ and C 2 by one standard deviation, or using the minimization method without the factor f , neither the phase nor the branching fraction changes. Taking differences of various fit results as the systematic uncertainties, the phase Φ γ,cont is determined to be (−2 ± 36) • or (−22 ± 36) • , and the B(J/ψ → ηπ + π − ) is (3.78 ± 0.68) × 10 −4 .