Cross section measurement of $e^+e^- \rightarrow \eta'J/\psi$ from $\sqrt{s} = 4.178$ to $4.600$ GeV

M. Ablikim, M. N. Achasov, P. Adlarson, S. Ahmed, M. Albrecht, M. Alekseev , A. Amoroso , F. F. An, Q. An, Y. Bai, O. Bakina, R. Baldini Ferroli , I. Balossino , Y. Ban, K. Begzsuren, J. V. Bennett, N. Berger , M. Bertani, D. Bettoni, F. Bianchi , J Biernat, J. Bloms , I. Boyko, R. A. Briere, H. Cai , X. Cai , A. Calcaterra , G. F. Cao, N. Cao, S. A. Cetin , J. Chai , J. F. Chang, W. L. Chang, G. Chelkov, D. Y. Chen, G. Chen, H. S. Chen, J. Chen, M. L. Chen, S. J. Chen, Y. B. Chen, W. Cheng , G. Cibinetto, F. Cossio , X. F. Cui, H. L. Dai, J. P. Dai, X. C. Dai, A. Dbeyssi, D. Dedovich, Z. Y. Deng, A. Denig, I. Denysenko, M. Destefanis , F. De Mori , Y. Ding, C. Dong, J. Dong, L. Y. Dong, M. Y. Dong, Z. L. Dou, S. X. Du, J. Z. Fan, J. Fang, S. S. Fang, Y. Fang, R. Farinelli , L. Fava , F. Feldbauer, G. Felici, C. Q. Feng, M. Fritsch, C. D. Fu, Y. Fu, Q. Gao, X. L. Gao , Y. Gao, Y. Gao, Y. G. Gao, B. Garillon , I. Garzia , E. M. Gersabeck, A. Gilman, K. Goetzen, L. Gong, W. X. Gong , W. Gradl, M. Greco , L. M. Gu, M. H. Gu, S. Gu, Y. T. Gu, A. Q. Guo, L. B. Guo, R. P. Guo, Y. P. Guo, A. Guskov, S. Han, X. Q. Hao, F. A. Harris, K. L. He, F. H. Heinsius, T. Held, Y. K. Heng, M. Himmelreich , Y. R. Hou, Z. L. Hou, H. M. Hu, J. F. Hu, T. Hu, Y. Hu, G. S. Huang, J. S. Huang, X. T. Huang, X. Z. Huang, N. Huesken, T. Hussain, W. Ikegami Andersson, W. Imoehl, M. Irshad, Q. Ji, Q. P. Ji, X. B. Ji, X. L. Ji, H. L. Jiang, X. S. Jiang , X. Y. Jiang, J. B. Jiao, Z. Jiao , D. P. Jin , S. Jin, Y. Jin, T. Johansson, N. Kalantar-Nayestanaki, X. S. Kang, R. Kappert, M. Kavatsyuk, B. C. Ke, I. K. Keshk, A. Khoukaz, P. Kiese, R. Kiuchi, R. Kliemt, L. Koch, O. B. Kolcu , B. Kopf, M. Kuemmel, M. Kuessner, A. Kupsc, M. Kurth, M. G. Kurth, W. Kühn, J. S. Lange, P. Larin, L. Lavezzi , H. Leithoff, T. Lenz, C. Li, C. H. Li, Cheng Li, D. M. Li, F. Li, G. Li, H. B. Li, H. J. Li , J. C. Li, Ke Li, L. K. Li, Lei Li, P. L. Li, P. R. Li, W. D. Li, W. G. Li, X. H. Li , X. L. Li, X. N. Li, Z. B. Li, Z. Y. Li, H. Liang, H. Liang , Y. F. Liang, Y. T. Liang, G. R. Liao, L. Z. Liao , J. Libby, C. X. Lin, D. X. Lin, Y. J. Lin, B. Liu, B. J. Liu, C. X. Liu, D. Liu, D. Y. Liu, F. H. Liu, Fang Liu, Feng Liu, H. B. Liu, H. M. Liu, Huanhuan Liu, Huihui Liu, J. B. Liu, J. Y. Liu, K. Liu, K. Y. Liu, Ke Liu, L. Y. Liu, Q. Liu, S. B. Liu, T. Liu, X. Liu, X. Y. Liu, Y. B. Liu, Z. A. Liu , Zhiqing Liu, Y. F. Long, X. C. Lou, H. J. Lu, J. D. Lu, J. G. Lu, Y. Lu, Y. P. Lu, C. L. Luo, M. X. Luo, P. W. Luo, T. Luo , X. L. Luo, S. Lusso , X. R. Lyu, F. C. Ma, H. L. Ma, L. L. Ma, M. M. Ma , Q. M. Ma, X. N. Ma, X. X. Ma, X. Y. Ma, Y. M. Ma, F. E. Maas, M. Maggiora , S. Maldaner, S. Malde, Q. A. Malik, A. Mangoni , Y. J. Mao, Z. P. Mao, S. Marcello , Z. X. Meng, J. G. Messchendorp, G. Mezzadri, J. Min, T. J. Min, R. E. Mitchell, X. H. Mo , Y. J. Mo, C. Morales Morales , N. Yu. Muchnoi, H. Muramatsu, A. Mustafa, S. Nakhoul , Y. Nefedov, F. Nerling , I. B. Nikolaev, Z. Ning, S. Nisar, S. L. Niu, S. L. Olsen, Q. Ouyang, S. Pacetti , Y. Pan, M. Papenbrock, P. Patteri, M. Pelizaeus, H. P. Peng, K. Peters , J. Pettersson, J. L. Ping, R. G. Ping, A. Pitka, R. Poling , V. Prasad , M. Qi, S. Qian, C. F. Qiao, X. P. Qin, X. S. Qin, Z. H. Qin, J. F. Qiu, S. Q. Qu, K. H. Rashid, K. Ravindran, C. F. Redmer, M. Richter, A. Rivetti , V. Rodin, M. Rolo , G. Rong, Ch. Rosner, M. Rump, A. Sarantsev, M. Savrié , Y. Schelhaas, K. Schoenning, W. Shan, X. Y. Shan, M. Shao, C. P. Shen, P. X. Shen, X. Y. Shen, H. Y. Sheng, X. Shi, X. D Shi, J. J. Song, Q. Q. Song, X. Y. Song, S. Sosio , C. Sowa, S. Spataro , F. F. Sui, G. X. Sun, J. F. Sun, L. Sun, S. S. Sun, X. H. Sun, Y. J. Sun, Y. K Sun, Y. Z. Sun, Z. J. Sun, Z. T. Sun, Y. T Tan, C. J. Tang, G. Y. Tang, X. Tang, V. Thoren, B. Tsednee, I. Uman, B. Wang, B. L. Wang, C. W. Wang, D. Y. Wang, K. Wang, L. L. Wang, L. S. Wang, M. Wang, M. Z. Wang, Meng Wang, P. L. Wang, R. M. Wang, W. P. Wang, X. Wang, X. F. Wang, X. L. Wang , Y. Wang, Y. Wang, Y. F. Wang, Y. Q. Wang, Z. Wang, Z. G. Wang, Z. Y. Wang, Z. Y. Wang, Zongyuan Wang, T. Weber, D. H. Wei, P. Weidenkaff, F. Weidner, H. W. Wen, S. P. Wen, U. Wiedner, G. Wilkinson, M. Wolke, L. H. Wu, L. J. Wu, Z. Wu, L. Xia, Y. Xia, S. Y. Xiao, Y. J. Xiao, Z. J. Xiao, Y. G. Xie, Y. H. Xie, T. Y. Xing, X. A. Xiong, Q. L. Xiu, G. F. Xu, J. J. Xu, L. Xu, Q. J. Xu, W. Xu, X. P. Xu, F. Yan, L. Yan , W. B. Yan, W. C. Yan, Y. H. Yan, H. J. Yang, H. X. Yang, L. Yang, R. X. Yang, S. L. Yang, Y. H. Yang, Y. X. Yang, Yifan Yang, Z. Q. Yang, M. Ye, M. H. Ye, J. H. Yin, Z. Y. You, B. X. Yu, C. X. Yu, J. S. Yu, T. Yu, C. Z. Yuan, X. Q. Yuan, Y. Yuan, C. X. Yue, A. Yuncu, A. A. Zafar, Y. Zeng, B. X. Zhang, B. Y. Zhang, C. C. Zhang, D. H. Zhang, H. H. Zhang, H. Y. Zhang, J. Zhang, J. L. Zhang, J. Q. Zhang, J. W. Zhang, J. Y. Zhang, J. Z. Zhang, K. Zhang, L. Zhang, L. Zhang, S. F. Zhang, T. J. Zhang, X. Y. Zhang, Y. Zhang, Y. H. Zhang, Y. T. Zhang, Yang Zhang, Yao Zhang, Yi Zhang , Yu Zhang, Z. H. Zhang, Z. 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The cross section of the process e + e − → η ′ J/ψ is measured at center-of-mass energies from √ s = 4.178 to 4.600 GeV using data samples corresponding to a total integrated luminosity of 11 fb −1 collected with the BESIII detector operating at the BEPCII storage ring. The dependence of the cross section on √ s shows an enhancement around 4.2 GeV. While the shape of the cross section cannot be fully explained with a single ψ(4160) or ψ(4260) state, a coherent sum of the two states does provide a reasonable description of the data.

I. INTRODUCTION
The Belle collaboration recently observed the transition Υ(4S) → η ′ Υ(1S) [1]. It is therefore likely that a similar transition exists in the charmonium sector. Moreover, CLEO-c, BESIII, and Belle measured the cross section as a function of √ s for the reaction e + e − → ηJ/ψ [2][3][4], which apparently shows a significant contribution from ψ(4160) decays. In Ref. [5], the authors reproduce the measured e + e − → ηJ/ψ line shape and predict the cross section of e + e − → η ′ J/ψ. A measurement of the cross sections of e + e − → η ′ J/ψ and ηJ/ψ can thus help the development of related theories. The measured cross section of e + e − → η ′ J/ψ can also be compared with that of e + e − → ηJ/ψ, which can provide more information to study charmonium(-like) states. BESIII recently observed the process e + e − → η ′ J/ψ using data collected at √ s = 4.226 and 4.258 GeV. Due to limited statistics, no significant signal was observed at other energy values in the range from 4.189 to 4.600 GeV [6]. The line shape of the measured cross section could be reasonably described by a single ψ(4160) state, supporting the hypothesis that the ψ(4160) decays to η ′ J/ψ. However, since the process e + e − → η ′ J/ψ was only observed at two energy points, no conclusions could be drawn regarding possible additional states decaying to η ′ J/ψ. Now that BESIII has collected more e + e − annihilation data samples around 4.2 GeV in 2016 and 2017, it is a good opportunity to search for the η ′ transition ψ(4160) → η ′ J/ψ or ψ(4260) → η ′ J/ψ, which will add another tile to our effort to understand the puzzle of the exotic states observed in the charmonium sector [7][8][9][10][11].

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector is a magnetic spectrometer [12] located at the Beijing Electron Positron Collider (BEPCII) [14]. The cylindrical core of the BESIII detector 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 identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over the 4π solid angle. The chargedparticle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for the electrons from Bhabha scattering. 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. The end cap TOF system was upgraded in 2015 with multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [15].
Simulated data samples produced with the geant4based [16] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate the background contributions. The simulation includes the beam energy spread and initial-state radiation (ISR) in the e + e − annihilations modeled with the generator kkmc [17]. Signal MC samples for e + e − → η ′ J/ψ are generated at each centerof-mass energy point assuming that the cross section follows a coherent sum of a ψ(4160) Breit-Wigner (BW) function and a ψ(4260) BW function, with masses and widths are fixed to their Particle Data Group (PDG) values [18]. The inclusive MC samples consist of the production of open charm processes, the ISR production of vector charmonium(-like) states, and the continuum processes incorporated in kkmc [17]. The known decay modes are modeled with evtgen [19] using branching fractions summarized and averaged by the PDG [18], and the remaining unknown decays from the charmonium states are generated with lundcharm [20]. Final state radiation from charged final state particles is incorporated with the photos package [21].

III. EVENT SELECTION
For each charged track, the distance of closest approach to the interaction point (IP) is required to be within 10 cm in the beam direction and within 1 cm in the plane perpendicular to the beam direction. The polar angles (θ) of the tracks must be within the fiducial volume of MDC (| cos θ| < 0.93). Photons are reconstructed from isolated showers in EMC, which are at least 20 • away from the nearest charged track. The photon energy is required to be at least 25 MeV in the barrel region (| cos θ| < 0.8) or 50 MeV in the end cap region (0.86 < | cos θ| < 0.92). To suppress electronic noise and energy depositions unrelated to the event, the EMC cluster timing from the reconstructed event start time is further required to satisfy 0 ≤ t ≤ 700 ns.
Since the reaction e + e − → η ′ J/ψ results in the final states γγπ + π − e + e − /µ + µ − and γπ + π − e + e − /µ + µ − , candidate events are required to have four tracks with zero net charge, at least two good photons for η ′ → π + π − η, and at least one for η ′ → γπ + π − . Tracks with momenta larger than 1 GeV/c are assigned as leptons from the decay of the J/ψ; otherwise, they are considered as pions from η ′ decays. Leptons from the J/ψ decay with ener-gy deposited in EMC larger than 1.0 GeV are identified as electrons, and those less than 0.4 GeV as muons. To reduce the background contributions and to improve the mass resolution, a four-constraint (4C) kinematic fit is performed for the η ′ → γπ + π − decay mode, constraining the total four-momentum of the final state particles to the total initial four-momentum of the colliding beams. A five-constraint (5C) kinematic fit is performed for the η ′ → π + π − η decay mode both to constrain the total fourmomentum of the final state particles to the total initial four-momentum of the colliding beams and to constrain the invariant mass of the two photons from the decay of the η to its nominal mass [18]. If there is more than one combination in an event, the one with the smallest χ 2 4C or χ 2 5C of the kinematic fit is selected. The χ 2 4C or χ 2

IV. BORN CROSS SECTION MEASUREMENT
Scatter plots of the ℓ + ℓ − invariant mass, M (ℓ + ℓ − ), and the π + π − η/γπ + π − invariant masses, M (π + π − η)/M (γπ + π − ), are shown in Fig. 1 for data taken at √ s = 4.178 GeV and combined data taken at other 13 energy points. A high-density area can be observed originating from the e + e − → η ′ J/ψ decay. The J/ψ signal region is defined by the mass range [3.07, 3.13] GeV/c 2 in M (ℓ + ℓ − ) and is indicated by horizontal dashed lines. Sideband regions, defined by the ranges [3.00, 3.06] GeV/c 2 and [3.14, 3.20] GeV/c 2 , are used to study the non-resonant background. The nominal η ′ mass is indicated by the vertical dashed lines. Figure 2 shows the distributions of M (π + π − η) and M (γπ + π − ) for data in the J/ψ signal region. Signals for the η ′ meson are observed. The shaded histograms correspond to the normalized events from the J/ψ sideband region. In order to extract the signal yield, a simultaneous maximum likelihood fit is performed to the two η ′ decay modes. The η ′ signal is modeled by the MCdetermined shape, and the background is described with a 1st order polynomial. In the fit, the total signal yield is a free parameter, the ratio of the number of η ′ → π + π − η signal events to the number of η ′ → γπ + π − signal events  is fixed to , where ǫ π + π − η and ǫ γπ + π − are the efficiencies for the π + π − η and γπ + π − decay modes, respectively. B(η ′ → π + π − η), B(η → γγ) and B(η ′ → γπ + π − ) are the branching fractions, and are taken from PDG [18]. The solid curves in Fig. 2 show the fit results. Data taken at all center-of-mass energies are analyzed using the same method and the fit results are summarized in Table I. The Born cross section is calculated with where N sig is the total number of signal events, L is the integrated luminosity obtained using the same method in Ref. [22], 1 + δ(s) is the ISR correction factor obtained from a Quantum Electrodynamics calculation [17,23], 1 |1−Π| 2 is the correction factor for vacuum polarization [24], B 1 is the product of branching fractions B(J/ψ → ℓ + ℓ − ) × B(η ′ → π + π − η) × B(η → γγ), and B 2 is the product of branching fractions B(J/ψ → ℓ + ℓ − ) × B(η ′ → γπ + π − ). For data at √ s = 4.278, 4.358 and 4.600 GeV, which have no significant signals, we calculate upper limits at 90% confidence level (C.L.) using the Bayesian method assuming a uniform prior distribution. The upper limit on the number of η ′ signal events N up η ′ at 90% C.L. is obtained by solving the equation  terior distribution (of signal events), which is the likelihood function multiplied by the prior distribution. The systematic uncertainty is taken into account by smearing the posterior distribution. The Born cross sections (or upper limits at 90% C.L.) at each energy point for e + e − → η ′ J/ψ are listed in Table I.

V. SYSTEMATIC UNCERTAINTY
The systematic uncertainties of the Born cross section measurement originate from the luminosity determination, the tracking efficiency, the photon detection efficiency, the kinematic fit, the J/ψ mass window, the radiative correction, the fit range, the signal and the background modeling, and the input branching fractions.
The luminosities are measured with a precision of 1.0% using the Bhabha process [22]. The uncertainty in the tracking efficiency is 1.0% per track [25]. Since the two decay channels have the same number of charged tracks in the same region of momenta, their tracking efficiencies are fully correlated. Therefore, a 4.0% uncertainty is introduced to the final results.
The uncertainty in photon reconstruction is 1.0% per photon [26]. There are two photons for the η ′ → π + π − η mode and one photon for η ′ → γπ + π − . Therefore, we vary the values ǫ π + π − η and ǫ γπ + π − up or down by 1% × N γ and refit the data, where N γ is the number of photons in the final state. The maximum change of the measured upper at 90% C.L.) for the reaction e + e − → η ′ J/ψ at different center-ofmass energies √ s, together with integrated luminosities L, number of signal events N sig , radiative correction factors 1 + δ(s), vacuum polarization factors 1 |1−Π| 2 , and efficiencies ǫ π + π − η and ǫ γπ + π − . The first uncertainties are statistical, and the second systematic.
√ s (GeV) cross section is taken as the systematic uncertainty.
The uncertainty due to the kinematic fit is estimated by correcting the helix parameters of charged tracks according to the method described in Ref. [27]. The difference between detection efficiencies obtained from MC samples with and without this correction is taken as the uncertainty.
The line shape of the e + e − → η ′ J/ψ cross section will affect the radiative correction factor and the efficiency. In the nominal results, we use a coherent sum of ψ(4160) and ψ(4260) resonances [18] as the line shape. To estimate the uncertainty from the radiative correction, we change the line shape to a coherent sum of ψ(4160), ψ(4260), and ψ(4415) resonances, a coherent sum of ψ(4160), Y (4220), and Y (4320) resonances [8], a coherent sum of ψ(4160), ψ(4260), and a continuum component, and take the largest difference of the cross section measurement to the nominal one as the systematic uncertainty.
Due to limited statistics, we add all data together to estimate the uncertainties from the fit range, the signal shape, and the background shape. The uncertainty from the fit range is obtained by varying the boundary of the fit range by ±0.01 GeV/c 2 . We take the largest difference of the cross section measurement to the nominal one as the systematic uncertainty. For the uncertainty from the signal shape, we use the MC-determined shape convolved with a Gaussian function to refit the data. The Gaussian function compensates for a possible mass resolution discrepancy between data and MC simulations, and its parameters are free. The systematic uncertainty due to the background shape is estimated by changing the background shape from a 1st-order polynomial to a 2nd-order polynomial, and taking the difference as the uncertainty. The uncertainties from the input branching fractions are taken from PDG [18]. Table II summarizes all the systematic uncertainties related to the cross section measurement of the e + e − → η ′ J/ψ process for each center-of-mass energy. The overall systematic uncertainties are obtained by adding all the sources of systematic uncertainties in quadrature assuming they are uncorrelated. Figure 3 shows the dressed cross sections (σ = σ B |1−Π| 2 ) for the e + e − → η ′ J/ψ reaction at different energy points. We observe an enhancement in the cross section around 4.2 GeV. By assuming that the η ′ J/ψ signals come from a single resonance ψ(4160) or ψ(4260), with mass M and width Γ that are fixed to their PDG values [18], we use a least χ 2 method to fit the cross section data with the following formula :

VI. DISCUSSION
where Φ( √ s) = p/ √ s is the two-body phase space factor, p is the η ′ momentum in the e + e − center-of-mass frame, and Γ ee is the electronic width of the ψ(4160) or ψ(4260). The χ 2 function is constructed as:   where σ data i and σ fit i are the measured and fitted cross sections of the i-th energy point, respectively, and ∆ i is the corresponding statistical uncertainty. The goodness of fit is χ 2 /NDF = 38/13, corresponding to a confidence level of 2.9 × 10 −4 , for a single resonance ψ(4160) and χ 2 /NDF = 63/13, corresponding to a confidence level of 1.5 × 10 −8 , for a single resonance ψ(4260), where NDF is the number of degrees of freedom. The fit qualities indicate that the data cannot be described well by a single ψ(4160) or ψ(4260) resonance.
Then we try to use a coherent sum of ψ(4160) and ψ(4260) resonances to fit the e + e − → η ′ J/ψ cross section, where the resonances' parameters are fixed to those from PDG [18]. The fit result is shown in Fig. 3 and Table III. The goodness of fit is χ 2 /NDF = 19/11, corresponding to a confidence level of 6.1%, indicating that the e + e − → η ′ J/ψ cross section can be described by a coherent sum of ψ(4160) and ψ(4260). The significances for the ψ(4160) and ψ(4260) are 6.3σ and 4.0σ. Significance of ψ(4160) is to compare to single ψ(4260) fit, while vice versa. In additional, we also try to use a coherent sum of ψ(4160), Y (4220) and Y (4320) resonances to fit, where Y (4220) and Y (4320)'s parameters are fixed to the results in Ref. [8]. The goodness of fit is χ 2 /NDF = 14/9, corresponding to a confidence level of 12.2%. The contribution of the continuum process is studied by means of a phase space function Φ 3 ( √ s) or a 1 s parametrization, and the cross section is fitted again taking into account this additional factor. We find that the additional contribution of the continuum is not statistically significant.

VII. SUMMARY
The process e + e − → η ′ J/ψ has been studied using 14 data samples collected at center-of-mass energies from √ s = 4.178 to 4.600 GeV. The √ s-dependence of the cross section has been measured. In the previous study, the process e + e − → η ′ J/ψ was only observed at √ s = 4.226 and 4.258 GeV, which is not sufficient to constrain the parameterization of the line shape of e + e − → η ′ J/ψ around √ s = 4.2 GeV. In this study, the cross section of e + e − → η ′ J/ψ is measured by adding more data samples at nine energy points in the range 4.178 √ s 4.278 GeV, which improves our understanding of the line shape of e + e − → η ′ J/ψ around √ s = 4.2 GeV. The results of the data samples at the previous five energy points are also updated. The e + e − → η ′ J/ψ cross section cannot be properly described by a single ψ(4160) or ψ(4260) resonance, while a coherent sum of ψ(4160) and ψ(4260) offers a better description. Further experimental studies with higher statistics are needed to draw a clearer conclusion on the structures in the e + e − → η ′ J/ψ process. The cross section of e + e − → η ′ J/ψ is about an order of magnitude lower than that of e + e − → ηJ/ψ [3], and the line shape of e + e − → η ′ J/ψ is relatively flat from √ s = 4.2 to 4.26 GeV, while that of e + e − → ηJ/ψ is sharply drop. The precise measurements of e + e − → η ′ J/ψ and ηJ/ψ in the future may be useful inputs for a study of η − η ′ mixing.