First Measurements of $\chi_{cJ}\rightarrow \Sigma^{-} \bar{\Sigma}^{+} (J = 0, 1, 2)$ Decays

M. Ablikim, M. N. Achasov, P. Adlarson, S. Ahmed, M. Albrecht, A. Amoroso , Q. An, Anita, 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, A. Bortone , I. Boyko, R. A. Briere, H. Cai, X. Cai , A. Calcaterra , G. F. Cao, N. Cao , S. A. Cetin , J. F. Chang, W. L. Chang, G. Chelkov, D. Y. Chen, G. Chen, H. S. Chen, M. L. Chen, S. J. Chen, X. R. Chen, Y. B. Chen, W. Cheng , G. Cibinetto, F. Cossio , X. F. Cui, H. L. Dai, J. P. Dai, X. C. Dai, A. Dbeyssi, R. B. de Boer, 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, S. X. Du, 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, X. L. Gao , Y. Gao, Y. Gao, Y. G. Gao, 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, C. Y Guan, A. Q. Guo, L. B. Guo, R. P. Guo, Y. P. Guo, Y. P. Guo, A. Guskov, S. Han, T. T. Han, T. Z. Han, X. Q. Hao, F. A. Harris, K. L. He, F. H. Heinsius, T. Held, Y. K. Heng, M. Himmelreich , T. Holtmann, Y. R. Hou, Z. L. Hou, H. M. Hu, J. F. Hu, T. Hu, Y. Hu, G. S. Huang, L. Q. Huang, X. T. Huang, Z. Huang, N. Huesken, T. Hussain, W. Ikegami Andersson, W. Imoehl, M. Irshad, S. Jaeger, S. Janchiv, Q. Ji, Q. P. Ji, X. B. Ji , X. L. Ji, H. B. Jiang, X. S. Jiang , X. Y. Jiang, J. B. Jiao , Z. Jiao, 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. G. Kurth, W. Kühn, J. J. Lane, J. S. Lange, P. Larin, L. Lavezzi , H. Leithoff, M. Lellmann, T. Lenz, C. Li, C. H. Li, Cheng Li, D. M. Li, F. Li, G. Li, H. B. Li, H. J. Li, J. L. Li, J. Q. Li, Ke Li, L. K. Li, Lei Li, P. L. Li, P. R. Li, S. Y. Li, W. D. Li, W. G. Li, X. H. Li , X. L. Li, Z. B. Li, Z. Y. Li, H. Liang , H. Liang, Y. F. Liang, Y. T. Liang, L. Z. Liao, J. Libby, C. X. 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. Liu, Q. Liu, S. B. Liu, Shuai Liu, T. Liu, X. Liu, Y. B. Liu, Z. A. Liu, Z. Q. Liu, Y. F. Long, X. C. Lou, H. J. Lu, J. D. Lu, J. G. Lu, X. L. 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, R. Q. Ma, R. T. 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 , T. J. Min, R. E. Mitchell , X. H. Mo , Y. J. Mo, N. Yu. Muchnoi, H. Muramatsu, S. Nakhoul , Y. Nefedov, F. Nerling , I. B. Nikolaev, Z. Ning, S. Nisar , S. L. Olsen, Q. Ouyang, S. Pacetti , X. Pan, Y. Pan, A. Pathak, P. Patteri, M. Pelizaeus, H. P. Peng, K. Peters , J. Pettersson, J. L. Ping, R. G. Ping, A. Pitka, R. Poling , V. Prasad , H. Qi, H. R. Qi, M. Qi, T. Y. Qi, S. Qian, W.-B. Qian, Z. Qian, C. F. Qiao, L. Q. Qin, X. P. Qin, X. S. Qin, Z. H. Qin, J. F. Qiu, S. Q. Qu, K. H. Rashid, K. Ravindran, C. F. Redmer, A. Rivetti , V. Rodin, M. Rolo , G. Rong, Ch. Rosner, M. Rump, A. Sarantsev, M. Savrié , Y. Schelhaas, C. Schnier, K. Schoenning, D. C. Shan, W. Shan, X. Y. Shan, M. Shao, C. P. Shen, P. X. Shen, X. Y. Shen, H. C. Shi, R. S. Shi, X. Shi, X. D Shi, J. J. Song, Q. Q. Song, W. M. Song, Y. X. Song, S. Sosio , S. Spataro , F. F. Sui, G. X. Sun, J. F. Sun, L. Sun, S. S. Sun, T. Sun, W. Y. Sun, Y. J. Sun, Y. K Sun, Y. Z. Sun, Z. T. Sun, Y. H. Tan, Y. X. Tan, C. J. Tang, G. Y. Tang, J. Tang, V. Thoren, B. Tsednee, I. Uman, B. Wang, B. L. Wang, C. W. Wang, D. Y. Wang, H. P. Wang, K. Wang, L. L. Wang, M. Wang, M. Z. Wang, Meng Wang, W. H. Wang, W. P. Wang, X. Wang, X. F. Wang, X. L. Wang, Y. Wang, Y. Wang, Y. D. Wang, Y. F. Wang, Y. Q. Wang, Z. Wang, Z. Y. Wang, Ziyi Wang, Zongyuan Wang, T. Weber, D. H. Wei, P. Weidenkaff, F. Weidner, S. P. Wen, D. J. White, U. Wiedner, G. Wilkinson, M. Wolke, L. Wollenberg, J. F. Wu, L. H. Wu, L. J. Wu, X. Wu, Z. Wu, L. Xia, H. Xiao, S. Y. Xiao, Y. J. Xiao, Z. J. Xiao, X. H. Xie, Y. G. Xie, Y. H. Xie, T. Y. Xing, X. A. Xiong, G. F. Xu, J. J. Xu, Q. J. Xu, W. Xu, X. P. Xu, L. Yan, L. Yan , W. B. Yan, W. C. Yan, Xu Yan, H. J. Yang, H. X. Yang, L. Yang, R. X. Yang, S. L. Yang, Y. H. Yang, Y. X. Yang, Yifan Yang, Zhi Yang, M. Ye, M. H. Ye, J. H. Yin, Z. Y. You, B. X. Yu, C. X. Yu, G. Yu, J. S. Yu, T. Yu, C. Z. Yuan, W. Yuan , X. Q. Yuan, Y. Yuan, Z. Y. Yuan, C. X. Yue, A. Yuncu, A. A. Zafar , Y. Zeng, B. X. Zhang, Guangyi Zhang, H. H. Zhang, H. Y. Zhang, J. L. Zhang, J. Q. Zhang, J. W. Zhang, J. Y. Zhang, J. Z. Zhang, Jianyu Zhang, Jiawei Zhang, L. Zhang, Lei Zhang, S. Zhang, S. F. Zhang, T. J. Zhang, X. Y. Zhang, Y. Zhang, Y. H. Zhang, Y. T. Zhang, Yan Zhang, Yao Zhang, Yi Zhang, Z. H. Zhang, Z. Y. Zhang, G. Zhao, J. Zhao, J. Y. Zhao, J. Z. Zhao, Lei Zhao , Ling Zhao, M. G. Zhao, Q. Zhao, S. J. Zhao, Y. B. Zhao, Y. X. Zhao Zhao, Z. G. Zhao, A. Zhemchugov, B. Zheng, J. P. Zheng, Y. Zheng, Y. H. Zheng, B. Zhong, C. Zhong, L. P. Zhou, Q. Zhou, X. Zhou, X. K. Zhou, X. R. Zhou, A. N. Zhu, J. Zhu, K. Zhu, K. J. Zhu, S. H. Zhu, W. J. Zhu, X. L. Zhu, Y. C. Zhu, Z. A. Zhu, B. S. Zou, J. H. Zou


I. INTRODUCTION
Experimental studies of the χ cJ (J = 0, 1, 2) states are important for testing models that are based on nonperturbative Quantum Chromodynamics (QCD). The χ cJ mesons are P -wave cc triple-states with a spin parity J ++ , and cannot be produced directly in e + e − annihilation. However, they can be produced in the radiative decays of the vector charmonium state ψ(3686) with considerable branching fractions (BFs) of ∼ 9% [1]. A large sample of ψ(3686) decays has been collected at BESIII, which provides a good opportunity to investigate the Pwave χ cJ states [2].
Many theoretical calculations show that the color octet mechanism (COM) could have a large contribution in describing P -wave quarkonium decays [3][4][5]. The predictions for χ cJ decays to meson pairs are in agreement with the experimental results [6], while contradictions are observed in the χ cJ decays to baryon pairs (BB) [4,5]. For example, the BFs of χ cJ → ΛΛ disagree with measured values [7]. In addition, the study of χ c0 → BB is helpful to test the validity of the helicity selection rule (HSR) [8,9], which prohibits χ c0 → BB. Measured BFs for χ c0 → pp, ΛΛ and Ξ −Ξ+ do not vanish [7,10], demonstrating a strong violation of HSR in charmonium decay. The quark creation model (QCM) [11] is developed to explain the strengthened decays of χ c0 → BB and it predicts the rate of χ c0,2 → Ξ + Ξ − [10] well. However, the same model is unable to accurately reproduce the ob-served decay rates to other BB final states [7]. Recent BF data for χ c1,2 → Σ +Σ− and Σ 0Σ0 [12] are in good agreement with COM predictions [4], while measured BFs of χ c0 → Σ +Σ− and Σ 0Σ0 [12,13] are inconsistent with COM models based on the charm-meson-loop mechanism [5,14], and violate the HSR, too. Experimentally, there are no BF data of χ cJ → Σ −Σ+ , and therefore those measurements are necessary to further test the validity of COM, HSR and QCM.

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector operating at the Beijing electronpositron collider (BEPCII), is a double-ring e + e − collider with a peak luminosity of 1 × 10 33 cm −2 s −1 at center-ofmass energy √ s = 3.77 GeV [2,16]. The BESIII detector has a geometrical acceptance of 93% over 4π solid angle. The cylindrical core of the BESIII detector consists of a small-cell, helium-gas-based (60% He, 40% C 3 H 8 ) main drift chamber (MDC) which is used to track the charged particles. The MDC is surrounded by a time-of-flight (TOF) system built from plastic scintillators that is used for charged-particle identification (PID). Photons are detected and their energies and positions are measured with an electromagnetic calorimeter (EMC) consisting of 6240 CsI(TI) crystals. The sub-detectors are enclosed in a superconducting solenoid magnet with a field strength of 1 T. Outside the magnet coil, the muon detector consists of 1000 m 2 resistive plate chambers in nine barrel and eight end-cap layers, providing a spatial resolution of better than 2 cm. The momentum resolution of charged particle is 0.5% at 1 GeV. The energy loss (dE/dx) measurement provided by the MDC has a resolution of 6%, and the time resolution of the TOF is 80 ps (110 ps) in the barrel (end-caps). The energy resolution for photons is 2.5% (5%) at 1 GeV in the barrel (end-caps) of the EMC.
A dedicated Monte Carlo (MC) simulation of the BESIII detector based on geant4 [17] is used for the optimization of event selection criteria, the determination of the detection efficiencies, and to estimate the contributions of backgrounds. A generic MC sample with 5.06 × 10 8 events is generated, where the production of the ψ(3686) resonance is simulated by the MC event generator kkmc [18]. Particle decays are generated by evtgen [19] for the known decay modes with BFs taken from Particle Data Group (PDG), and by lundcharm [20] for the remaining unknown decays. For the MC simulation of the signal process, the decay of ψ(3686) → γχ cJ is generated by following the angular distributions taken from Ref. [21], where the polar angles θ of radiation photons are distributed according to (1 + cos 2 θ), (1 − 1 3 cos 2 θ), (1 + 1 13 cos 2 θ) for J = 0, 1, 2. The χ cJ → Σ −Σ+ decays are generated with the angsam [19] model, with helicity angles of the Σ satisfying the angular distribution 1 + α cos 2 θ. Note that α = 0 for the decay of the χ c0 because the helicity angular distribution of a scalar particle is isotropic. The subsequent decays Σ − → nπ − andΣ + →nπ + are generated with uniform momentum distribution in the phase space (PHSP) [22].

III. EVENT SELECTION AND BACKGROUND ANALYSIS
We reconstruct the candidate events from the decay chain ψ(3686) → γχ cJ followed by χ cJ → Σ −Σ+ with subsequent decays Σ − → nπ − andΣ + →nπ + . The charged tracks are reconstructed with the hit information from the MDC. The polar angles of charged tracks in the MDC have to fulfill | cos θ| < 0.93. A loose vertex requirement is applied for charged-track candidates to implement the sizable decay lengths of Σ − andΣ + , and each charged track is required to have a point of closest approach to e + e − interaction point that is within 10 cm in the plane perpendicular to the beam axis and within ±30 cm in the beam direction. The combined information of dE/dx and TOF is used to calculate PID probabilities for the pion, kaon and proton hypothesis, respec-tively, and the particle type with the highest probability is assigned to the corresponding track. In this analysis, candidate events are required to have two charged tracks identified as π + and π − .
There are three neutral particles in the final states of the signal process, the radiative photon γ, anti-neutron n and neutron n. The radiative photon deposits most of its energy in the EMC with a high efficiency. Then annihilates in the EMC and produces several secondary particles with a total energy deposition up to 2 GeV. The n, on the other hand, is not identifiable due to its low interaction efficiency and its small energy deposition. Therefore, then and radiative photon are selected in this process. The most energetic shower in the EMC is assigned to be then candidate. To discriminaten from photons and to suppress the electronic noise, several selection criteria are used. Firstly, the deposited energy ofn is required to be in the range 0.2−2.0 GeV. Secondly, the second moment of candidate shower, de- where E i is the energy deposited in the i th crystal of the shower and r i is the distance from the center of that crystal to the center of the shower [23]. Furthermore, the number of EMC hits in a 40 • cone seen from the vertex around then shower direction is required to be greater than 20. After applying these selection criteria, then candidates have a purity of more than 98% estimated from signal MC sample.
To avoid the secondary showers originating fromn annihilation, the radiative photon is selected from EMC showers that have an opening angle with respect to thē n direction that is greater than 40 • . Good photon candidates are selected by requiring a minimum energy deposition of 80 MeV in the EMC, and are isolated from all charged tracks by a minimum angle of 10 • . The time information of the EMC is used to further suppress electronic noise and energy depositions unrelated to the event. At least one good photon candidate is required in an event.
The momentum or direction information of candidate particles are subjected to a kinematic fit that assumes the ψ(3686) → γnnπ + π − hypothesis, where the direction of n in the fit is involved and n is treated as a missing particle. The kinematic fit is then applied by imposing energy and momentum conservation at the IP and by constraining thenπ + invariant mass to match the nominalΣ + mass [1]. For events with more than one photon candidate, the combination with a minimum χ 2 kfit is chosen with the requirement that χ 2 kfit < 20. After applying the kinematic fit, the backgrounds from ψ(3686) → π 0 π 0 J/ψ followed by decays of J/ψ → BB and π 0 → γγ are suppressed by reconstructing events with two photon candidates. An event is then discarded when the invariant mass of any two photons are located within 120 MeV/c 2 and 150 MeV/c 2 . The contamination of the channel ψ(3686) → π + π − J/ψ with J/ψ → nn is removed by requiring |M rec (π + π − ) − m(J/ψ)| > 10 MeV/c 2 , where M rec (π + π − ) is the recoil mass of the π + π − pair and m(J/ψ) is the world average mass of the J/ψ meson [1]. Another sources of backgrounds are from events containing a K 0 S . These events are removed by requiring |M (π + π − ) − m(K 0 S )| > 10 MeV/c 2 , whereby M (π + π − ) and m(K 0 S ) are the reconstructed π + π − invariant mass and world average mass of the K 0 S [1], respectively. The signal could be contaminated with background from ψ(3686) → Σ −Σ+ whereby one fake photon has been reconstructed. To remove such background, events are rejected for which the χ 2 kfit (Σ −Σ+ ) is smaller than χ 2 kfit (γΣ −Σ+ ). The invariant-mass spectrum of nπ − and the recoil mass spectrum of the γ are shown in Fig. 1 for both data and MC simulations, where Σ − and χ cJ signals can be observed. The MC results represent the main characteristics of the various background sources. However, they cannot fully describe the data due to missing or improper modeling of background processes involving BB, especially when the final states contain nn. Using the topology technique [24], we have categorized the main background sources into three kinds: a) the process ψ(3686) → γχ cJ whereby the χ cJ decays to hadronic final states, which shows a peak in M rec (γ) and no peaking structure in M (nπ − ); b) the process ψ(3686) → BB or J/ψ → BB via the hadronic transition from ψ(3686), which is not peaking in M rec (γ) but shows a wide bump in M (nπ − ); c) the decays ψ(3686) to hadronic final states, which are non-peaking in both M rec (γ) and M (nπ − ). Besides, a two-dimensional (2D) distribution of M (nπ − ) and M rec (γ) is shown in Fig. 2 for the data. Clear accumulations of candidate events of the signal process χ c0 → Σ −Σ+ are observed around the intersections of the χ c0 and Σ − mass regions, and a signature of the process χ c1,2 → Σ −Σ+ can be observed. A data sample corresponding to an integrated luminosity of 44 pb −1 , taken at √ s = 3.65 GeV, is used to estimate the continuum background arising from quantum electrodynamics (QED) processes. No peaking backgrounds are observed in the mass spectrum of M rec (γ) for the continuum data sample, therefore the contribution from QED background can be neglected.

IV. EXTRACTION OF THE SIGNAL
To extract the signal yields for χ cJ → Σ −Σ+ , unbinned maximum-likelihood fits to the M rec (γ) distributions as a function of M (nπ − ) are performed, noted as bin-by-bin fit. The bin width for M (nπ − ) is determined by testing the MC samples, where the MC samples include events from MC-generated background sources, and events randomly sampled from signal MC events with the same amount events as observed in data as signal. The bin width is determined when the minimum input-output difference is obtained for the extraction of the signal and it is found to be 10 MeV/c 2 .
In the fit of M rec (γ) in each nπ − bin, the χ cJ signals are described by the MC shapes convoluted with Gaussian functions to compensate for a possible resolution difference between the data and MC. For a proper modeling of the lineshape of the signal, thereby suppressing photon misidentification, we selected signal MC events for which the opening angle of the reconstructed photon matches the value given by the generator. A second-order Chebychev polynomial function is used to describe the non-χ cJ background. It should be noted that the M rec (γ) resolution of the process ψ(3686) → γχ cJ , with inclusive decays of the χ cJ , is the same as observed in the signal MC data. Figure 3 shows the results of a bin-by-bin fit of one of the M rec (γ) distributions selected for a bin in M (nπ − ) at the Σ − peak position. Figure 4 shows the fitted signal yields of ψ(3686) → γχ cJ as a function of M (nπ − ). Clear signatures of Σ − decays can be observed. Binned least-χ 2 fits are subsequently performed to these spectra. The signal shapes are described by MC-simulated responses convoluted with Gaussian distributions and backgrounds are described by second-order Chebychev polynomials. The fit results are shown by the lines in Fig. 4. The statistical significances of the signal for the three χ cJ cases are found to be 30σ, 5.8σ and 3.6σ, respectively. The significances are calculated from the χ 2 differences between fits with and without the signal processes. The corresponding signal yields are summarized in Table I. The BFs are obtained from: where N obs is the number of signal events obtained from the bin-by-bin fit; ǫ is the detection efficiency obtained from signal MC after the photon matching; B i is the product of BFs for the ψ(3686) → γχ cJ , Σ − → nπ − and Σ + →nπ + channels; and N ψ(3686) is the total number of ψ(3686) events. The corresponding detection efficiencies and the resultant BFs are summarized in Table I. We note that due to the low-energy radiative photon of χ cJ (J = 1, 2), the detection efficiency tends to get smaller due to the rejection of π 0 -mass requirement.

V. ESTIMATION OF SYSTEMATIC UNCERTAINTIES
Various sources of systematic uncertainties are studied and summarized in Table II. The investigated uncertainties are discussed in detail in the following: a. MDC Tracking: The tracking efficiencies for π + /π − as functions of the transverse momentum have been studied with the process J/ψ → Σ * −Σ+ → π − Λn π + (Λ → π − p). The efficiency difference between data and MC is 1.4% for each charged-pion track.
b. Photon Reconstruction: The uncertainty of the photon-detection efficiency is estimated to be 1.0% per photon [25].

c.n Selection and Kinematic Fit:
The systematic uncertainties of then selection and the kinematic fit involving then is studied using the control sample of J/ψ → Σ * Σ+ . The relative difference of 5.8% in efficiency between MC and data is assigned as the corresponding systematic uncertainty.
d. Mass Window Requirement: Various cuts in the mass spectra have been used to select events, namely on M (γγ), M (π + π − ) and M rec. (π + π − ). Cross checks of systematic effects for these mass window requirements are considered following the procedure described in Ref. [26]. The consistency of the results is checked by comparing the uncorrelated differences between the parameter values, x test ± σ test , obtained from the fits to the nominal results, x nom. ± σ nom. . The systematic sources cannot be discarded when the significance of uncorrelated differences, ∆x uncor. = |x nom. − x test |/ |σ 2 nom. − σ 2 test | > 2. By comparing the results of various selections taken within a proper range with the nominal result, the one with the largest difference is taken as an estimate of the corresponding uncertainty. For the χ c0 case, the π 0 veto is tested by varying the rejection windows, |M (γγ) − m(π 0 )| from 3 to 18 MeV/c 2 . The largest deviation ∆x uncor. is found when the veto is applied at 12 MeV/c 2 . Similar attempts are performed for the mass windows of M rec (π + π − ) and M(π + π − ). The largest deviations are found when the windows are |M rec (π + π − ) − m(J/ψ)| > 16 MeV/c 2 and |M (π + π − ) -m(K 0 S )| > 12 MeV/c 2 . The differences to the nominal results are then taken as an estimate of the systematic uncertainty. In all cases, we observe no tendency of ∆x uncor. along with the selection variations, indicating no bias in these selection criteria. For χ c1,2 , it is found that the ∆x uncor. for all the tests are less than 2σ. Therefore, no systematic uncertainties are considered in that case.   e. Fitting Process: To estimate the uncertainties from the fitting process, the following three studies are made.
(i) Bin Width: The bin width in the bin-by-bin fit is determined to be 10 MeV/c 2 by testing a series of MC samples as described in Sec. IV. The systematic uncertainties are determined by taking the difference between the determined branching fractions and their input values for χ cJ → Σ −Σ+ .
(ii). Fit of χ cJ : To extract the uncertainties associated with the fit procedure on M rec (γ), alternative fits are performed by replacing the second-order polynomial function with a third-order function for the background description, fixing the width of the Gaussian functions for the signal description, and by varying the fitting range. All the relative changes in the results are taken as the uncertainties from the fit.
(iii) Fit of M (Σ − ): Similarly, alternative fits are applied by varying the MC-simulated signal and background shapes and fit ranges. The differences are treated as a systematic uncertainty.
f. Generator: For the χ c0 case, the angular distribution of the Σ − in the χ c0 rest frame is isotropic since the χ c0 is a scalar particle. Therefore, no systematic uncertainty needs to be considered for the χ c0 . For χ c1,2 , on the other hand, we considered two extreme cases in the analysis, namely with α = 1 and −1, respectively. The resulting differences in efficiency with a factor of √ 12 are then assigned as the source of a systematic uncertainty.
g. MC Truth Matching Angle: Since in the analysis of the signal MC data sample only events are selected whereby the difference between the angle of the reconstructed photon and the generated one (MC truth angle) is less than 10 • , it might lead to a systematic error in the efficiency determination. Several differences with MC truth angles are considered ranging from 10 • to 20 • . The largest difference on the efficiencies are considered as the source of systematic uncertainty.
Other Uncertainties: The total number of ψ(3686) decays is determined by analyzing the inclusive hadronic events from ψ(3686) decays with an uncertainty of 0.6% [15]. The uncertainties due to the BFs ψ(3686) → γχ cJ are quoted from the PDG [1]. The systematic error due to uncertainties in the trigger efficiency is negligible for this analysis.
Total Systematic Uncertainty: We assume that all systematic uncertainties given above are independent and we add them in quadrature to obtain the total systemat-TABLE III. Results of the BFs (in units of 10 −5 ) for the measurement of χcJ → Σ −Σ+ , compared with the χcJ → Σ +Σ− results from BESIII [13] and theoretical predictions [4][5] [11]. The first errors are statistical and the second systematic.

VI. SUMMARY
Based on (448.1 ± 2.9) × 10 6 ψ(3686) events collected with the BESIII detector, the BFs of the processes χ cJ → Σ −Σ+ are measured and the results are summarized in Table III. This is the first BF measurement of χ cJ → Σ −Σ+ . The results of χ cJ → Σ −Σ+ are consistent with χ cJ → Σ +Σ− [13] from BESIII within the uncertainties, which confirm the prediction of isospin symmetry. The BF of χ c0 → Σ −Σ+ does not vanish, which demonstrates a strong violation of the HSR. Both predictions based on the COM [5] and QCM [11] fail to describe our measured result. The measured BFs of χ c1,2 → Σ −Σ+ are in good agreement with the theoretical predictions based on COM [4] and consistent within 1σ with the prediction based on QCM [11] for χ c2 → ΣΣ.

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