Measurements of the center-of-mass energies of e^{+}e^{-} collisions at BESIII
BESIII Collaboration, M. Ablikim, M. N. Achasov, P. Adlarson, S. Ahmed, M. Albrecht, R. Aliberti, A. Amoroso, M. R. An, Q. An, X. H. Bai, Y. Bai, O. Bakina, R. Baldini Ferroli, I. Balossino, Y. Ban, K. Begzsuren, N. Berger, M. Bertani, D. Bettoni, F. Bianchi, 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, Z. J Chen, W. S. Cheng, G. Cibinetto, F. Cossio, X. F. Cui, H. L. Dai, X. C. Dai, A. Dbeyssi, R. E. 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, X. Dong, S. X. Du, Y. L. Fan, J. Fang, S. S. Fang, Y. Fang, R. Farinelli, L. Fava, F. Feldbauer, G. Felici, C. Q. Feng, J. H. Feng, M. Fritsch, C. D. Fu, Y. Gao, Y. Gao, Y. Gao, Y. G. Gao, I. Garzia, P. T. Ge, C. Geng, 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, A. Guskov, T. T. Han, et al. (419 additional authors not shown)
aa r X i v : . [ h e p - e x ] F e b Measurements of the center-of-mass energies of e + e − collisions at BESIII M. Ablikim , M. N. Achasov ,c , P. Adlarson , S. Ahmed , M. Albrecht , R. Aliberti ,A. Amoroso A, C , M. R. An , Q. An , , X. H. Bai , Y. Bai , O. Bakina , R. Baldini Ferroli A ,I. Balossino A, , Y. Ban ,k , K. Begzsuren , N. Berger , M. Bertani A , D. Bettoni A ,F. Bianchi A, C , J. Bloms , A. Bortone A, C , I. Boyko , R. A. Briere , H. Cai , X. Cai , ,A. Calcaterra A , G. F. Cao , , N. Cao , , S. A. Cetin B , J. F. Chang , , W. L. Chang , ,G. Chelkov ,b , D. Y. Chen , G. Chen , H. S. Chen , , M. L. Chen , , S. J. Chen , X. R. Chen ,Y. B. Chen , , Z. J Chen ,l , W. S. Cheng C , G. Cibinetto A , F. Cossio C , X. F. Cui , H. L. Dai , ,X. C. Dai , , A. Dbeyssi , R. E. de Boer , D. Dedovich , Z. Y. Deng , A. Denig , I. Denysenko ,M. Destefanis A, C , F. De Mori A, C , Y. Ding , C. Dong , J. Dong , , L. Y. Dong , ,M. Y. Dong , X. Dong , S. X. Du , Y. L. Fan , J. Fang , , S. S. Fang , , Y. Fang , R. Farinelli A ,L. Fava B, C , F. Feldbauer , G. Felici A , C. Q. Feng , , J. H. Feng , M. Fritsch , C. D. Fu ,Y. Gao ,k , Y. Gao , Y. Gao , , Y. G. Gao , I. Garzia A, B , P. T. Ge , C. Geng , E. M. Gersabeck ,A Gilman , K. Goetzen , L. Gong , W. X. Gong , , W. Gradl , M. Greco A, C , 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 ,h ,A. Guskov , T. T. Han , W. Y. Han , X. Q. Hao , F. A. Harris , N H ¨usken , , K. L. He , ,F. H. Heinsius , C. H. Heinz , T. Held , Y. K. Heng , C. Herold , M. Himmelreich ,f , T. Holtmann ,Y. R. Hou , Z. L. Hou , H. M. Hu , , J. F. Hu ,m , T. Hu , Y. Hu , G. S. Huang , , L. Q. Huang ,X. T. Huang , Y. P. Huang , Z. Huang ,k , T. Hussain , W. Ikegami Andersson , W. Imoehl ,M. Irshad , , S. Jaeger , S. Janchiv ,j , Q. Ji , Q. P. Ji , X. B. Ji , , X. L. Ji , , Y. Y. Ji ,H. B. Jiang , X. S. 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,e , B. Kopf ,M. Kuemmel , M. Kuessner , A. Kupsc , M. G. Kurth , , W. K ¨uhn , J. J. Lane , J. S. Lange , P.Larin , A. Lavania , L. Lavezzi A, C, , Z. H. Lei , , H. Leithoff , M. Lellmann , T. Lenz ,C. Li , C. H. Li , Cheng Li , , D. M. Li , F. Li , , G. Li , H. Li , , H. Li , H. B. Li , , H. J. Li ,H. J. Li ,h , J. L. Li , J. Q. Li , J. S. Li , Ke Li , L. K. Li , Lei Li , P. R. Li , S. Y. Li , W. D. Li , ,W. G. Li , X. H. Li , , X. L. Li , Xiaoyu Li , , Z. Y. Li , H. Liang , , H. Liang , , H. Liang ,Y. F. Liang , Y. T. Liang , G. R. Liao , L. Z. Liao , , J. Libby , C. X. Lin , B. J. Liu , C. X. Liu ,D. Liu , , F. H. Liu , Fang Liu , Feng Liu , H. B. Liu , H. M. Liu , , Huanhuan Liu , Huihui Liu ,J. B. Liu , , J. L. Liu , J. Y. Liu , , K. Liu , K. Y. Liu , Ke Liu , L. Liu , , M. H. Liu ,h , P. L. Liu ,Q. Liu , Q. Liu , S. B. Liu , , Shuai Liu , T. Liu , , W. M. Liu , , X. Liu , Y. Liu , Y. B. Liu ,Z. A. Liu , Z. Q. Liu , X. C. Lou , F. X. Lu , F. X. Lu , 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 ,h , X. L. Luo , , S. Lusso C ,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. X. Ma , , X. Y. Ma , , F. E. Maas , M. Maggiora A, C , S. Maldaner , S. Malde , Q. A. Malik ,A. Mangoni B , Y. J. Mao ,k , Z. P. Mao , S. Marcello A, C , Z. X. Meng , J. G. Messchendorp ,G. Mezzadri A, , T. J. Min , R. E. Mitchell , X. H. Mo , Y. J. Mo , N. Yu. Muchnoi ,c ,H. Muramatsu , S. Nakhoul ,f , Y. Nefedov , F. Nerling ,f , I. B. Nikolaev ,c , Z. Ning , , S. Nisar ,i ,S. L. Olsen , Q. Ouyang , S. Pacetti B, C , X. Pan ,h , Y. Pan , A. Pathak , P. Patteri A , M. Pelizaeus ,H. P. Peng , , K. Peters ,f , J. Pettersson , J. L. Ping , R. G. Ping , , R. Poling , V. Prasad , ,H. Qi , , H. R. Qi , K. H. Qi , M. Qi , T. Y. 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 C , V. Rodin , M. Rolo C , G. Rong , , Ch. Rosner , . Rump , H. S. Sang , A. Sarantsev ,d , Y. Schelhaas , C. Schnier , K. Schoenning ,M. Scodeggio A, B , D. C. Shan , W. Shan , X. Y. Shan , , J. F. Shangguan , 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 , W. M. Song , , Y. X. Song ,k , S. Sosio A, C , S. Spataro A, C , K. X. Su , P. P. Su ,F. F. Sui , G. X. Sun , H. K. Sun , J. F. Sun , L. Sun , S. S. Sun , , T. Sun , , W. Y. Sun ,W. Y. Sun , X Sun ,l , 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 , J. X. Teng , , V. Thoren , Y. T. Tian , I. Uman D , B. Wang ,C. W. Wang , D. Y. Wang ,k , H. J. Wang , H. P. Wang , , K. Wang , , L. L. Wang , M. Wang ,M. Z. Wang ,k , Meng Wang , , W. Wang , W. H. Wang , W. P. Wang , , X. Wang ,k , X. F. Wang ,X. L. Wang ,h , Y. Wang , Y. Wang , , Y. D. Wang , Y. F. Wang , Y. Q. Wang , Y. Y. Wang ,Z. Wang , , Z. Y. Wang , Ziyi Wang , Zongyuan Wang , , 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 ,h , Z. Wu , , L. Xia , , H. Xiao ,h , S. Y. Xiao , Z. J. Xiao ,X. H. Xie ,k , Y. G. Xie , , Y. H. Xie , T. Y. Xing , , G. F. Xu , Q. J. Xu , W. Xu , , X. P. Xu ,Y. C. Xu , F. Yan ,h , L. Yan ,h , W. B. Yan , , W. C. Yan , Xu Yan , H. J. Yang ,g , H. X. Yang ,L. Yang , S. L. 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 ,l , T. Yu , C. Z. Yuan , , L. Yuan ,X. Q. Yuan ,k , Y. Yuan , Z. Y. Yuan , C. X. Yue , A. Yuncu B,a , A. A. Zafar , Y. Zeng ,l ,B. X. Zhang , Guangyi Zhang , H. Zhang , H. H. Zhang , H. H. Zhang , H. Y. Zhang , ,J. J. Zhang , J. L. Zhang , J. Q. Zhang , J. W. Zhang , J. Y. Zhang , J. Z. Zhang , , Jianyu Zhang , ,Jiawei Zhang , , L. M. Zhang , L. Q. Zhang , Lei Zhang , S. Zhang , S. F. Zhang ,Shulei Zhang ,l , X. D. Zhang , X. Y. Zhang , Y. Zhang , Y. H. Zhang , , Y. T. Zhang , ,Yan Zhang , , Yao Zhang , Yi Zhang ,h , 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 , Z. G. Zhao , , A. Zhemchugov ,b , B. Zheng , J. P. Zheng , ,Y. Zheng ,k , Y. H. Zheng , B. Zhong , C. Zhong , L. P. Zhou , , Q. Zhou , , X. Zhou ,X. K. Zhou , X. R. Zhou , , X. Y. Zhou , A. N. Zhu , , J. Zhu , K. Zhu , K. J. Zhu , S. H. Zhu ,T. J. Zhu , W. J. Zhu , W. J. Zhu ,h , Y. C. Zhu , , Z. A. Zhu , , B. S. Zou , J. H. Zou (BESIII Collaboration) Institute of High Energy Physics, Beijing 100049, People’s Republic of China Beihang University, Beijing 100191, People’s Republic of China Beijing Institute of Petrochemical Technology, Beijing 102617, People’s Republic of China Bochum Ruhr-University, D-44780 Bochum, Germany Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Central China Normal University, Wuhan 430079, People’s Republic of China China Center of Advanced Science and Technology, Beijing 100190, People’s Republic of China COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore,Pakistan Fudan University, Shanghai 200443, People’s Republic of China G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany Guangxi Normal University, Guilin 541004, People’s Republic of China Guangxi University, Nanning 530004, People’s Republic of China Hangzhou Normal University, Hangzhou 310036, People’s Republic of China Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Henan Normal University, Xinxiang 453007, People’s Republic of China Henan University of Science and Technology, Luoyang 471003, People’s Republic of China Huangshan College, Huangshan 245000, People’s Republic of China Hunan Normal University, Changsha 410081, People’s Republic of China Hunan University, Changsha 410082, People’s Republic of China Indian Institute of Technology Madras, Chennai 600036, India Indiana University, Bloomington, Indiana 47405, USA (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia,I-06100, Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of Ferrara, I-44122, Ferrara, Italy Institute of Modern Physics, Lanzhou 730000, People’s Republic of China Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330, Mongolia Jilin University, Changchun 130012, People’s Republic of China Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392Giessen, Germany Lanzhou University, Lanzhou 730000, People’s Republic of China Liaoning Normal University, Dalian 116029, People’s Republic of China Liaoning University, Shenyang 110036, People’s Republic of China Nanjing Normal University, Nanjing 210023, People’s Republic of China Nanjing University, Nanjing 210093, People’s Republic of China Nankai University, Tianjin 300071, People’s Republic of China North China Electric Power University, Beijing 102206, People’s Republic of China Peking University, Beijing 100871, People’s Republic of China Qufu Normal University, Qufu 273165, People’s Republic of China Shandong Normal University, Jinan 250014, People’s Republic of China Shandong University, Jinan 250100, People’s Republic of China Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China Shanxi Normal University, Linfen 041004, People’s Republic of China Shanxi University, Taiyuan 030006, People’s Republic of China Sichuan University, Chengdu 610064, People’s Republic of China Soochow University, Suzhou 215006, People’s Republic of China South China Normal University, Guangzhou 510006, People’s Republic of China Southeast University, Nanjing 211100, People’s Republic of China State Key Laboratory of Particle Detection and Electronics, Beijing 100049, Hefei 230026, People’sRepublic of China Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand Tsinghua University, Beijing 100084, People’s Republic of China (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi University, 34060 Eyup,Istanbul, Turkey; (C)Uludag University, 16059 Bursa, Turkey; (D)Near East University, Nicosia, NorthCyprus, Mersin 10, Turkey University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China University of Groningen, NL-9747 AA Groningen, The Netherlands University of Hawaii, Honolulu, Hawaii 96822, USA University of Jinan, Jinan 250022, People’s Republic of China University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom University of Minnesota, Minneapolis, Minnesota 55455, USA University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany University of Oxford, Keble Rd, Oxford, UK OX13RH University of Science and Technology Liaoning, Anshan 114051, People’s Republic of China University of Science and Technology of China, Hefei 230026, People’s Republic of China University of South China, Hengyang 421001, People’s Republic of China University of the Punjab, Lahore-54590, Pakistan (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern Piedmont, I-15121, Alessandria,Italy; (C)INFN, I-10125, Turin, Italy Uppsala University, Box 516, SE-75120 Uppsala, Sweden Wuhan University, Wuhan 430072, People’s Republic of China Xinyang Normal University, Xinyang 464000, People’s Republic of China Zhejiang University, Hangzhou 310027, People’s Republic of China Zhengzhou University, Zhengzhou 450001, People’s Republic of China a Also at Bogazici University, 34342 Istanbul, Turkey b Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia d Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia e Also at Istanbul Arel University, 34295 Istanbul, Turkey f Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany g Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education;Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics,Shanghai 200240, People’s Republic of China h Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of ModernPhysics, Fudan University, Shanghai 200443, People’s Republic of China i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA j Currently at: Institute of Physics and Technology, Peace Ave.54B, Ulaanbaatar 13330, Mongolia k Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871,People’s Republic of China l School of Physics and Electronics, Hunan University, Changsha 410082, China m Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, SouthChina Normal University, Guangzhou 510006, China (Dated: February 10, 2021) bstract During the 2016-17 and 2018-19 running periods, the BESIII experiment collected 7.5 fb − of e + e − collision data at center-of-mass energies ranging from 4.13 to 4.44 GeV. These data samples are primarilyused for the study of excited charmonium and charmoniumlike states. By analyzing the di-muon process e + e − → ( γ ISR / FSR ) µ + µ − , we measure the center-of-mass energies of the data samples with a precision of0.6 MeV. Through a run-by-run study, we find that the center-of-mass energies were stable throughout mostof the data-taking period. I. INTRODUCTION
The BESIII experiment [1] was designed to study physics in the τ -charm energy region (2.0– 4.9 GeV) [2] with e + e − annihilation produced by the BEPCII storage ring [3]. Since it startedrunning in 2008, a variety of data samples have been collected at different center-of-mass (CM)energies for the study of light hadron spectroscopy, charmonium and charmoniumlike states (alsocalled XYZ states), charm physics, τ physics, various QCD-related studies, and the search for newphysics beyond the standard model [4].The Beam Energy Measurement System (BEMS) [5] was designed to precisely measure BE-SIII CM energies ( E cm ) using a method based on Compton back-scattered photons. However, itscapability at high energy ( E cm above 4 GeV) is degraded by its detection efficiency and limited cal-ibration sources for high-energy gamma rays. Therefore, an alternative algorithm was developedto measure the E cm for data samples above 4 GeV. This method uses the well-understood QEDprocess e + e − → ( γ ISR / FSR ) µ + µ − (the di-muon process), where γ ISR / FSR is a radiative photon dueto initial state radiation (ISR) and/or final state radiation (FSR). Using this method, a precision of0.8 MeV was previously achieved for the data taken from 2011 to 2014 [6].In this paper, we present the E cm measurement for the XYZ data samples taken at BESIIIfrom 2017 to 2019. The method used in Ref. [6] is followed, but the precision of the momentumcalibration is improved, and the E cm is measured with an uncertainty of 0.6 MeV.Using the selected di-muon events, e + e − → ( γ ISR / FSR ) µ + µ − , we determine E cm with E cm = ( M p ( µ + µ − ) + ∆ M ISR / FSR + ∆ M cal ) × c , (1)where M p ( µ + µ − ) is the peak position of the µ + µ − invariant mass of selected di-muon events; ∆ M ISR / FSR is the mass shift due to the emission of ISR or FSR photons, estimated from MonteCarlo (MC) simulation of the di-muon process by turning on and off the ISR/FSR processes inMC generation; and ∆ M cal is the correction introduced by the momentum calibration of the µ + µ − tracks, obtained from an analysis of the process e + e − → γ ISR
J/ψ . II. THE BESIII DETECTOR AND DATA SETS
The BESIII detector is described in detail in Ref. [1]. The cylindrical core of the detector cov-ers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), aplastic 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 iden-tification modules interleaved with steel. The charged-particle momentum resolution at /c . , and the dE/dx resolution is for electrons from Bhabha scattering. The EMC mea-sures photon energies with a resolution of . ( ) at GeV in the barrel (end cap) region. Thetime resolution in the TOF barrel region is 68 ps, while that in the end cap region is 110 ps. Theend cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology,providing a time resolution of 60 ps [7].The data samples analyzed in this work are listed in Table I. They include 16 different CM ener-gies from 4.13 to 4.44 GeV and were collected in two running years: from December 2016 to May2017 (labelled as “2017XYZ” hereafter, the integrated luminosities are measured in Ref. [8]); andfrom February 2019 to June 2019 (labelled as “2019XYZ” hereafter, the integrated luminositiesare estimated by using online monitoring information). The column “Sample” shows the nomi-nal CM energy in MeV used during online data taking. The true CM energy is usually within afew MeV of the nominal value. Run numbers are used to divde the data into subsamples. Othercolumns, such as L ( pb − ), will be illustrated below. TABLE I. Summary of the data samples, including run numbers, integrated luminosity L [8], the measured J/ψ mass after FSR correction M cor ( J/ψ ) (in MeV/ c ), M p ( µ + µ − ) (in MeV/ c ), and E cm . Superscriptsrepresent data from different periods: “1” stands for 2017XYZ data, and “2” stands for 2019XYZ data. Thefirst uncertainties are statistical, and the second systematic.Sample Run Number L ( pb − ) M cor ( J/ψ ) M p ( µ + µ − ) E cm (MeV)4130 . ± .
30 4130 . ± .
05 4128 . ± . ± . . ± .
29 4158 . ± .
05 4157 . ± . ± . . ± .
16 3097 . ± .
28 4187 . ± .
05 4189 . ± . ± . . ± .
05 3098 . ± .
27 4198 . ± .
05 4199 . ± . ± . . ± .
81 3097 . ± .
29 4207 . ± .
06 4209 . ± . ± . . ± .
80 3097 . ± .
26 4217 . ± .
05 4218 . ± . ± . . ± .
39 3097 . ± .
24 4233 . ± .
04 4235 . ± . ± . . ± .
69 3097 . ± .
24 4242 . ± .
04 4243 . ± . ± . . ± .
13 3098 . ± .
26 4265 . ± .
04 4266 . ± . ± . . ± .
97 3097 . ± .
48 4277 . ± .
04 4277 . ± . ± . . ± .
28 4289 . ± .
06 4288 . ± . ± . . ± .
30 4313 . ± .
06 4312 . ± . ± . . ± .
29 4338 . ± .
06 4337 . ± . ± . . ± .
30 4378 . ± .
06 4377 . ± . ± . . ± .
31 4398 . ± .
06 4396 . ± . ± . . ± .
29 4437 . ± .
06 4437 . ± . ± . A GEANT
BABAYAGA e + e − → γ ISR
J/ψ sample is generated with
KKMC [11]. One million events are generated foreach process at each CM energy.
III. EVENT SELECTION AND MEASUREMENT OF M p ( µ + µ − ) The di-muon process e + e − → ( γ ISR / FSR ) µ + µ − is selected by requiring two oppositely chargedtracks in the detector, each positively identified as a muon. Both charged tracks are reconstructed6rom hits in the MDC within the polar angle range | cos θ | < . and are required to pass interactionpoint (IP) within 10 cm along the beam direction and within 1 cm in the plane perpendicular tothe beam. The energy deposition in the EMC for each charged track is required to be less than0.4 GeV to suppress backgrounds from radiative Bhabha events.The sample after these selections includes di-muon events with no photon emission or withvery low-energy radiative photons, ISR J/ψ with
J/ψ → µ + µ − , and ISR µ + µ − events with asmooth µ + µ − invariant mass ( M ( µ + µ − ) ) distribution. The events in the J/ψ mass region areused for track momentum calibration and those with high invariant mass are used to measure the E cm after the additional selections shown below.To suppress di-muon events with high energy radiative photons, a requirement on the cosineof the opening angle between the two tracks, cos θ µ + µ − < − . . To further remove cosmicray events, the TOF time difference between the two tracks is required to be | ∆ t | < ns. Thebackground contribution after the above selection criteria is less than 0.1% compared with signaland is therefore neglected in the following analysis.The M ( µ + µ − ) distribution for the 4190 data sample is shown in Fig. 1 as an example. Thedistributions of the other samples are very similar. The distribution is a Gaussian due to themomentum resolution of the µ + µ − but is distorted by ISR and FSR effects which produce a tailon the left side of the peak. The central part of the distribution can be approximated with aGaussian function. We measure the peak position of the distribution ( M p ( µ + µ − ) ) by fitting itwith a Gaussian function in a range of ( − σ, + 1 . σ ) around the peak, where σ is the standarddeviation of the Gaussian. If the goodness of the fit, χ /ndf > . ( ndf is the number of degreesof freedom of the fit), we slightly reduce the fit range until χ /ndf < . to guarantee a good fitquality. The fit result for the 4190 data sample is shown in Fig. 1. The values of M p ( µ + µ − ) forthe other data samples are obtained in a similar way and are shown in Table I. ) c ) (GeV/ - µ + µ M( c E v en t s / . M e V / DataFit
FIG. 1. The µ + µ − invariant mass distribution and the fit result of the 4190 sample. Dots with error bars aredata, and the red solid curve is the fit. To examine the stability of the E cm over the data-taking period for each data sample, the fitprocedure is repeated for each run of the data sample. The measured peak values of the µ + µ − invariant mass distribution versus run number for all 16 samples are shown in Fig. 2. There aresmall jumps of less than 1 MeV in the 4200, 4210, 4246, 4380, and 4400 samples. Before and after7he jumps, the energy is stable. We fit each stable part of the distribution with a linear functionand Table II summarizes the average, M ave ( µ + µ − ) , for each period of time. The deviation of M ave ( µ + µ − ) from the peak position obtained in the full data sample is taken as one source ofsystematic uncertainty. RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN Number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) )( G e V / { i t } - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M RUN number ) c )( G e V / - µ + µ ( p M FIG. 2. Measured run-by-run values for the M p ( µ + µ − ) of di-muon events in each data sample. The redsolid lines show the fit results for the data samples of each stable period of time. The green dotted lines arethe fit results of the entire sample when there is an energy jump. IV. MOMENTUM CALIBRATION WITH ISR
J/ψ
SIGNAL
The momentum measurement of the muon tracks is validated with
J/ψ → µ + µ − candidatesproduced via the process e + e − → γ ISR
J/ψ selected in the previous section. The distribution of M ( µ + µ − ) of each sample is fitted with a crystal-ball function [12] for the J/ψ signal and a linearfunction to model the background from continuum production of e + e − → γµ + µ − . Figure 3(a)shows the fit result for the 4190 data sample as an example. The peak position of the J/ψ signal, M obs ( J/ψ ) , is used to calibrate the momentum measurement of the muon tracks.Due to FSR, J/ψ → µ + µ − γ FSR , the measured M obs ( J/ψ ) is slightly lower than the world av-erage J/ψ mass ( m J/ψ ) given by the PDG [13]. The mass shift due to the FSR photon(s) ∆ M γJ/ψ FSR of the process e + e − → γ ISR
J/ψ at each E cm is obtained by using the generator PHOTOS [14] withFSR turned on or off. The shift is around 0.3 MeV/ c with little dependence on the CM energy ofthe data sample. 8 ABLE II. Average value of M ave ( µ + µ − ) (in MeV/ c ) for each stable data-taking period within each datasample. Sample Run Number M ave ( µ + µ − ) Run Number M ave ( µ + µ − ) . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . ) c ) (GeV/ - µ + µ M( ) c E v en t s / ( . M e V / dataFitting resultBackgroundsignal (a) Energy (MeV) ) c ) ( M e V / ψ ( J / c o r M ∆ (b) FIG. 3. (a) Fit to the M ( µ + µ − ) distribution in the J/ψ signal region for the 4190 data sample. Black dotswith error bars are data, the red curve shows the fit result, the blue curve is for signal, and the green dashedline is for background. (b) The difference between M cor ( J/ψ ) and the world average mass of J/ψ [13], ∆ M cor ( J/ψ ) for each data sample. Comparing the M cor ( J/ψ ) = M obs ( J/ψ ) + ∆ M γJ/ψ FSR (as shown in Table I) with
J/ψ nominalmass value m J/ψ in Particle Data Book (PDG), we measure the bias in the
J/ψ mass measurement( ∆ M cor ( J/ψ ) ) due to the muon track momentum calibration, as shown in Fig. 3(b). It can be seenthat the bias in J/ψ invariant mass is stable throughout one running year, but is quite different inthe 2017XYZ and 2019XYZ samples. This may indicate that the calibrations in these two periodsof time have significant differences.Through MC simulation we find that the bias in M ( µ + µ − ) measurement depends on ( M ( µ + µ − ) − J/ψ ) linearly (see Fig. 4), so the correction to the M ( µ + µ − ) due to calibration is expressed as ∆ M cal = − ( k × ( E cm − m J/ψ ) + ∆ M cor ( J/ψ )) , (2)where the slopes k = (7 . ± . × − and (7 . ± . × − are for the 2017XYZ and2019XYZ samples, respectively. They agree within the statistical uncertainties in the MC samples,which indicates that the momentum dependence of the calibration constants is very similar in the2017XYZ and 2019XYZ samples. (MeV) cm E ) c ( M e V / i n c m - E ou t c m E FIG. 4. The dependence of the bias in M p ( µ + µ − ) due to the bias in track momentum calibration withdi-moun events and it should be the same for data and MC (slope k ). The ( E outcm − E incm ) are the differencebetween output and input E cm (the output E cm is equal to the M p ( µ + µ − ) if the events without radiation.)at different CM energies and simulated by di-muon events without radiation. V. THE MASS SHIFT ∆ M ISR / FSR
The E cm of the initial e + e − pair is measured via the di-muon process e + e − → ( γ ISR / FSR ) µ + µ − .However, due to the emission of radiative photons, the invariant mass of the µ + µ − pair is smallerthan E cm by ∆ M ISR / FSR . This correction is estimated with MC simulation using
BABAYAGA M ( µ + µ − ) from the samples with ISR/FSR on and off with a Gaussian functionin a range around the peak (same as for data in Sec. III). The difference in M p ( µ + µ − ) is takenas the mass shift ∆ M ISR / FSR caused by ISR or FSR. The ∆ M ISR / FSR versus E cm shown in Fig. 5indicates that the ISR/FSR effect depends on E cm linearly. The data are fitted with a linear functionto have an improved precision measurement of the correction. The fit gives ∆ M ISR / FSR = (1 . ± . × − × E cm + ( − . ± . (3)with a correlation factor of − . between the slope and the intercept, and the goodness of the fitis χ /ndf = 6 . / . 10 (MeV) cm E ) c ( M e V / I S R / F S R M ∆ FIG. 5. The mass shift ∆ M ISR / FSR versus CM energy for e + e − → ( γ ISR / FSR ) µ + µ − MC samples. Thered solid line is the fit with a linear function.
VI. SYSTEMATIC UNCERTAINTIES
The systematic uncertainty in E cm is from the momentum calibration of µ ± , the estimation ofthe mass shift ∆ M ISR / FSR due to ISR/FSR, the open angle cut of cos θ µ + µ − , the corresponding fitprocedure, and the generator. The bias of the momentum measurement of µ ± and the estimationof the mass shift ∆ M ISR / FSR due to ISR/FSR both have a linear relationship with E cm , and theuncertainty caused by the uncertainty of the parameters is regarded as the systematic uncertainties.In order to reduce the influence of the events with large radiation, we have required cos θ µ + µ − < − . . Different cut values will give different M p ( µ + µ − ) and corresponding radiation correc-tion values ∆ M ISR / FSR . The changes in these two parts cancel each other out. The biggest dif-ference comes from the data between − . and − . , and is . ± . MeV. We take0.14 MeV as the uncertainty due to this requirement.The M p ( µ + µ − ) is measured by fitting with a Gaussian function in the range of ( − σ, + 1 . σ ) around the peak with fit quality χ /ndf < . . If the fit range is smaller than the standard range,the difference in fit results is less than 0.1 MeV. We take this as the uncertainty due to the fitmethod.The contribution to the systematic uncertainty of the ISR/FSR correction from the generator isnegligibly small, as claimed in Ref. [10]. The uncertainties from other sources, such as backgroundand other event selection criteria, are negligible.Assuming all the sources of systematic uncertainty are independent, the total systematic uncer-tainty is obtained by adding all the items in quadrature, which is listed in Table I. The uncertaintyis smaller than 0.6 MeV for all the data samples. VII. SUMMARY
The center-of-mass energies, E cm , of the data samples are obtained by using Eq. (1), withthe correction factors in Eqs. (2) and (3). The final results are listed in Table I including the11tatistical and systematic uncertainties. The corresponding statistical uncertainty is very small,and the systematic uncertainty is found to be less than 0.6 MeV. The stability of E cm over time forthe data samples is also examined.The results presented in this work are essential for the discovery of new states and the inves-tigation of the transitions of charmonium and charmoniumlike states [15] using the BESIII data.Some of the analyses have been presented in Refs. [16–21]. ACKNOWLEDGMENTS
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for theirstrong support. This work is supported in part by National Key Research and Development Pro-gram of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400; National NaturalScience Foundation of China (NSFC) under Contracts Nos. 11625523, 11635010, 11735014,11822506, 11835012, 11935015, 11935016, 11935018, 11961141012; the Chinese Academyof Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facil-ity Funds of the NSFC and CAS under Contracts Nos. U1732263, U1832207; CAS Key Re-search Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physicsand Cosmology; ERC under Contract No. 758462; European Union Horizon 2020 research andinnovation programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790;German Research Foundation DFG under Contracts Nos. 443159800, Collaborative ResearchCenter CRC 1044, FOR 2359, FOR 2359, GRK 214; Istituto Nazionale di Fisica Nucleare, Italy;Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science andTechnology fund; Olle Engkvist Foundation under Contract No. 200-0605; STFC (United King-dom); The Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; TheRoyal Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research Council;U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0012069. [1] M. Ablikim et al. [BESIII Collaboration], Nucl. Instrum. Meth. A , 345 (2010).[2] D. M. Asner et al. , Int. J. Mod. Phys. A , 499 (2009).[3] C. H. Yu et al. , Proceedings of IPAC2016, Busan, Korea, 2016, doi:10.18429/JACoW-IPAC2016-TUYA01.[4] M. Ablikim et al. [BESIII Collaboration], Chin. Phys. C , 040001 (2020).[5] E. V. Abakumova et al. , Nucl. Instrum. Meth. A , 21 (2011).[6] M. Ablikim et al. [BESIII Collaboration], Chin. Phys. C , 063001 (2016).[7] X. Li et al. , Radiat. Detect. Technol. Methods , 13 (2017); Y. X. Guo et al. , Radiat. Detect. Technol.Methods , 15 (2017); P. Cao et al. , Nucl. Instrum. Meth. A , 163053 (2020).[8] Yifan Yang, “The Study of M1 Transitions of Charmonia at BE-SIII”, Ph.D thesis, Institute of High Energy Physics, 2019. .[9] S. Agostinelli et al. [GEANT4 Collaboration], Nucl. Instrum. Meth. A , 250 (2003).[10] G. Balossini, C. M. Carloni Calame, G. Montagna, O. Nicrosini and F. Piccinini, Nucl. Phys. B ,227 (2006).
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