# Measurement of the total and leptonic decay widths of the $J/\psi$ resonance
with an energy scan method at BESIII

M. Ablikim1, M. N. Achasov11,b, P. Adlarson70, M. Albrecht4, R. Aliberti31, A.
Amoroso69A,69C, M. R. An35, Q. An66,53, X. H. Bai61, Y. Bai52, O. Bakina32, R.
Baldini Ferroli26A, I. Balossino27A, Y. Ban42,g, V. Batozskaya1,40, D.
Becker31, K. Begzsuren29, N. Berger31, M. Bertani26A, D. Bettoni27A, F.
Bianchi69A,69C, J. Bloms63, A. Bortone69A,69C, I. Boyko32, R. A. Briere5, A.
Brueggemann63, H. Cai71, X. Cai1,53, A. Calcaterra26A, G. F. Cao1,58, N.
Cao1,58, S. A. Cetin57A, J. F. Chang1,53, W. L. Chang1,58, G. Chelkov32,a, C.
Chen39, Chao Chen50, G. Chen1, H. S. Chen1,58, M. L. Chen1,53, S. J. Chen38,
S. M. Chen56, T. Chen1, X. R. Chen28,58, X. T. Chen1, Y. B. Chen1,53, Z. J.
Chen23,h, W. S. Cheng69C, S. K. Choi 50, X. Chu39, G. Cibinetto27A, F.
Cossio69C, J. J. Cui45, H. L. Dai1,53, J. P. Dai73, A. Dbeyssi17, R. E. de
Boer4, D. Dedovich32, Z. Y. Deng1, A. Denig31, I. Denysenko32, M.
Destefanis69A,69C, F. De Mori69A,69C, Y. Ding36, J. Dong1,53, L. Y. Dong1,58,
M. Y. Dong1,53,58, X. Dong71, S. X. Du75, P. Egorov32,a, Y. L. Fan71, J.
Fang1,53, S. S. Fang1,58, W. X. Fang1, Y. Fang1, R. Farinelli27A, L.
Fava69B,69C, F. Feldbauer4, G. Felici26A, C. Q. Feng66,53, J. H. Feng54, K
Fischer64, M. Fritsch4, C. Fritzsch63, C. D. Fu1, H. Gao58, Y. N. Gao42,g,
Yang Gao66,53, S. Garbolino69C, I. Garzia27A,27B, P. T. Ge71, Z. W. Ge38, C.
Geng54, E. M. Gersabeck62, A Gilman64, K. Goetzen12, L. Gong36, W. X.
Gong1,53, W. Gradl31, M. Greco69A,69C, L. M. Gu38, M. H. Gu1,53, Y. T. Gu14,
C. Y Guan1,58, A. Q. Guo28,58, L. B. Guo37, R. P. Guo44, Y. P. Guo10,f, A.
Guskov32,a, T. T. Han45, W. Y. Han35, X. Q. Hao18, F. A. Harris60, K. K. He50,
K. L. He1,58, F. H. Heinsius4, C. H. Heinz31, Y. K. Heng1,53,58, C. Herold55,
M. Himmelreich31,d, G. Y. Hou1,58, Y. R. Hou58, Z. L. Hou1, H. M. Hu1,58, J.
F. Hu51,i, T. Hu1,53,58, Y. Hu1, G. S. Huang66,53, K. X. Huang54, L. Q.
Huang28,58, X. T. Huang45, Y. P. Huang1, Z. Huang42,g, T. Hussain68, N
Hüsken25,31, W. Imoehl25, M. Irshad66,53, J. Jackson25, S. Jaeger4, S.
Janchiv29, E. Jang50, J. H. Jeong50, Q. Ji1, Q. P. Ji18, X. B. Ji1,58, X. L.
Ji1,53, Y. Y. Ji45, Z. K. Jia66,53, H. B. Jiang45, S. S. Jiang35, X. S.
Jiang1,53,58, Y. Jiang58, J. B. Jiao45, Z. Jiao21, S. Jin38, Y. Jin61, M. Q.
Jing1,58, T. Johansson70, N. Kalantar-Nayestanaki59, X. S. Kang36, R.
Kappert59, B. C. Ke75, I. K. Keshk4, A. Khoukaz63, R. Kiuchi1, R. Kliemt12, L.
Koch33, O. B. Kolcu57A, B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc40,70, W.
Kühn33, J. J. Lane62, J. S. Lange33, P. Larin17, A. Lavania24, L.
Lavezzi69A,69C, Z. H. Lei66,53, H. Leithoff31, M. Lellmann31, T. Lenz31, C.
Li43, C. Li39, C. H. Li35, Cheng Li66,53, D. M. Li75, F. Li1,53, G. Li1, H.
Li47, H. Li66,53, H. B. Li1,58, H. J. Li18, H. N. Li51,i, J. Q. Li4, J. S.
Li54, J. W. Li45, Ke Li1, L. J Li1, L. K. Li1, Lei Li3, M. H. Li39, P. R.
Li34,j,k, S. X. Li10, S. Y. Li56, T. Li45, W. D. Li1,58, W. G. Li1, X. H.
Li66,53, X. L. Li45, Xiaoyu Li1,58, Z. X. Li14, H. Liang66,53, H. Liang1,58,
H. Liang30, Y. F. Liang49, Y. T. Liang28,58, G. R. Liao13, L. Z. Liao45, J.
Libby24, A. Limphirat55, C. X. Lin54, D. X. Lin28,58, T. Lin1, B. J. Liu1, C.
X. Liu1, D. Liu17,66, F. H. Liu48, Fang Liu1, Feng Liu6, G. M. Liu51,i, H.
Liu34,j,k, H. B. Liu14, H. M. Liu1,58, Huanhuan Liu1, Huihui Liu19, J. B.
Liu66,53, J. L. Liu67, J. Y. Liu1,58, K. Liu1, K. Y. Liu36, Ke Liu20, L.
Liu66,53, Lu Liu39, M. H. Liu10,f, P. L. Liu1, Q. Liu58, S. B. Liu66,53, T.
Liu10,f, W. K. Liu39, W. M. Liu66,53, X. Liu34,j,k, Y. Liu34,j,k, Y. B. Liu39,
Z. A. Liu1,53,58, Z. Q. Liu45, X. C. Lou1,53,58, F. X. Lu54, H. J. Lu21, J. G.
Lu1,53, X. L. Lu1, Y. Lu7, Y. P. Lu1,53, Z. H. Lu1, C. L. Luo37, M. X. Luo74,
T. Luo10,f, X. L. Luo1,53, X. R. Lyu58, Y. F. Lyu39, F. C. Ma36, H. L. Ma1, L.
L. Ma45, M. M. Ma1,58, Q. M. Ma1, R. Q. Ma1,58, R. T. Ma58, X. Y. Ma1,53, Y.
Ma42,g, F. E. Maas17, M. Maggiora69A,69C, S. Maldaner4, S. Malde64, Q. A.
Malik68, A. Mangoni26B, Y. J. Mao42,g, Z. P. Mao1, S. Marcello69A,69C, Z. X.
Meng61, G. Mezzadri27A, H. Miao1, T. J. Min38, R. E. Mitchell25, X. H.
Mo1,53,58, N. Yu. Muchnoi11,b, Y. Nefedov32, F. Nerling17,d, I. B.
Nikolaev11,b, Z. Ning1,53, S. Nisar9,l, Y. Niu 45, S. L. Olsen58, Q.
Ouyang1,53,58, S. Pacetti26B,26C, X. Pan10,f, Y. Pan52, A. Pathak30, M.
Pelizaeus4, H. P. Peng66,53, K. Peters12,d, J. L. Ping37, R. G. Ping1,58, S.
Plura31, S. Pogodin32, V. Prasad66,53, F. Z. Qi1, H. Qi66,53, H. R. Qi56, M.
Qi38, T. Y. Qi10,f, S. Qian1,53, W. B. Qian58, Z. Qian54, C. F. Qiao58, J. J.
Qin67, L. Q. Qin13, X. P. Qin10,f, X. S. Qin45, Z. H. Qin1,53, J. F. Qiu1, S.
Q. Qu39, S. Q. Qu56, K. H. Rashid68, C. F. Redmer31, K. J. Ren35, A.
Rivetti69C, V. Rodin59, M. Rolo69C, G. Rong1,58, Ch. Rosner17, S. N. Ruan39,
H. S. Sang66, A. Sarantsev32,c, Y. Schelhaas31, C. Schnier4, K. Schoenning70,
M. Scodeggio27A,27B, K. Y. Shan10,f, W. Shan22, X. Y. Shan66,53, J. F.
Shangguan50, L. G. Shao1,58, M. Shao66,53, C. P. Shen10,f, H. F. Shen1,58, X.
Y. Shen1,58, B. A. Shi58, H. C. Shi66,53, J. Y. Shi1, q. q. Shi50, R. S.
Shi1,58, X. Shi1,53, X. D Shi66,53, J. J. Song18, W. M. Song30,1, Y. X.
Song42,g, S. Sosio69A,69C, S. Spataro69A,69C, F. Stieler31, K. X. Su71, P. P.
Su50, Y. J. Su58, G. X. Sun1, H. Sun58, H. K. Sun1, J. F. Sun18, L. Sun71, S.
S. Sun1,58, T. Sun1,58, W. Y. Sun30, X Sun23,h, Y. J. Sun66,53, Y. Z. Sun1, Z.
T. Sun45, Y. H. Tan71, Y. X. Tan66,53, C. J. Tang49, G. Y. Tang1, J. Tang54,
L. Y Tao67, Q. T. Tao23,h, M. Tat64, J. X. Teng66,53, V. Thoren70, W. H.
Tian47, Y. Tian28,58, I. Uman57B, B. Wang1, B. L. Wang58, C. W. Wang38, D. Y.
Wang42,g, F. Wang67, H. J. Wang34,j,k, H. P. Wang1,58, K. Wang1,53, L. L.
Wang1, M. Wang45, M. Z. Wang42,g, Meng Wang1,58, S. Wang13, S. Wang10,f, T.
Wang10,f, T. J. Wang39, W. Wang54, W. H. Wang71, W. P. Wang66,53, X. Wang42,g,
X. F. Wang34,j,k, X. L. Wang10,f, Y. Wang56, Y. D. Wang41, Y. F. Wang1,53,58,
Y. H. Wang43, Y. Q. Wang1, Yaqian Wang16,1, Z. Wang1,53, Z. Y. Wang1,58, Ziyi
Wang58, D. H. Wei13, F. Weidner63, S. P. Wen1, D. J. White62, U. Wiedner4, G.
Wilkinson64, M. Wolke70, L. Wollenberg4, J. F. Wu1,58, L. H. Wu1, L. J.
Wu1,58, X. Wu10,f, X. H. Wu30, Y. Wu66, Z. Wu1,53, L. Xia66,53, T. Xiang42,g,
D. Xiao34,j,k, G. Y. Xiao38, H. Xiao10,f, S. Y. Xiao1, Y. L. Xiao10,f, Z. J.
Xiao37, C. Xie38, X. H. Xie42,g, Y. Xie45, Y. G. Xie1,53, Y. H. Xie6, Z. P.
Xie66,53, T. Y. Xing1,58, C. F. Xu1, C. J. Xu54, G. F. Xu1, H. Y. Xu61, Q. J.
Xu15, X. P. Xu50, Y. C. Xu58, Z. P. Xu38, F. Yan10,f, L. Yan10,f, W. B.
Yan66,53, W. C. Yan75, H. J. Yang46,e, H. L. Yang30, H. X. Yang1, L. Yang47,
S. L. Yang58, Tao Yang1, Y. F. Yang39, Y. X. Yang1,58, Yifan Yang1,58, M.
Ye1,53, M. H. Ye8, J. H. Yin1, Z. Y. You54, B. X. Yu1,53,58, C. X. Yu39, G.
Yu1,58, T. Yu67, X. D. Yu42,g, C. Z. Yuan1,58, L. Yuan2, S. C. Yuan1, X. Q.
Yuan1, Y. Yuan1,58, Z. Y. Yuan54, C. X. Yue35, A. A. Zafar68, F. R. Zeng45, X.
Zeng Zeng6, Y. Zeng23,h, Y. H. Zhan54, A. Q. Zhang1, B. L. Zhang1, B. X.
Zhang1, D. H. Zhang39, G. Y. Zhang18, H. Zhang66, H. H. Zhang30, H. H.
Zhang54, H. Y. Zhang1,53, J. L. Zhang72, J. Q. Zhang37, J. W. Zhang1,53,58, J.
X. Zhang34,j,k, J. Y. Zhang1, J. Z. Zhang1,58, Jianyu Zhang1,58, Jiawei
Zhang1,58, L. M. Zhang56, L. Q. Zhang54, Lei Zhang38, P. Zhang1, Q. Y.
Zhang35,75, Shuihan Zhang1,58, Shulei Zhang23,h, X. D. Zhang41, X. M. Zhang1,
X. Y. Zhang45, X. Y. Zhang50, Y. Zhang64, Y. T. Zhang75, Y. H. Zhang1,53, Yan
Zhang66,53, Yao Zhang1, Z. H. Zhang1, Z. Y. Zhang71, Z. Y. Zhang39, G. Zhao1,
J. Zhao35, J. Y. Zhao1,58, J. Z. Zhao1,53, Lei Zhao66,53, Ling Zhao1, M. G.
Zhao39, Q. Zhao1, S. J. Zhao75, Y. B. Zhao1,53, Y. X. Zhao28,58, Z. G.
Zhao66,53, A. Zhemchugov32,a, B. Zheng67, J. P. Zheng1,53, Y. H. Zheng58, B.
Zhong37, C. Zhong67, X. Zhong54, H. Zhou45, L. P. Zhou1,58, X. Zhou71, X. K.
Zhou58, X. R. Zhou66,53, X. Y. Zhou35, Y. Z. Zhou10,f, J. Zhu39, K. Zhu1, K.
J. Zhu1,53,58, L. X. Zhu58, S. H. Zhu65, S. Q. Zhu38, T. J. Zhu72, W. J.
Zhu10,f, Y. C. Zhu66,53, Z. A. Zhu1,58, B. S. Zou1, J. H. Zou1 (BESIII
Collaboration) 1 Institute of High Energy Physics, Beijing 100049, People’s
Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
10 Fudan University, Shanghai 200433, People’s Republic of China
11 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
12 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
13 Guangxi Normal University, Guilin 541004, People’s Republic of China
14 Guangxi University, Nanning 530004, People’s Republic of China
15 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
16 Hebei University, Baoding 071002, People’s Republic of China
17 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
18 Henan Normal University, Xinxiang 453007, People’s Republic of China
19 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
20 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
21 Huangshan College, Huangshan 245000, People’s Republic of China
22 Hunan Normal University, Changsha 410081, People’s Republic of China
23 Hunan University, Changsha 410082, People’s Republic of China
24 Indian Institute of Technology Madras, Chennai 600036, India
25 Indiana University, Bloomington, Indiana 47405, USA
26 INFN Laboratori Nazionali di Frascati , (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
27 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
28 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
29 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
30 Jilin University, Changchun 130012, People’s Republic of China
31 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
32 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
33 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
34 Lanzhou University, Lanzhou 730000, People’s Republic of China
35 Liaoning Normal University, Dalian 116029, People’s Republic of China
36 Liaoning University, Shenyang 110036, People’s Republic of China
37 Nanjing Normal University, Nanjing 210023, People’s Republic of China
38 Nanjing University, Nanjing 210093, People’s Republic of China
39 Nankai University, Tianjin 300071, People’s Republic of China
40 National Centre for Nuclear Research, Warsaw 02-093, Poland
41 North China Electric Power University, Beijing 102206, People’s Republic of
China
42 Peking University, Beijing 100871, People’s Republic of China
43 Qufu Normal University, Qufu 273165, People’s Republic of China
44 Shandong Normal University, Jinan 250014, People’s Republic of China
45 Shandong University, Jinan 250100, People’s Republic of China
46 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
47 Shanxi Normal University, Linfen 041004, People’s Republic of China
48 Shanxi University, Taiyuan 030006, People’s Republic of China
49 Sichuan University, Chengdu 610064, People’s Republic of China
50 Soochow University, Suzhou 215006, People’s Republic of China
51 South China Normal University, Guangzhou 510006, People’s Republic of China
52 Southeast University, Nanjing 211100, People’s Republic of China
53 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
54 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
55 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
56 Tsinghua University, Beijing 100084, People’s Republic of China
57 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
58 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
59 University of Groningen, NL-9747 AA Groningen, The Netherlands
60 University of Hawaii, Honolulu, Hawaii 96822, USA
61 University of Jinan, Jinan 250022, People’s Republic of China
62 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
63 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
64 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
65 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
66 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
67 University of South China, Hengyang 421001, People’s Republic of China
68 University of the Punjab, Lahore-54590, Pakistan
69 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
70 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
71 Wuhan University, Wuhan 430072, People’s Republic of China
72 Xinyang Normal University, Xinyang 464000, People’s Republic of China
73 Yunnan University, Kunming 650500, People’s Republic of China
74 Zhejiang University, Hangzhou 310027, People’s Republic of China
75 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e 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
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan

###### Abstract

Using $e^{+}e^{-}$ annihilation data sets collected with the BESIII detector,
we measure the cross sections of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ at fifteen center-of-mass energy points in the
vicinity of the $J/\psi$ resonance. By a simultaneous fit to the measured,
center-of-mass energy dependent cross sections of the two processes, the
combined quantities $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ are determined to be ($0.346\pm
0.009$) and ($0.335\pm 0.006$) keV, respectively, where $\Gamma_{ee}$,
$\Gamma_{\mu\mu}$, and $\Gamma_{\rm tot}$ are the electronic, muonic, and
total decay widths of the $J/\psi$ resonance, respectively. Using the
resultant $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$, the ratio
$\Gamma_{ee}/\Gamma_{\mu\mu}$ is calculated to be $1.031\pm 0.015$, which is
consistent with the expectation of lepton universality within about two
standard deviations. Assuming lepton universality and using the branching
fraction of the $J/\psi$ leptonic decay measured by BESIII in 2013,
$\Gamma_{\rm tot}$ and $\Gamma_{ll}$ are determined to be ($93.0\pm 2.1$) and
($5.56\pm 0.11$) keV, respectively, where $\Gamma_{ll}$ is the average
leptonic decay width of the $J/\psi$ resonance.

Keywords: $J/\psi$ , decay width, lepton universality, energy scan, BESIII

††journal: Physics Letters B

##  1 Introduction

The total and electronic decay widths $\Gamma_{\rm tot}$ and $\Gamma_{ee}$ of
the $J/\psi$ resonance, present in the Breit-Wigner formulae for all the decay
modes of $J/\psi$ produced in $e^{+}e^{-}$ collisions [1], are among its most
important parameters. Theoretically, these decay widths, reflecting $J/\psi$
internal interactions, are predicted by various potential models [2, 3, 4, 5,
6] and lattice quantum chromodynamics [7]. Measurements of these decay widths
and comparisons of the experimental results with different theoretical
calculations can help us gain a better understanding of the underlying
physics.

Furthermore, the ratio of the electronic to muonic decay widths of the
$J/\psi$ resonance, $\Gamma_{ee}/\Gamma_{\mu\mu}$, can be used to test the
lepton universality assumption [8]. Based on the assumption, the ratio is
derived to be [9]

$\Gamma_{ee}/\Gamma_{\mu\mu}=\frac{\beta_{e}(3-\beta_{e}^{2})}{\beta_{\mu}(3-\beta_{\mu}^{2})},$
(1)

with

$\beta_{l}=\sqrt{1-(2m_{l}/M)^{2}}\text{\ \ \ \ \ }l=e,\mu,$ (2)

where $m_{e}$, $m_{\mu}$, and $M$ are the masses of electron, muon, and the
$J/\psi$ resonance, respectively. Taking the values of $m_{e}$, $m_{\mu}$, and
$M$ from the Particle Data Group (PDG) [1], $\Gamma_{ee}/\Gamma_{\mu\mu}$ is
calculated to be 1.00000814211(6), which has a deviation from 1 that is far
less than the experimental precision at present. Thus, any observed,
significant deviation of $\Gamma_{ee}/\Gamma_{\mu\mu}$ from 1 will be a hint
of physics beyond the Standard Model [10].

Since the discovery of the $J/\psi$ resonance in 1974 [11, 12], its decay
widths have been measured by many experiments [13, 14, 15, 16, 17]. The
precision of the measurements has been improved significantly in the past two
decades. In 2004 and 2006, the $J/\psi$ decay widths have been measured by
studying the $J/\psi$ samples produced in the initial state radiation (ISR)
return process $e^{+}e^{-}\to\gamma^{\rm ISR}\mu^{+}\mu^{-}$ collected at the
$\Upsilon(4S)$ and $\psi(3770)$ peaks by BaBar [18] and CLEO [19],
respectively. In 2010, KEDR improved the measurement precision by performing
an energy scan (ES) around the $J/\psi$ peak and studying the $J/\psi$
production in the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ [20]. In 2018 and 2020, KEDR presented new
results with the $J/\psi$ production in the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\rm inclusive\ \rm hadrons$ [21].

Operating in the $\tau$-charm energy region, the high luminosity of the BEPCII
collider [22] and the excellent performance of the BESIII detector [23] offer
us a good opportunity for the precision measurements of the $J/\psi$ decay
widths. In 2016, BESIII measured the $J/\psi$ decay widths by applying the ISR
return technique to the data sample collected at the $\psi(3770)$ peak, and
obtained a result with improved precision [24]. In this Letter, we report a
new precision measurement of the $J/\psi$ decay widths with the ES method,
confirming and complementing the above measurement.

Since the $J/\psi$ resonance contributes to the vacuum polarization in the
time-like region, the cross sections of the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ are functions of the $J/\psi$
decay widths [25, 26]. Specifically, the cross section ($\sigma_{\rm 0}$) of
each process with respect to the center-of-mass (CM) energy ($W_{\rm 0}$) can
be written as

$\sigma_{\rm 0}(W_{\rm 0})=\sigma_{\rm 0}^{\rm C}(W_{\rm 0})+\sigma_{\rm
0}^{\rm R}(W_{\rm 0})+\sigma_{\rm 0}^{\rm I}(W_{\rm 0}),$ (3)

where $\sigma_{\rm 0}^{\rm C}$, $\sigma_{\rm 0}^{\rm R}$ and $\sigma_{\rm
0}^{\rm I}$ are the continuum, resonance and interference terms, respectively.
The formula still holds after considering the ISR effect, and we take
$\sigma_{\rm 0}$, $\sigma_{\rm 0}^{\rm C}$, $\sigma_{\rm 0}^{\rm R}$ and
$\sigma_{\rm 0}^{\rm I}$ here as the quantities with ISR considered. Unlike
the term $\sigma_{\rm 0}^{\rm C}$, the terms $\sigma_{\rm 0}^{\rm R}$ and
$\sigma_{\rm 0}^{\rm I}$ depend on the $J/\psi$ decay widths, and their
analytic forms are derived in Ref. [27] using the structure function method
[28, 29].

In Eq. (3), $\sigma_{\rm 0}^{\rm R}$ is the primary term related to the
$J/\psi$ decay widths, and its major subterm is proportional to
$\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ for the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$, respectively [27]. Therefore,
we can determine $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ by fitting to the measured, CM
energy dependent cross sections of the two processes. Then,
$\Gamma_{ee}/\Gamma_{\mu\mu}$ can be evaluated as the ratio of
$\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ to
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$. Combined with the branching
fraction of the $J/\psi$ leptonic decay measured by BESIII in 2013 [30], the
total and leptonic $J/\psi$ decay widths can be obtained from
$\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ as well.

##  2 Experimental facilities and data sets

The data used in this work were collected with the BESIII detector [23], which
operates at the south crossing point of the BEPCII collider [22]. BEPCII is a
double-ring $e^{+}e^{-}$ collider operating in the $\tau$-charm energy region
(2.0-4.9 GeV) with its achieved peak luminosity of $10^{33}\text{
cm}^{-2}\text{s}^{-1}$ at the CM energy $\sqrt{s}=3.773$ GeV. The BESIII
detector with a geometrical acceptance of 93% of 4$\pi$, consists of the
following main components: (1) a small-cell, helium-based main drift chamber
(MDC) measuring the momenta of charged tracks in a 1 T magnetic field with a
resolution of 0.5% for 1 GeV/$c$ transverse momentum and the specific energy
loss ($dE/dx$) with a resolution of 6%; (2) a time-of-flight (TOF) system for
particle identification composed of a barrel and two endcaps made of plastic
scintillators; the time resolution is 80 ps in the barrel, and 110 ps in the
endcaps; (3) an electromagnetic calorimeter (EMC) made of CsI(Tl) crystals
arranged in a cylindrical shape (barrel) and two endcaps; for 1.0 GeV photons,
the energy resolution is 2.5% in the barrel and 5% in the endcaps; (4) a
superconducting solenoid magnet providing a nominal magnetic field of 1 T (0.9
T in 2012) parallel to the beam direction; (5) a muon chamber system made of
resistive plate chambers with position resolution about 2 cm.

In addition, a beam energy measurement system (BEMS), located at the north
crossing point of the BEPCII storage rings, is used to determine the BEPCII
beam energies by measuring the energies of Compton back-scattered photons
[31].

In 2012, an ES experiment was performed at fifteen CM energy points in the
vicinity of the $J/\psi$ resonance. The measured CM energies and integrated
luminosities are summarized in Table 1. The CM energies are

Table 1: Basic information of the data samples at all individual CM energy points, including the proposed CM energies (Prop. $\sqrt{s}$) before the data taking, the CM energies measured by the BEMS and calibrated with the $J/\psi$ mass from the PDG (Calib. BEMS $\sqrt{s}$), and the integrated luminosities (Int. $L$). For Calib. BEMS $\sqrt{s}$ and Int. $L$, the first uncertainties are statistical and the second ones are systematic. Prop. $\sqrt{s}$ (MeV) | Calib. BEMS $\sqrt{s}$ (MeV) | Int. $L$ (pb-1)
---|---|---
3050.0 | 3049.642$\pm$0.026$\pm$0.033 | 14.919$\pm$0.029$\pm$0.158
3060.0 | 3058.693$\pm$0.028$\pm$0.033 | 15.060$\pm$0.029$\pm$0.158
3083.0 | 3082.496$\pm$0.023$\pm$0.033 | 4.769$\pm$0.017$\pm$0.052
3090.0 | 3088.854$\pm$0.022$\pm$0.033 | 15.558$\pm$0.030$\pm$0.162
3093.0 | 3091.760$\pm$0.025$\pm$0.033 | 14.910$\pm$0.030$\pm$0.157
3094.3 | 3094.697$\pm$0.084$\pm$0.033 | 2.143$\pm$0.011$\pm$0.023
3095.2 | 3095.430$\pm$0.081$\pm$0.033 | 1.816$\pm$0.010$\pm$0.019
3095.8 | 3095.826$\pm$0.075$\pm$0.033 | 2.135$\pm$0.011$\pm$0.023
3096.9 | 3097.213$\pm$0.076$\pm$0.033 | 2.069$\pm$0.011$\pm$0.024
3098.2 | 3098.340$\pm$0.075$\pm$0.033 | 2.203$\pm$0.011$\pm$0.023
3099.0 | 3099.042$\pm$0.093$\pm$0.033 | 0.756$\pm$0.007$\pm$0.008
3101.5 | 3101.359$\pm$0.106$\pm$0.033 | 1.612$\pm$0.010$\pm$0.018
3105.5 | 3105.580$\pm$0.090$\pm$0.033 | 2.106$\pm$0.011$\pm$0.022
3112.0 | 3112.051$\pm$0.093$\pm$0.033 | 1.720$\pm$0.010$\pm$0.019
3120.0 | 3119.878$\pm$0.115$\pm$0.033 | 1.264$\pm$0.009$\pm$0.013

measured by the BEMS and calibrated according to the $J/\psi$ mass value given
by the PDG [1]. As a consequence, the $J/\psi$ mass can not be determined in
this work. The CM energy calibration process fits to the $J/\psi$ lineshapes
in the $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ final states simultaneously to their
preliminary measured, CM energy dependent cross sections. Adding the
uncertainties of the $J/\psi$ masses from the fit and from the PDG in
quadrature gives a total calibration uncertainty of 0.033 MeV, which is
comparable with the systematic uncertainty (0.043 MeV) of the calibration (via
the inclusive hadronic decay mode) to the small $J/\psi$ scan data used for
the $\tau$ mass measurement [32]. The corresponding integrated luminosities
are measured offline with $e^{+}e^{-}\to\gamma\gamma$ events [33].

To determine the signal detection efficiencies, Monte Carlo (MC) simulated
events of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ in the polar angle ranges of 34∘-146∘ and
0∘-180∘, respectively, incorporating the ISR and final state radiation (FSR)
effects, are simulated with a revised version of the Babayaga-3.5 [34]
generator, which is modified by the authors to explicitly involve the $J/\psi$
resonance in the vacuum polarization. In addition, MC events of the processes
$e^{+}e^{-}\to\rm inclusive\ \rm hadrons$ and $\gamma^{*}\gamma^{*}\to X$
($e^{+}e^{-}e^{+}e^{-}$, $e^{+}e^{-}\mu^{+}\mu^{-}$, etc.) are generated for
background studies with the ConExc [35] and BesTwogam [36] generators,
respectively. When generating these events, the calibrated BEMS CM energies
are used.

A geant4 [37] based MC simulation program including the geometric description
and response of the detector is used to simulate the interaction of final
state particles in the detector. Both the experimetal data and simulated MC
are reconstructed and analysed under the gaudi [38] based offline software
system.

##  3 Event selection

The signal candidates of $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ events are required to have two oppositely
charged tracks in the MDC. Each charged track has to fulfill the following
requirements: it must originate from the interaction region of $|V_{\rm r}|<1$
cm and $|V_{\rm z}|<10$ cm, where $|V_{\rm r}|$ and $|V_{\rm z}|$ are its
closest approach relative to the collision point in the x-y plane and along
the z axis (taken as the axis of the MDC), respectively; it must hit the
detector in the barrel region of $|\cos\theta|<0.8$, where $\theta$ is the
polar angle of the reconstructed momentum vector with respect to the z axis.

For $e^{+}e^{-}\to e^{+}e^{-}$ candidates, the next two criteria are applied
further to each of the selected tracks: its momentum ($P$) is required to be
larger than 0.7 times the beam energy ($E_{\rm beam}$), and its energy
deposited in the EMC ($E$) has to be larger than 0.6 times the momentum.

For $e^{+}e^{-}\to\mu^{+}\mu^{-}$ candidates, the following conditions are
required in addition: for each of the selected tracks, $P$ must be larger than
0.8$E_{\rm beam}$, $E$ has to be larger than 25 MeV and less than 0.25$P$, and
valid timing information is required to be left in the TOF; at the event
level, no neutral showers with a deposited energy above 25 MeV are allowed in
the EMC, and the difference of the flight time of the two charged tracks
($\Delta t_{\rm TOF}^{\mu}$) obtained with the TOF has to be less than $1.5$
ns to suppress cosmic rays.

Figure 1 shows the comparison between data and MC simulation of variables used
in the event selection. The data shown in the figure are those of surviving
candidate events subtracting the residual background estimated with the MC
simulation. In general, the MC simulation provides a good description of the
data although minor discrepancies between them are visible. The effects of
these discrepancies are taken into account in the systematic uncertainty
estimation (see Section 4.2 for details).

For the process $e^{+}e^{-}\to e^{+}e^{-}$ ($\mu^{+}\mu^{-}$), the
efficiencies obtained from the signal MC samples are about 70% (80%), and the
background levels esitimated with the MC simulation are less than 0.05%
(0.5%). Closer examination with a generic event type analysis tool, TopoAna
[39], shows that the backgrounds mainly arise from events with
$\pi^{+}\pi^{-}$, $K^{+}K^{-}$ or $e^{+}e^{-}e^{+}e^{-}$
($e^{+}e^{-}\mu^{+}\mu^{-}$) final states.

##  4 Cross section measurement

###  4.1 Nominal results with statistical uncertainties

Usually, the cross section $\sigma$ is determined from

$\sigma=\frac{N_{\rm sig}-N_{\rm bkgs}}{L\cdot\epsilon_{\rm
trg}\cdot\epsilon_{\rm recsel}}\cdot f,$ (4)

where $N_{\rm sig}$ is the number of signal events selected from data, $N_{\rm
bkgs}$ is the number of the residual background events, $L$ is the integrated
luminosity, $\epsilon_{\rm trg}$ is the trigger efficiency, $\epsilon_{\rm
recsel}$ is the reconstruction-selection efficiency and $f$ is a
reconstruction efficiency correction factor.

The trigger efficiency is taken as 100% in this work [40]. The correction
factor $f$ is related to the imperfection of the detector simulation. In
practice, the reconstruction efficiencies (including the tracking efficiency
in the MDC and the cluster reconstruction efficiency in the EMC) from MC
simulation deviate those from data. We study the corresponding reconstruction

Fig. 1: Comparison between data and MC simulation of the distributions of the
variables used in the selection criteria for the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ at $\sqrt{s}=3096.9$ MeV. For
all the plots, dots with error bars show the background-subtracted data (the
background level is evaluated with the MC simulation) and the histograms
denote the signal MC. The small discrepancy in the last plot is due to the
imperfection of the MC simulation, and it has a negligible effect on the cross
section measurement of $e^{+}e^{-}\to\mu^{+}\mu^{-}$.

efficiencies for leptons in different $\cos\theta$ bins. To compensate the
deviation of the reconstruction-selection efficiency, the correction factor
$f$ is introduced as

$\displaystyle f$ $\displaystyle=N^{\rm MC}_{\rm
obs}\Bigg{/}\Bigg{(}\sum\limits_{m}\sum\limits_{n}N^{\rm MC}_{\rm
obs}(m,n)\cdot\frac{\epsilon_{\rm trk}^{\rm data}(m)}{\epsilon_{\rm trk}^{\rm
MC}(m)}$ $\displaystyle\cdot\frac{\epsilon_{\rm trk}^{\rm
data}(n)}{\epsilon_{\rm trk}^{\rm MC}(n)}\cdot\frac{\epsilon_{\rm clst}^{\rm
data}(m)}{\epsilon_{\rm clst}^{\rm MC}(m)}\cdot\frac{\epsilon_{\rm clst}^{\rm
data}(n)}{\epsilon_{\rm clst}^{\rm MC}(n)}\Bigg{)}.$ (5)

Here, $N^{\rm MC}_{\rm obs}$ stands for the number of surviving events of
signal MC samples, $m$ ($n$) for the $m^{\rm th}$ ($n^{\rm th}$) $\cos\theta$
bin of positively (negatively) charged leptons, $\epsilon_{\rm trk}^{\rm
data}$ and $\epsilon_{\rm trk}^{\rm MC}$ ($\epsilon_{\rm clst}^{\rm data}$ and
$\epsilon_{\rm clst}^{\rm MC}$) for the MDC tracking efficiency (EMC cluster
reconstruction efficiency) of leptons from data and MC simulation,
respectively.

The measured cross sections and related input quantities at all individual CM
energy points of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ are summarized in Tables 2 and 3, respectively.

###  4.2 Systematic uncertainties

The systematic uncertainties of the measured cross sections arise mainly from
the integrated luminosities, trigger efficiencies, CM energies, reconstruction
and selection efficiencies, efficiency correction factors and residual
backgrounds.

The uncertainties due to the integrated luminosities are estimated to be less
than 1.40% (1.26% at $\sqrt{s}=3096.9$ MeV) [33], while those resulting from
trigger efficiencies are evaluated as 0.10% [40].

Table 2: Measured cross sections and related input quantities of the process $e^{+}e^{-}\to e^{+}e^{-}$ at all individual CM energy points. The uncertainties of the input quantities are statistical, while the first and second uncertainties of cross sections are statistical and systematic, respectively. Notably, the cross sections here are confined in the polar angle range of 34∘–146∘. Prop. $\sqrt{s}$ (MeV) | $N_{\rm sig}$ | $N_{\rm bkgs}$ | $\epsilon_{\rm recsel}$ | $f$ |  $\sigma$ (nb)
---|---|---|---|---|---
3.0500 | $2274639\pm 1508$ | $164\pm 13$ | $0.6909\pm 0.0002$ | $0.9989\pm 0.0002$ | $220.4\pm 0.2\pm 2.5$
3.0600 | $2286953\pm 1512$ | $145\pm 12$ | $0.6910\pm 0.0002$ | $0.9989\pm 0.0002$ | $219.5\pm 0.2\pm 2.5$
3.0830 | $710011\pm 843$ | $57\pm 8$ | $0.6913\pm 0.0002$ | $0.9989\pm 0.0002$ | $215.1\pm 0.3\pm 2.6$
3.0900 | $2341309\pm 1530$ | $182\pm 13$ | $0.6918\pm 0.0002$ | $0.9989\pm 0.0002$ | $217.3\pm 0.2\pm 2.4$
3.0930 | $2240003\pm 1497$ | $181\pm 13$ | $0.6918\pm 0.0002$ | $0.9989\pm 0.0002$ | $216.9\pm 0.2\pm 2.5$
3.0943 | $345449\pm 588$ | $34\pm 6$ | $0.6989\pm 0.0002$ | $0.9990\pm 0.0002$ | $230.4\pm 0.4\pm 2.9$
3.0952 | $335948\pm 580$ | $52\pm 7$ | $0.7180\pm 0.0002$ | $0.9991\pm 0.0002$ | $257.3\pm 0.4\pm 3.4$
3.0958 | $487256\pm 698$ | $61\pm 8$ | $0.7329\pm 0.0002$ | $0.9992\pm 0.0002$ | $311.2\pm 0.4\pm 6.9$
3.0969 | $577995\pm 760$ | $174\pm 13$ | $0.7580\pm 0.0002$ | $0.9994\pm 0.0002$ | $368.2\pm 0.5\pm 5.2$
3.0982 | $443694\pm 666$ | $104\pm 10$ | $0.7286\pm 0.0002$ | $0.9992\pm 0.0002$ | $276.2\pm 0.4\pm 4.2$
3.0990 | $127932\pm 358$ | $19\pm 4$ | $0.7082\pm 0.0002$ | $0.9991\pm 0.0002$ | $238.6\pm 0.7\pm 4.3$
3.1015 | $242275\pm 492$ | $26\pm 5$ | $0.6958\pm 0.0002$ | $0.9990\pm 0.0002$ | $215.7\pm 0.4\pm 2.8$
3.1055 | $313080\pm 560$ | $32\pm 6$ | $0.6929\pm 0.0002$ | $0.9990\pm 0.0002$ | $214.3\pm 0.4\pm 2.7$
3.1120 | $251731\pm 502$ | $21\pm 5$ | $0.6919\pm 0.0002$ | $0.9990\pm 0.0002$ | $211.3\pm 0.4\pm 2.8$
3.1200 | $185572\pm 431$ | $13\pm 4$ | $0.6913\pm 0.0002$ | $0.9990\pm 0.0002$ | $212.1\pm 0.5\pm 2.8$
Table 3: Measured cross sections and related input quantities of the process $e^{+}e^{-}\to\mu^{+}\mu^{-}$ at all individual CM energy points. The uncertainties of the input quantities are statistical, while the first and second uncertainties of cross sections are statistical and systematic, respectively. Here, the cross sections are defined in the polar angle range of 0∘–180∘. Prop. $\sqrt{s}$ (MeV) | $N_{\rm sig}$ | $N_{\rm bkgs}$ | $\epsilon_{\rm recsel}$ | $f$ |  $\sigma$ (nb)
---|---|---|---|---|---
3.0500 | $85997\pm 293$ | $313\pm 18$ | $0.5359\pm 0.0002$ | $1.0094\pm 0.0003$ | $10.82\pm 0.04\pm 0.12$
3.0600 | $85639\pm 293$ | $316\pm 18$ | $0.5360\pm 0.0002$ | $1.0094\pm 0.0003$ | $10.67\pm 0.04\pm 0.12$
3.0830 | $24413\pm 156$ | $93\pm 10$ | $0.5286\pm 0.0002$ | $1.0094\pm 0.0003$ | $9.74\pm 0.06\pm 0.12$
3.0900 | $72967\pm 270$ | $344\pm 19$ | $0.5202\pm 0.0002$ | $1.0094\pm 0.0003$ | $9.06\pm 0.03\pm 0.11$
3.0930 | $60436\pm 246$ | $336\pm 18$ | $0.5056\pm 0.0002$ | $1.0094\pm 0.0003$ | $8.05\pm 0.03\pm 0.10$
3.0943 | $17432\pm 132$ | $82\pm 9$ | $0.5781\pm 0.0002$ | $1.0093\pm 0.0003$ | $14.13\pm 0.11\pm 0.35$
3.0952 | $53866\pm 232$ | $176\pm 13$ | $0.6299\pm 0.0002$ | $1.0093\pm 0.0003$ | $47.36\pm 0.20\pm 0.66$
3.0958 | $158653\pm 398$ | $318\pm 18$ | $0.6366\pm 0.0002$ | $1.0093\pm 0.0003$ | $117.59\pm 0.29\pm 1.51$
3.0969 | $287418\pm 536$ | $728\pm 27$ | $0.6418\pm 0.0002$ | $1.0093\pm 0.0003$ | $217.88\pm 0.41\pm 2.81$
3.0982 | $158804\pm 399$ | $294\pm 17$ | $0.6369\pm 0.0002$ | $1.0093\pm 0.0003$ | $114.03\pm 0.29\pm 1.39$
3.0990 | $27505\pm 166$ | $58\pm 8$ | $0.6216\pm 0.0002$ | $1.0093\pm 0.0003$ | $58.92\pm 0.35\pm 0.85$
3.1015 | $23087\pm 152$ | $119\pm 11$ | $0.5981\pm 0.0002$ | $1.0093\pm 0.0003$ | $24.04\pm 0.16\pm 0.36$
3.1055 | $21595\pm 147$ | $139\pm 12$ | $0.5786\pm 0.0002$ | $1.0093\pm 0.0003$ | $17.77\pm 0.12\pm 0.26$
3.1120 | $14357\pm 120$ | $69\pm 8$ | $0.5674\pm 0.0002$ | $1.0093\pm 0.0003$ | $14.78\pm 0.12\pm 0.21$
3.1200 | $9352\pm 97$ | $53\pm 7$ | $0.5570\pm 0.0002$ | $1.0093\pm 0.0003$ | $13.33\pm 0.14\pm 0.19$
Table 4: Relative systematic uncertainties of the measured cross sections of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ at $\sqrt{s}=3096.9$ MeV. Source | Uncertainty (%)
---|---
$e^{+}e^{-}\to e^{+}e^{-}$ | $e^{+}e^{-}\to\mu^{+}\mu^{-}$
Integrated luminosity measurement | 1.26 | 1.26
Trigger efficiency determination | 0.10 | 0.10
CM energy measurement | 0.17 | 0.11
$\cos\theta$ requirement | 0.55 | 0.01
$P$ requirement | 0.13 | 0.02
$E/P$ requirement | 0.05 | 0.01
$\Delta t_{\rm TOF}^{\mu}$ requirement | — | 0.05
Efficiency correction factor $f$ | 0.10 | 0.10
Residual backgrounds | 0.03 | 0.25
Total | 1.40 | 1.30

To estimate the uncertainties due to the CM energies, two additional sets of
MC samples are generated by increasing or decreasing the CM energies by one
standard deviation with respect to their nominal values. The largest changes
of the efficiencies with respect to their nominal values are taken as the
uncertainties.

The uncertainties associated with the momentum requirement for the process
$e^{+}e^{-}\to e^{+}e^{-}$ are estimated by changing the selection criteria
from $P>0.7E_{\rm beam}$ to $P>0.6E_{\rm beam}$. The resultant changes in the
calculated cross sections are taken as the uncertainties. The uncertainties
related to other requirements are estimated with the similar method.
Specifically, for the selection of $e^{+}e^{-}\to e^{+}e^{-}$ events, the
analysis is carried out with the alternative criteria of $|\cos\theta|<0.7$
and $E/P<0.7$, individually, while for $e^{+}e^{-}\to\mu^{+}\mu^{-}$, the
analysis is repeated with the alternative criteria of $P>0.9E_{\rm beam}$,
$|\cos\theta|<0.7$, $E/P<0.35$ and $\Delta t_{\rm TOF}^{\mu}<2.5$ ns,
individually.

As shown in Tables 2 and 3, the statistical uncertainties of the efficiency
correction factors are 0.02% and 0.03% for the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$, respectively, which are
determined from the statistics of the samples used to study the reconstruction
efficiencies. On the other hand, detailed studies show that the purities of
the control samples for the electron tracking, electron clustering, muon
tracking, and muon clustering efficiencies in data are about 99.99%, 99.81%,
98.45%, and 99.52%, respectively. Considering other factors, such as the
background contaminations, the uncertainties resulting from the efficiency
correction factors can be roughly and conservatively estimated to be 0.10% for
both the $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$
processes.

The numbers of residual background events, estimated with the MC simulation,
are subtracted from the numbers of surviving events in the calculation of the
cross sections, and hence the uncertainties of background levels need be taken
into account. Since the uncertainties of the cross sections for some dominant
background channels (for example, $e^{+}e^{-}\to K^{+}K^{-}$) set in the
generator are as large as 100%, we therefore take the background levels
themselves as the related uncertainties. As a result, the uncertainties for
the processes $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ at
$\sqrt{s}=3096.9$ MeV are 0.03% and 0.25%, respectively.

Table 4 shows a summary of the systematic uncertainties of the measured cross
sections of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ at $\sqrt{s}=3096.9$ MeV. The total systematic
uncertainties, 1.40% and 1.29% for the two processes individually, are the
square root of the quadratic sum of the individual uncertainties and dominated
by those associated with the integrated luminosities. The systematic
uncertainties of the measured cross sections at other CM energy points are
estimated with the same method, and they are summarized in Tables 2 and 3
together with the statistical uncertainties.

###  4.3 Correlation analysis

To consider the correlations between the measured cross sections of the same
process at different CM energy points, the corresponding covariance matrices
are estimated. To estimate such a covariance matrix, contributions from all
related uncertainty sources are analysed and estimated according to their
nature and the method of uncertainty propagation. To get an impression of the
strength of these correlations, the correlation coefficient matrices of the
measured cross sections of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ are shown in Fig. 4.3. We find that the
correlations are strong and can not be neglected.

In the covariance matrix analysis above, the corresponding covariance matrix
of the measured luminosities at different CM energy points is estimated in
advance with the similar method. This matrix is required when estimating the
covariance matrices of the measured cross sections and constructing the global
$\chi^{2}$ function for the simultaneous fit of the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ (see Section 5.2).

Fig. 2: Correlation coefficient matrices of the measured cross sections of the
processes $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$.

##  5 Decay width determination

###  5.1 Energy spread and final state radiation

To determine the $J/\psi$ decay widths, a simultaneous fit to the measured, CM
energy dependent cross sections of the processes $e^{+}e^{-}\to e^{+}e^{-}$
and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ is required. In the theoretical formulae
used in the fit, the effects of the beam energy spread and FSR are taken into
account as well.

By assuming the CM energy spread follows a Gaussian distribution, the
theoretical cross section is

$\sigma(W)=\int\sigma_{\rm 0}(W_{\rm 0})\left(\frac{1}{\sqrt{2\pi}S_{\rm
W}}\text{e}^{-\frac{(W_{\rm 0}-W)^{2}}{2S_{\rm W}^{2}}}\right)dW_{\rm 0}.$ (6)

Here, $W$ ($=\sqrt{s}$) and $S_{\rm W}$ are the mean and standard deviation of
the CM energy distribution, respectively. According to the formula and the
expression of $\sigma_{\rm 0}$ in Eq. (3), $\sigma$ can also be divided into
three terms: the continuum term ($\sigma^{\rm C}$), the resonance term
($\sigma^{\rm R}$) and interference term ($\sigma^{\rm I}$). In practice,
$\sigma^{\rm C}$ is evaluated with the Babayaga-3.5 generator [34] with the
effects of $J/\psi$ and FSR switched off, while $\sigma^{\rm R}+\sigma^{\rm
I}$ is calculated using the analytic formulae for $\sigma_{\rm 0}^{\rm
R}+\sigma_{\rm 0}^{\rm I}$ in Ref. [27].

In Eq. (3), only the ISR effect is involved in $\sigma_{\rm 0}$. To take into
account the FSR effect, we introduce a correction factor $R^{\rm FSR}$ into
the theoretical cross section:

$\sigma_{\rm 0}^{\rm theor}(W)=\sigma(W)\cdot R^{\rm FSR}(W).$ (7)

In practice, $R^{\rm FSR}$ is obtained with the Babayaga-3.5 generator [34] as
the ratio of the calculated cross sections with and without the FSR effect.
For example, at $\sqrt{s}=3096.9$ MeV, $R^{\rm FSR}$ is 0.980 and 0.998 for
the processes $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$,
respectively.

Due to the high-order corrections related to the FSR effect, the cross
sections of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ are calculated by the Babayaga-3.5 generator
with the uncertainties of 0.5% and 1.0%, respectively [34]. Thus, systematic
deviations of $R^{\rm FSR}(W)$ from their truth values probably appear in the
vicinity of the $J/\psi$ resonance. To take the possible deviations into
consideration, we implement one free scaling parameter in the theoretical
cross section formula of each process for the simultaneous fit,

$\sigma^{\rm theor}(W)=\sigma_{\rm 0}^{\rm theor}(W)\cdot F.$ (8)

Specifically, the free scaling parameters for the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ are referred to as $F_{ee}$ and
$F_{\mu\mu}$, respectively.

###  5.2 Global $\chi^{2}$ function

To take into account the correlations between the measured cross sections of
different processes and/or at different CM energy points, as well as the
uncertainties of the CM energies, we construct a global $\chi^{2}$ function
for the simultaneous fit according to the standard covariance matrix method as

$\chi^{2}=\Delta\sigma^{\rm T}\cdot V^{-1}\cdot\Delta\sigma,$ (9)

where

$\Delta\sigma(i)=\left\\{\begin{array}[]{ll}\displaystyle\sigma_{ee}^{\rm
exper}(i)-\sigma_{ee}^{\rm theor}(i)&\displaystyle i=1,2\cdots 14,15\\\
\displaystyle\sigma_{\rm\mu\mu}^{\rm exper}(i-15)-\sigma_{\rm\mu\mu}^{\rm
theor}(i-15)&\displaystyle i=16,17\cdots 29,30\\\ \end{array}\right.$ (10)

and

$V(i,j)=\left\\{\begin{array}[]{ll}\displaystyle
V_{ee}(i,j)+\delta(i,j)\,\left(\frac{\partial\sigma_{ee}^{\rm theor}}{\partial
W}(i)\,\Delta W(i)\right)^{2}&\displaystyle\footnotesize{1}⃝\\\
\displaystyle\frac{\sigma_{ee}^{\rm exper}(i)\,\sigma_{\rm\mu\mu}^{\rm
exper}(j-15)}{L(i)\,L(j-15)}\,V_{\rm
L}(i,j-15)+\delta(i,j-15)\,\frac{\partial\sigma_{ee}^{\rm theor}}{\partial
W}(i)\,\frac{\partial\sigma_{\rm\mu\mu}^{\rm theor}}{\partial W}(i)\,(\Delta
W(i))^{2}&\displaystyle\footnotesize{2}⃝\\\
\displaystyle\frac{\sigma_{ee}^{\rm exper}(j)\,\sigma_{\rm\mu\mu}^{\rm
exper}(i-15)}{L(i-15)\,L(j)}V_{\rm
L}(i-15,j)+\delta(i-15,j)\,\frac{\partial\sigma_{ee}^{\rm theor}}{\partial
W}(j)\,\frac{\partial\sigma_{\rm\mu\mu}^{\rm theor}}{\partial W}(j)\,(\Delta
W(j))^{2}&\displaystyle\footnotesize{3}⃝\\\ \displaystyle
V_{\rm\mu\mu}(i-15,j-15)+\delta(i-15,j-15)\,\left(\frac{\partial\sigma_{\rm\mu\mu}^{\rm
theor}}{\partial W}(i-15)\,\Delta
W(i-15)\right)^{2}&\displaystyle\footnotesize{4}⃝\\\ \end{array}\right.$ (11)

with

$\left\\{\begin{array}[]{l}\displaystyle\footnotesize{1}⃝\text{ denotes
}i=1,2\cdots 14,15,\ j=1,2\cdots 14,15\\\ \displaystyle\footnotesize{2}⃝\text{
denotes }i=1,2\cdots 14,15,\ j=16,17\cdots 29,30\\\
\displaystyle\footnotesize{3}⃝\text{ denotes }i=16,17\cdots 29,30,\
j=1,2\cdots 14,15\\\ \displaystyle\footnotesize{4}⃝\text{ denotes
}i=16,17\cdots 29,30,\ j=16,17\cdots 29,30.\\\ \end{array}\right.$ (12)

Here, $\sigma_{ee}^{\rm exper}$ and $\sigma_{ee}^{\rm theor}$
($\sigma_{\rm\mu\mu}^{\rm exper}$ and $\sigma_{\rm\mu\mu}^{\rm theor}$) are
the experimental measured and theoretical predicted cross sections of the
process $e^{+}e^{-}\to e^{+}e^{-}$ ($\mu^{+}\mu^{-}$), $V_{ee}$
($V_{\rm\mu\mu}$) is the covariance matrix of the measured cross sections of
$e^{+}e^{-}\to e^{+}e^{-}$ ($\mu^{+}\mu^{-}$), $V_{\rm L}$ is the covariance
matrix of the measured luminosities, $i$ and $j$ are the horizontal and
vertical indices of the $30\times 30$ covariance matrix $V$, $\delta$ is
Kronecker delta function, and $\Delta W$ is the statistcal uncertainty of the
CM energy as listed in Table 1, whose systematic uncertainty, 0.033 MeV, will
be taken into account by examining the changes of the fit result due to the
changes of the CM energies by 0.033 MeV.

###  5.3 Simultaneous fit and parameter transformation

By minimizing the global $\chi^{2}$ function, the simultaneous fit to the
measured, CM energy dependent cross sections of the processes $e^{+}e^{-}\to
e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ is carried out. In the fit, the
following six parameters $M$, $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$,
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$, $S_{\rm W}$, $F_{ee}$ and
$F_{\mu\mu}$ are float, while $\sigma^{\rm C}(W)$ and $R^{\rm FSR}(W)$ are
expressed as piecewise linear interpolation functions based on hundreds of
pairs of ($W$,$\sigma^{\rm C}$) and ($W$,$R^{\rm FSR}$) values obtained with
the Babayaga-3.5 generator. The resultant fit curves are shown in Fig. 5.3,
and the corresponding fit quality is $\chi^{2}_{\rm min}/ndf\approx
23.0/24\approx 1.0$, where $\chi^{2}_{\rm min}$ and $ndf$ are the minimized
global chisquare and the number of degrees of freedom, respectively.

Fig. 3: Simultaneous fit to the measured, CM energy dependent cross sections
of the processes $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$.
For clear display of the interference effect [denoted by $\sigma_{\rm 0}^{\rm
I}$ in Eq. (3) and mainly illustrated by the small dip in front of the peak],
the plot for the process $e^{+}e^{-}\to\mu^{+}\mu^{-}$ is drawn with a
logarithmic vertical axis, while the plot for $e^{+}e^{-}\to e^{+}e^{-}$ is
drawn with a linear vertical axis, because in this process the interference
effect is less noticeable due to the existence of the scattering channel. In
the plots, the red points with error bars and blue curves are from data and
fitting, respectively.

Table 5: Comparison of the $J/\psi$ decay widths obtained in this work with those from other experiments, the PDG and the BESIII work using the ISR return technique. Collab. | Method | Year |  $\Gamma_{\rm tot}$ (keV) |  $\Gamma_{ll}$ (keV) | Ref.
---|---|---|---|---|---
BaBar | ISR return | 2004 | 94.7$\pm$4.4 | 5.61$\pm$0.21 | [18]
CLEO | ISR return | 2006 | 96.1$\pm$3.2 | 5.71$\pm$0.16 | [19]
KEDR | ES | 2010 | 94.1$\pm$2.7 | 5.59$\pm$0.12 | [20]
BESIII | ISR return | 2016 | — | 5.58$\pm$0.09 | [24]
KEDR | ES | 2018 | 92.5$\pm$2.0 | 5.55$\pm$0.11 | [21]
PDG | — | 2020 | 92.9$\pm$2.8 | 5.53$\pm$0.10 | [1]
This work | ES | 2021 | 93.0$\pm$2.1 | 5.56$\pm$0.11 | —

The fit result of $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ are ($0.346\pm 0.009$) and
($0.335\pm 0.006$) keV with the covariance and correlation coefficient between
them as 0.000046 keV2 and 0.83, respectively. Taking into account the
correlation term, we evaluate $\Gamma_{ee}/\Gamma_{\mu\mu}$ to be $1.031\pm
0.015$, which is consistent with the expectation of lepton universality within
about 2$\sigma$. Besides, $S_{\rm W}$ is fitted to be ($0.916\pm 0.018$) MeV,
which is consistent with the designed energy spread of the BEPCII collider,
and $F_{ee}$ and $F_{\mu\mu}$ are fitted to be $0.995\pm 0.009$ and $1.015\pm
0.011$, respectively, which are compatible with the precision levels of the
Babayaga-3.5 generator within uncertainties.

Assuming lepton universality, $\Gamma_{ee}=\Gamma_{\mu\mu}$, and referring to
$\Gamma_{ee}$ and $\Gamma_{\mu\mu}$ as $\Gamma_{ll}$,
$\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ are combined and referred to as
$\Gamma_{ll}\Gamma_{ll}/\Gamma_{\rm tot}$. The resultant
$\Gamma_{ll}\Gamma_{ll}/\Gamma_{\rm tot}$ is ($0.332\pm 0.006$) keV, which is
smaller than the individual values of $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm
tot}$ and $\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ because of the
correlation between the latter two.

Combining the resultant $\Gamma_{ll}\Gamma_{ll}/\Gamma_{\rm tot}$ with the
branching ratio of the $J/\psi$ leptonic decay measured by BESIII in 2013,
$B(J/\psi\to l^{+}l^{-})=\Gamma_{ll}/\Gamma_{\rm tot}=(5.978\pm 0.040)$ %
[30], $\Gamma_{\rm tot}$ and $\Gamma_{ll}$ are determined to be ($93.0\pm
2.1$) and ($5.56\pm 0.11$) keV, respectively.

As mentioned previously, the impact of the systematic uncertainty (0.033 MeV)
of the CM energies requires additional consideration. By increasing and
decreasing the CM energies by 0.033 MeV, we repeat the entire simultaneous fit
process twice, the relative changes of the results are less than 0.1%, and are
neglected.

###  5.4 Result and comparison

The results obtained in this work are summarized as follows:

$\displaystyle\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}=(0.346\pm 0.009)\text{
keV,}$ $\displaystyle\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}=(0.335\pm
0.006)\text{ keV,}$ $\displaystyle\Gamma_{ll}\Gamma_{ll}/\Gamma_{\rm
tot}=(0.332\pm 0.006)\text{ keV,}$
$\displaystyle\Gamma_{ee}/\Gamma_{\mu\mu}=1.031\pm 0.015\text{,}$
$\displaystyle\Gamma_{\rm tot}=(93.0\pm 2.1)\text{ keV,}$
$\displaystyle\Gamma_{ll}=(5.56\pm 0.11)\text{ keV.}$

The uncertainties quoted here are total uncertainties, which are obtained with
all the statistical and systematic uncertainties of the input quantities taken
into consideration.

The result of $\Gamma_{ee}/\Gamma_{\mu\mu}$ is consistent with and more
precise than the result ($1.002\pm 0.025$) given by KEDR with the same method
[20]. It is also in agreement with but less precise than the previous BESIII
result ($1.0017\pm 0.0037$) obtained with a different approach [30]. Table 5
shows a comparison of the $\Gamma_{\rm tot}$ and $\Gamma_{ll}$ obtained in
this work with those from other works and the PDG. The results given by this
work agree with all other results; they come up to a new precision level,
together with previous results obtained with the ISR return technique at
BESIII and ES method at KEDR.

##  6 Summary

Based on the data samples collected with the BESIII detector at fifteen CM
energy points in the vicinity of the $J/\psi$ resonance, the cross sections of
the processes $e^{+}e^{-}\to e^{+}e^{-}$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ are
measured and summarized in Tables 2 and 3, respectively. By performing a
simultaneous fit of the cross sections of the two processes as functions of
the center-of-mass energy, $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$ of the $J/\psi$ resonance are
determined to be ($0.346\pm 0.009$) and ($0.335\pm 0.006$) keV, respectively.

Using the obtained $\Gamma_{ee}\Gamma_{ee}/\Gamma_{\rm tot}$ and
$\Gamma_{ee}\Gamma_{\mu\mu}/\Gamma_{\rm tot}$, the ratio of the $J/\psi$
leptonic decay widths, $\Gamma_{ee}/\Gamma_{\mu\mu}$, is evaluated to be
$1.031\pm 0.015$, which agrees with the expectation of lepton universality
within about two standard deviations. Under the assumption of lepton
universality and combining with $B(J/\psi\to l^{+}l^{-})$ measured by BESIII
[30], the total and leptonic decay widths of the $J/\psi$ resonance,
$\Gamma_{\rm tot}$ and $\Gamma_{ll}$, are determined to be ($93.0\pm 2.1$) and
($5.56\pm 0.11$) keV, respectively. These results are consistent with the
previous results and reach the world leading level.

##  7 Acknowledgments

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11275211, 11475090, 11335008, 11635010, 11735014, 11835012, 11935015,
11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013,
12122509, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program;
Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract
No. U1832207; 100 Talents Program of CAS; The Institute of Nuclear and
Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and
Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020
research and innovation programme under Marie Sklodowska-Curie grant agreement
under Contract No. 894790; German Research Foundation DFG under Contracts Nos.
443159800, Collaborative Research Center CRC 1044, FOR 2359, GRK 2149;
Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of
Turkey under Contract No. DPT2006K-120470; National Science and Technology
fund; National Science Research and Innovation Fund (NSRF) via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation under Contract No. B16F640076; STFC (United Kingdom); Suranaree
University of Technology (SUT), Thailand Science Research and Innovation
(TSRI), and National Science Research and Innovation Fund (NSRF) under
Contract No. 160355; The Royal Society, UK under Contracts Nos. DH140054,
DH160214; The Swedish Research Council; U. S. Department of Energy under
Contract No. DE-FG02-05ER41374

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