# Search for $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ at center-of-mass energies
from 4.66 to 4.95 GeV

M. Ablikim1, M. N. Achasov4,c, P. Adlarson76, O. Afedulidis3, X. C. Ai81, R.
Aliberti35, A. Amoroso75A,75C, Q. An72,58,a, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,h, H.-R. Bao64, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi75A,75C, E.
Bianco75A,75C, A. Bortone75A,75C, I. Boyko36, R. A. Briere5, A. Brueggemann69,
H. Cai77, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,64, N. Cao1,64, S. A.
Cetin62A, J. F. Chang1,58, G. R. Che43, G. Chelkov36,b, C. Chen43, C. H.
Chen9, Chao Chen55, G. Chen1, H. S. Chen1,64, H. Y. Chen20, M. L. Chen1,58,64,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,64, X. R. Chen31,64, X. T.
Chen1,64, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,i, Z. Y. Chen1,64, S. K.
Choi10A, G. Cibinetto29A, F. Cossio75C, J. J. Cui50, H. L. Dai1,58, J. P.
Dai79, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng73, Z. Y. Deng1,
A. Denig35, I. Denysenko36, M. Destefanis75A,75C, F. De Mori75A,75C, B.
Ding67,1, X. X. Ding46,h, Y. Ding40, Y. Ding34, J. Dong1,58, L. Y. Dong1,64,
M. Y. Dong1,58,64, X. Dong77, M. C. Du1, S. X. Du81, Y. Y. Duan55, Z. H.
Duan42, P. Egorov36,b, Y. H. Fan45, J. Fang1,58, J. Fang59, S. S. Fang1,64, W.
X. Fang1, Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava75B,75C, F.
Feldbauer3, G. Felici28A, C. Q. Feng72,58, J. H. Feng59, Y. T. Feng72,58, M.
Fritsch3, C. D. Fu1, J. L. Fu64, Y. W. Fu1,64, H. Gao64, X. B. Gao41, Y. N.
Gao46,h, Yang Gao72,58, S. Garbolino75C, I. Garzia29A,29B, L. Ge81, P. T.
Ge19, Z. W. Ge42, C. Geng59, E. M. Gersabeck68, A. Gilman70, K. Goetzen13, L.
Gong40, W. X. Gong1,58, W. Gradl35, S. Gramigna29A,29B, M. Greco75A,75C, M. H.
Gu1,58, Y. T. Gu15, C. Y. Guan1,64, A. Q. Guo31,64, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,g, A. Guskov36,b, J. Gutierrez27, K. L. Han64, T. T.
Han1, F. Hanisch3, X. Q. Hao19, F. A. Harris66, K. K. He55, K. L. He1,64, F.
H. Heinsius3, C. H. Heinz35, Y. K. Heng1,58,64, C. Herold60, T. Holtmann3, P.
C. Hong34, G. Y. Hou1,64, X. T. Hou1,64, Y. R. Hou64, Z. L. Hou1, B. Y. Hu59,
H. M. Hu1,64, J. F. Hu56,j, S. L. Hu12,g, T. Hu1,58,64, Y. Hu1, G. S.
Huang72,58, K. X. Huang59, L. Q. Huang31,64, X. T. Huang50, Y. P. Huang1, Y.
S. Huang59, T. Hussain74, F. Hölzken3, N. Hüsken35, N. in der Wiesche69, J.
Jackson27, S. Janchiv32, J. H. Jeong10A, Q. Ji1, Q. P. Ji19, W. Ji1,64, X. B.
Ji1,64, X. L. Ji1,58, Y. Y. Ji50, X. Q. Jia50, Z. K. Jia72,58, D. Jiang1,64,
H. B. Jiang77, P. C. Jiang46,h, S. S. Jiang39, T. J. Jiang16, X. S.
Jiang1,58,64, Y. Jiang64, J. B. Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y.
Jin67, M. Q. Jing1,64, X. M. Jing64, T. Johansson76, S. Kabana33, N. Kalantar-
Nayestanaki65, X. L. Kang9, X. S. Kang40, M. Kavatsyuk65, B. C. Ke81, V.
Khachatryan27, A. Khoukaz69, R. Kiuchi1, O. B. Kolcu62A, B. Kopf3, M.
Kuessner3, X. Kui1,64, N. Kumar26, A. Kupsc44,76, W. Kühn37, J. J. Lane68, L.
Lavezzi75A,75C, T. T. Lei72,58, Z. H. Lei72,58, M. Lellmann35, T. Lenz35, C.
Li47, C. Li43, C. H. Li39, Cheng Li72,58, D. M. Li81, F. Li1,58, G. Li1, H. B.
Li1,64, H. J. Li19, H. N. Li56,j, Hui Li43, J. R. Li61, J. S. Li59, K. Li1, L.
J. Li1,64, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k,l, Q. M. Li1,64, Q.
X. Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,64, W. G. Li1,a, X.
Li1,64, X. H. Li72,58, X. L. Li50, X. Y. Li1,64, X. Z. Li59, Y. G. Li46,h, Z.
J. Li59, Z. Y. Li79, C. Liang42, H. Liang72,58, H. Liang1,64, Y. F. Liang54,
Y. T. Liang31,64, G. R. Liao14, Y. P. Liao1,64, J. Libby26, A. Limphirat60, C.
C. Lin55, D. X. Lin31,64, T. Lin1, B. J. Liu1, B. X. Liu77, C. Liu34, C. X.
Liu1, F. Liu1, F. H. Liu53, Feng Liu6, G. M. Liu56,j, H. Liu38,k,l, H. B.
Liu15, H. H. Liu1, H. M. Liu1,64, Huihui Liu21, J. B. Liu72,58, J. Y. Liu1,64,
K. Liu38,k,l, K. Y. Liu40, Ke Liu22, L. Liu72,58, L. C. Liu43, Lu Liu43, M. H.
Liu12,g, P. L. Liu1, Q. Liu64, S. B. Liu72,58, T. Liu12,g, W. K. Liu43, W. M.
Liu72,58, X. Liu39, X. Liu38,k,l, Y. Liu81, Y. Liu38,k,l, Y. B. Liu43, Z. A.
Liu1,58,64, Z. D. Liu9, Z. Q. Liu50, X. C. Lou1,58,64, F. X. Lu59, H. J. Lu23,
J. G. Lu1,58, X. L. Lu1, Y. Lu7, Y. P. Lu1,58, Z. H. Lu1,64, C. L. Luo41, J.
R. Luo59, M. X. Luo80, T. Luo12,g, X. L. Luo1,58, X. R. Lyu64, Y. F. Lyu43, F.
C. Ma40, H. Ma79, H. L. Ma1, J. L. Ma1,64, L. L. Ma50, L. R. Ma67, M. M.
Ma1,64, Q. M. Ma1, R. Q. Ma1,64, T. Ma72,58, X. T. Ma1,64, X. Y. Ma1,58, Y.
Ma46,h, Y. M. Ma31, F. E. Maas18, M. Maggiora75A,75C, S. Malde70, Y. J.
Mao46,h, Z. P. Mao1, S. Marcello75A,75C, Z. X. Meng67, J. G.
Messchendorp13,65, G. Mezzadri29A, H. Miao1,64, T. J. Min42, R. E. Mitchell27,
X. H. Mo1,58,64, B. Moses27, N. Yu. Muchnoi4,c, J. Muskalla35, Y. Nefedov36,
F. Nerling18,e, L. S. Nie20, I. B. Nikolaev4,c, Z. Ning1,58, S. Nisar11,m, Q.
L. Niu38,k,l, W. D. Niu55, Y. Niu 50, S. L. Olsen64, Q. Ouyang1,58,64, S.
Pacetti28B,28C, X. Pan55, Y. Pan57, A. Pathak34, Y. P. Pei72,58, M.
Pelizaeus3, H. P. Peng72,58, Y. Y. Peng38,k,l, K. Peters13,e, J. L. Ping41, R.
G. Ping1,64, S. Plura35, V. Prasad33, F. Z. Qi1, H. Qi72,58, H. R. Qi61, M.
Qi42, T. Y. Qi12,g, S. Qian1,58, W. B. Qian64, C. F. Qiao64, X. K. Qiao81, J.
J. Qin73, L. Q. Qin14, L. Y. Qin72,58, X. P. Qin12,g, X. S. Qin50, Z. H.
Qin1,58, J. F. Qiu1, Z. H. Qu73, C. F. Redmer35, K. J. Ren39, A. Rivetti75C,
M. Rolo75C, G. Rong1,64, Ch. Rosner18, S. N. Ruan43, N. Salone44, A.
Sarantsev36,d, Y. Schelhaas35, K. Schoenning76, M. Scodeggio29A, K. Y.
Shan12,g, W. Shan24, X. Y. Shan72,58, Z. J. Shang38,k,l, J. F. Shangguan16, L.
G. Shao1,64, M. Shao72,58, C. P. Shen12,g, H. F. Shen1,8, W. H. Shen64, X. Y.
Shen1,64, B. A. Shi64, H. Shi72,58, H. C. Shi72,58, J. L. Shi12,g, J. Y. Shi1,
Q. Q. Shi55, S. Y. Shi73, X. Shi1,58, J. J. Song19, T. Z. Song59, W. M.
Song34,1, Y. J. Song12,g, Y. X. Song46,h,n, S. Sosio75A,75C, S.
Spataro75A,75C, F. Stieler35, Y. J. Su64, G. B. Sun77, G. X. Sun1, H. Sun64,
H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun77, S. S. Sun1,64, T. Sun51,f, W. Y.
Sun34, Y. Sun9, Y. J. Sun72,58, Y. Z. Sun1, Z. Q. Sun1,64, Z. T. Sun50, C. J.
Tang54, G. Y. Tang1, J. Tang59, M. Tang72,58, Y. A. Tang77, L. Y. Tao73, Q. T.
Tao25,i, M. Tat70, J. X. Teng72,58, V. Thoren76, W. H. Tian59, Y. Tian31,64,
Z. F. Tian77, I. Uman62B, Y. Wan55, S. J. Wang 50, B. Wang1, B. L. Wang64, Bo
Wang72,58, D. Y. Wang46,h, F. Wang73, H. J. Wang38,k,l, J. J. Wang77, J. P.
Wang 50, K. Wang1,58, L. L. Wang1, M. Wang50, N. Y. Wang64, S. Wang12,g, S.
Wang38,k,l, T. Wang12,g, T. J. Wang43, W. Wang73, W. Wang59, W. P.
Wang35,72,o, W. P. Wang72,58, X. Wang46,h, X. F. Wang38,k,l, X. J. Wang39, X.
L. Wang12,g, X. N. Wang1, Y. Wang61, Y. D. Wang45, Y. F. Wang1,58,64, Y. L.
Wang19, Y. N. Wang45, Y. Q. Wang1, Yaqian Wang17, Yi Wang61, Z. Wang1,58, Z.
L. Wang73, Z. Y. Wang1,64, Ziyi Wang64, D. H. Wei14, F. Weidner69, S. P. Wen1,
Y. R. Wen39, U. Wiedner3, G. Wilkinson70, M. Wolke76, L. Wollenberg3, C. Wu39,
J. F. Wu1,8, L. H. Wu1, L. J. Wu1,64, X. Wu12,g, X. H. Wu34, Y. Wu72,58, Y. H.
Wu55, Y. J. Wu31, Z. Wu1,58, L. Xia72,58, X. M. Xian39, B. H. Xiang1,64, T.
Xiang46,h, D. Xiao38,k,l, G. Y. Xiao42, S. Y. Xiao1, Y. L. Xiao12,g, Z. J.
Xiao41, C. Xie42, X. H. Xie46,h, Y. Xie50, Y. G. Xie1,58, Y. H. Xie6, Z. P.
Xie72,58, T. Y. Xing1,64, C. F. Xu1,64, C. J. Xu59, G. F. Xu1, H. Y. Xu67,2,p,
M. Xu72,58, Q. J. Xu16, Q. N. Xu30, W. Xu1, W. L. Xu67, X. P. Xu55, Y. C.
Xu78, Z. S. Xu64, F. Yan12,g, L. Yan12,g, W. B. Yan72,58, W. C. Yan81, X. Q.
Yan1,64, H. J. Yang51,f, H. L. Yang34, H. X. Yang1, T. Yang1, Y. Yang12,g, Y.
F. Yang1,64, Y. F. Yang43, Y. X. Yang1,64, Z. W. Yang38,k,l, Z. P. Yao50, M.
Ye1,58, M. H. Ye8, J. H. Yin1, Junhao Yin43, Z. Y. You59, B. X. Yu1,58,64, C.
X. Yu43, G. Yu1,64, J. S. Yu25,i, T. Yu73, X. D. Yu46,h, Y. C. Yu81, C. Z.
Yuan1,64, J. Yuan45, J. Yuan34, L. Yuan2, S. C. Yuan1,64, Y. Yuan1,64, Z. Y.
Yuan59, C. X. Yue39, A. A. Zafar74, F. R. Zeng50, S. H. Zeng63A,63B,63C,63D,
X. Zeng12,g, Y. Zeng25,i, Y. J. Zeng59, Y. J. Zeng1,64, X. Y. Zhai34, Y. C.
Zhai50, Y. H. Zhan59, A. Q. Zhang1,64, B. L. Zhang1,64, B. X. Zhang1, D. H.
Zhang43, G. Y. Zhang19, H. Zhang81, H. Zhang72,58, H. C. Zhang1,58,64, H. H.
Zhang59, H. H. Zhang34, H. Q. Zhang1,58,64, H. R. Zhang72,58, H. Y. Zhang1,58,
J. Zhang81, J. Zhang59, J. J. Zhang52, J. L. Zhang20, J. Q. Zhang41, J. S.
Zhang12,g, J. W. Zhang1,58,64, J. X. Zhang38,k,l, J. Y. Zhang1, J. Z.
Zhang1,64, Jianyu Zhang64, L. M. Zhang61, Lei Zhang42, P. Zhang1,64, Q. Y.
Zhang34, R. Y. Zhang38,k,l, S. H. Zhang1,64, Shulei Zhang25,i, X. D. Zhang45,
X. M. Zhang1, X. Y. Zhang50, Y. Zhang73, Y. Zhang1, Y. T. Zhang81, Y. H.
Zhang1,58, Y. M. Zhang39, Yan Zhang72,58, Z. D. Zhang1, Z. H. Zhang1, Z. L.
Zhang34, Z. Y. Zhang43, Z. Y. Zhang77, Z. Z. Zhang45, G. Zhao1, J. Y.
Zhao1,64, J. Z. Zhao1,58, L. Zhao1, Lei Zhao72,58, M. G. Zhao43, N. Zhao79, R.
P. Zhao64, S. J. Zhao81, Y. B. Zhao1,58, Y. X. Zhao31,64, Z. G. Zhao72,58, A.
Zhemchugov36,b, B. Zheng73, B. M. Zheng34, J. P. Zheng1,58, W. J. Zheng1,64,
Y. H. Zheng64, B. Zhong41, X. Zhong59, H. Zhou50, J. Y. Zhou34, L. P.
Zhou1,64, S. Zhou6, X. Zhou77, X. K. Zhou6, X. R. Zhou72,58, X. Y. Zhou39, Y.
Z. Zhou12,g, A. N. Zhu64, J. Zhu43, K. Zhu1, K. J. Zhu1,58,64, K. S. Zhu12,g,
L. Zhu34, L. X. Zhu64, S. H. Zhu71, T. J. Zhu12,g, W. D. Zhu41, Y. C.
Zhu72,58, Z. A. Zhu1,64, J. H. Zou1, J. Zu72,58 (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 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
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 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University, Kaifeng 475004, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 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
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Inner Mongolia University, Hohhot 010021, People’s Republic of China
31 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica
1000000, Chile
34 Jilin University, Changchun 130012, People’s Republic of China
35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
38 Lanzhou University, Lanzhou 730000, People’s Republic of China
39 Liaoning Normal University, Dalian 116029, People’s Republic of China
40 Liaoning University, Shenyang 110036, People’s Republic of China
41 Nanjing Normal University, Nanjing 210023, People’s Republic of China
42 Nanjing University, Nanjing 210093, People’s Republic of China
43 Nankai University, Tianjin 300071, People’s Republic of China
44 National Centre for Nuclear Research, Warsaw 02-093, Poland
45 North China Electric Power University, Beijing 102206, People’s Republic of
China
46 Peking University, Beijing 100871, People’s Republic of China
47 Qufu Normal University, Qufu 273165, People’s Republic of China
48 Renmin University of China, Beijing 100872, People’s Republic of China
49 Shandong Normal University, Jinan 250014, People’s Republic of China
50 Shandong University, Jinan 250100, People’s Republic of China
51 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
52 Shanxi Normal University, Linfen 041004, People’s Republic of China
53 Shanxi University, Taiyuan 030006, People’s Republic of China
54 Sichuan University, Chengdu 610064, People’s Republic of China
55 Soochow University, Suzhou 215006, People’s Republic of China
56 South China Normal University, Guangzhou 510006, People’s Republic of China
57 Southeast University, Nanjing 211100, People’s Republic of China
58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
59 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
60 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
61 Tsinghua University, Beijing 100084, People’s Republic of China
62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
63 University of Bristol, (A)H H Wills Physics Laboratory; (B)Tyndall Avenue;
(C)Bristol; (D)BS8 1TL
64 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
65 University of Groningen, NL-9747 AA Groningen, The Netherlands
66 University of Hawaii, Honolulu, Hawaii 96822, USA
67 University of Jinan, Jinan 250022, People’s Republic of China
68 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
69 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
70 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
71 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
72 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
73 University of South China, Hengyang 421001, People’s Republic of China
74 University of the Punjab, Lahore-54590, Pakistan
75 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
76 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
77 Wuhan University, Wuhan 430072, People’s Republic of China
78 Yantai University, Yantai 264005, People’s Republic of China
79 Yunnan University, Kunming 650500, People’s Republic of China
80 Zhejiang University, Hangzhou 310027, People’s Republic of China
81 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Deceased
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 Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
f 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
g 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
h Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
i Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
k Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
m Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan
n Also at Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland
o Also at Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
p Also at School of Physics, Beihang University, Beijing 100191 , China

###### Abstract

Using data samples with an integrated luminosity of $4.67$ fb-1collected by
the BESIII detector operating at the BEPCII collider, we search for the
process $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ at center-of-mass energies from
$4.66$ to $4.95~{}\mathrm{GeV}$. No significant signal is observed, and upper
limits for the Born cross sections
$\sigma^{B}(e^{+}e^{-}\to\eta^{\prime}\psi(2S))$ at the 90% confidence level
are determined.

##  I Introduction

The charmonium sector with bound states of charm and anti-charm quarks,
provides an ideal laboratory for exploring the interplay between perturbative
and nonperturbative effects in Quantum Chromodynamics (QCD) Brambilla:2014jmp
. Since the discovery of the $\chi_{c1}(3872)$ in 2003 Belle:2003nnu ,
numerous charmonium-like states have been observed. After their properties do
not align with those expected for conventional charmonium states, they are
promising candidates for exotic hadrons. The vector states (labelled with
$\psi(\mathrm{mass})$) BaBar:2005hhc ; BaBar:2006ait ; BESIII:2016adj  are
directly observable in $e^{+}e^{-}$ annihilations or initial state radiation
(ISR) processes.

The $\psi(4660)$ was initially reported by the Belle experiment and later by
BaBar in the ISR process $e^{+}e^{-}\to\gamma_{\rm ISR}\pi^{+}\pi^{-}\psi(2S)$
Belle:2007umv ; BaBar:2012hpr . BESIII measured the
$e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ cross sections at center-of-mass (c. m.)
energies ($\sqrt{s}$) between $4.0076$ and $4.6984~{}\mathrm{GeV}$ using 35
scan samples with a total integrated luminosity of $20.1~{}\mbox{fb${}^{-1}$}$
BESIII:2021njb . The results are consistent with those of the earlier BaBar
and Belle experiments, but with significantly improved precision. The BESIII
collaboration has measured a mass of $(4651.0\pm 37.8\pm 2.1)~{}\mathrm{MeV}$
and width of $(155.4\pm 24.8\pm 0.8)~{}\mathrm{MeV}$ for the $\psi(4660)$.
Using the ISR method, the Belle experiment identified a structure close to the
$\psi(4660)$ in the $e^{+}e^{-}\to\Lambda_{c}\bar{\Lambda}_{c}$ cross-section
shape Belle:2008xmh . Measured by the BESIII collaboration, the cross section
of the same process reveals a different line shape BESIII:2017kqg . Various
theoretical interpretations have been proposed to explain the nature of the
$\psi(4660)$, such as hybrid meson Chen:2016qju , molecular state Guo:2010tk ,
hadrocharmonium Dubynskiy:2008mq , and tetraquark state Cotugno:2009ys .
Besides the $\psi(4660)$, two structures around $4.75~{}\mathrm{GeV}$ were
reported in recent studies of $e^{+}e^{-}\to K\bar{K}J/\psi$ BESIII:2023wqy ;
BESIII:2022kcv  and $e^{+}e^{-}\to D_{s}^{*+}D_{s}^{*-}$ BESIII:2023wsc
processes at BESIII. Further exploration of the decay properties is essential
for a comprehensive understanding of the internal structure of these vector
structures.

In heavy $Q\bar{Q}$ systems, the hadronic transitions serve as a crucial probe
of their internal structures and help to establish the understanding of light
quark coupling with a heavy degree of freedom Anwar:2016mxo . The hadronic
transition between two vector quarkonia via an $\eta$ meson is suppressed by
$SU(3)$ flavor symmetry and heavy quark spin symmetry Voloshin:2011hw
compared to the transition producing two pions. A notable example for this is
the low transition ratio between $\Gamma(\psi(2S)\to J/\psi\eta)$ and
$\Gamma(\psi(2S)\to J/\psi\pi\pi)$. Inspired by the Nambu-Jona-Lasinio model,
the authors of Ref. Anwar:2016mxo  estimated the partial widths $\Gamma(Y\to
J/\psi\eta)$ of $\psi(4360)$, $\psi(4390)$, and $\psi(4660)$ by assuming them
as $c\bar{c}$ bound states. The $e^{+}e^{-}\to\eta J/\psi$ BESIII:2023tll ,
$e^{+}e^{-}\to\eta^{\prime}J/\psi$ BESIII:2019nmu , and
$e^{+}e^{-}\to\eta\psi(2S)$ BESIII:2024jzt  processes have been studied by the
BESIII experiment. Here, the $\psi(4040)$, $\psi(4230)$, and $\psi(4360)$ were
observed in the $\eta J/\psi$ process. An enhancement around
$4.2~{}\mathrm{GeV}$ is found in the cross section of
$e^{+}e^{-}\to\eta^{\prime}J/\psi$. Using data samples collected between
$\sqrt{s}=4.288~{}\mathrm{GeV}$ and $4.951~{}\mathrm{GeV}$, the
$e^{+}e^{-}\to\eta\psi(2S)$ process is observed with a statistical
significance above $5.0\sigma$. The $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$
process has not been investigated yet.

In this article, we analyze data samples collected at c. m. energies ranging
from $4.66$ to $4.95~{}\mathrm{GeV}$ to study the
$e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ process and investigate possible vector
charmonium or charmonium-like states in its cross section. The $\psi(2S)$ is
reconstructed via the pion transition process
$\psi(2S)\to\pi^{+}\pi^{-}J/\psi$, and $J/\psi\to e^{+}e^{-}$ or
$\mu^{+}\mu^{-}$. The $\eta^{\prime}$ is reconstructed by two decay modes
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$ (Mode I) and
$\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ with $\eta\to\gamma\gamma$ (Mode II) to
collect more statistics.

##  II Experiment setup and MC simulation

The BESIII detector Ablikim:2009aa  records symmetric $e^{+}e^{-}$ collisions
provided by the BEPCII storage ring Yu:IPAC2016-TUYA01  in the c. m. energy
range $2.0<\sqrt{s}<4.95~{}\mathrm{GeV}$, with a peak luminosity of $1\times
10^{33}\;\text{cm}^{-2}\text{s}^{-1}$ achieved at $\sqrt{s}=3.77\;\text{GeV}$.
BESIII has collected large data samples in this energy region Ablikim:2019hff
; EventFilter . The cylindrical core of the BESIII detector covers 93% of the
full solid angle and consists of a helium-based multilayer drift chamber
(MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl)
electromagnetic calorimeter (EMC), which are all enclosed in a superconducting
solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported
by an octagonal flux-return yoke with resistive plate counter muon
identification modules interleaved with steel. The charged-particle momentum
resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the ${\rm d}E/{\rm d}x$
resolution is $6\%$ for electrons from Bhabha scattering. The EMC measures
photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel
(end cap) region. The time resolution in the TOF barrel region is 68 ps, while
that in the end cap region was 110 ps. The end cap TOF system was upgraded in
2015 using multigap resistive plate chamber technology, providing a time
resolution of 60 ps.

Monte Carlo (MC) simulated events are used to determine the detection
efficiency, optimize selection criteria, and investigate possible background
contributions. These samples are produced with a geant4-based geant4  package,
which includes the geometric description of the BESIII detector and the
detector response. The simulation models the beam energy spread and initial
state radiation in the $e^{+}e^{-}$ annihilations with the generator kkmc
ref:kkmc . The inclusive MC sample includes the production of open charm
processes, the ISR production of vector charmonium(-like) states, and the
continuum processes incorporated in kkmc ref:kkmc . All particle decays are
modeled with evtgen ref:evtgen  using branching fractions either taken from
the Particle Data Group (PDG) ParticleDataGroup:2022pth , when available, or
otherwise estimated with lundcharm ref:lundcharm . Final state radiation (FSR)
from charged final state particles is incorporated using the photos package
photos . The signal MC samples of $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ are
generated for each c. m. energy assuming the Born cross sections following the
line shape of the $\psi(4660)$, which is parameterized with a Breit-Wigner
function using the mass and width taken from the PDG ParticleDataGroup:2022pth
.

##  III Signal selection and background modeling

The signal process of $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ comprises six
charged tracks and one photon for Mode I or two photons for Mode II. Charged
tracks detected in the MDC must be within a polar angle ($\theta$) range of
$|\rm{cos\theta}|<0.93$, where $\theta$ is the polar angle with respect to the
positive direction of the $z$-axis, the symmetry axis of the MDC. For charged
tracks, the distance of closest approach to the interaction point (IP) must be
less than 10 cm along the $z$-axis, and less than 1 cm in the transverse
plane. Photon candidates are identified by EMC showers. The deposited energy
of each shower must be more than 25 MeV in the barrel region where $|\rm
cos\theta|<0.80$ and more than 50 MeV in the end cap regions where $|\rm
cos\theta|$ is between $0.86$ and $0.92$. To exclude showers originating from
the charged tracks, the angle subtended by the EMC shower and the position of
the closest charged track at the EMC must be greater than 10 degrees as
measured from the IP. To suppress electronic noise and showers unrelated to
the event, the difference between the EMC time and the event start time is
required to be within [0, 700] ns.

To enhance the event selection efficiency, a candidate event should meet
either of the following criteria: it must consist of five charged tracks with
a net charge of $\pm 1$, accounting for the possibility of one missing charged
pion. Alternatively, it should involve six charged tracks with a net charge of
zero. In Mode I, there must be at least one good photon, and Mode II requires
at least two photons.

MC simulations indicate a distinct kinematic separation between pions and
leptons. The charged tracks with momentum above $1.0~{}\mathrm{GeV}/c$ are
categorized as leptons while those with momenta below $0.8~{}\mathrm{GeV}/c$
are classified as pions. Electrons and muons are distinguished using the ratio
of their energies ($E$) deposited in the EMC to momenta ($p$): the charged
tracks are labeled as electrons if $E/p>0.7$, and as muons if $E/p<0.7$ and
$E<0.45~{}\mathrm{GeV}$.

In order to further suppress background contributions and improve the mass
resolution, a four-constraint (4C) kinematic fit is performed for six-track
events by requiring the four-momenta of the final state particles to be
consistent with the initial one. For five-track events that account for a
missing pion, a kinematic fit with one degree of freedom (1C) is applied by
constraining the mass of the missing particle to the pion nominal mass. The
optimal photon(s) is(are) those that yield the minimum $\chi^{2}$ value. The
four-momenta of the final state particles used in the following analysis are
from the kinematic fit. The corresponding $\chi^{2}$ values are required to
satisfy $\chi^{2}_{4\rm C}<30$ ($\chi^{2}_{4\rm C}<60$) or $\chi^{2}_{1\rm
C}<10$ ($\chi^{2}_{1\rm C}<25$) for Mode I (Mode II), respectively. These
criteria are optimized by maximizing ${\epsilon}/(a/2+\sqrt{B})$, where
$\epsilon$ denotes the signal MC efficiency, $B$ is normalized number of
background events based on the inclusive MC sample, and $a$ is the number of
$\sigma$ corresponding to one-sided Gaussian tests at the desired significance
level Punzi:2003bu . In Mode I, to suppress the background contribution with
one less photon, we require the $\chi^{2}$ value of the 4C kinematic fit with
one less photon to be greater than that of the signal process. This
requirement results in an efficiency loss of $9.6\%$, while $69.7\%$ of
background events can be removed. This requirement is not applied in Mode II
due to the low background level.

The signal candidates are required to be within the $J/\psi$ mass region,
defined as $[3.083,3.111]~{}\mathrm{GeV}/c^{2}$ and
$[3.073,3.121]~{}\mathrm{GeV}/c^{2}$ for the invariant mass of lepton pairs in
Mode I and Mode II, respectively. The mass requirement of the $\eta$ candidate
is set to be $[0.482,0.604]~{}\mathrm{GeV}/c^{2}$ in Mode II. The mass
requirements of the $J/\psi$ and $\eta$ candidates are optimized by maximizing
$\epsilon/(a/2+\sqrt{B})$. This analysis involves four pion tracks in the
final states, yielding four potential combinations of $\psi(2S)$ candidates
for each event. A mass window of $[3.680,3.693]~{}\mathrm{GeV}/c^{2}$ around
the $\psi(2S)$ peak is applied to the combinations of invariant mass
$M(\pi^{+}\pi^{-}J/\psi)$. In order to improve the mass resolution, the
invariant mass is calculated using
$M(\pi^{+}\pi^{-}l^{+}l^{-})-M(l^{+}l^{-})+m(J/\psi)$, where $m(J/\psi)$ is
the nominal mass of $J/\psi$. For those candidates that survived, the one with
minimal value of
$|\frac{M(\gamma\pi^{+}\pi^{-}/\eta\pi^{+}\pi^{-})-m(\eta^{\prime})}{\sigma_{\eta^{\prime}}}|^{2}+|\frac{{|M(\pi^{+}\pi^{-}J/\psi)-m(\psi(2S))}}{\sigma_{\psi(2S)}}|^{2}$
is selected. Here $m(\eta^{\prime})$ and $m(\psi(2S))$ represent the nominal
masses of $\eta^{\prime}$ and $\psi(2S)$ quoted from the PDG
ParticleDataGroup:2022pth , while $\sigma_{\eta^{\prime}}$ and
$\sigma_{\psi(2S)}$ are the mass resolutions obtained from signal MC
simulation of the corresponding invariant mass distributions.

After applying the above selection criteria, the possible remaining background
contributions are studied based on the inclusive MC samples with an integrated
luminosity equivalent to 10 times of the real data. For Mode I, the dominant
remaining background is the $e^{+}e^{-}\to\pi\pi\psi(2S)$ processes for both
charged and neutral modes. To better estimate their contributions, 100,000
exclusive MC events are generated at each c. m. energy point with modeling of
the intermediate states and the cross section line shape taken from Ref.
BESIII:2021njb . For Mode II, the background contribution estimated by using
inclusive MC samples are $0.6$, $0.8$, $0.6$, $0$, $0.1$, $0.1$, $0$, $0$, and
$0$ for the nine data samples, which is negligible.

##  IV Measurement of the cross section

The Born cross section for studied signal process is determined by

$\small\sigma^{B}=\frac{N_{\rm sig}}{\mathcal{L}_{\rm
int}\cdot(1+\delta)\cdot\frac{1}{\left|1-\Pi\right|^{2}}\cdot\epsilon\cdot\mathcal{B}},$
(1)

where $N_{\rm sig}$ represents the observed signal yield, $\mathcal{L}_{\rm
int}$ is the integrated luminosity of the data, $1+\delta$ denotes the
radiative correction factor obtained from the kkmc generator,
$\frac{1}{\left|1-\Pi\right|^{2}}$ stands for the vacuum polarization factor
derived from QED calculations WorkingGrouponRadiativeCorrections:2010bjp ,
$\epsilon$ is the detection efficiency estimated through MC simulation, and
$\mathcal{B}$ represents the product of the branching fractions of
intermediate states in the decay cascade, quoted from the PDG
ParticleDataGroup:2022pth , which is 1.22% for Mode I and 0.692% for Mode II.

The signal yields are obtained by performing simultaneous unbinned maximum
likelihood fits to the corresponding $M(\gamma\pi^{+}\pi^{-})$ and
$M(\eta\pi^{+}\pi^{-})$ mass spectra, where the Born cross section value for
$e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ is shared between the two decay modes at
the same c. m. energy. The signal shapes for both modes are modeled by MC-
simulated shapes. In Mode I, the background shapes originating from
$e^{+}e^{-}\rightarrow\pi\pi\psi(2S)$ processes are taken from the line shapes
of the corresponding simulated MC samples. Other background contributions are
described by the inclusive MC line shape after excluding the aforementioned
processes. Regarding Mode II, only the signal component is considered since
the background contribution is negligible. This assumption results in a
conservative estimation of the upper limit of the Born cross section, as well
as the upper limit of $\Gamma_{ee}\cdot\mathcal{B}$ which will be introduced
later. In the fitting procedure, the number of
$e^{+}e^{-}\rightarrow\pi\pi\psi(2S)$ events is fixed to the values determined
by using the corresponding cross section and detection efficiency, and the
number of other background contributions is a free parameter. The fit results
for the data sample collected at $4.78~{}\mathrm{GeV}$ are shown in Fig. 1,
the results for other energy points can be found in Appendix A. Due to low
statistics, the upper limits on the cross section at the 90% confidence level
(C. L.) are provided using the profile likelihood method incorporating the
systematic uncertainties Stenson:2006gwf . The cross sections and signal
yields are listed in Table 1 for the different c. m. energies.

Table 1: The numerical results for $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$. $\mathcal{L}$ is the integrated luminosity, $(1+\delta)$ is the ISR correction factor, $\frac{1}{|1-\Pi|^{2}}$ is the VP correction factor, $\epsilon_{1}$ and $\epsilon_{2}$ are the selection efficiencies for Mode I and Mode II respectively, $N_{\rm sig}$ denotes the number of signal events determined from the simultaneous fit, $\sigma^{B}$ represents Born cross sections, $\alpha$ is the significance of the signal events, $N^{\rm ul}$ denotes the upper limits of signal events, and $\sigma^{\rm ul}$ is the upper limit of the Born cross section. $\sqrt{s}(\mathrm{MeV})$ | $\mathcal{L}({\rm pb}^{-1})$ | $(1+\delta)$ | $\frac{1}{|1-\Pi|^{2}}$ | $\epsilon_{1}(\%)$ | $\epsilon_{2}(\%)$ | $N_{\rm sig}$ |  $\sigma^{B}$(pb) | $\alpha$ | $N^{\rm ul}$ |  $\sigma^{\rm ul}$(pb)
---|---|---|---|---|---|---|---|---|---|---
4661.24 | 529.43 | 0.764 | 1.054 | 25.6 | 21.6 | $0.0^{+0.7}_{-0.0}$ | $0.0^{+0.5}_{-0.0}$ | 0.0 | 3.3 | 2.5
4681.92 | 1667.39 | 0.865 | 1.054 | 22.8 | 20.5 | $0.0^{+1.9}_{-0.0}$ | $0.0^{+0.4}_{-0.0}$ | 0.0 | 6.4 | 1.5
4698.82 | 535.54 | 0.935 | 1.055 | 21.4 | 19.3 | $2.9^{+1.8}_{-1.4}$ | $2.1^{+1.3}_{-1.0}$ | 1.3 | 7.1 | 5.1
4739.70 | 163.87 | 1.092 | 1.055 | 18.6 | 17.0 | $1.1^{+1.2}_{-0.7}$ | $2.6^{+2.8}_{-1.6}$ | 1.4 | 4.0 | 9.3
4750.05 | 366.55 | 1.133 | 1.055 | 18.0 | 16.3 | $0.0^{+0.5}_{-0.0}$ | $0.0^{+0.5}_{-0.0}$ | 0.0 | 2.5 | 2.6
4780.54 | 511.47 | 1.241 | 1.055 | 16.4 | 14.4 | $2.8^{+1.8}_{-1.3}$ | $2.1^{+1.3}_{-1.0}$ | 0.1 | 7.7 | 5.7
4843.07 | 525.16 | 1.438 | 1.056 | 14.0 | 11.8 | $1.6^{+1.4}_{-0.9}$ | $1.2^{+1.0}_{-0.7}$ | 1.7 | 5.0 | 3.7
4918.02 | 207.82 | 1.664 | 1.056 | 11.5 | 9.8 | $0.9^{+1.2}_{-0.7}$ | $1.8^{+2.3}_{-1.4}$ | 0.8 | 3.7 | 7.2
4950.93 | 159.28 | 1.771 | 1.056 | 10.8 | 8.7 | $0.0^{+0.7}_{-0.0}$ | $0.0^{+1.8}_{-0.0}$ | 0.0 | 4.3 | 10.9
Figure 1:  Distributions of $M(\gamma\pi^{+}\pi^{-})$ for Mode I (left) and
$M(\eta\pi^{+}\pi^{-})$ for Mode II (right) for data at a c. m. energy of
$4.78~{}\mathrm{GeV}$. The black dots with error bars are the data, the solid
blue lines denote the best fit results, the red dashed lines represent the
possible signal contribution, and the dashed curves in other colors (cyan,
green, and yellow) indicate the background contributions.

To study possible charmonium-like states in the
$e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ cross section, a maximum likelihood fit
is performed to the dressed cross section. The likelihood is defined as

$\small
L\left(\mu^{\mathrm{sig}};p\right)=\prod_{i=1}^{9}L_{i}\left(\mu_{i}^{\mathrm{sig}};p_{i}\right),$
(2)

where $\mu_{i}^{\text{sig }}$ is the expected number of signal events and
$p_{i}$ are the parameters of the fit models. $L_{i}$, as a function of
$\mu_{i}^{\text{sig }}$, is obtained by scanning the number of $\eta^{\prime}$
signals in the simultaneous fit at each c. m. energy point. The systematic
uncertainties of the Born cross section, which will be discussed in Section V,
have been considered in the construction of $L_{i}$. Assuming that the
$\eta^{\prime}\psi(2S)$ signal originates from one resonance, the dressed
cross section is parameterized as a relativistic Breit-Wigner function:

$\small\sigma(m)=\bigg{|}\frac{\sqrt{12\pi\mathcal{B}\Gamma_{ee}\Gamma}}{s-M^{2}+iM\Gamma}\sqrt{\frac{\Phi(m)}{\Phi\left(M\right)}}\bigg{|}^{2},$
(3)

where $\Phi(m)$ is the phase space factor, $M$ and $\Gamma$ represent the mass
and width of the resonance, respectively, $\Gamma_{ee}$ denotes the electronic
partial width, and $\mathcal{B}$ is the branching fraction to
$\eta^{\prime}\psi(2S)$. Two alternative resonance assumptions of the
$\psi(4660)$ and $\psi(4710)$ are used to describe the line shape of the
dressed cross section. The mass and width of the $\psi(4660)$ are fixed to the
PDG ParticleDataGroup:2022pth  values and the ones of $\psi(4710)$ are fixed
to the BESIII results BESIII:2023wqy . The fit results are shown in Fig. 2.
The upper limits of $\Gamma_{ee}\cdot\mathcal{B}$ are determined at the 90% C.
L. with the likelihood profile method, and are less than $0.22$ eV and $0.54$
eV for the $\psi(4660)$ and $\psi(4710)$, respectively.

Figure 2: Fit to the measured dressed cross section with the Breit-Wigner
function. The dots with error bars are the measured cross sections at nine c.
m. energy points. The green and red lines represent the fit results with the
$\psi(4660)$ and $\psi(4710)$ assumptions, respectively.

##  V Systematic uncertainties

The sources of the systematic uncertainties in the measurements of the Born
cross section are classified into multiplicative and additive terms.

The multiplicative terms include the luminosity, the branching fractions from
intermediate states, the radiative correction, the vacuum polarization, and
the detection efficiency. They are described in detail below.

  1. (1)

The uncertainty of the integrated luminosity is 0.6% as obtained by analyzing
the large-angle Bhabha scattering events BESIII:2022ulv .

  2. (2)

The input branching fractions of $\psi(2S)$ and $\eta^{\prime}$ are quoted
from the PDG ParticleDataGroup:2022pth .

  3. (3)

To evaluate the uncertainty of the ISR factor, two sources are considered.
First, the input line shape is changed to either a constant term or $q^{3}/s$,
instead of the Breit-Wigner function of the $\psi(4660)$. Here $q^{3}$ is the
P-wave phase space factor with $q$ denoting the momentum of $\eta^{\prime}$ in
the $e^{+}e^{-}$ c. m. frame. Second, considering the uncertainties of the
parameters of the $\psi(4660)$, we change the mass and width by one standard
deviation. The largest difference on the resulting cross section is assigned
as the uncertainty.

  4. (4)

The uncertainty of the vacuum polarization factor is taken as 0.5% from the
QED calculation Burkhardt:1989ky .

  5. (5)

For photon and pion detection efficiencies, the uncertainties are $1.0\%$ per
photon BESIII:2010ank  and $1.0\%$ per pion Yuan:2015wga , respectively. The
uncertainties of photons are $1.0\%$ for Mode I and $2.0\%$ for Mode II.
Events with three or four pions are kept and they are combined in each mode
according to the detection efficiencies. Then the uncertainties from the two
modes are combined considering the associated branching fractions.

  6. (6)

The uncertainty of the lepton tracking is estimated to be $1.0\%$ per track
BESIII:2013ris . The total uncertainty is $2.0\%$.

  7. (7)

The uncertainties of the 4C and 1C kinematic fit is estimated by correcting
the helix parameters of the charged tracks according to the method in Ref.
BESIII:2012mpj . The difference in the selection efficiency with or without
the correction is taken as the systematic uncertainty.

  8. (8)

To evaluate the uncertainty of the mass windows, the MC shape is smeared with
a Gaussian function to get better consistency with data. The parameters of the
Gaussian function are taken from Ref. BESIII:2021njb  for $J/\psi$ and
$\psi(2S)$, and Ref. BESIII:2022yoo  for $\eta$. The difference on the
detection efficiencies before and after smearing the Gaussian function is
taken as the uncertainty.

The additive terms originate from the fit to the invariant mass distributions
of the $\eta^{\prime}$ candidates, consisting of the signal shape, the
background shape, and the background size. To estimate the uncertainty from
the signal shape arising from the mass resolution difference between data and
MC simulation, the MC shape is smeared with a Gaussian function taken from
Ref. BESIII:2019nmu . The uncertainty of the background shape is estimated by
replacing the inclusive MC shape with a polynomial. The systematic uncertainty
from the fixed background contribution from
$e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ and
$e^{+}e^{-}\to\pi^{0}\pi^{0}\psi(2S)$ is estimated by varying the expected
number of background events by one standard deviation. The maximum number of
signal events among the above conditions is used to calculate the upper limit
of the cross section. Additionally, the multiplicative terms are convoluted
with the likelihood distributions in the form of a Gaussian function.

The systematic uncertainties in two modes are combined according to the
branching fractions and detection efficiencies. The combined systematic
uncertainties are summarized in Table 2. Assuming all sources are independent,
the total uncertainties are obtained by adding the individual values in
quadrature.

Table 2: Relative systematic uncertainties (in unit of %) at different c. m. energies. Energy point(MeV) | 4660 | 4680 | 4700 | 4740 | 4750 | 4780 | 4840 | 4914 | 4946
---|---|---|---|---|---|---|---|---|---
Luminosity | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | 0.6
Branching ratio $\psi(2S)$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
Branching ratio $\eta^{\prime}$ | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3
ISR factor 1 | 15.2 | 18.1 | 17.9 | 18.0 | 18.1 | 18.0 | 15.0 | 13.0 | 13.1
ISR factor 2 | 6.7 | 5.4 | 6.1 | 8.1 | 8.1 | 8.7 | 8.3 | 8.7 | 9.4
VP factor | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
Lepton efficiency | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0
Pions efficiency | 3.6 | 3.6 | 3.6 | 3.6 | 3.6 | 3.6 | 3.6 | 3.6 | 3.6
Photon efficiency | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4
Kinematic fit | 3.0 | 2.8 | 2.9 | 2.7 | 2.5 | 2.4 | 2.3 | 2.8 | 2.7
Mass window $\eta$ | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3
Mass window $J/\psi$ | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3
Mass window $\psi(2S)$ | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5
Total | 17.6 | 19.7 | 19.7 | 20.5 | 20.5 | 20.7 | 17.9 | 16.6 | 17.0

##  VI Summary

In summary, by analyzing data samples with an integrated luminosity of
$4.67~{}\mbox{fb${}^{-1}$}$ collected by the BESIII detector operating at the
BEPCII collider, the $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ process is
investigated at c. m. energies between $4.66$ and $4.946~{}\mathrm{GeV}$. No
significant signal is observed from the data, and the upper limits of the Born
cross section $\sigma^{B}(e^{+}e^{-}\to\eta^{\prime}\psi(2S))$ at the 90% C.
L. are determined at the nine c. m. energy points. The vector resonance
contribution is investigated by the analysis of the dressed cross section. No
statistically significant contribution from the vector states $\psi(4660)$ or
$\psi(4710)$ is found. The corresponding upper limit of the product of
electronic partial width and branching fraction to $\eta^{\prime}\psi(2S)$ at
the 90% C. L. are estimated to be smaller than 0.22 eV or 0.54 eV,
respectively. The cross sections of $e^{+}e^{-}\to\eta^{\prime}\psi(2S)$ are
found to be of the same order of magnitude as those of
$e^{+}e^{-}\to\eta\psi(2S)$ BESIII:2024jzt , while the cross section ratio of
$e^{+}e^{-}\to\eta^{\prime}J/\psi$ to $e^{+}e^{-}\to\eta J/\psi$ is about
$0.1$ in the $\psi(4230)$ mass region. Future investigations using data
samples with higher luminosity, such as the upgraded BEPCII BESIII:2020nme ;
Geng:2021qqj  and STCF Achasov:2023gey , are necessary to provide further
insight.

##  VII Acknowledgement

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. 2020YFA0406300, 2020YFA0406400,
2023YFA1606000; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 12375070, 11635010, 11735014, 11935015, 11935016, 11935018,
12025502, 12035009, 12035013, 12061131003, 12192260, 12192261, 12192262,
12192263, 12192264, 12192265, 12221005, 12225509, 12235017, 12361141819; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the
CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U2032108, No.
U1832207; 100 Talents Program of CAS; The Institute of Nuclear and Particle
Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and
Cosmology; German Research Foundation DFG under Contracts Nos. 455635585,
FOR5327, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of
Development of Turkey under Contract No. DPT2006K-120470; National Research
Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National Science
and Technology fund of Mongolia; National Science Research and Innovation Fund
(NSRF) via the Program Management Unit for Human Resources & Institutional
Development, Research and Innovation of Thailand under Contract No.
B16F640076; Polish National Science Centre under Contract No.
2019/35/O/ST2/02907; The Swedish Research Council; U. S. Department of Energy
under Contract No. DE-FG02-05ER41374

##  Appendix A The $\eta^{\prime}$ mass distribution at other c. m. energies

The fitting outcomes for each energy point and the combined data are presented
in Fig. 3. The combined data fitting, incorporating the background component
in Mode II and modeled as a 1st-order polynomial, yields a value of $N_{\rm
sig}=7.2^{+3.9}_{-3.4}$ with a statistical significance of $2.1$ and an upper
limit at $N^{\rm ul}=12.8$.

(a) $4.66~{}\mathrm{GeV}$ (b) $4.68~{}\mathrm{GeV}$ (c) $4.70~{}\mathrm{GeV}$
(d) $4.74~{}\mathrm{GeV}$

(e) $4.75~{}\mathrm{GeV}$ (f) $4.78~{}\mathrm{GeV}$ (g) $4.84~{}\mathrm{GeV}$
(h) $4.914~{}\mathrm{GeV}$

(i) $4.946~{}\mathrm{GeV}$ (j) Combined data

Figure 3:  Distributions of $M(\gamma\pi^{+}\pi^{-})$ for Mode I (left) and
$M(\eta\pi^{+}\pi^{-})$ for Mode II (right) for data at all c. m. energies.

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