# Search for $\eta_{c}(2S)$ $\rightarrow$ $K^{+}K^{-}\eta^{\prime}$ decay

M. Ablikim1, M. N. Achasov4,c, P. Adlarson75, O. Afedulidis3, X. C. Ai80, R.
Aliberti35, A. Amoroso74A,74C, Q. An71,58,a, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,h, H.-R. Bao63, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi74A,74C, E.
Bianco74A,74C, A. Bortone74A,74C, I. Boyko36, R. A. Briere5, A. Brueggemann68,
H. Cai76, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,63, N. Cao1,63, S. A.
Cetin62A, J. F. Chang1,58, G. R. Che43, G. Chelkov36,b, C. Chen43, C. H.
Chen9, Chao Chen55, G. Chen1, H. S. Chen1,63, H. Y. Chen20, M. L. Chen1,58,63,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,63, X. R. Chen31,63, X. T.
Chen1,63, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,i, Z. Y. Chen1,63, S. K.
Choi10A, G. Cibinetto29A, F. Cossio74C, J. J. Cui50, H. L. Dai1,58, J. P.
Dai78, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng72, Z. Y. Deng1,
A. Denig35, I. Denysenko36, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding46,h, Y. Ding40, Y. Ding34, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, M. C. Du1, S. X. Du80, Y. Y. Duan55, Z. H.
Duan42, P. Egorov36,b, Y. H. Fan45, J. Fang1,58, J. Fang59, S. S. Fang1,63, W.
X. Fang1, Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava74B,74C, F.
Feldbauer3, G. Felici28A, C. Q. Feng71,58, J. H. Feng59, Y. T. Feng71,58, M.
Fritsch3, C. D. Fu1, J. L. Fu63, Y. W. Fu1,63, H. Gao63, X. B. Gao41, Y. N.
Gao46,h, Yang Gao71,58, S. Garbolino74C, I. Garzia29A,29B, L. Ge80, P. T.
Ge76, Z. W. Ge42, C. Geng59, E. M. Gersabeck67, A. Gilman69, K. Goetzen13, L.
Gong40, W. X. Gong1,58, W. Gradl35, S. Gramigna29A,29B, M. Greco74A,74C, M. H.
Gu1,58, Y. T. Gu15, C. Y. Guan1,63, A. Q. Guo31,63, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,g, A. Guskov36,b, J. Gutierrez27, K. L. Han63, T. T.
Han1, F. Hanisch3, X. Q. Hao19, F. A. Harris65, K. K. He55, K. L. He1,63, F.
H. Heinsius3, C. H. Heinz35, Y. K. Heng1,58,63, C. Herold60, T. Holtmann3, P.
C. Hong34, G. Y. Hou1,63, X. T. Hou1,63, Y. R. Hou63, Z. L. Hou1, B. Y. Hu59,
H. M. Hu1,63, J. F. Hu56,j, S. L. Hu12,g, T. Hu1,58,63, Y. Hu1, G. S.
Huang71,58, K. X. Huang59, L. Q. Huang31,63, X. T. Huang50, Y. P. Huang1, Y.
S. Huang59, T. Hussain73, F. Hölzken3, N. Hüsken35, N. in der Wiesche68, J.
Jackson27, S. Janchiv32, J. H. Jeong10A, Q. Ji1, Q. P. Ji19, W. Ji1,63, X. B.
Ji1,63, X. L. Ji1,58, Y. Y. Ji50, X. Q. Jia50, Z. K. Jia71,58, D. Jiang1,63,
H. B. Jiang76, P. C. Jiang46,h, S. S. Jiang39, T. J. Jiang16, X. S.
Jiang1,58,63, Y. Jiang63, J. B. Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y.
Jin66, M. Q. Jing1,63, X. M. Jing63, T. Johansson75, S. Kabana33, N. Kalantar-
Nayestanaki64, X. L. Kang9, X. S. Kang40, M. Kavatsyuk64, B. C. Ke80, V.
Khachatryan27, A. Khoukaz68, R. Kiuchi1, O. B. Kolcu62A, B. Kopf3, M.
Kuessner3, X. Kui1,63, N. Kumar26, A. Kupsc44,75, W. Kühn37, J. J. Lane67, P.
Larin18, L. Lavezzi74A,74C, T. T. Lei71,58, Z. H. Lei71,58, M. Lellmann35, T.
Lenz35, C. Li43, C. Li47, C. H. Li39, Cheng Li71,58, D. M. Li80, F. Li1,58, G.
Li1, H. B. Li1,63, H. J. Li19, H. N. Li56,j, Hui Li43, J. R. Li61, J. S. Li59,
K. Li1, L. J. Li1,63, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k,l, Q. M.
Li1,63, Q. X. Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,63, W. G.
Li1,a, X. Li1,63, X. H. Li71,58, X. L. Li50, X. Y. Li1,63, X. Z. Li59, Y. G.
Li46,h, Z. J. Li59, Z. Y. Li78, C. Liang42, H. Liang71,58, H. Liang1,63, Y. F.
Liang54, Y. T. Liang31,63, G. R. Liao14, L. Z. Liao50, Y. P. Liao1,63, J.
Libby26, A. Limphirat60, C. C. Lin55, D. X. Lin31,63, T. Lin1, B. J. Liu1, B.
X. Liu76, 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. Liu1,63, Huihui Liu21,
J. B. Liu71,58, J. Y. Liu1,63, K. Liu38,k,l, K. Y. Liu40, Ke Liu22, L.
Liu71,58, L. C. Liu43, Lu Liu43, M. H. Liu12,g, P. L. Liu1, Q. Liu63, S. B.
Liu71,58, T. Liu12,g, W. K. Liu43, W. M. Liu71,58, X. Liu38,k,l, X. Liu39, Y.
Liu38,k,l, Y. Liu80, Y. B. Liu43, Z. A. Liu1,58,63, Z. D. Liu9, Z. Q. Liu50,
X. C. Lou1,58,63, F. X. Lu59, H. J. Lu23, J. G. Lu1,58, X. L. Lu1, Y. Lu7, Y.
P. Lu1,58, Z. H. Lu1,63, C. L. Luo41, J. R. Luo59, M. X. Luo79, T. Luo12,g, X.
L. Luo1,58, X. R. Lyu63, Y. F. Lyu43, F. C. Ma40, H. Ma78, H. L. Ma1, J. L.
Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M. Ma1, R. Q. Ma1,63, T. Ma71,58, X. T.
Ma1,63, X. Y. Ma1,58, Y. Ma46,h, Y. M. Ma31, F. E. Maas18, M. Maggiora74A,74C,
S. Malde69, Y. J. Mao46,h, Z. P. Mao1, S. Marcello74A,74C, Z. X. Meng66, J. G.
Messchendorp13,64, G. Mezzadri29A, H. Miao1,63, T. J. Min42, R. E. Mitchell27,
X. H. Mo1,58,63, 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. Olsen63, Q. Ouyang1,58,63, S.
Pacetti28B,28C, X. Pan55, Y. Pan57, A. Pathak34, P. Patteri28A, Y. P.
Pei71,58, M. Pelizaeus3, H. P. Peng71,58, Y. Y. Peng38,k,l, K. Peters13,e, J.
L. Ping41, R. G. Ping1,63, S. Plura35, V. Prasad33, F. Z. Qi1, H. Qi71,58, H.
R. Qi61, M. Qi42, T. Y. Qi12,g, S. Qian1,58, W. B. Qian63, C. F. Qiao63, X. K.
Qiao80, J. J. Qin72, L. Q. Qin14, L. Y. Qin71,58, X. P. Qin12,g, X. S. Qin50,
Z. H. Qin1,58, J. F. Qiu1, Z. H. Qu72, C. F. Redmer35, K. J. Ren39, A.
Rivetti74C, M. Rolo74C, G. Rong1,63, Ch. Rosner18, S. N. Ruan43, N. Salone44,
A. Sarantsev36,d, Y. Schelhaas35, K. Schoenning75, M. Scodeggio29A, K. Y.
Shan12,g, W. Shan24, X. Y. Shan71,58, Z. J. Shang38,k,l, J. F. Shangguan16, L.
G. Shao1,63, M. Shao71,58, C. P. Shen12,g, H. F. Shen1,8, W. H. Shen63, X. Y.
Shen1,63, B. A. Shi63, H. Shi71,58, H. C. Shi71,58, J. L. Shi12,g, J. Y. Shi1,
Q. Q. Shi55, S. Y. Shi72, X. Shi1,58, J. J. Song19, T. Z. Song59, W. M.
Song34,1, Y. J. Song12,g, Y. X. Song46,h,n, S. Sosio74A,74C, S.
Spataro74A,74C, F. Stieler35, Y. J. Su63, G. B. Sun76, G. X. Sun1, H. Sun63,
H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun76, S. S. Sun1,63, T. Sun51,f, W. Y.
Sun34, Y. Sun9, Y. J. Sun71,58, Y. Z. Sun1, Z. Q. Sun1,63, Z. T. Sun50, C. J.
Tang54, G. Y. Tang1, J. Tang59, M. Tang71,58, Y. A. Tang76, L. Y. Tao72, Q. T.
Tao25,i, M. Tat69, J. X. Teng71,58, V. Thoren75, W. H. Tian59, Y. Tian31,63,
Z. F. Tian76, I. Uman62B, Y. Wan55, S. J. Wang 50, B. Wang1, B. L. Wang63, Bo
Wang71,58, D. Y. Wang46,h, F. Wang72, H. J. Wang38,k,l, J. J. Wang76, J. P.
Wang 50, K. Wang1,58, L. L. Wang1, M. Wang50, N. Y. Wang63, S. Wang12,g, S.
Wang38,k,l, T. Wang12,g, T. J. Wang43, W. Wang59, W. Wang72, W. P.
Wang35,71,o, 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. Wang1,58,63, Y. L. Wang19, Y. N.
Wang45, Y. Q. Wang1, Yaqian Wang17, Yi Wang61, Z. Wang1,58, Z. L. Wang72, Z.
Y. Wang1,63, Ziyi Wang63, D. H. Wei14, F. Weidner68, S. P. Wen1, Y. R. Wen39,
U. Wiedner3, G. Wilkinson69, M. Wolke75, L. Wollenberg3, C. Wu39, J. F. Wu1,8,
L. H. Wu1, L. J. Wu1,63, X. Wu12,g, X. H. Wu34, Y. Wu71,58, Y. H. Wu55, Y. J.
Wu31, Z. Wu1,58, L. Xia71,58, X. M. Xian39, B. H. Xiang1,63, 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. Xie71,58, T. Y.
Xing1,63, C. F. Xu1,63, C. J. Xu59, G. F. Xu1, H. Y. Xu66,2,p, M. Xu71,58, Q.
J. Xu16, Q. N. Xu30, W. Xu1, W. L. Xu66, X. P. Xu55, Y. C. Xu77, Z. P. Xu42,
Z. S. Xu63, F. Yan12,g, L. Yan12,g, W. B. Yan71,58, W. C. Yan80, X. Q. Yan1,
H. J. Yang51,f, H. L. Yang34, H. X. Yang1, T. Yang1, Y. Yang12,g, Y. F.
Yang1,63, Y. F. Yang43, Y. X. Yang1,63, Z. W. Yang38,k,l, Z. P. Yao50, M.
Ye1,58, M. H. Ye8, J. H. Yin1, Z. Y. You59, B. X. Yu1,58,63, C. X. Yu43, G.
Yu1,63, J. S. Yu25,i, T. Yu72, X. D. Yu46,h, Y. C. Yu80, C. Z. Yuan1,63, J.
Yuan34, J. Yuan45, L. Yuan2, S. C. Yuan1,63, Y. Yuan1,63, Z. Y. Yuan59, C. X.
Yue39, A. A. Zafar73, F. R. Zeng50, S. H. Zeng72, X. Zeng12,g, Y. Zeng25,i, Y.
J. Zeng1,63, Y. J. Zeng59, X. Y. Zhai34, Y. C. Zhai50, Y. H. Zhan59, A. Q.
Zhang1,63, B. L. Zhang1,63, B. X. Zhang1, D. H. Zhang43, G. Y. Zhang19, H.
Zhang80, H. Zhang71,58, H. C. Zhang1,58,63, H. H. Zhang34, H. H. Zhang59, H.
Q. Zhang1,58,63, H. R. Zhang71,58, H. Y. Zhang1,58, J. Zhang80, J. Zhang59, J.
J. Zhang52, J. L. Zhang20, J. Q. Zhang41, J. S. Zhang12,g, J. W. Zhang1,58,63,
J. X. Zhang38,k,l, J. Y. Zhang1, J. Z. Zhang1,63, Jianyu Zhang63, L. M.
Zhang61, Lei Zhang42, P. Zhang1,63, Q. Y. Zhang34, R. Y. Zhang38,k,l, S. H.
Zhang1,63, Shulei Zhang25,i, X. D. Zhang45, X. M. Zhang1, X. Y. Zhang50, Y.
Zhang72, Y. Zhang1, Y. T. Zhang80, Y. H. Zhang1,58, Y. M. Zhang39, Yan
Zhang71,58, Z. D. Zhang1, Z. H. Zhang1, Z. L. Zhang34, Z. Y. Zhang76, Z. Y.
Zhang43, Z. Z. Zhang45, G. Zhao1, J. Y. Zhao1,63, J. Z. Zhao1,58, L. Zhao1,
Lei Zhao71,58, M. G. Zhao43, N. Zhao78, R. P. Zhao63, S. J. Zhao80, Y. B.
Zhao1,58, Y. X. Zhao31,63, Z. G. Zhao71,58, A. Zhemchugov36,b, B. Zheng72, B.
M. Zheng34, J. P. Zheng1,58, W. J. Zheng1,63, Y. H. Zheng63, B. Zhong41, X.
Zhong59, H. Zhou50, J. Y. Zhou34, L. P. Zhou1,63, S. Zhou6, X. Zhou76, X. K.
Zhou6, X. R. Zhou71,58, X. Y. Zhou39, Y. Z. Zhou12,g, J. Zhu43, K. Zhu1, K. J.
Zhu1,58,63, K. S. Zhu12,g, L. Zhu34, L. X. Zhu63, S. H. Zhu70, S. Q. Zhu42, T.
J. Zhu12,g, W. D. Zhu41, Y. C. Zhu71,58, Z. A. Zhu1,63, J. H. Zou1, J. Zu71,58
(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 Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
64 University of Groningen, NL-9747 AA Groningen, The Netherlands
65 University of Hawaii, Honolulu, Hawaii 96822, USA
66 University of Jinan, Jinan 250022, People’s Republic of China
67 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
68 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
69 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
70 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
71 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
72 University of South China, Hengyang 421001, People’s Republic of China
73 University of the Punjab, Lahore-54590, Pakistan
74 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
75 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
76 Wuhan University, Wuhan 430072, People’s Republic of China
77 Yantai University, Yantai 264005, People’s Republic of China
78 Yunnan University, Kunming 650500, People’s Republic of China
79 Zhejiang University, Hangzhou 310027, People’s Republic of China
80 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 $(2.712\pm 0.014)\times 10^{9}$ $\psi(3686)$ events collected with the
BESIII detector operating at the BEPCII, we find an evidence of the
$\eta_{c}(2S)\to K^{+}K^{-}\eta^{\prime}$ decay with a statistical
significance of 3.1$\sigma$. Its decay branching fraction is measured to be
$(12.24\pm 4.60(\mathrm{stat.})\pm 2.37(\mathrm{syst.})\pm
4.68(\mathrm{extr.}))\times 10^{-4}$, where the first uncertainty is
statistical, the second is systematic, and the third uncertainty is from the
branching fraction of the $\psi(3686)\to\gamma\eta_{c}(2S)$ decay. The upper
limit on the product branching fraction
$B[\psi(3686)\to\gamma\eta_{c}(2S)]\times$ $B[\eta_{c}(2S)\to
K^{+}K^{-}\eta^{\prime}]$ is set to be $1.14\times 10^{-6}$ at 90% confidence
level. In addition, the branching fractions of $\chi_{c1}\to
K^{+}K^{-}\eta^{\prime}$ and $\chi_{c2}\to K^{+}K^{-}\eta^{\prime}$ are
updated to be $(8.47\pm 0.09(\mathrm{stat.})\pm 0.47(\mathrm{syst.}))\times
10^{-4}$ and $(1.53\pm 0.04(\mathrm{stat.})\pm 0.08(\mathrm{syst.}))\times
10^{-4}$, respectively. The precision is improved by twofold.

##  I INTRODUCTION

Charmonium states are composed of a pair of charm and anti-charm quarks. Their
decay dynamics can be used to probe the strong interaction and to test models
of Quantum Chromodynamics (QCD). It is well predicted that the ratio of
branching fractions of $\psi(3686)$ to $J/\psi$ decays into the same light
hadron final state is around 13%, the so-called “12% rule” which was first
proposed by Appelquist and Politzer using perturbative QCD intro1:qcd .
Although this rule has been verified for several hadronic channels intro2 ,
there are exceptions for the $\rho\pi$ and $K^{*}\bar{K}$ final states, where
the ratio is suppressed at least by an order of magnitude intro3 , known as
the “$\rho\pi$ puzzle”. The $\eta_{c}(2S)$ and $\eta_{c}(1S)$ are the spin-
singlet partners of $\psi(3686)$ and $J/\psi$, respectively. The corresponding
ratio for these two states was calculated in Refs. intro4  and intro5 . In
Ref. intro4 , the ratio is predicted to be 13%, while the authors of Ref.
intro5  argue that the ratio is 1, considering that the spin-singlet states
are dominantly decaying into light hadron final states via two-gluon
intermediate states. Using the related experimental information from light
hadron final states pdg , the authors of Ref. in6:example  recently tested
this branching fraction ratio in several decay modes in6:example  and found
that the experimental data significantly deviates from both theoretical
predictions.

The available data on the measurement of the branching fractions of the
$\eta_{c}(2S)$ decays is currently still limited. Although various experiments
have searched for nineteen decay modes so far, only seven decay branching
fractions have been measured pdg . Looking for new decay modes of
$\eta_{c}(2S)$ will provide a better understanding of its decay properties.
Considering the radiative magnetic dipole transition,
$\psi(3686)\to\gamma\eta_{c}(2S)$ in13:3c4c , the $\eta_{c}(2S)$ decays into
$3(\pi^{+}\pi^{-})$ lsx , $\pi^{+}\pi^{-}\eta$ etacp-ppeta , and
$K_{S}^{0}K\pi\pi\pi$ etacp-ppetac  were studied at BESIII. Obvious
$\eta_{c}(2S)$ signal events were observed in the $3(\pi^{+}\pi^{-})$ lsx  and
$K_{S}^{0}K\pi\pi\pi$ etacp-ppetac  decay modes with statistical significances
larger than $5\sigma$. The decay of $\eta_{c}(2S)\to K^{+}K^{-}\eta^{\prime}$
has not been measured yet, while the branching fraction of $\eta_{c}(1S)\to
K^{+}K^{-}\eta^{\prime}$ has been measured to be $(0.87\pm 0.18)\%$ pdg .

The $\chi_{\textit{cJ}}\rightarrow K^{+}K^{-}\eta^{\prime}$($J=1,2$) decays
were first observed by BESIII in the
$\psi(3686)\rightarrow\gamma\chi_{\textit{cJ}}$ decay with $(106.4\pm
0.9)\times 10^{6}$ $\psi(3686)$ events. The branching fractions of
$\chi_{c1}\rightarrow K^{+}K^{-}\eta^{\prime}$ and $\chi_{c2}\rightarrow
K^{+}K^{-}\eta^{\prime}$ were measured to be $(8.75\pm 0.87)\times 10^{-4}$
and $(1.94\pm 0.34)\times 10^{-4}$, respectively szt .

In this paper, we search for the $\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime}$ decay via the
$\psi(3686)\rightarrow\gamma\eta_{c}(2S)$ decay, and report the updated
measurements of the branching fractions of $\chi_{\textit{cJ}}\rightarrow
K^{+}K^{-}\eta^{\prime}$ ($J=1$ and $2$). The decay of $\chi_{c0}\rightarrow
K^{+}K^{-}\eta^{\prime}$ is forbidden by spin-parity conservation.

##  II BESIII DETECTOR, DATA SAMPLE, AND MONTE CARLO SIMULATION

The BESIII detector bes3:detector  records symmetric $e^{+}e^{-}$ collisions
provided by the BEPCII storage ring bes3:detector2  in the center-of-mass
(c.m.) energy ($\sqrt{s}$) range from 1.84 to 4.95 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 bes3:detector3 ; EcmsMea ; 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 bes3:detector4 ,
which benefits 83% of the data used in this analysis.

$(2.712\pm 0.014)\times 10^{9}$ $\psi(3686)$ liucheng  events collected by the
BESIII detector are used to search for
$\eta_{c}(2S)/\chi_{\textit{cJ}}\rightarrow K^{+}K^{-}\eta^{\prime}$ signal
events, including $(107.7\pm 0.6)\times 10^{6}$ $\psi(3686)$ events collected
in 2009, $(345.4\pm 2.6)\times 10^{6}$ $\psi(3686)$ events collected in 2012,
and $(2259.3\pm 11.1)\times 10^{6}$ $\psi(3686)$ events collected in 2021. An
additional continuum data set recorded at $\sqrt{s}=3.65$ GeV with an
integrated luminosity of 401 $\rm{pb}^{-1}$ liucheng  is used to determine the
nonresonant continuum background contributions.

Simulated samples produced with a geant4-based bes3:g4  Monte Carlo (MC)
package which includes the geometric description of the BESIII detector and
the detector response, are used to determine the detection efficiency and to
estimate the backgrounds. The production of the $\psi(3686)$ resonance is
simulated with the kkmc generator bes3:kkmc  taking into account the beam
energy spread and initial state radiation in the $e^{+}e^{-}$ annihilations.
Subsequent decays of the $\psi(3686)$ are modeled with evtgen bes3:evtgen
using branching fractions either taken from the Particle Data Group pdg , when
available, or otherwise estimated with lundcharm bes3:lundcharm . Final state
radiation (FSR) from charged final state particles is incorporated using the
photos package bes3:photon .

In this analysis, exclusive MC simulations of the signal reactions
$\psi(3686)\rightarrow\gamma\eta_{c}(2S)$, $\gamma\chi_{c1,2}$ with
$\eta_{c}(2S)$, $\chi_{c1,2}\rightarrow K^{+}K^{-}\eta^{\prime}$ are
generated, where the $\eta^{\prime}$ is reconstructed via its two main decay
modes $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ and
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ with
$\eta\rightarrow\gamma\gamma$. The radiative transition
$\psi(3686)\rightarrow\gamma\eta_{c}(2S)$, $\gamma\chi_{c1,2}$ is modeled by
following the angular distribution of $(1+\lambda\rm{cos}^{2}\theta)$, where
$\theta$ is the polar angle of the radiative photon in the rest frame of the
$\psi(3686)$ state with $\lambda$ set to be 1 for $\eta_{c}(2S)$ and -1/3,
1/13 for $\chi_{c1}$ and $\chi_{c2}$, respectively dec-etacp-chicj . The
dynamics in the $\chi_{\textit{cJ}}\rightarrow K^{+}K^{-}\eta^{\prime}$ decay
is considered by using Dalitz plots as input for the MC generator. For the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ decay, a model that takes into
account the $\rho-\omega$ interference and box anomaly is used modelgpipi .
All other decays are modeled evenly distributed in phase space with evtgen
bes3:evtgen . An inclusive MC simulation is used to study the background
contributions, which includes the production of the $\psi(3686)$ resonance,
the ISR production of the $J/\psi$, and the continuum processes incorporated
in kkmc bes3:kkmc . An additional exclusive MC simulation of the reaction
$\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$ is generated with evtgen
bes3:evtgen  to better investigate background contribution of this process.

##  III EVENT SELECTION

Charged tracks detected in the MDC are required to be within a polar angle
($\theta$) range of $|\cos\theta|<0.93$, where $\theta$ is defined with
respect to the $z$-axis, which is the symmetry axis of the MDC. The distance
of the closest approach to the interaction point must be less than 10 cm along
the $z$-axis, $|V_{z}|$, and less than 1 cm in the transverse plane,
$|V_{xy}|$. The helix parameters of the charged tracks in MC simulation are
corrected to improve the consistency between data and MC simulation helix .

Photon candidates are identified using showers in the EMC. The deposited
energy of each shower must be more than $25~{}\rm MeV$ in both the barrel
($|\cos\theta|<0.8$) and end cap ($0.86<|\cos\theta|<0.92$) regions. To
suppress electronic noise and energy depositions unrelated to the event, the
difference between the EMC cluster time and the reconstructed event start time
is required to be within $[0,~{}700]$ ns.

Charged particle identification (PID) is based on combining the d$E$/d$x$ and
TOF information to construct a $\chi^{2}_{\rm PID}$. The values
$\chi^{2}_{h-\mathrm{PID}}(i)$ are calculated for each charged track $i$ for
each particle hypothesis $h$ ($h=\pi$ or $K$). If there is no valid PID
information for the charged track, the $\chi^{2}_{h-\mathrm{PID}}(i)$ value is
set to $0$.

For the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode, four charged
tracks with net charge zero and at least two good photon candidates are
required. A four-constraint (4C) kinematic fit is performed to constrain the
four-momenta of the final state particles to the initial one. The best photon
candidates and the species of the charged tracks in the final state are
determined with the minimum $\chi^{2}_{\rm tot}=\chi^{2}_{\rm
4C}+\sum_{i=1}^{4}\chi^{2}_{h-\mathrm{PID}}(i)$, where $\chi^{2}_{\rm 4C}$ is
given by the 4C kinematic fit and $\chi^{2}_{h-\mathrm{PID}}(i)$ is taken from
PID. To suppress background events and to select
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ candidates, only the events
satisfying $\chi^{2}_{\rm{4C}}<25$ and $0.94~{}{\rm
GeV}/c^{2}<M_{\pi^{+}\pi^{-}\gamma}<0.97~{}{\rm GeV}/c^{2}$ are kept for the
further analysis. Here, $M_{\pi^{+}\pi^{-}\gamma}$ denotes the invariant mass
of $\pi^{+}\pi^{-}\gamma$. If both $\pi^{+}\pi^{-}\gamma$ combinations can
meet the selection requirements, both are retained. The fraction of events
with two entries is about 1.6% in data.

For the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$,
$\eta\rightarrow\gamma\gamma$ mode, four charged tracks with net charge zero
and at least three good photon candidates are required. The photon pair
passing a 1C kinematic fit is regarded as $\eta$ candidate, in which the
$\gamma\gamma$ invariant mass, $M_{\gamma\gamma}$, is constrained to the
nominal mass of $\eta$ pdg . If there is more than one $\eta$ candidate in the
event, all possible combinations are kept. Similarly, a 4C kinematic fit is
performed, and the best photon candidate, $\eta$ candidate, and the species of
the charged tracks in the final state are determined with the minimum
$\chi^{2}_{\rm tot}$ = $\chi^{2}_{\rm 1C}$ \+ $\chi^{2}_{\rm 4C}$ \+
$\sum_{i=1}^{4}$ $\chi^{2}_{h-\mathrm{PID}}(i)$. To suppress background
contributions from other decays and to select the
$\eta\rightarrow\gamma\gamma$ and $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$
candidates, only the events satisfying $\chi^{2}_{\rm{4C}}<20$, $0.52~{}{\rm
GeV}/c^{2}<M_{\gamma\gamma}<0.57~{}{\rm GeV}/c^{2}$, and $0.93~{}{\rm
GeV}/c^{2}<M_{\pi^{+}\pi^{-}\eta}<0.98~{}{\rm GeV}/c^{2}$ are kept for the
further analysis, where $M_{\pi^{+}\pi^{-}\eta}$ is the invariant mass of the
$\pi^{+}\pi^{-}\eta$ system.

The requirements of $\chi^{2}_{\rm 4C}$ in these two modes are optimized
taking the maximum of the figure-of-merit (FOM) defined as $S/\sqrt{S+B}$,
where $S$ and $B$ are the expected yields of the $\eta_{c}(2S)$ signal and
background events, respectively, in the $\eta_{c}(2S)$ signal region defined
as (3.60, 3.70) ${\rm GeV}/c^{2}$. $S$ is calculated based on an assumed
branching fraction of $B(\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime})=B(\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta)=5\times
10^{-3}$ pdg . $B$ is taken from the inclusive MC sample and scaled to data
according to the total number of $\psi(3686)$ event. The mass window
requirements for $\eta$ and $\eta^{\prime}$ candidates correspond to three
times the mass resolution determined via fitting the corresponding invariant
mass distributions $M_{\gamma\gamma}$ and $M_{\pi\pi\eta,~{}\gamma}$ from data
with a sum of two Gaussian functions.

Special requirements are placed to suppress various background contributions,
the details are listed in Table 1. For the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode, the background events
from $\psi(3686)\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}/\eta$ are excluded
by requiring $M_{\gamma\gamma}$ to be outside the $\pi^{0}$ and $\eta$ mass
windows. The background events from
$\psi(3686)\rightarrow\gamma\chi_{\textit{cJ}}$
($\chi_{\textit{cJ}}\rightarrow\gamma J/\psi,J/\psi\rightarrow
K^{+}K^{-}\pi^{+}\pi^{-}$ or $\chi_{\textit{cJ}}\rightarrow
K^{+}K^{-}\pi^{+}\pi^{-}$) are excluded by requiring the invariant mass of the
$K^{+}K^{-}\pi^{+}\pi^{-}$ system ($M_{K^{+}K^{-}\pi^{+}\pi^{-}}$) to be
outside the $J/\psi$ and $\chi_{\textit{cJ}}$ ($J=0,1,2$) mass windows. The
background events from $\psi(3686)\rightarrow\pi^{+}\pi^{-}J/\psi$ are
excluded by requiring the recoil mass of the $\pi^{+}\pi^{-}$ pairs
($M_{\pi^{+}\pi^{-}}^{\rm recoil}$) to be outside the $J/\psi$ mass window.
The background events from
$\psi(3686)\rightarrow\eta^{\prime}\phi,~{}\phi\rightarrow K^{+}K^{-}$ are
excluded by requiring the invariant mass of the $K^{+}K^{-}$ system
($M_{K^{+}K^{-}}$) to be outside the $\phi$ mass window. These requirements
are named $\pi^{0}$ veto, $\eta$ veto, $J/\psi$ veto, $\chi_{\textit{cJ}}$
($J=0,1,2$) veto, $\pi^{+}\pi^{-}J/\psi$ veto, and $\phi$ veto, respectively.

Table 1: The mass windows used to veto background contributions.
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode
---
Veto |  Mass window (${\rm GeV}/c^{2}$)
$\pi^{0}$ |  $0.122<M_{\gamma\gamma}<0.146$
$\eta$ |  $0.526<M_{\gamma\gamma}<0.566$
$J/\psi$ |  $3.08<M_{K^{+}K^{-}\pi^{+}\pi^{-}}<3.12$
$\chi_{\textit{cJ}}$($J=0$) |  $3.37<M_{K^{+}K^{-}\pi^{+}\pi^{-}}<3.43$
$\chi_{\textit{cJ}}$($J=1$) |  $3.48<M_{K^{+}K^{-}\pi^{+}\pi^{-}}<3.51$
$\chi_{\textit{cJ}}$($J=2$) |  $3.53<M_{K^{+}K^{-}\pi^{+}\pi^{-}}<3.54$
$\pi^{+}\pi^{-}J/\psi$ |  $3.091<M_{\pi^{+}\pi^{-}}^{\rm recoil}<3.103$
$\phi$ |  $M_{K^{+}K^{-}}<1.03$
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode
Veto |  Mass window (${\rm GeV}/c^{2}$)
$\pi^{0}(\gamma\gamma_{\eta})$ |  $0.118<M_{\gamma\gamma_{\eta}}<0.150$
$\eta J/\psi$ |  $3.070<M_{\eta}^{\rm recoil}<3.133$
$\phi$ |  $M_{K^{+}K^{-}}<1.03$

For the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode, the background
events from $\psi(3686)\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$ are
excluded by requiring $M_{\gamma\gamma_{\eta}}$ to be outside the $\pi^{0}$
mass window. Here $\gamma_{\eta}$ refers to the photon used to reconstruct
$\eta$. The background events from $\psi(3686)\rightarrow\eta J/\psi$ are
excluded by requiring the recoil mass of $\eta$ ($M_{\eta}^{\rm recoil}$) to
be outside the $J/\psi$ mass window. The background events from
$\psi(3686)\rightarrow\eta^{\prime}\phi,~{}\phi\rightarrow K^{+}K^{-}$ are
excluded by requiring $M_{K^{+}K^{-}}$ to be outside the $\phi$ mass window.
These are named $\pi^{0}$ veto, $\eta J/\psi$ veto, and $\phi$ veto,
respectively.

After above selection and by using the signal MC simulations, the signal
detection efficiencies for the $\chi_{c1}\rightarrow K^{+}K^{-}\eta^{\prime}$,
$\chi_{c2}\rightarrow K^{+}K^{-}\eta^{\prime}$, and $\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime}$ signal channels are ($17.7\pm 0.1$)%, ($19.7\pm
0.1$)%, and ($14.3\pm 0.1$)% for the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode, and ($9.9\pm 0.1$)%,
($10.9\pm 0.1$)%, and ($8.9\pm 0.1$)% for the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode.

##  IV BACKGROUND ESTIMATION

The background contribution is studied with the inclusive MC sample and the
continuum data. In the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode,
about $99.7\%$ of the background events come from processes without
$\eta^{\prime}$ as an intermediate state (non-$\eta^{\prime}$ processes). In
the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode, the dominant background
contribution is also from the non-$\eta^{\prime}$ processes, which is about
$90.1\%$ of the background events. Among the remaining background processes,
$\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$ events are accumulated close
to the $\eta_{c}(2S)$ signal and are treated separately.

###  IV.1 $\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$ background

Using the 4C kinematic fit information for the final state particles,
background contributions from $\psi(3686)\rightarrow
K^{+}K^{-}\eta^{\prime}(\gamma_{\rm{FSR}})$ processes create a peaking
structure close to the $\eta_{c}(2S)$ mass in the invariant mass distribution
of the $K^{+}K^{-}\eta^{\prime}$ system, that also contaminates the
$\eta_{c}(2S)$ signal. In order to better differentiate between the
$\eta_{c}(2S)$ signal and the $\psi(3686)\rightarrow
K^{+}K^{-}\eta^{\prime}(\gamma_{\rm{FSR}})$ background contributions, a 3C
kinematic fit is performed, in which the energy of the radiative photon is not
used as input in13:3c4c . In Fig. 1, a comparison between the results of the
3C and 4C fits are presented for $M_{K^{+}K^{-}\eta^{\prime}}$ showing that
the peak from the $\psi(3686)\rightarrow
K^{+}K^{-}\eta^{\prime}(\gamma_{\rm{FSR}})$ events is pulled towards the
$\psi(3686)$ peak in the 3C fit.

Figure 1: Comparison of the $M_{K^{+}K^{-}\eta^{\prime}}$ distributions of the
signal MC events from the 3C fit (blue solid line) and 4C fit (red dashed
line), and those of the exclusive MC simulation of $\psi(3686)\rightarrow
K^{+}K^{-}\eta^{\prime}$ from the 3C fit (black dashed-dotted line) and 4C fit
(black dashed line).

\begin{overpic}[width=208.13574pt]{fig2-1.pdf} \put(50.0,1.0){(a)}
\end{overpic}\begin{overpic}[width=221.14381pt]{fig2-2.pdf}
\put(50.0,1.0){(b)}
\end{overpic}\begin{overpic}[width=208.13574pt]{fig2-3.pdf}
\put(50.0,0.0){(c)}
\end{overpic}\begin{overpic}[width=221.14381pt]{fig2-4.pdf}
\put(50.0,0.0){(d)} \end{overpic}

Figure 2: The left figures show fits to the (a) $M_{\pi^{+}\pi^{-}\gamma}$ and
(c) $M_{\pi^{+}\pi^{-}\eta}$ distributions of the accepted candidate events in
data (black dots with error bars). The red solid curves are the fit result,
the dashed lines are the signal component, and the blue dashed lines represent
the background contribution. The pair of blue arrows indicate the
$\eta^{\prime}$ signal region, and the two pairs of green arrows are the
sideband regions. The right figures show comparisons of the
$M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ distributions between data and MC
simulations for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode (b)
and for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode (d). The red
dashed histograms are the non-$\eta^{\prime}$ background events estimated with
the $\eta^{\prime}$ sideband events in data, the blue solid histograms
represent non-$\eta^{\prime}$ background events estimated with the inclusive
MC sample, the purple solid histograms show the background contribution
containing $\eta^{\prime}$ in the final state estimated using the inclusive MC
sample, and the green filled histograms show the background contribution from
$\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$ estimated using the MC
simulation. The red dashed histograms and the green filled histograms are
stacked.

The tail on the left side of the black dashed-dotted curve of the 3C case in
Fig. 1 for the background contribution is caused by FSR. The difference of the
FSR ratio between data and MC simulation has been studied in a previous work
using
$\psi(3686)\rightarrow\gamma\chi_{c0},~{}\chi_{c0}\rightarrow\pi^{+}\pi^{-}K^{+}K^{-}$
wyaqian . There the FSR correction factor

$\displaystyle f_{\rm FSR}=R_{\rm FSR}^{\rm data}/R_{\rm FSR}^{\rm MC}$ (1)

is determined to be $1.39\pm 0.08\pm 0.04$, where $R_{\rm{FSR}}^{\rm data}$ is
the ratio of events with and without FSR in data, and $R_{\rm{FSR}}^{\rm{MC}}$
is the same ratio for MC simulations.

The fraction of MC events with and without FSR is corrected with $f$, and the
line shape of $M_{K^{+}K^{-}\eta^{\prime}}^{\rm{3C}}$ from the corrected MC
sample, as shown in Fig. 1, is used in the fit to the
$M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ distribution from data to extract the
signal yield.

###  IV.2 Non-$\eta^{\prime}$ background

The non-$\eta^{\prime}$ background contribution is estimated using events in
$\eta^{\prime}$ sideband regions. The $M_{\pi^{+}\pi^{-}\gamma}$ distribution
is fitted to determine the signal and sideband regions and the scale factor
between the $\eta^{\prime}$ signal and sideband regions. The scale factor is
calculated using the number of background events in the signal region and the
sum of the number of background events in both sideband regions. These number
of background events are determined using the background line shape with
parameters fixed to the fitted values.

For the $\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ mode, the sideband regions are
defined as $[0.8,0.9]~{}{\rm GeV}/c^{2}$ and $[1.0,1.1]~{}{\rm GeV}/c^{2}$.
The scale factor is determined to be 0.194. In the fit, the $\eta^{\prime}$
signal is described by an MC simulated shape convolved with a Gaussian
function to account for differences in the detector resolution between data
and MC, and the background is described by a sixth-order polynomial function.
The fit result is shown in Fig. 2(a). The fit quality, obtained by performing
a $\chi^{2}$ test of the fitted curve and the binned mass spectrum, is
$\chi^{2}/\mathrm{ndf}=247.2/190=1.3$. Here $\mathrm{ndf}$ is the number of
degrees of freedom. There is a small and non-smooth background contribution on
the right of the $\eta^{\prime}$ peak caused by the misselection of the
photon. The effect on the sideband estimation is taken into account in the
systematic uncertainty. For the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$
mode, the sideband regions are $[0.85,0.90]~{}{\rm GeV}/c^{2}$ and
$[1.0,1.1]~{}{\rm GeV}/c^{2}$, and the scale factor is 0.162. They are
determined using the same method as for the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode, except that the
background is described by a third-order polynomial function. The fit result
is shown in Fig. 2(c) and the fit quality is
$\chi^{2}/\mathrm{ndf}=137.7/168=0.8$.

Figures 2(b) and 2(d) show the comparison of the
$M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ distributions from data together with
selected background components estimated either with the $\eta^{\prime}$
sideband events from data, or with the inclusive and exclusive MC simulations.
The comparison shows that the non-$\eta^{\prime}$ background events estimated
from the $\eta^{\prime}$ sideband region (red dashed curve) can describe the
non-$\eta^{\prime}$ background of inclusive MC (blue curve).

###  IV.3 Background including $\eta^{\prime}$

The background contribution containing $\eta^{\prime}$ in the final state is
about 0.3% for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode and
9.9% for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode, estimated
using the inclusive MC sample. In Figs. 2(b) and 2(d), the purple solid
histograms show the $M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ distributions of
this background contribution. They are smoothly distributed, thus could be
fitted to a polynomial function.

###  IV.4 Continuum contribution

The background contribution from the continuum process is estimated using the
continuum data set at $\sqrt{s}=$3.650 GeV. The event selections applied to
this data sample are the same as for the $\psi(3686)$ data sample, except that
the c.m. energy is changed from 3.686 GeV to 3.650 GeV in the kinematic fit.
Considering the difference of the c.m. energies, the
$M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ ($m$ in Eq.(2)) from the continuum data
is shifted and corrected by

$m_{\rm shifted}\rightarrow a(m-m_{0})+m_{0},$ (2)

where $m_{0}$ = 1.945 ${\rm GeV}/c^{2}$ is the mass threshold for
$K^{+}K^{-}\eta^{\prime}$ which should not be shifted, $a=(3.686~{}{\rm
GeV}/c^{2}-m_{0})/(3.650~{}{\rm GeV}/c^{2}-m_{0})=1.021$ is the shift
coefficient for $\sqrt{s}=3.650$ GeV to ensure that the events at 3.650 GeV
are shifted to 3.686 GeV. The $M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$
distributions before and after the shift are shown in Fig. 3.

\begin{overpic}[width=216.81pt]{fig3-1.pdf} \put(50.0,1.0){(a)} \end{overpic}
\begin{overpic}[width=216.81pt]{fig3-2.pdf} \put(50.0,0.0){(b)} \end{overpic}

Figure 3: The $M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ distributions for the (a)
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ and (b)
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ modes from the continuum data at
3.650 GeV. The blue dots with error bars are before the mass shift, the black
dots with error bars are after the mass shift.

The mass spectrum of $M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ is normalized
based on the differences in the integrated luminosity and cross section. For
the continuum data set at 3.650 GeV, the scale factor is calculated via

$f_{\rm
continuum}=\frac{\mathcal{L}_{3.686}}{\mathcal{L}_{3.650}}\times\frac{\sigma_{3.686}}{\sigma_{3.650}}=8.31,$
(3)

where $\mathcal{L}_{3.686}=3.4~{}\mathrm{fb}^{-1}$ and
$\mathcal{L}_{3.650}=401~{}\mathrm{pb}^{-1}$ liucheng  are the integrated
luminosities at 3.686 GeV and 3.650 GeV, respectively, and $\sigma$ is the
production cross section, which is assumed to be proportional to $1/s$. The
shape and the estimated continuum background yield of the continuum data
sample with applied scaling factor are used to fit the data.

\begin{overpic}[width=214.63977pt]{fig4-1.pdf} \put(50.0,1.0){}
\end{overpic}\begin{overpic}[width=214.63977pt]{fig4-2.pdf} \put(50.0,1.0){}
\end{overpic}\begin{overpic}[width=214.63977pt]{fig4-3.pdf} \put(50.0,0.0){}
\end{overpic}\begin{overpic}[width=214.63977pt]{fig4-4.pdf} \put(50.0,0.0){}
\end{overpic}

Figure 4: The simultaneous fit to the $M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$
distributions in the whole fit range in (a) and (c), and the range only
containing $\eta_{c}(2S)$ signal in (b) and (d). The dots with error bars are
data, the red solid curves correspond to the best fit result, the blue dashed
lines show the $\eta_{c}(2S)$ and $\chi_{\textit{cJ}}~{}$($J=1,2$) signal
shapes, the purple dashed lines represent the contribution from
$\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$, the green dashed lines show
the non-$\eta^{\prime}$ contribution from the $\eta^{\prime}$-sideand events
from data, the black dashed lines are the continuum contribution, and the cyan
dashed lines represent the remaining background.

##  V SIGNAL EXTRACTION

The signal yields are determined from an unbinned maximum likelihood fit to
the $M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}$ spectra of the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ and
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode simultaneously. The function
used in the fit is defined as

$\mathrm{PDF}_{\rm total}=N^{\rm sig}\cdot S+\sum_{i=1}^{4}N^{\rm
bkg}_{i}\cdot B_{i}.$ (4)

Here, $N^{\rm sig}$ and $N^{\rm bkg}_{i}$ represent the numbers of signal
events and background events, and $S$ and $B_{i}$ are the signal and
background PDFs.

The signal PDF is given by

$S=[(E_{\gamma}^{3}\times f_{d}(E_{\gamma})\times BW(m))\otimes
DG(m)\times\epsilon(m)]\otimes G(\delta m,\delta\sigma),$ (5)

where $m$ corresponds to the $M_{K^{+}K^{-}\eta^{\prime}}^{3C}$ distributions
after the 3C kinematic fit, $E_{\gamma}$ is the energy of the radiative photon
in the rest frame of the $\psi(3686)$, and $BW(m)$ is the relativistic Breit-
Wigner function used to describe the resonance. The mass and total width are
fixed to the PDG values for $\eta_{c}(2S)$ pdg  and are free parameters for
$\chi_{c1,2}$. The factor $E_{\gamma}^{3}$ is introduced due to the energy
dependence of the transition matrix element from the radiative photon energy
in the radiative magnetic diplole transition process. The function
$f_{d}(E_{\gamma})$ damps the diverging tail raised by $E_{\gamma}^{3}$. It is
adapted from previous work by the KEDR Collaboration kedr  with
$f_{d}(E_{\gamma})=\dfrac{E_{0}^{2}}{E_{\gamma}E_{0}+(E_{\gamma}-E_{0})^{2}}$,
where
$E_{0}=\dfrac{M_{\psi(3686)}^{2}-M_{\eta_{c}(2S)/\chi_{\textit{cJ}}}^{2}}{2M_{\psi(3686)}}$
is the most probable energy of the radiative photon. $DG(m)$ represents the
mass resolution, determined by fitting $\Delta
M=M_{K^{+}K^{-}\eta^{\prime}}^{\rm 3C}-M_{K^{+}K^{-}\eta^{\prime}}^{\rm true}$
with a sum of two Guassian functions. Here, $M_{K^{+}K^{-}\eta^{\prime}}^{\rm
3C}$ is the invariant mass of $K^{+}K^{-}\eta^{\prime}$ from the 3C kinematic
fit and $M_{K^{+}K^{-}\eta^{\prime}}^{\rm true}$ is the mass at generator
level. $\epsilon(m)$ denotes the efficiency curve, which is obtained from the
signal MC simulations and parameterized by a generalized ARGUS function ARGUS
, defined as

$\epsilon(m)=m(1-(\frac{m}{m_{0}})^{2})^{p}\times
e^{c(1-(\frac{m}{m_{0}})^{2})},$ (6)

where $m_{0}=3.67~{}{\rm GeV}/c^{2}$ is the kinematic limit (cut-off)
corresponding to the $\eta_{c}(2S)$ mass including detector resolution. The
parameters $p$ and $c$ are determined to be $p=0.311,~{}c=-2.74$ for
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ and $p=0.238,~{}c=-2.04$ for
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$. To account for the mass
resolution difference between data and MC simulations, an additional Gaussian
function, $G(\delta m,\delta\sigma)$ is included. For the $\chi_{\textit{cJ}}$
signals, the parameters of this Gaussian function are floated, while for the
$\eta_{c}(2S)$ signal they are fixed to the values extrapolated from
$\chi_{\textit{cJ}}$ with a linear assumption, which are $\delta m=(0.12\pm
0.18)~{}{\rm MeV}/c^{2}$ and $\delta\sigma=(-0.87\pm 1.87)~{}{\rm MeV}/c^{2}$
for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode, $\delta
m=(-6.52\pm 2.73)~{}{\rm MeV}/c^{2}$ and $\delta\sigma=(5.39\pm 5.06)~{}{\rm
MeV}/c^{2}$ for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode. The
mass and width of $\chi_{c1,2}$ determined from the fit are consistent with
the results from PDG.

In the simultaneous fit, the branching fraction of
$\eta_{c}(2S)/\chi_{\textit{cJ}}\rightarrow K^{+}K^{-}\eta^{\prime}$,
$\mathcal{B}^{\rm sig}$ is the same for the two $\eta^{\prime}$ modes, it is
connected to the number of signal events via

$\mathcal{B}^{\rm sig}=\frac{N^{\rm
sig}}{N_{\psi(3686)}\cdot\mathcal{B}(\psi(3686)\to\gamma\eta_{c}(2S)/\chi_{\textit{cJ}})\cdot\bar{\epsilon}},$
(7)

where
$\bar{\epsilon}=\mathcal{B}(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma)\cdot\epsilon_{1}+\mathcal{B}(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta)\cdot\mathcal{B}(\eta\to\gamma\gamma)\cdot\epsilon_{2}$
is the average detection efficiency, $\mathcal{B}$ is the branching fraction
of the corresponding process, $\epsilon_{1}$ and $\epsilon_{2}$ are the signal
detection efficiencies for the $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$
and $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ modes, respectively.

The background contributions have been well estimated in section IV. They are
decomposed into four components: the $\psi(3686)\rightarrow
K^{+}K^{-}\eta^{\prime}$ background ($B_{1}$), the non-$\eta^{\prime}$
background ($B_{2}$), the continuum contribution ($B_{3}$), and the remaining
backgrounds ($B_{4}$). The $B_{1}$ shape is taken from the
$\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$ MC simulation, corrected by
the FSR ratio $f$, the $B_{2(3)}$ line shape is directly determined from data,
and $B_{4}$ is a second-order polynomial function. $N_{1}^{\rm bkg}$ and
$N_{4}^{\rm bkg}$ are free parameters in the fit, while $N_{2}^{\rm bkg}$ and
$N_{3}^{\rm bkg}$ are fixed. We added a truncation at $3.686~{}{\rm
GeV}/c^{2}$ to the line shape of the continuum sample and
$\eta^{\prime}$-sideband to account for the kinematic boundary.

Table 2: Branching fractions for the three different signal reactions. The first uncertainty is statistical and the second is systematic. For $\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}$, the third uncertainty is from $\mathcal{B}[\psi(3686)\rightarrow\gamma\eta_{c}(2S)]$.  Channel |  $\mathcal{B}~{}(\times 10^{-4})$ |  $\rm{PDG~{}}(\times 10^{-4})$
---|---|---
$\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}$ |  $\rm 12.24\pm 4.60(stat.)\pm 2.37(syst.)\pm 4.68(extr.)$ |  -
$\chi_{c1}\rightarrow K^{+}K^{-}\eta^{\prime}$ |  $\rm 8.47\pm 0.09(stat.)\pm 0.47(syst.)$ |  $8.8\pm 0.9$
$\chi_{c2}\rightarrow K^{+}K^{-}\eta^{\prime}$ |  $\rm 1.53\pm 0.04(stat.)\pm 0.08(syst.)$ |  $1.94\pm 0.34$
Table 3:  Systematic uncertainties for the branching fraction measurements [$\%$]. For $\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}$, the value in parentheses is the total systematic uncertainty without the uncertainty from $\mathcal{B}(\psi(3686)\rightarrow\gamma\eta_{c}(2S))$.  Source |  $\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}$ |  $\chi_{c1}\rightarrow K^{+}K^{-}\eta^{\prime}$ |  $\chi_{c2}\rightarrow K^{+}K^{-}\eta^{\prime}$
---|---|---|---
Multiplicative term
Total number of $\psi(3686)$ events |  0.5 |  0.5 |  0.5
Tracking |  4.0 |  4.0 |  4.0
Photon reconstruction |  2.3 |  2.3 |  2.3
$\mathcal{B}(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma/\eta)$ |  1.5 |  1.5 |  1.5
$\mathcal{B}(\psi(3686)\rightarrow\gamma\eta_{c}(2S)/\chi_{\textit{cJ}})$ |  38.2 |  2.5 |  2.1
MC statistics |  0.1 |  0.1 |  0.1
MC generator |  - |  0.6 |  0.0
Kinematic fit |  1.1 |  0.8 |  1.1
$\eta$ mass window |  0.0 |  0.0 |  0.0
$\eta^{\prime}$ mass window |  0.2 |  0.2 |  0.2
All background veto |  3.7 |  0.0 |  0.3
Additive term
FSR factor |  1.3 |  0.1 |  0.2
Number of continuum events |  3.8 |  0.0 |  0.0
Shape of continuum |  13.4 |  0.2 |  0.3
Shape of non-$\eta^{\prime}$ events |  4.3 |  0.1 |  0.2
Shape of $\psi(3686)\rightarrow K^{+}K^{-}\eta^{\prime}$ |  5.2 |  0.0 |  0.1
Shape of other background |  0.2 |  0.2 |  0.1
Region of $\eta^{\prime}$ sideband |  4.4 |  0.1 |  0.2
Number of non-$\eta^{\prime}$ events |  1.8 |  0.0 |  0.2
Damping function form |  2.9 |  0.0 |  0.2
First kind of resolution |  3.2 |  0.5 |  0.1
Second kind of resolution($\delta m$) |  5.5 |  - |  -
Second kind of resolution($\delta\sigma$) |  3.7 |  - |  -
Efficiency curve |  1.3 |  0.2 |  0.3
Mass of $\eta_{c}(2S)$ |  3.1 |  0.1 |  0.1
Width of $\eta_{c}(2S)$ |  1.1 |  0.0 |  0.1
Total |  42.9 (19.4) |  5.6 |  5.5

The fit results are shown in Fig. 4, and the fit quality is
$\chi^{2}/\rm{ndf}=228.2/145=1.6$. The results for the branching ratios are
summarized in Table 2. The spectra from the two decay modes are also fitted
separately, the results are consistent with each other. An input and output
check is performed to check the bias of the fit procedure, the output values
are consistent with the input ones. The statistical significance of the
$\eta_{c}(2S)$ signal is estimated to be 3.1$\sigma$, calculated using the
difference of the logarithmic likelihoods sigma ,
-2ln$(\mathcal{L}_{0}/\mathcal{L}_{\rm max})$. Here $\mathcal{L}_{\rm max}$
and $\mathcal{L}_{0}$ are the maximized likelihoods with and without the
$\eta_{c}(2S)$ signal component, respectively. The difference in the number of
degrees of freedom ($\Delta$ndf=1) has been taken into account.

##  VI SYSTEMATIC UNCERTAINTY ESTIMATION

The systematic uncertainties in the branching fraction measurements are
summarized in Table 3. They are classified in two categories: the
multiplicative terms and the additive terms.

The multiplicative terms refer to the uncertainties due to the total number of
$\psi(3686)$ events, the detector efficiency, and the branching fractions.
They are estimated separately for the two $\eta^{\prime}$ decay modes. Varying
the corresponding values by $\pm 1\sigma$, the changes on the average
detection efficiency is assigned as the systematic uncertainty. They are
described in detail in the following.

The total number of $\psi(3686)$ event is determined to be $(27.12\pm
0.14)\times 10^{8}$ liucheng . Its uncertainty is 0.5%.

The uncertainty from the tracking is assigned to be 1% per track using the
control samples of $J/\psi\rightarrow\pi^{0}\pi^{+}\pi^{-}$ or
$J/\psi\rightarrow K_{S}^{0}K\pi+c.c$ sys-track . The uncertainty from the
photon reconstruction is studied using the control samples
$J/\psi\rightarrow\rho^{0}\pi^{0}$ and $e^{+}e^{-}\rightarrow\gamma\gamma$
sys-photon , and is assigned to be 1.0% per photon.

The uncertainties from the branching fractions of
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$,
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, and
$\eta\rightarrow\gamma\gamma$ are 1.4%, 1.2%, and 0.5%, respectively. The
uncertainties from the branching fractions of
$\psi(3686)\rightarrow\gamma\eta_{c}(2S)$, $\chi_{c1}$, and $\chi_{c2}$ are
38.2% psiptogametacp , 2.5%, and 2.1% pdg , respectively.

The Dalitz plot, momentum, and cos$\theta$ distributions of $K^{+}$, $K^{-}$,
and $\eta^{\prime}$ in the signal MC samples produced with the body3 model are
basically consistent with the distributions in data. Therefore, the
uncertainty from the MC generator is estimated by changing the number of bins
of the input Dalitz plot. Due to limited statistics of the MC samples used, a
0.1% uncertainty is taken for each decay.

In the kinematic fit, the helix parameters of charged tracks in MC samples
have been corrected to improve the consistency between data and MC simulations
helix . Half of the differences of efficiencies with and without the helix
parameter correction are taken as the systematic uncertainties. This is a
conservative estimation, as the uncertainties of the correction factors are at
10% level, and the difference between data and MC simulation in the
$\chi^{2}_{\rm 4C}$ distributions after the correction is much smaller than
that before.

For the uncertainty arising from the mass window requirements of $\eta$ and
$\eta^{\prime}$, the $M_{\eta}$ and $M_{\eta^{\prime}}$ distributions from
data are fitted with the corresponding simulated shapes convolved with a
Gaussian resolution function. The $\eta$ and $\eta^{\prime}$ distributions
from the signal MC samples are smeared with the resultant Gaussian function.
The change of the signal detection efficiency is taken as the uncertainty.

The uncertainties due to different mass vetoes are examined via a Barlow test
barlowtest . A significant deviation ($\zeta$) between the nominal result and
the systematic test is defined as

$\displaystyle\zeta=\dfrac{|\mathcal{B}_{\rm nominal}^{\rm
sig}-\mathcal{B}_{\rm test}^{\rm sig}|}{\sqrt{|\sigma_{\mathcal{B}_{\rm
nominal}^{\rm sig}}^{2}-\sigma_{\mathcal{B}_{\rm test}^{\rm sig}}^{2}|}},$ (8)

where $\sigma_{\mathcal{B}}$ is the statistical error of the branching
fraction. We vary the veto mass windows used and repeat the simultaneous fit.
The obtained $\zeta$ distribution shows no significant deviation. As a
conservative estimation, the maximum difference in the branching fraction is
taken as the systematic uncertainty. The systematic uncertainty caused by
different vetoes estimated by the Barlow tests is mainly caused by statistical
fluctuations. Therefore, for vetoes where there is no obvious trend and
$\zeta$ is less than 2, the systematic uncertainty of $\eta_{c}(2S)$ and
$\chi_{c1,2}$ is set to be 0. Taking the $\pi^{0}$ veto in the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode for example, we vary the
requirement on $M_{\gamma\gamma}$ to be (0.110, 0.158), (0.112, 0.156),
(0.114, 0.154), (0.116, 0.152), (0.118, 0.150), (0.120, 0.148), (0.122,
0.146), (0.124, 0.144) ${\rm GeV}/c^{2}$. For other mass vetoes, the mass
windows are changed using the similar method, as detailed in Table 4.

Table 4: The mass window variations (in unit of ${\rm GeV}/c^{2}$) used to
estimate the uncertainties for all mass vetoes.
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode
---
Veto |  Lower limit |  Upper limit |  Step
$\pi^{0}$ |  (0.110, 0.124) |  (0.144, 0.158) |  0.002
$\eta$ |  (0.518, 0.532) |  (0.560, 0.574) |  0.002
$J/\psi$ |  (3.072, 3.086) |  (3.114, 3.128) |  0.002
$\chi_{\textit{cJ}}$($J=0$) |  (3.365, 3.372) |  (3.428, 3.435) |  0.001
$\chi_{\textit{cJ}}$($J=1$) |  (3.475, 3.482) |  (3.508, 3.515) |  0.001
$\chi_{\textit{cJ}}$($J=2$) |  (3.525, 3.532) |  (3.538, 3.545) |  0.001
$\pi^{+}\pi^{-}J/\psi$ |  (3.086, 3.093) |  (3.101, 3.108) |  0.001
$\phi$ |  - |  (1.028, 1.035) |  0.001
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ mode
Veto |  Lower limit |  Upper limit |  Step
$\pi^{0}(\gamma\gamma_{\eta})$ |  (0.114, 0.121) |  (0.147, 0.154) |  0.001
$\eta J/\psi$ |  (3.060, 3.074) |  (3.129, 3.143) |  0.002
$\phi$ |  - |  (1.028, 1.035) |  0.001

The additive terms are related to the determination of $N_{\rm sig}$ from the
fit. Different fit conditions are tested in the simultaneous fit, and the
difference on the branching fraction is taken as systematic uncertainty. These
are introduced in the following.

The uncertainty due to the FSR is estimated by varying the factor of
$f_{\rm{FSR}}=1.39\pm 0.08\pm 0.04$ wyaqian  by $\pm 1\sigma$.

The uncertainty from the number of continuum events introduced by the $\sigma$
in Eq. (3) is estimated by changing $1/s$ to $1/s^{n}$, where $n$=1.5 or 3.
Here the value of $n$ is taken from Ref. zhaojingyi , where several
$e^{+}e^{-}\to light~{}hadron$ processes are measured, and the dependency of
the cross section on $s$ varies from 1.5 to 3. The number of continuum events
is fixed to the values estimated using the new scale factor, the changes on
the branching fraction is taken as uncertainty.

In the nominal fit, the line shapes of $B_{1,2,3}$ are extracted from MC
simulated sample or data samples using RooKeysPdf keyspdf . The uncertainty
from the corresponding line shape is estimated by replacing it with RooHistPdf
histpdf . As for $B_{4}$, it is changed from a second-order polynomial
function to the line shape extracted from the background events including
$\eta^{\prime}$ from the inclusive MC sample using RooKeysPdf.

The uncertainty due to the scale factor of the non-$\eta^{\prime}$ background
events is estimated by replacing the parameters of the background line shape
with alternative values, which are obtained from a multi-dimensional Gaussian
sampling. In the sampling, the covariance matrix from the fit to
$M_{\pi^{+}\pi^{-}\gamma}$ ($M_{\pi^{+}\pi^{-}\eta}$) is used as input. A
total of 10000 samplings are performed, thereby giving 10000 different scale
factors and numbers of non-$\eta^{\prime}$ background events. A Gaussian fit
is performed on this distribution, the obtained standard deviation of the
Gaussian distribution is regarded as the uncertainty of the number of
non-$\eta^{\prime}$ background events. The number of non-$\eta^{\prime}$
background events is varied by $\pm 1\sigma$ in the simultaneous fit to
estimate the systematic uncertainty. Additional systematic uncertainty from
non-$\eta^{\prime}$ background events is considered in the
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ mode by changing the right
sideband region from [1.0, 1.1]$~{}{\rm GeV}/c^{2}$ to [1.002, 1.098]$~{}{\rm
GeV}/c^{2}$ and [0.998, 1.102]$~{}{\rm GeV}/c^{2}$, given the fit shown in
Fig. 2(a) is not perfect on the right side of the $\eta^{\prime}$ peak.

The uncertainty from the damping function is evaluated with an alternative
damping function used by the CLEO cleo  Collaboration,
$f_{d}(E_{\gamma})=\exp(-E_{\gamma}^{2}/8\beta^{2})$. The uncertainty from
parametrizing the efficiency curve is estimated by changing the ARGUS function
into RooHistPDF. The uncertainty from the fixed mass and width of
$\eta_{c}(2S)$ is estimated by varying them by $\pm 1\sigma$. To estimate the
uncertainty from the parameterization of the mass resolution of signal, we use
RooKeysPdf to replace the double Gaussian function. The systematic uncertainty
from the mass resolution difference between data and MC simulations is
considered by varying $\delta m$ and $\delta\sigma$ by $\pm 1\sigma$ for
$\eta_{c}(2S)$. There is no such term for $\chi_{c1}$ and $\chi_{c2}$, since
the parameters of the Gaussian function are free parameters in the fit.

##  VII Upper limit of the branching fraction

Since the significance of $\eta_{c}(2S)\to K^{+}K^{-}\eta^{\prime}$ is just
above $3\sigma$, we also determine the upper limit on the product branching
fraction of $B[\psi(3686)\rightarrow\gamma\eta_{c}(2S)]\times
B[\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}]$ at 90% confidence level
(C.L.) using a Bayesian method bayesian .

Figure 5: The likelihood distribution as a function of the branching fraction
of $B(\psi(3686)\rightarrow\gamma\eta_{c}(2S)\times\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime})$. The blue solid line shows the likelihood
distribution considering additive uncertainties, and the red solid line shows
the likelihood distribution considering all uncertainties.

The systematic uncertainty effects on the upper limit of the product branching
fraction are considered in two steps. For additive terms, the largest upper
limit of the branching fraction from different fit conditions is chosen. The
multiplicative systematic uncertainties are incorporated by convolving the
likelihood distribution with a Gaussian function upper . The likelihood
distributions are shown in Fig. 5. The upper limit on the product branching
fraction of $B[\psi(3686)\rightarrow\gamma\eta_{c}(2S)]\times
B[\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}]$ at 90% C.L is determined
to be $1.14\times 10^{-6}$.

##  VIII SUMMARY

Using $(2.712\pm 0.014)\times 10^{9}$ $\psi(3686)$ events collected by the
BESIII detector, we have searched for the decay $\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime}$ in $\psi(3686)$ radiative decays with the two decay
modes $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\gamma$ and
$\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$. The branching fraction of
$B[\psi(3686)\rightarrow\gamma\eta_{c}(2S)]\times B[\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime}]$ is $\rm(6.37\pm 2.39(stat.)\pm 1.24(syst.))\times
10^{-7}$. The branching fraction of $\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime}$ is measured to be $\rm(12.24\pm 4.60(stat.)\pm
2.37(syst.)\pm 4.68(extr.))\times 10^{-4}$, using the branching fraction of
$\psi(3686)\rightarrow\gamma\eta_{c}(2S)=(5.2\pm 0.3\pm
0.5^{+1.9}_{-1.4})\times 10^{-4}$ psiptogametacp . The statistical
significance of $\eta_{c}(2S)\rightarrow K^{+}K^{-}\eta^{\prime}$ is
3.1$\sigma$. The upper limit on the product branching fraction of
$B(\psi(3686)\rightarrow\gamma\eta_{c}(2S)\times\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime})$ at 90% C.L is determined to be $1.14\times 10^{-6}$.

The branching fractions of $\chi_{c1,2}\rightarrow K^{+}K^{-}\eta^{\prime}$
are determined to be $\rm(8.47\pm 0.09(stat.)\pm 0.47(syst.))\times 10^{-4}$
and $\rm(1.53\pm 0.04(stat.)\pm 0.08(syst.))\times 10^{-4}$ for $\chi_{c1}$
and $\chi_{c2}$. These results are consistent with the previous measurement
pdg  (considering the correlation between this and the previous measurement),
but with significantly improved precision.

Figure 6: The comparison of $\rm Q$ between this study and other hadronic
decay modes in6:example ; lsx ; etacp-ppeta . Dots with error bars are data
and the vertical lines show $\rm Q=0.133$ and $\rm Q=1$.

Combining our branching fraction of $\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime}$ and that of $\eta_{c}(1S)\rightarrow
K^{+}K^{-}\eta^{\prime}$ from the BaBar Collaboration etactokketap , we
determine the branching fraction ratio to be

$\displaystyle Q$ $\displaystyle=\frac{\mathcal{B}(\eta_{c}(2S)\rightarrow
K^{+}K^{-}\eta^{\prime})}{\mathcal{B}(\eta_{c}(1S)\to
K^{+}K^{-}\eta^{\prime})}=\frac{(12.24\pm 6.98)\times 10^{-4}}{(0.87\pm
0.18)\times 10^{-2}},$ (9) $\displaystyle=0.141\pm 0.086,$

where the uncertainty includes the statistical and systematic uncertainties.
The correlated uncertainties, e.g. those from the branching fractions of the
$\eta^{\prime}$ decays are canceled. The comparison of $Q$ from this study and
in other hadronic decay modes are shown in Fig. 6. The here obtained $Q$ is
closer to the predicted value of 12% from Ref. intro4 , although the
uncertainty is large. Especially for the decay channels $\eta_{c}(2S)\to
p\bar{p}$, $\eta_{c}(2S)\to K\bar{K}\pi$,
$\eta_{c}(2S)\to\pi\pi\eta^{\prime}$, and $\eta_{c}(2S)\to\pi\pi\eta$, the
central values of $Q$ is closer to 12%. More investigations on other decay
modes and with improved precision are desired to unveil the underlying
mechanism.

##  IX 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. 2020YFA0406300, 2020YFA0406400,
2023YFA1606000; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11635010, 11735014, 11835012, 11935015, 11935016, 11935018,
11961141012, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017,
12150004; Program of Science and Technology Development Plan of Jilin Province
of China under Contract No. 20210508047RQ and 20230101021JC; 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,
U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos.
QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; The Institute
of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for
Particle Physics and Cosmology; 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.
455635585, Collaborative Research Center CRC 1044, 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

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