# Search for $C$-even states decaying to $D_{s}^{\pm}D_{s}^{*\mp}$ with masses
between $4.08$ and $4.32$ $\rm GeV/{\it c}^{2}$

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. Ding34, Y. Ding40, 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. Fang59, J. Fang1,58, 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, Z. L. Guan22, 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, T. Hussain73, F. Hölzken3, N Hüsken27,35, 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. Li47, C. Li43, 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,
Ke Li1, L. J Li1,63, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,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. Z. Li59, Xiaoyu Li1,63, Y. G.
Li46,h, Z. J. Li59, Z. X. Li15, Z. Y. Li78, C. Liang42, H. Liang71,58, H.
Liang1,63, Y. F. Liang54, Y. T. Liang31,63, G. R. Liao14, L. Z. Liao50, 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. H. Liu53, Fang Liu1, Feng Liu6, G. M.
Liu56,j, H. Liu38,k,l, H. B. Liu15, H. M. Liu1,63, Huanhuan Liu1, 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. Liu39, X. Liu38,k,l, 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, 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. 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. Shangguan55, 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, Meng Wang1,63, N. Y. Wang63, S.
Wang38,k,l, S. Wang12,g, T. Wang12,g, T. J. Wang43, W. Wang72, W. Wang59, 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, 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, Tao Yang1, Y. Yang12,g, Y. F. Yang43,
Y. X. Yang1,63, Yifan 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, L.
Yuan2, S. C. Yuan1, Y. Yuan1,63, Y. J. Yuan45, Z. Y. Yuan59, C. X. Yue39, A.
A. Zafar73, F. R. Zeng50, S. H. Zeng72, X. Zeng12,g, Y. Zeng25,i, 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. Zhang59, H. H. Zhang34, 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, Shuihan Zhang1,63,
Shulei Zhang25,i, X. D. Zhang45, X. M. Zhang1, X. Y. Zhang50, Y. Zhang72, Y.
T. Zhang80, Y. H. Zhang1,58, Y. M. Zhang39, Yan Zhang71,58, Yao Zhang1, 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, Lei Zhao71,58, Ling Zhao1,
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

###### Abstract

Six $C$-even states, denoted as $X$, with quantum numbers $J^{PC}=0^{-+}$,
$1^{\pm+}$, or $2^{\pm+}$, are searched for via the $e^{+}e^{-}\to\gamma
D_{s}^{\pm}D_{s}^{*\mp}$ process using $(1667.39\pm 8.84)~{}\mathrm{pb}^{-1}$
of $e^{+}e^{-}$ collision data collected with the BESIII detector operating at
the BEPCII storage ring at center-of-mass energy of $\sqrt{s}=(4681.92\pm
0.30)~{}\mathrm{MeV}$. No statistically significant signal is observed in the
mass range from $4.08$ to $4.32~{}\mbox{\,GeV}/c^{2}$. The upper limits of
$\sigma[e^{+}e^{-}\to\gamma X]\cdot\mathcal{B}[X\to D_{s}^{\pm}D_{s}^{*\mp}]$
at a $90\%$ confidence level are determined.

##  I I. INTRODUCTION

The charmonium system is a good laboratory for studying the nonperturbative
behavior of quantum chromodynamics (QCD). In the past two decades, a few
resonant structures have been observed that cannot be explained by a simple
mesonic ($q\bar{q}$) or baryonic ($qqq$) configuration, which has resulted in
great experimental and theoretical interest.

Charmonium states with mass above the open-charm threshold are expected to
decay dominantly to open-charm final states ozione ; ozitwo ; ozithree . In
Ref. Higher_Charmonia , the masses and total widths of the $\chi_{c2}(3P)$,
$\chi_{c1}(3P)$, and $\eta_{c2}(2D)$ are calculated according to the
nonrelativistic potential model or the relativized Godfrey-Isgur model
Higher_Charmonia1 . The open-charm strong decay widths are determined assuming
harmonic oscillator wave functions and the ${}^{3}{P}_{0}$ decay model. These
states are predicted to have sizeable decay widths to the
$D_{s}^{\pm}D_{s}^{*\mp}$ final state, as listed in Table 1. None of them have
so far been observed in experiment. In an amplitude analysis of the $B^{+}\to
J/\psi\phi K^{+}$ decay, a $1^{++}$ state was observed in the $\phi J/\psi$
final state lhca ; lhcb ; lhcc , the average mass of which,
$4286^{+8}_{-9}~{}{\rm MeV}/c^{\rm 2}$, is close to that of the predicted
$\chi_{c1}(3P)$. In the hybrid meson configuration, several states are
predicted by lattice QCD QED1 ; QED2 . The strong decay widths of these hybrid
states are computed based on the constituent gluon model constituent_gluon .
The $D_{\rm s}^{\pm}D_{\rm s}^{*\mp}$ mode is expected to be the leading decay
channel for the $X(4217)$ with $J^{PC}=1^{-+}$ and the $X(4279)$ with
$J^{PC}=0^{-+}$, where the numbers in the brackets denote the mass of the
hybrid states from theoretical predictions. The resonant parameters of the two
states are summarized in Table 1.

Many of the reported structures have masses close to the thresholds of open-
charm meson pairs. For example, the $X(3872)$ x3872 , $Z_{c}(3900)^{\pm}$
zc3900a ; zc3900b ; zc3900c , $Z_{c}(4020)^{\pm}$ zc4020a ; zc4020b , and
$Z_{cs}(3985)$ zcs3985a ; zcs3985b are near the thresholds of
$D^{0}\bar{D}^{*0}$, $D\bar{D}^{*}$, $D^{*}\bar{D}^{*}$, and
$D_{s}^{(*)}\bar{D}^{(*)}$, respectively. In the meson molecule picture, Ref.
heavy-antiheavy predicted abundant hadron spectra near open-charm thresholds
by solving the Bethe-Salpeter equation. In addition, a virtual state, referred
to as the $X(4080)$ in this paper, with a mass close to the
$D_{s}^{\pm}D_{s}^{*\mp}$ threshold, is expected to decay predominantly to
$D_{s}^{\pm}D_{s}^{*\mp}$. Its quantum numbers are $J^{PC}=1^{++}$ and its
total width is assumed to be $5~{}\mathrm{MeV}$.

Table 1: Masses ($M$), total widths ($\Gamma$), partial widths
($\Gamma_{D_{s}^{\pm}D_{s}^{*\mp}}$), and quantum numbers of the six $C$-even
states.

State |  $M$ ($\mathrm{MeV}/c^{2}$) |  $\Gamma$ ($\rm MeV$) |  $\Gamma_{D_{s}^{\pm}D_{s}^{*\mp}}$ ($\mathrm{MeV}$) | $J^{PC}$
---|---|---|---|---
$\eta_{c2}(2D)$ Higher_Charmonia | $4158$ | $111$ | $18$ | $2^{-+}$
$\chi_{c1}(3P)$ Higher_Charmonia | $4271$ | $39$ | $9.7$ | $1^{++}$
$\chi_{c2}(3P)$ Higher_Charmonia | $4317$ | $66$ | $11$ | $2^{++}$
$X(4080)$ heavy-antiheavy | $4082.55$ | $5$ | - | $1^{++}$
$X(4217)$ QED1 ; QED2 ; constituent_gluon | $4217$ | $6$ | $6$ | $1^{-+}$
$X(4279)$ QED1 ; QED2 ; constituent_gluon | $4279$ | $110$ | $34$ | $0^{-+}$

It has been shown that $C$-even states can be produced in $e^{+}e^{-}$
collisions accompanied by photon emission. For example, the
$e^{+}e^{-}\to\gamma X(3872)$ gamma3872 and $\gamma\chi_{c1,2}(1P)$ gammachicj
processes have been observed by the BESIII experiment. In this paper, a study
of $e^{+}e^{-}\to\gamma D_{s}^{\pm}D_{s}^{*\mp}$ is presented to search for
the $C$-even structures mentioned above. Throughout this paper, the charge
conjugated mode is implied. This analysis is performed using a data sample
collected at center-of-mass energy $\sqrt{s}=(4681.92\pm 0.08\pm
0.29)~{}\mathrm{MeV}$ with an integrated luminosity of $(1667.39\pm 0.21\pm
8.84)~{}\mathrm{pb}^{-1}$ BESIII:lum . The $D_{s}^{*}$ meson is reconstructed
using its radiative decay into the $\gamma D_{s}$ final state. The $D_{s}$
meson is reconstructed with the subsequent $D_{s}\to K^{+}K^{-}\pi$ and
$K^{0}_{S}K$ decays, with $K^{0}_{S}\to\pi^{+}\pi^{-}$. The two decay modes of
the $D_{s}$ are combined into four decay chains (DCs):
DC-I: $D_{s}^{+}D_{s}^{-}\to(K^{+}K^{-}\pi^{+})(K^{+}K^{-}\pi^{-})$,
DC-II: $D_{s}^{+}D_{s}^{-}\to(K^{+}K^{-}\pi^{+})(K^{0}_{S}K^{-})$,
DC-III: $D_{s}^{+}D_{s}^{-}\to(K^{0}_{S}K^{+})(K^{+}K^{-}\pi^{-})$, and
DC-IV: $D_{s}^{+}D_{s}^{-}\to(K^{0}_{S}K^{+})(K^{0}_{S}K^{-})$.

##  II II. BESIII DETECTOR AND MONTE CARLO SIMULATION

The BESIII detector bes records symmetric $e^{+}e^{-}$ collisions provided by
the BEPCII storage ring Yu:IPAC2016-TUYA01 , which operates with a peak
luminosity of $1\times 10^{33}$ cm-2s-1 in the center-of-mass energy range
from $2.0$ to $4.95$ GeV Ablikim:2019hff . 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 at
$1~{}\mbox{\,GeV}$. The EMC measures photon energies with a resolution of
$2.5\%$ ($5\%$) at $1~{}\mbox{\,GeV}$ in the barrel (end cap) region. The time
resolution in the TOF barrel region is 68 ps, while that in the end cap region
is 60 ps etofa ; etofb ; etofc .

Simulated events produced with a geant4-based geant4 Monte Carlo (MC) package,
which includes the geometric description of the BESIII detector and the
detector response, are used to determine detection efficiencies and to
estimate background contributions. The simulation models the beam energy
spread and initial state radiation (ISR) in the $e^{+}e^{-}$ annihilations
with the generator kkmc ref:kkmca ; ref:kkmcb . The generic MC samples include
the production of open-charm processes, the ISR production of vector
charmonium(-like) states, and the continuum processes implemented in kkmc
ref:kkmca ; ref:kkmcb . The known decay modes are modelled with evtgen
ref:evtgen ; ref:evtgena using branching fractions taken from the Particle
Data Group (PDG) PDG2018 , and the remaining unknown charmonium decays are
modelled with lundcharm ref:lundcharma ; ref:lundcharmb . Final state
radiation for charged final state particles is incorporated using the photos
package ref:photos .

The signal processes $e^{+}e^{-}\to\gamma X,~{}X\to D_{s}^{\pm}D_{s}^{*\mp}$
and their subsequent decays are modelled with evtgen ref:evtgen ; ref:evtgena
. The $D_{s}\to K^{+}K^{-}\pi$ decay is generated according to the amplitude
models reported in Refs. ref:da1 ; ref:da2 . The input cross section line
shape of $e^{+}e^{-}\to\gamma X$ is assumed to follow those of the
$\psi(\rm{4660})$, and is parameterized using a Breit-Wigner function with
parameters taken from the PDG PDG2018 . Two groups of samples are generated
for the signal processes, Group-I and Group-II.Group-I is produced with the
predicted masses and widths of the $X$, while Group-II is with the masses
ranging from $4080$ to $4320~{}\mathrm{MeV}/c^{2}$ in steps of
$40~{}\mathrm{MeV}/c^{2}$ and widths varying from a set of values:
$5,~{}10,~{}50,~{}80,\rm{\rm and}~{}100~{}\mathrm{MeV}$. The quantum numbers
$J^{PC}$ are listed in Table 1.

Samples of $e^{+}e^{-}\to(\gamma_{\rm ISR})D_{s}^{(*)\pm}D_{s}^{*\mp}$ events
are generated to estimate the background contamination. They are modelled
using helicity-amplitude models in evtgen ref:evtgen ; ref:evtgena . The
production cross section is taken from a previous BESIII measurement
ref:dsstardsstar .

##  III III. EVENT SELECTION AND BACKGROUND STUDY

A full reconstruction method is used to reconstruct the signal process $\gamma
D_{s}^{\pm}D_{s}^{*\mp}$($\to\gamma D_{s}^{\mp}$), $D_{s}\to K^{+}K^{-}\pi$ or
$K^{0}_{S}(\to\pi^{+}\pi^{-})K$. 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 symmetry axis of the MDC
(defined as the $z$-axis). For charged tracks not originating from $K^{0}_{S}$
decays, the distance of the closest approach to the interaction point (IP)
must be less than $10~{}\mathrm{cm}$ along the $z$-axis ($V_{z}$), and less
than $1~{}\mathrm{cm}$ in the transverse plane ($V_{xy}$). Particle
identification (PID) for charged tracks combines measurements of the energy
deposited (${\rm d}E/{\rm d}x$) in the MDC and the flight time in the TOF to
form a likelihood for each hadron hypothesis. Tracks are identified as kaons
(pions) when the kaon (pion) hypothesis has larger probability. Tracks without
PID information are rejected.

Each $K^{0}_{S}$ candidate is reconstructed from two oppositely charged tracks
satisfying $|V_{z}|<20~{}\mathrm{cm}$ and $|\cos\theta|<0.93$. The two charged
tracks are assigned as $\pi^{+}\pi^{-}$ without imposing PID criteria. They
are constrained to originate from a common vertex and are required to have an
invariant mass
$(M_{\pi^{+}\pi^{-}})~{}\rm{within~{}the~{}interval}~{}(0.478,~{}0.518)$
GeV/$c^{2}$. The $\chi^{2}$ of the vertex fit is required to be less than
$100$. A secondary vertex fit is performed to ensure that the $K^{0}_{S}$
momentum points back to the IP. The decay length of the $K^{0}_{S}$ candidate
is required to be greater than twice the vertex resolution.

Photon candidates are identified using isolated showers not associated with
charged tracks in the EMC. The deposited energy of each shower must be greater
than $25~{}\mathrm{MeV}$ in the barrel region ($|\cos\theta|<0.8$) and greater
than $50~{}\mathrm{MeV}$ in the end cap regions ($0.86<|\cos\theta|<0.92$).
The difference between the EMC time and the event start time is required to be
within $[0,~{}700]~{}\mathrm{ns}$ to suppress electronic noise and showers
unrelated to the event.

The selected $K^{\pm}$, $\pi^{\pm}$, and $K^{0}_{S}$ candidates in the event
are combined to reconstruct $D_{s}\to K^{+}K^{-}\pi$ or $D_{s}\to K^{0}_{S}K$.
The $D_{s}$ candidates are kept if the invariant masses of the $K^{+}K^{-}\pi$
or $K^{0}_{S}K$ systems, denoted as $M_{D_{s}}$, are within a $\pm
150~{}\mathrm{MeV}/c^{2}$ mass window around the nominal $D_{s}$ mass. The
$D_{s}^{+}D_{s}^{-}$ pairs are selected requiring that each $K^{\pm}$,
$\pi^{\pm}$, and $K^{0}_{S}$ candidate is used in the event reconstruction at
most once. If there is more than one combination in an event, the one with
$(M_{D_{s}^{+}}+M_{D_{s}^{-}})/2$ closest to the nominal mass of the $D_{s}$
meson PDG2018 is chosen.

In addition to the $D_{s}^{+}D_{s}^{-}$ pair, each signal candidate is
required to have at least two extra photons. A six-constraint (6C) kinematic
fit is performed, where the four-momentum of the final state particles is
constrained to that of the initial $e^{+}e^{-}$ system and the masses of the
reconstructed $D_{s}^{+}$ and $D_{s}^{-}$ are constrained to their nominal
masses. If more than two photons are found or there is more than one pair of
$D_{s}^{+}D_{s}^{-}$ decaying in different DCs in an event, the combination
with the minimum $\chi^{2}_{\rm 6C}$ is retained and $\chi^{2}_{\rm 6C}$ is
further required to be less than 200, where $\chi^{2}_{\rm 6C}$ is the
$\chi^{2}$ from the 6C kinematic fit. To study the background contributions
from non-$D_{s}$ processes, a four-constraint kinematic (4C) fit without the
two $D_{s}$ mass constraints is applied and the four-momenta of the final
state particles are computed using the momenta determined by the 4C kinematic
fit, unless otherwise specified.

To suppress background contributions from non-$D_{s}$ processes, a $D_{s}$
mass window of $|M_{D_{s}}-m_{D_{s}}|<21~{}\mathrm{MeV}/c^{2}$, which is about
three times the mass resolution, is applied, where $m_{D_{s}}$ is the nominal
$D_{s}$ mass. The two selected photon candidates and the two $D_{s}$ mesons
can form four combinations of $\gamma D_{s}$. The one with invariant mass
($M_{\gamma D_{s}})$ closest to the nominal mass of the $D_{s}^{*}$ is
retained for further analysis. The invariant mass $M^{\prime}_{\gamma
D_{s}}=(M_{\gamma D_{s}}-M_{D_{s}}+m_{D_{s}})$ is required to be within the
interval $(m_{D_{s}^{*}}\pm 27)~{}\mathrm{MeV}/c^{2}$, where $m_{D_{s}^{*}}$
is the nominal $D_{s}^{*\pm}$ mass taken from the PDG PDG2018 . The variable
$M_{\gamma D_{s}}$ formed with the unused photon and $D_{s}$ candidates is
required to be outside of this same mass window. The photon not from the
$D_{s}^{*}$ decay is labelled $\gamma_{1}$ and its recoil mass spectrum,
$M_{\gamma_{1}}^{\rm rec}$, is used to extract the signal yields.
$M_{\gamma_{1}}^{\rm rec}$ is calculated by $\sqrt{(E_{\rm
cms}-E_{\gamma_{1}})^{2}-(\vec{P}_{\rm cms}-\vec{P}_{\gamma_{1}})^{2}}$, where
($E_{\rm cms},\vec{P}_{\rm cms}$) and ($E_{\gamma_{1}},\vec{P}_{\gamma_{1}}$)
are the four-momenta of the initial state BESIII:lum and $\gamma_{1}$,
respectively. After imposing all the selection requirements, the dominant
background processes are $e^{+}e^{-}\to(\gamma_{\rm
ISR})D_{s}^{\pm}D_{s}^{*\mp}$ and $(\gamma_{\rm
ISR})D_{s}^{*\pm}D_{s}^{*\mp}$. The contributions from other background
processes are less than $10\%$ of the total backgrounds. Events with a
$D^{(*)}\bar{D}^{(*)}$ pair in the final state are found to be linearly
distributed in the recoil mass spectrum of $\gamma_{1}$ by analyzing the
generic MC samples.

##  IV IV. SIGNAL YIELD EXTRACTION

The $X$ signal yields are extracted using an unbinned maximum likelihood fit
to the $M_{\gamma_{1}}^{\rm rec}$ spectrum. The events from the four DCs are
combined due to limited statistics. The fit probability density function (PDF)
contains four components: signal ($S$), $e^{+}e^{-}\to(\gamma_{\rm
ISR})D_{s}^{\pm}D_{s}^{*\mp}$ background ($B_{1}$), $e^{+}e^{-}\to(\gamma_{\rm
ISR})D_{s}^{*\pm}D_{s}^{*\mp}$ background ($B_{2}$), and other remaining
backgrounds ($B_{3}$). The function used in the fit is defined as

$\displaystyle f_{\rm sum}=N_{\rm sig}\cdot S+N_{\rm bkg1}\cdot B_{1}+N_{\rm
bkg2}\cdot B_{2}+N_{\rm bkg3}\cdot B_{3}.$

Here, $N_{\rm sig}$ and $N_{\rm bkg(1,2,3)}$ represent the numbers of signal
events and background events, and they are free parameters in the fit.

The signal PDF is a Breit-Wigner function (BW) with mass $M$ and width
$\Gamma$, convoluted with a Double Gaussian and a Gaussian function. The
Double Gaussian function, with the parameters
$m_{2},m_{1},~{}\sigma_{2},\sigma_{1}$, and $f$, describes the mass resolution
and mass shift in the simulation, and the Gaussian function accounts for the
mass resolution difference between data and simulation. To get the parameters
of the Double Gaussian function, the $M_{\gamma_{1}}^{\rm
rec}-M_{\gamma_{1}}^{\rm truth}$ distribution from the MC samples is fitted,
where $M_{\gamma_{1}}^{\rm truth}$ is the recoil mass spectrum of the
$\gamma_{1}$ distribution at generator level. The parameters of the Gaussian
function are obtained by using a control sample, $e^{+}e^{-}\to
D_{s}^{\pm}D_{s}^{*\mp}$. The selection of the control sample is similar to
that of the signal sample but only one photon is required instead of two. The
$D_{s}$ and $D_{s}^{*}$ mass distributions from data are fitted with the line
shape from MC samples convoluted with Gaussian functions. The parameters of
the Gaussian function representing the difference of the mass resolutions
between data and MC simulation are then used as the parameters of the Gaussian
function in the extraction of the $X$ signal yield. The $B_{1(2)}$ line shape
is determined from the $e^{+}e^{-}\to(\gamma_{\rm
ISR})D_{s}^{\pm}D_{s}^{*\mp}$ ($e^{+}e^{-}\to(\gamma_{\rm
ISR})D_{s}^{*\pm}D_{s}^{*\mp}$) MC simulation and $B_{3}$ is a polynomial
function.

Figure 1 shows the fit results where the masses and widths of the $X$ states
are fixed to the predictions listed in Table 1. The signal yields are
summarized in Table 2. Since no events are retained near the $D_{s}D_{s}^{*}$
mass threshold, the number of $X(4080)$ singal yields is determined using
$N_{\rm obs}-N_{\rm bkg(1,2,3)}$, where $N_{\rm obs}=0$ is the number of
observed events, $N_{\rm bkg(1,2,3)}$ are the numbers of background events
obtained from the fit with $N_{\rm sig}$ set to zero. Here, all the numbers
are counted within $[4.068,~{}4.098]~{}{\rm GeV}/c^{2}$, which is about
$3\sigma$ around the $X(4080)$ nominal mass. The statistical significance is
calculated from the logarithmic likelihoods,
$-2\ln(\mathcal{L}_{0}/\mathcal{L}_{\rm max})$, where $\mathcal{L}_{\rm max}$
and $\mathcal{L}_{0}$ are the maximum likelihood values with and without the
signal component in the fits. The difference in the number of degrees of
freedom ($\Delta\mathrm{n.d.f}=1$) has been taken into account. The largest
statistical significance among the $X$ is $3.3\sigma$ and corresponds to the
$X(4279)$, which has a large predicted width. The upper limits of the number
of signal events are determined using a Bayesian method ref:Bayesianmethod
with a prior. The likelihood distribution, $\mathcal{L}(x)$, is determined by
repeating the fit to the $M_{\gamma_{1}}^{\rm rec}$ spectrum with a series of
fixed values for the number of signal events. The upper limits at 90%
confidence level (C.L.) are determined from $\int_{0}^{N_{\rm sig}^{\rm
UL}}L(x)dx/\int_{0}^{\infty}L(x)dx=0.9$, as listed in Table 2.

Fig. 1: Fits to the $M_{\gamma_{1}}^{\rm rec}$ spectrum with masses and widths
fixed to those of the $\eta_{c2}(2D)$, $\chi_{c1}(3P)$, $\chi_{c2}(3P)$,
$X(4080)$, $X(4217)$, and $X(4279)$ as listed in Table 1. The dots with error
bars are the data distribution and the blue solid lines are the best fit
results.

In addition, upper limits for the signal cross sections as a function of the
$X$ masses in the range $[4.08,~{}4.32]~{}\mathrm{GeV}/c^{2}$ are also
provided as listed in Table 1. From studies of simulation samples (Group-II),
it is found the parameters $m_{2}-m_{1}$, $\sigma_{2}/\sigma_{1}$, and $f$ of
the resolution function are very similar for various widths and $J^{PC}$. The
parameters are found to be

$\displaystyle m_{2}-m_{1}$ $\displaystyle=(15.6\pm 0.2)~{}\rm{MeV/c^{2}},$
(1) $\displaystyle\sigma_{2}/\sigma_{1}$ $\displaystyle=3.01\pm 0.02,$
$\displaystyle f$ $\displaystyle=0.692\pm 0.005,$

using the MC sample with $\Gamma=50~{}\mathrm{MeV}$ and $J^{PC}=0^{-+}$, and
they are fixed in the latter fit to $M_{\gamma_{1}}^{\rm rec}$. The $m_{1}$
and $\sigma_{1}$ dependence on the mass of the $X$ state are shown in Fig. 2.
A parameterization of the resolution, as a function of $m_{\rm 1}$ and
$\sigma_{\rm 1}$ vs. $M_{\rm X}$ distributions are fitted with a first order
polynomial functions. The $m_{\rm 1}$ and $\sigma_{\rm 1}$ dependences for
other widths and $J^{PC}$ are determined separately. The trends are similar to
those shown in Fig. 2. The mass resolutions of the $X$ in the region
$[4.08,~{}4.32]~{}{\rm GeV}/c^{2}$ are determined from these fitted function.
The upper limits of the number of signal events as a function of the $X$ mass
for different $X$ widths and quantum numbers are shown in Fig. 3.

Fig. 2: Distributions of $m_{1}$ (upper) and $\sigma_{1}$ (lower) for
$\Gamma=50~{}\mathrm{MeV}$ and $J^{PC}$ = $0^{-+}$ according to MC simulation
studies. The solid lines are the fit results. Fig. 3: Upper limits on the
number of $X$ signal events as a function of $X$ mass for different quantum
numbers ($J^{PC}=0^{-+}$, $1^{-+}$, $1^{++}$, $2^{-+}$, and $2^{++}$) and
widths of the $X$ ($\Gamma=5$, $10$, $50$, $80$, and $100~{}\mathrm{MeV}$).

##  V V. CROSS SECTION CALCULATION

The Born cross section of $e^{+}e^{-}\to\gamma X$ times the branching fraction
of $X\to D_{s}^{\pm}D_{s}^{*\mp}$ is calculated as

$\displaystyle\sigma\cdot\mathcal{B}[X\to
D_{s}^{\pm}D_{s}^{*\mp}]=\frac{N_{\rm sig}}{\mathcal{L}_{\rm
int}f^{r}f^{v}\mathcal{B}[D_{s}^{*}\to\gamma D_{s}]\bar{\epsilon}},$ (2)

where $N_{\rm sig}$ is the number of signal events obtained from the fit;
$\bar{\epsilon}=\sum_{j,k=1}^{2}\epsilon_{jk}\mathcal{B}_{j}\mathcal{B}_{k}$
represents the average detection efficiency, where $\epsilon_{jk}$ is the
detection efficiency for $e^{+}e^{-}\to\gamma X,X\to
D_{s}^{\pm}D_{s}^{*\mp},D_{s}^{*\mp}\to\gamma D_{s}^{\mp},$ $D_{s}^{+}\to
j,D_{s}^{-}\to k$; and $\mathcal{B}_{j}$ ($\mathcal{B}_{k}$) is the branching
fraction of $D_{s}^{+}\to j$ ($D_{s}^{-}\to k$). Here $j$ and $k$ are the
first and second decay modes of the $D_{s}$. The ISR correction factor
($f^{r}=(1+\delta)$) is calculated following the procedure as described in
Refs. ref:kkmca ; ref:kkmcb . The vacuum polarization correction factor is
$f^{v}=|1+\Pi|^{2}$ Jegerlehner:2011mw . $\mathcal{B}[{D_{s}^{*}\to\gamma
D_{s}]}$ is the branching fraction of $D_{s}^{*}\to\gamma D_{s}$. The results
are summarized in Table 2.

Table 2: The upper limits of Born cross section of $e^{+}e^{-}\to\gamma X$ times the branching fraction of $X\to D_{s}^{\pm}D_{s}^{*\mp}$ at 90% C.L. for each candidate $C$-even state. Here, $f^{r}$, $f^{v}$, $N_{\rm sig}$, $N_{\rm sig}^{\rm UL}$, and $\bar{\epsilon}$ are the ISR correction factor, the vacuum polarization correction factor, the number of signal events, the upper limit on the number of signal events, and the average detection efficiency, respectively. “$\sigma^{\mathrm{UL}}\cdot\mathcal{B}$ with sys.” stands for the upper limits of the cross section times the branching ratio with systematic uncertainties. “Significance” represents statistical significance. | $\eta_{c2}(2D)$ | $\chi_{c1}(3P)$ | $\chi_{c2}(3P)$ | $X(4080)$ | $X(4217)$ | $X(4279)$
---|---|---|---|---|---|---
$f^{r}$ | $1.06$ | $1.06$ | $1.06$ | $1.06$ | $1.06$ | $1.06$
$f^{v}$ | $1.05$ | $1.05$ | $1.05$ | $1.05$ | $1.05$ | $1.05$
$N_{\rm sig}^{\rm UL}$ | $6.7$ | $16.3$ | $18.7$ | $2.4$ | $7.6$ | $19.6$
$N_{\rm sig}$ | $-5.6^{+4.2}_{-3.2}$ | $9.8^{+5.2}_{-4.4}$ | $13.0^{+4.5}_{-3.9}$ | $-0.9^{+0.3}_{-0.2}$ | $2.3^{+3.0}_{-2.4}$ | $13.8^{+4.5}_{-3.8}$
Significance $(\sigma)$ | $1.3$ | $2.6$ | $3.1$ | $-$ | $0.9$ | $3.3$
$\bar{\epsilon}~{}(10^{-4})$ | $3.73$ | $3.48$ | $3.26$ | $4.21$ | $3.50$ | $3.11$
$\sigma^{\rm UL}\cdot\mathcal{B}$ with sys. (pb) | $13.3$ | $36.3$ | $45.5$ | $4.1$ | $15.7$ | $51.7$

Figure 4 illustrates the upper limits of $\sigma^{\rm UL}\cdot\mathcal{B}[X\to
D_{s}^{\pm}D_{s}^{*\mp}]$ under different assumptions of the mass, width, and
$J^{PC}$ of the $X$ states after considering all the systematic uncertainties.
Detailed descriptions of the systematic uncertainties are provided in the
subsequent sections. The detection efficiency as a function of $M$ for
different width and $J^{PC}$ assumptions of the $X$ state is determined from
Group-II MC simulations. The distributions are fitted with a first order
polynomial function, as shown in Fig. 5.

Fig. 4: Upper limits on $\sigma[e^{+}e^{-}\to\gamma X]\cdot\mathcal{B}[X\to
D_{s}^{\pm}D_{s}^{*\mp}]$ as a function of $X$ mass for different quantum
numbers ($J^{PC}=0^{-+}$, $1^{-+}$, $1^{++}$, $2^{-+}$, and $2^{++}$) and
widths of the $X$ ($\Gamma=5$, $10$, $50$, $80$, and $100~{}\mathrm{MeV}$)
after considering the systematic uncertainties. Fig. 5: Distributions of the
efficiencies for the four channels for different $J^{PC}$ and width
assumptions. The lines are the fit results.

In the calculation of the upper limits of Born cross sections displayed in
Fig. 4, the detection efficiencies are extracted from the fitted curves and
combined into the average detection efficiency $\bar{\epsilon}$.

##  VI VII. SYSTEMATIC UNCERTAINTY ESTIMATION

The systematic uncertainties on $\sigma\cdot\mathcal{B}$ are mainly from the
luminosity measurement, the detection efficiency, the quoted branching
fractions, the ISR correction factor, and the fit models. The uncertainty from
the vacuum polarization correction is negligible. A summary of systematic
uncertainties is given in Table 3. The total systematic uncertainty is taken
as a quadrature sum of each contribution.

Table 3: Summary of systematic uncertainties (in $\%$) for the Born cross section measurements. Source | $J^{PC}=0^{-+}$ | $1^{-+}$ | $1^{++}$ | $2^{-+}$ | $2^{++}$
---|---|---|---|---|---
Integrated luminosity | $1.0$ | $1.0$ | $1.0$ | $1.0$ | $1.0$
PID | $5.7$ | $5.7$ | $5.7$ | $5.7$ | $5.7$
Tracking | $5.7$ | $5.7$ | $5.7$ | $5.7$ | $5.7$
Photon reconstruction | $2.0$ | $2.0$ | $2.0$ | $2.0$ | $2.0$
$K_{S}^{0}$ reconstruction | $0.4$ | $0.4$ | $0.4$ | $0.4$ | $0.4$
Kinematic fit | $0.7$ | $0.7$ | $0.8$ | $0.8$ | $0.7$
$D_{s}$ mass window | $0.3$ | $0.3$ | $0.3$ | $0.3$ | $0.3$
$D_{s}^{*}$ mass window | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
ISR correction factor | $24.6$ | $24.4$ | $23.8$ | $23.5$ | $23.7$
Efficiency curve parameters | $0.7$ | $0.8$ | $0.7$ | $0.8$ | $0.7$
Efficiency curve parameterization | $0.7$ | $1.4$ | $1.5$ | $1.0$ | $0.9$
$\mathcal{B}(D_{s}^{*}\to\gamma D_{s})$ | $0.8$ | $0.8$ | $0.8$ | $0.8$ | $0.8$
$\mathcal{B}(D_{s}\to K^{+}K^{-}\pi)$ | $3.3$ | $3.3$ | $3.3$ | $3.3$ | $3.3$
$\mathcal{B}(D_{s}\to K^{0}_{S}K)$ | $0.8$ | $0.8$ | $0.8$ | $0.8$ | $0.8$
$\mathcal{B}(K^{0}_{S}\to\pi^{+}\pi^{-})$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
Total | $26.3$ | $26.1$ | $25.5$ | $25.2$ | $25.4$

(a) The integrated luminosity is measured using Bhabha scattering events, with
an uncertainty of $1.0\%$ BESIII:lum .

(b) The difference in the tracking efficiencies between data and MC simulation
is estimated to be $1.0\%$ per track Ablikim:092009 ; Ablikim:2011kv ;
error_ppb2pi . There are six charged tracks in DC-I, four tracks in DC-II and
DC-III, and two tracks in DC-IV. The detection efficiencies of the four
channels are changed by $\pm m\%$ to recalculate the Born cross section, where
$m$ is the number of charged tracks in the channel. The larger difference to
the nominal result, $5.7\%$, is taken as the systematic uncertainty.

(c) The systematic uncertainty from PID is $1.0\%$ per $\pi/K$ track
Ablikim:092009 ; Ablikim:2011kv ; error_ppb2pi . This uncertainty is
calculated using the same method as used for tracking, and $5.7\%$ is taken as
the systematic uncertainty.

(d) The systematic uncertainty of the photon reconstruction efficiency is
$1.0\%$ per photon ref:gamma based on studies of the processes
$\chi_{0,2}\to\pi^{0}\pi^{0},\eta\eta$. Because there are two photons in the
final state, $2.0\%$ is assigned as the systematic uncertainty.

(e) The $K^{0}_{S}$ reconstruction efficiency includes the geometric
acceptance, tracking efficiency, and the candidate selection. The difference
in the $K^{0}_{S}$ reconstruction efficiency between data and MC simulation is
estimated with the control samples $J/\psi\rightarrow
K^{*}(892)^{\pm}K^{\mp},K^{*}(892)^{\pm}\rightarrow K^{0}_{S}\pi^{\pm}$, and
$J/\psi\to\phi K^{0}_{S}K^{\pm}\pi^{\mp}$ ks_err and is determined to be
1.2$\%$. There is one $K^{0}_{S}$ candidate in DC-II and DC-III and two
$K^{0}_{S}$ candidates in DC-IV. The detection efficiencies of the channels
with $K^{0}_{S}$ in the final state are varied by $\pm n\cdot 1.2\%$ and the
resulting detection efficiencies are used to calculate the Born cross section.
Here $n$ is the number of $K^{0}_{S}$ in the channel. The larger difference in
the resulting Born cross section, $0.4\%$, is assigned as the systematic
uncertainty for the $K^{0}_{S}$ reconstruction.

(f) The systematic uncertainty caused by the 6C kinematic fit is estimated by
correcting the helix parameters of the simulated charged tracks ref:kfit . The
result after correction is taken as the nominal one, and the difference
between the results with and without this correction, $0.7\%$ to $0.8\%$, is
taken as the systematic uncertainty.

(g) The systematic uncertainties from the $D_{s}$ and $D_{s}^{*}$ mass window
requirements are due to the mass resolution difference between data and MC. It
is estimated by correcting the mass resolution in MC to improve the
consistency between data and MC simulation. The correction is applied by
smearing the MC distribution with a Gaussian function, and the parameters of
the Gaussian function are obtained from the control sample of $e^{+}e^{-}\to
D_{s}^{\pm}D_{s}^{*\mp},~{}D_{s}^{*\mp}\to\gamma D_{s}^{\mp}$. The mean and
standard deviation of the Gaussian function are $(0.3\pm
0.6)~{}\mathrm{MeV}/c^{2}$ and $(3.7\pm 0.9)~{}\mathrm{MeV}/c^{2}$ for the
$D_{s}$, $(1.5\pm 1.1)~{}\mathrm{MeV}/c^{2}$ and $(0.5\pm
4.2)~{}\mathrm{MeV}/c^{2}$ for the $D_{s}^{*}$. The selection efficiency
difference with or without the smearing procedure is taken as the systematic
uncertainty, which is $0.3\%$ and $0.1\%$ for the $D_{s}$ and $D_{s}^{*}$,
respectively.

(h) In the nominal results, the detection efficiency and the ISR correction
factor are obtained by assuming the Born cross section of $e^{+}e^{-}\to\gamma
X$ follows the line shape of the $\psi(4660)$, which is a well-established
vector state in this energy region. Different shapes of the input cross
sections result in different probabilities of generating events with the ISR
photon. Assuming the branching fraction of $\psi(4660)\to\gamma X$ is small,
the input cross section line shape is changed to $E_{\gamma}^{3}/s$ (similar
to the pure continuum process), where $E_{\gamma}$ is energy of the radiative
photon and $s$ is the square of the center-of-mass energy. There is a large
difference between the two input line shapes, which leads to large differences
in the detection efficiencies and the ISR correction factors. The changes in
the resultant Born cross sections are taken as the systematic uncertainty from
the ISR correction factor ref:weight .

(i) For the calculation of the Born cross section of $e^{+}e^{-}\to\gamma
X,X\to D_{s}^{\pm}D_{s}^{*\mp}$ with different $M$ under assumed $\Gamma$ and
$J^{PC}$, the detection efficiency is determined from the fitted curves. The
uncertainty in the efficiency induced by the fitted parameters is
$\Delta\epsilon_{jk}=\sqrt{(m\cdot\Delta a_{jk})^{2}+(\Delta
b_{jk})^{2}+2\cdot m\cdot\mathrm{cov}(a_{jk},b_{jk})}$, where $a_{jk}$ and
$b_{jk}$ are the parameters of the polynomial function, $\Delta a_{jk}$ and
$\Delta b_{jk}$ are the corresponding uncertainties, and
$\mathrm{cov}(a_{jk},b_{jk})$ is the covariance between $a_{jk}$ and $b_{jk}$
given by the corresponding fit. The detection efficiency, $\epsilon_{jk}$, is
varied by $\pm\Delta\epsilon_{jk}$, and the difference on the efficiency is
taken as the systematic uncertainty.

(j) The systematic uncertainty from the parameterization of the efficiency
curves is estimated by changing the fitted function from a first order to a
second order polynomial function. The average detection efficiency is
recalculated with parameters read from the alternative fitted curves. The
largest difference is taken as the systematic uncertainty.

(k) The branching fractions of $D_{s}\to K^{+}K^{-}\pi$, $D_{s}\to
K^{0}_{S}K$, $D_{s}^{*}\to\gamma D_{s}$, and $K^{0}_{S}\to\pi^{+}\pi^{-}$ are
taken from the PDG PDG2018 , and their uncertainties are propagated to the
Born cross section measurement. The systematic uncertainty caused by the
branching fractions of $D_{s}\to K^{+}K^{-}\pi$, $D_{s}\to K^{0}_{S}K$,
$D_{s}^{*}\to\gamma D_{s}$, and $K^{0}_{S}\to\pi^{+}\pi^{-}$ are $3.3\%$,
$0.8\%$, $0.8\%$, and $0.1\%$, respectively.

(l) The fit-related systematic uncertainties are estimated by varying the fit
models. The largest upper limit on the number of signal events is taken as the
upper limit of the Born cross section with the systematic uncertainty
considered. The line shapes of the background contributions from
$e^{+}e^{-}\to(\gamma_{\rm ISR})D_{s}^{\pm(*)}D_{s}^{*\mp}$ are determined
from the MC simulations with the latest precise measurements as input. The
uncertainties are found to be negligible. The uncertainty from the description
of the remaining background events is estimated by changing the constant
polynomial function to the line shape extracted from generic MC simulation.
The value listed under $1^{++}$ is determined from $\chi_{c1}(3P)$. In the
determination of the upper limit of the number of signal events with different
$M$ under assumed width and $J^{PC}$, the mass resolutions are read from the
fit curves shown in Fig. 2. The systematic uncertainties from
$m_{1}-m_{2},\sigma_{2}/\sigma_{1}$, and $f$ are considered by varying the
parameters within their uncertainties, while the systematic uncertainties from
$m_{1}~{}{\rm and}~{}\sigma_{1}$, are considered by varying $\Delta m_{1}$ and
$\Delta\sigma_{1}$. The largest upper limit on the number of signal events is
taken. Here $\Delta m_{1}$ and $\Delta\sigma_{1}$ are calculated using the
same formula as used to calculate $\Delta\epsilon_{jk}$. To take into account
the systematic uncertainties in the upper limit calculation, the likelihood
distributions are convolved with Gaussian functions as

$L^{\prime}(x)=\int_{0}^{1}L(x;N_{\rm
sig}\bar{\epsilon}/\hat{\bar{\epsilon}})\exp[-\frac{(\bar{\epsilon}-\hat{\bar{\epsilon}})^{2}}{2\sigma_{\rm
sys.}^{2}}]d\bar{\epsilon},$

where $\hat{\bar{\epsilon}}$ is the nominal average detection efficiency, and
$\sigma_{\rm sys.}$ is the systematic uncertainty ref:Bayesianmethod ; sigma .
The upper limit of the Born cross section times the branching fraction with
fixed masses and widths are listed in Table 2. The upper limits of
$\sigma[e^{+}e^{-}\to\gamma X]\cdot\mathcal{B}[X\to D_{s}^{\pm}D_{s}^{*\mp}]$
as a function of $X$ mass after considering all the systematic uncertainties
are displayed in Fig. 4.

##  VII VIII. SUMMARY

In summary, based on a data sample collected at $\sqrt{s}=(4681.92\pm 0.08\pm
0.29)~{}\mathrm{MeV}$ with an integrated luminosity of $(1667.39\pm 0.21\pm
8.84)~{}\mathrm{pb}^{-1}$, a search for $C$-even states is performed via the
process $e^{+}e^{-}\to\gamma D_{s}^{\pm}D_{s}^{*\mp}$. No statistically
significant signals are observed. Upper limits on $\sigma[e^{+}e^{-}\to\gamma
X]\cdot\mathcal{B}[X\to D_{s}^{\pm}D_{s}^{*\mp}]$ ($X$= $\eta_{c2}(2D)$,
$\chi_{c1}(3P)$, $\chi_{c2}(3P)$, $X(4080)$, $X(4217)$, and $X(4279)$) at
$90\%$ C.L. are determined to be $13.3$ pb, $36.3$ pb, $45.5$ pb, $4.1$ pb,
$15.7$ pb, and $51.7$ pb, respectively. Upper limits with other mass and width
assumptions are also determined and shown in Fig. 4. These upper limits can be
used to constrain theoretical models.

##  VIII 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;
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; 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; 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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