# First Observation of a Three-Resonance Structure in
$e^{+}e^{-}\rightarrow$Nonopen Charm Hadrons

M. Ablikim1, M. N. Achasov5,b, P. Adlarson75, X. C. Ai81, R. Aliberti36, A.
Amoroso74A,74C, M. R. An40, Q. An71,58, Y. Bai57, O. Bakina37, I.
Balossino30A, Y. Ban47,g, V. Batozskaya1,45, K. Begzsuren33, N. Berger36, M.
Berlowski45, M. Bertani29A, D. Bettoni30A, F. Bianchi74A,74C, E.
Bianco74A,74C, A. Bortone74A,74C, I. Boyko37, R. A. Briere6, A. Brueggemann68,
H. Cai76, X. Cai1,58, A. Calcaterra29A, G. F. Cao1,63, N. Cao1,63, S. A.
Cetin62A, J. F. Chang1,58, T. T. Chang77, W. L. Chang1,63, G. R. Che44, G.
Chelkov37,a, C. Chen44, Chao Chen55, G. Chen1, H. S. Chen1,63, M. L.
Chen1,58,63, S. J. Chen43, S. M. Chen61, T. Chen1,63, X. R. Chen32,63, X. T.
Chen1,63, Y. B. Chen1,58, Y. Q. Chen35, Z. J. Chen26,h, W. S. Cheng74C, S. K.
Choi11A, X. Chu44, G. Cibinetto30A, S. C. Coen4, F. Cossio74C, J. J. Cui50, H.
L. Dai1,58, J. P. Dai79, A. Dbeyssi19, R. E. de Boer4, D. Dedovich37, Z. Y.
Deng1, A. Denig36, I. Denysenko37, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding47,g, Y. Ding41, Y. Ding35, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, M. C. Du1, S. X. Du81, Z. H. Duan43, P.
Egorov37,a, Y. L. Fan76, J. Fang1,58, S. S. Fang1,63, W. X. Fang1, Y. Fang1,
R. Farinelli30A, L. Fava74B,74C, F. Feldbauer4, G. Felici29A, C. Q. Feng71,58,
J. H. Feng59, K Fischer69, M. Fritsch4, C. Fritzsch68, C. D. Fu1, J. L. Fu63,
Y. W. Fu1, H. Gao63, Y. N. Gao47,g, Yang Gao71,58, S. Garbolino74C, I.
Garzia30A,30B, P. T. Ge76, Z. W. Ge43, C. Geng59, E. M. Gersabeck67, A
Gilman69, K. Goetzen14, L. Gong41, W. X. Gong1,58, W. Gradl36, S.
Gramigna30A,30B, M. Greco74A,74C, M. H. Gu1,58, Y. T. Gu16, C. Y Guan1,63, Z.
L. Guan23, A. Q. Guo32,63, L. B. Guo42, M. J. Guo50, R. P. Guo49, Y. P.
Guo13,f, A. Guskov37,a, T. T. Han50, W. Y. Han40, X. Q. Hao20, F. A. Harris65,
K. K. He55, K. L. He1,63, F. H H.. Heinsius4, C. H. Heinz36, Y. K.
Heng1,58,63, C. Herold60, T. Holtmann4, P. C. Hong13,f, G. Y. Hou1,63, X. T.
Hou1,63, Y. R. Hou63, Z. L. Hou1, H. M. Hu1,63, J. F. Hu56,i, T. Hu1,58,63, Y.
Hu1, G. S. Huang71,58, K. X. Huang59, L. Q. Huang32,63, X. T. Huang50, Y. P.
Huang1, T. Hussain73, N Hüsken28,36, W. Imoehl28, M. Irshad71,58, J.
Jackson28, S. Jaeger4, S. Janchiv33, J. H. Jeong11A, Q. Ji1, Q. P. Ji20, X. B.
Ji1,63, X. L. Ji1,58, Y. Y. Ji50, X. Q. Jia50, Z. K. Jia71,58, H. J. Jiang76,
L. L. Jiang1, P. C. Jiang47,g, S. S. Jiang40, T. J. Jiang17, X. S.
Jiang1,58,63, Y. Jiang63, J. B. Jiao50, Z. Jiao24, S. Jin43, Y. Jin66, M. Q.
Jing1,63, T. Johansson75, X. Kui1, S. Kabana34, N. Kalantar-Nayestanaki64, X.
L. Kang10, X. S. Kang41, R. Kappert64, M. Kavatsyuk64, B. C. Ke81, A.
Khoukaz68, R. Kiuchi1, R. Kliemt14, O. B. Kolcu62A, B. Kopf4, M. K. Kuessner4,
A. Kupsc45,75, W. Kühn38, J. J. Lane67, P. Larin19, A. Lavania27, L.
Lavezzi74A,74C, T. T. Lei71,k, Z. H. Lei71,58, H. Leithoff36, M. Lellmann36,
T. Lenz36, C. Li44, C. Li48, C. H. Li40, Cheng Li71,58, D. M. Li81, F. Li1,58,
G. Li1, H. Li71,58, H. B. Li1,63, H. J. Li20, H. N. Li56,i, Hui Li44, J. R.
Li61, J. S. Li59, J. W. Li50, K. L. Li20, Ke Li1, L. J Li1,63, L. K. Li1, Lei
Li3, M. H. Li44, P. R. Li39,j,k, Q. X. Li50, S. X. Li13, T. Li50, W. D.
Li1,63, W. G. Li1, X. H. Li71,58, X. L. Li50, Xiaoyu Li1,63, Y. G. Li47,g, Z.
J. Li59, Z. X. Li16, C. Liang43, H. Liang35, H. Liang1,63, H. Liang71,58, Y.
F. Liang54, Y. T. Liang32,63, G. R. Liao15, L. Z. Liao50, Y. P. Liao1,63, J.
Libby27, A. Limphirat60, D. X. Lin32,63, T. Lin1, B. J. Liu1, B. X. Liu76, C.
Liu35, C. X. Liu1, F. H. Liu53, Fang Liu1, Feng Liu7, G. M. Liu56,i, H.
Liu39,j,k, H. B. Liu16, H. M. Liu1,63, Huanhuan Liu1, Huihui Liu22, J. B.
Liu71,58, J. L. Liu72, J. Y. Liu1,63, K. Liu1, K. Y. Liu41, Ke Liu23, L.
Liu71,58, L. C. Liu44, Lu Liu44, M. H. Liu13,f, P. L. Liu1, Q. Liu63, S. B.
Liu71,58, T. Liu13,f, W. K. Liu44, W. M. Liu71,58, X. Liu39,j,k, Y. Liu39,j,k,
Y. Liu81, Y. B. Liu44, Z. A. Liu1,58,63, Z. Q. Liu50, X. C. Lou1,58,63, F. X.
Lu59, H. J. Lu24, J. G. Lu1,58, X. L. Lu1, Y. Lu8, Y. P. Lu1,58, Z. H. Lu1,63,
C. L. Luo42, M. X. Luo80, T. Luo13,f, X. L. Luo1,58, X. R. Lyu63, Y. F. Lyu44,
F. C. Ma41, H. L. Ma1, J. L. Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M. Ma1, R.
Q. Ma1,63, R. T. Ma63, X. Y. Ma1,58, Y. Ma47,g, Y. M. Ma32, F. E. Maas19, M.
Maggiora74A,74C, S. Malde69, Q. A. Malik73, A. Mangoni29B, Y. J. Mao47,g, Z.
P. Mao1, S. Marcello74A,74C, Z. X. Meng66, J. G. Messchendorp14,64, G.
Mezzadri30A, H. Miao1,63, T. J. Min43, R. E. Mitchell28, X. H. Mo1,58,63, N.
Yu. Muchnoi5,b, Y. Nefedov37, F. Nerling19,d, I. B. Nikolaev5,b, Z. Ning1,58,
S. Nisar12,l, Y. Niu 50, S. L. Olsen63, Q. Ouyang1,58,63, S. Pacetti29B,29C,
X. Pan55, Y. Pan57, A. Pathak35, P. Patteri29A, Y. P. Pei71,58, M. Pelizaeus4,
H. P. Peng71,58, K. Peters14,d, J. L. Ping42, R. G. Ping1,63, S. Plura36, S.
Pogodin37, V. Prasad34, F. Z. Qi1, H. Qi71,58, H. R. Qi61, M. Qi43, T. Y.
Qi13,f, S. Qian1,58, W. B. Qian63, C. F. Qiao63, J. J. Qin72, L. Q. Qin15, X.
P. Qin13,f, X. S. Qin50, Z. H. Qin1,58, J. F. Qiu1, S. Q. Qu61, C. F.
Redmer36, K. J. Ren40, A. Rivetti74C, V. Rodin64, M. Rolo74C, G. Rong1,63, Ch.
Rosner19, S. N. Ruan44, N. Salone45, A. Sarantsev37,c, Y. Schelhaas36, K.
Schoenning75, M. Scodeggio30A,30B, K. Y. Shan13,f, W. Shan25, X. Y. Shan71,58,
J. F. Shangguan55, L. G. Shao1,63, M. Shao71,58, C. P. Shen13,f, H. F.
Shen1,63, W. H. Shen63, X. Y. Shen1,63, B. A. Shi63, H. C. Shi71,58, J. L.
Shi13, J. Y. Shi1, Q. Q. Shi55, R. S. Shi1,63, X. Shi1,58, J. J. Song20, T. Z.
Song59, W. M. Song35,1, Y. J. Song13, Y. X. Song47,g, S. Sosio74A,74C, S.
Spataro74A,74C, F. Stieler36, Y. J. Su63, G. B. Sun76, G. X. Sun1, H. Sun63,
H. K. Sun1, J. F. Sun20, K. Sun61, L. Sun76, S. S. Sun1,63, T. Sun1,63, W. Y.
Sun35, Y. Sun10, Y. J. Sun71,58, Y. Z. Sun1, Z. T. Sun50, Y. X. Tan71,58, C.
J. Tang54, G. Y. Tang1, J. Tang59, Y. A. Tang76, L. Y Tao72, Q. T. Tao26,h, M.
Tat69, J. X. Teng71,58, V. Thoren75, W. H. Tian59, W. H. Tian52, Y. Tian32,63,
Z. F. Tian76, I. Uman62B, S. J. Wang 50, B. Wang1, B. L. Wang63, Bo Wang71,58,
C. W. Wang43, D. Y. Wang47,g, F. Wang72, H. J. Wang39,j,k, H. P. Wang1,63, J.
P. Wang 50, K. Wang1,58, L. L. Wang1, M. Wang50, Meng Wang1,63, S. Wang39,j,k,
S. Wang13,f, T. Wang13,f, T. J. Wang44, W. Wang72, W. Wang59, W. P. Wang71,58,
X. Wang47,g, X. F. Wang39,j,k, X. J. Wang40, X. L. Wang13,f, Y. Wang61, Y. D.
Wang46, Y. F. Wang1,58,63, Y. H. Wang48, Y. N. Wang46, Y. Q. Wang1, Yaqian
Wang18,1, Yi Wang61, Z. Wang1,58, Z. L. Wang72, Z. Y. Wang1,63, Ziyi Wang63,
D. Wei70, D. H. Wei15, F. Weidner68, S. P. Wen1, C. W. Wenzel4, U. W.
Wiedner4, G. Wilkinson69, M. Wolke75, L. Wollenberg4, C. Wu40, J. F. Wu1,63,
L. H. Wu1, L. J. Wu1,63, X. Wu13,f, X. H. Wu35, Y. Wu71, Y. J. Wu32, Z.
Wu1,58, L. Xia71,58, X. M. Xian40, T. Xiang47,g, D. Xiao39,j,k, G. Y. Xiao43,
H. Xiao13,f, S. Y. Xiao1, Y. L. Xiao13,f, Z. J. Xiao42, C. Xie43, X. H.
Xie47,g, Y. Xie50, Y. G. Xie1,58, Y. H. Xie7, Z. P. Xie71,58, T. Y. Xing1,63,
C. F. Xu1,63, C. J. Xu59, G. F. Xu1, H. Y. Xu66, Q. J. Xu17, Q. N. Xu31, W.
Xu1,63, W. L. Xu66, X. P. Xu55, Y. C. Xu78, Z. P. Xu43, Z. S. Xu63, F.
Yan13,f, L. Yan13,f, W. B. Yan71,58, W. C. Yan81, X. Q. Yan1, H. J. Yang51,e,
H. L. Yang35, H. X. Yang1, Tao Yang1, Y. Yang13,f, Y. F. Yang44, Y. X.
Yang1,63, Yifan Yang1,63, Z. W. Yang39,j,k, Z. P. Yao50, M. Ye1,58, M. H. Ye9,
J. H. Yin1, Z. Y. You59, B. X. Yu1,58,63, C. X. Yu44, G. Yu1,63, J. S. Yu26,h,
T. Yu72, X. D. Yu47,g, C. Z. Yuan1,63, L. Yuan2, S. C. Yuan1, X. Q. Yuan1, Y.
Yuan1,63, Z. Y. Yuan59, C. X. Yue40, A. A. Zafar73, F. R. Zeng50, X. Zeng13,f,
Y. Zeng26,h, Y. J. Zeng1,63, X. Y. Zhai35, Y. C. Zhai50, Y. H. Zhan59, A. Q.
Zhang1,63, B. L. Zhang1,63, B. X. Zhang1, D. H. Zhang44, G. Y. Zhang20, H.
Zhang71, H. H. Zhang59, H. H. Zhang35, H. Q. Zhang1,58,63, H. Y. Zhang1,58, J.
J. Zhang52, J. L. Zhang21, J. Q. Zhang42, J. W. Zhang1,58,63, J. X.
Zhang39,j,k, J. Y. Zhang1, J. Z. Zhang1,63, Jianyu Zhang63, Jiawei Zhang1,63,
L. M. Zhang61, L. Q. Zhang59, Lei Zhang43, P. Zhang1, Q. Y. Zhang40,81,
Shuihan Zhang1,63, Shulei Zhang26,h, X. D. Zhang46, X. M. Zhang1, X. Y.
Zhang50, Xuyan Zhang55, Y. Zhang72, Y. Zhang69, Y. T. Zhang81, Y. H.
Zhang1,58, Yan Zhang71,58, Yao Zhang1, Z. H. Zhang1, Z. L. Zhang35, Z. Y.
Zhang44, Z. Y. Zhang76, G. Zhao1, J. Zhao40, J. Y. Zhao1,63, J. Z. Zhao1,58,
Lei Zhao71,58, Ling Zhao1, M. G. Zhao44, S. J. Zhao81, Y. B. Zhao1,58, Y. X.
Zhao32,63, Z. G. Zhao71,58, A. Zhemchugov37,a, B. Zheng72, J. P. Zheng1,58, W.
J. Zheng1,63, Y. H. Zheng63, B. Zhong42, X. Zhong59, H. Zhou50, L. P.
Zhou1,63, X. Zhou76, X. K. Zhou7, X. R. Zhou71,58, X. Y. Zhou40, Y. Z.
Zhou13,f, J. Zhu44, K. Zhu1, K. J. Zhu1,58,63, L. Zhu35, L. X. Zhu63, S. H.
Zhu70, S. Q. Zhu43, T. J. Zhu13,f, W. J. Zhu13,f, 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 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
6 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
7 Central China Normal University, Wuhan 430079, People’s Republic of China
8 Central South University, Changsha 410083, People’s Republic of China
9 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
10 China University of Geosciences, Wuhan 430074, People’s Republic of China
11 Chung-Ang University, Seoul, 06974, Republic of Korea
12 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
13 Fudan University, Shanghai 200433, People’s Republic of China
14 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
15 Guangxi Normal University, Guilin 541004, People’s Republic of China
16 Guangxi University, Nanning 530004, People’s Republic of China
17 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
18 Hebei University, Baoding 071002, People’s Republic of China
19 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
20 Henan Normal University, Xinxiang 453007, People’s Republic of China
21 Henan University, Kaifeng 475004, People’s Republic of China
22 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
23 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
24 Huangshan College, Huangshan 245000, People’s Republic of China
25 Hunan Normal University, Changsha 410081, People’s Republic of China
26 Hunan University, Changsha 410082, People’s Republic of China
27 Indian Institute of Technology Madras, Chennai 600036, India
28 Indiana University, Bloomington, Indiana 47405, USA
29 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
30 INFN Sezione di Ferrara, (a)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (b)University of Ferrara, I-44122, Ferrara, Italy
31 Inner Mongolia University, Hohhot 010021, People’s Republic of China
32 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
33 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
34 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica
1000000, Chile
35 Jilin University, Changchun 130012, People’s Republic of China
36 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
37 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
38 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
39 Lanzhou University, Lanzhou 730000, People’s Republic of China
40 Liaoning Normal University, Dalian 116029, People’s Republic of China
41 Liaoning University, Shenyang 110036, People’s Republic of China
42 Nanjing Normal University, Nanjing 210023, People’s Republic of China
43 Nanjing University, Nanjing 210093, People’s Republic of China
44 Nankai University, Tianjin 300071, People’s Republic of China
45 National Centre for Nuclear Research, Warsaw 02-093, Poland
46 North China Electric Power University, Beijing 102206, People’s Republic of
China
47 Peking University, Beijing 100871, People’s Republic of China
48 Qufu Normal University, Qufu 273165, 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 Xinyang Normal University, Xinyang 464000, People’s Republic of China
78 Yantai University, Yantai 264005, People’s Republic of China
79 Yunnan University, Kunming 650500, People’s Republic of China
80 Zhejiang University, Hangzhou 310027, People’s Republic of China
81 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan

###### Abstract

We report the measurement of the inclusive cross sections for
$e^{+}e^{-}$$\rightarrow$nOCH (where nOCH denotes non-open charm hadrons) with
improved precision at center-of-mass (c.m.) energies from 3.645 to 3.871 GeV.
We observe three resonances: $\mathcal{R}(3760)$, $\mathcal{R}(3780)$, and
$\mathcal{R}(3810)$ with significances of $8.1\sigma$, $13.7\sigma$, and
$8.8\sigma$, respectively. The $\mathcal{R}(3810)$ state is observed for the
first time, while the $\mathcal{R}(3760)$ and $\mathcal{R}(3780)$ states are
observed for the first time in the nOCH cross sections. Two sets of resonance
parameters describe the energy-dependent line shape of the cross sections
well. In set I [set II], the $\mathcal{R}(3810)$ state has mass $(3805.7\pm
1.1\pm 2.7)$ [$(3805.7\pm 1.1\pm 2.7)$] MeV/$c^{2}$, total width $(11.6\pm
2.9\pm 1.9)$ [$(11.5\pm 2.8\pm 1.9)$] MeV, and an electronic width multiplied
by the nOCH decay branching fraction of $(10.9\pm 3.8\pm 2.5)$ [$(11.0\pm
3.4\pm 2.5)$] eV. In addition, we measure the branching fractions
${\mathcal{B}}[{\mathcal{R}}(3760)$$\rightarrow$nOCH$]=(25.2\pm 16.1\pm
30.4)\%[(6.4\pm 4.8\pm 7.7)\%]$ and
${\mathcal{B}}[\mathcal{R}(3780)$$\rightarrow$nOCH$]=(12.3\pm 6.6\pm
8.3)\%[(10.4\pm 4.8\pm 7.0)\%]$ for the first time. The $\mathcal{R}(3760)$
state can be interpreted as an open-charm (OC) molecular state, but containing
a simple four-quark state component. The $\mathcal{R}(3810)$ state can be
interpreted as a hadrocharmonium state.

Until two decades ago, it was widely believed that hadron resonances with
masses higher than the open-charm (OC) pair thresholds decay entirely to OC
final states via the strong interaction. However, in July, 2003, BES reported
the observation of seven events of hadron resonance(s) in this mass regime
decaying to nOCH hep_ex_0307028v1 ; Psi3770AndItsRelatedPhysicsAtBESII_2019 ;
PhysRevD102_112009_Y2020 ; PRL127_082002_Y2021 . This discovery overturned the
understanding of resonance decays and opened up a new era in hadron
spectroscopy. In this Letter, we denote these resonances as $X_{\rm aboveOC}$
PhysRevD102_112009_Y2020 , which encompasses both supposed pure quark-
antiquark $c\bar{c}$ states, i.e. $\psi(3770)$, $\psi(4040)$, $\psi(4160)$,
and $\psi(4415)$, and nonpure quark-antiquark $c\bar{c}$ states (hereafter
referred to as ‘non-$c\bar{c}$’), such as four-quark states, OC-pair molecular
states, hadrocharmonium states, and hybrid charmonium states
F_E_Close_PLB578_119_y2004 ; M_B_Voloshin_JETP_23_333_y1976 ;
HQ_EPJC_72_1534_y2011 ; Voloshin_PLB66_344_y2008 . Quantum chromodynamics
(QCD) expects that the non-$c\bar{c}$ states exist in nature, and thus a
discovery of these systems would be an important validation of the QCD
predictions. The first nOCH final-state decay of a $X_{\rm aboveOC}$ resonance
to be observed after the 2002 discoveries was $J/\psi{\pi^{+}}{\pi^{-}}$
Psi3770AndItsRelatedPhysicsAtBESII_2019 ; hep_ex_0307028v1 ; PLB605_Y2005_63 ,
seen by BES-II. This final state can originate from a $c\bar{c}$ state, a
non-$c\bar{c}$ state, or a combination of both PRL127_082002_Y2021 . This
discovery stimulated strong interest in using $J/\psi\pi^{+}\pi^{-}$ or
similar final states as golden channels to probe other nOC decays from the
$X_{\rm aboveOC}$, and led to the discovery of several exotic states
X3872_PhysRevLett91_262001_Y2003 ; X4260_PRL95_142001_Y2005 ;
X4260_PRD74_091194_In2006 ; X4260_PRL96_162003_In2006 ; PRL110_252001_In2013 ;
PRL112_092001_In2013 ; PRL111_242001_In2013 ; PRL99_142002_In2007 ;
PRL98_212001_In2005 ; PRL118_092001_2017 ; PRL118_092002_2017 .

Subsequent studies of the $\psi(3770)$ resonance showed that it has a
branching fraction of about $15\%$ into nOCH final states PLB641_145_2006 ;
PRL97_121801_2006 ; PRD76_122002_2007 ; PLB659_74_2008 . This fact indicates
the contribution of some undiscovered states RongG_CPC_34_778_Y2010  with
masses around 3.773 GeV/$c^{2}$, which predominantly decay into nOCH. In 2008,
the BES-II experiment observed for the first time a double-peaked structure
named the $\mathcal{R}s(3770)$ in $e^{+}e^{-}$$\rightarrow$hadrons at c.m.
energies around 3.76 GeV bes2_prl_2structures , which is composed of two
states labeled the $\mathcal{R}(3760)$ and $\mathcal{R}(3780)$. The existence
of the $\mathcal{R}(3760)$ state was confirmed in $e^{+}e^{-}\rightarrow
J/\psi{X}$ by BESIII PRL127_082002_Y2021 . It is seen that the study of the
inclusive nOCH decays of $X_{\rm aboveOC}$ both helps the understanding of
known states, and is a sensitive probe for undiscovered resonances,
particularly non-$c\bar{c}$ states.

In this Letter, we report a measurement of the cross sections for
$e^{+}e^{-}\rightarrow$nOCH at c.m. energies from 3.645 to 3.871 GeV, studies
of $\mathcal{R}(3760)$ and $\mathcal{R}(3780)$ production and decays, and a
search for an additional resonant state in this energy region. The data
samples used in this analysis were collected at 42 c.m. energies in 2010 and
correspond to a total integrated luminosity of 75.5 pb-1.

The BESIII detector bes3  response is studied using Monte Carlo (MC) samples.
The simulations are performed using a Geant4-based geant4  software package.
Simulated samples for $q\bar{q}$ vector states (i.e. $u\bar{u}$, $d\bar{d}$,
$s\bar{s}$, and $c\bar{c}$) and their decays to hadrons are generated using
the MC event generators kkmc kkmc , evtgen BesEvtGen , and lundcharm
LundCharm_CJC_et_al_2000 . Background sources are estimated with MC samples
generated using kkmc, and the MC event generators babayaga babayaga  and
twogam twogam .

We select the inclusive nOCH events from the events with charged particles or
charged and neutral particles. In order to reject background contributions
from $e^{+}e^{-}\rightarrow({\gamma})e^{+}e^{-}$ and
$e^{+}e^{-}\rightarrow({\gamma})\mu^{+}\mu^{-}$, we require the events to have
more than two charged tracks ($N_{\rm CTrk}$), and impose the following
selection criteria for each track: (i) the distance (${R_{xy}}$) of the point
of the closest approach to the beam pipe must satisfy the condition
${R_{xy}}\leq 1.0$ cm; (ii) the polar angle $\theta$ must satisfy the
condition $|\\!\cos\theta|$$<0.93$; (iii) the momentum $p$ must be less than
$E_{\rm b}$+$0.02E_{\rm b}$$\sqrt{1+E^{2}_{\rm b}}$, where $E_{\rm b}$ is the
beam energy; (iv) the time-of-flight $t_{\rm TOF}$ must satisfy 2.0$<t_{\rm
TOF}$$<$20.0 ns and $|t_{\rm TOF}$-$t_{\rm p}|$$<2.0$ ns, where $t_{\rm p}$ is
the expected time-of-flight of protons; (v) the energy $E_{\rm EMC}$ deposited
in the electromagnetic calorimeter (EMC) must be less than $1$ GeV; (vi) the
penetration depth in the muon-chamber system must be less than $30$ cm.

For the selection of photons, we require the deposited energy of a neutral
cluster in the EMC to be greater than 25 MeV in the barrel and 50 MeV in the
end caps. We do not apply any requirements on the number of photons in the
event. To suppress electronic noise and showers unrelated to the event, we
impose the condition that the difference between the EMC time and the event
start time be within [0,700] ns. To reduce the beam-associated events (beam
interactions with gas or material), we demand that at least one charged track
or photon must point into each hemisphere of $\cos\theta<0$ and
$\cos\theta>0$. In addition, for each event, we require the total energy
($E_{\rm EMC}^{\rm tot}$) deposited in the EMC by the charged and neutral
particles to be greater than $0.28E_{\rm b}$.

Some beam-associated background sources still survive this selection. These
background sources are produced at random $z$ positions PLB641_145_2006 ,
while genuine nOCH events are produced around $z=0$, where $z$ is the distance
to the interaction point along the BESIII axis. To distinguish the nOCH events
from the background sources, we calculate the averaged $z$ ($Z_{\rm AVRG}$) of
the charged tracks in each event. Figure 1 shows the distribution of the
averaged $Z_{\rm AVRG}$ of the accepted events from the data sample collected
at $\sqrt{s}=3.779$ GeV. Using a double-Gaussian function to describe the
signal shape plus a second-order Chebychev function to parameterize the
background shape, we fit the $Z_{\rm AVRG}$ distribution of event vertices to
extract the number of nOCH candidates, $N^{\rm fit}_{\rm had}$ at each energy
point.

Figure 1:  $Z_{\rm AVRG}$ distribution of the vertices of the selected events
from the data sample collected at $\sqrt{s}=3.779$ GeV, where the dots with
error bar represent data, the solid line in blue is the best fit, and the
dashed line in red is the fitted background shape.

The background to the $N^{{\rm fit}}_{\rm had}$ distribution comes from
various sources, e.g. (i) $e^{+}e^{-}\rightarrow(\gamma)e^{+}e^{-}$, (ii)
$e^{+}e^{-}\rightarrow(\gamma)\mu^{+}\mu^{-}$, (iii)
$e^{+}e^{-}\rightarrow\gamma\gamma$, (iv)
$e^{+}e^{-}\rightarrow(\gamma)\tau^{+}\tau^{-}$, (v)
$e^{+}e^{-}\rightarrow(\gamma)e^{+}e^{-}\ell^{+}\ell^{-}$ ($\ell=e$, $\mu$ or
$\tau$), (vi) $e^{+}e^{-}\rightarrow(\gamma)e^{+}e^{-}X_{\rm had}$ (where
$X_{\rm had}$ denotes hadrons), and (vii)
$e^{+}e^{-}\rightarrow(\gamma)$$D\bar{D}$. The total amount of background
$N_{\rm b}$ at each c.m. energy $\sqrt{s}$ is determined by $N_{\rm
b}=\sum^{i=7}_{i=1}N_{{\rm b},i}$ with $N_{{\rm
b},i}=\mathcal{L}\times\sigma_{{\rm b},i}\times\eta_{i}$, where $\mathcal{L}$
is the integrated luminosity of the data sample, $\sigma_{{\rm b},i}$ is the
cross section for the $i$th background source and $\eta_{i}$ is the
probability of misidentifying a candidate from the $i$th background source as
a nOCH event, which is determined by analyzing the large background MC
samples. The cross sections $\sigma_{\rm b}$ for sources (i), (ii) and (iii)
are taken from babayaga. For source (iv), $\sigma_{\rm b}$ is calculated using
the formulae given in Refs. calc_ditau1 ; calc_ditau2 . The cross sections for
sources (v) and (vi) are taken from twogam, while that for source (vii) is
taken from the observed cross sections $\sigma^{\rm o}_{D\bar{D}}(s)$ for
$e^{+}e^{-}$$\rightarrow$$D\bar{D}$ determined using the same data samples.
For example, at $\sqrt{s}=3.7731$ GeV, $N^{\rm fit}_{\rm
had}=(35235\pm{225})$, $N_{{\rm
b},{\ell^{+}\ell^{-},\gamma\gamma}}=(1206\pm{7})$, $N_{{\rm
b},{e^{+}e^{-}{\ell^{+}\ell^{-}},{e^{+}e^{-}{\rm X}_{\rm had}}}}=(341\pm{6})$,
and $N_{{\rm b},{D\bar{D}}}=(9458\pm 186)$, where the uncertainties on the
first two backgrounds arise from the statistical uncertainties on
$\mathcal{L}$, $\sigma_{{\rm b},{i}}$, and $\eta_{i}$, and that on the third
is due to the statistical uncertainty on $\sigma^{\rm o}_{D\bar{D}}(s)$. To
provide the most conservative signal significance for the resonance searches
that are discussed below, we directly subtract $N_{{b}}$ from $N^{{\rm
fit}}_{\rm had}$ and allow the statistical uncertainty of $N_{{\rm
b},{D\bar{D}}}$ to fully contribute to the uncertainty in the difference
between $N^{{\rm fit}}_{\rm had}$ and $N_{{b}}$. This procedure yields
$N^{{\rm o}}_{\rm nOCH}=(24230\pm{292})$, where the uncertainty is statistical
and includes the contribution from the background estimates.

We determine the detection efficiency using MC simulated events for the four
components of the process $e^{+}e^{-}$$\rightarrow\,$nOCH: (i)
$e^{+}e^{-}$$\rightarrow\,$light-hadron (LH) continuum processes including
lower-mass resonances (LMRs) with masses below 2 GeV (CPLMRs) (ii)
$J/\psi$$\rightarrow\,$hadrons, (iii) $\psi(3686)$$\rightarrow\,$hadrons, and
(iv) ${\mathcal{R}s(3770)}$$\rightarrow\,$nOCH bes2_prl_2structures . We
generate simulated samples for these events with the kkmc package. These
events include initial-state radiation (ISR) and final-state radiation (FSR)
processes. For the subsequent decays of the $J/\psi$, $\psi(3686)$ and
${\mathcal{R}s(3770)}$, we use evtgen to generate the known final states,
given in Ref. PDG2022 , and use lundcharm to generate the remaining unknown
final states. The resulting selection efficiency is determined using
$\epsilon=\sum^{4}_{1}w_{i}\epsilon_{i}$, where $w_{i}$ is the number of nOCH
events simulated for the $i$th component over the total number of MC simulated
nOCH events for all four components, and $\epsilon_{i}$ is the corresponding
efficiency for selection of the nOCH events from the $i$th component. Figure 2
(top) shows the efficiencies determined at the 42 energy points.

Figure 2:  The efficiency $\epsilon$ (top) versus the c.m energy; the ISR
correction factor $[1+\delta(s)]$ (bottom) versus the c.m energy.

At $\sqrt{s}=3.7731$ GeV, the efficiency is $\epsilon=62.11\%$. The integrated
luminosity corresponding to the data collected at this energy is
$\mathcal{L}=(1831.63\pm 4.49)$ nb-1. Dividing the number $N^{{\rm o}}_{\rm
nOCH}=(24230\pm{292})$ by both the integrated luminosity and the efficiency
yields the observed nOCH cross section $\sigma^{\rm o}_{\rm nOCH}=(21.299\pm
0.262)$ nb, where the uncertainty arises from the statistical uncertainty on
the $N^{\rm o}_{\rm OCH}$, the size of the MC samples, and the statistical
uncertainty of the luminosity measurement. Similarly, we measure $\sigma^{\rm
o}_{\rm nOCH}(s)$ at the remaining 41 energy points.

Table 1 summarizes the systematic uncertainties assigned to the $\sigma^{\rm
o}_{\rm nOCH}(s)$ measurements. To determine the systematic uncertainties due
to the choice of the selection criteria, we vary each criterion from its
baseline value to an alternative setting, as given in Table 1, and assign the
resulting change in $\sigma^{\rm o}_{\rm nOCH}(s)$ as the systematic
uncertainty. Removing the requirement that tracks must point into different
hemispheres in $z$ changes $\sigma^{\rm o}_{\rm nOCH}(s)$ by $0.80\%$, which
is taken as the corresponding uncertainty associated with this possible source
of bias. To estimate the uncertainty associated with the fit to the $Z_{\rm
AVRG}$ distribution, we reperform the fit changing the background shape from a
second-order to third-order, and then from a third-order to fourth-order
Chebychev function. These result in $0.83\%$ change in the cross sections,
which is assigned as the corresponding systematic uncertainty. The uncertainty
on the number of background sources is dominated by the estimation of the two-
photon contribution, which induces a $0.75\%$ uncertainty of the $\sigma^{\rm
o}_{\rm nOCH}(s)$.

The choice of the MC generator for $e^{+}e^{-}\rightarrow{\rm nOCH}$ impacts
the selection efficiency. We recalculate the efficiency with different MC
packages and take the $1.70\%$ variation observed as the associated
uncertainty. There is an uncertainty on $\sigma^{\rm o}_{\rm nOCH}(s)$ of
$1.00\%$ arising from the corresponding uncertainty on the luminosity. Adding
these contributions in quadrature yields a total systematic uncertainty of
$2.89\%$ on $\sigma^{\rm o}_{\rm nOCH}(s)$. This total does not include an
energy-dependent uncertainty on $\sigma^{\rm o}_{\rm nOCH}(s)$, caused by the
$1.43\%$ uncertainty on the $\sigma^{\rm o}_{D\bar{D}}(s)$ shape, which is
accounted for when considering the systematic uncertainties on the fitted
parameter values, as discussed below.

Table 1: Systematic uncertainties on $\sigma^{\rm o}_{\rm nOCH}(s)$. Source | Variation range | Uncertainty (%)
---|---|---
${R_{xy}}$ cut | [0.9, 1.1] | 0.20
cos$\theta$ cut | [0.875, 0.930] | 0.40
$p<$$E_{\rm b}$+0.02$E_{\rm b}$$\sqrt{1+E_{\rm b}}$ | [nominal value, $\infty$] | 0.01
$\rm t_{TOF}$ cut and ${\rm t_{TOF}}$-${\rm t_{p}}$ | [0, $\infty$] and [2,$\infty$] | 0.09
$E_{\rm EMC}$ cut | [1, $\infty$] | 0.39
Penetration depth | [30, $\infty$] | 0.04
$E_{\rm EMC}^{\rm tot}$ cut | [0,0.28] | 0.34
$N_{\rm CTrk}$ cut | [2,3] | 1.45
Different hemisphere in $z$ |  | 0.80
Fitting $Z_{\rm AVRG}$ distribution |  | 0.83
$N_{\rm b}$ |  | 0.75
MC signal model |  | 1.70
Integrated luminosity |  | 1.00
Total |  | 2.89

To investigate whether the $\mathcal{R}(3760)$ and $\mathcal{R}(3780)$ states
decay to nOCH and if a new resonance $\mathcal{R}$ exists in this energy
region, we perform a least-$\chi^{2}$ fit to the nOCH cross sections. The
dressed nOCH cross section is modeled by:

$\sigma^{\rm D}_{\rm nOCH}(s^{\prime})=\sigma^{\rm D}_{\rm
LH}(s^{\prime})+\sigma^{\rm D}_{J/\psi}(s^{\prime})+\sigma^{\rm
D}_{{\mathcal{R}}_{{\mathcal{S}}\rm up{3680}}}(s^{\prime}),$ (1)

where $s^{\prime}=s(1-x)$, $x$ is the radiative-photon energy fraction,
$\sigma^{\rm D}_{\rm LH}(s^{\prime})$, $\sigma^{\rm D}_{J/\psi}(s^{\prime})$
and $\sigma^{\rm D}_{{\mathcal{R}}_{{\mathcal{S}}\rm up{\rm
3680}}}(s^{\prime})$ are the cross sections of $e^{+}e^{-}$$\rightarrow\,$LH,
$J/\psi$$\rightarrow$hadrons, and ${\mathcal{R}}_{{\mathcal{S}}\rm
up{3680}}$$\rightarrow$nOCH, respectively. Here
${\mathcal{R}}_{{\mathcal{S}}\rm up{3680}}$ indicates the states with masses
above 3.680 GeV/$c^{2}$. The cross sections for CPLMRs are taken to be
$\sigma^{\rm D}_{\rm LH}(s^{\prime})$=$f\sigma^{\rm
B}_{\mu^{+}\mu^{-}}(s^{\prime})+\sigma^{\rm D}_{\rm LMRs}(s^{\prime})$, where
$\sigma^{\rm B}_{\mu^{+}\mu^{-}}(s^{\prime})$ is the Born cross section for
continuum $e^{+}e^{-}$$\rightarrow$$\mu^{+}\mu^{-}$ production, $f$ is a free
parameter, $f\sigma^{\rm B}_{\mu^{+}\mu^{-}}(s^{\prime})$ gives the cross
section for continuum $e^{+}e^{-}$$\rightarrow$hadrons production
PRL97_121801_2006 , and $\sigma^{\rm D}_{\rm LMRs}(s^{\prime})$ is the cross
section for the production of LMRs decaying into LH, which is determined using
the zeroth-order cross sections PDG2022  multiplied by the vacuum-polarization
correction factor $\frac{1}{|1-\Pi(s)|^{2}}$ Structure_Function ;
PRD74_054012_Y2006  at energies below 2 GeV. The cross sections for $J/\psi$
and ${\mathcal{R}}_{{\mathcal{S}}\rm up{3680}}$ decaying into nOCH are,
respectively, taken as $\sigma^{\rm
D}_{J/\psi}(s^{\prime})$=$|A_{J/\psi}(s^{\prime})|^{2}$ and $\sigma^{\rm
D}_{{\mathcal{R}}_{{\mathcal{S}}\rm
up{3680}}}(s^{\prime})$=$|A_{\psi(3686)}(s^{\prime})$+$\sum^{3}_{1}A_{k}e^{i\phi_{k}}(s^{\prime})|^{2}$,
where $k=1,2,3$ are for the $\mathcal{R}(3760)$ state, $\mathcal{R}(3780)$
state, and $\mathcal{R}$, respectively, and $\phi_{k}$ are relative phases.

In the above formulations, $A_{\mathcal{S}}$ are the generic decay amplitudes
for these states, which are parameterized by
$A_{\mathcal{S}}(s^{\prime})={\sqrt{12\pi{\Gamma^{ee}_{\mathcal{S}}\Gamma^{\rm
tot}_{\mathcal{S}}{\mathcal{B}(\mathcal{S}\rightarrow{\rm
nOCH})}}}}/{[(s^{\prime}-M^{2}_{\mathcal{S}})+i(\Gamma^{\rm
tot}_{\mathcal{S}}M_{\mathcal{S}})]}$, in which $\mathcal{S}$ stands for the
$J/\psi$, $\psi(3686)$, $\mathcal{R}(3760)$, and $\mathcal{R}(3780)$ states,
as well as resonance $\mathcal{R}$; $M_{\mathcal{S}}$, $\Gamma^{\rm
tot}_{\mathcal{S}}$ and $\Gamma^{ee}_{\mathcal{S}}$ are, respectively, the
mass, total and electronic widths of the $\mathcal{S}$, and
$\mathcal{B}(\mathcal{S}\rightarrow{\rm nOCH})$ is the decay branching
fraction of the $\mathcal{S}$. For the $\mathcal{R}(3780)$ state, the total
width is set to be energy dependent, as in Ref. PRL97_121801_2006 .

In the fit, the masses, the total widths, the electronic widths, and the
hadronic decay branching fractions of $J/\psi$ and $\psi(3686)$ resonances are
fixed to those given in Ref. PDG2022 . The $\mathcal{R}(3780)$ mass and total
width are fixed to $(3781.0\pm 1.3\pm 0.5)$ MeV/$c^{2}$ and $(19.3\pm 3.1\pm
0.1)$ MeV bes2_prl_2structures , respectively. The remaining parameters are
left as free parameters in the fit.

The observed cross section is described by $\sigma^{\rm O}_{\rm
nOCH}(s)=\int_{0}^{1-(4m_{\pi}^{2}/s)}dx~{}{\sigma}^{\rm D}_{\rm
nOCH}[s(1-x)]~{}{\mathcal{F}(x,s)}$, where $\mathcal{F}(x,s)$ is the sampling
function Structure_Function ; PRL97_121801_2006 , and $m_{\pi}$ is the pion
mass. The cross section as a function of energy is determined by a fit to
$\sigma^{\rm o}_{\rm nOCH}(s)$ at the 42 energy points. Using $\sigma^{\rm
D}_{\rm nOCH}(s)$ obtained from the parameter values, we determine the ISR
correction factors $[1+\delta(s)]=\sigma^{\rm O}_{\rm nOCH}(s)/\sigma^{\rm
D}_{\rm nOCH}(s)$ at the 42 energy points, which are shown in Fig. 2 (bottom).
Dividing the observed nOCH cross section $\sigma^{\rm o}_{\rm nOCH}(s)$ by
$(1+\delta(s))$ yields the observed dressed nOCH cross section $\sigma^{\rm
d}_{\rm nOCH}(s)$, where the lower case superscripts are used to distinguish
the measured from the predicted (upper case superscripts) quantities.

The circles with error bars in Fig. 3 show the $\sigma^{\rm d}_{\rm nOCH}(s)$
measurements, where the uncertainties are statistical. Using the same fit
procedure described above, we fit $\sigma^{\rm d}_{\rm nOCH}(s)$ with the
function $\sigma^{\rm D}_{\rm nOCH}(s)$ given in Eq. (1), with $x$ fixed to
zero. The fit converges at two sets of acceptable parameter values. We denote
these as Result I and Result II. Table 2 summarizes the parameter values,
where the first uncertainties are from the fit to the $\sigma^{\rm d}_{\rm
nOCH}(s)$, and the second are systematic. The two sets of results have the fit
quality of $\chi^{2}/{\rm ndof}$ of $22.1/31$, and $22.0/31$, respectively. As
the mass of the fitted resonance is close to 3810 MeV, we denote it as
$\mathcal{R}(3810)$. The cross section described by Result I is superimposed
on the fit results in Fig. 3. Also shown are the fit results including one
contribution to $\sigma^{\rm D}_{{\mathcal{R}}_{{\mathcal{S}}\rm
up{3680}}}(s)$ included at a time. The measured mass and total width of the
$\mathcal{R}(3760)$ state are consistent within $1.8\sigma$ and $0.9\sigma$,
respectively, with those measured by the BES-II experiment
bes2_prl_2structures .

Figure 3:  The dressed cross sections for $e^{+}e^{-}\rightarrow$nOCH, showing also the Result-I fit (line 1) and the contributions to this fit including separate contributions to $\sigma^{\rm D}_{{\mathcal{R}}_{{\mathcal{S}}\rm up{3680}}}(s)$ (lines 2–6).  Table 2: Results of the fit to the cross sections for $e^{+}e^{-}\rightarrow$nOCH showing the values of the mass $M_{k}$ [MeV/$c^{2}$], total width $\Gamma^{\rm tot}_{k}$ [MeV], the product of electronic width ($\Gamma^{ee}_{k}$) and nOCH branching fraction (${\mathcal{B}}_{k}$) $\Gamma^{ee}_{k}{\mathcal{B}}_{k}$ [eV], and relative phase $\phi_{k}$ [degree], where $k$ represents $\mathcal{R}(3760)$, $\mathcal{R}(3780)$, and $\mathcal{R}(3810)$. ${\mathcal{B}_{k}}$ is the branching fractions [$\%$] for $\mathcal{R}(3760)$ and $\mathcal{R}(3780)$ decays into nOCH and OC. Parameters | Result I | Result II
---|---|---
$M_{\mathcal{R}(3760)}$ | $3739.9\pm 4.2\pm 2.6$ | $3739.7\pm 3.9\pm 2.6$
$\Gamma^{\rm tot}_{\mathcal{R}(3760)}$ | $23.9\pm 8.2\pm 4.8$ | $22.5\pm 8.3\pm 4.5$
$\Gamma^{ee}_{\mathcal{R}(3760)}{{\mathcal{B}}_{\mathcal{R}(3760)}}$ | $46.8\pm 29.9\pm 25.1$ | $11.9\pm 9.0\pm 6.4$
$\phi_{\mathcal{R}(3760)}$ | $228\pm 52\pm 58$ | $113\pm 51\pm 29$
$\Gamma^{ee}_{\mathcal{R}(3780)}{{\mathcal{B}}_{\mathcal{R}(3780)}}$ | $29.9\pm 16.1\pm 3.7$ | $25.3\pm 11.6\pm 3.1$
$\phi_{\mathcal{R}(3780)}$ | $82\pm 126\pm 17$ | $250\pm 119\pm 52$
${\rm M}_{\mathcal{R}(3810)}$ | $3805.7\pm 1.1\pm 2.7$ | $3805.7\pm 1.1\pm 2.7$
${\rm\Gamma^{\rm tot}_{\mathcal{R}(3810)}}$ | $11.6\pm 2.9\pm 1.9$ | $11.5\pm 2.8\pm 1.9$
$\Gamma^{ee}_{\mathcal{R}(3810)}{{\mathcal{B}}_{\mathcal{R}(3810)}}$ | $10.9\pm 3.8\pm 2.5$ | $11.0\pm 3.4\pm 2.5$
$\phi_{\mathcal{R}(3810)}$ | $52\pm 149\pm 25$ | $215\pm 148\pm 103$
$f$ | $2.28\pm 0.01\pm 0.07$ | $2.28\pm 0.01\pm 0.07$
${\mathcal{B}}[\mathcal{R}(3760)\rightarrow$${\rm nOCH}]$ | $25.2\pm 16.1\pm 30.4$ | $6.4\pm 4.8\pm 7.7$
${\mathcal{B}}[\mathcal{R}(3780)\rightarrow$${\rm nOCH}]$ | $12.3\pm 6.6\pm 8.3$ | $10.4\pm 4.8\pm 7.0$
${\mathcal{B}}[\mathcal{R}(3760)\rightarrow$${\rm OC}]$ | $74.8\pm 16.1\pm 30.4$ | $93.6\pm 4.8\pm 7.7$
${\mathcal{B}}[\mathcal{R}(3780)\rightarrow$${\rm OC}]$ | $87.7\pm 6.6\pm 8.3$ | $89.6\pm 4.8\pm 7.0$

To estimate the systematic uncertainties of the fitted parameter in Table 2,
we vary the values of the $\sigma^{\rm d}_{\rm nOCH}(s)$, the values of the
$\sigma^{\rm o}_{D\bar{D}}(s)$, and the fixed parameters by $\pm 1\sigma$,
refit the $\sigma^{\rm d}_{\rm nOCH}(s)$, and take the difference between the
refitted parameter value and the baseline fit result as the corresponding
systematic uncertainty. The estimation of the systematic uncertainty due to
the uncertainty of the c.m. energies is similar to that described in Ref.
PRL127_082002_Y2021 . Adding these uncertainties in quadrature yields the
total systematic uncertainty.

Dividing the measured values for
$\Gamma^{ee}_{\mathcal{R}(3760)}{\mathcal{B}}_{\mathcal{R}(3760)}$ and
$\Gamma^{ee}_{\mathcal{R}(3780)}{\mathcal{B}}_{\mathcal{R}(3780)}$ in Table 2
by $\Gamma^{ee}_{\mathcal{R}(3760)}=(186\pm 201\pm 8){~{}\rm eV}$
bes2_prl_2structures  and $\Gamma^{ee}_{\mathcal{R}(3780)}=(243\pm 160\pm
9)~{}{\rm eV}$ bes2_prl_2structures , respectively, yields the nOCH branching
fractions for the decays $\mathcal{R}(3760)\rightarrow{\rm nOCH}$ and
$\mathcal{R}(3780)\rightarrow{\rm nOCH}$, which are shown in Table 2. Their
corresponding OC branching fractions are also presented in Tab. 2. These nOCH
branching fractions of $\mathcal{R}(3780)$ decays are in good agreement with
$\mathcal{B}[\psi(3770)\rightarrow$non-$D\bar{D}$$)]=(15.1\pm 5.6\pm 1.8)\%$
PLB641_145_2006 ; PRL97_121801_2006 ; PRD76_122002_2007 ; PLB659_74_2008
measured by the BES-II experiment.

By removing the $\mathcal{R}(3810)$ from the $\sigma^{\rm
D}_{{\mathcal{R}}_{{\mathcal{S}}\rm up{3680}}}(s)$ as discussed above, the
$\chi^{2}/{\rm ndof}$ of the fit changes from $22.1/31$ to $116.1/36$,
indicating that the significance of the $\mathcal{R}(3810)$ signal is
$8.8\sigma$. Similarly, the significance of the $\mathcal{R}(3760)$ signal is
determined to be $8.1\sigma$ by comparing the difference of $\chi^{2}/{\rm
ndof}$ relative to the number degrees of freedom with and without including
the $\mathcal{R}(3760)$ component in the fit. A similar procedure is applied
to determine the $\mathcal{R}(3780)$ significance, which is $13.7\sigma$.
These significances include the systematic uncertainties.

The charmonium model Eithtin_chmonuim_prd1978  predicts only the $1^{3}D_{1}$
state existing in the c.m. energy range from 3.733 to 3.870 GeV, which is
generally assumed to be the $\psi(3770)$, and so the $\mathcal{R}(3760)$ and
$\mathcal{R}(3810)$ state are presumably two non-$c\bar{c}$ states. The
$\mathcal{R}(3760)$ state can be explained as a $p$-wave resonance of a four-
quark ($c\bar{c}q\bar{q}$) state A_De_Rujule_PRL38_317_Year1977 ;
Rafe_Hyam_Schindler_SLAC219 . It can be thought of either as an OC molecular
state, or as a four-quark bound state Rafe_Hyam_Schindler_SLAC219 . Reference
S_Dubynskiy_M_B_Voloshin_PRD78_116014_Y2008  interprets $\mathcal{R}(3760)$ as
a possible molecular OC threshold resonance. The most salient features of the
$\mathcal{R}(3760)$ state is that its mass, which is $(3739.9\pm 4.2\pm 2.6)$
MeV/$c^{2}$ for Result I [$3739.7\pm 3.9\pm 2.6)$ MeV/$c^{2}$ for Result II],
is just at the $D^{+}D^{-}$ threshold $(3739.3\pm 0.1)$ MeV/$c^{2}$, and its
branching fraction is ${\mathcal{B}}[\mathcal{R}(3760)\rightarrow{\rm
OC}]=(74.8\pm 16.1\pm 30.4)\%$
[${\mathcal{B}}[\mathcal{R}(3760)\rightarrow{\rm OC}]=(93.6\pm 4.8\pm
7.7)\%$]. These experimental facts can lead one naturally to interpret the
$\mathcal{R}(3760)$ state as an OC pair molecular state, but containing a
simple four-quark state component. As no signal for the decay
$\mathcal{R}(3810)$$\rightarrow$$D\bar{D}$ is observed in the cross sections
for $e^{+}e^{-}$$\rightarrow$$D\bar{D}$ at c.m. energies around 3.810 GeV
BaBar_PRD76_111105_Y2007 ; Belle_PRL77_011103_Y2008 , and the
$\mathcal{R}(3810)$ mass $(3805.7\pm 1.1\pm 2.7)$ MeV/$c^{2}$ for Result I
[$(3805.7\pm 1.1\pm 2.7)$ MeV/$c^{2}$ for Result II] is exactly at the
${h_{c}}\pi^{+}\pi^{-}$ threshold $(3804.5\pm 0.1)$ MeV/$c^{2}$, the
$\mathcal{R}(3810)$ state can be interpreted as a hadrocharmonium resonance
Voloshin_PLB66_344_y2008 .

In summary, we have measured the inclusive nOCH cross sections of
$e^{+}e^{-}$$\rightarrow$ nOCH with improved precision at c.m. energies from
3.645 to 3.871 GeV. We observe three resonances: $\mathcal{R}(3760)$,
$\mathcal{R}(3780)$, and $\mathcal{R}(3810)$ in the energy-dependent line
shape of the nOCH cross sections with significances of $8.1\sigma$,
$13.7\sigma$, and $8.8\sigma$, respectively. The $\mathcal{R}(3760)$ and
$\mathcal{R}(3780)$ states are observed for the first time in the nOCH cross
sections, while the $\mathcal{R}(3810)$ state is observed for the first time
with mass $(3805.7\pm 1.1\pm 2.7)$ [$(3805.7\pm 1.1\pm 2.7)$] MeV/$c^{2}$,
total width $(11.6\pm 2.9\pm 1.9)$ [$(11.5\pm 2.8\pm 1.9)$] MeV, and the
product of the electronic width and the nOCH decay branching fraction
$(10.9\pm 3.8\pm 2.5)$ [$(11.0\pm 3.4\pm 2.5)$] eV for Result I [Result II].
In addition, for the first time, we measure
${\mathcal{B}}[\mathcal{R}(3760)$$\rightarrow$nOCH]=$(25.2\pm 16.1\pm
30.4)\%[(6.4\pm 4.8\pm 7.7)\%]$ and
${\mathcal{B}}[\mathcal{R}(3780)$$\rightarrow$nOCH]=$(12.3\pm 6.6\pm
8.3)\%[(10.4\pm 4.8\pm 7.0)\%]$. The $\mathcal{R}(3760)$ state can be
interpreted as an OC pair molecular state, but containing a simple four-quark
state component. The $\mathcal{R}(3810)$ state can be interpreted as a
hadrocharmonium state Voloshin_PLB66_344_y2008 .

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 No. 2009CB825204, No.
2020YFA0406300, No. 2020YFA0406400; National Natural Science Foundation of
China (NSFC) under Contracts No. 10935007, No. 11635010, No. 11735014, No.
11835012, No. 11935015, No. 11935016, No. 11935018, No. 11961141012, No.
12022510, No. 12025502, No. 12035009, No. 12035013, No. 12061131003, No.
12192260, No. 12192261, No. 12192262, No. 12192263, No. 12192264, No.
12192265, No. 12221005, No. 12225509, No. 12235017; the Chinese Academy of
Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for
Excellence in Particle Physics (CCEPP); CAS Key Research Program of Frontier
Sciences under Contracts No. QYZDJ-SSW-SLH003, No. QYZDJ-SSW-SLH040; 100
Talents Program of CAS; The CAS Research Program under Code No. Y41G1010Y1;
The CAS Other Research Program under Code No. Y129360; The Institute of
Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle
Physics and Cosmology; ERC under Contract No. 758462; European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement under Contract No. 894790; German Research Foundation DFG under
Contracts No. 443159800, No. 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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