# Observation of Three Charmonium-like States with $J^{PC}=1^{--}$ in
$e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ Process

M. Ablikim1, M. N. Achasov13,b, P. Adlarson73, R. Aliberti34, A.
Amoroso72A,72C, M. R. An38, Q. An69,56, Y. Bai55, O. Bakina35, I.
Balossino29A, Y. Ban45,g, V. Batozskaya1,43, K. Begzsuren31, N. Berger34, M.
Bertani28A, D. Bettoni29A, F. Bianchi72A,72C, E. Bianco72A,72C, J. Bloms66, A.
Bortone72A,72C, I. Boyko35, R. A. Briere5, A. Brueggemann66, H. Cai74, X.
Cai1,56, A. Calcaterra28A, G. F. Cao1,61, N. Cao1,61, S. A. Cetin60A, J. F.
Chang1,56, T. T. Chang75, W. L. Chang1,61, G. R. Che42, G. Chelkov35,a, C.
Chen42, Chao Chen53, G. Chen1, H. S. Chen1,61, M. L. Chen1,56,61, S. J.
Chen41, S. M. Chen59, T. Chen1,61, X. R. Chen30,61, X. T. Chen1,61, Y. B.
Chen1,56, Y. Q. Chen33, Z. J. Chen25,h, W. S. Cheng72C, S. K. Choi10A, X.
Chu42, G. Cibinetto29A, S. C. Coen4, F. Cossio72C, J. J. Cui48, H. L. Dai1,56,
J. P. Dai77, A. Dbeyssi19, R. E. de Boer4, D. Dedovich35, Z. Y. Deng1, A.
Denig34, I. Denysenko35, M. Destefanis72A,72C, F. De Mori72A,72C, B. Ding64,1,
X. X. Ding45,g, Y. Ding39, Y. Ding33, J. Dong1,56, L. Y. Dong1,61, M. Y.
Dong1,56,61, X. Dong74, S. X. Du79, Z. H. Duan41, P. Egorov35,a, Y. L. Fan74,
J. Fang1,56, S. S. Fang1,61, W. X. Fang1, Y. Fang1, R. Farinelli29A, L.
Fava72B,72C, F. Feldbauer4, G. Felici28A, C. Q. Feng69,56, J. H. Feng57, K
Fischer67, M. Fritsch4, C. Fritzsch66, C. D. Fu1, Y. W. Fu1, H. Gao61, Y. N.
Gao45,g, Yang Gao69,56, S. Garbolino72C, I. Garzia29A,29B, P. T. Ge74, Z. W.
Ge41, C. Geng57, E. M. Gersabeck65, A Gilman67, K. Goetzen14, L. Gong39, W. X.
Gong1,56, W. Gradl34, S. Gramigna29A,29B, M. Greco72A,72C, M. H. Gu1,56, Y. T.
Gu16, C. Y Guan1,61, Z. L. Guan22, A. Q. Guo30,61, L. B. Guo40, R. P. Guo47,
Y. P. Guo12,f, A. Guskov35,a, X. T. H.1,61, W. Y. Han38, X. Q. Hao20, F. A.
Harris63, K. K. He53, K. L. He1,61, F. H. Heinsius4, C. H. Heinz34, Y. K.
Heng1,56,61, C. Herold58, T. Holtmann4, P. C. Hong12,f, G. Y. Hou1,61, Y. R.
Hou61, Z. L. Hou1, H. M. Hu1,61, J. F. Hu54,i, T. Hu1,56,61, Y. Hu1, G. S.
Huang69,56, K. X. Huang57, L. Q. Huang30,61, X. T. Huang48, Y. P. Huang1, T.
Hussain71, N Hüsken27,34, W. Imoehl27, M. Irshad69,56, J. Jackson27, S.
Jaeger4, S. Janchiv31, J. H. Jeong10A, Q. Ji1, Q. P. Ji20, X. B. Ji1,61, X. L.
Ji1,56, Y. Y. Ji48, Z. K. Jia69,56, P. C. Jiang45,g, S. S. Jiang38, T. J.
Jiang17, X. S. Jiang1,56,61, Y. Jiang61, J. B. Jiao48, Z. Jiao23, S. Jin41, Y.
Jin64, M. Q. Jing1,61, T. Johansson73, X. K.1, S. Kabana32, N. Kalantar-
Nayestanaki62, X. L. Kang9, X. S. Kang39, R. Kappert62, M. Kavatsyuk62, B. C.
Ke79, A. Khoukaz66, R. Kiuchi1, R. Kliemt14, L. Koch36, O. B. Kolcu60A, B.
Kopf4, M. Kuessner4, A. Kupsc43,73, W. Kühn36, J. J. Lane65, J. S. Lange36, P.
Larin19, A. Lavania26, L. Lavezzi72A,72C, T. T. Lei69,k, Z. H. Lei69,56, H.
Leithoff34, M. Lellmann34, T. Lenz34, C. Li46, C. Li42, C. H. Li38, Cheng
Li69,56, D. M. Li79, F. Li1,56, G. Li1, H. Li69,56, H. B. Li1,61, H. J. Li20,
H. N. Li54,i, Hui Li42, J. R. Li59, J. S. Li57, J. W. Li48, Ke Li1, L. J
Li1,61, L. K. Li1, Lei Li3, M. H. Li42, P. R. Li37,j,k, S. X. Li12, T. Li48,
W. D. Li1,61, W. G. Li1, X. H. Li69,56, X. L. Li48, Xiaoyu Li1,61, Y. G.
Li45,g, Z. J. Li57, Z. X. Li16, Z. Y. Li57, C. Liang41, H. Liang69,56, H.
Liang33, H. Liang1,61, Y. F. Liang52, Y. T. Liang30,61, G. R. Liao15, L. Z.
Liao48, J. Libby26, A. Limphirat58, D. X. Lin30,61, T. Lin1, B. X. Liu74, B.
J. Liu1, C. Liu33, C. X. Liu1, D. Liu19,69, F. H. Liu51, Fang Liu1, Feng Liu6,
G. M. Liu54,i, H. Liu37,j,k, H. B. Liu16, H. M. Liu1,61, Huanhuan Liu1, Huihui
Liu21, J. B. Liu69,56, J. L. Liu70, J. Y. Liu1,61, K. Liu1, K. Y. Liu39, Ke
Liu22, L. Liu69,56, L. C. Liu42, Lu Liu42, M. H. Liu12,f, P. L. Liu1, Q.
Liu61, S. B. Liu69,56, T. Liu12,f, W. K. Liu42, W. M. Liu69,56, X. Liu37,j,k,
Y. Liu37,j,k, Y. B. Liu42, Z. A. Liu1,56,61, Z. Q. Liu48, X. C. Lou1,56,61, F.
X. Lu57, H. J. Lu23, J. G. Lu1,56, X. L. Lu1, Y. Lu7, Y. P. Lu1,56, Z. H.
Lu1,61, C. L. Luo40, M. X. Luo78, T. Luo12,f, X. L. Luo1,56, X. R. Lyu61, Y.
F. Lyu42, F. C. Ma39, H. L. Ma1, J. L. Ma1,61, L. L. Ma48, M. M. Ma1,61, Q. M.
Ma1, R. Q. Ma1,61, R. T. Ma61, X. Y. Ma1,56, Y. Ma45,g, F. E. Maas19, M.
Maggiora72A,72C, S. Maldaner4, S. Malde67, A. Mangoni28B, Y. J. Mao45,g, Z. P.
Mao1, S. Marcello72A,72C, Z. X. Meng64, J. G. Messchendorp14,62, G.
Mezzadri29A, H. Miao1,61, T. J. Min41, R. E. Mitchell27, X. H. Mo1,56,61, N.
Yu. Muchnoi13,b, Y. Nefedov35, F. Nerling19,d, I. B. Nikolaev13,b, Z.
Ning1,56, S. Nisar11,l, Y. Niu 48, S. L. Olsen61, Q. Ouyang1,56,61, S.
Pacetti28B,28C, X. Pan53, Y. Pan55, A. Pathak33, Y. P. Pei69,56, M.
Pelizaeus4, H. P. Peng69,56, K. Peters14,d, J. L. Ping40, R. G. Ping1,61, S.
Plura34, S. Pogodin35, V. Prasad32, F. Z. Qi1, H. Qi69,56, H. R. Qi59, M.
Qi41, T. Y. Qi12,f, S. Qian1,56, W. B. Qian61, C. F. Qiao61, J. J. Qin70, L.
Q. Qin15, X. P. Qin12,f, X. S. Qin48, Z. H. Qin1,56, J. F. Qiu1, S. Q. Qu59,
C. F. Redmer34, K. J. Ren38, A. Rivetti72C, V. Rodin62, M. Rolo72C, G.
Rong1,61, Ch. Rosner19, S. N. Ruan42, N. Salone43, A. Sarantsev35,c, Y.
Schelhaas34, K. Schoenning73, M. Scodeggio29A,29B, K. Y. Shan12,f, W. Shan24,
X. Y. Shan69,56, J. F. Shangguan53, L. G. Shao1,61, M. Shao69,56, C. P.
Shen12,f, H. F. Shen1,61, W. H. Shen61, X. Y. Shen1,61, B. A. Shi61, H. C.
Shi69,56, J. Y. Shi1, Q. Q. Shi53, R. S. Shi1,61, X. Shi1,56, J. J. Song20, T.
Z. Song57, W. M. Song33,1, Y. X. Song45,g, S. Sosio72A,72C, S. Spataro72A,72C,
F. Stieler34, Y. J. Su61, G. B. Sun74, G. X. Sun1, H. Sun61, H. K. Sun1, J. F.
Sun20, K. Sun59, L. Sun74, S. S. Sun1,61, T. Sun1,61, W. Y. Sun33, Y. Sun9, Y.
J. Sun69,56, Y. Z. Sun1, Z. T. Sun48, Y. X. Tan69,56, C. J. Tang52, G. Y.
Tang1, J. Tang57, Y. A. Tang74, L. Y Tao70, Q. T. Tao25,h, M. Tat67, J. X.
Teng69,56, V. Thoren73, W. H. Tian57, W. H. Tian50, Y. Tian30,61, Z. F.
Tian74, I. Uman60B, B. Wang1, B. L. Wang61, Bo Wang69,56, C. W. Wang41, D. Y.
Wang45,g, F. Wang70, H. J. Wang37,j,k, H. P. Wang1,61, K. Wang1,56, L. L.
Wang1, M. Wang48, Meng Wang1,61, S. Wang12,f, T. Wang12,f, T. J. Wang42, W.
Wang57, W. Wang70, W. H. Wang74, W. P. Wang69,56, X. Wang45,g, X. F.
Wang37,j,k, X. J. Wang38, X. L. Wang12,f, Y. Wang59, Y. D. Wang44, Y. F.
Wang1,56,61, Y. H. Wang46, Y. N. Wang44, Y. Q. Wang1, Yaqian Wang18,1, Yi
Wang59, Z. Wang1,56, Z. L. Wang70, Z. Y. Wang1,61, Ziyi Wang61, D. Wei68, D.
H. Wei15, F. Weidner66, S. P. Wen1, C. W. Wenzel4, U. Wiedner4, G.
Wilkinson67, M. Wolke73, L. Wollenberg4, C. Wu38, J. F. Wu1,61, L. H. Wu1, L.
J. Wu1,61, X. Wu12,f, X. H. Wu33, Y. Wu69, Y. J Wu30, Z. Wu1,56, L. Xia69,56,
X. M. Xian38, T. Xiang45,g, D. Xiao37,j,k, G. Y. Xiao41, H. Xiao12,f, S. Y.
Xiao1, Y. L. Xiao12,f, Z. J. Xiao40, C. Xie41, X. H. Xie45,g, Y. Xie48, Y. G.
Xie1,56, Y. H. Xie6, Z. P. Xie69,56, T. Y. Xing1,61, C. F. Xu1,61, C. J. Xu57,
G. F. Xu1, H. Y. Xu64, Q. J. Xu17, W. L. Xu64, X. P. Xu53, Y. C. Xu76, Z. P.
Xu41, Z. S. Xu61, F. Yan12,f, L. Yan12,f, W. B. Yan69,56, W. C. Yan79, X. Q
Yan1, H. J. Yang49,e, H. L. Yang33, H. X. Yang1, Tao Yang1, Y. Yang12,f, Y. F.
Yang42, Y. X. Yang1,61, Yifan Yang1,61, M. Ye1,56, M. H. Ye8, J. H. Yin1, Z.
Y. You57, B. X. Yu1,56,61, C. X. Yu42, G. Yu1,61, T. Yu70, X. D. Yu45,g, C. Z.
Yuan1,61, L. Yuan2, S. C. Yuan1, X. Q. Yuan1, Y. Yuan1,61, Z. Y. Yuan57, C. X.
Yue38, A. A. Zafar71, F. R. Zeng48, X. Zeng12,f, Y. Zeng25,h, Y. J. Zeng1,61,
X. Y. Zhai33, Y. H. Zhan57, A. Q. Zhang1,61, B. L. Zhang1,61, B. X. Zhang1, D.
H. Zhang42, G. Y. Zhang20, H. Zhang69, H. H. Zhang33, H. H. Zhang57, H. Q.
Zhang1,56,61, H. Y. Zhang1,56, J. J. Zhang50, J. L. Zhang75, J. Q. Zhang40, J.
W. Zhang1,56,61, J. X. Zhang37,j,k, J. Y. Zhang1, J. Z. Zhang1,61, Jiawei
Zhang1,61, L. M. Zhang59, L. Q. Zhang57, Lei Zhang41, P. Zhang1, Q. Y.
Zhang38,79, Shuihan Zhang1,61, Shulei Zhang25,h, X. D. Zhang44, X. M. Zhang1,
X. Y. Zhang53, X. Y. Zhang48, Y. Zhang67, Y. T. Zhang79, Y. H. Zhang1,56, Yan
Zhang69,56, Yao Zhang1, Z. H. Zhang1, Z. L. Zhang33, Z. Y. Zhang74, Z. Y.
Zhang42, G. Zhao1, J. Zhao38, J. Y. Zhao1,61, J. Z. Zhao1,56, Lei Zhao69,56,
Ling Zhao1, M. G. Zhao42, S. J. Zhao79, Y. B. Zhao1,56, Y. X. Zhao30,61, Z. G.
Zhao69,56, A. Zhemchugov35,a, B. Zheng70, J. P. Zheng1,56, W. J. Zheng1,61, Y.
H. Zheng61, B. Zhong40, X. Zhong57, H. Zhou48, L. P. Zhou1,61, X. Zhou74, X.
K. Zhou6, X. R. Zhou69,56, X. Y. Zhou38, Y. Z. Zhou12,f, J. Zhu42, K. Zhu1, K.
J. Zhu1,56,61, L. Zhu33, L. X. Zhu61, S. H. Zhu68, S. Q. Zhu41, T. J. Zhu12,f,
W. J. Zhu12,f, Y. C. Zhu69,56, Z. A. Zhu1,61, J. H. Zou1, J. Zu69,56 (BESIII
Collaboration) 1 Institute of High Energy Physics, Beijing 100049, People’s
Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 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 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
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 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 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
31 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
32 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D,
Arica, Chile
33 Jilin University, Changchun 130012, People’s Republic of China
34 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
35 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
36 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
37 Lanzhou University, Lanzhou 730000, People’s Republic of China
38 Liaoning Normal University, Dalian 116029, People’s Republic of China
39 Liaoning University, Shenyang 110036, People’s Republic of China
40 Nanjing Normal University, Nanjing 210023, People’s Republic of China
41 Nanjing University, Nanjing 210093, People’s Republic of China
42 Nankai University, Tianjin 300071, People’s Republic of China
43 National Centre for Nuclear Research, Warsaw 02-093, Poland
44 North China Electric Power University, Beijing 102206, People’s Republic of
China
45 Peking University, Beijing 100871, People’s Republic of China
46 Qufu Normal University, Qufu 273165, People’s Republic of China
47 Shandong Normal University, Jinan 250014, People’s Republic of China
48 Shandong University, Jinan 250100, People’s Republic of China
49 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
50 Shanxi Normal University, Linfen 041004, People’s Republic of China
51 Shanxi University, Taiyuan 030006, People’s Republic of China
52 Sichuan University, Chengdu 610064, People’s Republic of China
53 Soochow University, Suzhou 215006, People’s Republic of China
54 South China Normal University, Guangzhou 510006, People’s Republic of China
55 Southeast University, Nanjing 211100, People’s Republic of China
56 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
57 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
58 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
59 Tsinghua University, Beijing 100084, People’s Republic of China
60 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
61 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
62 University of Groningen, NL-9747 AA Groningen, The Netherlands
63 University of Hawaii, Honolulu, Hawaii 96822, USA
64 University of Jinan, Jinan 250022, People’s Republic of China
65 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
66 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
67 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
68 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
69 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
70 University of South China, Hengyang 421001, People’s Republic of China
71 University of the Punjab, Lahore-54590, Pakistan
72 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
73 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
74 Wuhan University, Wuhan 430072, People’s Republic of China
75 Xinyang Normal University, Xinyang 464000, People’s Republic of China
76 Yantai University, Yantai 264005, People’s Republic of China
77 Yunnan University, Kunming 650500, People’s Republic of China
78 Zhejiang University, Hangzhou 310027, People’s Republic of China
79 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan

###### Abstract

The Born cross sections of the process $e^{+}e^{-}\to
D^{*0}D^{*-}\pi^{+}+c.c.$ at center-of-mass energies from 4.189 to 4.951 GeV
are measured for the first time. The data samples used correspond to an
integrated luminosity of $17.9\,\mathrm{fb}^{-1}$ and were collected by the
BESIII detector operating at the BEPCII storage ring. Three enhancements
around 4.20, 4.47 and 4.67 GeV are visible. The resonances have masses of
$4209.6\pm 4.7\pm 5.9\,\mathrm{MeV}/c^{2}$, $4469.1\pm 26.2\pm
3.6\,\mathrm{MeV}/c^{2}$ and $4675.3\pm 29.5\pm 3.5\,\mathrm{MeV}/c^{2}$ and
widths of $81.6\pm 17.8\pm 9.0\,\mathrm{MeV}$, $246.3\pm 36.7\pm
9.4\,\mathrm{MeV}$ and $218.3\pm 72.9\pm 9.3\,\mathrm{MeV}$, respectively,
where the first uncertainties are statistical and the second systematic. The
first and third resonances are consistent with the $Y(4230)$ and $Y(4660)$
states, respectively, while the second one is compatible with the $Y(4500)$
observed in the $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ process. These three $Y$
states are observed in $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ process for
the first time.

Over the past years, a series of charmonium-like vector meson states with
$J^{PC}=1^{--}$, referred to as $Y$ states, have been observed via electron-
positron annihilation in numerous experiments [1]. Dedicated studies of $Y$
states were initially triggered by the discovery of the $Y(4230)$, previously
called the $Y(4260)$, in the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ process at
BaBar [2], which was confirmed at CLEO [3], Belle [4] and BESIII [5, 6, 7].
Later, the $Y(4360)$ and $Y(4660)$ states were established in
$e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ at BaBar [8, 9], Belle [10, 11] and
BESIII [12, 13, 14]. Some similar resonance enhancements around 4.23 GeV, 4.36
GeV or 4.66 GeV are also reported in $\pi^{+}\pi^{-}h_{c}$ [15],
$\omega\chi_{c0}$ [16, 17], $\eta J/\psi$ [18], $\eta^{\prime}J/\psi$ [19],
$D^{0}D^{*-}\pi^{+}$ [20], $\pi\pi\psi_{2}(3823)$ [21],
$\pi^{+}\pi^{-}D^{+}D^{-}$ [22, 23] and $K^{+}K^{-}J/\psi$ [24] final states
at BESIII. In addition, a new $Y$ state, the $Y(4500)$, is observed in
$e^{+}e^{-}\to K^{+}K^{-}J/\psi$, recently [24]. Due to no corresponding
charmonia predicted in the quark model [25], these $Y$ states are widely
recognized as candidates for exotic multiquark hadrons. Possible
interpretations include tetra-quarks, hybrid mesons, hadron molecules,
hadrocharmonium and threshold effects [26, 27].

One striking feature of these $Y$ states is their large coupling to charmonium
final states [28]. In contrast, there is a dip around the known $Y(4230)$ mass
in the cross section of $e^{+}e^{-}\to\text{inclusive hadrons}$ [29] and in
the exclusive two-body production of $e^{+}e^{-}\to D^{(*)}\bar{D}^{(*)}$ [30,
31]. This is opposite to the behavior of conventional $c\bar{c}$ charmonium
states lying above the $D\bar{D}$ mass threshold, which dominantly decays to
open-charm final states [1]. Therefore, it is essential to investigate the
coupling of the $Y$ states to different open-charm channels to help identify
their nature.

In the process $e^{+}e^{-}\to D^{0}D^{*-}\pi^{+}$ [20], BESIII first
determined a sizable coupling of the $Y(4230)$ with the open-charm
$D^{0}D^{*-}\pi^{+}$ decay, which is consistent with the hypothesis of a
$D_{1}(2420)\bar{D}$ molecular state [32, 33, 34, 35, 36, 37, 38]. In lattice
QCD, the leptonic partial width of the $Y(4230)$, $\Gamma^{ee}_{Y(4230)}$,
using a hybrid scenario, is predicted to be less than $40\,\mathrm{eV}$ [39].
To date, $\Gamma^{ee}_{Y(4230)}$ is evaluated to be $36.4\pm
4.7\,\mathrm{eV}$, based on the combined analysis of the known $Y(4230)$ decay
channels [40]. Study of the $Y(4230)$ in the open-charm process $e^{+}e^{-}\to
D^{*}\bar{D}^{*}\pi$ provides new input to $\Gamma^{ee}_{Y(4230)}$ and to the
relative size of different decay modes, which can be used to test the
theoretical explanation of the hybrid and $D\bar{D}_{1}$ molecular state
models.

The $Y(4500)$ is reported in the $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ process
[24]. Its spin and mass agree with the predicted baryonium state around 4560
MeV [41]. Furthermore, other theoretical models explain it as a
$D_{s}\bar{D}_{s1}$ molecule, which is a hidden-strangeness partner of the
$Y(4230)$ state under the $D\bar{D}_{1}$ molecule assumption [42], and a
charmonium $5S$-$4D$ mixing state [43]. Thus, an independent confirmation in
another channel would be important to explore its nature. In addition, since
the Born cross section of $e^{+}e^{-}\to KZ_{cs}(3985)$ peaks around the
$Y(4660)$ mass [44, 45], the $Y(4660)$ is considered to be a hidden-
strangeness state. Given that the heavier $Y(4500)$ and $Y(4660)$ states have
not been observed in open-charm decay, it is also highly desirable to explore
these states in the process $e^{+}e^{-}\to D^{*}\bar{D}^{*}\pi$ to help
determine their quark constituents.

In this Letter, the Born cross sections of the $e^{+}e^{-}\to
D^{*0}D^{*-}\pi^{+}+c.c.$ processes are measured at 86 center-of-mass energies
from 4.189 to 4.951 GeV for the first time. The data sets used are accumulated
with the BESIII detector at the BEPCII collider and correspond to an
integrated luminosity of $17.9\,\mathrm{fb}^{-1}$ [46, 47, 48]. Details about
BEPCII and BESIII can be found in Refs. [49, 50, 51]. The data sets include 49
energy points with integrated luminosities less than $10\,\mathrm{pb}^{-1}$
(“Scan data”) and another 37 energy points with larger integrated luminosities
(“XYZ data”). Details of the data sets can be found in Tables I and II of the
supplemental material [52]. By fitting the line-shape of dressed cross
sections with various hypotheses, we report the observation of three vector
charmonium-like $Y$ states in $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$.

Simulated data samples are produced with geant4-based [55] Monte Carlo (MC)
software, which includes the geometric description [56] of the BESIII detector
and the detector response, as detailed in Ref. [51]. The simulation models the
beam energy spread and initial state radiation (ISR) in the $e^{+}e^{-}$
annihilation with the generator kkmc [57]. The signal MC samples of the
$e^{+}e^{-}\to\pi^{+}D^{*0}D^{*-}$ process are generated according to the
partial-wave-analysis results at each energy point. Possible background
contributions are estimated by inclusive MC simulation samples, which include
the production of open charm processes, the ISR production of vector
charmonium(-like) states, and the continuum processes incorporated in kkmc.
All particle decays are modeled with evtgen [58] using branching fractions
(BFs) either taken from the Particle Data Group (PDG) [1], when available, or
otherwise estimated with lundcharm [59]. Final state radiation from charged
final state particles is incorporated using photos [60].

To improve the signal selection efficiency, a partial-reconstruction technique
is employed to identify the $D^{*0}D^{*-}\pi^{+}$ final states, in which two
tagging methods, the $D^{0}$-tag and the $D^{-}$-tag, are performed. In the
$D^{0}(D^{-})$-tag method, the bachelor charged $\pi^{+}$ from primary
production, the $D^{0}(D^{-})$ meson and at least one soft
$\pi^{0}(\to\gamma\gamma)$ from $D^{*0}(D^{*-})\to D^{0}(D^{-})\pi^{0}$ decay
are reconstructed. Here and elsewhere, charge-conjugate modes are always
implied. To improve the signal purity, only the decays $D^{0}\to
K^{-}\pi^{+}$, $K^{-}\pi^{+}\pi^{0}$, and $K^{-}\pi^{+}\pi^{+}\pi^{-}$
($D^{-}\to K^{+}\pi^{-}\pi^{-}$), which have relatively large BFs, are
reconstructed. By reconstructing the $D^{*0}(D^{*-})$ and the bachelor
$\pi^{+}$, the flavor of the missing $D^{*-}(D^{*0})$ meson is fixed. All the
charged tracks and $\pi^{0}$ candidates are selected following the criteria in
Ref. [61]. To form candidates for $D^{0}\to K^{-}\pi^{+}$, $D^{0}\to
K^{-}\pi^{+}\pi^{0}$, $D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}$ and $D^{-}\to
K^{+}\pi^{-}\pi^{-}$ decays, the reconstructed final state invariant masses
are required to be within $(1.835,1.887)$, $(1.827,1.882)$, $(1.855,1.874)$
and $(1.856,1.883)\,\mathrm{GeV}/c^{2}$, respectively. Here, the different
mass regions are due to the various momentum resolutions. The $\pi^{0}$
candidates from $D^{*}$ decays can be either from the reconstructed or missing
$D^{*}$ candidates. For the $\pi^{0}$ from missing $D^{*}$ candidates, its
momentum in the reconstructed $D\pi^{+}$ recoil system, $P^{*}(\pi^{0})$,
peaks around $40\,\mathrm{MeV}/c$. To distinguish the source of $\pi^{0}$ with
reconstructed $D^{*}$ candidates, the reconstructed invariant masses are
required to satisfy $M(D^{0}\pi^{0})\in(2.004,2.009)\,\mathrm{GeV}/c^{2}$ with
$P^{*}(\pi^{0})\notin(0.025,0.050)\,\mathrm{GeV}/c$ in the $D^{0}$-tag method,
and $M(D^{-}\pi^{0})\in(2.008,2.013)\,\mathrm{GeV}/c^{2}$ with
$P^{*}(\pi^{0})\notin(0.030,0.055)\,\mathrm{GeV}/c$ in the $D^{-}$-tag method,
as shown in Fig. 1 for data at $\sqrt{s}=4.600\,\mathrm{GeV}$. Moreover, the
$\pi^{+}D^{0}$ invariant mass must be greater than $2.02\,\mathrm{GeV}/c^{2}$
in the $D^{0}$-tag method to reject background for the bachelor $\pi^{+}$ from
$D^{*+}\to\pi^{+}D^{0}$.

To improve the resolution and further suppress the background, a kinematic fit
(3C) is performed to constrain the reconstructed $\pi^{0}$, $D^{0}(D^{-})$,
and $D^{*0}(D^{*-})$ mesons to their individual known masses [1]. Candidate
events are required to have $\chi_{\rm 3C}^{2}<50$ and the fitted four-momenta
of all related particles are used for further analysis. If there is more than
one $\pi^{0}D^{0}(D^{-})$ candidate in an event, only the one with the minimum
$\chi_{\rm 3C}^{2}$ is retained. Furthermore, if one event survives in both
tag methods, only the combination in the $D^{0}$-tag method is kept to avoid
double counting in the simultaneous fit.

Figure 1:  Two-dimensional distributions of $P^{*}(\pi^{0})$ versus
$M(\pi^{0}D)$ for the $D^{0}$-tag (left) and $D^{-}$-tag (right) methods in
data at $\sqrt{s}=4.600\,\mathrm{GeV}$. The events between the vertical solid
lines are kept and those between the horizontal dashed lines are vetoed based
on the requirements for $M(\pi^{0}D)$ and $P^{*}(\pi^{0})$. The distributions
for signal MC simulation samples are shown in Fig. 1 of the supplemental
material [52].

Figure 2 shows the distributions of the recoil masses of reconstructed
$\pi^{+}$, $\pi^{0}$ and $D$ mesons, $RM(\pi^{+}\pi^{0}D^{0})$ and
$RM(\pi^{+}\pi^{0}D^{-})$. The background study based on inclusive MC
simulation samples shows that the shape of the background at each energy point
is smooth and can be well described by a second-order Chebyshev function. A
peaking background is found in the signal MC simulation due to the mis-
combination of particles from the missing and tagged sides. Its shape can be
obtained by selecting the unmatched events from inclusive MC simulation
samples, in which the missing $D^{*}$ candidate decays inclusively while the
tagged $D^{*}$ candidate decays into the signal process final state. Its
contribution will be fixed in the fit according to the ratio between matched
and unmatched events in the MC simulation.

An unbinned extended maximum likelihood fit is performed on the distributions
of $RM(\pi^{+}\pi^{0}D^{0})$ and $RM(\pi^{+}\pi^{0}D^{-})$ simultaneously to
determine the Born cross section at each energy point. Figure 2 shows the fit
results at $\sqrt{s}=4.600\,\mathrm{GeV}$, as an example. The signal shape is
derived from MC simulation convolved with a Gaussian function with free
parameters to account for the resolution difference between data and MC
simulation. The background shape is parameterized as a sum of the shape from
unmatched MC samples and a second-order Chebyshev function. The Born cross
sections ($\sigma^{\rm Born}$) at the individual energy points are calculated
by

$\displaystyle\begin{aligned} \sigma^{\rm Born}=&\frac{\sigma^{\rm
dressed}}{\frac{1}{|1-\Pi|^{2}}}\\\ =&\frac{N^{\rm
obs}_{D^{0(-)}\text{-tag}}}{\mathcal{L}_{\rm
int}\cdot\epsilon_{D^{0(-)}\text{-tag}}\cdot\hat{\mathcal{B}}_{D^{0(-)}\text{-tag}}\cdot(1+\delta^{\rm
ISR})\cdot\frac{1}{|1-\Pi|^{2}}}.\end{aligned}$

Here, $N^{\rm obs}_{D^{0(-)}\text{-tag}}$ is calculated according to
$\sigma^{\rm Born}$ which is taken as a common parameter in the simultaneous
fit, $\epsilon_{{D^{0(-)}\text{-tag}}}$ is the detection efficiency,
$\mathcal{L}_{\rm int}$ is the integral luminosity measured by Refs. [46, 47,
48], $\hat{\mathcal{B}}_{{D^{0(-)}\text{-tag}}}$ stands for an equivalent BF
including all the related products of BF obtained from the PDG [1],
$(1+\delta^{\rm ISR})$ and $\frac{1}{|1-\Pi|^{2}}$ are the correction factors
for ISR and vacuum polarization [62]. To estimate the ISR factors and consider
the correlation effect on detection efficiencies, an iterative weighting
method [63] is performed to correct the corresponding dressed cross section
values. All the numerical results from the fits are summarized in Tables I and
II of Ref. [52] for the XYZ and Scan data samples, respectively.

Figure 2:  The distributions of the recoil masses $RM(\pi^{+}\pi^{0}D^{0})$
(left) and $RM(\pi^{+}\pi^{0}D^{-})$ (right) for data at
$\sqrt{s}=4.600\,\mathrm{GeV}$ with simultaneous fit results overlaid. The red
solid-curve is the signal shape. The pink and blue dashed-curve are the
peaking and smooth background, respectively. The light green shadowed
histogram is the simulated inclusive background MC samples.

The systematic uncertainties in the Born cross section measurements, as
detailed in Ref. [52], are divided into three parts. The first part relates to
the determination of the detection efficiency, including the tracking,
particle identification, $\pi^{0}$ reconstruction, signal region requirements,
signal decay model and ISR correction factor. The second part relates to the
estimation of signal yields from the fit, consisting of the signal and
background shapes as well as the fit range. The last part includes the
uncertainties from the luminosities and the intermediate BFs. The items in the
first and third parts are completely correlated between different energy
points, except for the uncertainties due to signal region requirements and the
signal decay model. For the second part at low-yield ($<300$ events) energy
points, the systematic uncertainties obtained at their nearest energy point in
high-yield ($>300$ events) XYZ data are used. All the systematic uncertainties
are studied for each tag method and combined to obtain the total systematic
uncertainties according to their signal yields. The total relative systematic
uncertainties at different energy points are between $6.7\%$ to $9.6\%$.

The dressed cross sections obtained at various energy points are shown in Fig.
3. Three possible enhancements around 4.20, 4.47, and 4.67 GeV are observed.
To fit this line-shape, we use the coherent sum of a continuum amplitude for
$e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ and three resonance amplitudes
described by relativistic Breit-Wigner (BW) functions

$\displaystyle\sigma^{\rm
dressed}(\sqrt{s})=C_{0}\Big{|}C_{1}\sqrt{\Phi(\sqrt{s})}+\sum^{3}_{k=1}{\rm
BW}_{k}(\sqrt{s})e^{i\phi_{k}}\Big{|}^{2},$

where $C_{0}=3.894\times 10^{5}\,\mathrm{nb}\cdot\,\mathrm{GeV}^{2}$ is a unit
conversion factor, $C_{1}$ is the continuum free parameter, and $\phi_{k}$ is
the phase angle among different components. The relativistic BW amplitude for
a resonance $R_{k}\to D^{*0}D^{*-}\pi^{+}$ is written as

$\displaystyle{\rm
BW}_{k}(\sqrt{s})=\frac{m_{k}}{\sqrt{s}}\cdot\frac{\sqrt{12\pi\cdot\Gamma^{ee}_{k}\cdot\mathcal{B}_{k}\cdot\Gamma^{\rm
tot}_{k}}}{s-m_{k}^{2}+im_{k}\Gamma^{\rm
tot}_{k}}\cdot\sqrt{\frac{\Phi(\sqrt{s})}{\Phi(m_{k})}},$

where $m_{k}$ and $\Gamma^{\rm tot}_{k}$ are the $k$-th resonance mass and
total width, respectively, $\Gamma^{ee}_{k}\cdot\mathcal{B}_{k}$ is the
leptonic width of the $k$-th resonance times the BF of $R_{k}\to
D^{*0}D^{*-}\pi^{+}+c.c.$, and $\Phi(\sqrt{s})$ is the three body phase space
contribution defined as
$\Phi(\sqrt{s})=\iint\frac{1}{(2\pi)^{3}32(\sqrt{s})^{3}}dm_{23}^{2}dm_{12}^{2}$
[1].

The $\chi^{2}$ of the fit to the dressed cross section line-shape is
constructed according to the method in Ref. [64] by incorporating both the
statistical and systematic uncertainty and considering both the correlated and
uncorrelated terms. To avoid the bias from correlated systematic uncertainties
in the $\chi^{2}$ minimization [65], the correlated uncertainties are
calculated according to the predicted cross section values times the
corresponding relative uncertainties when constructing the covariance matrix.

The fit result is shown in Fig. 3. There are eight solutions with the same fit
quality with identical continuum contributions as well as masses and widths
for the resonances [66]. However, the resulting product
$\Gamma^{ee}_{k}\mathcal{B}_{k}$ and phases $\phi_{k}$ are different, as
plotted in Fig. 2 of Ref. [52]. The numerical results are listed in Table 1.
The dressed cross sections are also fitted under the assumption of only two
resonances plus the continuum component. The relative changes in the
$\chi^{2}$ value ($\Delta{\rm\chi^{2}}=130.5$) and the number of degrees of
freedom ($\Delta{\rm ndof}=4$) are used to estimate the significance of the
three-resonance hypothesis over the two-resonance hypothesis as $10.8\sigma$.
The significance of the two-resonance hypothesis over the one-resonance
hypothesis is $22.8\sigma$ according to the changes of
$\Delta{\rm\chi^{2}}=537.1$ and $\Delta{\rm ndof}=4$.

(I)

Figure 3:  The fit results (solution I) of the dressed cross section line-shape of $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$. The black points are data, with statistical and systematic uncertainties. The blue curve is the total fit. The green, azure and orange dashed curves describe three BW functions, and the pink dashed-curve is the three body phase space contribution. Table 1:  The fit results of the dressed cross section line-shape of $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ with eight different solutions. For the uncertainties of the masses and widths, those from the solutions with maximum uncertainties are adopted.  | I | II | III | IV | V | VI | VII | VIII
---|---|---|---|---|---|---|---|---
$C_{1}\,(10^{-3})$ | $4.2\pm 1.5$
$m_{1}\,(\mathrm{MeV}/c^{2})$ | $4209.6\pm 4.7$
$\Gamma^{\rm tot}_{1}\,(\mathrm{MeV})$ | $81.6\pm 17.8$
$\Gamma^{ee}_{1}\mathcal{B}_{1}\,(\mathrm{eV})$ | $5.4\pm 1.1$ | $6.0\pm 1.3$ | $4.8\pm 0.9$ | $5.3\pm 1.1$ | $17.9\pm 7.2$ | $19.8\pm 6.6$ | $20.2\pm 7.4$ | $22.4\pm 9.0$
$\phi_{1}\,(\mathrm{rad})$ | $3.1\pm 0.5$ | $3.8\pm 0.4$ | $1.9\pm 0.7$ | $2.6\pm 0.6$ | $4.2\pm 0.3$ | $4.8\pm 0.2$ | $5.4\pm 0.3$ | $6.0\pm 0.3$
$m_{2}\,(\mathrm{MeV}/c^{2})$ | $4469.1\pm 26.2$
$\Gamma^{\rm tot}_{2}\,(\mathrm{MeV})$ | $246.3\pm 36.7$
$\Gamma^{ee}_{2}\mathcal{B}_{2}\,(\mathrm{eV})$ | $243.3\pm 83.5$ | $832.5\pm 716.5$ | $107.4\pm 50.6$ | $367.4\pm 370.8$ | $225.5\pm 94.9$ | $770.8\pm 383.8$ | $510.1\pm 202.3$ | $1744.3\pm 926.9$
$\phi_{2}\,(\mathrm{rad})$ | $4.4\pm 0.3$ | $-0.9\pm 0.3$ | $2.6\pm 0.6$ | $3.7\pm 0.8$ | $1.9\pm 0.8$ | $3.0\pm 0.4$ | $3.7\pm 0.3$ | $-1.5\pm 0.3$
$m_{3}\,(\mathrm{MeV}/c^{2})$ | $4675.3\pm 29.5$
$\Gamma^{\rm tot}_{3}\,(\mathrm{MeV})$ | $218.3\pm 72.9$
$\Gamma^{ee}_{3}\mathcal{B}_{3}\,(\mathrm{eV})$ | $75.8\pm 148.8$ | $1601.9\pm 1152.6$ | $19.4\pm 27.1$ | $411.6\pm 230.5$ | $24.4\pm 34.5$ | $515.6\pm 244.6$ | $95.1\pm 173.1$ | $2005.3\pm 1166.1$
$\phi_{3}\,(\mathrm{rad})$ | $4.9\pm 1.4$ | $-2.9\pm 0.4$ | $2.1\pm 0.4$ | $0.6\pm 1.1$ | $1.7\pm 0.5$ | $6.5\pm 0.5$ | $4.5\pm 1.3$ | $-3.3\pm 0.3$

The systematic uncertainties of the resonance parameters are mainly from the
uncertainties of the center-of-mass energy calibration, beam energy spread and
parameterization of the continuum contribution. Other uncertainties from the
measured cross sections have been included in the line-shape fit. The
uncertainty from the center-of-mass energy measurement is estimated by
propagating the largest uncertainty of the measured energies
($0.8\,\mathrm{MeV}/c^{2}$) to the Y-state mass parameter. The uncertainty
from beam energy spread is considered by smearing the energy with its spread
value at each energy point. The differences of resonance parameters determined
from fits using nominal and smeared line-shapes are taken as the systematic
uncertainties. To estimate the uncertainty related to the fit model, the three
body continuum contribution is replaced by a third-order polynomial
parameterized function. The resulting differences in the masses and widths of
resonances are taken as systematic uncertainties. Assuming the individual
systematic uncertainties are uncorrelated, the total systematic uncertainty is
obtained by summing the individual values in quadrature, as listed in Table 2.

Table 2:  The systematic uncertainties in the measurements of the Y-state parameters.  Source | Energy | Beam spread | Fit model | Total
---|---|---|---|---
$m_{1}$$(\,\mathrm{MeV}/c^{2})$ | 0.8 | 5.5 | 2.0 | 5.9
$\Gamma_{1}^{\rm tot}(\,\mathrm{MeV})$ | - | 1.7 | 8.8 | 9.0
$m_{2}$$(\,\mathrm{MeV}/c^{2})$ | 0.8 | 3.5 | 0.7 | 3.6
$\Gamma_{2}^{\rm tot}(\,\mathrm{MeV})$ | - | 6.9 | 6.4 | 9.4
$m_{3}$$(\,\mathrm{MeV}/c^{2})$ | 0.8 | 1.5 | 3.1 | 3.5
$\Gamma_{3}^{\rm tot}(\,\mathrm{MeV})$ | - | 7.4 | 5.7 | 9.3

In summary, the Born cross sections of the process $e^{+}e^{-}\to
D^{*0}D^{*-}\pi^{+}+c.c.$ at 86 center-of-mass energies from $\sqrt{s}=4.189$
to $4.951\,\mathrm{GeV}$ are measured for the first time with the data samples
collected by the BESIII detector. Fitting the dressed cross sections with a
three-resonance hypothesis, their masses and widths are determined to be
$m_{1}=4209.6\pm 4.7\pm 5.9\,\mathrm{MeV}/c^{2}$, and $\Gamma_{1}=81.6\pm
17.8\pm 9.0\,\mathrm{MeV}$ (denoted as $Y(4210)$), $m_{2}=4469.1\pm 26.2\pm
3.6\,\mathrm{MeV}/c^{2}$, and $\Gamma_{2}=246.3\pm 36.7\pm 9.4\,\mathrm{MeV}$
(denoted as $Y(4470)$), $m_{3}=4675.3\pm 29.5\pm 3.5\,\mathrm{MeV}/c^{2}$, and
$\Gamma_{3}=218.3\pm 72.9\pm 9.3\,\mathrm{MeV}$ (denoted as $Y(4660)$), where
the first uncertainties are statistical and the second are systematic. The
significance of the three-resonance hypothesis compared with the two-resonance
one is greater than $10\sigma$. The mass of $Y(4210)$ is consistent with the
mass of $Y(4230)$ from the combined fit in Ref. [40]. If we assume they are
the same resonance, $\Gamma^{ee}_{Y(4230)}$ becomes greater than 40 eV, which
disfavors the hybrid interpretation under the lattice QCD calculation [39]. In
addition, we find the couplings of $Y(4230)$ to $D^{*0}D^{*-}\pi^{+}$ and
$D^{0}D^{*-}\pi^{+}$ are at the same order of magnitude. This is the first
observation of the state $Y(4470)$ in an open-charm process, and its resonance
parameters are compatible with those of the $Y(4500)$ state observed in
$e^{+}e^{-}\to K^{+}K^{-}J/\psi$ [24]. Assuming the $Y(4470)$ and $Y(4500)$
are the same state, the rate of its decay to $D^{*}\bar{D}^{*}\pi$ is two
orders of magnitude greater than that to $K\bar{K}J/\psi$, which is
inconsistent with the conjecture of hidden-strangeness enhancement of the
Y(4500) [42]. We confirm for the first time the existence of the resonance
$Y(4660)$ in open-charm final states with resonance parameters consistent with
the latest results derived in $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(2S)$ at BESIII
[14]. However, the relative size of their couplings can not be constrained by
current data, as different fit solutions result in large variations of the
product $\Gamma^{ee}_{Y(4660)}\mathcal{B}_{Y(4660)}$. Further amplitude
analyses of different open- and hidden-charm final states are desired to
advance our knowledge of the nature of these $Y$ states.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012,
12022510, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265, 12221005; 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. U1832207; CAS Key Research
Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-
SLH040; 100 Talents Program of CAS; Fundamental Research Funds for the Central
Universities, Lanzhou University, University of Chinese Academy of Sciences;
The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key
Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462;
European Union’s Horizon 2020 research and innovation programme under Marie
Sklodowska-Curie grant agreement under Contract No. 894790; German Research
Foundation DFG under Contracts Nos. 443159800, 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; National Science
Research and Innovation Fund (NSRF) via the Program Management Unit for Human
Resources & Institutional Development, Research and Innovation under Contract
No. B16F640076; Polish National Science Centre under Contract No.
2019/35/O/ST2/02907; Suranaree University of Technology (SUT), Thailand
Science Research and Innovation (TSRI), and National Science Research and
Innovation Fund (NSRF) under Contract No. 160355; The Royal Society, UK under
Contract No. DH160214; The Swedish Research Council; U. S. Department of
Energy under Contract No. DE-FG02-05ER41374.

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Supplemental Material for “Observation of Three Charmonium-like States with
$J^{PC}=1^{--}$ in $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ Process”

##  Appendix A Details on event selection

Two-dimensional distributions of $P^{*}(\pi^{0})$ versus $M(\pi^{0}D)$ for MC
samples with $D^{0}$-tag and $D^{-}$-tag methods at
$\sqrt{s}=4.600\,\mathrm{GeV}$ are shown in Fig. 1, respectively.

Figure 1:  Two-dimensions distributions of $P^{*}(\pi^{0})$ versus
$M(\pi^{0}D)$ for MC simulation with $D^{0}$-tag and $D^{-}$-tag methods at
$\sqrt{s}=4.600\,\mathrm{GeV}$. The events are kept inside the vertical solid
lines and vetoed inside the horizontal dashed lines based on the requirements
for $M(\pi^{0}D)$ and $P^{*}(\pi^{0})$.

##  Appendix B Signal yields and Born cross section

A simultaneous fit of $D^{0}$-tag and $D^{-}$-tag is performed at each energy
point according to the calculation of Born cross section.

$\displaystyle\begin{aligned} \sigma^{\rm Born}=\frac{\sigma^{\rm
dressed}}{\frac{1}{|1-\Pi|^{2}}}=\frac{N^{\rm
obs}_{D^{0(-)}\text{-tag}}}{\mathcal{L}_{\rm
int}\cdot\epsilon_{D^{0(-)}\text{-tag}}\cdot\hat{\mathcal{B}}_{D^{0(-)}\text{-tag}}\cdot(1+\delta^{\rm
ISR})\cdot\frac{1}{|1-\Pi|^{2}}}.\end{aligned}$

The the integral luminosities $\mathcal{L}_{\rm int}$ are measured by Refs.
[46, 47, 48]. The $\sigma^{\rm Born}$ is taken as a common parameter in the
fitting while detection efficiencies, ISR and vacuum polarization factors are
estimated based on MC simulation. With the fit results shown in Tables 1 and
2, $N_{D^{0}\text{-tag}}^{\rm obs}$ and $N_{D^{-}\text{-tag}}^{\rm obs}$ can
be calculated directly.

Table 1:  The Born cross section of $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ for XYZ data sets. The first uncertainties are statistical and the second ones systematic.  |  $\sqrt{s}$ (GeV) |  $\mathcal{L}_{\rm int}$ (pb-1) | $1+\delta^{\rm ISR}$ | $\frac{1}{|1-\Pi|^{2}}$ | $N_{D^{0}\text{-tag}}^{\rm obs}$ |  $\epsilon_{D^{0}\text{-tag}}$ (%) | $N_{D^{-}\text{-tag}}^{\rm obs}$ |  $\epsilon_{D^{-}\text{-tag}}$ (%) |  $\sigma^{\rm Born}$ (pb)
---|---|---|---|---|---|---|---|---|---
| 4.189 | 570.0 | 0.667 | 1.056 | $32.0\pm 5.5$ | 1.1 | $8.3\pm 1.4$ | 1.6 | $44.9\pm 7.8\pm 3.5$
| 4.199 | 526.6 | 0.675 | 1.056 | $62.6\pm 8.6$ | 2.0 | $13.4\pm 1.8$ | 2.6 | $48.5\pm 6.7\pm 3.8$
| 4.209 | 572.1 | 0.696 | 1.057 | $193.3\pm 14.2$ | 2.9 | $37.5\pm 2.7$ | 3.3 | $94.4\pm 6.9\pm 7.5$
| 4.219 | 569.2 | 0.726 | 1.056 | $291.0\pm 17.1$ | 3.6 | $55.0\pm 3.2$ | 4.0 | $110.2\pm 6.5\pm 8.7$
| 4.226 | 1111.9 | 0.749 | 1.056 | $1173.5\pm 34.9$ | 4.3 | $173.9\pm 5.2$ | 4.8 | $145.6\pm 4.3\pm 11.5$
| 4.236 | 530.3 | 0.770 | 1.056 | $468.0\pm 24.1$ | 4.2 | $88.5\pm 4.6$ | 4.8 | $151.7\pm 7.8\pm 11.5$
| 4.242 | 55.9 | 0.780 | 1.055 | $51.6\pm 8.9$ | 5.4 | $9.7\pm 1.7$ | 6.1 | $122.3\pm 21.2\pm 8.5$
| 4.244 | 538.1 | 0.784 | 1.056 | $522.4\pm 26.6$ | 5.3 | $98.5\pm 5.0$ | 6.0 | $130.4\pm 6.6\pm 9.0$
| 4.258 | 828.4 | 0.795 | 1.054 | $1077.9\pm 37.9$ | 5.5 | $162.8\pm 5.7$ | 6.2 | $132.6\pm 4.7\pm 10.8$
| 4.267 | 531.1 | 0.797 | 1.053 | $571.3\pm 28.2$ | 4.6 | $109.5\pm 5.4$ | 5.3 | $163.6\pm 8.1\pm 11.6$
| 4.278 | 175.7 | 0.802 | 1.053 | $177.7\pm 16.1$ | 4.6 | $34.1\pm 3.1$ | 5.2 | $153.7\pm 13.9\pm 10.9$
| 4.287 | 494.2 | 0.802 | 1.053 | $578.5\pm 27.7$ | 5.8 | $109.9\pm 5.3$ | 6.5 | $141.8\pm 6.8\pm 10.4$
| 4.308 | 45.1 | 0.802 | 1.052 | $85.0\pm 11.1$ | 5.5 | $16.3\pm 2.1$ | 6.3 | $239.0\pm 31.3\pm 19.1$
| 4.311 | 494.3 | 0.802 | 1.052 | $745.5\pm 33.2$ | 5.5 | $143.8\pm 6.4$ | 6.3 | $193.7\pm 8.6\pm 15.5$
| 4.337 | 506.1 | 0.799 | 1.051 | $1078.9\pm 42.2$ | 7.4 | $201.5\pm 7.9$ | 8.2 | $203.7\pm 8.0\pm 19.6$
| 4.358 | 543.9 | 0.798 | 1.051 | $1604.8\pm 50.2$ | 8.0 | $304.4\pm 9.5$ | 9.0 | $262.3\pm 8.2\pm 20.1$
| 4.377 | 524.7 | 0.796 | 1.051 | $1909.7\pm 55.2$ | 6.0 | $367.6\pm 10.6$ | 6.8 | $431.5\pm 12.5\pm 35.9$
| 4.387 | 55.6 | 0.794 | 1.051 | $234.1\pm 19.2$ | 7.8 | $45.2\pm 3.7$ | 8.9 | $384.3\pm 31.5\pm 28.0$
| 4.395 | 508.2 | 0.792 | 1.051 | $2421.6\pm 60.0$ | 7.6 | $466.1\pm 11.5$ | 8.7 | $446.5\pm 11.1\pm 32.5$
| 4.416 | 1090.7 | 0.794 | 1.052 | $6225.0\pm 95.7$ | 7.4 | $1185.2\pm 18.2$ | 8.3 | $547.7\pm 8.4\pm 38.5$
| 4.436 | 570.6 | 0.796 | 1.054 | $3798.9\pm 74.6$ | 7.6 | $755.4\pm 14.8$ | 9.0 | $619.3\pm 12.2\pm 42.2$
| 4.467 | 111.1 | 0.810 | 1.055 | $929.9\pm 35.9$ | 7.8 | $176.2\pm 6.8$ | 8.8 | $742.9\pm 28.7\pm 50.8$
| 4.527 | 112.1 | 0.863 | 1.054 | $1128.2\pm 39.2$ | 8.4 | $217.5\pm 7.5$ | 9.7 | $776.7\pm 27.0\pm 60.5$
| 4.575 | 48.9 | 0.900 | 1.054 | $430.3\pm 25.1$ | 10.2 | $86.1\pm 5.0$ | 12.1 | $537.1\pm 31.4\pm 36.4$
| 4.600 | 586.9 | 0.905 | 1.055 | $5964.6\pm 95.1$ | 10.4 | $1206.9\pm 19.2$ | 12.5 | $606.4\pm 9.7\pm 41.2$
| 4.613 | 103.7 | 0.906 | 1.055 | $1019.2\pm 38.9$ | 9.9 | $210.1\pm 8.0$ | 12.1 | $613.4\pm 23.4\pm 41.2$
| 4.628 | 521.5 | 0.902 | 1.054 | $5115.1\pm 88.6$ | 10.2 | $1026.4\pm 17.8$ | 12.1 | $599.2\pm 10.4\pm 40.5$
| 4.641 | 551.7 | 0.901 | 1.054 | $5404.6\pm 90.2$ | 10.0 | $1123.5\pm 18.7$ | 12.4 | $608.4\pm 10.1\pm 44.4$
| 4.661 | 529.4 | 0.895 | 1.054 | $5434.9\pm 91.8$ | 10.1 | $1111.6\pm 18.8$ | 12.2 | $641.7\pm 10.8\pm 47.1$
| 4.682 | 1667.4 | 0.896 | 1.054 | $18143.7\pm 168.0$ | 10.6 | $3743.9\pm 34.7$ | 12.9 | $647.2\pm 6.0\pm 45.3$
| 4.699 | 535.5 | 0.900 | 1.055 | $6088.0\pm 97.2$ | 10.4 | $1265.0\pm 20.2$ | 12.9 | $680.3\pm 10.9\pm 48.9$
| 4.740 | 163.9 | 0.925 | 1.055 | $1761.7\pm 54.3$ | 11.0 | $374.8\pm 11.5$ | 13.9 | $595.0\pm 18.3\pm 43.3$
| 4.750 | 366.6 | 0.934 | 1.055 | $3902.4\pm 80.9$ | 11.0 | $824.4\pm 17.1$ | 13.8 | $580.0\pm 12.0\pm 40.9$
| 4.781 | 511.5 | 0.957 | 1.055 | $5185.2\pm 95.2$ | 10.9 | $1113.7\pm 20.4$ | 13.9 | $546.8\pm 10.0\pm 36.6$
| 4.843 | 525.2 | 0.979 | 1.056 | $4539.6\pm 91.7$ | 10.6 | $972.9\pm 19.7$ | 13.4 | $468.4\pm 9.5\pm 33.1$
| 4.918 | 207.8 | 0.973 | 1.056 | $1674.8\pm 55.2$ | 10.4 | $370.3\pm 12.2$ | 13.6 | $448.7\pm 14.8\pm 31.8$
| 4.951 | 159.3 | 0.965 | 1.056 | $1273.9\pm 39.5$ | 10.3 | $283.3\pm 8.8$ | 13.5 | $452.9\pm 14.1\pm 31.1$
Table 2:  The Born cross section of $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ for Scan data sets. The first uncertainties are statistical and the second ones systematic.  |  $\sqrt{s}$ (GeV) |  $\mathcal{L}_{\rm int}$ (pb-1) | $1+\delta^{\rm ISR}$ | $\frac{1}{|1-\Pi|^{2}}$ | $N_{D^{0}\text{-tag}}^{\rm obs}$ |  $\epsilon_{D^{0}\text{-tag}}$ (%) | $N_{D^{-}\text{-tag}}^{\rm obs}$ |  $\epsilon_{D^{-}\text{-tag}}$ (%) |  $\sigma^{\rm Born}$ (pb)
---|---|---|---|---|---|---|---|---|---
| 4.200 | 6.8 | 0.677 | 1.057 | $2.9\pm 1.5$ | 2.3 | $0.6\pm 0.3$ | 2.8 | $156.2\pm 81.6\pm 12.3$
| 4.203 | 7.6 | 0.681 | 1.057 | $3.4\pm 1.6$ | 2.5 | $0.7\pm 0.3$ | 3.0 | $146.3\pm 71.0\pm 11.5$
| 4.207 | 7.7 | 0.691 | 1.057 | $3.7\pm 1.9$ | 2.9 | $0.7\pm 0.4$ | 3.3 | $134.3\pm 68.0\pm 10.6$
| 4.212 | 7.8 | 0.705 | 1.057 | $4.2\pm 1.9$ | 3.2 | $0.8\pm 0.4$ | 3.8 | $134.6\pm 61.2\pm 10.6$
| 4.217 | 7.9 | 0.720 | 1.056 | $6.0\pm 2.2$ | 3.8 | $1.1\pm 0.4$ | 4.1 | $158.1\pm 58.7\pm 13.0$
| 4.222 | 8.2 | 0.738 | 1.056 | $6.5\pm 2.8$ | 4.0 | $1.2\pm 0.5$ | 4.4 | $150.2\pm 64.5\pm 11.8$
| 4.227 | 8.2 | 0.751 | 1.056 | $5.9\pm 2.3$ | 4.2 | $1.1\pm 0.4$ | 4.7 | $128.8\pm 50.7\pm 10.1$
| 4.232 | 8.3 | 0.762 | 1.056 | $7.0\pm 2.7$ | 4.2 | $1.3\pm 0.5$ | 4.7 | $148.8\pm 58.0\pm 11.7$
| 4.237 | 7.8 | 0.773 | 1.055 | $3.5\pm 2.6$ | 5.0 | $0.7\pm 0.5$ | 5.7 | $65.0\pm 47.7\pm 4.5$
| 4.240 | 8.6 | 0.778 | 1.055 | $10.0\pm 3.8$ | 5.2 | $1.8\pm 0.7$ | 5.6 | $163.5\pm 61.9\pm 11.3$
| 4.242 | 8.5 | 0.781 | 1.056 | $8.1\pm 3.0$ | 5.5 | $1.5\pm 0.5$ | 5.9 | $125.4\pm 46.9\pm 8.7$
| 4.245 | 8.6 | 0.784 | 1.056 | $11.7\pm 3.5$ | 5.5 | $2.2\pm 0.6$ | 6.0 | $179.1\pm 52.8\pm 12.4$
| 4.247 | 8.6 | 0.787 | 1.055 | $14.9\pm 3.5$ | 5.6 | $2.7\pm 0.6$ | 6.2 | $219.5\pm 51.4\pm 15.2$
| 4.252 | 8.7 | 0.792 | 1.054 | $5.3\pm 2.6$ | 5.3 | $1.0\pm 0.5$ | 5.9 | $81.9\pm 40.7\pm 6.6$
| 4.257 | 8.9 | 0.795 | 1.054 | $10.7\pm 3.9$ | 5.4 | $2.1\pm 0.8$ | 6.2 | $157.6\pm 57.9\pm 12.8$
| 4.262 | 8.6 | 0.796 | 1.053 | $16.9\pm 3.6$ | 4.6 | $3.2\pm 0.7$ | 5.1 | $300.1\pm 64.4\pm 21.2$
| 4.267 | 8.6 | 0.798 | 1.053 | $5.2\pm 3.1$ | 4.7 | $1.0\pm 0.6$ | 5.2 | $90.8\pm 54.2\pm 6.4$
| 4.272 | 8.6 | 0.800 | 1.053 | $7.7\pm 2.9$ | 4.7 | $1.5\pm 0.6$ | 5.4 | $133.5\pm 51.1\pm 9.4$
| 4.277 | 8.7 | 0.800 | 1.053 | $10.2\pm 3.9$ | 5.9 | $1.9\pm 0.7$ | 6.6 | $139.0\pm 53.3\pm 9.8$
| 4.282 | 8.6 | 0.803 | 1.053 | $16.0\pm 3.6$ | 6.1 | $3.0\pm 0.7$ | 6.7 | $214.0\pm 48.9\pm 20.8$
| 4.287 | 9.0 | 0.802 | 1.053 | $9.1\pm 3.7$ | 6.1 | $1.7\pm 0.7$ | 6.7 | $117.6\pm 47.5\pm 11.4$
| 4.297 | 8.5 | 0.801 | 1.052 | $15.9\pm 4.6$ | 5.5 | $2.9\pm 0.8$ | 5.9 | $242.0\pm 70.1\pm 19.3$
| 4.307 | 8.6 | 0.803 | 1.052 | $10.9\pm 3.9$ | 5.4 | $2.1\pm 0.7$ | 6.1 | $163.2\pm 57.9\pm 13.0$
| 4.317 | 9.3 | 0.801 | 1.052 | $3.7\pm 3.6$ | 5.9 | $0.7\pm 0.7$ | 6.5 | $53.4\pm 46.1\pm 3.7$
| 4.327 | 8.7 | 0.800 | 1.051 | $30.2\pm 6.6$ | 7.4 | $5.7\pm 1.2$ | 8.2 | $333.1\pm 72.2\pm 31.9$
| 4.337 | 8.7 | 0.798 | 1.051 | $18.2\pm 5.6$ | 7.4 | $3.4\pm 1.1$ | 8.3 | $199.9\pm 61.8\pm 19.2$
| 4.347 | 8.5 | 0.798 | 1.051 | $23.8\pm 6.1$ | 7.8 | $4.3\pm 1.1$ | 8.4 | $251.9\pm 64.7\pm 19.2$
| 4.357 | 8.1 | 0.797 | 1.051 | $36.9\pm 6.8$ | 7.6 | $6.9\pm 1.3$ | 8.4 | $428.1\pm 79.2\pm 32.7$
| 4.367 | 8.5 | 0.796 | 1.051 | $23.0\pm 6.9$ | 6.1 | $4.2\pm 1.3$ | 6.6 | $316.5\pm 94.4\pm 26.3$
| 4.377 | 8.2 | 0.796 | 1.051 | $29.9\pm 6.7$ | 6.2 | $5.7\pm 1.3$ | 7.0 | $421.9\pm 94.8\pm 35.0$
| 4.387 | 7.5 | 0.794 | 1.051 | $40.2\pm 7.8$ | 7.9 | $7.3\pm 1.4$ | 8.5 | $487.6\pm 94.2\pm 35.4$
| 4.392 | 7.4 | 0.794 | 1.051 | $37.5\pm 7.4$ | 7.9 | $7.2\pm 1.4$ | 8.9 | $454.6\pm 89.3\pm 33.0$
| 4.397 | 7.2 | 0.793 | 1.051 | $38.3\pm 7.5$ | 8.0 | $7.2\pm 1.4$ | 9.0 | $471.3\pm 92.1\pm 34.2$
| 4.407 | 6.4 | 0.793 | 1.052 | $36.3\pm 7.4$ | 7.3 | $6.9\pm 1.4$ | 8.2 | $558.4\pm 113.5\pm 39.0$
| 4.417 | 7.5 | 0.793 | 1.052 | $37.4\pm 7.8$ | 7.5 | $7.1\pm 1.5$ | 8.4 | $473.1\pm 99.3\pm 33.1$
| 4.422 | 7.4 | 0.793 | 1.052 | $43.9\pm 7.9$ | 7.4 | $8.3\pm 1.5$ | 8.3 | $564.9\pm 101.8\pm 39.5$
| 4.427 | 6.8 | 0.794 | 1.053 | $42.0\pm 7.7$ | 8.7 | $8.1\pm 1.5$ | 9.9 | $503.3\pm 91.9\pm 34.2$
| 4.437 | 7.6 | 0.796 | 1.054 | $52.5\pm 9.0$ | 8.0 | $10.3\pm 1.8$ | 9.3 | $609.9\pm 104.1\pm 41.4$
| 4.447 | 7.7 | 0.800 | 1.054 | $67.7\pm 9.5$ | 7.3 | $12.8\pm 1.8$ | 8.2 | $845.9\pm 118.8\pm 57.5$
| 4.457 | 8.7 | 0.803 | 1.055 | $69.9\pm 9.9$ | 7.4 | $13.5\pm 1.9$ | 8.5 | $756.7\pm 107.2\pm 51.5$
| 4.477 | 8.2 | 0.816 | 1.055 | $85.4\pm 10.6$ | 7.9 | $15.6\pm 1.9$ | 8.6 | $909.9\pm 113.2\pm 61.9$
| 4.497 | 8.0 | 0.833 | 1.055 | $81.6\pm 10.5$ | 7.9 | $15.6\pm 2.0$ | 8.9 | $874.1\pm 112.4\pm 59.5$
| 4.517 | 8.7 | 0.855 | 1.055 | $73.6\pm 10.5$ | 8.2 | $13.9\pm 2.0$ | 9.2 | $682.5\pm 97.1\pm 53.0$
| 4.537 | 9.3 | 0.875 | 1.054 | $101.3\pm 11.9$ | 9.4 | $20.1\pm 2.4$ | 11.1 | $739.8\pm 87.2\pm 57.5$
| 4.547 | 8.8 | 0.884 | 1.054 | $92.0\pm 11.5$ | 9.6 | $18.2\pm 2.3$ | 11.2 | $697.0\pm 87.3\pm 54.2$
| 4.557 | 8.3 | 0.892 | 1.054 | $90.9\pm 11.0$ | 9.8 | $17.9\pm 2.2$ | 11.5 | $704.7\pm 85.2\pm 47.7$
| 4.567 | 8.4 | 0.897 | 1.054 | $96.5\pm 12.0$ | 9.8 | $19.0\pm 2.4$ | 11.4 | $737.6\pm 91.5\pm 49.9$
| 4.577 | 8.5 | 0.901 | 1.055 | $78.7\pm 10.9$ | 9.7 | $15.8\pm 2.2$ | 11.6 | $591.7\pm 82.3\pm 40.0$
| 4.587 | 8.2 | 0.905 | 1.055 | $83.3\pm 11.3$ | 9.9 | $16.6\pm 2.2$ | 11.7 | $639.5\pm 86.4\pm 43.3$

##  Appendix C Multiple solutions of line-shape fit

The dressed cross section is parameterized as the coherent sum of a continuum
amplitude for $e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$ and three resonance
amplitudes described by relativistic Breit-Wigner functions:

$\displaystyle\sigma^{\rm
dressed}(\sqrt{s})=C_{0}|C_{1}\sqrt{\Phi(\sqrt{s})}+\sum^{3}_{k=1}\text{BW}_{k}(\sqrt{s})e^{i\phi_{k}}|^{2},$

with relativistic Breit-Wigner amplitude for a resonance $R_{k}\to
D^{*0}D^{*-}\pi^{+}$,

$\displaystyle\text{BW}_{k}(\sqrt{s})=\frac{m_{k}}{\sqrt{s}}\cdot\frac{\sqrt{12\pi\cdot\Gamma^{ee}_{k}\mathcal{B}_{k}\cdot\Gamma^{\rm
tot}_{k}}}{s-m_{k}^{2}+im_{k}\Gamma^{\rm
tot}_{k}}\cdot\sqrt{\frac{\Phi(\sqrt{s})}{\Phi(m_{k})}},$

and 3-body phase space contribution
$\Phi(\sqrt{s})=\iint\frac{1}{(2\pi)^{3}32(\sqrt{s})^{3}}dm_{23}^{2}dm_{12}^{2}$.
All the parameters in the fit are free except for $C_{0}=3.894\times
10^{5}\,\mathrm{nb}\cdot\,\mathrm{GeV}^{2}$ as a unit conversion factor. There
are eight solutions with the same fit quality, as shown in Fig. 2.

(I)(II)
(III)(IV)
(V)(VI)
(VII)(VIII)

Figure 2: The fit results of the dressed cross section line-shape of
$e^{+}e^{-}\to D^{*0}D^{*-}\pi^{+}+c.c.$. The blue curve is the total fit. The
green, azure and orange dashed curves describe three BW functions, and the
pink dashed-curve is the three body phase space contribution.

##  Appendix D Systematic uncertainties

The systematic uncertainty studies are performed at energy points where the
signal yield greater than 500 events. For other points suffering limited
statistics, the uncertainty from the closest point is taken as its systematic
uncertainty. All the systematic uncertainties are studied on each tag method
separately and then combined together according to their yields. To consider
all the related systematic uncertainties in the measurement of the Born cross
sections, the sources of systematic uncertainty are divided into three
categories.

The first category includes uncertainties associated with the detection
efficiencies, such as tracking, particle identification, $\pi^{0}$
reconstruction, signal region requirements of the reconstructed unstable
particles (i.e., the $P^{*}(\pi^{0})$ momentum rejection region,
$D^{0}(D^{-})$ and $D^{*0}(D^{*-})$ mass window requirements), MC simulation
model and ISR correction factors. The uncertainty of detection and PID
efficiency is 1.0% for each charged track [53], and 2.0% for $\pi^{0}$
reconstruction [54]. The uncertainties associated with the $P^{*}(\pi^{0})$,
$M(D^{0}(D^{-}))$ and $M(D^{*0}(D^{*-}))$ windows are estimated by re-
extracting the detection efficiencies with Gaussian-smeared MC samples where
the Gaussian parameters are obtained from the discrepancies between data and
MC simulation. The MC samples are corrected according to the partial-wave-
analysis (PWA) results at each energy points. To estimate the uncertainties
from PWA-corrected MC samples, the samples with different PWA results are re-
corrected by changing the possible components used in the PWA. The differences
of efficiencies extracted from nominal and re-corrected MC samples are taken
as the systematic uncertainty of the signal model. The ISR correction factors
can have an impact on the detection efficiencies by affecting the slope of the
line-shape and can be treated together, as $(1+\delta^{\rm
ISR})\cdot\epsilon$, to estimate the uncertainty from ISR correction. Since
the ISR correction factors are estimated by the fitting-iteration method, the
uncertainty of this item comes from four different parts: the differences
between the last two iterations are taken as the uncertainty of the iteration
method itself; the uncertainty from the line-shape fitting model used in the
iteration are estimated by replacing the PHSP model with a parameterized
function; the uncertainty of $(1+\delta^{\rm ISR})\cdot\epsilon$ is estimated
by 500 groups of cross-section toys which are re-sampled according to the
uncertainty of the parameters and the corresponding covariance matrix; the
vacuum polarization factors are taken from QED calculation at each energy
point and affect the slope of the line-shape, its uncertainty is estimated to
be $0.5\%$.

The second category includes uncertainties associated with the signal shape,
background shape and fit range, which affect the estimation of signal yields.
The uncertainty of the signal shape is estimated by convolving it with a
double-Gaussian function in the fit instead of a single Gaussian function in
the nominal results. The description of background shape is changed from a
$2^{\rm nd}$ order Chebyshev function to a linear function in the fit and the
differences between the two cases are considered as the uncertainty. The
uncertainty associated with the fit range is determined by altering the fit
range from $(1.91,2.12)\,\mathrm{GeV}/c^{2}$ to
$(1.911,2.12)\,\mathrm{GeV}/c^{2}$.

The last category includes the luminosity, quoted BFs where the uncertainty in
luminosity is 1.0% at each energy point [46, 47, 48], and the uncertainty from
quoted BFs of intermediate states in the subsequent decays, which are taken
from the PDG [1].

Assuming no significant correlations between sources, the total systematic
uncertainty is obtained as the sum in quadrature. Table 3 summarizes the
systematic uncertainties of the cross section at various energy points.

Table 3:  The summary of all the systematic uncertainties in the Born cross section measurement. The items marked with ‘$\dagger$’ are treated as fully correlated uncertainties, the others are uncorrelated uncertainties. (unit: %)  $\sqrt{s}$ (GeV) | Track† | PID† |  $\pi^{0}$† | Signal region | Decay Model | $(1+\delta^{\rm ISR})^{\dagger}$ | Signal shape | Bkg. shape | Fit range | $\mathcal{L}_{\rm int}^{\dagger}$ | $\mathcal{B}^{\dagger}$ | Total
---|---|---|---|---|---|---|---|---|---|---|---|---
4.226 | 3.7 | 3.7 | 2.8 | 0.5 | 0.9 | 1.4 | 2.5 | 0.9 | 3.1 | 1.0 | 2.7 | 7.9
4.236 | 3.7 | 3.7 | 2.7 | 0.4 | 0.3 | 1.0 | 2.3 | 0.5 | 2.6 | 1.0 | 2.7 | 7.6
4.244 | 3.6 | 3.6 | 2.8 | 0.6 | 0.6 | 0.8 | 1.0 | 1.2 | 1.3 | 1.0 | 2.7 | 6.9
4.258 | 3.6 | 3.6 | 2.8 | 0.6 | 0.0 | 0.7 | 2.6 | 2.5 | 3.2 | 1.0 | 2.7 | 8.1
4.267 | 3.6 | 3.6 | 2.8 | 0.7 | 0.1 | 0.7 | 1.8 | 0.5 | 1.8 | 1.0 | 2.7 | 7.1
4.288 | 3.7 | 3.7 | 2.8 | 0.2 | 0.1 | 1.9 | 1.6 | 1.6 | 1.6 | 1.0 | 2.7 | 7.3
4.312 | 3.6 | 3.6 | 2.7 | 0.3 | 0.6 | 2.9 | 1.6 | 3.0 | 1.5 | 1.0 | 2.6 | 8.0
4.337 | 3.6 | 3.6 | 2.7 | 0.4 | 0.5 | 2.3 | 1.9 | 6.0 | 2.1 | 1.0 | 2.7 | 9.6
4.358 | 3.7 | 3.7 | 2.8 | 0.5 | 0.1 | 2.3 | 1.3 | 2.5 | 1.7 | 1.0 | 2.7 | 7.7
4.377 | 3.7 | 3.7 | 2.7 | 0.4 | 0.4 | 1.2 | 0.9 | 4.6 | 1.7 | 1.0 | 2.7 | 8.3
4.397 | 3.7 | 3.7 | 2.8 | 0.3 | 0.6 | 0.6 | 0.9 | 2.7 | 0.8 | 1.0 | 2.7 | 7.3
4.416 | 3.7 | 3.7 | 2.7 | 0.6 | 0.4 | 0.8 | 0.2 | 2.3 | 0.5 | 1.0 | 2.7 | 7.0
4.436 | 3.7 | 3.7 | 2.8 | 0.3 | 0.2 | 0.9 | 1.2 | 1.3 | 0.0 | 1.0 | 2.7 | 6.8
4.467 | 3.7 | 3.7 | 2.7 | 0.5 | 0.6 | 1.0 | 0.3 | 1.5 | 0.4 | 1.0 | 2.7 | 6.8
4.527 | 3.7 | 3.7 | 2.7 | 0.4 | 0.4 | 0.9 | 3.1 | 1.0 | 2.5 | 1.0 | 2.7 | 7.8
4.575 | 3.7 | 3.7 | 2.8 | 0.5 | 0.3 | 0.6 | 1.1 | 0.6 | 0.4 | 1.0 | 2.7 | 6.8
4.600 | 3.7 | 3.7 | 2.7 | 0.5 | 1.2 | 0.9 | 0.5 | 0.3 | 0.7 | 1.0 | 2.6 | 6.8
4.612 | 3.7 | 3.7 | 2.8 | 0.2 | 0.4 | 1.1 | 0.5 | 0.7 | 0.5 | 1.0 | 2.7 | 6.7
4.628 | 3.7 | 3.7 | 2.7 | 0.4 | 0.1 | 1.3 | 0.7 | 0.9 | 0.4 | 1.0 | 2.6 | 6.8
4.641 | 3.7 | 3.7 | 2.7 | 0.3 | 1.0 | 2.7 | 0.7 | 1.1 | 0.8 | 1.0 | 2.7 | 7.3
4.661 | 3.7 | 3.7 | 2.8 | 0.3 | 2.0 | 2.6 | 0.6 | 0.3 | 0.1 | 1.0 | 2.7 | 7.3
4.681 | 3.7 | 3.7 | 2.7 | 0.3 | 0.5 | 2.4 | 0.2 | 0.5 | 0.1 | 1.0 | 2.7 | 7.0
4.698 | 3.7 | 3.7 | 2.7 | 0.2 | 1.0 | 2.5 | 0.1 | 1.1 | 0.4 | 1.0 | 2.7 | 7.2
4.740 | 3.7 | 3.7 | 2.7 | 0.5 | 2.5 | 1.5 | 0.5 | 0.8 | 0.8 | 1.0 | 2.6 | 7.3
4.750 | 3.7 | 3.7 | 2.7 | 0.3 | 2.1 | 1.0 | 0.1 | 0.6 | 0.5 | 1.0 | 2.6 | 7.0
4.781 | 3.7 | 3.7 | 2.7 | 0.4 | 0.4 | 0.8 | 0.2 | 0.4 | 0.9 | 1.0 | 2.6 | 6.7
4.843 | 3.7 | 3.7 | 2.7 | 0.2 | 2.1 | 1.2 | 0.4 | 0.9 | 0.6 | 1.0 | 2.6 | 7.1
4.918 | 3.7 | 3.7 | 2.7 | 0.4 | 1.8 | 1.2 | 0.4 | 1.4 | 0.5 | 1.0 | 2.6 | 7.1
4.951 | 3.7 | 3.7 | 2.8 | 0.4 | 0.6 | 0.8 | 0.6 | 1.6 | 0.6 | 1.0 | 2.7 | 6.9