# Measurement of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ cross
sections at center-of-mass energies from 4.190 to 4.946 GeV

M. Ablikim1, M. N. Achasov11,b, P. Adlarson69, M. Albrecht4, R. Aliberti30, A.
Amoroso68A,68C, M. R. An34, Q. An65,52, X. H. Bai60, Y. Bai51, O. Bakina31, R.
Baldini Ferroli25A, I. Balossino26A, Y. Ban41,g, V. Batozskaya1,39, D.
Becker30, K. Begzsuren28, N. Berger30, M. Bertani25A, D. Bettoni26A, F.
Bianchi68A,68C, J. Bloms62, A. Bortone68A,68C, I. Boyko31, R. A. Briere5, A.
Brueggemann62, H. Cai70, X. Cai1,52, A. Calcaterra25A, G. F. Cao1,57, N.
Cao1,57, S. A. Cetin56A, J. F. Chang1,52, W. L. Chang1,57, G. Chelkov31,a, C.
Chen38, Chao Chen49, G. Chen1, H. S. Chen1,57, M. L. Chen1,52, S. J. Chen37,
S. M. Chen55, T. Chen1, X. R. Chen27,57, X. T. Chen1, Y. B. Chen1,52, Z. J.
Chen22,h, W. S. Cheng68C, S. K. Choi 49, X. Chu38, G. Cibinetto26A, F.
Cossio68C, J. J. Cui44, H. L. Dai1,52, J. P. Dai72, A. Dbeyssi16, R. E. de
Boer4, D. Dedovich31, Z. Y. Deng1, A. Denig30, I. Denysenko31, M.
Destefanis68A,68C, F. De Mori68A,68C, Y. Ding35, J. Dong1,52, L. Y. Dong1,57,
M. Y. Dong1,52,57, X. Dong70, S. X. Du74, P. Egorov31,a, Y. L. Fan70, J.
Fang1,52, S. S. Fang1,57, W. X. Fang1, Y. Fang1, R. Farinelli26A, L.
Fava68B,68C, F. Feldbauer4, G. Felici25A, C. Q. Feng65,52, J. H. Feng53, K
Fischer63, M. Fritsch4, C. Fritzsch62, C. D. Fu1, H. Gao57, Y. N. Gao41,g,
Yang Gao65,52, S. Garbolino68C, I. Garzia26A,26B, P. T. Ge70, Z. W. Ge37, C.
Geng53, E. M. Gersabeck61, A Gilman63, L. Gong35, W. X. Gong1,52, W. Gradl30,
M. Greco68A,68C, L. M. Gu37, M. H. Gu1,52, Y. T. Gu13, C. Y Guan1,57, A. Q.
Guo27,57, L. B. Guo36, R. P. Guo43, Y. P. Guo10,f, A. Guskov31,a, T. T. Han44,
W. Y. Han34, X. Q. Hao17, F. A. Harris59, K. K. He49, K. L. He1,57, F. H.
Heinsius4, C. H. Heinz30, Y. K. Heng1,52,57, C. Herold54, Himmelreich30,d, G.
Y. Hou1,57, Y. R. Hou57, Z. L. Hou1, H. M. Hu1,57, J. F. Hu50,i, T. Hu1,52,57,
Y. Hu1, G. S. Huang65,52, K. X. Huang53, L. Q. Huang27,57, L. Q. Huang66, X.
T. Huang44, Y. P. Huang1, Z. Huang41,g, T. Hussain67, N Hüsken24,30, W.
Imoehl24, M. Irshad65,52, J. Jackson24, S. Jaeger4, S. Janchiv28, E. Jang49,
J. H. Jeong49, Q. Ji1, Q. P. Ji17, X. B. Ji1,57, X. L. Ji1,52, Y. Y. Ji44, Z.
K. Jia65,52, H. B. Jiang44, S. S. Jiang34, X. S. Jiang1,52,57, Y. Jiang57, J.
B. Jiao44, Z. Jiao20, S. Jin37, Y. Jin60, M. Q. Jing1,57, T. Johansson69, N.
Kalantar-Nayestanaki58, X. S. Kang35, R. Kappert58, M. Kavatsyuk58, B. C.
Ke74, I. K. Keshk4, A. Khoukaz62, P. Kiese30, R. Kiuchi1, L. Koch32, O. B.
Kolcu56A, B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc39,69, W. Kühn32, J. J.
Lane61, J. S. Lange32, P. Larin16, A. Lavania23, L. Lavezzi68A,68C, Z. H.
Lei65,52, H. Leithoff30, M. Lellmann30, T. Lenz30, C. Li42, C. Li38, C. H.
Li34, Cheng Li65,52, D. M. Li74, F. Li1,52, G. Li1, H. Li46, H. Li65,52, H. B.
Li1,57, H. J. Li17, H. N. Li50,i, J. Q. Li4, J. S. Li53, J. W. Li44, Ke Li1,
L. J Li1, L. K. Li1, Lei Li3, M. H. Li38, P. R. Li33,j,k, S. X. Li10, S. Y.
Li55, T. Li44, W. D. Li1,57, W. G. Li1, X. H. Li65,52, X. L. Li44, Xiaoyu
Li1,57, H. Liang1,57, H. Liang65,52, H. Liang29, Y. F. Liang48, Y. T.
Liang27,57, G. R. Liao12, L. Z. Liao44, J. Libby23, A. Limphirat54, C. X.
Lin53, D. X. Lin27,57, T. Lin1, B. J. Liu1, C. X. Liu1, D. Liu16,65, F. H.
Liu47, Fang Liu1, Feng Liu6, G. M. Liu50,i, H. Liu33,j,k, H. B. Liu13, H. M.
Liu1,57, Huanhuan Liu1, Huihui Liu18, J. B. Liu65,52, J. L. Liu66, J. Y.
Liu1,57, K. Liu1, K. Y. Liu35, Ke Liu19, L. Liu65,52, Lu Liu38, M. H. Liu10,f,
P. L. Liu1, Q. Liu57, S. B. Liu65,52, T. Liu10,f, W. K. Liu38, W. M. Liu65,52,
X. Liu33,j,k, Y. Liu33,j,k, Y. B. Liu38, Z. A. Liu1,52,57, Z. Q. Liu44, X. C.
Lou1,52,57, F. X. Lu53, H. J. Lu20, J. G. Lu1,52, X. L. Lu1, Y. Lu7, Y. P.
Lu1,52, Z. H. Lu1, C. L. Luo36, M. X. Luo73, T. Luo10,f, X. L. Luo1,52, X. R.
Lyu57, Y. F. Lyu38, F. C. Ma35, H. L. Ma1, L. L. Ma44, M. M. Ma1,57, Q. M.
Ma1, R. Q. Ma1,57, R. T. Ma57, X. Y. Ma1,52, Y. Ma41,g, F. E. Maas16, M.
Maggiora68A,68C, S. Maldaner4, S. Malde63, Q. A. Malik67, A. Mangoni25B, Y. J.
Mao41,g, Z. P. Mao1, S. Marcello68A,68C, Z. X. Meng60, G. Mezzadri26A, H.
Miao1, T. J. Min37, R. E. Mitchell24, X. H. Mo1,52,57, N. Yu. Muchnoi11,b, Y.
Nefedov31, F. Nerling16,d, I. B. Nikolaev11,b, Z. Ning1,52, S. Nisar9,l, Y.
Niu 44, S. L. Olsen57, Q. Ouyang1,52,57, S. Pacetti25B,25C, X. Pan10,f, Y.
Pan51, A. Pathak1, A. Pathak29, M. Pelizaeus4, H. P. Peng65,52, J.
Pettersson69, J. L. Ping36, R. G. Ping1,57, S. Plura30, S. Pogodin31, V.
Prasad65,52, F. Z. Qi1, H. Qi65,52, H. R. Qi55, M. Qi37, T. Y. Qi10,f, S.
Qian1,52, W. B. Qian57, Z. Qian53, C. F. Qiao57, J. J. Qin66, L. Q. Qin12, X.
P. Qin10,f, X. S. Qin44, Z. H. Qin1,52, J. F. Qiu1, S. Q. Qu38, K. H.
Rashid67, C. F. Redmer30, K. J. Ren34, A. Rivetti68C, V. Rodin58, M. Rolo68C,
G. Rong1,57, Ch. Rosner16, S. N. Ruan38, H. S. Sang65, A. Sarantsev31,c, Y.
Schelhaas30, C. Schnier4, K. Schoenning69, M. Scodeggio26A,26B, K. Y.
Shan10,f, W. Shan21, X. Y. Shan65,52, J. F. Shangguan49, L. G. Shao1,57, M.
Shao65,52, C. P. Shen10,f, H. F. Shen1,57, X. Y. Shen1,57, B. A. Shi57, H. C.
Shi65,52, J. Y. Shi1, q. q. Shi49, R. S. Shi1,57, X. Shi1,52, X. D Shi65,52,
J. J. Song17, W. M. Song29,1, Y. X. Song41,g, S. Sosio68A,68C, S.
Spataro68A,68C, F. Stieler30, K. X. Su70, P. P. Su49, Y. J. Su57, G. X. Sun1,
H. Sun57, H. K. Sun1, J. F. Sun17, L. Sun70, S. S. Sun1,57, T. Sun1,57, W. Y.
Sun29, X Sun22,h, Y. J. Sun65,52, Y. Z. Sun1, Z. T. Sun44, Y. H. Tan70, Y. X.
Tan65,52, C. J. Tang48, G. Y. Tang1, J. Tang53, L. Y Tao66, Q. T. Tao22,h, M.
Tat63, J. X. Teng65,52, V. Thoren69, W. H. Tian46, Y. Tian27,57, I. Uman56B,
B. Wang1, B. L. Wang57, C. W. Wang37, D. Y. Wang41,g, F. Wang66, H. J.
Wang33,j,k, H. P. Wang1,57, K. Wang1,52, L. L. Wang1, M. Wang44, M. Z.
Wang41,g, Meng Wang1,57, S. Wang10,f, S. Wang12, T. Wang10,f, T. J. Wang38, W.
Wang53, W. H. Wang70, W. P. Wang65,52, X. Wang41,g, X. F. Wang33,j,k, X. L.
Wang10,f, Y. D. Wang40, Y. F. Wang1,52,57, Y. H. Wang42, Y. Q. Wang1, Yaqian
Wang15,1, Yi2020 Wang55, Z. Wang1,52, Z. Y. Wang1,57, Ziyi Wang57, D. H.
Wei12, F. Weidner62, S. P. Wen1, D. J. White61, U. Wiedner4, G. Wilkinson63,
M. Wolke69, L. Wollenberg4, J. F. Wu1,57, L. H. Wu1, L. J. Wu1,57, X. Wu10,f,
X. H. Wu29, Y. Wu65, Z. Wu1,52, L. Xia65,52, T. Xiang41,g, D. Xiao33,j,k, G.
Y. Xiao37, H. Xiao10,f, S. Y. Xiao1, Y. L. Xiao10,f, Z. J. Xiao36, C. Xie37,
X. H. Xie41,g, Y. Xie44, Y. G. Xie1,52, Y. H. Xie6, Z. P. Xie65,52, T. Y.
Xing1,57, C. F. Xu1, C. J. Xu53, G. F. Xu1, H. Y. Xu60, Q. J. Xu14, S. Y.
Xu64, X. P. Xu49, Y. C. Xu57, Z. P. Xu37, F. Yan10,f, L. Yan10,f, W. B.
Yan65,52, W. C. Yan74, H. J. Yang45,e, H. L. Yang29, H. X. Yang1, L. Yang46,
S. L. Yang57, Tao Yang1, Y. F. Yang38, Y. X. Yang1,57, Yifan Yang1,57, M.
Ye1,52, M. H. Ye8, J. H. Yin1, Z. Y. You53, B. X. Yu1,52,57, C. X. Yu38, G.
Yu1,57, T. Yu66, X. D. Yu41,g, C. Z. Yuan1,57, L. Yuan2, S. C. Yuan1, X. Q.
Yuan1, Y. Yuan1,57, Z. Y. Yuan53, C. X. Yue34, A. A. Zafar67, F. R. Zeng44, X.
Zeng6, Y. Zeng22,h, Y. H. Zhan53, A. Q. Zhang1, B. L. Zhang1, B. X. Zhang1, D.
H. Zhang38, G. Y. Zhang17, H. Zhang65, H. H. Zhang53, H. H. Zhang29, H. Y.
Zhang1,52, J. L. Zhang71, J. Q. Zhang36, J. W. Zhang1,52,57, J. X.
Zhang33,j,k, J. Y. Zhang1, J. Z. Zhang1,57, Jianyu Zhang1,57, Jiawei
Zhang1,57, L. M. Zhang55, L. Q. Zhang53, Lei Zhang37, P. Zhang1, Q. Y.
Zhang34,74, Shulei Zhang22,h, X. D. Zhang40, X. M. Zhang1, X. Y. Zhang44, X.
Y. Zhang49, Y. Zhang63, Y. T. Zhang74, Y. H. Zhang1,52, Yan Zhang65,52, Yao
Zhang1, Z. H. Zhang1, Z. Y. Zhang38, Z. Y. Zhang70, G. Zhao1, J. Zhao34, J. Y.
Zhao1,57, J. Z. Zhao1,52, Lei Zhao65,52, Ling Zhao1, M. G. Zhao38, Q. Zhao1,
S. J. Zhao74, Y. B. Zhao1,52, Y. X. Zhao27,57, Z. G. Zhao65,52, A.
Zhemchugov31,a, B. Zheng66, J. P. Zheng1,52, Y. H. Zheng57, B. Zhong36, C.
Zhong66, X. Zhong53, H. Zhou44, L. P. Zhou1,57, X. Zhou70, X. K. Zhou57, X. R.
Zhou65,52, X. Y. Zhou34, Y. Z. Zhou10,f, J. Zhu38, K. Zhu1, K. J. Zhu1,52,57,
L. X. Zhu57, S. H. Zhu64, S. Q. Zhu37, T. J. Zhu71, W. J. Zhu10,f, Y. C.
Zhu65,52, Z. A. Zhu1,57, B. S. Zou1, J. H. Zou1 (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 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
10 Fudan University, Shanghai 200433, People’s Republic of China
11 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Hebei University, Baoding 071002, People’s Republic of China
16 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
17 Henan Normal University, Xinxiang 453007, People’s Republic of China
18 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
19 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
20 Huangshan College, Huangshan 245000, People’s Republic of China
21 Hunan Normal University, Changsha 410081, People’s Republic of China
22 Hunan University, Changsha 410082, People’s Republic of China
23 Indian Institute of Technology Madras, Chennai 600036, India
24 Indiana University, Bloomington, Indiana 47405, USA
25 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
26 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
27 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
28 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
29 Jilin University, Changchun 130012, People’s Republic of China
30 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
31 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
32 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
33 Lanzhou University, Lanzhou 730000, People’s Republic of China
34 Liaoning Normal University, Dalian 116029, People’s Republic of China
35 Liaoning University, Shenyang 110036, People’s Republic of China
36 Nanjing Normal University, Nanjing 210023, People’s Republic of China
37 Nanjing University, Nanjing 210093, People’s Republic of China
38 Nankai University, Tianjin 300071, People’s Republic of China
39 National Centre for Nuclear Research, Warsaw 02-093, Poland
40 North China Electric Power University, Beijing 102206, People’s Republic of
China
41 Peking University, Beijing 100871, People’s Republic of China
42 Qufu Normal University, Qufu 273165, People’s Republic of China
43 Shandong Normal University, Jinan 250014, People’s Republic of China
44 Shandong University, Jinan 250100, People’s Republic of China
45 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
46 Shanxi Normal University, Linfen 041004, People’s Republic of China
47 Shanxi University, Taiyuan 030006, People’s Republic of China
48 Sichuan University, Chengdu 610064, People’s Republic of China
49 Soochow University, Suzhou 215006, People’s Republic of China
50 South China Normal University, Guangzhou 510006, People’s Republic of China
51 Southeast University, Nanjing 211100, People’s Republic of China
52 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
53 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
54 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
55 Tsinghua University, Beijing 100084, People’s Republic of China
56 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
57 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
58 University of Groningen, NL-9747 AA Groningen, The Netherlands
59 University of Hawaii, Honolulu, Hawaii 96822, USA
60 University of Jinan, Jinan 250022, People’s Republic of China
61 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
62 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
63 University of Oxford, Keble Rd, Oxford, UK OX13RH
64 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
65 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
66 University of South China, Hengyang 421001, People’s Republic of China
67 University of the Punjab, Lahore-54590, Pakistan
68 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
69 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
70 Wuhan University, Wuhan 430072, People’s Republic of China
71 Xinyang Normal University, Xinyang 464000, People’s Republic of China
72 Yunnan University, Kunming 650500, People’s Republic of China
73 Zhejiang University, Hangzhou 310027, People’s Republic of China
74 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

Using data samples collected with the BESIII detector operating at the BEPCII
storage ring, we measure the cross sections of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process at center-of-mass
energies from 4.190 to 4.946 GeV with a partial reconstruction method. Two
resonance structures are seen and the resonance parameters are determined from
a fit to the cross section line shape. The first resonance we observe has a
mass of (4373.1 $\pm$ 4.0 $\pm$ 2.2) MeV/$c^{2}$ and a width of (146.5 $\pm$
7.4 $\pm$ 1.3) MeV, in agreement with those of the $Y(4390)$ state; the other
resonance has a mass of (4706 $\pm$ 11 $\pm$ 4) MeV/$c^{2}$, a width of (45
$\pm$ 28 $\pm$ 9) MeV, and a statistical significance of $4.1$ standard
deviations ($\sigma$). This is the first evidence for a vector state at this
mass value. The spin-$3$ $D$-wave charmonium state $X(3842)$ is searched for
through the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}X(3842)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
process, and evidence with a significance of $4.2\sigma$ is found in the data
samples with center-of-mass energies from 4.600 to 4.700 GeV.

##  I Introduction

The charmonium states with masses below the open charm threshold and a few
vector states above the open charm threshold are well-established pdg , and
they agree well with theoretical calculations based on QCD Brambilla:2004jw ;
review4 ; Brambilla:2014jmp  and QCD-inspired potential models eichten ;
godfrey ; barnes . The vector charmonia $\psi(4040)$, $\psi(4160)$, and
$\psi(4415)$ were assigned as the $3^{3}S_{1}$, $2^{3}D_{1}$, and $4^{3}S_{1}$
states, respectively, since only these three structures were observed in the
total $e^{+}e^{-}$ annihilation cross section bes2_psis .

However, a few more vector states, the $Y$ states, were discovered by the
BaBar and Belle $B$-factory experiments PBFB . These include the $Y(4260)$
babar_y4260 , the $Y(4360)$ babar_y4360 ; belle_y4660 , and the $Y(4660)$
belle_y4660 . They are produced via the initial state radiation (ISR) process
in $e^{+}e^{-}$ annihilation and, thus, are vector states with quantum numbers
$J^{PC}=1^{--}$, the same as the excited $\psi$ states listed above. These
states were observed in hidden-charm final states in contrast to the excited
$\psi$ states peaking in the inclusive hadronic cross section bes2_psis ;
uglov-kmatrix . The final states in the latter are dominated by open-charm
meson pairs.

In potential models, five vector charmonium states with masses between 4.0 and
4.7 GeV/$c^{2}$ are expected, namely the $\psi(3^{3}S_{1})$,
$\psi(2^{3}D_{1})$, $\psi(4^{3}S_{1})$, $\psi(3^{3}D_{1})$, and
$\psi(5^{3}S_{1})$. The first three are often identified as the $\psi(4040)$,
$\psi(4160)$, and $\psi(4415)$ states, respectively. The masses of the as yet
undiscovered $\psi(3D)$ and $\psi(5S)$ are expected to be higher than 4.4
GeV/$c^{2}$. However, six vector states have been identified in the mass
region between 4.0 and 4.7 GeV/$c^{2}$, as listed above. This makes the
$Y(4260)$, the $Y(4360)$, and perhaps the $Y(4660)$ states good candidates for
new types of exotic particles, and has stimulated theoretical work regarding
their interpretation. They have been variously considered as candidates for
tetraquark states, molecular states, hybrid states, and hadro-charmonia
review1 ; review2 ; review3 ; review4 .

With masses above the open-charm thresholds, both $Y$ and excited $\psi$
states should couple to open-charm final states, and many studies have been
performed to measure the cross sections of two-body final states with a pair
of charmed mesons DD ; 2018DD ; 2020DsstDs1 ; 2021DsstDsj  and three-body
final states with a pair of charmed mesons and a light meson bes3_ddstarpi .
Although four-body final states with a pair of charmed mesons and a pair of
light mesons 2019HY ; 2020ZY  have also been studied, and the production of
intermediate two-body ($D_{1}(2420)\bar{D}+c.c.$) and three-body
($\pi^{+}\pi^{-}\psi(3770)$) states have been observed, the total cross
section of the four-body final states has not been reported. In such final
states, new exotic particles and new decay modes of known $Y$ and excited
$\psi$ states can be searched for.

In this paper, we report the first measurement of the cross sections of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process with the data samples
taken at 37 center-of-mass energies ($\sqrt{s}$) from 4.190 to 4.946 GeV, the
study of the decays of the excited $\psi$ and $Y$ states into this final
state, and the observation of a new resonant structure in the cross section
line shape.

Two of the $D$-wave spin-triplet states $\psi(1^{3}D_{1})$ ($\psi(3770)$) and
$\psi(1^{3}D_{2})$ ($\psi_{2}(3823)$) have been observed in the $e^{+}e^{-}$
annihilation process $e^{+}e^{-}\to\pi^{+}\pi^{-}\psi(1D)$ 2019HY ; 2020ZY ;
2015X3823 . As their spin partner, the $\psi(1^{3}D_{3})$ ($X(3842)$) observed
by LHCb 2019X3842  can also be produced in a similar process and can be
searched for in the $\pi^{+}\pi^{-}D^{+}D^{-}$ final state, since the
$X(3842)$ decays to $D\bar{D}$.

##  II Detector and data samples

The BESIII detector Ablikim:2009aa  records $e^{+}e^{-}$ collisions provided
by the BEPCII storage ring Yu:IPAC2016-TUYA01 . The cylindrical core of the
BESIII detector covers 93% of the full solid angle and consists of a helium-
based multilayer drift chamber (MDC), a plastic scintillator time-of-flight
system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all
enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic
field. The solenoid is supported by an octagonal flux-return yoke with
resistive plate counter muon identification modules interleaved with steel.
The charged-particle momentum resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and
the $dE/dx$ resolution is $6\%$ for electrons from Bhabha scattering. The EMC
measures photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in
the barrel (end cap) region. The time resolution in the TOF barrel region is
68 ps, while that in the end cap region is 110 ps. The end cap TOF system was
upgraded in 2015 using multi-gap resistive plate chamber technology, providing
a time resolution of 60 ps etof .

In this analysis, the experimental data samples used are listed in Table 1.
The center-of-mass energy is measured using dimuon events with a precision of
0.8 MeV for data samples with $\sqrt{s}$ smaller than 4.610 GeV 2016mumu ;
2021mumu  and using $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ events with a
precision of 0.6 MeV for data samples with $\sqrt{s}$ larger than or equal to
4.610 GeV 2022LcLc . The integrated luminosity is determined by analyzing
large angle Bhabha scattering events with an uncertainty of 1.0% 2019SWM ;
2022lum ; 2022LcLc . The integrated luminosity of the total data sample is
$17.4~{}\rm{fb}^{-1}$.

Table 1: Yields and cross sections results for the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process at different center-
of-mass energies. Here, $\sigma$ is the cross section of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process, where the first
uncertainties are statistical and the second systematic; $\mathscr{L}$, $S$,
and $\sigma_{\rm ul}$ are the integrated luminosity, statistical significance,
and upper limit of the cross section at 90% confidence level, respectively.
$N_{\rm signal}$ and $N_{\rm sideband}$ are the number of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ events from fits to
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distributions in $M(K^{-}\pi^{+}\pi^{+})$
signal and sideband regions, respectively.

$\sqrt{s}$ nominal value (GeV) |  $\sqrt{s}$ (MeV) |  $\mathscr{L}$ ($\rm{pb}^{-1}$) | $N_{\rm{signal}}$ | $N_{\rm{sideband}}$ | $\sigma\ (\rm{pb})$ | $S$ |  $\sigma_{\rm{ul}}$ (pb)
---|---|---|---|---|---|---|---
4.190 | $4188.59\pm 0.15\pm 0.68$ |  570.0 |  $-$8$\ \pm\ $ | 10 |  $-$17$\ \pm\ $ | 11 | 0.1$\ \pm\ $ | 0.9 $\ \pm\ $ | 0.0 | - | 1.0
4.200 | $4199.15\pm 0.05\pm 0.34$ |  526.0 |  $-$5$\ \pm\ $ | 11 |  $-$15$\ \pm\ $ | 12 | 0.2$\ \pm\ $ | 1.0 $\ \pm\ $ | 0.0 | - | 1.2
4.210 | $4207.73\pm 0.14\pm 0.61$ |  572.1 | 15$\ \pm\ $ | 13 | 19$\ \pm\ $ | 14 | 0.3$\ \pm\ $ | 1.0 $\ \pm\ $ | 0.1 | $\ \ \ 1.2\sigma$ | 2.6
4.220 | $4217.13\pm 0.14\pm 0.67$ |  569.2 | 17$\ \pm\ $ | 12 | 14$\ \pm\ $ | 13 | 0.7$\ \pm\ $ | 0.9 $\ \pm\ $ | 0.1 | $\ \ \ 1.5\sigma$ | 2.6
4.230 | $4225.54\pm 0.05\pm 0.65$ | 1100.9 | 119$\ \pm\ $ | 25 | 12$\ \pm\ $ | 20 | 3.4$\ \pm\ $ | 0.8 $\ \pm\ $ | 0.3 | $\ \ \ 5.9\sigma$ | -
4.237 | $4235.77\pm 0.04\pm 0.30$ |  530.3 | 25$\ \pm\ $ | 14 |  $-$29$\ \pm\ $ | 13 | 2.6$\ \pm\ $ | 1.0 $\ \pm\ $ | 0.2 | $\ \ \ 1.9\sigma$ | 3.5
4.245 | $4241.66\pm 0.12\pm 0.73$ |  55.9 | 5$\ \pm\ $ | 6 |  $-$3$\ \pm\ $ | 4 | 4.0$\ \pm\ $ | 3.7 $\ \pm\ $ | 0.3 | $\ \ \ 0.9\sigma$ | 9.0
4.246 | $4243.97\pm 0.04\pm 0.30$ |  538.1 | 101$\ \pm\ $ | 19 | 1$\ \pm\ $ | 15 | 6.1$\ \pm\ $ | 1.3 $\ \pm\ $ | 0.7 | $\ \ \ 6.6\sigma$ | -
4.260 | $4258.00\pm 0.06\pm 0.60$ |  825.7 | 159$\ \pm\ $ | 26 | 17$\ \pm\ $ | 22 | 5.6$\ \pm\ $ | 1.1 $\ \pm\ $ | 0.5 | $\ \ \ 7.5\sigma$ | -
4.270 | $4266.81\pm 0.04\pm 0.32$ |  531.1 | 61$\ \pm\ $ | 18 |  $-$27$\ \pm\ $ | 17 | 4.3$\ \pm\ $ | 1.2 $\ \pm\ $ | 0.4 | $\ \ \ 3.6\sigma$ | 6.7
4.280 | $4277.78\pm 0.11\pm 0.52$ |  175.7 | 25$\ \pm\ $ | 12 | 2$\ \pm\ $ | 11 | 4.2$\ \pm\ $ | 2.4 $\ \pm\ $ | 0.4 | $\ \ \ 2.2\sigma$ | 9.0
4.290 | $4288.43\pm 0.06\pm 0.34$ |  502.4 | 140$\ \pm\ $ | 23 | 4$\ \pm\ $ | 20 | 8.6$\ \pm\ $ | 1.6 $\ \pm\ $ | 0.7 | $\ \ \ 7.1\sigma$ | -
4.310 | $4307.89\pm 0.17\pm 0.63$ |  45.1 | 25$\ \pm\ $ | 8 |  $-$4$\ \pm\ $ | 7 | 17.1$\ \pm\ $ | 5.5 $\ \pm\ $ | 1.5 | $\ \ \ 3.4\sigma$ | 30
4.315 | $4312.68\pm 0.06\pm 0.35$ |  501.2 | 263$\ \pm\ $ | 29 | 9$\ \pm\ $ | 23 | 15.4$\ \pm\ $ | 1.9 $\ \pm\ $ | 1.3 | $\ \ \ \ 11\sigma$ | -
4.340 | $4337.93\pm 0.06\pm 0.35$ |  505.0 | 666$\ \pm\ $ | 42 | 20$\ \pm\ $ | 27 | 36.9$\ \pm\ $ | 2.5 $\ \pm\ $ | 3.1 | $\ \ \ \ \ 21\sigma$ | -
4.360 | $4358.26\pm 0.05\pm 0.62$ |  544.0 | 1038$\ \pm\ $ | 53 | 8$\ \pm\ $ | 34 | 48.2$\ \pm\ $ | 2.6 $\ \pm\ $ | 4.1 | $\ \ \ \ \ 26\sigma$ | -
4.380 | $4377.88\pm 0.06\pm 0.35$ |  522.7 | 1184$\ \pm\ $ | 67 |  $-$35$\ \pm\ $ | 37 | 61.6$\ \pm\ $ | 3.6 $\ \pm\ $ | 5.2 | $\ \ \ \ \ 25\sigma$ | -
4.390 | $4387.40\pm 0.17\pm 0.65$ |  55.6 | 111$\ \pm\ $ | 18 | 19$\ \pm\ $ | 13 | 46.2$\ \pm\ $ | 8.8 $\ \pm\ $ | 3.9 | $\ \ \ 7.4\sigma$ | -
4.400 | $4396.83\pm 0.06\pm 0.36$ |  507.8 | 1217$\ \pm\ $ | 62 | 61$\ \pm\ $ | 43 | 61.9$\ \pm\ $ | 3.5 $\ \pm\ $ | 5.2 | $\ \ \ \ \ 24\sigma$ | -
4.420 | $4415.94\pm 0.04\pm 0.62$ | 1090.7 | 3144$\ \pm\ $ | 112 | 216$\ \pm\ $ | 71 | 67.7$\ \pm\ $ | 2.6 $\ \pm\ $ | 5.8 | $\ \ \ \ \ 37\sigma$ | -
4.440 | $4437.59\pm 0.06\pm 0.35$ |  569.9 | 1588$\ \pm\ $ | 85 | 140$\ \pm\ $ | 59 | 65.1$\ \pm\ $ | 3.9 $\ \pm\ $ | 5.9 | $\ \ \ \ \ 23\sigma$ | -
4.470 | $4467.06\pm 0.11\pm 0.73$ |  111.1 | 192$\ \pm\ $ | 35 | 36$\ \pm\ $ | 25 | 36.0$\ \pm\ $ | 7.8 $\ \pm\ $ | 3.9 | $\ \ \ 6.8\sigma$ | -
4.530 | $4527.14\pm 0.11\pm 0.72$ |  112.1 | 141$\ \pm\ $ | 34 | 17$\ \pm\ $ | 28 | 30.4$\ \pm\ $ | 8.6 $\ \pm\ $ | 3.1 | $\ \ \ 4.1\sigma$ | 41
4.575 | $4574.50\pm 0.18\pm 0.70$ |  48.9 | 39$\ \pm\ $ | 18 | 12$\ \pm\ $ | 19 | 15.5$\ \pm\ $ | 9.9 $\ \pm\ $ | 1.4 | $\ \ \ 2.2\sigma$ | 38
4.600 | $4599.53\pm 0.07\pm 0.74$ |  586.9 | 811$\ \pm\ $ | 74 |  $-$16$\ \pm\ $ | 69 | 31.2$\ \pm\ $ | 3.1 $\ \pm\ $ | 2.8 | $\ \ \ \ \ 12\sigma$ | -
4.612 | $4611.86\pm 0.12\pm 0.32$ |  103.8 | 139$\ \pm\ $ | 31 | 42$\ \pm\ $ | 29 | 27.3$\ \pm\ $ | 8.1 $\ \pm\ $ | 2.3 | $\ \ \ 4.9\sigma$ | 40
4.620 | $4628.00\pm 0.06\pm 0.32$ |  521.5 | 758$\ \pm\ $ | 90 | 30$\ \pm\ $ | 72 | 33.7$\ \pm\ $ | 4.4 $\ \pm\ $ | 2.9 | $\ \ \ \ \ 11\sigma$ | -
4.640 | $4640.91\pm 0.06\pm 0.38$ |  552.4 | 725$\ \pm\ $ | 85 |  $-$65$\ \pm\ $ | 71 | 32.2$\ \pm\ $ | 3.9 $\ \pm\ $ | 2.8 | $\ \ \ \ \ 10\sigma$ | -
4.660 | $4661.24\pm 0.06\pm 0.29$ |  529.6 | 814$\ \pm\ $ | 93 |  $-$51$\ \pm\ $ | 73 | 38.1$\ \pm\ $ | 4.6 $\ \pm\ $ | 3.4 | $\ \ \ \ \ 11\sigma$ | -
4.680 | $4681.92\pm 0.08\pm 0.29$ | 1669.3 | 2427$\ \pm\ $ | 156 |  $-$12$\ \pm\ $ | 128 | 33.7$\ \pm\ $ | 2.3 $\ \pm\ $ | 2.8 | $\ \ \ \ \ 19\sigma$ | -
4.700 | $4698.82\pm 0.10\pm 0.39$ |  536.5 | 1020$\ \pm\ $ | 85 |  $-$58$\ \pm\ $ | 76 | 45.7$\ \pm\ $ | 4.1 $\ \pm\ $ | 4.0 | $\ \ \ \ \ 13\sigma$ | -
4.740 | $4739.70\pm 0.20\pm 0.30$ |  164.3 | 330$\ \pm\ $ | 45 | 47$\ \pm\ $ | 41 | 39.8$\ \pm\ $ | 6.5 $\ \pm\ $ | 3.4 | $\ \ \ 8.2\sigma$ | -
4.750 | $4750.05\pm 0.12\pm 0.29$ |  367.2 | 781$\ \pm\ $ | 71 | 71$\ \pm\ $ | 59 | 43.2$\ \pm\ $ | 4.5 $\ \pm\ $ | 3.8 | $\ \ \ \ \ 13\sigma$ | -
4.780 | $4780.54\pm 0.12\pm 0.33$ |  512.8 | 1042$\ \pm\ $ | 94 | 217$\ \pm\ $ | 78 | 39.6$\ \pm\ $ | 4.3 $\ \pm\ $ | 3.3 | $\ \ \ \ \ 14\sigma$ | -
4.840 | $4843.07\pm 0.20\pm 0.31$ |  527.3 | 1050$\ \pm\ $ | 100 | 10$\ \pm\ $ | 81 | 43.4$\ \pm\ $ | 4.5 $\ \pm\ $ | 3.7 | $\ \ \ \ \ 13\sigma$ | -
4.914 | $4918.02\pm 0.34\pm 0.35$ |  208.1 | 471$\ \pm\ $ | 67 | 40$\ \pm\ $ | 58 | 48.6$\ \pm\ $ | 7.9 $\ \pm\ $ | 4.2 | $\ \ \ 8.2\sigma$ | -
4.946 | $4950.93\pm 0.36\pm 0.44$ |  160.4 | 247$\ \pm\ $ | 51 | 80$\ \pm\ $ | 51 | 29.4$\ \pm\ $ | 8.2 $\ \pm\ $ | 2.5 | $\ \ \ 5.0\sigma$ | -

To increase signal yields, a partial reconstruction method is employed for the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process. A $D^{+}$ meson is
reconstructed via its high branching fraction (9.38%) decay mode,
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, and an additional $\pi^{+}\pi^{-}$
pair is selected from the remaining charged tracks. The recoil mass of the
$\pi^{+}\pi^{-}D^{+}$ system is used to identify the $D^{-}$ meson. Unless
explicitly mentioned, the inclusion of charge conjugate modes is implied
throughout the context.

Simulated data samples produced with a geant4-based geant4  Monte Carlo (MC)
package, which includes the geometric description of the BESIII detector and
the detector response, are used to determine detection efficiencies and to
estimate backgrounds. The simulation models the beam energy spread and ISR in
the $e^{+}e^{-}$ annihilations with the generator kkmc kkmc .

In order to estimate the potential background contributions, inclusive MC
samples generated at $\sqrt{s}=$ 4.230, 4.360, 4.420, and 4.600 GeV are used.
The inclusive MC sample includes the production of open charm processes, the
ISR production of vector charmonium(-like) states, and the continuum processes
incorporated in kkmc kkmc . The known decay modes are modelled with evtgen
evtgen  using branching fractions taken from the Particle Data Group (PDG) pdg
, and the remaining unknown charmonium decays are modelled with lundcharm
lundcharm . Final state radiation (FSR) from charged final state particles is
incorporated using the photos package photos .

For the optimization of the selection criteria and signal extraction, the
following MC samples are produced at each $\sqrt{s}$: $e^{+}e^{-}\rightarrow
D_{1}(2420)^{+}D^{-}$, with $D_{1}(2420)^{+}\rightarrow D^{+}\pi^{+}\pi^{-}$,
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(3770)$, with $\psi(3770)\rightarrow
D^{+}D^{-}$, where $D_{1}(2420)^{+}D^{-}$ and $\pi^{+}\pi^{-}\psi(3770)$ are
uniformly distributed in the phase space, and
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}(\rm{PHSP})$ where the
$\pi^{+}\pi^{-}D^{+}D^{-}$ events are uniformly distributed in the phase space
to represent the processes with unknown intermediate states. For the
$D_{1}(2420)^{+}\rightarrow D^{+}\pi^{+}\pi^{-}$ process, the
$D^{+}\pi^{+}\pi^{-}$ events are also uniformly distributed in the phase
space.

##  III Event Selection

Charged tracks detected in the MDC are required to be within a polar angle
($\theta$) range of $|\cos\theta|<0.93$, where $\theta$ is defined with
respect to the z-axis, which is the symmetry axis of the MDC. For charged
tracks not originating from $K_{S}^{0}$ or $\Lambda$ decays, the distance of
closest approach to the interaction point (IP) must be less than 10 cm along
the $z$-axis, $|V_{z}|$, and less than 1 cm in the transverse plane,
$|V_{xy}|$. A charged track should have a good quality in the track fitting
and be within the angle coverage of the MDC, $|\cos\theta|<0.93$. A good
charged track (excluding those from $K_{S}^{0}$ or $\Lambda$ decays) is
required to be within 1 cm of the $e^{+}e^{-}$ annihilation interaction point
(IP) transverse to the beam line ($|V_{xy}|<1$ cm) and within 10 cm of the IP
along the beam axis ($|V_{z}|<10$ cm).

Particle identification (PID) for charged tracks combines measurements of the
energy deposited in the MDC (d$E$/d$x$) and the flight time in the TOF to form
likelihoods $\mathcal{L}(h)~{}(h=p,K,\pi)$ for each hadron $h$ hypothesis.
Tracks are identified as protons when the proton hypothesis has the greatest
likelihood ($\mathcal{L}(p)>\mathcal{L}(K)$ and
$\mathcal{L}(p)>\mathcal{L}(\pi)$), while charged kaons and pions are
identified by comparing the likelihoods for the kaon and pion hypotheses,
$\mathcal{L}(K)>\mathcal{L}(\pi)$ and $\mathcal{L}(\pi)>\mathcal{L}(K)$,
respectively.

Particle identification (PID) for charged tracks combines measurements of the
energy loss in the MDC ($dE/dx$) and the flight time in the TOF. Likelihoods
$\mathcal{L}(h)$ ($h=K$, $\pi$, $p$) for each hadron $h$ hypothesis are formed
and each track is assigned to the particle type corresponding to the
hypothesis with the greatest likelihood. A proton (or an anti-proton) is
identified if $\mathcal{L}(p)>\mathcal{L}(\pi)$ and
$\mathcal{L}(p)>\mathcal{L}(K)$. In order to suppress background from
$e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ and other possible
charmed baryons, events with proton or anti-proton tracks are rejected.
Charged kaons and pions are identified by comparing the likelihoods for the
kaon and pion hypotheses, $\mathcal{L}(K)>\mathcal{L}(\pi)$ and
$\mathcal{L}(\pi)>\mathcal{L}(K)$, respectively.

To reconstruct the $D^{+}$ meson, one $K^{-}$ and two $\pi^{+}$ candidate
tracks are selected. They are required to originate from a common vertex and
the quality of the vertex fit is required to satisfy $\chi^{2}_{VF}<100$. All
possible $K^{-}\pi^{+}\pi^{+}$ combinations in the event which satisfy these
criteria are kept as $D^{+}$ candidates for further analysis. There are 1.1
$D^{+}$ candidates per event on average after $M(K^{-}\pi^{+}\pi^{+})$ and
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ requirements mentioned in the following
paragraph. For each $D^{+}$ candidate, a $\pi^{+}\pi^{-}$ pair is selected
from the charged tracks not used in $D^{+}$ reconstruction (referred to as
$\pi_{d}^{+}$ and $\pi_{d}^{-}$) and the recoil mass of
$D^{+}\pi_{d}^{+}\pi_{d}^{-}$ ($RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$) is
calculated to identify the $D^{-}$ candidate.

Figure 1 shows $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ versus the invariant mass of
the $K^{-}\pi^{+}\pi^{+}$ ($M(K^{-}\pi^{+}\pi^{+})$) for data samples at
$\sqrt{s}=$ 4.230, 4.420, and 4.680 GeV. Clear $D^{-}$ and $D^{+}$ signal
peaks can be seen in the $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ and
$M(K^{-}\pi^{+}\pi^{+})$ distributions, respectively. The
$\pi^{+}\pi^{-}D^{+}D^{-}$ signal region is defined as
$|M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}|<d_{M}$ and
$|RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})-m_{D^{-}}|<d_{RM}$, and the sideband regions
as $-5d_{M}<M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}<-3d_{M}$ or
$3d_{M}<M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}<5d_{M}$ , and
$-5d_{RM}<RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})-m_{D^{-}}<-3d_{RM}$ or
$3d_{RM}<RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})-m_{D^{-}}<5d_{RM}$ , where
$m_{D^{\pm}}=1.86966$ GeV/$c^{2}$ is the known $D^{\pm}$ mass pdg . The signal
and sideband regions are indicated in Fig. 1. A linear mass or recoil mass
dependence is assumed in estimating the background level in the signal region.
The widths of the window are $d_{M}=11$ MeV/$c^{2}$ for all the data samples,
and $d_{RM}=6$ MeV/$c^{2}$ for data samples with $\sqrt{s}$ smaller than 4.310
GeV, and $d_{RM}=9$ MeV/$c^{2}$ for data samples with $\sqrt{s}$ greater than
or equal to 4.310 GeV. Each sideband has the same width as that of the signal
region.

Figure 1: Distributions of $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ versus
$M(K^{-}\pi^{+}\pi^{+})$ for data samples at $\sqrt{s}$ = 4.230 (a), 4.420
(b), and 4.680 (c) GeV. The red solid box shows the signal region, the blue
dashed boxes the sideband regions with one real $D^{+}$ or $D^{-}$ candidate,
and the blue dotted boxes the sideband regions with fake $D^{+}$ and $D^{-}$
candidates. The indices of the boxes from top to bottom and left to right are
($-$1, 1), (0, 1), (1, 1), ($-$1, 0), (0, 0), (1, 0), ($-$1, $-$1), (0, $-$1),
and (1, $-$1), respectively. The region with index (0, 0) is the signal
region, while the others are the sideband regions (color version online).

After requiring $|RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})-m_{D^{-}}|<d_{RM}$, the
$M(K^{-}\pi^{+}\pi^{+})$ distributions are shown in Fig. 2 for data samples at
$\sqrt{s}=$ 4.230, 4.420, and 4.680 GeV as examples. In the following
analysis, the $K^{-}\pi^{+}\pi^{+}$ combination in the signal region is
constrained to the known $D^{+}$ mass, $m_{D^{+}}$, with a kinematic fit to
improve its momentum resolution, and those in the sideband regions are
constrained to the central value of the corresponding sideband region.

Figure 2: The $K^{-}\pi^{+}\pi^{+}$ invariant mass distributions for data
samples at $\sqrt{s}=$ 4.230 (a), 4.420 (b), and 4.680 (c) GeV. The black dots
with error bars are data, the regions between the two red dashed arrows are
$D^{+}$ signal regions and those between blue dash-dotted arrows are sideband
regions (color version online).

Figure 3 shows the $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distributions after
requiring $|M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}|<d_{M}$ for data samples at
$\sqrt{s}=$ 4.230, 4.420, and 4.680 GeV. Clear $D^{-}$ signal peaks are
visible in all data samples. The $D^{-}$ signal and sideband regions are
indicated by the arrows.

Figure 3: Distributions of $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ in the
$M(K^{-}\pi^{+}\pi^{+})$ signal region for data samples at $\sqrt{s}=$ 4.230
(a), 4.420 (b), and 4.680 (c) GeV. The black dots with error bars are data,
the regions between the two red dashed arrows are $D^{-}$ signal regions and
those between blue dash-dotted arrows are sideband regions (color version
online).

The $e^{+}e^{-}\rightarrow D^{0}D^{-}\pi^{+}$ process produces a peaking
background in the $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distribution as shown in
Fig. 4(a). The peaking background may come from $e^{+}e^{-}\rightarrow
D^{0}D^{-}\pi^{+}$, with $D^{0}\rightarrow
K^{-}\pi_{d}^{+}\pi_{d}^{-}\pi^{+}$, where a directly produced $\pi^{+}$,
together with $\pi^{+}$ and $K^{-}$ from $D^{0}$, forms the tagged $D^{+}$.
Figure 4(b) shows the $M(K^{-}\pi_{d}^{+}\pi_{d}^{-}\pi^{+})$111Here,
$\pi^{+}$ could be either of the charged pions in the decay $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$. distribution, where a clear $D^{0}$ peak is seen. We
require $|M(K^{-}\pi_{d}^{+}\pi_{d}^{-}\pi^{+})-m_{D^{0}}|>0.01$ GeV/$c^{2}$
to veto these $D^{0}$ background contributions, where $m_{D^{0}}=$ 1.86484
GeV/$c^{2}$ pdg . The value of $0.01$ GeV/$c^{2}$ corresponds to twice the
resolution of $M(K^{-}\pi_{d}^{+}\pi_{d}^{-}\pi^{+})$, which is 0.0045
GeV/$c^{2}$. The effectiveness of this veto can be seen in Fig. 4(c). The
number of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ events from fits to
the $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distributions in the
$M(K^{-}\pi^{+}\pi^{+})$ sideband region before and after the veto are 614
$\pm$ 92 and 216 $\pm$ 72, respectively.

Figure 4: The $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distribution for combinations
in the $M(K^{-}\pi^{+}\pi^{+})$ sideband before (a) and after (c) the $D^{0}$
veto at $\sqrt{s}=$ 4.420 GeV. The $M(K^{-}\pi_{d}^{+}\pi_{d}^{-}\pi^{+})$
distribution in the $M(K^{-}\pi^{+}\pi^{+})$ signal region is shown in the
middle plot (b). The black dots with error bars are data and the red and green
histograms are MC simulations of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}(\rm{PHSP})$ and
$e^{+}e^{-}\rightarrow D^{0}D^{-}\pi^{+}$ processes, respectively, with
inclusive decays of both $D$ mesons (color version online). The normalizations
of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}(\rm{PHSP})$ and
$e^{+}e^{-}\rightarrow D^{0}D^{-}\pi^{+}$ processes are accroding to cross
sections of total $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process and
$e^{+}e^{-}\rightarrow D^{0}D^{-}\pi^{+}$ process measured from data samples,
respectively.

After applying all the above selection criteria, we compare distributions for
events in the $D^{+}$ and $D^{-}$ signal region ($S$ sample) and sideband
regions ($B$ sample) to further suppress non-$\pi^{+}\pi^{-}D^{+}D^{-}$
background. The $B$ sample is defined as

$\begin{split}B=f_{1}\cdot(B_{-1,0}+B_{1,0})+f_{2}\cdot(B_{0,-1}+B_{0,1})-\\\
f_{3}\cdot(B_{-1,-1}+B_{1,-1}+B_{-1,1}+B_{1,1}),\end{split}$ (1)

where $B_{i,j}$ is the sideband region defined in Fig. 1, $f_{1}=0.5$,
$f_{2}=0.5$, and $f_{3}=f_{1}f_{2}=0.25$ are the normalization factors
assuming a linear mass dependence in the background distributions. In order to
improve the momentum resolutions of the final state particles, $D^{+}$ and
$D^{-}$ mass constraints and a total four-momentum conservation constraint to
that of the initial $e^{+}e^{-}$ system are applied. For events in the $D^{+}$
or $D^{-}$ sidebands, the masses of the $D^{+}$ and $D^{-}$ combinations are
constrained to the central values of the corresponding sideband region.

The invariant mass distribution of the $\pi_{d}^{+}\pi_{d}^{-}$ pair is shown
in Fig. 5(a), where clear $K_{S}^{0}$ peaks can be seen in both the $S$ and
$B$ samples. In order to veto the $K_{S}^{0}\rightarrow\pi_{d}^{+}\pi_{d}^{-}$
background, a secondary vertex fit is performed on the
$\pi_{d}^{+}\pi_{d}^{-}$ pair. The decay length $L_{\pi_{d}^{+}\pi_{d}^{-}}$
divided by its uncertainty $\Delta_{L_{\pi_{d}^{+}\pi_{d}^{-}}}$ of the
combinations with $\pi_{d}^{+}\pi_{d}^{-}$ invariant mass between 491.0 and
503.5 MeV/$c^{2}$ is shown in Fig. 5(b). By requiring
$|L_{\pi_{d}^{+}\pi_{d}^{-}}/\Delta_{L_{\pi_{d}^{+}\pi_{d}^{-}}}|<2$ the
$K_{S}^{0}$ background is suppressed significantly as shown in Fig. 5(c).

Figure 5: Distributions of $M(\pi_{d}^{+}\pi_{d}^{-})$ before (a) and after
(c) the $|L_{\pi_{d}^{+}\pi_{d}^{-}}/\Delta_{L_{\pi_{d}^{+}\pi_{d}^{-}}}|<2$
requirement, and the
$L_{\pi_{d}^{+}\pi_{d}^{-}}/\Delta_{L_{\pi_{d}^{+}\pi_{d}^{-}}}$ distribution
(b) at $\sqrt{s}=$ 4.420 GeV. The black dots with error bars stand for the $S$
sample and the green shaded histograms for the $B$ sample (color version
online).

The $|V_{xy}|$ and $|V_{z}|$ distributions of the $K^{-}$ and $\pi^{+}$ tracks
used in $D^{+}$ tag, and the $\pi_{d}^{+}$ and $\pi_{d}^{-}$ tracks from
direct $e^{+}e^{-}$ annihilation are shown in Fig. 6. Compared with the
typical requirements of less than 1 cm and 10 cm for $|V_{xy}|$ and $|V_{z}|$,
respectively, a set of tighter selection criteria $|V_{xy}|<0.55$ cm and
$|V_{z}|<3$ cm is identified by optimizing the $\pi^{+}\pi^{-}D^{+}D^{-}$
signal significance.

Figure 6: Distributions of $|V_{xy}|$ for data samples at $\sqrt{s}=$ 4.230
(a), 4.420 (c), and 4.680 (e) GeV, and those of $|V_{z}|$ for data samples at
$\sqrt{s}=$ 4.230 (b), 4.420 (d), and 4.680 (f) GeV. The black dots with error
bars correspond to the $S$ sample and the shaded histograms to the $B$ sample
(color version online).

After applying all the above selection criteria, requiring
$|M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}|<d_{M}$, and constraining
$M(K^{-}\pi^{+}\pi^{+})$ to the $D^{+}$ mass, we obtain the
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distributions (Figs. 7(a, c, e)) for data
samples at $\sqrt{s}=$ 4.230, 4.420, and 4.680 GeV. Clear $D^{-}$ signal peaks
are observed in all these data samples. The non-$\pi^{+}\pi^{-}D^{+}D^{-}$
background is studied by examining the $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$
distributions (Figs. 7(b, d, f)) for $K^{-}\pi^{+}\pi^{+}$ combinations in the
$D^{+}$ mass sideband regions, defined as
$-5d_{M}<M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}<-3d_{M}$ or
$3d_{M}<M(K^{-}\pi^{+}\pi^{+})-m_{D^{+}}<5d_{M}$. No significant $D^{-}$
signal peaks are observed in the sideband samples. In calculating
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ for the sideband events, the
$M(K^{-}\pi^{+}\pi^{+})$ is constrained to the central values of the
corresponding sideband region rather than to the known mass $m_{D^{+}}$. The
number of $e^{+}e^{-}\to\pi^{+}\pi^{-}D^{+}D^{-}$ signal events is obtained by
subtracting the number of $D^{-}$ signal candidates in the $D^{+}$ sideband
regions from that in the $D^{+}$ signal region, as discussed in Sec. IV.

Figure 7: Distributions of $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ in
$M(K^{-}\pi^{+}\pi^{+})$ signal (a, c, e) and sideband (b, d, f) regions for
data samples at $\sqrt{s}=$ 4.230 (a, b), 4.420 (c, d), and 4.680 (e, f) GeV,
and the best fits to the distributions. The black dots with error bars are
data, the red dashed, green dash-dotted, and blue solid lines are the signal,
background, and total fit, respectively (color version online). The fit
qualities are tested using a $\chi^{2}$-test method, with $\chi^{2}/\rm{n.\
d.\ f.}=$93.72/91, 96.39/95, 83.74/93, 104.25/95, 98.69/93, and 85.12/95 for
(a), (b), (c), (d), (e), and (f), respectively.

##  IV Cross sections of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
process

The cross section of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
process is calculated with

$\sigma=\frac{N_{\rm signal}-N_{\rm
sideband}/2}{2f(\sum_{i}\omega_{i}\epsilon_{i}(1+\delta)_{i})\frac{1}{|1-\Pi|^{2}}\mathscr{B}\mathscr{L}},$
(2)

where $N_{\rm signal}$ and $N_{\rm sideband}$ are the number of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ events from fits to
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distributions (Fig. 7) in the
$M(K^{-}\pi^{+}\pi^{+})$ signal and sideband regions, respectively,
$\frac{1}{|1-\Pi|^{2}}$ is the vacuum polarization factor, $\mathscr{B}$ is
the branching fraction of the decay $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ pdg
, and $\mathscr{L}$ is the integrated luminosity of the data sample. $f$
denotes an efficiency correction factor

$\begin{split}f=f^{M(K^{-}\pi^{+}\pi^{+})}f^{K\nrightarrow p}f^{V_{xy,\
z}}f^{L/\Delta_{L}}\\\ f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})},\end{split}$ (3)

with $f^{v}$ referring to the efficiency correction factor caused by selection
criterion $v$, which includes $M(K^{-}\pi^{+}\pi^{+})$ and
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ mass window requirements, $p/\bar{p}$ veto
($K\nrightarrow p$), $V_{xy,\ z}$ requirements, and
$L_{\pi_{d}^{+}\pi_{d}^{-}}/\Delta_{L_{\pi_{d}^{+}\pi_{d}^{-}}}$ requirement
for $K_{S}^{0}$ background suppression. Details on the evaluation of $f^{v}$
can be found in Sec. VII.1. $(1+\delta)_{i}$ is the ISR correction factor, and
$\omega_{i}$ and $\epsilon_{i}$ are the fraction and the detection efficiency
of subprocess $i$, respectively, here, $i=0$, $1$, and $2$ correspond to
$e^{+}e^{-}\rightarrow
D_{1}(2420)^{+}D^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$,
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(3770)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$,
and $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}(\rm{PHSP})$ subprocess,
respectively. $\omega_{i}$ is estimated from the $S$ sample by a one
dimentional simultaneous extended-unbinned-likelihood fit to $RM(D^{+})$,
$RM(D_{\rm miss}^{-})$, and $RM(\pi_{d}^{+}\pi_{d}^{-})$ distributions, and
the background is estimated by the $B$ sample. For data samples with
$\sqrt{s}$ larger than $4.315$ GeV, $i=0$, $1$, and $2$, while for data
samples with $\sqrt{s}$ smaller than or equal to $4.315$ GeV, $i=1$ and $2$,
since the threshold of $D_{1}(2420)\bar{D}$ is 4291.75 MeV/$c^{2}$, and no
significant $D_{1}(2420)\bar{D}$ events are observed at $\sqrt{s}=$ 4.310 and
4.315 GeV.

Figure 7 shows the fit results of data samples at $\sqrt{s}=$ 4.230, 4.420,
and 4.680 GeV. The signal shape is modelled by the
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distributions in MC simulation of each
subprocess weighted according to $\omega_{i}$ and convolved with a Gaussian
function to take the resolution difference between data and MC simulation into
account. The background shape is described by a second-order Chebychev
polynomial function. At each $\sqrt{s}$, the signal shape for the fit in the
$M(K^{-}\pi^{+}\pi^{+})$ sideband regions is the same as that for the fit in
the $M(K^{-}\pi^{+}\pi^{+})$ signal region. The results for $N_{\rm signal}$
and $N_{\rm sideband}$ obtained from the fits are listed in Table 1, together
with the fit results for all other data samples. The calculated cross section
of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process is shown in
Fig. 8.

Figure 8: Cross section of the process $e^{+}e^{-}\to\pi^{+}\pi^{-}D^{+}D^{-}$
and fit with the coherent sum of two BW functions and a phase space term
(Solution IV) (a), and with the coherent sum of two BW functions only
(Solution II) (b). Other solutions of (a) and (b) could be found in Table 2
and 3, respectively. Dots with error bars are data with the statistical
uncertainties and the red lines show the best fit results. For (a), the black,
green, and pink solid lines describe $BW_{0}$, $BW_{1}$, and $\Phi_{4}$
components, respectively, and the red, green, and blue dashed lines describe
interferences between $BW_{0}$ and $BW_{1}$, $BW_{0}$ and $\Phi_{4}$, and
$BW_{1}$ and $\Phi_{4}$, respectively. For (b), the black and green solid
lines describe $BW_{0}$ and $BW_{1}$ components, respectively, and the red
dashed line describes the inteference between $BW_{0}$ and $BW_{1}$. (color
version online). The fit qualities are tested using a $\chi^{2}$-test method,
with $\chi^{2}/\rm{n.\ d.\ f.}=$47.1/28 and 46.1/30 for (a) and (b),
respectively. The $\rm{n.\ d.\ f.}$ denotes the number of degrees of freedom.
The $\chi^{2}$ function is constructed as
$\chi^{2}=\Sigma\frac{(\sigma_{i}-\sigma^{fit}_{i})^{2}}{\delta_{i}^{2}}$,
here, $\sigma_{i}$ and $\sigma^{fit}_{i}$ are the measured and fitted cross
section of the $i^{th}$ data sample, respectively, and $\delta_{i}$ is the
standard deviation of the measured cross section, which includes the
statistical uncertainties only.

For data samples where no significant
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ signal peaks are observed
(statistical significance smaller than $5\sigma$), the upper limits on the
cross section are calculated using a Bayesian method 2003UPL . By fitting the
$RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ distribution for the events in the $D^{+}$
signal region with fixed values for the signal yield, a scan of the likelihood
distribution as a function of the cross section is obtained. To take the total
systematic uncertainty (listed in Table 5) into consideration, the likelihood
distribution is convolved with a Gaussian function with a width corresponding
to the overall systematic uncertainty. The upper limit on the cross section at
90% C.L. is obtained from
$\int_{0}^{\sigma}L(x)dx/\int_{0}^{\infty}L(x)dx=0.9.$ The upper limits on the
cross sections are listed in Table 1.

##  V Resonances in the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ Cross
Section line shape

Clear resonant structures around $\sqrt{s}=$ 4.390 and 4.700 GeV can be seen
in Fig. 8, and there is no significant signal at other energies, including at
the expected masses of the $Y(4230)$, $Y(4360)$, and $Y(4660)$ states.

A fit to the measured $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ cross
section line shape is performed with a coherent sum of two Breit-Wigner (BW)
functions and a phase space term

$\begin{split}\sigma(\sqrt{s})=|BW_{0}(\sqrt{s})+BW_{1}(\sqrt{s})e^{i\phi_{1}}\\\
+ce^{i\phi_{2}}\Phi_{4}(\sqrt{s})|^{2},\end{split}$ (4)

with the BW function defined as

$BW_{j}(\sqrt{s})=\frac{\sqrt{12\pi{\Gamma^{e^{+}e^{-}}_{j}}\Gamma^{\rm
tot}_{j}\mathscr{B}_{j}}}{s-m_{j}^{2}+im_{j}\Gamma^{\rm
tot}_{j}}\sqrt{\frac{\Phi_{4}(\sqrt{s})}{\Phi_{4}(m_{j})}},$ (5)

where $m_{j}$, $\Gamma^{\rm tot}_{j}$, and ${\Gamma^{e^{+}e^{-}}_{j}}$ are the
mass, width, and electronic partial width of the $j^{th}$ resonance ($R_{j}$),
respectively; $\mathscr{B}_{j}$ is the branching fraction of the decay
$R_{j}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$, $\phi_{j}$ is the relative phase
between the $j^{th}$ resonance, as well as the phase space term,
$\Phi_{4}(\sqrt{s})$ is the phase space factor of the four-body decay
$R\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$, and $c$ is a constant describing the
magnitude of $\Phi_{4}(\sqrt{s})$.

There are four solutions with the same fit quality and identical resonance
parameters for $R_{0}$ and $R_{1}$ as well as $c$, but different
$\Gamma^{e^{+}e^{-}}_{j}\mathscr{B}_{j}$ and $\phi_{j}$, as listed in Table 2.
The fitted parameters for $R_{0}$ are in agreement with those of the $Y(4390)$
resonance observed by the BESIII Collaboration in the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$ process BESIII:2016adj . The
statistical significance of $R_{1}$ is determined to be $4.1\sigma$ by
comparing the likelihood of the baseline fit and that of the fit without
$R_{1}$.

Table 2: The fitted parameters of the cross sections of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ with the coherent sum of two Breit-Wigner functions and a phase space term. The first uncertainties are statistical and the second systematic. Parameters | Solution I | Solution II | Solution III | Solution IV |
---|---|---|---|---|---
$c\ (\rm{MeV}^{-3/2})$ | $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1.6\pm 0.9\pm 0.1)\times 10^{3}$ |
$m_{0}\ (\rm{MeV}/c^{2})$ | $4373.1\pm 4.0\pm 1.0$ |
$\Gamma_{0}\ (\rm{MeV})$ | $\ \ 146.5\pm 7.4\pm 1.1$ |
$m_{1}\ (\rm{MeV}/c^{2})$ | $\ \ 4706\ \pm 11\ \pm 4\ \ $ |
$\Gamma_{1}\ (\rm{MeV})$ | $\ \ \ \ \ \ 45\ \pm 28\ \pm 9\ \ $ |
$\Gamma^{e^{+}e^{-}}_{0}\mathscr{B}_{0}\ (\rm eV)$ | 9.3$\ \pm\ $0.8$\ \pm\ $1.8 | 9.3$\ \pm\ $0.8$\ \pm\ $1.3 | 13.0$\ \pm\ $1.5$\ \pm\ $1.7 | 9.9$\ \pm\ $1.1$\ \pm\ $1.4 |
$\Gamma^{e^{+}e^{-}}_{1}\mathscr{B}_{1}\ (\rm eV)$ | 11.9$\ \pm\ $6.5$\ \pm\ $3.2 | 0.2$\ \pm\ $0.1$\ \pm\ $0.1 | 0.2$\ \pm\ $0.1$\ \pm\ $0.1 | 10.8$\ \pm\ $5.3$\ \pm\ $2.8 |
$\phi_{1}\ (\rm{rad})$ | 4.9$\ \pm\ $0.2$\ \pm\ $0.0 | 0.3$\ \pm\ $0.4$\ \pm\ $0.0 | 1.1$\ \pm\ $0.7$\ \pm\ $0.0 | 4.1$\ \pm\ $0.3$\ \pm\ $0.0 |
$\phi_{2}\ (\rm{rad})$ | 4.6$\ \pm\ $0.3$\ \pm\ $1.0 | 1.6$\ \pm\ $0.3$\ \pm\ $1.0 | 1.6$\ \pm\ $0.3$\ \pm\ $1.0 | 4.5$\ \pm\ $0.3$\ \pm\ $1.0 |

If we omit the phase space term from the baseline fit, the fit quality becomes
slightly worse, indicating the statistical significance of this amplitude is
only $1.4\sigma$ and the solutions of this amplitude could be found in Table
3. However, the fit in the high energy region becomes very different, as shown
in Fig. 8(b). For the fit with the coherent sum of two BW functions only,
there are two solutions with the same fit quality and identical resonance
parameters for $R_{0}$ and $R_{1}$, but different $\Gamma^{e^{+}e^{-}}_{j}$
and $\phi_{j}$, as listed in Table 3. The statistical significance of $R_{1}$
is $7.0\sigma$.

Table 3: The fitted parameters of the cross sections of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ with the coherent sum of two Breit-Wigner functions. The uncertainties are statistical. Parameters | Solution I | Solution II |
---|---|---|---
$m_{0}\ (\rm{MeV}/c^{2})$ | $4378.0\pm 8.5$ |
$\Gamma_{0}\ (\rm{MeV})$ | $\ \ \ \ 152\pm 14$ |
$m_{1}\ (\rm{MeV}/c^{2})$ | $\ \ 4605\pm 90$ |
$\Gamma_{1}\ (\rm{MeV})$ | $\ \ \ \ 245\pm 67$ |
$\Gamma^{e^{+}e^{-}}_{0}\mathscr{B}_{0}\ (\rm{eV})$ | 21$\ \pm\ $12 | 12.2$\ \pm\ $5.8 |
$\Gamma^{e^{+}e^{-}}_{1}\mathscr{B}_{1}\ (\rm{eV})$ | 54$\ \pm\ $15 | 1.3$\ \pm\ $2.7 |
$\phi_{1}\ (\rm{rad})$ | 4.1$\ \pm\ $0.3 | 5.6$\ \pm\ $2.6 |

Other than the $R_{0}$ and $R_{1}$ contributions, we also tested the
statistical significances of the possible structures around $\sqrt{s}=$ 4.245
and 4.914 GeV. By adding the $Y(4230)$ amplitude to the fit, with the mass and
width fixed according to the world averaged values pdg , its significance is
found to be only $0.3\sigma$. By adding a new resonance at high energy with
free mass and width, the statistical significance is found to be $1.3\sigma$.
Therefore, such additional structures are not considered at the upper and
lower mass regions.

Note that there are four points ($\sqrt{s}$ from 4.400 to 4.600 GeV)
systematically below the fitted line. Since the integrated luminosities of
these data samples are very low, larger data samples are needed to draw a
conclusion.

##  VI Evidence for $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}X(3842)$

To search for the $X(3842)$ state, the $S$ sample defined in Sec. III with the
additional $K^{0}_{S}$ veto and stringent $|V_{xy}|$ and $|V_{z}|$
requirements ($|V_{xy}|<0.55$ cm and $|V_{z}|<3$ cm) is used. In order to
suppress the $e^{+}e^{-}\rightarrow D_{1}(2420)^{+}D^{-}$ background, the
$D_{1}(2420)$ signal is suppressed by requiring
$|RM(D^{+})-m_{D_{1}(2420)^{-}}|>0.01$ GeV/$c^{2}$ and
$|RM(D_{\rm{miss}}^{-})-m_{D_{1}(2420)^{+}}|>0.01$ GeV/$c^{2}$, where
$m_{D_{1}(2420)^{\pm}}=2.4221$ GeV/$c^{2}$ is the known $D_{1}(2420)^{\pm}$
mass pdg  222Here, the selection criteria correspond to the resolutions of
$RM(D^{+})$ and $RM(D_{\rm miss}^{-})$, both equal to 0.01 GeV/$c^{2}$..

The $RM(\pi_{d}^{+}\pi_{d}^{-})$ (equivalent to the invariant mass of
$D^{+}D_{\rm miss}^{-}$) distributions in all data samples are examined. While
no significant signal is observed at any single $\sqrt{s}$, there is evidence
for an $X(3842)$ resonance for $\sqrt{s}$ from 4.600 to 4.700 GeV. Figure 9
shows the $RM(\pi_{d}^{+}\pi_{d}^{-})$ distributions at $\sqrt{s}=$ 4.420,
4.680 GeV, and data samples with $\sqrt{s}$ from 4.600 to 4.700 GeV.

Figure 9: The $RM(\pi_{d}^{+}\pi_{d}^{-})$ distributions and the fits at
$\sqrt{s}=$ 4.420 (a), 4.680 (b) GeV, and data samples with $\sqrt{s}$ from
4.600 to 4.700 GeV (c). The black dots with error bars are the $S$ sample, and
the red dashed, green dash-dotted, and blue solid curves are the signal shape,
background shape, and total fit, respectively (color version online). The fit
qualities are tested using a $\chi^{2}$-test method, with $\chi^{2}/\rm{n.\
d.\ f.}=$26.3/45, 41.4/43, and 57.07/45 for (a), (b), and (c), respectively.

To fit the $RM(\pi_{d}^{+}\pi_{d}^{-})$ distributions, the $X(3842)$ signal
shape is obtained from MC simulation of the $e^{+}e^{-}\rightarrow
f_{0}(500)X(3842)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process333Two body decay
is assumed since it has the largest phase space and $f_{0}(500)$ is the meson
with $J^{P}=0^{+}$., and convolved with a Gaussian function to take the
resolution difference between data and MC simulation into account. The mean
and sigma values of the Gaussian function for other fits are fixed to the fit
values obtained at $\sqrt{s}=$ 4.680 GeV as this sample contains the largest
number of signal events. The mass and width of the $f_{0}(500)$ are taken from
Ref. 2001sigma  when generating MC events. The background is described with a
second-order Chebychev polynomial function.

Fit results of the $RM(\pi_{d}^{+}\pi_{d}^{-})$ distributions are shown in
Figs. 9(a, b, c). The signal yields (statistical significances) are $-$39
$\pm$ 18 ($-2.0\sigma$), 58 $\pm$ 24 ($1.8\sigma$), and 155 $\pm$ 38
($4.2\sigma$) at $\sqrt{s}=$ 4.420, 4.680 GeV and data samples with $\sqrt{s}$
from 4.600 to 4.700 GeV, respectively. Furthermore, for data samples with
$\sqrt{s}$ from 4.600 to 4.700 GeV, the fits are also performed by changing
the fit range, the signal shape, or the background shape. In all cases, the
minimum value of the $X(3842)$ resonance significance is $4.2\sigma$. The fit
results at other energies are listed in Table 4. The cross sections of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}X(3842)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
process are calculated in a similar way as for other processes, and the upper
limits of the cross sections are determined using a similar strategy to that
described in Sec. IV. The results are also listed in Table 4.

Table 4: Results for the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}X(3842)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process. Here, $\sigma$ is the cross section of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}X(3842)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process, where the first uncertainty is statistical and the second systematic; $S$ is the statistical significance; $\epsilon$, $(1+\delta)$, $N$, and $\sigma_{\rm ul}$ are the detection efficieny, ISR correction factor, signal yields, and the upper limit of cross section at 90% confidence level. $\sqrt{s}^{\rm nominal}$ | $\epsilon(\%)$ | $(1+\delta)$ |  $\mathscr{L}$ ($\rm{pb}^{-1}$) | $N$ | $\sigma\ (\rm{pb})$ | $S$ |  $\sigma_{\rm{ul}}$ (pb)
---|---|---|---|---|---|---|---
4.190 | 3.3 | 0.80 | 570.0 | 1$\pm$ | 2 | 0.5$\pm$ | 0.8 $\pm$ | 0.1 | $\ \ \ \ 0.6\sigma$ | 2.5
4.200 | 4.8 | 0.81 | 526.0 | 3$\pm$ | 3 | 0.8$\pm$ | 0.8 $\pm$ | 0.1 | $\ \ \ \ 1.1\sigma$ | 2.6
4.210 | 5.8 | 0.82 | 572.1 |  $-$1$\pm$ | 1 |  $-$0.2$\pm$ | 0.4 $\pm$ | 0.0 | - | 0.9
4.220 | 7.0 | 0.83 | 569.1 |  $-$2$\pm$ | 2 |  $-$0.4$\pm$ | 0.4 $\pm$ | 0.1 | - | 0.7
4.230 | 9.1 | 0.83 | 1100.9 | 0$\pm$ | 4 | 0.0$\pm$ | 0.3 $\pm$ | 0.0 | $\ \ \ \ 0.1\sigma$ | 0.6
4.237 | 9.7 | 0.84 | 530.0 | 0$\pm$ | 3 | 0.1$\pm$ | 0.4 $\pm$ | 0.0 | $\ \ \ \ 0.3\sigma$ | 1.0
4.246 | 10.4 | 0.84 | 538.1 |  $-$3$\pm$ | 2 |  $-$0.4$\pm$ | 0.3 $\pm$ | 0.1 | - | 0.5
4.269 | 11.8 | 0.85 | 825.7 |  $-$8$\pm$ | 4 |  $-$0.6$\pm$ | 0.3 $\pm$ | 0.1 | - | 0.3
4.270 | 12.2 | 0.85 | 531.1 | 6$\pm$ | 4 | 0.6$\pm$ | 0.4 $\pm$ | 0.1 | $\ \ \ \ 1.5\sigma$ | 1.5
4.290 | 12.2 | 0.86 | 502.4 | 0$\pm$ | 4 |  $-$0.0$\pm$ | 0.4 $\pm$ | 0.0 | $\ \ \ \ 0.1\sigma$ | 0.9
4.315 | 14.8 | 0.87 | 501.2 | 2$\pm$ | 6 | 0.2$\pm$ | 0.5 $\pm$ | 0.0 | $\ \ \ \ 0.4\sigma$ | 0.9
4.340 | 15.6 | 0.88 | 505.0 |  $-$8$\pm$ | 7 |  $-$0.7$\pm$ | 0.6 $\pm$ | 0.1 | - | 0.6
4.360 | 17.1 | 0.88 | 544.0 |  $-$7$\pm$ | 9 |  $-$0.5$\pm$ | 0.6 $\pm$ | 0.1 | - | 1.0
4.380 | 16.2 | 0.89 | 522.7 |  $-$19$\pm$ | 8 |  $-$1.3$\pm$ | 0.6 $\pm$ | 0.2 | - | 1.1
4.400 | 16.4 | 0.89 | 507.8 | 11$\pm$ | 12 | 0.8$\pm$ | 0.9 $\pm$ | 0.1 | $\ \ \ \ 1.0\sigma$ | 4.6
4.420 | 18.3 | 0.90 | 1090.7 |  $-$39$\pm$ | 18 |  $-$1.2$\pm$ | 0.6 $\pm$ | 0.2 | - | 0.6
4.440 | 16.7 | 0.90 | 569.9 | 7$\pm$ | 15 | 0.5$\pm$ | 0.9 $\pm$ | 0.1 | $\ \ \ \ 0.5\sigma$ | 3.1
4.600 | 19.6 | 0.92 | 586.9 | 31$\pm$ | 13 | 1.6$\pm$ | 0.7 $\pm$ | 0.2 | $\ \ \ \ 2.5\sigma$ | 3.3
4.620 | 18.5 | 0.93 | 521.5 | 27$\pm$ | 13 | 1.6$\pm$ | 0.8 $\pm$ | 0.2 | $\ \ \ \ 2.2\sigma$ | 3.4
4.640 | 18.8 | 0.93 | 552.4 | 17$\pm$ | 13 | 1.0$\pm$ | 0.7 $\pm$ | 0.1 | $\ \ \ \ 1.4\sigma$ | 2.3
4.660 | 19.1 | 0.93 | 529.6 | 13$\pm$ | 13 | 0.8$\pm$ | 0.7 $\pm$ | 0.1 | $\ \ \ \ 1.1\sigma$ | 2.0
4.680 | 19.1 | 0.93 | 1669.3 | 58$\pm$ | 24 | 1.0$\pm$ | 0.4 $\pm$ | 0.1 | $\ \ \ \ 1.8\sigma$ | 1.5
4.700 | 19.1 | 0.93 | 536.5 | 1$\pm$ | 13 | 0.1$\pm$ | 0.7 $\pm$ | 0.0 | $\ \ \ \ 0.1\sigma$ | 1.4
4.750 | 20.1 | 0.93 | 367.2 | 0$\pm$ | 10 |  $-$0.1$\pm$ | 0.8 $\pm$ | 0.0 | $\ \ \ \ 0.1\sigma$ | 1.4
4.780 | 20.0 | 0.94 | 512.8 | 15$\pm$ | 12 | 0.9$\pm$ | 0.7 $\pm$ | 0.1 | $\ \ \ \ 1.4\sigma$ | 2.0
4.840 | 20.5 | 0.94 | 527.3 |  $-$11$\pm$ | 10 |  $-$0.6$\pm$ | 0.5 $\pm$ | 0.1 | - | 0.7
4.916 | 20.4 | 0.95 | 208.1 |  $-$6$\pm$ | 5 |  $-$0.9$\pm$ | 0.8 $\pm$ | 0.1 | - | 1.1
4.946 | 20.0 | 0.95 | 160.4 |  $-$10$\pm$ | 3 |  $-$1.7$\pm$ | 0.6 $\pm$ | 0.2 | - | 0.8

##  VII Systematic Uncertainties

###  VII.1 Systematic uncertainties for the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ cross sections

The systematic uncertainties in the cross section measurement of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process stem from many
sources. The systematic uncertainties associated with the detection
efficiencies, including tracking TRACKPID  and PID TRACKPID , are estimated as
1% for each. The systematic uncertainty associated with the integrated
luminosity measurement using Bhabha ($e^{+}e^{-}\rightarrow e^{+}e^{-}$)
events is estimated as 1% Lum . For the vacuum polarization factor
calculation, the systematic uncertainty originates mainly from hadronic
contributions, and is estimated as 0.1% according to VP . The systematic
uncertainty coming from the input branching fraction of $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ is estimated as 1.7% pdg . Details of further systematic
uncertainties are given below.

The selection efficiency is obtained from MC simulation and corrected
according to the measurements with control samples selected from data
directly. The efficiency correction factor $f^{v}$ is defined as

$f^{v}=\epsilon^{v}_{\rm{data}}/\epsilon^{v}_{\rm{MC}},$ (6)

with

$\epsilon^{v}_{\rm{data}(\rm{MC})}=N^{v}_{\rm{signal}_{\rm{data}(\rm{MC})}}/N^{v}_{\rm{all}_{\rm{data}(\rm{MC})}},$
(7)

where the subscript “MC” represents MC simulation and the subscript “data”
represents the data sample, $N^{v}_{\rm{signal}_{\rm{data}(\rm{MC})}}$ is the
number of events in the signal region of a selection criterion $v$, and
$N^{v}_{\rm{all}_{\rm{data}(\rm{MC})}}$ is the number of events in the full
range of $v$.

The uncertainty of $\epsilon^{v}_{\rm{data}(\rm{MC})}$

$\sigma_{\epsilon^{v}_{\rm{data}(\rm{MC})}}=\sqrt{\frac{\epsilon^{v}_{\rm{data}(\rm{MC})}(1-\epsilon^{v}_{\rm{data}(\rm{MC})})}{N^{v}_{\rm{all}_{\rm{data}(\rm{MC})}}}},$
(8)

and the uncertainty of $f^{v}$ is

$\frac{\sigma^{2}_{f^{v}}}{{f^{v}}^{2}}=\frac{\sigma_{\epsilon^{v}_{\rm{data}}}^{2}}{{\epsilon_{\rm{data}}^{v}}^{2}}+\frac{\sigma_{\epsilon^{v}_{\rm{MC}}}^{2}}{{\epsilon^{v}_{\rm{MC}}}^{2}},$
(9)

since data and MC simulation are independent. For $f^{v}=(1\pm\Delta
f^{v})\pm\sigma_{f^{v}}$, if $|\frac{\Delta f^{v}}{\sigma_{f^{v}}}|<1.00$, no
correction will be applied and $|\Delta f^{v}|+\sigma_{f^{v}}$ will be taken
as the systematic uncertainty, where $\Delta f^{v}$ is the deviation of
$f^{v}$ from 1; while if $|\frac{\Delta f^{v}}{\sigma_{f^{v}}}|>1.00$, the MC
efficiency will be corrected as $\epsilon=\epsilon_{MC}\times f^{v}$, and
$\sigma_{f^{v}}$ will be taken as the systematic uncertainty.

In order to avoid effects from statistical uncertainty, only data samples at
$\sqrt{s}=$ 4.340, 4.360, 4.400, 4.420, 4.440, 4.600, and 4.680 GeV are used
to estimate the systematic uncertainty originating from the
$M(K^{-}\pi^{+}\pi^{+})$ ($RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$) mass window
requirement. A constant parameter is used to fit the distributions of
$f^{M(K^{-}\pi^{+}\pi^{+})}$ ($f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})}$) among the
data samples mentioned above, and the fitted $f^{M(K^{-}\pi^{+}\pi^{+})}$
($f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})}$) and
$\sigma_{f^{M(K^{-}\pi^{+}\pi^{+})}}$
($\sigma_{f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})}}$) values are 0.986 $\pm$ 0.003
(0.984 $\pm$ 0.005), the value of $\Big{|}\frac{\Delta
f^{M(K^{-}\pi^{+}\pi^{+})}}{\sigma_{f^{M(K^{-}\pi^{+}\pi^{+})}}}\Big{|}$
($\Big{|}\frac{\Delta
f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})}}{\sigma_{f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})}}}\Big{|}$)
is 5.6 (2.8), therefore, the systematic uncertainty is taken as 0.3% (0.5%)
and $f^{M(K^{-}\pi^{+}\pi^{+})}$ ($f^{RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})}$) is
set to be 0.986 (0.984).

In order to avoid effects from the statistical uncertainty, the same set of
data samples as mentioned in the previous paragraph is used to estimate the
systematic uncertainties originating from the fit range and background shape
of $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$. The systematic uncertainty coming from
the choice of the fit range is estimated by varying the limits of the fit
range from (1.75, 1.96) GeV/$c^{2}$ to (1.77, 1.97) GeV/$c^{2}$. The
background shape is varied from the first-order Chebychev polynomial function
to a second-order one at $\sqrt{s}=$ 4.340 and 4.360 GeV, and the second-order
Chebychev polynomial function to the first-order one at $\sqrt{s}=$ 4.400,
4.420, 4.440, 4.600, and 4.680 GeV. The largest difference of the cross
section compared with the baseline value among the data samples mentioned
above is taken as a systematic uncertainty of 1.6% (1.7%) for the fit range
(background shape).

The systematic uncertainty stemming from the $p/\bar{p}$ veto, which is caused
by the difference in mis-identification probability of $K$ to $p/\bar{p}$
between data and MC simulation, is estimated by the control sample of
$J/\psi\rightarrow K_{S}^{0}K^{-}\pi^{+}+c.c.$ with the BESIII $J/\psi$ sample
KsKpi . The values of $f^{K\nrightarrow p}$ and $\sigma_{f^{K\nrightarrow p}}$
are 0.996 $\pm$ 0.003, and the value of $|\frac{\Delta f^{K\nrightarrow
p}}{\sigma_{f^{K\nrightarrow p}}}|$ is 1.4, therefore, the systematic
uncertainty is taken as 0.3% and $f^{K\nrightarrow p}$ is set to 0.996.
Similarly, using the control sample of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ at $\sqrt{s}=4.260$ GeV
2013Zc_3900 , $f^{L/\Delta_{L}}$ and $f^{V_{xy,\ z}}$ are estimated by
performing a secondary vertex fit on $\pi^{+}$ and $\pi^{-}$ pair and
comparing $V_{xy,\ z}$ of $\pi^{+}$, $\pi^{-}$, and lepton pair from $J/\psi$
in data and MC simulation, respectively. The values of $f^{L/\Delta_{L}}$
($f^{V_{xy,\ z}}$) and $\sigma_{f^{L/\Delta_{L}}}$ ($\sigma_{f^{V_{xy,\ z}}}$)
are 0.992 $\pm$ 0.010 (0.997 $\pm$ 0.001), $f^{L/\Delta_{L}}$ ($f^{V_{xy,\
z}}$) is set as 0.992 (0.997), and the systematic uncertainty associated with
the $L_{\pi_{d}^{+}\pi_{d}^{-}}/\Delta_{L_{\pi_{d}^{+}\pi_{d}^{-}}}$
requirement for the $K_{S}^{0}$ veto ($V_{xy,\ z}$ requirements) is 1.0%
(0.1%).

Table 5: Systematic uncertainties (%) from the scale factors $f_{1}$ and $f_{2}$ ($f_{1}$ and $f_{2}$), $\psi(3770)$ and $D_{1}(2420)^{+}$ shapes, including a new Breit-Wigner shape in the high energy region when parameterizing each subprocess cross section line shape, uncertainty of $\omega_{i}$ ($\omega_{i}$), and angular distribution modeling of $e^{+}e^{-}\rightarrow D_{1}(2420)^{+}D^{-}$ decay (helamp). The last column shows the total systematic uncertainty obtained by summing up all sources of systematic uncertainties in quadrature assuming they are uncorrelated. $\sqrt{s}^{\rm nominal}$ |  $f_{1}$ and $f_{2}$ |  $\psi(3770)$ shape |  $D_{1}(2420)^{+}$ shape | New Breit-Wigner | $\omega_{i}$ | helamp | Total |
---|---|---|---|---|---|---|---|---
4.190 | 0.0 | 0.0 | - | 0.0 | 14.4 | - | 16.6 |
4.200 | 0.0 | 0.0 | - | 0.0 | 19.3 | - | 21.0 |
4.210 | 0.0 | 0.0 | - | 0.0 | 16.8 | - | 18.7 |
4.220 | 0.0 | 0.0 | - | 0.0 | 0.8 | - | 8.3 |
4.230 | 0.6 | 1.7 | - | 0.9 | 0.8 | - | 8.6 |
4.237 | 0.0 | 1.6 | - | 2.3 | 0.6 | - | 8.8 |
4.245 | 0.5 | 0.3 | - | 0.3 | 1.8 | - | 8.5 |
4.246 | 0.5 | 0.7 | - | 3.0 | 7.8 | - | 11.8 |
4.260 | 0.2 | 0.9 | - | 0.7 | 0.4 | - | 8.4 |
4.270 | 0.0 | 0.5 | - | 1.2 | 0.7 | - | 8.4 |
4.280 | 0.2 | 1.4 | - | 1.4 | 0.5 | - | 8.5 |
4.290 | 0.1 | 0.1 | - | 1.8 | 0.5 | - | 8.5 |
4.310 | 1.5 | 0.3 | - | 1.4 | 0.4 | - | 8.5 |
4.315 | 0.1 | 0.4 | - | 0.3 | 0.2 | - | 8.3 |
4.340 | 0.1 | 0.4 | 0.7 | 1.2 | 0.3 | 0.5 | 8.4 |
4.360 | 0.2 | 0.0 | 0.2 | 2.0 | 0.1 | 0.2 | 8.5 |
4.380 | 0.2 | 0.6 | 0.1 | 0.4 | 0.3 | 0.8 | 8.4 |
4.390 | 0.1 | 0.3 | 1.2 | 1.0 | 0.8 | 0.3 | 8.5 |
4.400 | 0.1 | 0.5 | 1.0 | 0.1 | 0.7 | 0.6 | 8.4 |
4.420 | 0.0 | 0.1 | 0.6 | 0.1 | 1.8 | 0.8 | 8.5 |
4.440 | 0.2 | 0.9 | 3.3 | 0.2 | 1.5 | 0.4 | 9.1 |
4.470 | 0.2 | 1.4 | 0.9 | 0.5 | 6.6 | 0.2 | 10.7 |
4.530 | 0.2 | 0.2 | 0.4 | 0.3 | 6.0 | 0.7 | 10.3 |
4.575 | 0.5 | 0.3 | 1.8 | 0.0 | 2.1 | 1.3 | 8.8 |
4.600 | 0.5 | 0.4 | 1.7 | 2.8 | 0.8 | 0.8 | 9.0 |
4.612 | 0.1 | 0.2 | 0.0 | 0.3 | 1.2 | 0.5 | 8.4 |
4.620 | 0.1 | 1.5 | 0.9 | 0.3 | 1.1 | 0.4 | 8.5 |
4.640 | 0.5 | 0.2 | 0.8 | 1.6 | 1.4 | 0.2 | 8.6 |
4.660 | 0.2 | 0.9 | 0.3 | 2.6 | 1.0 | 0.2 | 8.8 |
4.680 | 0.2 | 0.4 | 0.9 | 0.0 | 0.4 | 0.1 | 8.3 |
4.700 | 0.0 | 0.8 | 0.7 | 2.3 | 0.5 | 0.0 | 8.7 |
4.740 | 0.1 | 0.1 | 2.1 | 0.5 | 0.8 | 0.3 | 8.6 |
4.750 | 0.5 | 0.8 | 0.6 | 1.9 | 1.2 | 0.2 | 8.7 |
4.780 | 0.1 | 0.3 | 0.2 | 0.4 | 0.8 | 0.3 | 8.3 |
4.840 | 0.3 | 0.9 | 0.7 | 0.5 | 0.6 | 1.9 | 8.6 |
4.914 | 0.0 | 0.6 | 1.1 | 1.1 | 1.4 | 1.7 | 8.7 |
4.946 | 0.8 | 0.6 | 0.4 | 0.6 | 1.6 | 0.9 | 8.6 |

$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(3770)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
and $e^{+}e^{-}\rightarrow
D_{1}(2420)^{+}D^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ processes are
simulated when estimating $\omega_{i}$, for the estimation of the systematic
uncertainty stemming from the $\psi(3770)$ ($D_{1}(2420)^{+}$) shape,
alternative MC samples are produced by varying the width of $\psi(3770)$
($D_{1}(2420)^{+}$) by one standard deviation of its world average value pdg .
The difference of the cross section of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process compared with the
baseline value is taken as the systematic uncertainty as listed in Table 5.

In Sec. II, $e^{+}e^{-}$ is assumed to annihilate into $D_{1}(2420)^{+}D^{-}$
directly, and the systematic uncertainty stemming from modeling the angular
distribution of the $e^{+}e^{-}\rightarrow D_{1}(2420)^{+}D^{-}$ process is
estimated by repeating the analysis procedure with the new model. For the
$e^{+}e^{-}\rightarrow D_{1}(2420)^{+}D^{-}$ process, two extreme cases of the
angular distribution following $1+\cos^{2}\theta_{D_{1}}$ and
$1-\cos^{2}\theta_{D_{1}}$ are assumed, where $\theta_{D_{1}}$ is the helicity
angle of the $D_{1}(2420)^{+}$ in the rest frame of the initial $e^{+}e^{-}$
system. The fractions of these two cases are estimated by fitting to the
$\rm{cos}\ \theta_{D^{+}}$ distribution, where $\theta_{D^{+}}$ is the polar
angle of $D^{+}$ in the rest frame of the initial $e^{+}e^{-}$ system, the
detection efficiency of $e^{+}e^{-}\rightarrow
D_{1}(2420)^{+}D^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process is
recalculated according to the detection efficiencies and fractions of these
two cases, and the cross section of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process is recalculated as
well. The difference of the cross section of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process compared with the
baseline value is taken as the systematic uncertainty as listed in Table 5.

In Sec. III, the normalization factor $f_{1}$ ($f_{2}$) in the $B$ sample is
estimated by assuming a linear background distribution in
$M(K^{-}\pi^{+}\pi^{+})$ ($RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$). A second-order
Chebychev polynomial function is used as the background shape to fit the
$M(K^{-}\pi^{+}\pi^{+})$ ($RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$) distribution to
estimate $f_{1}$ ($f_{2}$). The signal shape is modelled by the
$M(K^{-}\pi^{+}\pi^{+})$ ($RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$) distributions in
MC simulation of each subprocess weighted according to fractions of each
subprocess, $\omega_{i}$, and convolved with a Gaussian function to take the
resolution difference between data and MC simulation into consideration.
$\omega_{i}$ is re-estimated according to the new $f_{1}$ and $f_{2}$, and the
cross section of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process
is recalculated. The difference from the baseline value is taken as the
systematic uncertainty originating from this source as listed in Table 5.

The systematic uncertainty due to the uncertainty of the fraction of each
subprocess, $\omega_{i}$, is estimated by varying $\omega_{i}$ 500 times
according to the convariant matrix in the simultaneous fit of $RM(D^{+})$,
$RM(D_{\rm{miss}}^{-})$, and $RM(\pi_{d}^{+}\pi_{d}^{-})$ distributions for
each $\sqrt{s}$. In each iteration, the difference between the cross section
of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process and the
baseline value is calculated, and the distribution of the differences is
sampled at each $\sqrt{s}$, the width of the distribution is taken as the
systematic uncertainty as listed in Table 5.

The systematic uncertainty of the radiative correction is calculated by using
the kkmc package. Initially, the observed signal events are assumed to
originate from the $Y(4260)$ resonance to obtain the efficiency and ISR
correction factor. Then, the measured line shape is used as input to calculate
the efficiency and ISR correction again. This procedure is repeated until the
difference between the subsequent iteration is comparable with the statistical
uncertainty. The systematic uncertainty due to the input line shapes of
subprocesses is estimated as described below.

The input line shape of each subprocess is varied 500 times according to the
convariant matrix when parametrizing, and the
$\sum\omega_{i}\epsilon_{i}(1+\delta)_{i}$ distribution is sampled at
$\sqrt{s}=$ 4.380, 4.390, 4.400, 4.420, and 4.440 GeV. The maximum fraction of
width and mean values of the distributions, 2.8%, is taken as the systematic
uncertainty due to the input line shapes in the ISR correction. Moreover, new
resonances around $\sqrt{s}=$ 4.700 GeV are added when parameterizing the line
shape of each subprocess since there is an evidence around $\sqrt{s}=4.700$
GeV in the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ cross section line
shape, and the difference is taken as the systematic uncertainty associated
with the new BW resonance in the high energy regions as listed in Table 5.

Table 5 summarizes the total systematic uncertainties. The total systematic
uncertainty at each $\sqrt{s}$ is obtained by summing up all sources of
systematic uncertainties in quadrature, assuming that they are uncorrelated.

###  VII.2 Systematic uncertainties in resonance parameters

The systematic uncertainties when parameterizing the resonances in the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ cross section line shape
mainly stem from the absolute $\sqrt{s}$ measurement, the $\sqrt{s}$ spread,
global shift of the $\sqrt{s}$ for data samples taken in the same period, and
the systematic uncertainty of the cross section measurement.

The absolute $\sqrt{s}$ of data samples with $\sqrt{s}$ smaller than 4.610 GeV
are measured with dimuon events, with an uncertainty of
$\pm$ 0.8 MeV, while those with $\sqrt{s}$ larger than or equal to 4.610 GeV
are measured with $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ events with an
uncertainty of $\pm$ 0.6 MeV.
Thus, 0.8 MeV is taken as the systematic uncertainty, and propagates to the
masses of the resonances by the same amount.

The systematic uncertainty from the $\sqrt{s}$ spread is estimated by
convolving the fit formula with a Gaussian function with a width of 1.6 MeV,
which is the beam spread, determined from measurement results of the Beam
Energy Measurement System 2011BEMS  at other $\sqrt{s}$.

The systematic uncertainty from global shift of the $\sqrt{s}$ for data
samples taken in the same period is estimated by shifting the $\sqrt{s}$ of
corresponding data samples by 3 MeV and deviations of parameters is taken as
the systematic uncertainties.

The systematic uncertainty from the cross section measurement is divided into
two parts. The first part covers uncorrelated systematic uncertainties among
the different data samples (those in Table 5). The corresponding systematic
uncertainty is estimated by including the uncertainty in the fit to the cross
section, and taking the differences on the parameters as the systematic
uncertainties. The second part includes all the other systematic uncertainties
(8.3%), which is common for all data samples, and only affects the
$\Gamma^{e^{+}e^{-}}_{j}\mathscr{B}_{j}$ measurement.

Table 6 summarizes the systematic uncertainties in the parameters of
resonances for the four solutions. The total systematic uncertainty is
obtained by summing up all sources of systematic uncertainties in quadrature,
assuming they are uncorrelated.

Table 6: Systematic uncertainties in the measurement of the resonances parameters. $\sqrt{s}$ represents the systematic uncertainty from the center-of-mass measurement. $\sqrt{s}$ shift represents the systematic uncertainty from the global shift of $\sqrt{s}$ for data samples taken in the same period. Cross $\rm{section}_{a(b)}$ represents the systematic uncertainty from the cross section measurements which are uncorrelated (common) in each data sample. The units of $m_{i}$, $\Gamma^{\rm{tot}_{i}}$, $c$, $\Gamma^{e^{+}e^{-}}_{j}\mathscr{B}_{j}$, and $\phi_{j}$ are MeV/$c^{2}$, MeV, MeV-3/2, eV and rad, respectively. | Sources | $m_{0}$ | $\Gamma_{0}$ | $m_{1}$ | $\Gamma_{1}$ | $c$ | $\Gamma^{e^{+}e^{-}}_{0}\mathscr{B}_{0}$ | $\Gamma^{e^{+}e^{-}}_{1}\mathscr{B}_{1}$ | $\phi_{0}$ | $\phi_{1}$ |
---|---|---|---|---|---|---|---|---|---|---|---
Solution I | $\sqrt{s}$ | 0.8 | - | 0.8 | - | - | - | - | - | - |
$\sqrt{s}$ shift | 2.0 | 0.7 | 0.6 | 0.0 | 70 | 0.1 | 0.0 | 0.0 | 1.0 |
$\sqrt{s}$ spread | 0.0 | 0.1 | 0.1 | 0.3 | 92 | 1.6 | 1.3 | 0.0 | 1.0 |
Cross $\rm{Section}_{a}$ | 0.6 | 1.1 | 3.8 | 8.8 | 58 | 0.1 | 2.9 | - | - |
Cross $\rm{Section}_{b}$ | - | - | - | - | - | 0.8 | 0.0 | - | - |
| Overall | 2.2 | 1.3 | 3.9 | 8.8 | 13$\times$10 | 1.8 | 3.2 | 0.0 | 1.4 |
Solution II | $\sqrt{s}$ | 0.8 | - | 0.8 | - | - | - | - | - | - |
$\sqrt{s}$ shift | 2.0 | 0.7 | 0.6 | 0.0 | 70 | 0.1 | 0.0 | 0.0 | 3.2 |
$\sqrt{s}$ spread | 0.0 | 0.1 | 0.1 | 0.3 | 92 | 1.0 | 0.0 | 0.0 | 0.0 |
Cross $\rm{Section}_{a}$ | 0.6 | 1.1 | 3.8 | 8.8 | 58 | 0.1 | 0.1 | - | - |
Cross $\rm{Section}_{b}$ | - | - | - | - | - | 0.8 | 0.0 | - | - |
| Overall | 2.2 | 1.3 | 3.9 | 8.8 | 13$\times$10 | 1.3 | 0.1 | 0.0 | 3.2 |
Solution III | $\sqrt{s}$ | 0.8 | - | 0.8 | - | - | - | - | - | - |
$\sqrt{s}$ shift | 2.0 | 0.7 | 0.6 | 0.0 | 70 | 0.1 | 0.0 | 0.1 | 3.1 |
$\sqrt{s}$ spread | 0.0 | 0.1 | 0.1 | 0.3 | 92 | 1.5 | 0.0 | 0.0 | 3.1 |
Cross $\rm{Section}_{a}$ | 0.6 | 1.1 | 3.8 | 8.8 | 58 | 0.3 | 0.1 | - | - |
Cross $\rm{Section}_{b}$ | - | - | - | - | - | 0.8 | 0.0 | - | - |
| Overall | 2.2 | 1.3 | 3.9 | 8.8 | 13$\times$10 | 1.7 | 0.1 | 0.1 | 4.4 |
Solution IV | $\sqrt{s}$ | 0.8 | - | 0.8 | - | - | - | - | - | - |
$\sqrt{s}$ shift | 2.0 | 0.7 | 0.6 | 0.0 | 70 | 0.1 | 0.0 | 0.0 | 1.0 |
$\sqrt{s}$ spread | 0.0 | 0.1 | 0.1 | 0.3 | 92 | 1.1 | 1.2 | 0.0 | 0.0 |
Cross $\rm{Section}_{a}$ | 0.6 | 1.1 | 3.8 | 8.8 | 58 | 0.0 | 2.5 | - | - |
Cross $\rm{Section}_{b}$ | - | - | - | - | - | 0.8 | 0.0 | - | - |
| Overall | 2.2 | 1.3 | 3.9 | 8.8 | 13$\times$10 | 1.4 | 2.8 | 0.0 | 1.0 |

###  VII.3 Systematic uncertainties in $X(3842)$ measurement

Except for the fit range and the background shape of the
$RM(\pi_{d}^{+}\pi_{d}^{-})$, $RM(D^{+})$ and $RM(D_{\rm{miss}}^{-})$ mass
window requirements, other sources of systematic uncertainties associated with
this measurement are the same as those in Sec. VII.1, but with the fit range
and background shape of $RM(D^{+}\pi_{d}^{+}\pi_{d}^{-})$ excluded.

The systematic uncertainty originating from the fit range of
$RM(\pi_{d}^{+}\pi_{d}^{-})$ is estimated by varying the limits of the fit
range from (3.79, 3.89) GeV/$c^{2}$ to (3.81, 3.91) GeV/$c^{2}$. The
difference of the cross section from the baseline value in the data sample at
$\sqrt{s}=4.680$ GeV is taken as the systematic uncertainty, and is 10.4%. The
background shape is varied from a second-order Chebychev polynomial function
to a first order one in the data sample taken at $\sqrt{s}=4.680$ GeV, the
difference of the cross section compared with the baseline value is taken as
the systematic uncertainty, and is 1.9%.

The systematic uncertainty stemming from the $RM(D^{+})$ and
$RM(D_{\rm{miss}}^{-})$ mass window requirements, which is mainly caused by
the difference between distributions of data and MC simulation in the
corresponding selection criterion ranges, is estimated by producing
alternative MC samples where the mass and width of $f_{0}(500)$ are varied by
one standard deviation in the data sample at $\sqrt{s}=4.680$ GeV. The
difference of the cross section compared with the baseline value is taken as
the systematic uncertainty, and is 1.9%.

The total systematic uncertainty for data samples with $\sqrt{s}$ smaller than
or equal to 4.315 GeV are equal to 12.9% , and for those with $\sqrt{s}$
larger than 4.315 GeV are equal to 13.1% by summing up all sources of
systematic uncertainties in quadrature, assuming they are uncorrelated.

##  VIII Summary

Using data samples taken at $\sqrt{s}$ from 4.190 to 4.946 GeV, the cross
section of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ process is
reported for the first time by a partial reconstruction method.

In the cross section of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
process, a structure with 0.3$\sigma$ significance is visible around
$\sqrt{s}=4.245$ GeV. This might be the $Y(4230)$ resonance, however, due to
its low significance, it is not possible to assign it to the $Y(4230)$ or the
$Y(4260)$ state.

By fitting the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$ cross section
line shape, we observe a resonance with a mass of (4373.1 $\pm$ 4.0 $\pm$ 2.2)
MeV/$c^{2}$ and a width of (146.5 $\pm$ 7.4 $\pm$ 1.3) MeV, which is in
agreement with the $Y(4390)$. There is evidence with a statistical
significance of 4.1$\sigma$ for a second resonance with a mass of (4706 $\pm$
11 $\pm$ 4) MeV/$c^{2}$ and a width of (45 $\pm$ 28 $\pm$ 9) MeV. The first
uncertainties are statistical and the second are systematic.

The $X(3842)$ resonance is searched for in the $RM(\pi_{d}^{+}\pi_{d}^{-})$
distribution and evidence is found in the $M(\pi_{d}^{+}\pi_{d}^{-})$
distribution in data samples with $\sqrt{s}$ from 4.600 to 4.700 GeV, and its
significance is $4.2\sigma$. By comparing this study with previous studies,
the cross section of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(3770)\rightarrow\pi^{+}\pi^{-}D^{+}D^{-}$
process peaks around 4.390 GeV which indicates this process might be produced
via the $Y(4390)$ state 2019HY ; 2020ZY ; the process
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi_{2}(3823)(\rightarrow\gamma\chi_{c1})$
peaks around $\sqrt{s}=4.360$ and $4.420$ GeV, which means this process might
be produced via the $Y(4360)$ and the $\psi(4415)$ 2015X3823  resonances.
There is evidence that the cross section of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}X(3842)$ process peaks at $\sqrt{s}$ from
4.600 to 4.700 GeV, but no significant signal is observed in samples collected
at $\sqrt{s}$ around 4.400 GeV. This indicates that the production mechanism
of the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(1D)$ processes might be
different and could proceed via different $Y$ or $\psi$ states. More data
samples and more precise measurements are needed to reveal the mechanism
2020BESWP .

###### Acknowledgements.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012,
12022510, 12025502, 12035009, 12035013, 12192260, 12192261, 12192262,
12192263, 12192264, 12192265; the Chinese Academy of Sciences (CAS) Large-
Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds
of the NSFC and CAS under Contract No. U1832207; CAS Key Research Program of
Frontier Sciences under Contract No. QYZDJ-SSW-SLH040; 100 Talents Program of
CAS; 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, Collaborative Research Center CRC 1044, GRK 2149; Istituto
Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; 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; STFC (United Kingdom); 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 Contracts Nos. DH140054, DH160214; The
Swedish Research Council; U. S. Department of Energy under Contract No. DE-
FG02-05ER41374.

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