# Search for the production of deuterons and antideuterons in $e^{+}e^{-}$
annihilation at center-of-mass energies between 4.13 and 4.70 GeV

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,
Y. Ding33, Y. Ding39, 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. Liang33, H.
Liang69,56, 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, 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. Tian50, W. H. Tian57, Y. Tian30,61, Z. F. Tian74, I.
Uman60B, B. Wang69,56, B. Wang1, B. L. Wang61, 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, 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, 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. Zhang42, Z. Y. Zhang74, 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. 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, Krachi, Pakistan

###### Abstract

Using a data sample of $e^{+}e^{-}$ collision data corresponding to an
integrated luminosity of 19 fb-1 collected with the BESIII detector at the
BEPCII collider, we search for the production of deuterons and antideuterons
via $e^{+}e^{-}\to pp\pi^{-}\bar{d}+c.c.$ for the first time at center-of-mass
energies between 4.13 and 4.70 GeV. No significant signal is observed and the
upper limit of the $e^{+}e^{-}\to pp\pi^{-}\bar{d}+c.c.$ cross section is
determined to be from 9.0 to 145 fb depending on the center-of-mass energy at
the $90\%$ confidence level.

##  I Introduction

In the conventional quark model, mesons are composed of one quark and one
antiquark, while baryons are composed of three quarks. However, many states
with properties inconsistent with the conventional two or three quark models,
such as the X(3872) [1], Y(4260) [2], $Z_{c}$(3900) [3, 4], $\pi_{1}(1600)$
[5], and $P_{c}(4450)$ [6], have been discovered in the last two decades.
Numerous theoretical proposals and experimental investigations on the subject
of exotic states are reviewed in Refs. [7, 8, 9, 10, 11], and there is strong
evidence for the existence of tetraquark, pentaquark, and meson-meson and
meson-baryon molecular states [7, 8, 9].

The study of hadronic states with six quarks, either compact hexaquark states
or dibaryon states, has a long history [12, 11, 13, 14, 15, 16]. Among them
the $d^{*}(2380)$ has attracted substantial attention [17]. The $d^{*}(2380)$,
with a mass of about 2380 MeV and a width of about 70 MeV, was first observed
in the isoscalar double-pionic fusion process $pn\to d\pi^{0}\pi^{0}$ [18],
and was later confirmed in other double-pionic fusion processes $pn\to
d\pi^{+}\pi^{-}$ [19] and $pp\to d\pi^{+}\pi^{0}$ [20], and non-fusion
processes $pn\to pp\pi^{0}\pi^{-}$ [21] and $pn\to pn\pi^{0}\pi^{0}$ [22]. The
$d^{*}(2380)$ has been proposed to be the excited deuteron ($d$), a molecule
with large $\Delta\Delta$ component [23], or a hexaquark state which is
dominated by the hidden-color component [24].

The A2 collaboration observed a high spin polarization in the measurement of
the recoiling neutron in deuterium photodisintegration, which could related to
the excitation of the $d^{*}(2380)$ [25]. Apart from this measurement, most of
the results related to the $d^{*}(2380)$ have so far come from the WASA
experiment, and further studies from other experiments are needed to confirm
the existence of the $d^{*}(2380)$ and to search for other similar states. The
production of the antideuteron ($\bar{d}$) has been studied in several
$e^{+}e^{-}$ annihilation experiments. The ARGUS [26] and CLEO [27]
experiments observed $\bar{d}$ production at the level of $3\times 10^{-5}$
per $\Upsilon(1S)$ and $\Upsilon(2S)$ decays, and set limits on production in
$\Upsilon(4S)$ decays and $e^{+}e^{-}\to q\bar{q}$ at a center-of-mass energy
($\sqrt{s}$) of 10.6 GeV. The BaBar [28] experiment performed measurements of
inclusive antideuteron production in $\Upsilon(1S)$, $\Upsilon(2S)$,
$\Upsilon(3S)$ decays and in $e^{+}e^{-}$ annihilation into hadrons at
$\sqrt{s}\approx 10.58$ GeV. The ALEPH [29] experiment observed evidence at
$3\sigma$ significance for $\bar{d}$ production in $e^{+}e^{-}\to q\bar{q}$ at
$\sqrt{s}=91.2$ GeV. Technically, the $d^{*}(2380)$ state could be studied
with these data by combining the detected antideuteron with pions in the same
event.

The BESIII experiment [30] collects $e^{+}e^{-}$ collision data at $\sqrt{s}$
between 2 and 4.95 GeV. In recent years, the BESIII Collaboration has reported
the observation of $e^{+}e^{-}\to 2(p\bar{p})$ [31] and $e^{+}e^{-}\to
pp\bar{p}\bar{n}\pi^{-}+c.c.$ [32] with the production cross sections on the
order of 10 fb. This suggests the production of a deuteron or antideuteron
together with two other nucleons may also be observed. In this paper, we
search for the production of (anti)deuterons at the BESIII experiment for the
first time using the process $e^{+}e^{-}\to pp\pi^{-}\bar{d}+c.c.$ at
$\sqrt{s}$ from 4.13 to 4.70 GeV.

##  II The BESIII detector and data samples

The BESIII detector [30] records symmetric $e^{+}e^{-}$ collisions provided by
the BEPCII storage ring [33], which operates in the $\sqrt{s}$ range from 2.0
to 4.95 GeV. BESIII has collected large data samples in this energy region
[34]. 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 [35]. 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 specific energy loss (d$E$/d$x$)
resolution is $6\%$ for electrons from Bhabha scattering. The EMC measures
photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel
(end cap) region. The time resolution in the TOF barrel region is 68 ps, while
that in the end cap region 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 [36, 37, 38].

The data used in this work are listed in Table 1. The $\sqrt{s}$ for each data
set was measured using the di-muon process ($e^{+}e^{-}\to\mu^{+}\mu^{-}$)
with an uncertainty of less than 1.0 MeV [39, 40, 41], and the integrated
luminosities were measured using the Bhabha process ($e^{+}e^{-}\to
e^{+}e^{-}$) with an uncertainty of 1.0% [42, 41, 43]. The total integrated
luminosity of the data used in this work is approximately 19 fb-1, of which
about 15 fb-1 data samples were collected after the upgrade of the end cap TOF
system.

Table 1: The center-of-mass energies ($\sqrt{s}$), integrated luminosities ($\mathcal{L}_{\rm int}$), and signal yields for different cases. The uncertainties for signal yields are statistical only. $\sqrt{s}$ (MeV) | $\mathcal{L}_{\rm int}$ (pb-1) | Signal yield
---|---|---
case 1 | case 2 | case 3 | case 4
4128.5 | 401.5 | $\ \ \,0.4\pm 1.9$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4157.4 | 408.7 | $\ \ \,1.0\pm 2.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4178.0 | 3189.0 | $\ \ \ \;5.7\pm 11.7$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4188.8 | 526.7 | $\ \ \,4.7\pm 4.4$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4198.9 | 526.0 | $-0.1\pm 3.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4209.2 | 517.1 | $-1.4\pm 5.1$ | $0.0^{+0.9}$ | $1.0^{+1.4}$ | $0.0^{+0.9}$
4218.7 | 514.6 | $\ \,-2.1\pm 12.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4226.3 | 1100.9 | $-0.7\pm 4.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4235.7 | 530.3 | $\ \ \,3.4\pm 4.4$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4243.8 | 538.1 | $-2.1\pm 2.2$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4258.0 | 828.4 | $-2.1\pm 4.5$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4266.8 | 531.1 | $\ \ \,3.7\pm 4.8$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4277.7 | 175.5 | $\ \ \,0.0\pm 3.0$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4287.9 | 502.4 | $-2.1\pm 8.3$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4312.0 | 501.2 | $\ \ \,0.9\pm 3.4$ | $0.0^{+0.9}$ | $1.0^{+1.4}$ | $0.0^{+0.9}$
4337.4 | 505.0 | $\ \ \,3.6\pm 3.8$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4358.3 | 543.9 | $-2.0\pm 7.1$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4377.4 | 522.7 | $\ \ \,1.8\pm 3.8$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4396.4 | 507.8 | $-1.8\pm 6.3$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4415.6 | 1090.6 | $-1.0\pm 3.8$ | $1.0_{-0.7}^{+1.4}$ | $1.0_{-0.7}^{+1.4}$ | $0.0^{+0.9}$
4436.2 | 569.9 | $\ \ \,6.1\pm 5.1$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4467.1 | 111.1 | $-2.1\pm 0.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4527.1 | 112.1 | $-2.1\pm 1.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4599.5 | 586.9 | $\ \ \,1.4\pm 4.3$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4611.9 | 103.8 | $-2.1\pm 1.0$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4628.0 | 521.5 | $-2.1\pm 1.5$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4640.9 | 552.4 | $-0.4\pm 3.0$ | $1.0_{-0.7}^{+1.4}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4661.2 | 529.6 | $-2.1\pm 1.4$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4681.9 | 1669.3 | $-2.1\pm 2.6$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$
4698.8 | 536.4 | $\ \ \,3.3\pm 3.5$ | $0.0^{+0.9}$ | $0.0^{+0.9}$ | $0.0^{+0.9}$

Figure 1: Distributions of $RM(pp\pi^{-})$ and $RM(\bar{p}\bar{p}\pi^{+})$ in
data (dots with error bars), inclusive MC (shaded histograms) and the weighted
signal MC samples (red histograms) summed over all energy points and after all
the selections. The $\bar{d}$ cannot be simulated in the current BESIII
software system, so there is no signal MC shape in case 2. The region between
the two red arrows is the signal region.

To optimize the selection criteria, determine the detection efficiency and
estimate the background contributions, Monte Carlo (MC) simulated samples of
events are used. The MC samples are produced with a geant4-based [44] software
package, which includes the geometric description of the BESIII detector and
the detector response. The simulation models the beam energy spread and
initial state radiation (ISR) in the $e^{+}e^{-}$ annihilations with the
generator kkmc [45, 46]. 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. The known decay modes of
charmonium states are modeled with evtgen [47, 48] using branching fractions
taken from the Particle Data Group [49], and the remaining unknown charmonium
decays are modeled with lundcharm [50, 51]. Final state radiation from charged
final state particles is incorporated using the photos package [52]. The
signal MC samples of the process $e^{+}e^{-}\to pp\pi^{-}\bar{d}$ together
with the charge-conjugate process $e^{+}e^{-}\to\bar{p}\bar{p}\pi^{+}d$ are
generated with a phase space model at each $\sqrt{s}$.

##  III data analysis

To avoid possible bias in the (anti-)deuteron simulation and reconstruction
and to improve the selection efficiency, a partial-reconstruction technique is
implemented in which only the two protons (antiprotons) and charged $\pi^{-}$
($\pi^{+}$) are reconstructed, and the $\bar{d}$ ($d$) can be missed. Events
with three good charged tracks and with a net charge of one and events with
four good charged tracks and with zero net charge are selected as candidates.
Hereafter, the charge-conjugate mode is always implied, unless explicitly
stated otherwise.

For each good charged track, the polar angle $\theta$ is required to be within
a range of $|\cos\theta|<0.93$, where $\theta$ is defined with respect to the
symmetry axis of the MDC that is taken as the $z$ axis. The distance of
closest approach to the interaction point (IP) must be less than 10 cm along
the $z$-axis, $|V_{z}|<10$ cm, and less than 1 cm in the transverse plane,
$|V_{xy}|<1$ cm. For particle identification (PID), the specific energy loss
d$E$/d$x$ measured by the MDC and the flight time measured by the TOF are used
to form likelihoods $\mathcal{L}(h)~{}(h=p,K,\pi)$ for each hadron ($h$)
hypothesis. Tracks are identified as protons when $\mathcal{L}(p)>0.001$,
$\mathcal{L}(p)>\mathcal{L}(K)$ and $\mathcal{L}(p)>\mathcal{L}(\pi)$, while
charged pions are identified when $\mathcal{L}(\pi)>0.001$,
$\mathcal{L}(\pi)>\mathcal{L}(p)$ and $\mathcal{L}(\pi)>\mathcal{L}(K)$. To
suppress beam-related background contributions, we require the two protons and
the $\pi^{-}$ to originate from a common vertex and the $\chi^{2}$ of the
vertex fit, $\chi^{2}_{\rm VF}$, to be less than 70. The vertex position is
required to be within a range of $(R_{vx}-0.12$ cm)2+$(R_{vy}+0.14$
cm)${}^{2}\leq(0.5$ cm)2 to suppress backgrounds arising from interactions
between the beam particles and the beam pipe, where $R_{vx}$ and $R_{vy}$ are
the $x$ and $y$ coordinates of the vertex, respectively.

The data samples are divided into four mutually exclusive event classes
(‘cases’) that depend on the reconstruction method. For cases 1 and 2, the
signal mode is $e^{+}e^{-}\to pp\pi^{-}\bar{d}$ and the number of
reconstructed tracks is 3 or 4, respectively. Similarly, for cases 3 and 4,
the signal mode is $e^{+}e^{-}\to\bar{p}\bar{p}\pi^{+}d$ and the number of
tracks is 3 or 4, respectively. Different selection criteria are applied to
events in the four different classes.

For cases 1 and 3, the deuteron track is not reconstructed, either due to the
detector coverage or due to the large ionization of the low momentum track.
Based on the simulated MC samples, the transverse momentum recoiling against
the $pp\pi^{-}$ system, $p_{t}$, is required to be less than 0.35 GeV/$c$ when
$|\cos\theta_{r}|\leq 0.93$, where $\theta_{r}$ is the opening angle between
the recoiling system and the $z$-axis.

For cases 2 and 4, the deuteron track is reconstructed, and more information
can be used to suppress the background. The mass squared of the deuteron
candidate calculated using the TOF information, $m^{2}_{d}$, is required to be
within a range of $3.2<m^{2}_{d}<4.1$ (GeV/$c^{2}$)2. Here,
$m^{2}_{d}=p^{2}\cdot(\beta^{-2}-1)$, and $\beta=L_{\rm path}/(c\cdot T_{\rm
tof})$, where $p$ is the momentum of the deuteron track, $L_{\rm path}$ is the
length of the MDC track extrapolated to the TOF inner radius, $T_{\rm tof}$ is
the time of flight corresponding to $L_{\rm path}$ and $c$ is the speed of
light. The opening angle $\theta^{\prime}$ between the candidate deuteron
track and the recoil direction of the $pp\pi^{-}$ system, is required to
satisfy $\cos\theta^{\prime}>0.9$ to suppress background. Studies based on the
inclusive MC samples show that the main surviving background is $e^{+}e^{-}\to
pp\pi^{-}\bar{p}\bar{n}+c.c.$ To remove this background process, the number of
$\bar{p}$ is required to be zero for case 2 and the number of $p$ is required
to be zero for case 4.

##  IV background analysis and signal extraction

After imposing all the requirements mentioned above, the distribution of the
recoil mass of the $pp\pi^{-}$ ($RM(pp\pi^{-})$) system in data, inclusive MC,
and the weighted signal MC samples summed over all energy points is shown in
Fig. 1. For the inclusive MC samples, only a few background events survive,
and the remaining events do not form a peak in the $RM(pp\pi^{-})$
distribution. Only a few events are present in the data samples, except for
case 1. The two-dimensional distribution for $R_{vz}$ and $RM(pp\pi^{-})$ in
case 1 is shown in Fig. 2 at $\sqrt{s}=4235.7$ MeV as an example, where
$R_{vz}$ is the $z$ coordinate of the vertex. The flat $R_{vz}$ distribution
in data indicates that the main backgrounds in case 1 are not from
$e^{+}e^{-}$ annihilation, but from the interaction between beam particles and
detector material.

Figure 2: Two-dimensional distribution for $R_{vz}$ versus $RM(pp\pi^{-})$ (a)
and the projections of the two-dimensional unbinned fit on $RM(pp\pi)$ (b) and
$R_{vz}$ (c) in case 1 at $\sqrt{s}=4235.7$ MeV. Here, the red solid box is
the signal region and $N_{\rm sig}$ is the signal yield from the fit result.

Figure 3: Efficiencies for the $pp\pi^{-}\bar{d}$ (red triangles) and
$\bar{p}\bar{p}\pi^{+}d$ (blue dots) processes after applying part of the
selection criteria as described in Section V (a), and the proportions of case
3 (blue dots) and 4 (red triangles) events in the $\bar{p}\bar{p}\pi^{+}d$
process (b).

Different methods are used to determine the signal yields in the different
cases. For case 1, a two-dimensional unbinned maximum likelihood fit to the
$RM(pp\pi^{-})$ versus $R_{vz}$ distribution is performed to determine the
signal yields. Here, the signal shape is taken from the signal MC samples, and
a first or second order polynomial function is used to describe the background
shape. Figure 2 shows the fit result projected on $RM(pp\pi^{-})$ and $R_{vz}$
at $\sqrt{s}=4235.7$ MeV. For the other cases, a “counting” method is used to
determine the signal yields. The signal events are selected with both
$RM(pp\pi^{-})$ and $R_{vz}$ within a window of three standard deviations
($\pm 3\sigma$) around the mean values, _i.e._ , $1.863<RM(pp\pi^{-})<1.892$
GeV/$c^{2}$ and $-2.5<R_{vz}<3.5$ cm, referred to as the signal region. Here
the mean values and standard deviations of the $RM(pp\pi^{-})$ and $R_{vz}$
distributions are determined from the signal MC samples. The statistical
uncertainty of the number of signal events is estimated with the trolke [53,
54] package in the root [55] framework.

No significant signal is observed from the fit result for case 1, and almost
no events survive in the signal region for cases 2, 3 and 4. The signal yields
for the different cases are summarized in Table 1.

##  V Detection efficiency

The signal MC samples are used to determine the detection efficiency. In the
geant4 software used by the BESIII experiment, the $\bar{d}$ is not simulated.
Thus, for cases 1 and 2, which have a $\bar{d}$ in the final state, the
detection efficiencies for some of the selection criteria are estimated with
control samples selected from data. For cases 3 and 4, the processes can be
simulated well and the selection efficiency can be obtained from the signal MC
samples directly.

In order to estimate the selection efficiencies for cases 1 and 2, the part of
the selection criteria consisting of the requirements on the vertex fit, the
vertex position region, and the signal region of $RM(pp\pi^{-})$ and $R_{vz}$
are used to select $\bar{p}\bar{p}\pi^{+}d$ and $pp\pi^{-}\bar{d}$ processes
from the signal MC samples. For each process, the number of tracks can be 3 or
4. Studies based on the high-purity control sample of $e^{+}e^{-}\to
p\bar{p}\pi^{+}\pi^{-}$ show that the proportions of 3 or 4-track events in
$\pi^{+}\pi^{-}p$ tagged and $\pi^{+}\pi^{-}\bar{p}$ tagged samples are very
close. So we assume that the proportion of 3 or 4-track events in the
$\bar{p}\bar{p}\pi^{+}d$ process is also close to that in the
$pp\pi^{-}\bar{d}$ process in this analysis. The difference in the proportions
between $\pi^{+}\pi^{-}p$ tagged and $\pi^{+}\pi^{-}\bar{p}$ tagged samples
when the number of tracks equals to 3 or 4 in the control sample is taken as
the systematic uncertainty, which will be discussed in detail in the next
section. Figure 3(a) shows the efficiencies of the $pp\pi^{-}\bar{d}$ and
$\bar{p}\bar{p}\pi^{+}d$ processes after the above selection and Fig. 3(b)
shows the proportions of 3 and 4-track events in the $\bar{p}\bar{p}\pi^{+}d$
process. The efficiencies of cases 1 and 2 can be calculated with these
numbers accordingly.

Table 2 lists the selection efficiencies for the different event classes. The
efficiencies have been corrected for differences between data and MC
simulation, as shown by the total correction factors $W$ in this table. The
details on the correction factors are discussed in the next section.

Table 2: The total correction factor $w$ and the corresponding corrected efficiency $\epsilon$. $\sqrt{s}$ (MeV) | case 1 | case 2 | case 3 | case 4
---|---|---|---|---
$W$ | $\epsilon$ | $W$ | $\epsilon$ | $W$ | $\epsilon$ | $W$ | $\epsilon$
4128.5 | 0.925 | 0.056 | 1.036 | 0.032 | 1.023 | 0.047 | 1.034 | 0.025
4157.4 | 0.925 | 0.059 | 1.035 | 0.046 | 1.021 | 0.050 | 1.031 | 0.035
4178.0 | 0.927 | 0.060 | 1.035 | 0.056 | 1.019 | 0.050 | 1.027 | 0.043
4188.8 | 0.927 | 0.058 | 1.035 | 0.059 | 1.018 | 0.051 | 1.027 | 0.047
4198.9 | 0.927 | 0.059 | 1.034 | 0.064 | 1.019 | 0.050 | 1.025 | 0.049
4209.2 | 0.927 | 0.058 | 1.033 | 0.066 | 1.018 | 0.049 | 1.024 | 0.051
4218.7 | 0.928 | 0.056 | 1.034 | 0.071 | 1.018 | 0.049 | 1.023 | 0.056
4226.3 | 0.929 | 0.059 | 1.033 | 0.076 | 1.018 | 0.050 | 1.022 | 0.058
4235.7 | 0.928 | 0.057 | 1.034 | 0.079 | 1.017 | 0.049 | 1.023 | 0.062
4243.8 | 0.928 | 0.056 | 1.034 | 0.081 | 1.017 | 0.048 | 1.022 | 0.064
4258.0 | 0.929 | 0.056 | 1.034 | 0.084 | 1.017 | 0.049 | 1.022 | 0.068
4266.8 | 0.928 | 0.054 | 1.032 | 0.089 | 1.016 | 0.047 | 1.020 | 0.071
4277.7 | 0.928 | 0.053 | 1.033 | 0.088 | 1.016 | 0.047 | 1.021 | 0.070
4287.9 | 0.929 | 0.052 | 1.033 | 0.092 | 1.016 | 0.044 | 1.020 | 0.071
4312.0 | 0.927 | 0.050 | 1.033 | 0.010 | 1.013 | 0.043 | 1.019 | 0.077
4337.4 | 0.930 | 0.050 | 1.033 | 0.110 | 1.015 | 0.042 | 1.017 | 0.083
4358.3 | 0.931 | 0.048 | 1.033 | 0.116 | 1.015 | 0.042 | 1.017 | 0.093
4377.4 | 0.930 | 0.046 | 1.033 | 0.116 | 1.014 | 0.040 | 1.016 | 0.092
4396.4 | 0.930 | 0.045 | 1.032 | 0.123 | 1.013 | 0.038 | 1.015 | 0.094
4415.6 | 0.931 | 0.045 | 1.032 | 0.126 | 1.013 | 0.039 | 1.014 | 0.099
4436.2 | 0.931 | 0.043 | 1.032 | 0.129 | 1.013 | 0.038 | 1.014 | 0.104
4467.1 | 0.932 | 0.041 | 1.032 | 0.127 | 1.013 | 0.036 | 1.013 | 0.103
4527.1 | 0.931 | 0.038 | 1.032 | 0.137 | 1.012 | 0.034 | 1.013 | 0.113
4599.5 | 0.931 | 0.032 | 1.033 | 0.144 | 1.010 | 0.030 | 1.012 | 0.118
4611.9 | 0.931 | 0.034 | 1.033 | 0.142 | 1.010 | 0.030 | 1.012 | 0.115
4628.0 | 0.931 | 0.033 | 1.032 | 0.148 | 1.010 | 0.030 | 1.011 | 0.121
4640.9 | 0.932 | 0.032 | 1.033 | 0.149 | 1.010 | 0.029 | 1.011 | 0.120
4661.2 | 0.931 | 0.032 | 1.032 | 0.151 | 1.009 | 0.029 | 1.011 | 0.124
4681.9 | 0.931 | 0.031 | 1.032 | 0.150 | 1.008 | 0.028 | 1.010 | 0.124
4698.8 | 0.931 | 0.031 | 1.032 | 0.152 | 1.009 | 0.028 | 1.010 | 0.124

##  VI systematic uncertainty

The systematic uncertainties in the cross section measurement originate from
the integrated luminosity, the tracking and PID efficiencies, and the
determination of the signal yields, which includes the fit range, the signal
and background shape, etc. These systematic uncertainties are listed in Table
3 and discussed in more detail below.

Table 3: Relative systematic uncertainties (in units of %) in the cross section measurement of $e^{+}e^{-}\to pp\pi^{-}\bar{d}+c.c.$ Source | case 1 | case 2 | case 3 | case 4
---|---|---|---|---
Luminosity | 1.0 | 1.0 | 1.0 | 1.0
Tracking and PID efficiencies | 0.5-1.4 | 0.7-1.9 | 0.6-1.5 | 0.8-1.9
Vertex fit | 0.6 | 0.6 | 0.6 | 0.6
$R_{vx}$ and $R_{vy}$ | 0.3 | 0.3 | 0.3 | 0.3
$p_{t}$ and $\cos\theta_{r}$ | 2.2-3.1 | - | 2.2-3.1 | -
Open angle $\cos\theta^{\prime}$ | - | - | - | -
$m^{2}_{d}$ | - | 0.6-6.0 | - | 0.6-6.0
$p(\bar{p})$ veto | - | 0.4-2.1 | - | 0.4-2.1
Efficiency estimation | 3.0 | 1.0 | - | -
Signal yield extraction | 17.7 | 11.8 | 11.8 | 11.8
Sum | 18.1-18.3 | 12.0-13.4 | 12.1-12.3 | 11.9-13.4

The integrated luminosity is measured using Bhabha scattering events with an
uncertainty of 1.0% [43, 41, 42].

The tracking and PID efficiencies are studied with a high-purity control
sample of $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$ events, and the polar angle and
transverse momentum ($p_{t}$) dependent efficiencies are measured [56]. The
efficiency of MC events is corrected by the two-dimensional efficiency scale
factors, and the uncertainty is estimated by varying the efficiency scale
factors by one standard deviation for each $p_{t}$ versus $\cos\theta$ bin.
The differences between the new efficiencies and the nominal ones are taken as
the systematic uncertainties.

In this analysis, the selection efficiency is corrected according to the
measurements with control samples selected from data directly. The efficiency
correction factor $w^{\rm s}$ is defined as

$w^{\rm s}=\epsilon_{\rm data}^{\rm s}/\epsilon_{{\rm MC}}^{\rm s},$ (1)

where the subscripts “MC” and “data” represent MC simulation and data samples,
respectively, the superscript “s” represents the selection criterion “s” and
$\epsilon^{\rm s}$ is the efficiency for the selection “s” which is calculated
as

$\epsilon^{\rm s}=\frac{n_{\rm sig}^{\rm s}}{n_{\rm sig}^{\rm s}+n_{\rm
bkg}^{\rm s}},$ (2)

for both data and MC samples. Here $n_{\rm sig}^{\rm s}$ is the number of
events in the signal region with selection criterion “s” and $n_{\rm bkg}^{\rm
s}$ is the number of events out of the region of selection criterion “s”. The
uncertainty of $\epsilon^{\rm s}$ and the uncertainty of $w^{\rm s}$ is
calculated as

$\sigma_{\epsilon^{\rm s}}=\frac{n_{\rm bkg}^{\rm s}}{n_{\rm sig}^{\rm
s}}\cdot\frac{\sqrt{{(\frac{\sigma_{n_{\rm sig}^{\rm s}}}{n_{\rm bkg}^{\rm
s}})}^{2}+{{(\frac{\sigma_{n_{\rm bkg}^{\rm s}}}{n_{\rm bkg}^{\rm
s}})}^{2}}}}{{(1+\frac{n_{\rm bkg}^{\rm s}}{n_{\rm sig}^{\rm s}})}^{2}},$ (3)
$\sigma_{w^{\rm s}}=\frac{\epsilon_{\rm data}^{\rm s}}{\epsilon_{\rm MC}^{\rm
s}}\cdot\sqrt{{(\frac{\sigma_{\epsilon_{\rm data}^{\rm s}}}{\epsilon_{\rm
data}^{\rm s}})}^{2}+{(\frac{\sigma_{\epsilon_{\rm MC}^{\rm s}}}{\epsilon_{\rm
MC}^{\rm s}})}^{2}},$ (4)

where $\sigma_{n_{\rm sig}^{\rm s}}$ and $\sigma_{n_{\rm bkg}^{\rm s}}$ are
the uncertainties of $n_{\rm sig}^{\rm s}$ and $n_{\rm bkg}^{\rm s}$. For
$w^{\rm s}=(1+\Delta w^{\rm s})\pm\sigma_{w^{\rm s}}$, if $|\frac{\Delta
w^{\rm s}}{\sigma_{w^{\rm s}}}|<1.0$, the MC simulation is consistent with the
data, no correction will be applied, and $|\Delta w^{\rm s}|+\sigma_{w^{\rm
s}}$ is taken as the systematic uncertainty. On the other hand, if
$|\frac{\Delta w^{\rm s}}{\sigma_{w^{\rm s}}}|>1.0$, the MC efficiency will be
corrected to data, and the new efficiency after correction is
$\epsilon=\epsilon_{\rm MC}\cdot w^{\rm s}$, and $\sigma_{w^{\rm s}}$ is taken
as the systematic uncertainty. The total correction factor $W$ is defined as
$\prod\limits_{\rm s}\limits{w^{\rm s}}$, which is shown in Table 2.

The uncertainties of the vertex fit and vertex position are studied using a
control sample of $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$, $J/\psi\to
e^{+}e^{-}/\mu^{+}\mu^{-}$ events. The efficiency ratio between data and MC
simulation with the vertex fit requirement $\chi_{\rm VF}^{2}<70$ is $w^{\rm
VF}=1.002\pm 0.004$, and we take $\sigma^{\rm VF}=0.006$ as the systematic
uncertainty and set $w^{\rm VF}=1.0$. The efficiency ratio between data and MC
simulation with the vertex position requirement $(R_{vx}-0.12$
cm)2+$(R_{vy}+0.14$ cm)${}^{2}\leq(0.5$ cm)2 is $w^{\rm VP}=1.001\pm 0.002$,
and we take $\sigma^{\rm VP}=0.003$ as the systematic uncertainty and set
$w^{\rm VP}=1.0$.

The systematic uncertainty due to the $p_{t}$ and $\cos\theta_{r}$
requirements is estimated by loosening or tightening the nominal requirements.
The transverse momentum of the track recoiling against the $pp\pi^{-}$ system,
$p_{t}$, is required to be less than 0.33 or 0.37 GeV/$c$ when
$|\cos\theta_{r}|\leq 0.93$. The largest changes of the efficiency compared to
the nominal requirement range from 2.2% to 3.1% and are taken as the
corresponding uncertainties.

Because we do not have a pure deuteron sample, due to the different time
resolution of the TOF between data and MC simulation, the uncertainty from the
$m^{2}_{d}$ requirement is studied with the control sample $e^{+}e^{-}\to
p\bar{p}\pi^{+}\pi^{-}$. The $m^{2}$ requirement for the recoil mass of
$\pi^{+}\pi^{-}\bar{p}$ is applied at each energy point in the control sample.
Different $m^{2}$ ranges are chosen at each energy point to ensure that the
efficiency in the control sample is the same as that in this analysis. The
efficiency difference between data and MC simulation, which ranges from 0.6%
to 6.0%, is taken as the systematic uncertainty.

The systematic uncertainty of the $p(\bar{p})$ veto in the
$\bar{p}\bar{p}\pi^{+}d(pp\pi^{-}\bar{d})$ process is due to the difference in
proportion of deuterons misidentified as protons between data and signal MC
samples. The proportion of deuterons misidentified as protons in the
simulation can be calculated from the signal MC sample, which is taken as the
systematic uncertainty directly since the difference between data and MC
sample is small. The corresponding uncertainty ranges from 0.4% to 2.1%.

The efficiency of the $\cos\theta^{\prime}$ selection is very high due to the
loose requirement, and its systematic uncertainty is negligible.

Since the $\bar{d}$ cannot be simulated, the $\bar{p}\bar{p}\pi^{+}d$ process
is used to estimate the efficiency in the $pp\pi^{-}\bar{d}$ process, which is
described in detail in the previous section. To estimate the systematic
uncertainty, we take $e^{+}e^{-}\to p\bar{p}\pi^{+}\pi^{-}$ as the control
sample. The proportions of 3-track or 4-track events in the
$\pi^{+}\pi^{-}\bar{p}$ tagged and $\pi^{+}\pi^{-}p$ tagged samples are
obtained from the control sample and the difference in proportions between
$\pi^{+}\pi^{-}p$ tagged and $\pi^{+}\pi^{-}\bar{p}$ tagged samples when the
number of tracks equals 3 or 4 is taken as the systematic uncertainty. Based
on studies of the control sample, the correction factor and the systematic
uncertainty are calculated, where the correction factors are 0.93 and 1.03 for
cases 1 and 2, respectively, and the systematic uncertainties are 3% and 1%
for cases 1 and 2, respectively.

For the uncertainty of the signal yield extraction, alternative methods are
used to estimate the signal yield. For case 1, the uncertainty of the signal
yield extraction is estimated by varying the fit range, and changing the
signal and background shape. We vary the upper and lower bounds of $RM(pp\pi)$
and $R_{vz}$ by $\pm 5$ MeV/$c^{2}$ and $\pm 1$ cm, respectively, use a
Gaussian function to describe the signal shape, and use a third-order
polynomial function to describe the background shape. The largest difference
of the cross section compared with the nominal one is taken as the systematic
uncertainty, which is 17.7%. For cases 2, 3 and 4, the control samples,
$e^{+}e^{-}\to p\bar{p}\pi^{+}\pi^{-}$ and $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$,
are used to estimate the systematic uncertainties of the $R_{vz}$ and
$RM(pp\pi^{-})$ window selection, respectively. The largest difference in
efficiencies between data and simulated MC samples is taken as the systematic
uncertainty, which is 11.8%.

The total systematic uncertainties for the cross section measurement in
different cases are summarized in Table 3. Totals are obtained by summing the
individual values in quadrature under the assumption that all the sources are
independent.

##  VII Upper Limit of the Cross Section

The upper limit of the cross section at the $90\%$ confidence level (C.L.),
taking into account the systematic uncertainty, is calculated using a Bayesian
method. For case 1, a scan of the likelihood distribution ($\mathcal{L}$) as a
function of the signal yield ($N_{\rm sig}$) is obtained from fits to
$RM(pp\pi^{-})$ versus $R_{vz}$ with fixed values for the signal yield. To
take the systematic uncertainty into consideration, the likelihood
distribution is convolved with a Gaussian function with mean value zero and
standard deviation $N_{\rm sig}\cdot\sigma_{\rm sys}$. The upper limit on the
signal yield ($N_{\rm sig}^{\rm ul}$) at the 90% C.L. is determined as
${\int_{0}^{N_{\rm sig}^{\rm ul}}\mathcal{L}\,dN_{\rm
sig}}/{\int_{0}^{\infty}\mathcal{L}\,dN_{\rm sig}}$. For cases 2, 3, and 4, a
likelihood function is constructed to calculate the signal yield at the $90\%$
C.L. assuming that the numbers of signal ($N_{s}$) and background ($N_{b}$)
events obey a Poisson distribution, and the efficiency ($\epsilon$) obeys a
Gaussian distribution

$\displaystyle P(N_{s})=Pois(N_{s};\epsilon\mu),$ (5) $\displaystyle
P(N_{b})=Pois(N_{b};\tau b),$ $\displaystyle
P(\epsilon)=G(\epsilon;z,\sigma_{\epsilon}),$

where $\mu$ is the signal rate in the signal region, $b$ is the number of
background events in the signal region, $z$ is the efficiency obtained from MC
simulation, $\sigma_{\epsilon}$ is the absolute systematic uncertainty,
$\sigma_{\epsilon}=\epsilon\cdot\sigma_{\rm sys}$, where $\sigma_{\rm sys}$ is
the relative total systematic uncertainty, and $\tau$ is the ratio of the
sizes of the sideband regions and the signal regions.

Since there are very few events in the sideband region, we fix $b$ to 0 in the
likelihood function to get a more conservative upper limit. The likelihood
function is defined as

$L(\mu,\epsilon\ |\
N_{s},z)=\frac{(\epsilon\mu)^{N_{s}}}{N_{s}!}e^{-\epsilon\mu}\times\frac{1}{\sqrt{2\pi}\sigma_{\epsilon}}e^{-\frac{{(\epsilon-z)}^{2}}{2\sigma_{\epsilon}^{2}}}.$
(6)

The signal yield at the $90\%$ C.L. is determined using Eq. (6). Since the
data samples for the four cases are completely independent, the likelihood
functions for the different cases can be multiplied to get a combined
likelihood function. The signal yields at the $90\%$ C.L. for the processes of
$e^{+}e^{-}\to pp\pi^{-}\bar{d}$, $e^{+}e^{-}\to\bar{p}\bar{p}\pi^{+}d$, and
$e^{+}e^{-}\to pp\pi^{-}\bar{d}+c.c.$, which with the combination of cases 1
and 2, cases 3 and 4, and all four cases, are calculated. The corresponding
upper limit of the Born cross section at the $90\%$ C.L. is calculated as

$\sigma^{\rm ul}=\frac{N^{\rm ul}_{\rm
sig}}{\mathcal{L}\cdot\epsilon\cdot(1+\delta)\cdot\frac{1}{|1-\Pi|^{2}}},$ (7)

where $N^{\rm ul}_{\rm sig}$ is the upper limit of the signal yield at the
$90\%$ C.L., $\mathcal{L}$ is the integrated luminosity of the data set,
$\epsilon$ is the detection efficiency, $(1+\delta)$ and
$\frac{1}{|1-\Pi|^{2}}$ are the ISR and vacuum polarization [57] correction
factors, respectively. We use a flat dressed cross section line shape as the
input to generate the signal MC sample and obtain the corresponding ISR
correction factor $(1+\delta)$, since no significant signal is seen in data.
The numbers related to the Born cross section measurement are summarized in
Table 4. Figure 4 shows the upper limits of the Born cross sections at the 90%
C.L. for the processes of $e^{+}e^{-}\to pp\pi^{-}\bar{d}$,
$e^{+}e^{-}\to\bar{p}\bar{p}\pi^{+}d$, and $e^{+}e^{-}\to
pp\pi^{-}\bar{d}+c.c.$

Figure 4: Upper limits of the Born cross sections at the $90\%$ C.L.

##  VIII summary

We search for the process $e^{+}e^{-}\to pp\pi^{-}\bar{d}+c.c.$ with the
BESIII data at $\sqrt{s}$ from 4.13 to 4.70 GeV. No significant signal is
observed. The upper limits at the $90\%$ C.L on the Born cross sections of
$e^{+}e^{-}\to pp\pi^{-}\bar{d}$, $\bar{p}\bar{p}\pi^{+}d$ and
$pp\pi^{-}\bar{d}+c.c.$ are determined to be from 4.3 to 72 fb, 4.1 to 76 fb,
and 9.0 to 145 fb, respectively. The current BESIII sensitivity is in the same
order of magnitude as the cross section of inclusive $\bar{d}$ production
$\sigma(e^{+}e^{-}\to\bar{d}X)=(9.63\pm 0.41(\rm stat)^{+1.17}_{-1.01}(\rm
syst))$ fb at $\sqrt{s}\approx 10.58$ GeV from the BaBar experiment [28].

In the BESIII experiment, the low backgrounds guarantee a high sensitivity
measurement of the cross section in the process
$e^{+}e^{-}\to\bar{p}\bar{p}\pi^{+}d$ with the number of tracks equal to 3 or
4 and $e^{+}e^{-}\to pp\pi^{-}\bar{d}$ with the number of tracks equal to 3,
and an improved sensitivity may be obtained if the beam backgrounds can be
further suppressed in the process $e^{+}e^{-}\to pp\pi^{-}\bar{d}$ with the
number of tracks equal to 4. BEPCII is upgrading the luminosity performance
and increasing the maximum $\sqrt{s}$ [34], which will enable a further search
for the (anti)deuteron, the $d^{*}(2380)$, and other possible states with six
quarks at the BESIII experiment.

##  IX ACKNOWLEDGEMENT

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, 12061131003, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265; 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; 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.

Table 4: The upper limit of the Born cross section at the 90% C.L. and the numbers used in the measurement. $\sqrt{s}$ (MeV) | $1+\delta$ | $\frac{1}{|1-\Pi|^{2}}$ |  $\sigma^{\rm ul}(pp\pi^{-}\bar{d})$ (fb) |  $\sigma^{\rm ul}(\bar{p}\bar{p}\pi^{+}d)$ (fb) |  $\sigma^{\rm ul}(pp\pi^{-}\bar{d}+c.c.)$ (fb)
---|---|---|---|---|---
4128.5 | 0.886 | 1.052 | 64 | 43 | 104
4157.4 | 0.885 | 1.053 | 55 | 35 | 88
4178.0 | 0.897 | 1.054 | 6.3 | 4.1 | 9.9
4188.8 | 0.898 | 1.056 | 42 | 23 | 61
4198.9 | 0.900 | 1.056 | 33 | 23 | 55
4209.2 | 0.903 | 1.057 | 31 | 40 | 89
4218.7 | 0.904 | 1.056 | 27 | 22 | 49
4226.3 | 0.905 | 1.056 | 13 | 10 | 23
4235.7 | 0.908 | 1.056 | 28 | 20 | 47
4243.8 | 0.907 | 1.056 | 22 | 20 | 42
4258.0 | 0.909 | 1.054 | 16 | 12 | 28
4266.8 | 0.912 | 1.053 | 27 | 19 | 45
4277.7 | 0.912 | 1.053 | 72 | 58 | 129
4287.9 | 0.915 | 1.053 | 22 | 21 | 43
4312.0 | 0.917 | 1.052 | 23 | 34 | 72
4337.4 | 0.921 | 1.051 | 22 | 19 | 41
4358.3 | 0.925 | 1.051 | 17 | 16 | 33
4377.4 | 0.925 | 1.051 | 20 | 17 | 37
4396.4 | 0.927 | 1.051 | 18 | 17 | 35
4415.6 | 0.928 | 1.052 | 13 | 13 | 36
4436.2 | 0.932 | 1.054 | 17 | 14 | 31
4467.1 | 0.934 | 1.055 | 65 | 75 | 141
4527.1 | 0.939 | 1.054 | 67 | 70 | 137
4599.5 | 0.896 | 1.055 | 14 | 14 | 28
4611.9 | 0.943 | 1.054 | 69 | 76 | 145
4628.0 | 0.945 | 1.054 | 13 | 15 | 29
4640.9 | 0.946 | 1.054 | 23 | 14 | 48
4661.2 | 0.947 | 1.054 | 13 | 14 | 27
4681.9 | 0.948 | 1.054 | 4.3 | 4.5 | 9.0
4698.8 | 0.949 | 1.054 | 14 | 14 | 29

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