# Measurement of $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ from 3.5106
to 4.6988 GeV and study of $\Lambda\bar{\Lambda}$ mass threshold enhancement

M. Ablikim1, M. N. Achasov12,b, P. Adlarson72, M. Albrecht4, R. Aliberti33, A.
Amoroso71A,71C, M. R. An37, Q. An68,55, Y. Bai54, O. Bakina34, R. Baldini
Ferroli27A, I. Balossino28A, Y. Ban44,g, V. Batozskaya1,42, D. Becker33, K.
Begzsuren30, N. Berger33, M. Bertani27A, D. Bettoni28A, F. Bianchi71A,71C, E.
Bianco71A,71C, J. Bloms65, A. Bortone71A,71C, I. Boyko34, R. A. Briere5, A.
Brueggemann65, H. Cai73, X. Cai1,55, A. Calcaterra27A, G. F. Cao1,60, N.
Cao1,60, S. A. Cetin59A, J. F. Chang1,55, W. L. Chang1,60, G. R. Che41, G.
Chelkov34,a, C. Chen41, Chao Chen52, G. Chen1, H. S. Chen1,60, M. L.
Chen1,55,60, S. J. Chen40, S. M. Chen58, T. Chen1,60, X. R. Chen29,60, X. T.
Chen1,60, Y. B. Chen1,55, Z. J. Chen24,h, W. S. Cheng71C, S. K. Choi 52, X.
Chu41, G. Cibinetto28A, F. Cossio71C, J. J. Cui47, H. L. Dai1,55, J. P. Dai76,
A. Dbeyssi18, R. E. de Boer4, D. Dedovich34, Z. Y. Deng1, A. Denig33, I.
Denysenko34, M. Destefanis71A,71C, F. De Mori71A,71C, Y. Ding32, Y. Ding38, J.
Dong1,55, L. Y. Dong1,60, M. Y. Dong1,55,60, X. Dong73, S. X. Du78, Z. H.
Duan40, P. Egorov34,a, Y. L. Fan73, J. Fang1,55, S. S. Fang1,60, W. X. Fang1,
Y. Fang1, R. Farinelli28A, L. Fava71B,71C, F. Feldbauer4, G. Felici27A, C. Q.
Feng68,55, J. H. Feng56, K Fischer66, M. Fritsch4, C. Fritzsch65, C. D. Fu1,
H. Gao60, Y. N. Gao44,g, Yang Gao68,55, S. Garbolino71C, I. Garzia28A,28B, P.
T. Ge73, Z. W. Ge40, C. Geng56, E. M. Gersabeck64, A Gilman66, K. Goetzen13,
L. Gong38, W. X. Gong1,55, W. Gradl33, M. Greco71A,71C, L. M. Gu40, M. H.
Gu1,55, Y. T. Gu15, C. Y Guan1,60, A. Q. Guo29,60, L. B. Guo39, R. P. Guo46,
Y. P. Guo11,f, A. Guskov34,a, W. Y. Han37, X. Q. Hao19, F. A. Harris62, K. K.
He52, K. L. He1,60, F. H. Heinsius4, C. H. Heinz33, Y. K. Heng1,55,60, C.
Herold57, G. Y. Hou1,60, Y. R. Hou60, Z. L. Hou1, H. M. Hu1,60, J. F. Hu53,i,
T. Hu1,55,60, Y. Hu1, G. S. Huang68,55, K. X. Huang56, L. Q. Huang29,60, X. T.
Huang47, Y. P. Huang1, Z. Huang44,g, T. Hussain70, N Hüsken26,33, W. Imoehl26,
M. Irshad68,55, J. Jackson26, S. Jaeger4, S. Janchiv30, E. Jang52, J. H.
Jeong52, Q. Ji1, Q. P. Ji19, X. B. Ji1,60, X. L. Ji1,55, Y. Y. Ji47, Z. K.
Jia68,55, P. C. Jiang44,g, S. S. Jiang37, X. S. Jiang1,55,60, Y. Jiang60, J.
B. Jiao47, Z. Jiao22, S. Jin40, Y. Jin63, M. Q. Jing1,60, T. Johansson72, S.
Kabana31, N. Kalantar-Nayestanaki61, X. L. Kang9, X. S. Kang38, R. Kappert61,
M. Kavatsyuk61, B. C. Ke78, I. K. Keshk4, A. Khoukaz65, R. Kiuchi1, R.
Kliemt13, L. Koch35, O. B. Kolcu59A, B. Kopf4, M. Kuemmel4, M. Kuessner4, A.
Kupsc42,72, W. Kühn35, J. J. Lane64, J. S. Lange35, P. Larin18, A. Lavania25,
L. Lavezzi71A,71C, T. T. Lei68,k, Z. H. Lei68,55, H. Leithoff33, M.
Lellmann33, T. Lenz33, C. Li41, C. Li45, C. H. Li37, Cheng Li68,55, D. M.
Li78, F. Li1,55, G. Li1, H. Li68,55, H. B. Li1,60, H. J. Li19, H. N. Li53,i,
Hui Li41, J. Q. Li4, J. S. Li56, J. W. Li47, Ke Li1, L. J Li1,60, L. K. Li1,
Lei Li3, M. H. Li41, P. R. Li36,j,k, S. X. Li11, S. Y. Li58, T. Li47, W. D.
Li1,60, W. G. Li1, X. H. Li68,55, X. L. Li47, Xiaoyu Li1,60, Y. G. Li44,g, Z.
X. Li15, Z. Y. Li56, C. Liang40, H. Liang32, H. Liang1,60, H. Liang68,55, Y.
F. Liang51, Y. T. Liang29,60, G. R. Liao14, L. Z. Liao47, J. Libby25, A.
Limphirat57, D. X. Lin29,60, T. Lin1, B. J. Liu1, C. Liu32, C. X. Liu1, D.
Liu18,68, F. H. Liu50, Fang Liu1, Feng Liu6, G. M. Liu53,i, H. Liu36,j,k, H.
B. Liu15, H. M. Liu1,60, Huanhuan Liu1, Huihui Liu20, J. B. Liu68,55, J. L.
Liu69, J. Y. Liu1,60, K. Liu1, K. Y. Liu38, Ke Liu21, L. Liu68,55, L. C.
Liu21, Lu Liu41, M. H. Liu11,f, P. L. Liu1, Q. Liu60, S. B. Liu68,55, T.
Liu11,f, W. K. Liu41, W. M. Liu68,55, X. Liu36,j,k, Y. Liu36,j,k, Y. B. Liu41,
Z. A. Liu1,55,60, Z. Q. Liu47, X. C. Lou1,55,60, F. X. Lu56, H. J. Lu22, J. G.
Lu1,55, X. L. Lu1, Y. Lu7, Y. P. Lu1,55, Z. H. Lu1,60, C. L. Luo39, M. X.
Luo77, T. Luo11,f, X. L. Luo1,55, X. R. Lyu60, Y. F. Lyu41, F. C. Ma38, H. L.
Ma1, L. L. Ma47, M. M. Ma1,60, Q. M. Ma1, R. Q. Ma1,60, R. T. Ma60, X. Y.
Ma1,55, Y. Ma44,g, F. E. Maas18, M. Maggiora71A,71C, S. Maldaner4, S. Malde66,
Q. A. Malik70, A. Mangoni27B, Y. J. Mao44,g, Z. P. Mao1, S. Marcello71A,71C,
Z. X. Meng63, J. G. Messchendorp13,61, G. Mezzadri28A, H. Miao1,60, T. J.
Min40, R. E. Mitchell26, X. H. Mo1,55,60, N. Yu. Muchnoi12,b, Y. Nefedov34, F.
Nerling18,d, I. B. Nikolaev12,b, Z. Ning1,55, S. Nisar10,l, Y. Niu 47, S. L.
Olsen60, Q. Ouyang1,55,60, S. Pacetti27B,27C, X. Pan52, Y. Pan54, A. Pathak32,
Y. P. Pei68,55, M. Pelizaeus4, H. P. Peng68,55, K. Peters13,d, J. L. Ping39,
R. G. Ping1,60, S. Plura33, S. Pogodin34, V. Prasad68,55, F. Z. Qi1, H.
Qi68,55, H. R. Qi58, M. Qi40, T. Y. Qi11,f, S. Qian1,55, W. B. Qian60, Z.
Qian56, C. F. Qiao60, J. J. Qin69, L. Q. Qin14, X. P. Qin11,f, X. S. Qin47, Z.
H. Qin1,55, J. F. Qiu1, S. Q. Qu58, K. H. Rashid70, C. F. Redmer33, K. J.
Ren37, A. Rivetti71C, V. Rodin61, M. Rolo71C, G. Rong1,60, Ch. Rosner18, S. N.
Ruan41, A. Sarantsev34,c, Y. Schelhaas33, C. Schnier4, K. Schoenning72, M.
Scodeggio28A,28B, K. Y. Shan11,f, W. Shan23, X. Y. Shan68,55, J. F.
Shangguan52, L. G. Shao1,60, M. Shao68,55, C. P. Shen11,f, H. F. Shen1,60, W.
H. Shen60, X. Y. Shen1,60, B. A. Shi60, H. C. Shi68,55, J. Y. Shi1, Q. Q.
Shi52, R. S. Shi1,60, X. Shi1,55, J. J. Song19, W. M. Song32,1, Y. X.
Song44,g, S. Sosio71A,71C, S. Spataro71A,71C, F. Stieler33, P. P. Su52, Y. J.
Su60, G. X. Sun1, H. Sun60, H. K. Sun1, J. F. Sun19, L. Sun73, S. S. Sun1,60,
T. Sun1,60, W. Y. Sun32, Y. J. Sun68,55, Y. Z. Sun1, Z. T. Sun47, Y. X.
Tan68,55, C. J. Tang51, G. Y. Tang1, J. Tang56, Y. A. Tang73, L. Y Tao69, Q.
T. Tao24,h, M. Tat66, J. X. Teng68,55, V. Thoren72, W. H. Tian49, Y.
Tian29,60, I. Uman59B, B. Wang1, B. Wang68,55, B. L. Wang60, C. W. Wang40, D.
Y. Wang44,g, F. Wang69, H. J. Wang36,j,k, H. P. Wang1,60, K. Wang1,55, L. L.
Wang1, M. Wang47, Meng Wang1,60, S. Wang14, S. Wang11,f, T. Wang11,f, T. J.
Wang41, W. Wang56, W. H. Wang73, W. P. Wang68,55, X. Wang44,g, X. F.
Wang36,j,k, X. L. Wang11,f, Y. Wang58, Y. D. Wang43, Y. F. Wang1,55,60, Y. H.
Wang45, Y. Q. Wang1, Yaqian Wang17,1, Z. Wang1,55, Z. Y. Wang1,60, Ziyi
Wang60, D. H. Wei14, F. Weidner65, S. P. Wen1, D. J. White64, U. Wiedner4, G.
Wilkinson66, M. Wolke72, L. Wollenberg4, J. F. Wu1,60, L. H. Wu1, L. J.
Wu1,60, X. Wu11,f, X. H. Wu32, Y. Wu68, Y. J Wu29, Z. Wu1,55, L. Xia68,55, T.
Xiang44,g, D. Xiao36,j,k, G. Y. Xiao40, H. Xiao11,f, S. Y. Xiao1, Y. L.
Xiao11,f, Z. J. Xiao39, C. Xie40, X. H. Xie44,g, Y. Xie47, Y. G. Xie1,55, Y.
H. Xie6, Z. P. Xie68,55, T. Y. Xing1,60, C. F. Xu1,60, C. J. Xu56, G. F. Xu1,
H. Y. Xu63, Q. J. Xu16, X. P. Xu52, Y. C. Xu75, Z. P. Xu40, F. Yan11,f, L.
Yan11,f, W. B. Yan68,55, W. C. Yan78, H. J. Yang48,e, H. L. Yang32, H. X.
Yang1, Tao Yang1, Y. F. Yang41, Y. X. Yang1,60, Yifan Yang1,60, M. Ye1,55, M.
H. Ye8, J. H. Yin1, Z. Y. You56, B. X. Yu1,55,60, C. X. Yu41, G. Yu1,60, T.
Yu69, X. D. Yu44,g, C. Z. Yuan1,60, L. Yuan2, S. C. Yuan1, X. Q. Yuan1, Y.
Yuan1,60, Z. Y. Yuan56, C. X. Yue37, A. A. Zafar70, F. R. Zeng47, X. Zeng6, Y.
Zeng24,h, X. Y. Zhai32, Y. H. Zhan56, A. Q. Zhang1,60, B. L. Zhang1,60, B. X.
Zhang1, D. H. Zhang41, G. Y. Zhang19, H. Zhang68, H. H. Zhang56, H. H.
Zhang32, H. Q. Zhang1,55,60, H. Y. Zhang1,55, J. J. Zhang49, J. L. Zhang74, J.
Q. Zhang39, J. W. Zhang1,55,60, J. X. Zhang36,j,k, J. Y. Zhang1, J. Z.
Zhang1,60, Jianyu Zhang1,60, Jiawei Zhang1,60, L. M. Zhang58, L. Q. Zhang56,
Lei Zhang40, P. Zhang1, Q. Y. Zhang37,78, Shuihan Zhang1,60, Shulei Zhang24,h,
X. D. Zhang43, X. M. Zhang1, X. Y. Zhang47, X. Y. Zhang52, Y. Zhang66, Y. T.
Zhang78, Y. H. Zhang1,55, Yan Zhang68,55, Yao Zhang1, Z. H. Zhang1, Z. L.
Zhang32, Z. Y. Zhang41, Z. Y. Zhang73, G. Zhao1, J. Zhao37, J. Y. Zhao1,60, J.
Z. Zhao1,55, Lei Zhao68,55, Ling Zhao1, M. G. Zhao41, S. J. Zhao78, Y. B.
Zhao1,55, Y. X. Zhao29,60, Z. G. Zhao68,55, A. Zhemchugov34,a, B. Zheng69, J.
P. Zheng1,55, Y. H. Zheng60, B. Zhong39, C. Zhong69, X. Zhong56, H. Zhou47, L.
P. Zhou1,60, X. Zhou73, X. K. Zhou60, X. R. Zhou68,55, X. Y. Zhou37, Y. Z.
Zhou11,f, J. Zhu41, K. Zhu1, K. J. Zhu1,55,60, L. X. Zhu60, S. H. Zhu67, S. Q.
Zhu40, W. J. Zhu11,f, Y. C. Zhu68,55, Z. A. Zhu1,60, J. H. Zou1, J. Zu68,55
(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 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
11 Fudan University, Shanghai 200433, People’s Republic of China
12 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
21 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
22 Huangshan College, Huangshan 245000, People’s Republic of China
23 Hunan Normal University, Changsha 410081, People’s Republic of China
24 Hunan University, Changsha 410082, People’s Republic of China
25 Indian Institute of Technology Madras, Chennai 600036, India
26 Indiana University, Bloomington, Indiana 47405, USA
27 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
28 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
29 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
30 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
31 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D,
Arica, Chile
32 Jilin University, Changchun 130012, People’s Republic of China
33 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
34 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
35 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
36 Lanzhou University, Lanzhou 730000, People’s Republic of China
37 Liaoning Normal University, Dalian 116029, People’s Republic of China
38 Liaoning University, Shenyang 110036, People’s Republic of China
39 Nanjing Normal University, Nanjing 210023, People’s Republic of China
40 Nanjing University, Nanjing 210093, People’s Republic of China
41 Nankai University, Tianjin 300071, People’s Republic of China
42 National Centre for Nuclear Research, Warsaw 02-093, Poland
43 North China Electric Power University, Beijing 102206, People’s Republic of
China
44 Peking University, Beijing 100871, People’s Republic of China
45 Qufu Normal University, Qufu 273165, People’s Republic of China
46 Shandong Normal University, Jinan 250014, People’s Republic of China
47 Shandong University, Jinan 250100, People’s Republic of China
48 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
49 Shanxi Normal University, Linfen 041004, People’s Republic of China
50 Shanxi University, Taiyuan 030006, People’s Republic of China
51 Sichuan University, Chengdu 610064, People’s Republic of China
52 Soochow University, Suzhou 215006, People’s Republic of China
53 South China Normal University, Guangzhou 510006, People’s Republic of China
54 Southeast University, Nanjing 211100, People’s Republic of China
55 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
56 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
57 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
58 Tsinghua University, Beijing 100084, People’s Republic of China
59 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
60 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
61 University of Groningen, NL-9747 AA Groningen, The Netherlands
62 University of Hawaii, Honolulu, Hawaii 96822, USA
63 University of Jinan, Jinan 250022, People’s Republic of China
64 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
65 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
66 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
67 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
68 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
69 University of South China, Hengyang 421001, People’s Republic of China
70 University of the Punjab, Lahore-54590, Pakistan
71 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
72 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
73 Wuhan University, Wuhan 430072, People’s Republic of China
74 Xinyang Normal University, Xinyang 464000, People’s Republic of China
75 Yantai University, Yantai 264005, People’s Republic of China
76 Yunnan University, Kunming 650500, People’s Republic of China
77 Zhejiang University, Hangzhou 310027, People’s Republic of China
78 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 with a total integrated luminosity of approximately 18 fb-1
collected by the BESIII detector operating at the BEPCII, the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ is studied at center-of-mass
energies between 3.5106 and 4.6988 GeV. The Born cross section for the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ is measured. No significant
structure is observed in the Born cross section line shape. An enhancement
near the $\Lambda\bar{\Lambda}$ mass threshold is observed for the first time
in the process. The structure can be described by an $S$-wave Breit-Wigner
function. Neglecting contribution of excited $\Lambda$ states and potential
interferences, the mass and width are determined to be ($2356\pm 7\pm 17$)
MeV/$c^{2}$ and ($304\pm 28\pm 54$) MeV, respectively, where the first
uncertainties are statistical and the second are systematic.

###### pacs:

13.60.Rj, 13.66.De, 13.85.Lg, 14.40.Rt

##  I Introduction

In 2005, the BABAR collaboration reported a structure, the $Y(4260)$, with a
mass around 4.26 GeV/$c^{2}$ in the $\pi^{+}\pi^{-}J/\psi$ final state via the
initial state radiation (ISR) process
$e^{+}e^{-}\rightarrow\gamma_{\mathrm{ISR}}\pi^{+}\pi^{-}J/\psi$ [1]. The
observation was confirmed by the CLEO [2] and Belle [3] collaborations. Given
its strong coupling into the charmonium state, it must contain a charm-
anticharm quark pair, because of the strong suppression of the heavy quark-
antiquark pair creation within Quantum Chromodynamics [4]. Because the
$Y(4260)$ can be produced by a virtual photon from $e^{+}e^{-}$ annihilation,
it is a vector state with the spin parity $J^{PC}$ of $1^{--}$. Above the
$D\bar{D}$ threshold, the four vector charmonium states [$\psi(3770)$,
$\psi(4040)$, $\psi(4160)$, and $\psi(4415)$] predicted by the potential model
[5] have been established [6]. Moreover, the traditional charmonium states
dominantly couple to the ground-state open-charm meson pairs instead of hidden
charm states [6]. Therefore, the $Y(4260)$ cannot be explained as a
traditional charmonium state. A more precise measurement performed by BESIII
collaboration shows the $Y(4260)$ contains two structure $Y(4220)$ and
$Y(4330)$ [7]. Later, more states with similar properties were observed [8,
9]. These $Y$ states, named $\psi$ states in the Particle Data Group (PDG)
[6], are good candidates for exotic states, such as glueballs, compact
tetraquarks, hybrids, hadrocharmonia, etc. [4]. Some of these interpretations,
however, are disfavored by experiment [10]. Light hadron final states of $Y$
decays, such as $\pi^{0}p\overline{p}$ [11], $K^{0}_{S}K^{\pm}\pi^{\mp}$ [12],
$K_{S}^{0}K^{\pm}\pi^{\mp}\pi^{0}(\eta)$ [13], $\eta\phi\pi^{+}\pi^{-}$ [14]
$\Xi^{-}\bar{\Xi}^{+}$ [15], $2(p\overline{p})$ [16], $\Lambda\bar{\Lambda}$
[17], and $\omega\pi^{0}$ [18] have been searched for but not yet discovered.
Searching for new decay modes of $Y$ states, for instance
$Y\to\Lambda\bar{\Lambda}\eta$, can add more information to our understanding
of their inner structure.

Enhancements near the baryon-antibaryon mass threshold have been observed in
low energy $p\overline{p}$ collisions, charmonium decays, and $B$ meson
decays. Using PS185 low energy $p\overline{p}$ collision data, a structure
near the $\Lambda\bar{\Lambda}$ threshold was observed with a partial wave
analysis of $p\overline{p}\to\Lambda\bar{\Lambda}$ [19]. In charmonium decays,
enhancements are observed in $p\overline{p}$ final states of the processes
$J/\psi\rightarrow\gamma p\overline{p}$ [20]. In $B$ meson decays,
enhancements were found in the processes
$B^{0}\rightarrow\Lambda\bar{\Lambda}K^{0}(K^{*0})$ [21], $B^{0}\rightarrow
p\bar{\Lambda}\pi^{-}$ [22], $B^{+}\rightarrow
p\overline{p}\pi^{+},B^{0}\rightarrow p\overline{p}K^{0}$ [23],
$B^{+}\rightarrow\Lambda\bar{\Lambda}K^{+}$ [24], $B^{+(0)}\rightarrow
p\bar{p}K^{*+(0)}$ [25],
$B^{0}\rightarrow\bar{D}^{*0}p\overline{p},\bar{D}^{0}p\overline{p}$ [26],
$B^{0}\rightarrow p\bar{\Lambda}D^{(*)-}$ [27], $B^{0}_{(s)}\to
p\bar{p}h^{+}h^{\prime-}$ [28], $B^{+}\rightarrow p\bar{\Lambda}K^{+}K^{-}$
[29], $B^{0}\to p\bar{p}\pi^{0}$ [30], $B^{+}\to
p\overline{p}\mu^{+}\nu_{\mu}$ [31] etc. Strikingly, there is no corresponding
enhancement observed in $\psi(2S)\rightarrow\gamma p\overline{p}$ [32],
$\psi(2S)\rightarrow p\bar{p}\pi^{0}$ [33], $\Upsilon(1S)\to\gamma
p\overline{p}$ [34], $J/\psi\to\omega p\overline{p}$ [35],
$\psi(2S)\rightarrow p\overline{p}\eta$ [36], $J/\psi\text{,~{}}\psi(2S)\to
p\overline{p}\phi$ [37, 38],
$J/\psi\text{,~{}}\psi(2S)\rightarrow\eta\Sigma^{+}\bar{\Sigma}^{-}$ [39],
$\psi(2S)\rightarrow\Lambda\bar{\Lambda}\eta~{}(\pi^{0})$ [40],
$\psi(2S)\rightarrow\Lambda\bar{\Lambda}\omega$ [41], $B^{+}\to p\bar{p}K^{+}$
[42], $\bar{B}^{0}\to D^{0}\Lambda\bar{\Lambda}$ [43], $B\to p{\bar{p}}\pi\pi$
[44] $B_{s}^{0}\to J/\psi p\bar{p}$ [45], and $B^{-}\to
J/\psi\Lambda\overline{p}$ [46] decays. Many theory models have been proposed
for the interpretation of these enhancements, including a gluonic mechanism
[47], a fragmentation mechanism [47], baryonia [48], multiquark states [49],
and final state interactions [50]. Searching for enhancements with different
quantum numbers in different reactions can provide more constraints on these
models.

In this article, we report a study of the reaction
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ with a partial reconstruction
technique, based on the data recorded at center-of-mass energies from 3.5106
to 4.6988 GeV. The Born cross sections are measured, and possible resonances
are searched for by fitting the energy-dependent distribution of the Born
cross sections. In addition, a threshold enhancement is observed in the
$\Lambda\bar{\Lambda}$ mass spectrum. The mass and width of the structure are
determined by a fit with an $S$-wave Breit-Wigner function, and the angular
distribution of the structure is studied.

##  II Detector and data samples

The BESIII detector [51] records symmetric $e^{+}e^{-}$ collisions provided by
the BEPCII storage ring [52], which operates in the center-of-mass energy
range from 2.00 to 4.95 GeV. The cylindrical core of the BESIII detector
covers 93% of the full solid angle and consists of a helium-based multilayer
drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a
CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a
superconducting solenoidal magnet providing a 1.0 T magnetic field. The
solenoid is supported by an octagonal flux-return yoke with resistive plate
counter muon identification modules interleaved with steel. The charged-
particle momentum resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the ${\rm
d}E/{\rm d}x$ resolution is $6\%$ for electrons from Bhabha scattering. 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 [53].

Simulated data samples produced with geant4-based [54] Monte Carlo (MC)
software, which includes the geometric description of the BESIII detector and
the detector response, are used to determine detection efficiencies and to
estimate background contributions. The simulation includes the beam energy
spread and ISR in the $e^{+}e^{-}$ annihilations modeled with the generator
kkmc [55, *Jadach:1999vf]. 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 [55, *Jadach:1999vf]. All
particle decays are modeled with evtgen [57, *Ping:2008zz_BesGen] using
branching fractions either taken from the PDG [6], when available, or
otherwise estimated with lundcharm [59, *Yang:2014vra]. Final state radiation
from charged final state particles is incorporated using photos [61]. At each
energy point, we generate a signal MC sample of three-body process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ with
$\eta\rightarrow\gamma\gamma$ using a uniformly distributed phase space (PHSP)
model. The MC sample for the study of threshold enhancement is discussed in
Sec V.

##  III Event Selection

The signal candidates of the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ are selected with a partial
reconstruction technique, to achieve higher efficiency. This technique
requires that only the $\Lambda$ ($\bar{\Lambda}$) baryon and $\eta$ meson are
reconstructed, inferring the $\bar{\Lambda}$ ($\Lambda$) from energy-momentum
conservation. The $\Lambda$ ($\bar{\Lambda}$) candidates and the $\eta$
candidates are reconstructed with $p\pi^{-}$ ($\overline{p}\pi^{+}$) and
$\gamma\gamma$ decay modes, respectively.

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, the symmetry axis of the MDC. The distance of the
closest approach to the interaction point (IP) along the $z$-axis must be less
than 20 cm. Particle identification (PID) for charged tracks combines
measurements of ${\rm d}E/{\rm d}x$ in the MDC and the flight time in the TOF
to form likelihoods for each hadron (pion, kaon, and proton) hypothesis.
Tracks are identified as protons (pions) when the proton (pion) hypothesis has
the largest likelihood. Proton and pion track pairs with opposite charges are
constrained to originate from a common vertex to form the $\Lambda$
($\bar{\Lambda}$) candidates by a vertex fit [62]. The obtained $\chi^{2}$ in
the fit, denoted as $\chi^{2}_{\text{vtx}}$, are kept for further analysis.
The decay length of the $\Lambda$ ($\bar{\Lambda}$) candidate is required to
be greater than twice the vertex resolution away from the IP.

Photon candidates are identified using showers in the EMC. The deposited
energy of each shower must be more than 25 MeV in the barrel region
($|\\!\cos\theta|<0.80$) and more than 50 MeV in the end cap region
($0.86<|\\!\cos\theta|<0.92$). To exclude showers that originate from charged
tracks identified as antiprotons, the angle subtended by the EMC shower and
the position of the closest charged track at the EMC must be greater than 20
degrees as measured from the IP. The angle is required to be greater than 10
degrees for other types of charged tracks. To suppress electronic noise and
showers unrelated to the event, the difference between the EMC time and the
event start time is required to be within [0, 700] ns. To remove fake photons
from neutron-antineutron annihilations, $E_{3\times 3}/E_{5\times 5}$ and the
lateral moment [63] describing the shape of electromagnetic showers are
required to be less than $0.4$ and greater than $0.85$, respectively, where
$E_{3\times 3}(E_{5\times 5})$ is total energy deposited in the $3\times 3$
($5\times 5$) crystals around the seed of the shower [64]. Photon pairs are
used to reconstruct $\eta\to\gamma\gamma$ decays.

To suppress the background and improve the momentum resolution, a one-
constraint (1C) kinematic fit [65] under the hypothesis
$e^{+}e^{-}\to\gamma\gamma p\pi^{-}\overline{p}\pi^{+}$ is performed with the
recoil mass against $\gamma\gamma p\pi^{-}$
($\gamma\gamma\overline{p}\pi^{+}$) constrained to the known mass of
$\Lambda$, $M_{\Lambda}$ [6]. In case of multiple reconstructed $\gamma\gamma
p\pi^{-}$ ($\gamma\gamma\overline{p}\pi^{+}$) candidates, the candidate with
the smallest $\chi^{2}_{\text{vtx}}+\chi^{2}_{\text{1C}}$ is kept, where
$\chi^{2}_{\text{vtx}}$ and $\chi^{2}_{\text{1C}}$ are the $\chi^{2}$ of the
vertex fit and 1C kinematic fit, respectively. We require
$\chi^{2}_{\text{vtx}}<20$ and the $\chi^{2}_{\text{1C}}<10$. After the above
selection criteria are applied, the invariant mass of the $p\pi^{-}$
($\overline{p}\pi^{+}$) combination is required to satisfy
$1.112\text{~{}GeV}/c^{2}<M_{p\pi^{-}~{}(\overline{p}\pi^{+})}<1.120\text{~{}GeV}/c^{2}$.
A study with the inclusive MC sample using TopoAna [66] shows that the main
background contributions are from processes with $\Sigma^{0}$
($\bar{\Sigma}^{0}$) baryons in the final states. The $\Sigma^{0}$
($\bar{\Sigma}^{0}$) baryon decays to $\Lambda\gamma$ ($\bar{\Lambda}\gamma$)
with a branching fraction of 100% [6]. In order to veto processes with a soft
$\gamma_{L}$ from $\Sigma^{0}\to\Lambda\gamma_{L}$
($\bar{\Sigma}^{0}\to\bar{\Lambda}\gamma_{L}$) decays, the invariant mass of
the reconstructed $\Lambda$ ($\bar{\Lambda}$) and the selected photon with
lower energy $M_{\Lambda\gamma_{L}}$ ($M_{\bar{\Lambda}\gamma_{L}}$) is
required to be outside the $\Sigma^{0}$ ($\bar{\Sigma}^{0}$) mass region of
[1.113, 1.273] GeV/$c^{2}$. A study of the $M_{\gamma\gamma}$ distribution
with the inclusive MC indicates small peaking background contributions from
the processes
$e^{+}e^{-}\rightarrow\eta\Sigma^{0}\bar{\Lambda}+\text{c}.\text{c}$ and
$e^{+}e^{-}\rightarrow\Sigma^{0}\bar{\Sigma}^{0}\eta$ near the $\eta$ peak in
the $M_{\gamma\gamma}$ distribution. However, the process
$e^{+}e^{-}\rightarrow\eta\Sigma^{0}\bar{\Lambda}+\text{c}.\text{c}$ is
suppressed because of isospin violation. Due to the two extra photons in the
final state, the $e^{+}e^{-}\rightarrow\Sigma^{0}\bar{\Sigma}^{0}\eta$ process
is also suppressed by the 1C kinematic fit.

##  IV Cross Section Measurement

The Born cross section of the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ with contributions from
intermediate states at each center-of-mass energy is calculated as

$\displaystyle\sigma^{\mathrm{Born}}=\frac{N_{\mathrm{sig}}}{\mathcal{L}_{\mathrm{{int}}}\mathcal{B}\varepsilon(1+\delta)(1+\delta^{v})},$
(1)

where $N_{\mathrm{sig}}$ is the signal yield, $\mathcal{L}_{\mathrm{int}}$ is
the integrated luminosity measured using large-angle Bhabha events [67, 68,
69], $\mathcal{B}$ is the product of the branching fractions of
$\eta\rightarrow\gamma\gamma$ and $\Lambda\rightarrow p\pi^{-}$
($\bar{\Lambda}\rightarrow\overline{p}\pi^{+}$) decays from the PDG [6],
$\varepsilon$ is the reconstruction efficiency determined by MC simulation,
$(1+\delta)$ is the ISR correction factor, and $(1+\delta^{v})$ is the vacuum
polarization factor [70].

The signal yield is determined by an unbinned maximum-likelihood fit to the
$M_{\gamma\gamma}$ spectrum at each energy point. The background is modeled by
a linear function. The signal is described by an MC-simulated shape convolved
with a Gaussian function which is used to compensate for the differences in
resolution between the MC simulation and the data. The means of the Gaussian
functions at different energy points are free. The widths of the Gaussian
functions cannot be constrained by data samples with low luminosities.
Therefore, they are fixed to a common value obtained from the fit of the data
sample with the highest luminosity taken at $\sqrt{s}=$3.7730 GeV. Figure 1
shows the fit results for the data at $\sqrt{s}=$4.1784 GeV. The number of
signal events in the $\eta$ signal region ($0.521<M_{\gamma\gamma}<0.572$
GeV/$c^{2}$) for each energy point is determined by integrating the fitted
signal shape in the signal region and is summarized in Table 1. The
contribution of peaking background mentioned in Sec. III is estimated to be
1.34% at $\sqrt{s}=4.1784\text{~{}GeV}$ by adding the shape derived from the
MC simulation of the two processes to the nominal fit model. The effect of the
peaking background events on the determination of signal yields will be
discussed in Sec. VI.

\begin{overpic}[width=195.12767pt,percent]{./plots/ConVMggFitting3773.eps}\put(80.0,60.0){(b)}
\end{overpic}\begin{overpic}[width=195.12767pt,percent]{./plots/ConVMggFitting4180.eps}\put(80.0,60.0){(b)}
\end{overpic}

Figure 1: Fits to the invariant mass spectra of $\gamma\gamma$ for the events
surviving the selection criteria at (a) $\sqrt{s}=3.7730$ and (b)
$\sqrt{s}=4.1784$ GeV. The black points with error bars are data. The solid
blue, dashed red, and dashed green lines correspond to the fit result, signal,
and background, respectively.

As shown in Fig. 2, evident discrepancies between the background-subtracted
data and PHSP MC simulation are seen, where the background contributions are
subtracted using the events from the $\eta$ sidebands, defined as
$0.368<M_{\gamma\gamma}<0.470$ GeV/$c^{2}$ and
$0.622<M_{\gamma\gamma}<0.724\mathrm{~{}GeV}/c^{2}$. To estimate the
reconstruction efficiencies, the PHSP MC samples are weighted according to the
background-subtracted data distributions of $M_{\Lambda\bar{\Lambda}}$,
$M_{\Lambda\eta}$, and $M_{\bar{\Lambda}\eta}$ shown in Fig. 2. Because of
limited sample sizes, we can not determine the exact two-dimensional weighting
factors from the $e^{+}e^{-}\to\Lambda\bar{\Lambda}\eta$ Dalitz distributions
of data. We take the weight for each two-dimensional distribution as the
product of the one dimensional distributions. The distributions of
$M_{\Lambda\bar{\Lambda}}$, $M_{\Lambda\eta}$, and $M_{\bar{\Lambda}\eta}$ are
treated as independent. Because of this approximation, all combinations of
invariant masses among $\Lambda$, $\bar{\Lambda}$, and $\eta$ are taken into
account to make the MC simulation consistent with data. Therefore, the
weighting factor is calculated as the average of the three two-dimensional
distributions

$\displaystyle
w^{i=\\{m,n,o\\}}(M_{\Lambda\bar{\Lambda}},M_{\Lambda\eta},M_{\bar{\Lambda}\eta})=\frac{1}{3}\frac{N^{\text{data}}_{M_{\Lambda\bar{\Lambda}}}(m)}{N^{\mathrm{MC}}_{M_{\Lambda\bar{\Lambda}}}(m)}\frac{N^{\text{data}}_{M_{\Lambda\eta}}(n)}{N^{\mathrm{MC}}_{M_{\Lambda\eta}}(n)}$
(2)
$\displaystyle+\frac{1}{3}\frac{N^{\text{data}}_{M_{\Lambda\bar{\Lambda}}}(m)}{N^{\mathrm{MC}}_{M_{\Lambda\bar{\Lambda}}}(m)}\frac{N^{\text{data}}_{M_{\bar{\Lambda}\eta}}(o)}{N^{\mathrm{MC}}_{M_{\bar{\Lambda}\eta}}(o)}+\frac{1}{3}\frac{N^{\text{data}}_{M_{\Lambda\eta}}(n)}{N^{\mathrm{MC}}_{M_{\Lambda\eta}}(n)}\frac{N^{\text{data}}_{M_{\bar{\Lambda}\eta}}(o)}{N^{\mathrm{MC}}_{M_{\bar{\Lambda}\eta}}(o)},$

where $w^{i}$ is the weight for event $i$ corresponding to bins $m$, $n$, and
$o$ of the $M_{\Lambda\bar{\Lambda}}$, $M_{\bar{\Lambda}\eta}$ and
$M_{\bar{\Lambda}\eta}$ distributions,
$N^{\text{data}/\mathrm{MC}}_{M_{\Lambda\bar{\Lambda}}}(m)$,
$N^{\text{data}/\mathrm{MC}}_{M_{\Lambda\eta}}(n)$, and
$N^{\text{data}/\mathrm{MC}}_{M_{\bar{\Lambda}\eta}}(o)$ are numbers of events
in bins $m$, $n$, and $o$ of the $M_{\Lambda\bar{\Lambda}}$,
$M_{\bar{\Lambda}\eta}$ and $M_{\bar{\Lambda}\eta}$ distributions in data and
PHSP $\mathrm{MC}$ sample. The weighted MC sample is iteratively weighted
twice in the same way to achieve better agreement between data and MC
simulation. As shown in Fig. 2, the distributions of the weighted PHSP MC
sample (hatched histograms) are consistent with the background-subtracted data
(black points). The efficiencies from the weighted PHSP MC samples are
calculated as

$\varepsilon^{\text{wtd}}=\frac{\sum^{N_{\mathrm{rec}}}_{i=1}{w^{i}(M_{\Lambda\bar{\Lambda}},M_{\Lambda\eta},M_{\bar{\Lambda}\eta})}}{\sum^{N_{\mathrm{gen}}}_{j=1}{w^{j}(M_{\Lambda\bar{\Lambda}},M_{\Lambda\eta},M_{\bar{\Lambda}\eta})}},$
(3)

where $N_{\mathrm{rec}}$ and $N_{\mathrm{gen}}$ are the numbers of
reconstructed events and generated events and
$w^{i~{}(j)}(M_{\Lambda\bar{\Lambda}},M_{\Lambda\eta},M_{\bar{\Lambda}\eta})$
is the weighting factor for event $i$ ($j$).

\begin{overpic}[width=101.90096pt,percent]{./plots/plotBesStyleMLL_4180.eps}\put(80.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=101.90096pt,percent]{./plots/plotBesStyleMLE_4180.eps}
\put(80.0,60.0){(b)} \end{overpic}

\begin{overpic}[width=101.90096pt,percent]{./plots/plotBesStyleMLBE_4180.eps}
\put(80.0,60.0){(c)} \end{overpic}

Figure 2: Distributions of (a) $M_{\Lambda\bar{\Lambda}}$, (b)
$M_{\Lambda\eta}$, and (c) $M_{\bar{\Lambda}\eta}$ obtained from events within
the $\eta$ signal region at $\sqrt{s}=4.1784$ GeV. The black points with error
bars are background-subtracted data. The solid (red) histograms and hatched
(blue) histograms denote the PHSP MC and weighted PHSP MC sample,
respectively.

In Eq. (2), data is used in the calculation of the weighting factor. The
statistical uncertainties of the numbers of data events in the intervals of
the background-subtracted data distributions propagate to the weighting factor
and contribute to the uncertainty of the efficiency [see Eq. (3)] which will
be considered in Section VI.

To evaluate $\varepsilon$ after the ISR correction and $(1+\delta)$, an
iterative procedure is performed [71]. The initial value of $\varepsilon$ for
the initial line shape is obtained by the weighted PHSP MC sample. The initial
value of $(1+\delta)$ is determined by kkmc with the line shape assumed to be
flat. The LOWESS method [72] is used to smooth the line shape. The iterative
procedure is repeated until the ratio of the measured cross section to the
corresponding value from previous iteration is consistent with one within the
statistical uncertainty. Table 1 summarizes the $\varepsilon$ values after ISR
correction and $(1+\delta)$ for each energy point.

Table 1: Born cross sections at the thirty-one energy points for the process $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$, where $\sqrt{s}$ [73, 68, 69] is the center-of-mass energy, $\mathcal{L}_{\text{int}}$ is the integrated luminosity [67, 68, 69], signal yield is the number of signal events in the signal region where the uncertainty is statistical only, $\varepsilon$ is the reconstruction efficiency after the ISR correction where the uncertainty is from the limited data sample sizes, $(1+\delta)$ and $(1+\delta^{v})$ are the ISR correction factor and vacuum polarization factor, respectively, and $\sigma$ is the Born cross section where the first uncertainty is statistical and the second is systematic. $\sqrt{s}$ (GeV) |  $\mathcal{L}_{\text{int}}$ ($\mathrm{pb}^{-1}$) | Signal yield |  $\varepsilon$ (%) | ($1+\delta$) | ($1+\delta^{v}$) |  $\sigma$ ($\mathrm{pb}$)
---|---|---|---|---|---|---
3.5106 |  458.7 |  80.3$\pm$11.0 | $15.6\pm 0.1$ | 0.864 | 1.045 | $3.13\pm 0.43\pm 0.17$
3.7730 | 2916.9 | 378.9$\pm$24.1 | $18.4\pm 0.2$ | 0.948 | 1.059 | $1.77\pm 0.11\pm 0.10$
3.8720 |  219.2 |  15.8$\pm$ 5.1 | $17.4\pm 0.8$ | 0.991 | 1.050 | $1.00\pm 0.32\pm 0.07$
4.0076 |  482.0 |  64.1$\pm$ 9.0 | $18.8\pm 0.5$ | 0.979 | 1.046 | $1.77\pm 0.25\pm 0.10$
4.1285 |  401.5 |  36.4$\pm$ 7.0 | $18.0\pm 0.5$ | 1.054 | 1.053 | $1.17\pm 0.23\pm 0.08$
4.1574 |  408.7 |  37.1$\pm$ 7.0 | $19.6\pm 0.6$ | 1.057 | 1.054 | $1.09\pm 0.21\pm 0.07$
4.1784 | 3189.0 | 277.4$\pm$19.5 | $20.2\pm 0.3$ | 1.054 | 1.055 | $1.00\pm 0.07\pm 0.06$
4.1888 |  570.0 |  39.8$\pm$ 7.5 | $19.0\pm 0.6$ | 1.083 | 1.057 | $0.82\pm 0.15\pm 0.05$
4.1989 |  526.0 |  63.2$\pm$ 8.8 | $20.0\pm 0.5$ | 1.055 | 1.057 | $1.38\pm 0.19\pm 0.08$
4.2092 |  517.1 |  42.9$\pm$ 7.7 | $19.1\pm 0.6$ | 1.039 | 1.057 | $1.00\pm 0.18\pm 0.06$
4.2187 |  569.2 |  47.0$\pm$ 7.8 | $17.7\pm 0.6$ | 1.097 | 1.057 | $1.06\pm 0.17\pm 0.07$
4.2263 | 1100.9 |  80.6$\pm$10.3 | $20.7\pm 0.5$ | 1.048 | 1.057 | $0.81\pm 0.10\pm 0.05$
4.2357 |  530.3 |  41.5$\pm$ 7.5 | $20.3\pm 0.6$ | 1.060 | 1.055 | $0.89\pm 0.16\pm 0.06$
4.2438 |  538.1 |  48.6$\pm$ 8.0 | $20.0\pm 0.7$ | 1.085 | 1.056 | $1.01\pm 0.17\pm 0.07$
4.2580 |  828.4 |  58.7$\pm$ 8.9 | $19.9\pm 0.6$ | 1.060 | 1.054 | $0.79\pm 0.12\pm 0.05$
4.2668 |  531.1 |  39.1$\pm$ 7.4 | $19.1\pm 0.8$ | 1.029 | 1.053 | $0.91\pm 0.17\pm 0.07$
4.2879 |  502.4 |  46.9$\pm$ 7.5 | $20.9\pm 0.6$ | 1.000 | 1.053 | $1.05\pm 0.17\pm 0.07$
4.3121 |  501.2 |  40.0$\pm$ 7.3 | $18.6\pm 0.7$ | 1.026 | 1.052 | $1.04\pm 0.19\pm 0.07$
4.3374 |  505.0 |  44.6$\pm$ 7.4 | $20.0\pm 0.6$ | 1.020 | 1.051 | $1.06\pm 0.17\pm 0.07$
4.3583 |  543.9 |  53.6$\pm$ 7.8 | $20.3\pm 0.6$ | 1.058 | 1.051 | $1.11\pm 0.16\pm 0.07$
4.3774 |  522.7 |  27.1$\pm$ 6.0 | $20.4\pm 0.7$ | 1.114 | 1.051 | $0.58\pm 0.13\pm 0.04$
4.3964 |  507.8 |  25.2$\pm$ 5.9 | $17.7\pm 0.7$ | 1.141 | 1.052 | $0.58\pm 0.13\pm 0.05$
4.4156 | 1090.7 |  72.1$\pm$ 9.8 | $20.5\pm 0.6$ | 1.092 | 1.053 | $0.72\pm 0.10\pm 0.05$
4.4400 |  569.9 |  36.5$\pm$ 7.0 | $19.7\pm 0.6$ | 0.943 | 1.055 | $0.81\pm 0.15\pm 0.08$
4.4671 |  111.1 |  13.4$\pm$ 3.9 | $23.0\pm 1.1$ | 0.913 | 1.055 | $1.36\pm 0.39\pm 0.13$
4.5995 |  586.9 |  36.4$\pm$ 6.7 | $21.3\pm 0.8$ | 1.122 | 1.055 | $0.61\pm 0.11\pm 0.05$
4.6280 |  521.5 |  31.5$\pm$ 6.3 | $18.5\pm 0.7$ | 1.133 | 1.055 | $0.69\pm 0.14\pm 0.05$
4.6612 |  529.6 |  31.6$\pm$ 6.1 | $20.3\pm 0.9$ | 1.119 | 1.056 | $0.48\pm 0.12\pm 0.04$
4.6409 |  552.4 |  22.9$\pm$ 5.5 | $17.1\pm 0.6$ | 1.205 | 1.056 | $0.63\pm 0.12\pm 0.04$
4.6819 | 1669.3 |  74.0$\pm$ 9.6 | $17.2\pm 0.5$ | 1.258 | 1.056 | $0.49\pm 0.06\pm 0.03$
4.6988 |  536.5 |  23.1$\pm$ 5.5 | $19.2\pm 0.8$ | 1.188 | 1.056 | $0.45\pm 0.11\pm 0.04$
Figure 3: Measured Born cross sections of the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ at $\sqrt{s}$ from 3.5106 to
4.6988 GeV indicated by the black points with error bars (combined statistical
and systematic uncertainties). The dashed blue line is the fit result for the
fit model with the power-law function only. The solid red, yellow, green, and
pink lines are the fits with a coherent sum of the continuum amplitude and
Breit-Wigner amplitude of resonances $\psi(4160)$, $\psi(4220)$, $\psi(4360)$,
and $\psi(4415)$, respectively. The solid magenta line denotes the fit with a
coherent sum of the continuum amplitude, the resonance amplitude of resonance
$\psi(4360)$, and the resonance amplitude resonance for $\psi(4415)$. The
solid cyan line shows the fit with a coherent sum of a continuum amplitude and
three resonance amplitudes for $\psi(4220)$, $\psi(4360)$, and $\psi(4415)$.
The subplot shows the expanded region where the resonances are expected.

Table 1 also summarizes the measured Born cross sections at the thirty-one
energy points. To search for possible resonances decaying into
$\Lambda\bar{\Lambda}\eta$, the least-squares method is used to perform the
fit of the Born cross sections with both statistical and systematic
uncertainties taken into account, ignoring the correlation between the
different energy points. The continuum production of
$e^{+}e^{-}\to\Lambda\bar{\Lambda}\eta$ is described by a power-law function
$\sigma_{\mathrm{con}}=C/s^{\lambda}$ [74]. The possible resonances are added
coherently as

$\displaystyle\sigma(s)=$
$\displaystyle\left|\sqrt{\sigma_{\mathrm{con}}}\right.$ (4)
$\displaystyle+\sum_{k}\left.\sqrt{\sigma_{\mathrm{Reso}}^{k}}e^{i\phi^{k}}\frac{\mathit{m_{k}}\Gamma_{k}}{s-m_{k}^{2}+i\mathit{m_{k}}\Gamma_{k}}\right|^{2},$

where $m_{k}$ and $\Gamma_{k}$ are mass and width of the resonance $k$,
$\sigma_{\mathrm{Reso}}^{k}$ is production cross section of the resonance $k$
at the $m_{k}$, $\phi^{k}$ is the phase angle. Since most of the data samples
are taken at $\sqrt{s}$ from 4.2 to 4.6 GeV, as shown in Table 1, the four
well-established resonances, $\psi(4160)$, $\psi(4220)$, $\psi(4360)$, and
$\psi(4415)$ are taken into account in the fit. The masses and widths of these
resonances are fixed to the corresponding values in the PDG [6] and other
parameters are free. Figure 3 shows the fits to the Born cross section with
various fit strategies. The fit with the power-law function yields the
goodness of fit $\chi^{2}/\text{ndf}=30.3/29$, where ndf denotes the number of
degrees of freedom. The fitted parameters $C$ and $\lambda$ are $(4.1\pm
2.5)\times 10^{3}$ GeV2λ pb and $2.9\pm 0.2$, respectively. To search for the
potential resonance, we perform fit using coherent sum of the continuum
amplitude and one resonance amplitude. The fits with hypotheses of
$\psi(4160)$, $\psi(4220)$, $\psi(4360)$, and $\psi(4415)$ give
$\chi^{2}/\text{ndf}$ of 30.2/27, 29.9/27, 29.4/27, and 29.6/27, respectively.
Because of the two floated parameters ($\sigma_{\text{Reso}}^{k}$ and
$\phi^{k}$), the ndf for each fit is reduced by two compared to the fit with
the power-law function only. The significance for each hypothesis with the
resonance is less than 1$\sigma$, which is calculated with the changes of ndf
and $\chi^{2}$. According to the goodness of fit, it is preferable to describe
the cross section line shape by the power-law function. The power-law
function, however, cannot describe the region around $4.4\mathrm{~{}GeV}$.
Adding a single resonance amplitude cannot model the region well, either.
Therefore, we fit the line shape with a coherent sum of a continuum amplitude,
a resonance amplitude of $\psi(4360)$, and a resonance amplitude of
$\psi(4415)$, shown as the magenta line in Fig. 3. The fit with the two
resonances delivers a better description around 4.4 GeV with a goodness of fit
equal to 24.12/25 with corresponding $p$ value of 0.512. The global
significance with respect to the fit with the power-law function is
1.3$\sigma$. Since the cross section at 4.1989 GeV deviates from the fit
result by more than 1.0$\sigma$, we also fit the cross section line shape with
a coherent sum of a continuum amplitude and three resonance amplitudes of
$\psi(4220)$, $\psi(4360)$, and $\psi(4415)$ shown as the cyan line. The fit
quality is 22.63/23 corresponding to $p$ value of 0.483. The fit quality is
worse than the fit with two resonances because of the increasing number of
parameters.

##  V Study of $\Lambda\bar{\Lambda}$ Mass Threshold Enhancement

A clear enhancement is observed near the $\Lambda\bar{\Lambda}$ mass threshold
as shown in Fig. 6. This structure cannot be described by the PHSP MC
simulation (the hatched histograms in Fig. 2). As shown in Fig. 4,
examinations of the Dalitz plots of the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ indicate that the threshold
enhancement in ${\Lambda\bar{\Lambda}}$ mass spectra is the dominant
component. In addition, a small peak from the process $e^{+}e^{-}\rightarrow
J/\psi\eta\rightarrow\Lambda\bar{\Lambda}\eta$ at 3.1 GeV/$c^{2}$ can be seen
in Fig. 2.

\begin{overpic}[width=195.12767pt,percent]{./plots/mdallameta_3773.eps}
\put(77.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=195.12767pt,percent]{./plots/mdallameta_4180.eps}
\put(77.0,60.0){(b)} \end{overpic}

Figure 4: Dalitz plots of $M^{2}_{\Lambda\eta}$ versus
$M^{2}_{\Lambda\bar{\Lambda}}$ in data taken at (a) $\sqrt{s}=3.7730$ GeV and
(b) $\sqrt{s}=4.1784$ GeV.

To determine the mass and width of the enhancement, a simultaneous unbinned
maximum-likelihood fit is performed to the $\Lambda\bar{\Lambda}$ invariant
mass spectra of the data samples at the thirty-one energy points. The
enhancement is modeled by a Breit-Wigner function with mass-dependent width of
the following form [75]:

$\displaystyle\mathrm{d}N/\mathrm{d}M_{\Lambda\bar{\Lambda}}\propto$
$\displaystyle\varepsilon(M_{\Lambda\bar{\Lambda}})\left(p_{X}\right)^{2l_{X\eta}+1}f_{l_{X\eta}}^{2}\boldsymbol{(}\left(Rp_{X}\right)^{2}\boldsymbol{)}$
(5)
$\displaystyle\times\left|\text{BW}\left(M_{\Lambda\bar{\Lambda}}\right)\right|^{2}$
$\displaystyle\times\left(p_{\Lambda}\right)^{2l_{\Lambda\bar{\Lambda}}+1}f_{l_{\Lambda\bar{\Lambda}}}^{2}\boldsymbol{(}\left(Rp_{\Lambda}\right)^{2}\boldsymbol{)},$

where $X$ denotes the threshold enhancement, the mass-dependent reconstruction
efficiency $\varepsilon(M_{\Lambda\bar{\Lambda}})$ is determined by the MC
simulation, $p_{X}$ is the momentum of $X$ in the $e^{+}e^{-}$ center-of-mass
system, $l_{X\eta}$ is the orbital angular momentum between $X$ and $\eta$,
and $f_{l}(z)$ is the Blatt-Weisskopf barrier factor [76] with $f_{0}(z)=1$
and $f_{1}(z)=\frac{1}{\sqrt{1+z}}$, where $l$ is the orbital angular
momentum. The barrier radius factor $R$ is assumed to be
$1.0~{}(\mathrm{GeV}/c)^{-1}$ since it is poorly known for baryons. The Breit-
Wigner amplitude is defined as

$\text{BW}\propto\frac{1}{M^{2}_{X}-M^{2}_{\Lambda\bar{\Lambda}}-iM_{X}\Gamma},$
(6)

where $M_{X}$ is the mass of $X$. The mass dependent width $\Gamma$ is written
as

$\Gamma=\Gamma_{X}\left(\frac{p_{\Lambda}}{p^{X}_{\Lambda}}\right)^{2l_{\Lambda\bar{\Lambda}}+1}\frac{M_{X}}{M_{\Lambda\bar{\Lambda}}}\frac{f^{2}_{l_{\Lambda\bar{\Lambda}}}\boldsymbol{(}(Rp_{\Lambda})^{2}\boldsymbol{)}}{f^{2}_{l_{\Lambda\bar{\Lambda}}}\boldsymbol{(}(Rp^{X}_{\Lambda})^{2}\boldsymbol{)}},$
(7)

where $\Gamma_{X}$ is the width of $X$, $p_{\Lambda}$ is the momentum of
$\Lambda$ in the rest frame of $X$, $p_{\Lambda}^{X}$ is the momentum when
$M_{\Lambda\bar{\Lambda}}=M_{X}$, and $l_{\Lambda\bar{\Lambda}}$ is the
orbital angular momentum between $\Lambda$ and $\bar{\Lambda}$. In the
simultaneous fit, $M_{X}$ and $\Gamma_{X}$ are the common parameters. Based on
the conservation of angular momentum, parity, and charge conjugation, the
possible quantum numbers of the enhancement are listed in Table 2, which
summarizes the possible combinations of $l_{\Lambda\bar{\Lambda}}$ and
$l_{X\eta}$. We assume the structure has $J^{PC}$ equal to $1^{--}$ and
$l_{\Lambda\bar{\Lambda}}$ equal to 0 since the enhancement is near the
$\Lambda\bar{\Lambda}$ mass threshold.

Table 2: Possible quantum numbers of $X$ considering conservation of angular momenta, parity, and charge conjugation, where $S_{\Lambda\bar{\Lambda}}$ is the total spin of $\Lambda\bar{\Lambda}$ system $S_{\Lambda\bar{\Lambda}}$/$l_{\Lambda\bar{\Lambda}}$ of $\Lambda\bar{\Lambda}$ |  $J^{PC}$ of $X$ | $l_{X\eta}$
---|---|---
0/1 | $1^{+-}$ | 0
0/1 | $1^{+-}$ | 2
1/0 | $1^{--}$ | 1
1/2 | $1^{--}$ | 1
1/2 | $2^{--}$ | 1

We use MC simulations of the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ to determine
$\varepsilon(M_{\Lambda\bar{\Lambda}})$, where the mass distribution of
$\Lambda\bar{\Lambda}$ system follows a Breit-Wigner function with a
relatively large width so that the $M_{\Lambda\bar{\Lambda}}$ distribution is
uniform within the allowed region. The quantum numbers are configured
according to the assumption in the previous paragraph. Figure 5 shows
$\varepsilon(M_{\Lambda\bar{\Lambda}})$ as a function of
$M_{\Lambda\bar{\Lambda}}$ at $\sqrt{s}=4.1784\text{~{}GeV}$ with
$M_{\Lambda\bar{\Lambda}}<3.5$ GeV/$c^{2}$. We parametrize
$\varepsilon(M_{\Lambda\bar{\Lambda}})$ by a six-order polynomial function.
Because of the detector resolution, the reconstructed distributions exceed the
allowed region. As a result, the efficiency cannot be determined correctly in
the range of $M_{\Lambda\bar{\Lambda}}>3.5$ GeV/$c^{2}$, which is not included
in the parametrization. Instead, the efficiency in this region, which is far
away from the threshold enhancement, is estimated by a flat extrapolation
assuming the efficiency is constant in the region.

Figure 5: Fit of $\varepsilon(M_{\Lambda\bar{\Lambda}})$ to the mass-dependent
efficiency (the black points) at $\sqrt{s}=4.1784$ GeV. The red line in the
region $M_{\Lambda\bar{\Lambda}}<$ 3.5 GeV/$c^{2}$ represents the fit result
which is a six-order polynomial function. The red line in the region
$M_{\Lambda\bar{\Lambda}}>$ 3.5 GeV/$c^{2}$ is the extrapolation. The
efficiency is obtained from the distribution of the reconstructed value of
$M_{\Lambda\bar{\Lambda}}$ divided by the distribution of the generated value
of $M_{\Lambda\bar{\Lambda}}$.

The events from the nonresonant process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$ are described by shapes
derived from the PHSP MC samples. The background events are modeled by the
formula [77]

$f(M_{\Lambda\bar{\Lambda}})\propto\left(M_{\Lambda\bar{\Lambda}}-M_{\mathrm{\min}}\right)^{l}\left(M_{\mathrm{\max}}-M_{\Lambda\bar{\Lambda}}\right)^{h},$
(8)

where $M_{\mathrm{\min}}=2M_{\Lambda}$ is the minimum allowed
$M_{\Lambda\bar{\Lambda}}$, $M_{\mathrm{\max}}=\sqrt{s}-M_{\eta}$ is the
maximum allowed $M_{\Lambda\bar{\Lambda}}$ calculated with center-of-mass
energy $\sqrt{s}$ and known mass of $\eta$, $M_{\eta}$ [6], and $l$ and $h$
are determined by fitting the $M_{\Lambda\bar{\Lambda}}$ spectra of events
from the $\eta$ sidebands. The magnitudes of the background shapes are fixed
to the number of the fitted background events within the $\eta$ signal region.

Figure 6 shows the sum of the simultaneous fit to the
$M_{\Lambda\bar{\Lambda}}$ spectra at the thirty-one energy points. The mass
and width of $X$ obtained from the fit are $(2356\pm 7)\text{~{}MeV}/c^{2}$
and $(304\pm 28)$ MeV, respectively, where the uncertainties are statistical
only. In the fit, the excited $\Lambda$ states are not included because they
are not significant in the data (Fig. 4) and are not well-established [6].
Recently, an improved measurement of $\Lambda(1670)$ via the
$\psi(2S)\rightarrow\Lambda\bar{\Lambda}\eta$ decay has been reported by the
BESIII collaboration [40]. The effect of the $\Lambda(1670)$ on the
determination of mass and width will be discussed in Sec. VI.2 for the
estimation of systematic uncertainties. Because of the limited sample sizes,
the potential interferences among the threshold enhancement, excited $\Lambda$
states, and three-body PHSP component are neglected. The combined statistical
signal significance is greater than $10\sigma$, which is determined by the
difference of likelihood values between the fit with and without the model for
the enhancement [78].

Figure 6: Sum of the simultaneous fits to the $M_{\Lambda\bar{\Lambda}}$
distributions at the thirty-one energy points. The black points with error
bars are data. The solid blue curve shows the total fit. The dashed red line
is the enhancement. The cyan line shows PHSP component. The dashed green line
denotes the background component.

In the determination of mass and width of $X$, $l_{X\eta}$ is assumed to be
one. The $\cos\theta$ distributions of $\eta$ candidates should follow the
distribution of $\left(1+\cos^{2}\theta\right)$ when $l_{X\eta}=1$. Therefore,
the angular distributions of $X$ decays are studied to verify this assumption.
We perform a simultaneous unbinned maximum-likelihood fit to the $\cos\theta$
distributions at the thirty-one center-of-mass energies. The $\cos\theta$
distributions of $X$ decays are modeled by the function
$\varepsilon(\cos\theta)\left(1+\alpha\cos^{2}\theta\right)$, where the
parameter $\alpha$ is free and taken as a common parameter. The reconstruction
efficiency $\varepsilon(\cos\theta)$ is parametrized by a fourth-order
polynomial function. The shape of the PHSP process is modeled by the MC
simulations. The background contributions are described by the shapes derived
from the $\eta$ sidebands. In the fit, the numbers of events for the three
components are fixed to the values obtained from the simultaneous fit to the
$M_{\Lambda\bar{\Lambda}}$ distributions. Figure 7 shows the sum of the
simultaneous fit to the distributions of $\cos\theta$. The parameter $\alpha$
is determined to be $0.8\pm 0.3$, which is consistent with our assumption of
$\alpha=1$ considering the statistical uncertainty.

Figure 7: Sum of the simultaneous fits to the $\cos\theta$ distributions of
$\eta$ candidates at the thirty-one energy points. The black points with error
bars are the data. The solid blue line shows the total fit. The dashed red
line shows the $X$ component, and the dashed cyan line denotes the PHSP
component. The dashed green line is the background.

##  VI Systematic Uncertainties

###  VI.1 Born cross section measurement

According to Eq. (1), the uncertainties on the cross section measurement are
associated with the luminosity, branching fractions, reconstruction
efficiency, fit procedure, ISR correction, and vacuum polarization factor. The
uncertainty of the reconstruction efficiency includes contributions from the
$\Lambda$ reconstruction, PID of the pion, photon reconstruction, shower
requirements, $\Sigma^{0}$ mass window, and weighting procedure (limited data
samples sizes, binning of data distributions, $\eta$ mass window, and $\eta$
sidebands). The uncertainty of the fit procedure incorporates the fit range,
background shape, peaking background, and signal shape. To reduce the effect
of statistical uncertainty due to limited sample sizes, we take the samples at
3.7730 and 4.1784 GeV (the two samples with the highest luminosities) to
estimate the uncertainties from the $\Lambda$ reconstruction, $\Sigma^{0}$
mass window, weighting of PHSP MC samples (except for the limited data sample
size), and fit procedure.

  1. 1.

Luminosity. The uncertainties from the integrated luminosities measured using
the large-angle Bhabha events are assigned to be 1.0% [67, 68, 69].

  2. 2.

Branching fractions. The systematic uncertainty due to the branching fractions
$\mathcal{B}(\eta\rightarrow\gamma\gamma)$ and $\mathcal{B}(\Lambda\rightarrow
p\pi^{-})$ are 0.5% and 0.8%, respectively, taken from the PDG [6].

  3. 3.

$\Lambda$ reconstruction. The systematic uncertainty from the $\Lambda$
($\bar{\Lambda}$) reconstruction efficiency is estimated with the equation
[79]

$\left|\frac{\Sigma_{i}\frac{N_{i}}{\varepsilon_{i}^{\mathrm{Data}}}-\Sigma_{i}\frac{N_{i}}{\varepsilon_{i}^{\mathrm{MC}}}}{\Sigma_{i}\frac{N_{i}}{\varepsilon_{i}^{\mathrm{MC}}}}\right|,$
(9)

where $N_{i}$ is the number of data events in the interval $i$ with background
subtracted using the $\eta$ sidebands, and $\varepsilon_{i}^{\mathrm{Data}}$
and $\varepsilon_{i}^{\mathrm{MC}}$ are the efficiencies of $\Lambda$
$(\bar{\Lambda})$ reconstruction for the data and MC simulations in interval
$i$ of the two-dimensional (momentum and $\cos\theta$) distributions. The
efficiencies are determined by a control sample of
$J/\psi\rightarrow\overline{p}K^{+}\Lambda+\text{c}.\text{c}$. [80]. It takes
the tracking of proton and pion, PID of proton, mass window for $\Lambda$
$(\bar{\Lambda})$ candidates, and the requirement on the decay length into
account. The uncertainty is estimated to be 2.8%.

  4. 4.

PID of pion. The uncertainty introduced by the PID of the pion is estimated to
be 1.0% by a control sample of
${e}^{+}{e}^{-}\rightarrow{K}^{+}{K}^{-}\pi^{+}\pi^{-}$ [81].

  5. 5.

Photon reconstruction. A systematic uncertainty of 1.0% [82] is assigned to
the photon reconstruction efficiency. Hence, 2.0% is taken as the systematic
uncertainty for two photons.

  6. 6.

Shower requirements. The uncertainty associated with the requirements on the
lateral moment and $E_{3\times 3}/E_{5\times 5}$ is 0.8% [83].

  7. 7.

$\Sigma^{0}$ mass window. The requirement
$M_{\Lambda\gamma_{L}~{}(\bar{\Lambda}\gamma_{L})}\notin[1.113,1.273]$
GeV/$c^{2}$ is applied to suppress the background processes with $\Sigma^{0}$
$(\bar{\Sigma}^{0})$ in the final state. By changing the width of the nominal
mass window by $\pm$5 MeV/$c^{2}$, the corresponding uncertainty is estimated
to be 0.8%.

  8. 8.

Weighting procedure. Using the $M_{\Lambda\bar{\Lambda}}$, $M_{\Lambda\eta}$,
and $M_{\bar{\Lambda}\eta}$ distributions from the data events, the PHSP MC is
weighted to determine the reconstruction efficiency.

    1. (a)

Limited data sample size. To estimate the uncertainties caused by the
statistical fluctuation of the background-subtracted data distributions, we
first produce a series of $M_{\Lambda\bar{\Lambda}}$, $M_{\Lambda\eta}$, and
$M_{\bar{\Lambda}\eta}$ distributions for each energy point by sampling over
the three background-subtracted data distributions with Poisson functions. The
means of the Poisson functions are the number of events in the intervals of
the three background-subtracted data distributions. The PHSP MC samples are
then weighted by the generated $M_{\Lambda\bar{\Lambda}}$, $M_{\Lambda\eta}$,
and $M_{\bar{\Lambda}\eta}$ distributions. The resulting efficiency
distribution for each energy point is fitted with a double-shouldered crystal
ball function [84]. The fitted means and widths are taken as efficiencies and
corresponding uncertainties from limited data sample sizes, respectively.

    2. (b)

Binning of data distributions. The widths of intervals in the
$M_{\Lambda\bar{\Lambda}}$, $M_{\Lambda\eta}$, and $M_{\bar{\Lambda}\eta}$
distributions are increased by 10% to estimated corresponding uncertainties.
The resulting relative difference in the efficiency is assigned as
corresponding uncertainty, which is 0.5%.

    3. (c)

$\eta$ mass window. The width of the $\eta$ mass window is varied by $\pm$16
MeV/$c^{2}$ (twice the resolution of the invariant mass of the $\eta$
candidates) to estimate the corresponding uncertainty, which is 1.0%.

    4. (d)

$\eta$ sidebands. To estimate the uncertainty due to the sidebands, their
distances to the signal region are varied by $\pm 8$ MeV/$c^{2}$. The
corresponding uncertainty is found to be negligible.

  9. 9.

Fit procedure. The signal yield is determined by an unbinned maximum-
likelihood fit to the $M_{\gamma\gamma}$ spectrum. The following aspects are
considered when evaluating the systematic uncertainties associated with the
fit procedure.

    1. (a)

Fit range. The ranges of fits to the $M_{\gamma\gamma}$ distributions are
varied by $\pm 100$ MeV/$c^{2}$. The resulting relative difference in the
signal yield is taken as the corresponding uncertainty. The uncertainty is
taken as 0.9%.

    2. (b)

Background shape. The nominal background shape, a linear function, is replaced
by a second-order Chebychev polynomial. The estimated uncertainty is 1.6%.

    3. (c)

Peak background. For the peaking background from the processes
$e^{+}e^{-}\rightarrow\eta\Sigma^{0}\bar{\Lambda}+\text{c}.\text{c}$.
$\mathrm{and}$ $e^{+}e^{-}\rightarrow\eta\Sigma^{0}\bar{\Sigma}^{0}$, we add
the shapes of the two processes obtained from the MC simulations to the
nominal model. The corresponding uncertainty is estimated to be 2.9%.

    4. (d)

Signal shape. For the signal shape, we set the fixed width of the Gaussian
function to be free in the nominal fit. The corresponding uncertainty is
estimated to be 0.5%.

  10. 10.

ISR correction. For uncertainties from ISR correction factors, we replace the
smoothing method LOWESS [72] by the power-law function $C/s^{\lambda}$, where
the parameters $C$ and $\lambda$ are determined by fitting the measured Born
cross section. The relative changes with respect to the nominal cross sections
are assigned as the systematic uncertainties summarized in Table 3.

Table 3: Relative uncertainties (%) from ISR correction in the measurement of the Born cross section. $\sqrt{s}$ (GeV) | Uncertainty |  |  $\sqrt{s}$ (GeV) | Uncertainty
---|---|---|---|---
3.5106 | 0.1 |  | 4.2879 | 1.4
3.7730 | 0.3 |  | 4.3121 | 1.7
3.8720 | 1.4 |  | 4.3374 | 0.9
4.0076 | 0.5 |  | 4.3583 | 1.7
4.1285 | 2.1 |  | 4.3774 | 3.4
4.1574 | 2.3 |  | 4.3964 | 4.5
4.1784 | 3.0 |  | 4.4156 | 2.4
4.1888 | 2.4 |  | 4.4400 | 8.2
4.1989 | 1.4 |  | 4.4671 | 6.2
4.2092 | 0.8 |  | 4.5995 | 5.6
4.2187 | 0.7 |  | 4.6280 | 2.6
4.2263 | 1.2 |  | 4.6612 | 2.0
4.2357 | 1.0 |  | 4.6409 | 2.8
4.2438 | 1.2 |  | 4.6819 | 3.4
4.2580 | 1.2 |  | 4.6988 | 3.9
4.2668 | 3.8 |  |  |
  11. 11.

Vacuum polarization factor. The uncertainty introduced by the vacuum
polarization factor is less than 0.1% [70], which is negligible compared to
other sources of uncertainties.

Table 4 summarizes the uncertainties for all the samples.

Table 4: Systematic uncertainties in the measurement of the Born cross section. Source | Uncertainty
---|---
Luminosity | 1.0%
$\mathcal{B}(\eta\rightarrow\gamma\gamma)$ and $\mathcal{B}(\Lambda\rightarrow p\pi^{-})$ | 0.9%
$\Lambda$ reconstruction | 2.8%
PID of pion | 1.0%
Photon reconstruction | 2.0%
Shower requirements | 0.8%
$\Sigma^{0}$ mass window | 0.8%
Limited data sample size | see Table 1
Binning of data distributions | 0.5%
$\eta$ mass window | 1.0%
$\eta$ sidebands | negligible
Fit range | 0.9%
Background shape | 1.6%
Peaking background | 2.9%
Signal shape | 0.5%
ISR correction | see Table 3

###  VI.2 Mass and width of the threshold enhancement

The mass and width of the threshold enhancement in the $\Lambda\bar{\Lambda}$
mass spectra are determined by the simultaneous unbinned maximum-likelihood
fit. The uncertainties associated with the mass and width are assigned as
follows:

  1. 1.

Description of the threshold enhancement. In the simultaneous fit to the
$M_{\Lambda\bar{\Lambda}}$ distributions, the barrier radius is assumed to be
1.0. To determine the uncertainties from the assumption, the simultaneous fit
is repeated with the barrier radius being free, which results in a mass
difference of 1 MeV/$c^{2}$ and a width difference of 4 MeV. The uncertainties
due to the parametrization of efficiency curves
$\varepsilon(m_{\Lambda\bar{\Lambda}})$ are estimated by replacing the sixth-
order polynomial function with a seventh-order polynomial function. The
resulting differences in the mass and width are negligible.

  2. 2.

Background shape. The uncertainties related to the background description are
estimated by changing Eq. (8) to the shapes of histograms obtained from events
in $\eta$ sidebands of the data. A fit under the scenario yields a mass
difference of 5 MeV/$c^{2}$ and a width difference of 11 MeV.

  3. 3.

$e^{+}e^{-}\to J/\psi\eta\to\Lambda\bar{\Lambda}\eta$. In the
$\Lambda\bar{\Lambda}$ mass spectrum shown in Fig. 2, there is a small peak at
the $J/\psi$ mass from the process $e^{+}e^{-}\to
J/\psi\eta\to\Lambda\bar{\Lambda}\eta$. To estimate the effect of this process
on the determination of mass and width of the threshold enhancement, we add
the shape obtained from the MC simulation of $e^{+}e^{-}\to
J/\psi\eta\to\Lambda\bar{\Lambda}\eta$ to the nominal fit model. The
differences in the mass and width are $1\mathrm{~{}MeV}/c^{2}$ and
$1\text{~{}MeV}$, respectively.

  4. 4.

$e^{+}e^{-}\to\Lambda(1670)\bar{\Lambda}+\mathrm{c.c.}\to\Lambda\bar{\Lambda}\eta$.
To consider effect of the excited $\Lambda$ state $\Lambda(1670)$ on the
measurement of mass and width, we add the shape of the process obtained from
the MC simulation to the nominal model. The resulting differences in mass and
width, which are $11\mathrm{~{}MeV}/c^{2}$ and 48 MeV, to the nominal results
are assigned as corresponding uncertainties.

  5. 5.

$\Sigma^{0}$ mass window, $\eta$ mass window, and $\eta$ sidebands. The
$\Sigma^{0}$ mass window, $\eta$ mass window, and $\eta$ sidebands are altered
as mentioned in Sec. VI.1, to estimate corresponding uncertainties. The
maximum differences in mass are 6, 9, and 0.1 MeV/$c^{2}$, and the maximum
differences in width are 3, 20, and 1 MeV, which are assigned as corresponding
systematic uncertainties.

After adding these uncertainties in quadrature, the total systematic
uncertainties on the mass and width of the enhancement are determined to be 17
MeV/$c^{2}$ and $54\text{~{}MeV}$, respectively.

##  VII Summary

Based on the thirty-one data samples taken at the center-of-mass energies from
3.51 GeV to 4.70 GeV, we measure the Born cross section of the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$, as shown in Table 1. No
significant structure is observed in the line shape of the Born cross section,
which can be described by a power-law function $C/s^{\lambda}$ with $C=(4.1\pm
2.5)\times 10^{3}$ GeV2λpb and $\lambda=2.9\pm 0.2$.

Further, a clear enhancement is observed near the $\Lambda\bar{\Lambda}$ mass
threshold. The contribution of the excited $\Lambda$ are neglected in the
study of $M_{\Lambda\bar{\Lambda}}$ distribution since they are not
significant in the data and are not well-established [6]. Because of the
limited data sample sizes, the potential interferences among the structure,
excited $\Lambda$ states, and phase space process are also not considered. A
simultaneous fit to the $\Lambda\bar{\Lambda}$ mass spectra, assuming
$J^{PC}=1^{--}$, yields a mass of $(2356\pm 7\pm 17)\mathrm{~{}MeV}/c^{2}$ and
width of $(304\pm 28\pm 54)\text{~{}MeV}$ for the structure. The first
uncertainties are statistical and the second are systematic. The statistical
signal significance of the structure is larger than 10$\sigma$ over the
hypothesis of the pure contribution from the PHSP process of
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}\eta$. The $\cos\theta$
distributions of the structure can be described by a function of
$(1+\alpha\cos^{2}\theta)$. A simultaneous fit to the $\cos\theta$
distributions gives $\alpha=0.8\pm 0.3$ which is consistent with our
assumption of $\alpha=1$, where the uncertainty is statistical only. In the
PDG, there is no well-established resonance matching the structure in terms of
the resonance parameters. A vector hexaquark state decaying into
$\Lambda\bar{\Lambda}$ is proposed with a mass of 2200 MeV/${c}^{2}$ and a
width of $32\text{~{}MeV}$ [85]. The mass is close to our measurement but the
width of the hexaquark state is much narrower than the width of the observed
structure. In the framework of QCD sum rules, an investigation of light
baryonium states indicates that there exit possible light baryonium states,
including the $\Lambda\bar{\Lambda}$ state with $J^{PC}$ of $1^{--}$ [86]. The
mass for the $\Lambda\bar{\Lambda}$ state is evaluated to be $2.34\pm 0.12$
GeV/$c^{2}$, which is consistent with our measurement. For the structure with
$M=2290\pm 20\mathrm{~{}MeV}/c^{2}$ and $\Gamma=275\pm 35\mathrm{~{}MeV}$
observed with a partial wave analysis of PS185 data [19], the width is
consistent with our result within the uncertainties, while the mass difference
cannot be covered by the uncertainties. For the recently observed near-
threshold enhancement in the process
$e^{+}e^{-}\rightarrow\phi\Lambda\bar{\Lambda}$ [75], its $C$ parity is
opposite to the enhancement in our analysis. The measured values of mass and
width are $(2262\pm 4\pm 28)\text{ MeV}/c^{2}$ and $(72\pm 5\pm
43)\text{~{}MeV}$, respectively, which are smaller than the mass and width of
our measurements. A future larger data sample [87] and theories incorporating
a partial wave analysis could lead to a better understanding of the observed
structure.

###### 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; 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 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; Olle Engkvist Foundation under Contract No. 200-0605; 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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