# Observation of the $Y(4230)$ and evidence for a new vector charmonium-like
state $Y(4710)$ in $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$

M. Ablikim1, M. N. Achasov11,b, P. Adlarson70, M. Albrecht4, R. Aliberti31, A.
Amoroso69A,69C, M. R. An35, Q. An66,53, X. H. Bai61, Y. Bai52, O. Bakina32, R.
Baldini Ferroli26A, I. Balossino27A, Y. Ban42,g, V. Batozskaya1,40, D.
Becker31, K. Begzsuren29, N. Berger31, M. Bertani26A, D. Bettoni27A, F.
Bianchi69A,69C, J. Bloms63, A. Bortone69A,69C, I. Boyko32, R. A. Briere5, A.
Brueggemann63, H. Cai71, X. Cai1,53, A. Calcaterra26A, G. F. Cao1,58, N.
Cao1,58, S. A. Cetin57A, J. F. Chang1,53, W. L. Chang1,58, G. Chelkov32,a, C.
Chen39, Chao Chen50, G. Chen1, H. S. Chen1,58, M. L. Chen1,53, S. J. Chen38,
S. M. Chen56, T. Chen1, X. R. Chen28,58, X. T. Chen1, Y. B. Chen1,53, Z. J.
Chen23,h, W. S. Cheng69C, S. K. Choi 50, X. Chu39, G. Cibinetto27A, F.
Cossio69C, J. J. Cui45, H. L. Dai1,53, J. P. Dai73, A. Dbeyssi17, R. E. de
Boer4, D. Dedovich32, Z. Y. Deng1, A. Denig31, I. Denysenko32, M.
Destefanis69A,69C, F. De Mori69A,69C, Y. Ding36, J. Dong1,53, L. Y. Dong1,58,
M. Y. Dong1,53,58, X. Dong71, S. X. Du75, P. Egorov32,a, Y. L. Fan71, J.
Fang1,53, S. S. Fang1,58, W. X. Fang1, Y. Fang1, R. Farinelli27A, L.
Fava69B,69C, F. Feldbauer4, G. Felici26A, C. Q. Feng66,53, J. H. Feng54, K
Fischer64, M. Fritsch4, C. Fritzsch63, C. D. Fu1, H. Gao58, Y. N. Gao42,g,
Yang Gao66,53, S. Garbolino69C, I. Garzia27A,27B, P. T. Ge71, Z. W. Ge38, C.
Geng54, E. M. Gersabeck62, A Gilman64, K. Goetzen12, L. Gong36, W. X.
Gong1,53, W. Gradl31, M. Greco69A,69C, L. M. Gu38, M. H. Gu1,53, Y. T. Gu14,
C. Y Guan1,58, A. Q. Guo28,58, L. B. Guo37, R. P. Guo44, Y. P. Guo10,f, A.
Guskov32,a, T. T. Han45, W. Y. Han35, X. Q. Hao18, F. A. Harris60, K. K. He50,
K. L. He1,58, F. H. Heinsius4, C. H. Heinz31, Y. K. Heng1,53,58, C. Herold55,
G. Y. Hou1,58, Y. R. Hou58, Z. L. Hou1, H. M. Hu1,58, J. F. Hu51,i, T.
Hu1,53,58, Y. Hu1, G. S. Huang66,53, K. X. Huang54, L. Q. Huang28,58, X. T.
Huang45, Y. P. Huang1, Z. Huang42,g, T. Hussain68, N Hüsken25,31, W. Imoehl25,
M. Irshad66,53, J. Jackson25, S. Jaeger4, S. Janchiv29, E. Jang50, J. H.
Jeong50, Q. Ji1, Q. P. Ji18, X. B. Ji1,58, X. L. Ji1,53, Y. Y. Ji45, Z. K.
Jia66,53, H. B. Jiang45, S. S. Jiang35, X. S. Jiang1,53,58, Y. Jiang58, J. B.
Jiao45, Z. Jiao21, S. Jin38, Y. Jin61, M. Q. Jing1,58, T. Johansson70, N.
Kalantar-Nayestanaki59, X. S. Kang36, R. Kappert59, M. Kavatsyuk59, B. C.
Ke75, I. K. Keshk4, A. Khoukaz63, R. Kiuchi1, R. Kliemt12, L. Koch33, O. B.
Kolcu57A, B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc40,70, W. Kühn33, J. J.
Lane62, J. S. Lange33, P. Larin17, A. Lavania24, L. Lavezzi69A,69C, Z. H.
Lei66,53, H. Leithoff31, M. Lellmann31, T. Lenz31, C. Li39, C. Li43, C. H.
Li35, Cheng Li66,53, D. M. Li75, F. Li1,53, G. Li1, H. Li47, H. Li66,53, H. B.
Li1,58, H. J. Li18, H. N. Li51,i, J. Q. Li4, J. S. Li54, J. W. Li45, Ke Li1,
L. J Li1, L. K. Li1, Lei Li3, M. H. Li39, P. R. Li34,j,k, S. X. Li10, S. Y.
Li56, T. Li45, W. D. Li1,58, W. G. Li1, X. H. Li66,53, X. L. Li45, Xiaoyu
Li1,58, Y. G. Li42,g, Z. X. Li14, H. Liang66,53, H. Liang30, H. Liang1,58, Y.
F. Liang49, Y. T. Liang28,58, G. R. Liao13, L. Z. Liao45, J. Libby24, A.
Limphirat55, C. X. Lin54, D. X. Lin28,58, T. Lin1, B. J. Liu1, C. X. Liu1, D.
Liu17,66, F. H. Liu48, Fang Liu1, Feng Liu6, G. M. Liu51,i, H. Liu34,j,k, H.
B. Liu14, H. M. Liu1,58, Huanhuan Liu1, Huihui Liu19, J. B. Liu66,53, J. L.
Liu67, J. Y. Liu1,58, K. Liu1, K. Y. Liu36, Ke Liu20, L. Liu66,53, Lu Liu39,
M. H. Liu10,f, P. L. Liu1, Q. Liu58, S. B. Liu66,53, T. Liu10,f, W. K. Liu39,
W. M. Liu66,53, X. Liu34,j,k, Y. Liu34,j,k, Y. B. Liu39, Z. A. Liu1,53,58, Z.
Q. Liu45, X. C. Lou1,53,58, F. X. Lu54, H. J. Lu21, J. G. Lu1,53, X. L. Lu1,
Y. Lu7, Y. P. Lu1,53, Z. H. Lu1, C. L. Luo37, M. X. Luo74, T. Luo10,f, X. L.
Luo1,53, X. R. Lyu58, Y. F. Lyu39, F. C. Ma36, H. L. Ma1, L. L. Ma45, M. M.
Ma1,58, Q. M. Ma1, R. Q. Ma1,58, R. T. Ma58, X. Y. Ma1,53, Y. Ma42,g, F. E.
Maas17, M. Maggiora69A,69C, S. Maldaner4, S. Malde64, Q. A. Malik68, A.
Mangoni26B, Y. J. Mao42,g, Z. P. Mao1, S. Marcello69A,69C, Z. X. Meng61, J.
Messchendorp12,59, G. Mezzadri27A, H. Miao1, T. J. Min38, R. E. Mitchell25, X.
H. Mo1,53,58, N. Yu. Muchnoi11,b, Y. Nefedov32, F. Nerling17,d, I. B.
Nikolaev11,b, Z. Ning1,53, S. Nisar9,l, Y. Niu 45, S. L. Olsen58, Q.
Ouyang1,53,58, S. Pacetti26B,26C, X. Pan10,f, Y. Pan52, A. Pathak30, M.
Pelizaeus4, H. P. Peng66,53, K. Peters12,d, J. L. Ping37, R. G. Ping1,58, S.
Plura31, S. Pogodin32, V. Prasad66,53, F. Z. Qi1, H. Qi66,53, H. R. Qi56, M.
Qi38, T. Y. Qi10,f, S. Qian1,53, W. B. Qian58, Z. Qian54, C. F. Qiao58, J. J.
Qin67, L. Q. Qin13, X. P. Qin10,f, X. S. Qin45, Z. H. Qin1,53, J. F. Qiu1, S.
Q. Qu39, S. Q. Qu56, K. H. Rashid68, C. F. Redmer31, K. J. Ren35, A.
Rivetti69C, V. Rodin59, M. Rolo69C, G. Rong1,58, Ch. Rosner17, S. N. Ruan39,
H. S. Sang66, A. Sarantsev32,c, Y. Schelhaas31, C. Schnier4, K. Schoenning70,
M. Scodeggio27A,27B, K. Y. Shan10,f, W. Shan22, X. Y. Shan66,53, J. F.
Shangguan50, L. G. Shao1,58, M. Shao66,53, C. P. Shen10,f, H. F. Shen1,58, X.
Y. Shen1,58, B. A. Shi58, H. C. Shi66,53, J. Y. Shi1, q. q. Shi50, R. S.
Shi1,58, X. Shi1,53, X. D Shi66,53, J. J. Song18, W. M. Song30,1, Y. X.
Song42,g, S. Sosio69A,69C, S. Spataro69A,69C, F. Stieler31, K. X. Su71, P. P.
Su50, Y. J. Su58, G. X. Sun1, H. Sun58, H. K. Sun1, J. F. Sun18, L. Sun71, S.
S. Sun1,58, T. Sun1,58, W. Y. Sun30, X Sun23,h, Y. J. Sun66,53, Y. Z. Sun1, Z.
T. Sun45, Y. H. Tan71, Y. X. Tan66,53, C. J. Tang49, G. Y. Tang1, J. Tang54,
L. Y Tao67, Q. T. Tao23,h, M. Tat64, J. X. Teng66,53, V. Thoren70, W. H.
Tian47, Y. Tian28,58, I. Uman57B, B. Wang1, B. L. Wang58, C. W. Wang38, D. Y.
Wang42,g, F. Wang67, H. J. Wang34,j,k, H. P. Wang1,58, K. Wang1,53, L. L.
Wang1, M. Wang45, M. Z. Wang42,g, Meng Wang1,58, S. Wang13, S. Wang10,f, T.
Wang10,f, T. J. Wang39, W. Wang54, W. H. Wang71, W. P. Wang66,53, X. Wang42,g,
X. F. Wang34,j,k, X. L. Wang10,f, Y. Wang56, Y. D. Wang41, Y. F. Wang1,53,58,
Y. H. Wang43, Y. Q. Wang1, Yaqian Wang16,1, Z. Wang1,53, Z. Y. Wang1,58, Ziyi
Wang58, D. H. Wei13, F. Weidner63, S. P. Wen1, D. J. White62, U. Wiedner4, G.
Wilkinson64, M. Wolke70, L. Wollenberg4, J. F. Wu1,58, L. H. Wu1, L. J.
Wu1,58, X. Wu10,f, X. H. Wu30, Y. Wu66, Y. J Wu28, Z. Wu1,53, L. Xia66,53, T.
Xiang42,g, D. Xiao34,j,k, G. Y. Xiao38, H. Xiao10,f, S. Y. Xiao1, Y. L.
Xiao10,f, Z. J. Xiao37, C. Xie38, X. H. Xie42,g, Y. Xie45, Y. G. Xie1,53, Y.
H. Xie6, Z. P. Xie66,53, T. Y. Xing1,58, C. F. Xu1, C. J. Xu54, G. F. Xu1, H.
Y. Xu61, Q. J. Xu15, X. P. Xu50, Y. C. Xu58, Z. P. Xu38, F. Yan10,f, L.
Yan10,f, W. B. Yan66,53, W. C. Yan75, H. J. Yang46,e, H. L. Yang30, H. X.
Yang1, L. Yang47, S. L. Yang58, Tao Yang1, Y. F. Yang39, Y. X. Yang1,58, Yifan
Yang1,58, M. Ye1,53, M. H. Ye8, J. H. Yin1, Z. Y. You54, B. X. Yu1,53,58, C.
X. Yu39, G. Yu1,58, T. Yu67, X. D. Yu42,g, C. Z. Yuan1,58, L. Yuan2, S. C.
Yuan1, X. Q. Yuan1, Y. Yuan1,58, Z. Y. Yuan54, C. X. Yue35, A. A. Zafar68, F.
R. Zeng45, X. Zeng6, Y. Zeng23,h, Y. H. Zhan54, A. Q. Zhang1, B. L. Zhang1, B.
X. Zhang1, D. H. Zhang39, G. Y. Zhang18, H. Zhang66, H. H. Zhang30, H. H.
Zhang54, H. Y. Zhang1,53, J. L. Zhang72, J. Q. Zhang37, J. W. Zhang1,53,58, J.
X. Zhang34,j,k, J. Y. Zhang1, J. Z. Zhang1,58, Jianyu Zhang1,58, Jiawei
Zhang1,58, L. M. Zhang56, L. Q. Zhang54, Lei Zhang38, P. Zhang1, Q. Y.
Zhang35,75, Shuihan Zhang1,58, Shulei Zhang23,h, X. D. Zhang41, X. M. Zhang1,
X. Y. Zhang45, X. Y. Zhang50, Y. Zhang64, Y. T. Zhang75, Y. H. Zhang1,53, Yan
Zhang66,53, Yao Zhang1, Z. H. Zhang1, Z. Y. Zhang39, Z. Y. Zhang71, G. Zhao1,
J. Zhao35, J. Y. Zhao1,58, J. Z. Zhao1,53, Lei Zhao66,53, Ling Zhao1, M. G.
Zhao39, S. J. Zhao75, Y. B. Zhao1,53, Y. X. Zhao28,58, Z. G. Zhao66,53, A.
Zhemchugov32,a, B. Zheng67, J. P. Zheng1,53, Y. H. Zheng58, B. Zhong37, C.
Zhong67, X. Zhong54, H. Zhou45, L. P. Zhou1,58, X. Zhou71, X. K. Zhou58, X. R.
Zhou66,53, X. Y. Zhou35, Y. Z. Zhou10,f, J. Zhu39, K. Zhu1, K. J. Zhu1,53,58,
L. X. Zhu58, S. H. Zhu65, S. Q. Zhu38, T. J. Zhu72, W. J. Zhu10,f, Y. C.
Zhu66,53, Z. A. Zhu1,58, J. H. Zou1 (BESIII Collaboration) 1 Institute of High
Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
10 Fudan University, Shanghai 200433, People’s Republic of China
11 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
12 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
13 Guangxi Normal University, Guilin 541004, People’s Republic of China
14 Guangxi University, Nanning 530004, People’s Republic of China
15 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
16 Hebei University, Baoding 071002, People’s Republic of China
17 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
18 Henan Normal University, Xinxiang 453007, People’s Republic of China
19 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
20 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
21 Huangshan College, Huangshan 245000, People’s Republic of China
22 Hunan Normal University, Changsha 410081, People’s Republic of China
23 Hunan University, Changsha 410082, People’s Republic of China
24 Indian Institute of Technology Madras, Chennai 600036, India
25 Indiana University, Bloomington, Indiana 47405, USA
26 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
27 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
28 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
29 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
30 Jilin University, Changchun 130012, People’s Republic of China
31 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
32 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
33 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
34 Lanzhou University, Lanzhou 730000, People’s Republic of China
35 Liaoning Normal University, Dalian 116029, People’s Republic of China
36 Liaoning University, Shenyang 110036, People’s Republic of China
37 Nanjing Normal University, Nanjing 210023, People’s Republic of China
38 Nanjing University, Nanjing 210093, People’s Republic of China
39 Nankai University, Tianjin 300071, People’s Republic of China
40 National Centre for Nuclear Research, Warsaw 02-093, Poland
41 North China Electric Power University, Beijing 102206, People’s Republic of
China
42 Peking University, Beijing 100871, People’s Republic of China
43 Qufu Normal University, Qufu 273165, People’s Republic of China
44 Shandong Normal University, Jinan 250014, People’s Republic of China
45 Shandong University, Jinan 250100, People’s Republic of China
46 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
47 Shanxi Normal University, Linfen 041004, People’s Republic of China
48 Shanxi University, Taiyuan 030006, People’s Republic of China
49 Sichuan University, Chengdu 610064, People’s Republic of China
50 Soochow University, Suzhou 215006, People’s Republic of China
51 South China Normal University, Guangzhou 510006, People’s Republic of China
52 Southeast University, Nanjing 211100, People’s Republic of China
53 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
54 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
55 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
56 Tsinghua University, Beijing 100084, People’s Republic of China
57 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
58 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
59 University of Groningen, NL-9747 AA Groningen, The Netherlands
60 University of Hawaii, Honolulu, Hawaii 96822, USA
61 University of Jinan, Jinan 250022, People’s Republic of China
62 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
63 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
64 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
65 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
66 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
67 University of South China, Hengyang 421001, People’s Republic of China
68 University of the Punjab, Lahore-54590, Pakistan
69 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
70 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
71 Wuhan University, Wuhan 430072, People’s Republic of China
72 Xinyang Normal University, Xinyang 464000, People’s Republic of China
73 Yunnan University, Kunming 650500, People’s Republic of China
74 Zhejiang University, Hangzhou 310027, People’s Republic of China
75 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

Cross sections for the process $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ at
center-of-mass energies from $4.128$ to $4.950$ GeV are measured using data
samples with a total integrated luminosity of 21.2 fb-1 collected by the
BESIII detector operating at the BEPCII storage ring. The $Y(4230)$ state is
observed in the energy dependence of the $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ cross section for the first time with a statistical
significance of 26.0$\sigma$. In addition, an enhancement around $4.710$ GeV,
called the $Y(4710)$, is seen with a statistical significance of 4.2$\sigma$.
There is no clear structure around $4.484$ GeV. Using a fit with a coherent
sum of three Breit-Wigner functions, we determine the mass and width of the
$Y(4230)$ state to be $4226.9\pm 6.6\pm 22.0$ MeV/$c^{2}$ and $71.7\pm 16.2\pm
32.8$ MeV, respectively, and the mass and width of the $Y(4710)$ state to be
$4704.0\pm 52.3\pm 69.5$ MeV/$c^{2}$ and $183.2\pm 114.0\pm 96.1$ MeV,
respectively, where the first uncertainties are statistical and the second are
systematic. In addition, the average Born cross section ratio of
$e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ to $e^{+}e^{-}\to K^{+}K^{-}J/\psi$
is measured to be $0.388_{-0.028}^{+0.035}\pm 0.016$, or
$0.426_{-0.031}^{+0.038}\pm 0.018$ if three-body phase space is considered.

###### pacs:

14.20.Pt, 14.40.Lb, 13.25.Gv, 13.25.Es

##  I Introduction

In the past two decades, a series of charmonium-like states have been
discovered that do not fit into the spectrum predicted by the conventional
quark model Barnes  _et al._ (2005). The existence of these states challenges
our understanding of both charmonium spectroscopy and QCD calculations
Brambilla  _et al._ (2011); Briceno  _et al._ (2016). In particular, the
number of observed vector states with masses above open-charm threshold is
more than that expected for conventional charmonium states, and this implies
the existence of exotic states beyond the quark-antiquark meson picture. In
addition to the three well-established charmonium states above the
$\psi(3770)$, the $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$ Workman  _et
al._ (2022), other experimentally discovered $Y$ states, such as the
$Y(4230)$, overpopulate the conventional charmonium spectrum. These $Y$ states
have not yet been found to decay to $D\bar{D}$, although their masses are
above $D\bar{D}$ threshold Aubert  _et al._ (2007); Pakhlova  _et al._ (2008).
The $Y(4230)$ state was discovered via the process
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ by the BaBar experiment Aubert  _et al._
(2005) using initial state radiation (ISR) and then confirmed by the CLEO Coan
_et al._ (2006) and Belle experiments Yuan  _et al._ (2007). Several
theoretical interpretations have been proposed for the $Y(4230)$ state, such
as tetraquark Maiani  _et al._ (2005), meson molecule Ding (2009); Wang  _et
al._ (2013), hadroquarkonium Alberti  _et al._ (2017); Li and Voloshin (2014),
hybrid meson Zhu (2005); Close and Page (2005); Kou and Pene (2005), and
others Guo  _et al._ (2013); Maiani  _et al._ (2013); Braaten (2013); Liu and
Li (2013).

The BESIII experiment has previously studied $Y$ states via $e^{+}e^{-}$ cross
section measurements using various hidden-charm decay modes, such as
$e^{+}e^{-}\to\pi\pi J/\psi$ Ablikim  _et al._ (2017a); Ablikim (2020),
$\pi^{+}\pi^{-}h_{c}$ Ablikim  _et al._ (2017b), $\pi\pi\psi(2S)$ Ablikim  _et
al._ (2017c, 2018a, 2021a), $\omega\chi_{c0}$ Ablikim  _et al._ (2015a, 2019),
$K^{+}K^{-}J/\psi$ Ablikim  _et al._ (2018b, 2022a), and so on. Recently, in a
study of the cross sections of the process $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ at
center-of-mass (CM) energies ($\sqrt{s}$) below 4.600 GeV, the BESIII
experiment reported two structures, the $Y(4230)$ and $Y(4500)$ Ablikim  _et
al._ (2022a). The $K^{+}K^{-}J/\psi$ decay mode of the $Y(4230)$ state was
first seen by the CLEO experiment Coan  _et al._ (2006) in 2006. Later, the
Belle experiment measured the Born cross section of $K^{+}K^{-}J/\psi$ via ISR
Yuan  _et al._ (2008); Shen  _et al._ (2014), but failed to confirm the decay
$Y(4230)\to K^{+}K^{-}J/\psi$ due to limited data sample size. The $Y(4500)$
is a new structure that was first observed at BESIII with a statistical
significance of $8\sigma$ and its mass and width were measured to be
$4484.7\pm 13.3\pm 24.1$ MeV/$c^{2}$ and $111.1\pm 30.1\pm 15.2$ MeV,
respectively Ablikim  _et al._ (2022a). The neutral process $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ is a good probe to investigate the $Y(4500)$ state.
The Belle experiment measured the cross sections of $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ Yuan  _et al._ (2008); Shen  _et al._ (2014), but
only upper limits were given. Also, using data samples from $\sqrt{s}=4.189$
to $4.600$ MeV, corresponding to an integrated luminosity of 4.7 fb-1, the
BESIII experiment performed a measurement of the Born cross sections of
$e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ at fourteen energy points Ablikim
_et al._ (2018b), and measured non-zero Born cross sections at seven of those
energy points Ablikim  _et al._ (2018b). No structure was observed in the
measured Born cross sections of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$. In
addition, Ref. Ablikim  _et al._ (2018b) reported the Born cross section ratio
of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ to $e^{+}e^{-}\to
K^{+}K^{-}J/\psi$ to be $\frac{\sigma^{\rm
Born}(K_{S}^{0}K_{S}^{0}J/\psi)}{\sigma^{\rm
Born}(K^{+}K^{-}J/\psi)}=0.370^{+0.064}_{-0.058}\pm 0.042$, where the first
uncertainty is statistical and the second is systematic.

In this paper, we present a follow-up study of $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ at $\sqrt{s}$ from 4.128 to 4.950 GeV using data
samples corresponding to a total integrated luminosity ($\mathcal{L}$) of 21.2
fb-1 Ablikim  _et al._ (2015b, 2021b, 2022b, 2022c) collected at thirty-six
energy points by the BESIII detector Ablikim  _et al._ (2010).

##  II The BESIII detector and data samples

The BESIII detector Ablikim  _et al._ (2010) records symmetric $e^{+}e^{-}$
collisions provided by the BEPCII storage ring Yu  _et al._ (2016) in the CM
energy range from 2.0 to 4.95 GeV, with a peak luminosity of $1\times
10^{33}\;\text{cm}^{-2}\text{s}^{-1}$ achieved at $\sqrt{s}=3.77\;\text{GeV}$.
BESIII has collected large data samples in this energy region Ablikim  _et
al._ (2020). 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 Huang  _et al._ (2022). 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 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 multigap resistive plate chamber technology, providing a time resolution
of 60 ps Cao  _et al._ (2020).

Simulated data samples produced with a geant4-based Agostinelli  _et al._
(2003) Monte Carlo (MC) toolkit, which includes the geometric description of
the BESIII detector and the detector response, are used to determine detection
efficiencies and to estimate backgrounds. The simulation models the beam
energy spread and ISR in the $e^{+}e^{-}$ annihilations with the generator
kkmc Jadach  _et al._ (2000, 2001). 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
Jadach  _et al._ (2000, 2001). All particle decays are modelled with evtgen
Lange (2001); Ping (2008) using branching fractions either taken from the
Particle Data Group (PDG) Workman  _et al._ (2022), when available, or
otherwise estimated with lundcharm Chen  _et al._ (2000); Yang  _et al._
(2014). Final state radiation from charged final state particles is
incorporated using photos Richter-Was (1993). Signal MC samples are generated
for $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ using evtgen Lange (2001); Ping
(2008) and assuming a uniform distribution in the available phase space. kkmc
Jadach  _et al._ (2000, 2001) is used to calculate the ISR correction factors
needed to convert an observed cross section to a Born cross section Ping
(2014); Sun  _et al._ (2021).

##  III Data analysis

We reconstruct the final state $K_{S}^{0}K_{S}^{0}J/\psi$, where the $J/\psi$
decays into a lepton pair ($e^{+}e^{-}$ or $\mu^{+}\mu^{-}$), and one
$K_{S}^{0}$ decays into $\pi^{+}\pi^{-}$ while the other $K_{S}^{0}$ decays
into either $\pi^{+}\pi^{-}$ or $\pi^{0}\pi^{0}$. For events with six charged
tracks, full reconstruction is performed (the “two-$K_{S}^{0}$” reconstruction
method). For events with four or five charged tracks, partial reconstruction
is performed (the ”one-$K_{S}^{0}$” reconstruction method).

A charged lepton candidate ($e$ or $\mu$) must have a distance of closest
approach to the interaction point (IP) less than $10$ cm along the $z$-axis
($|V_{z}|<10.0$ cm) and less than $1$ cm in the transverse plane ($|V_{xy}|<1$
cm), and a polar angle ($\theta$) range of $|\\!\cos\theta|<0.93$. The
$z$-axis is the symmetry axis of the MDC, and $\theta$ is defined with respect
to the $z$-axis. In addition, the absolute momentum of a lepton candidate is
required to be greater than $0.95$ GeV/$c$ ($P>0.95$ GeV/$c$). A lepton with
energy deposited in the calorimeter greater than 0.95 GeV is assigned as an
electron, otherwise a muon.

\begin{overpic}[width=173.44534pt]{mks.pdf} \put(10.0,92.0){(a)}
\put(10.0,62.0){(b)} \put(10.0,32.0){(c)} \end{overpic} Figure 1: The
reconstructed $K_{S}^{0}$ mass $M(K_{S}^{0})$ for (a) the two-$K_{S}^{0}$
reconstruction method, (b) the one-$K_{S}^{0}$ reconstruction method, and (c)
both methods. Data from all CM energies are combined. The dots with error bars
are data, the green filled histograms are the background from inclusive MC
samples, and the red histograms are signal MC samples. The average signal
regions are shown by the red arrows.

A $K_{S}^{0}$ candidate is reconstructed from two oppositely charged tracks
satisfying $|V_{z}|<20$ cm and $P<0.95$ GeV/$c$. The two charged tracks are
assigned as $\pi^{+}\pi^{-}$ without imposing further particle identification
criteria. They are constrained to originate from a common vertex, and the
decay length of the $K_{S}^{0}$ candidate is required to be greater than twice
the vertex resolution away from the IP. The $K_{S}^{0}$ candidate is required
to have an invariant mass of $\pi^{+}\pi^{-}$ within
($m_{K_{S}^{0}}-3\sigma_{K_{S}^{0}}$, $m_{K_{S}^{0}}+3\sigma_{K_{S}^{0}}$),
where $m_{K_{S}^{0}}$ and $\sigma_{K_{S}^{0}}$ are the fitted mean and width
of a signal Gaussian function fit to the signal MC samples. The value of
$m_{K_{S}^{0}}$ increases from 497.6 to 498.0 MeV/c2 for different CM
energies, while $\sigma_{K_{S}^{0}}$ varies from 1.3 to 2.4 MeV/$c^{2}$. The
reconstructed $K_{S}^{0}$ mass spectra $M(K_{S}^{0})$ are shown in Fig. 1.

A photon candidate, originating from $\pi^{0}$ decay, is identified using
showers in the EMC. The deposited energy of the shower must be more than 25
MeV in the barrel region ($|\\!\cos\theta|<0.8$), and more than 50 MeV in the
end cap region ($0.86<|\\!\cos\theta|<0.92$). To exclude a shower that
originates from a charged track, the angle subtended by the EMC shower and the
position of the closest charged track at the EMC must be greater than 10
degrees as measured from the IP. 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. For the case with four
charged tracks (two charged pions and two leptons), the number of photons
($N_{\gamma}$) is required to be not less than two.

To suppress potential background and improve resolution, a four-constraint
(4C) kinematic fit is performed in the case of the two-$K_{S}^{0}$
reconstruction method, while a one-constraint (1C) kinematic fit is used in
the one-$K_{S}^{0}$ reconstruction method. For the 4C fit, the the total
final-state four-momentum is constrained to the initial CM system. For the 1C
fit, the missing mass is constrained to the known $K_{S}^{0}$ mass. The
$\chi^{2}$ of the kinematic fit is required to be less than 200 for the 4C fit
and less than 20 for the 1C fit. Both requirements on the $\chi^{2}$ of the
kinematic fit have been optimized using the figure-of-merit (FOM)
$\frac{S}{\sqrt{S+B}}$, where $S$ is the signal yield that is estimated by the
signal MC samples and normalized by the previously measured cross sections,
and $B$ is the background yield that is estimated by the inclusive MC samples
and normalized according to the luminosities. The optimization is performed in
the signal region, which is defined as
$M(l^{+}l^{-})\in(m_{J/\psi}-3\sigma_{J/\psi},m_{J/\psi}+3\sigma_{J/\psi})$
for both the one-$K_{S}^{0}$ and two-$K_{S}^{0}$ reconstruction methods, where
$M(l^{+}l^{-})$ is the invariant mass of the lepton pair, and $m_{J/\psi}$ and
$\sigma_{J/\psi}$ are the mean and width of a signal Gaussian function fit to
the signal MC samples. The value of $m_{J/\psi}$ is around 3097.3 MeV/c2,
while $\sigma_{J/\psi}$ increases from 3.4 to 6.8 MeV/$c^{2}$ for different CM
energies. In addition, the sideband regions of $M(l^{+}l^{-})$ are defined as
($m_{J/\psi}-13\sigma_{J/\psi}$, $m_{J/\psi}-7\sigma_{J/\psi}$) and
($m_{J/\psi}+7\sigma_{J/\psi}$, $m_{J/\psi}+13\sigma_{J/\psi}$). Therefore,
the normalized ratio of sideband to signal regions is 0.50.

\begin{overpic}[width=173.44534pt]{invmll.pdf} \put(10.0,92.0){(a)}
\put(10.0,62.0){(b)} \put(10.0,32.0){(c)} \end{overpic} Figure 2: The
invariant mass of the lepton pair $M(l^{+}l^{-})$ for (a) the two-$K_{S}^{0}$
reconstruction method, (b) the one-$K_{S}^{0}$ reconstruction method, and (c)
both methods. Data from all CM energies are combined. The dots with error bars
are data, the green filled histograms are the background from inclusive MC
samples, and the red histograms are signal MC samples. The average signal
regions are shown by the red arrows, and the average sideband regions are
shown by the green arrows.

After imposing the above requirements, the $M(l^{+}l^{-})$ distributions for
all CM energies combined are shown in Fig. 2. Obvious $J/\psi$ signal peaks
can be seen. Figure 2(a) and 2(b) are for the two-$K_{S}^{0}$ and
one-$K_{S}^{0}$ reconstruction methods, respectively. The numbers of signal
events for all CM energies are estimated to be $107.4\pm 10.4$ for the
two-$K_{S}^{0}$ reconstruction method and $237.6\pm 16.9$ for the
one-$K_{S}^{0}$ reconstruction method, where the uncertainties are statistical
only. The total number of signal events is then taken as the sum of those from
the two reconstruction methods. According to studies based on all inclusive MC
samples, there is no significant peaking background.

##  IV Cross section

\begin{overpic}[width=173.44534pt]{bcsratio_kpkm_ksks.pdf} \put(15.0,91.0){(a)} \put(15.0,47.0){(b)} \end{overpic} Figure 3: (a) The Born cross sections and upper limits for $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$. (b) The ratio of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ to $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ Born cross sections. The green and black dots with error bars in (a) are the Born cross sections from the previous measurement of the BESIII experiment Ablikim  _et al._ (2018b) and this work, respectively, while the red triangles are the upper limits from this work. The blue dashed line in (b) is the value expected from isospin symmetry, and the black dots with error bars are from this study. The statistical and systematic uncertainties are included. Table 1: The CM energies ($\sqrt{s}$), integrated luminosities ($\mathcal{L}$), signal yields ($N_{\rm sig}$), upper limits of signal yields at the 90% C.L. ($N_{\rm ul}$), background yields ($N_{\rm bkg}$), event selection efficiencies ($\epsilon$), ISR correction factors ($(1+\delta)$), vacuum polarization factors ($\delta^{\rm VP}$), Born cross sections ($\sigma^{\rm Born}$), upper limits of cross section at the 90% C.L. for the energy pionts with statistical significance less than 3$\sigma$ ($\sigma^{\rm ul}$), and the Born cross section ratios ($\frac{\sigma^{\rm Born}(K_{S}^{0}K_{S}^{0}J/\psi)}{\sigma^{\rm Born}(K^{+}K^{-}J/\psi)}$). The first uncertainties of $\sigma^{\rm Born}$ are statistical, and the second ones systematic. The uncertainties of $N_{\rm sig}$ are only statistical. $\sqrt{s}$ (GeV) |  $\mathcal{L}$ (pb-1) | $N_{\rm sig}$ | $N_{\rm ul}$ | $N_{\rm bkg}$ |  $\epsilon$ (%) | $(1+\delta)$ | $\delta^{\rm VP}$ |  $\sigma^{\rm Born}$ (pb) |  $\sigma^{\rm ul}$ (pb) | $\frac{\sigma^{\rm Born}(K_{S}^{0}K_{S}^{0}J/\psi)}{\sigma^{\rm Born}(K^{+}K^{-}J/\psi)}$
---|---|---|---|---|---|---|---|---|---|---
4.128 | 401.5 | $0.0_{-0.7}^{+1.0}$ | 2.4 | 0.0 | 16.3 | 0.691 | 1.052 | $0.00_{-0.12}^{+0.18}\pm 0.01$ | 0.5 | –
4.157 | 408.7 | $2.0_{-1.1}^{+1.8}$ | 5.9 | 0.0 | 18.4 | 0.706 | 1.053 | $0.30_{-0.16}^{+0.27}\pm 0.03$ | 1.0 | $0.532_{-0.352}^{+0.538}\pm 0.022$
4.178 | 3194.5 | $15.0_{-3.8}^{+4.4}$ | – | 1.0 | 19.8 | 0.711 | 1.054 | $0.26_{-0.07}^{+0.08}\pm 0.03$ | – | $0.380_{-0.108}^{+0.123}\pm 0.016$
4.189 | 526.7 | $4.0_{-1.7}^{+2.3}$ | – | 0.0 | 20.2 | 0.712 | 1.056 | $0.42_{-0.18}^{+0.24}\pm 0.04$ | – | $0.431_{-0.207}^{+0.271}\pm 0.018$
4.199 | 526.0 | $1.5_{-1.5}^{+2.2}$ | 6.1 | 1.5 | 20.6 | 0.713 | 1.056 | $0.15_{-0.15}^{+0.23}\pm 0.01$ | 0.7 | $0.115_{-0.116}^{+0.170}\pm 0.005$
4.209 | 517.1 | $7.0_{-2.3}^{+3.0}$ | – | 0.0 | 20.4 | 0.716 | 1.057 | $0.74_{-0.24}^{+0.32}\pm 0.07$ | – | $0.431_{-0.160}^{+0.202}\pm 0.018$
4.219 | 514.6 | $19.0_{-4.0}^{+4.7}$ | – | 0.0 | 20.9 | 0.721 | 1.056 | $1.94_{-0.41}^{+0.48}\pm 0.18$ | – | $0.535_{-0.129}^{+0.148}\pm 0.022$
4.226 | 1100.9 | $39.0_{-6.0}^{+6.7}$ | – | 1.0 | 21.5 | 0.733 | 1.056 | $1.79_{-0.28}^{+0.31}\pm 0.17$ | – | $0.414_{-0.070}^{+0.078}\pm 0.017$
4.236 | 530.3 | $10.0_{-3.1}^{+3.7}$ | – | 1.0 | 21.3 | 0.751 | 1.056 | $0.94_{-0.29}^{+0.35}\pm 0.09$ | – | $0.296_{-0.099}^{+0.117}\pm 0.012$
4.244 | 538.1 | $19.0_{-4.2}^{+4.9}$ | – | 1.0 | 20.9 | 0.774 | 1.056 | $1.73_{-0.38}^{+0.45}\pm 0.16$ | – | $0.505_{-0.126}^{+0.145}\pm 0.021$
4.258 | 828.4 | $26.0_{-4.9}^{+5.6}$ | – | 1.0 | 20.4 | 0.821 | 1.054 | $1.49_{-0.28}^{+0.32}\pm 0.14$ | – | $0.496_{-0.105}^{+0.118}\pm 0.021$
4.267 | 531.1 | $14.0_{-3.4}^{+4.1}$ | – | 0.0 | 19.9 | 0.851 | 1.053 | $1.24_{-0.30}^{+0.36}\pm 0.12$ | – | $0.654_{-0.186}^{+0.220}\pm 0.027$
4.278 | 175.7 | $3.0_{-1.4}^{+2.1}$ | – | 0.0 | 19.0 | 0.885 | 1.053 | $0.81_{-0.38}^{+0.56}\pm 0.08$ | – | $0.376_{-0.198}^{+0.284}\pm 0.016$
4.288 | 502.4 | $7.0_{-2.3}^{+3.0}$ | – | 0.0 | 18.0 | 0.914 | 1.053 | $0.67_{-0.22}^{+0.29}\pm 0.06$ | – | $0.413_{-0.155}^{+0.195}\pm 0.017$
4.312 | 501.0 | $5.0_{-2.9}^{+3.4}$ | 12.0 | 3.0 | 17.2 | 0.968 | 1.052 | $0.48_{-0.28}^{+0.32}\pm 0.05$ | 1.2 | $0.442_{-0.276}^{+0.321}\pm 0.018$
4.337 | 505.0 | $4.5_{-2.0}^{+2.6}$ | – | 0.5 | 17.1 | 1.002 | 1.051 | $0.41_{-0.18}^{+0.24}\pm 0.04$ | – | $0.496_{-0.267}^{+0.330}\pm 0.021$
4.358 | 543.9 | $5.5_{-2.5}^{+3.1}$ | – | 1.5 | 18.0 | 1.011 | 1.051 | $0.44_{-0.20}^{+0.25}\pm 0.04$ | – | $0.403_{-0.203}^{+0.246}\pm 0.017$
4.377 | 522.7 | $5.0_{-2.3}^{+2.9}$ | – | 1.0 | 17.4 | 0.997 | 1.051 | $0.44_{-0.20}^{+0.26}\pm 0.04$ | – | $0.296_{-0.150}^{+0.184}\pm 0.012$
4.396 | 507.8 | $10.5_{-3.0}^{+3.7}$ | – | 0.5 | 18.3 | 0.966 | 1.051 | $0.93_{-0.27}^{+0.33}\pm 0.09$ | – | $1.278_{-0.609}^{+0.693}\pm 0.053$
4.416 | 1090.7 | $8.0_{-3.3}^{+3.8}$ | – | 3.0 | 19.9 | 0.915 | 1.052 | $0.32_{-0.13}^{+0.15}\pm 0.03$ | – | $0.262_{-0.115}^{+0.132}\pm 0.011$
4.436 | 569.9 | $12.5_{-3.5}^{+4.2}$ | – | 1.5 | 20.7 | 0.862 | 1.054 | $0.98_{-0.27}^{+0.33}\pm 0.09$ | – | $0.429_{-0.136}^{+0.160}\pm 0.018$
4.467 | 111.1 | $1.0_{-0.7}^{+1.4}$ | 4.0 | 0.0 | 22.6 | 0.816 | 1.055 | $0.39_{-0.27}^{+0.54}\pm 0.04$ | 1.7 | $0.129_{-0.097}^{+0.185}\pm 0.005$
4.527 | 112.1 | $3.0_{-1.4}^{+2.1}$ | – | 0.0 | 21.7 | 0.913 | 1.054 | $1.08_{-0.50}^{+0.75}\pm 0.10$ | – | $0.223_{-0.114}^{+0.164}\pm 0.009$
4.600 | 586.9 | $10.5_{-3.0}^{+3.7}$ | – | 0.5 | 19.4 | 1.031 | 1.055 | $0.71_{-0.20}^{+0.25}\pm 0.07$ | – | $0.364_{-0.119}^{+0.142}\pm 0.015$
4.612 | 103.8 | $1.0_{-0.7}^{+1.4}$ | 4.0 | 0.0 | 19.0 | 1.013 | 1.055 | $0.40_{-0.28}^{+0.56}\pm 0.04$ | 1.7 | –
4.628 | 521.5 | $1.5_{-2.0}^{+2.8}$ | 7.2 | 3.5 | 19.4 | 0.977 | 1.054 | $0.12_{-0.16}^{+0.22}\pm 0.01$ | 0.6 | –
4.641 | 552.4 | $10.0_{-3.8}^{+4.3}$ | – | 4.0 | 20.1 | 0.944 | 1.054 | $0.76_{-0.29}^{+0.33}\pm 0.07$ | – | –
4.661 | 529.6 | $9.5_{-3.6}^{+4.1}$ | – | 3.5 | 21.3 | 0.897 | 1.054 | $0.75_{-0.28}^{+0.32}\pm 0.07$ | – | –
4.682 | 1669.3 | $37.5_{-6.7}^{+7.3}$ | – | 7.5 | 22.0 | 0.864 | 1.054 | $0.94_{-0.17}^{+0.18}\pm 0.09$ | – | –
4.698 | 536.5 | $14.5_{-3.6}^{+4.2}$ | – | 0.5 | 22.7 | 0.858 | 1.055 | $1.10_{-0.27}^{+0.32}\pm 0.10$ | – | –
4.740 | 164.3 | $1.5_{-1.3}^{+1.8}$ | 5.4 | 0.5 | 23.6 | 0.880 | 1.055 | $0.35_{-0.30}^{+0.42}\pm 0.03$ | 1.4 | –
4.750 | 367.2 | $12.5_{-3.8}^{+4.3}$ | – | 2.5 | 23.5 | 0.892 | 1.055 | $1.29_{-0.39}^{+0.44}\pm 0.12$ | – | –
4.780 | 512.8 | $14.0_{-4.5}^{+5.1}$ | – | 6.0 | 22.6 | 0.927 | 1.055 | $1.03_{-0.33}^{+0.38}\pm 0.10$ | – | –
4.842 | 527.3 | $10.0_{-3.3}^{+3.9}$ | – | 2.0 | 21.5 | 1.000 | 1.056 | $0.70_{-0.23}^{+0.27}\pm 0.07$ | – | –
4.918 | 208.1 | $3.5_{-1.8}^{+2.4}$ | 8.6 | 0.5 | 20.3 | 1.063 | 1.056 | $0.62_{-0.32}^{+0.42}\pm 0.06$ | 1.7 | –
4.950 | 160.4 | $1.0_{-1.1}^{+1.9}$ | 4.9 | 1.0 | 19.8 | 1.083 | 1.056 | $0.23_{-0.25}^{+0.44}\pm 0.02$ | 1.2 | –

The Born cross section $\sigma^{\rm Born}$ at each CM energy is calculated
using:

$\sigma^{\rm Born}\equiv\frac{N_{\rm
sig}}{\mathcal{L}\epsilon\mathcal{B}_{J/\psi\to
l^{+}l^{-}}(1+\delta)\delta^{\rm VP}},$ (1)

where $N_{\rm sig}$ is the signal yield, calculated by subtracting the number
of $J/\psi$ sideband events from the number of $J/\psi$ signal events;
$\epsilon$ is the event selection efficiency obtained from phase-space modeled
signal MC simulations, which includes the $K_{S}^{0}$ decay branching
fractions; $\mathcal{B}_{J/\psi\to l^{+}l^{-}}$ is the PDG value of the
branching fraction of the $J/\psi$ decaying into a lepton pair Workman  _et
al._ (2022); $(1+\delta)$ is the ISR correction factor; and $\delta^{\rm VP}$
is the vacuum polarization factor taken from Ref. Actis  _et al._ (2010).
Statistical uncertainties on the numbers of signal events are calculated using
the Rolke method Rolke  _et al._ (2005). In the Rolke method Rolke  _et al._
(2005), the background is assumed to obey a Poisson distribution. The values
for $\epsilon$ and $(1+\delta)$ are estimated based on signal MC samples using
an iterative weighting method Sun  _et al._ (2021). In the iterations, we
describe the dressed cross sections $\sigma^{\rm dress}=\sigma^{\rm
Born}\delta^{\rm VP}$ by a coherent sum of three Breit-Wigner functions, as
described in the next paragraph. For energy points with statistical
significance less than 3$\sigma$, upper limits are calculated at a 90%
confidence level (C.L.) and include systematic uncertainties determined using
the Rolke method Rolke  _et al._ (2005) with an additional uncertainty on the
efficiency. The measured Born cross sections are shown in Fig. 3(a), and the
detailed quantities are shown in Table 1. The Born cross section ratios
$\frac{\sigma^{\rm Born}(K_{S}^{0}K_{S}^{0}J/\psi)}{\sigma^{\rm
Born}(K^{+}K^{-}J/\psi)}$ are shown in Fig. 3(b).

\begin{overpic}[width=173.44534pt]{fit_dress_cross_section_ksks_3bw_fix4500.pdf} \put(43.0,93.0){(a)} \put(43.0,70.0){(b)} \put(43.0,47.0){(c)} \put(43.0,24.0){(d)} \end{overpic} Figure 4: Maximum likelihood fits to the dressed cross sections of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$. The four solutions are shown separately in panels (a)-(d). The dots with error bars are the dressed cross sections of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$; the red solid curves are the fit results using a coherent sum of three Breit-Wigner functions; and the green, brown, and the cyan dashed curves show the contributions of the $Y(4230)$, $Y(4500)$, and $Y(4710)$ states, respectively. The mass and width of the $Y(4500)$ state are fixed. The statistical and uncorrelated systematic uncertainties are included. Table 2: The fitted parameters of the measured cross sections of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$, where the first uncertainties are statistical and the second are systematic. The $M_{1}$, $\Gamma_{1}$, and $(\Gamma_{ee}\mathcal{B})_{1}$ parameters are the mass, width, and $\Gamma_{ee}\mathcal{B}$ for the $Y(4230)$ state; the $M_{2}$, $\Gamma_{2}$, $(\Gamma_{ee}\mathcal{B})_{2}$, and $\phi_{2}$ parameters are for the $Y(4500)$ state; and the $M_{3}$, $\Gamma_{3}$, $(\Gamma_{ee}\mathcal{B})_{3}$, and $\phi_{3}$ parameters are for the $Y(4710)$ state. The mass and width of the $Y(4500)$ state are quoted from Ref. Ablikim  _et al._ (2022a). Parameter |  Solution I |  Solution II |  Solution III |  Solution IV
---|---|---|---|---
$M_{1}$ (MeV/$c^{2}$) | $4226.9\pm 6.6\pm 22.0$
$\Gamma_{1}$ (MeV) | $71.7\pm 16.2\pm 32.8$
$(\Gamma_{ee}\mathcal{B})_{1}$ (eV) |  $0.13\pm 0.02\pm 0.05$ |  $0.14\pm 0.03\pm 0.06$ |  $0.18\pm 0.05\pm 0.07$ |  $0.20\pm 0.04\pm 0.07$
$M_{2}$ (MeV/$c^{2}$) (fixed) |  $4484.7\pm 13.3\pm 24.1$ [Ref. Ablikim  _et al._ (2022a)]
$\Gamma_{2}$ (MeV) (fixed) |  $111.1\pm 30.1\pm 15.2$ [Ref. Ablikim  _et al._ (2022a)]
$(\Gamma_{ee}\mathcal{B})_{2}$ (eV) |  $0.08\pm 0.09\pm 0.04$ |  $0.17\pm 0.14\pm 0.05$ |  $0.31\pm 0.26\pm 0.11$ |  $0.68\pm 0.24\pm 0.18$
$\phi_{2}$ (rad) |  $1.02\pm 0.57\pm 0.56$ |  $1.74\pm 1.11\pm 0.46$ |  $4.26\pm 0.76\pm 0.91$ |  $4.98\pm 0.31\pm 0.74$
$M_{3}$ (MeV/$c^{2}$) | $4704.0\pm 52.3\pm 69.5$
$\Gamma_{3}$ (MeV) | $183.2\pm 114.0\pm 96.1$
$(\Gamma_{ee}\mathcal{B})_{3}$ (eV) |  $0.12\pm 0.09\pm 0.11$ |  $0.68\pm 0.26\pm 0.21$ |  $0.18\pm 0.20\pm 0.10$ |  $1.04\pm 0.60\pm 0.35$
$\phi_{3}$ (rad) |  $0.92\pm 0.99\pm 0.84$ |  $5.37\pm 0.46\pm 0.95$ |  $5.38\pm 1.02\pm 0.80$ |  $3.55\pm 0.27\pm 1.03$

A maximum likelihood method is used to fit the dressed cross sections and
determine the parameters of the resonant structures. Assuming
$K_{S}^{0}K_{S}^{0}J/\psi$ is produced from three resonances, the cross
section is parameterized as a coherent sum of three relativistic Breit-Wigner
functions

$\sigma^{\rm
dress}\equiv|BW_{1}+BW_{2}e^{i\phi_{2}}+BW_{3}e^{i\phi_{3}}|^{2},$ (2)

where
$BW_{j}\equiv\frac{M_{j}}{\sqrt{s}}\frac{\sqrt{12\pi(\Gamma_{ee}\mathcal{B})_{j}\Gamma_{j}}}{s-M_{j}^{2}+iM_{j}\Gamma_{j}}\sqrt{\frac{\Phi(\sqrt{s})}{\Phi(M_{j})}}$
is the relativistic Breit-Wigner function with $j=1$, $2$, or $3$, and
$\Phi(\sqrt{s})$ is the three-body phase-space factor. The mass $M_{2}$ and
total width $\Gamma_{2}$ are fixed to those of the $Y(4500)$ Ablikim  _et al._
(2022a), while the other parameters, $i.e.$, the masses $M_{j}$, the total
widths $\Gamma_{j}$, the products of the electronic partial width and the
branching fraction to $K_{S}^{0}K_{S}^{0}J/\psi$
$(\Gamma_{ee}\mathcal{B})_{j}$, and the relative phase $\phi_{j}$ between the
three Breit-Wigner functions, are free. In the fit, only the statistical and
uncorrelated systematic uncertainties are taken into consideration. There are
four solutions for the parameters $(\Gamma_{ee}\mathcal{B})_{j}$ and
$\phi_{j}$ as shown in Table 2, and all the fit results are shown in Fig. 4.

Fitting the dressed cross sections with only two resonances ($Y(4230)$ and
$Y(4710)$) yields a worse result, and the change of the likelihood value is
$|\\!\Delta(-2\ln L)|=5.0$ compared to the three-resonance hypothesis. Taking
the change in the number of degrees of freedom (2) into account, the
statistical significance for the assumption of three resonant structures over
the assumption of two resonant structures is 1.4$\sigma$, which indicates the
$Y(4500)$ has little influence on the fits. To estimate the statistical
significance of the third resonant structure, we fit the dressed cross section
with the coherent sum of $BW_{1}$ and $BW_{2}$. This fit gives a worse result
as well, and the change of the likelihood value is $|\\!\Delta(-2\ln
L)|=26.3$. Taking the change in the number of degrees of freedom (4) into
account, the statistical significance of the third resonant structure is
4.2$\sigma$. The significance of the third resonant structure becomes
4.0$\sigma$ after taking into consideration the systematic uncertainties of
the dressed cross sections.

##  V Systematic uncertainty

Systematic uncertainties in the cross section measurement mainly come from the
luminosity, the branching fraction of $J/\psi\to e^{+}e^{-}/\mu^{+}\mu^{-}$,
the lepton pair mass resolution, the ISR correction factor, and the detection
efficiency. Systematic uncertainty from the luminosity is 0.6% based on
studies of Bhabha events Ablikim  _et al._ (2015b, 2022b, 2022c). Systematic
uncertainties from the subsequent branching fractions are quoted from the PDG
Workman  _et al._ (2022). Systematic uncertainty from the $J/\psi$ mass window
is caused by the $M(l^{+}l^{-})$ resolution differences of data and MC
simulations. To account for the differences in $J/\psi$ mass resolution, we
smear the width of the $J/\psi$ peak in the signal MC samples, and the changes
in the event selection efficiencies are less than 1.0%, which is assigned as
the systematic uncertainty due to the $J/\psi$ mass window.

To estimate the systematic uncertainty from the ISR correction factor, we
compare the dressed cross sections between the last two ISR correction
iterations. Due to the convergence of the iterations, the maximum difference
of the dressed cross sections is negligible. On the other hand, in the ISR
correction iterations, we also replace the description of the default dressed
cross section line shape with the coherent sum of three Breit-Wigner functions
and a phase-space function, or the coherent sum of two Breit-Wigner functions.
In addition, we free all parameters in the coherent sum of three Breit-Wigner
functions, and describe the background by a second order polynomial function.
The maximum difference due to the line shape is 2.7%, which is assigned as the
systematic uncertainty due to the ISR correction factor.

Sources of systematic uncertainty from the detection efficiency include
systematic uncertainties in $K_{S}^{0}$ reconstruction (2.0% per $K_{S}^{0}$)
Ablikim  _et al._ (2018b), tracking efficiency (1.0% per track), and photon
reconstruction (1.0% per photon) Ablikim  _et al._ (2015c). In addition, to
account for the systematic uncertainty from the requirement on the number of
photons in the four charged tracks reconstruction case, we compare the ratio
of the number of events from MC simulations to data for different $N_{\gamma}$
requirements and assign the maximum difference of 1.1% as the systematic
uncertainty. The uncertainties from the two reconstruction methods are summed
according to error propagation. The uncertainty due to the kinematic fit is
estimated by correcting the helix parameters of the simulated charged tracks
to match the resolution found in data, and changing the requirement on the
$\chi^{2}$ from the kinematic fit. To reduce the influence of data sample
sizes, we add all the data together to estimate the differences of the ratios
of signal number of events to weighted efficiency when changing the $\chi^{2}$
requirement, and the maximum difference (3.7%) is assigned as the systematic
uncertainty due to the kinematic fit. The systematic uncertainty due to the MC
simulation model is assigned as the maximum difference between the nominal and
data-weighted efficiency. Using the control sample $e^{+}e^{-}\to
K^{+}K^{-}J/\psi$, we calculate the nominal efficiency directly based on the
simulated exclusive MC samples. To estimate the data-weighted efficiency, we
divide the invariant mass of the $K^{+}K^{-}$ or $KJ/\psi$ system into 10
intervals, and define the data-weighted efficiency as $\sum N_{i}^{\rm
data}/\sum(N_{i}^{\rm data}w_{i})$, where $w_{i}$ is $\epsilon_{i}^{-1}$, and
$N_{i}^{\rm data}$ and $\epsilon_{i}$ are the number of signal events from
data and the efficiency from signal MC in the $i$-th interval, respectively.
We find a maximum efficiency difference of 6.8%.

Table 3: The relative systematic uncertainties for the cross sections of the process $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$. Source | Systematic uncertainty (%)
---|---
Luminosity | 0.6
Branching fraction | 0.4
$J/\psi$ mass window | 1.0
$(1+\delta)$ | 2.7
$K_{S}^{0}$ reconstruction | 3.6
Tracking | 2.0
Photon reconstruction | 2.0
Kinematic fit | 3.7
MC model | 6.8
Sum | 9.5

Assuming all sources of uncertainty are independent, the total systematic
uncertainty in the $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ cross section
measurement is determined to be the quadratic sum of the common systematic
uncertainties. The relative systematic uncertainties and their sum are shown
in Table 3.

Table 4: Systematic uncertainties in the measurement of resonance parameters, including those due to the CM energy ($\sqrt{s}$), the form of the fit function (FF), the parameterization of the fit function ($\Gamma_{\rm tot}$), the CM energy spread (ES), and the uncorrelated ($\sigma^{\rm dress}_{1}$) and correlated ($\sigma^{\rm dress}_{2}$) systematic uncertainties from the cross section measurements. The symbol “–” represents negligible uncertainties.  Parameter | $\sqrt{s}$ | FF | $\Gamma_{\rm tot}$ | ES | $\sigma^{\rm dress}_{1}$ | $\sigma^{\rm dress}_{2}$ | Sum
---|---|---|---|---|---|---|---
$M_{1}$ (MeV/$c^{2}$) | 0.9 | 2.2 | 21.9 | 0.1 | 0.1 | – | 22.0
$M_{3}$ (MeV/$c^{2}$) | 1.2 | 5.3 | 69.3 | 0.3 | 0.2 | – | 69.5
$\Gamma_{1}$ (MeV) | 0.5 | 9.5 | 31.4 | 0.2 | 0.7 | – | 32.8
$\Gamma_{3}$ (MeV) | 0.8 | 33.9 | 89.9 | 0.4 | 0.2 | – | 96.1
$(\Gamma_{ee}\mathcal{B})_{1}$ (eV) | – | 0.02 | 0.05 | – | – | 0.01 | 0.05
– | 0.02 | 0.06 | – | – | 0.01 | 0.06
– | 0.03 | 0.06 | – | – | 0.01 | 0.07
– | 0.03 | 0.06 | – | – | 0.01 | 0.07
$(\Gamma_{ee}\mathcal{B})_{2}$ (eV) | – | 0.02 | 0.03 | – | – | – | 0.04
– | 0.03 | 0.04 | – | – | 0.01 | 0.05
– | 0.07 | 0.08 | – | – | 0.02 | 0.11
– | 0.13 | 0.12 | – | – | 0.04 | 0.18
$(\Gamma_{ee}\mathcal{B})_{3}$ (eV) | – | 0.03 | 0.11 | – | – | 0.01 | 0.11
– | 0.21 | 0.01 | – | – | 0.04 | 0.21
– | 0.06 | 0.08 | – | – | 0.01 | 0.10
– | 0.34 | 0.06 | – | – | 0.06 | 0.35
$\phi_{2}$ (rad) | – | 0.34 | 0.44 | 0.01 | 0.01 | 0.06 | 0.56
– | 0.42 | 0.15 | 0.01 | 0.01 | 0.10 | 0.46
– | 0.40 | 0.78 | – | – | 0.26 | 0.91
– | 0.34 | 0.58 | – | – | 0.30 | 0.74
$\phi_{3}$ (rad) | – | 0.84 | 0.05 | – | – | 0.06 | 0.84
– | 0.65 | 0.61 | 0.01 | 0.01 | 0.32 | 0.95
– | 0.70 | 0.21 | 0.01 | 0.01 | 0.32 | 0.80
– | 0.73 | 0.70 | – | – | 0.21 | 1.03

The systematic uncertainties in the resonance parameters mainly come from the
absolute CM energy measurement, the form and parameterization of the fit
function, the CM energy spread, and the systematic uncertainty on the cross
section measurement. The absolute CM energy has been measured Ablikim  _et
al._ (2015b, 2021b, 2022b, 2022c), and the associated systematic uncertainty
is estimated by varying the absolute CM energies in the fits. The uncertainty
from the form of the fit function is estimated by replacing the nominal
function with the coherent sum of two Breit-Wigner functions, or the coherent
sum of three Breit-Wigner functions and a phase-space function. To estimate
the uncertainty from the form of the Breit-Wigner function, the $\Gamma_{j}$
in the denominator of the Breit-Wigner function is replaced with a mass-
dependent width $\Gamma_{j}\frac{\Phi(\sqrt{s})}{\Phi(M_{j})}$. The
uncertainty from the CM energy spread is estimated by convolving the fit
formula with a Gaussian function, whose width is set as the mass-dependent
beam spread Abakumova  _et al._ (2011). The uncertainty from the cross section
measurement is divided into two parts. The first one is uncorrelated
uncertainties of the cross sections among the different CM energy points, and
comes mainly from the fit to the $M(l^{+}l^{-})$ spectrum to determine the
signal yields. The corresponding uncertainty is estimated by including the
uncorrelated uncertainties in the dressed cross section fits, and the
differences on the parameters are taken as the corresponding uncertainties.
The second part, including all other uncertainties of the cross sections, is
common for all data points (6.0%), and only affects the parameter
$(\Gamma_{ee}\mathcal{B})$. The systematic uncertainties in the resonance
parameters are shown in Table 4.

##  VI Summary

In summary, we measure the Born cross sections of the process $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ at CM energies from 4.128 to 4.950 GeV. The measured
Born cross sections are consistent with the previous measurements of the
BESIII experiment Ablikim  _et al._ (2018b), as shown in Fig. 3.

A clear resonant structure for the $Y(4230)$ is observed via $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ for the first time, and the mass and width of the
$Y(4230)$ are determined to be $M(Y(4230))=4226.9\pm 6.6\pm 22.0$ MeV/$c^{2}$
and $\Gamma(Y(4230))=71.7\pm 16.2\pm 32.8$ MeV, where the first uncertainties
are statistical and the second are systematic. In addition, we see another
enhanced structure with a statistical significance 4.2$\sigma$, called the
$Y(4710)$. The mass and width of the $Y(4710)$ are determined to be
$M(Y(4710))=4704.0\pm 52.3\pm 69.5$ MeV/$c^{2}$ and $\Gamma(Y(4710))=183.2\pm
114.0\pm 96.1$ MeV, respectively. If this structure is the $\psi(5S)$, the
measured mass will be in favor of the linear potential model predictions Gui
_et al._ (2018). It is also in agreement with the results based on the BaBar
experiment van Beveren and Rupp (2011, 2009); van Beveren  _et al._ (2009).

The average Born cross section ratio $\frac{\sigma^{\rm
Born}(K_{S}^{0}K_{S}^{0}J/\psi)}{\sigma^{\rm Born}(K^{+}K^{-}J/\psi)}$ is
determined to be $0.388_{-0.028}^{+0.035}\pm 0.016$ based on an asymmetric
Gaussian fit of combined ratio likelihood simulations, where the statistical
uncertainty is obtained by the fit, and the common items of the systematic
uncertainties have been canceled. The P-value of the ratio likelihood
distribution greater than 0.5 is 0.0011 which indicates a $3.1\sigma$
significance isospin violation effect in $e^{+}e^{-}\to K\bar{K}J/\psi$. When
taking the three-body phase space into consideration, the average Born cross
section ratio becomes $0.426_{-0.031}^{+0.038}\pm 0.018$, with a P-value of
0.0304, which indicates an isospin violation effect in $e^{+}e^{-}\to
K\bar{K}J/\psi$ with $1.9\sigma$ significance.

Using the measured products of the electronic partial width and the decay
branching fractions $\Gamma(e^{+}e^{-}\to Y(4230))\mathcal{B}(Y(4230)\to
K_{S}^{0}K_{S}^{0}J/\psi)$ (Table 2) and $\Gamma(e^{+}e^{-}\to
Y(4230))\mathcal{B}(Y(4230)\to K^{+}K^{-}J/\psi)$ Ablikim  _et al._ (2022a),
the branching fraction ratio of $Y(4230)\to K_{S}^{0}K_{S}^{0}J/\psi$ to
$Y(4230)\to K^{+}K^{-}J/\psi$ is determined to be
$0.3<\frac{\mathcal{B}(Y(4230)\to
K_{S}^{0}K_{S}^{0}J/\psi)}{\mathcal{B}(Y(4230)\to K^{+}K^{-}J/\psi)}<0.7$,
where the large range of values is due to multiple fit solutions. It is hard
to make a clear conclusion about the consistency of the experimental
measurements with the theoretically expected value of $0.5$ assuming the
events are all from an isospin zero initial state.

We do not see a significant $Y(4500)$ contribution in the measured cross
sections of $e^{+}e^{-}\to K_{S}^{0}K_{S}^{0}J/\psi$ due to a lack of data
samples around 4.500 GeV. The maximum likelihood fit gives a statistical
significance of the $Y(4500)$ of less than 1.4$\sigma$. Larger data samples
are required to draw clear conclusions about the existence of the $Y(4500)$
and $Y(4710)$ states.

##  VII Acknowledgements

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012,
12022510, 12025502, 12035009, 12035013, 12192260, 12192261, 12192262,
12192263, 12192264, 12192265; the Chinese Academy of Sciences (CAS) Large-
Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds
of the NSFC and CAS under Contract No. U1832207; 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, Collaborative Research Center
CRC 1044, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of
Development of Turkey under Contract No. DPT2006K-120470; National Science and
Technology fund; National Science Research and Innovation Fund (NSRF) via the
Program Management Unit for Human Resources & Institutional Development,
Research and Innovation under Contract No. B16F640076; STFC (United Kingdom);
Suranaree University of Technology (SUT), Thailand Science Research and
Innovation (TSRI), and National Science Research and Innovation Fund (NSRF)
under Contract No. 160355; The Royal Society, UK under Contracts Nos.
DH140054, DH160214; The Swedish Research Council; U. S. Department of Energy
under Contract No. DE-FG02-05ER41374.

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