# Observation of the $Y(4230)$ and a new structure in $e^{+}e^{-}\rightarrow
K^{+}K^{-}J/\psi$

M. Ablikim1, M. N. Achasov10,b, P. Adlarson68, M. Albrecht4, R. Aliberti28, A.
Amoroso67A,67C, M. R. An32, Q. An64,50, X. H. Bai58, Y. Bai49, O. Bakina29, R.
Baldini Ferroli23A, I. Balossino24A, Y. Ban39,h, V. Batozskaya1,37, D.
Becker28, K. Begzsuren26, N. Berger28, M. Bertani23A, D. Bettoni24A, F.
Bianchi67A,67C, J. Bloms61, A. Bortone67A,67C, I. Boyko29, R. A. Briere5, A.
Brueggemann61, H. Cai69, X. Cai1,50, A. Calcaterra23A, G. F. Cao1,55, N.
Cao1,55, S. A. Cetin54A, J. F. Chang1,50, W. L. Chang1,55, G. Chelkov29,a, C.
Chen36, G. Chen1, H. S. Chen1,55, M. L. Chen1,50, S. J. Chen35, T. Chen1, X.
R. Chen25,55, X. T. Chen1, Y. B. Chen1,50, Z. J. Chen20,i, W. S. Cheng67C, X.
Chu36, G. Cibinetto24A, F. Cossio67C, J. J. Cui42, H. L. Dai1,50, J. P. Dai71,
A. Dbeyssi14, R. E. de Boer4, D. Dedovich29, Z. Y. Deng1, A. Denig28, I.
Denysenko29, M. Destefanis67A,67C, F. De Mori67A,67C, Y. Ding33, J. Dong1,50,
L. Y. Dong1,55, M. Y. Dong1,50,55, X. Dong69, S. X. Du73, P. Egorov29,a, Y. L.
Fan69, J. Fang1,50, S. S. Fang1,55, W. X. Fang1, Y. Fang1, R. Farinelli24A, L.
Fava67B,67C, F. Feldbauer4, G. Felici23A, C. Q. Feng64,50, J. H. Feng51, K
Fischer62, M. Fritsch4, C. Fritzsch61, C. D. Fu1, H. Gao55, Y. N. Gao39,h,
Yang Gao64,50, S. Garbolino67C, I. Garzia24A,24B, P. T. Ge69, C. Geng51, E. M.
Gersabeck59, A Gilman62, K. Goetzen11, L. Gong33, W. X. Gong1,50, W. Gradl28,
M. Greco67A,67C, M. H. Gu1,50, C. Y Guan1,55, A. Q. Guo25,55, L. B. Guo34, R.
P. Guo41, Y. P. Guo9,g, A. Guskov29,a, T. T. Han42, W. Y. Han32, X. Q. Hao15,
F. A. Harris57, K. K. He47, K. L. He1,55, F. H. Heinsius4, C. H. Heinz28, Y.
K. Heng1,50,55, C. Herold52, M. Himmelreich11,e, T. Holtmann4, G. Y. Hou1,55,
Y. R. Hou55, Z. L. Hou1, H. M. Hu1,55, J. F. Hu48,j, T. Hu1,50,55, Y. Hu1, G.
S. Huang64,50, K. X. Huang51, L. Q. Huang65, L. Q. Huang25,55, X. T. Huang42,
Y. P. Huang1, Z. Huang39,h, T. Hussain66, N Hüsken22,28, W. Imoehl22, M.
Irshad64,50, J. Jackson22, S. Jaeger4, S. Janchiv26, Q. Ji1, Q. P. Ji15, X. B.
Ji1,55, X. L. Ji1,50, Y. Y. Ji42, Z. K. Jia64,50, H. B. Jiang42, S. S.
Jiang32, X. S. Jiang1,50,55, Y. Jiang55, J. B. Jiao42, Z. Jiao18, S. Jin35, Y.
Jin58, M. Q. Jing1,55, T. Johansson68, N. Kalantar-Nayestanaki56, X. S.
Kang33, R. Kappert56, M. Kavatsyuk56, B. C. Ke73, I. K. Keshk4, A. Khoukaz61,
P. Kiese28, R. Kiuchi1, R. Kliemt11, L. Koch30, O. B. Kolcu54A, B. Kopf4, M.
Kuemmel4, M. Kuessner4, A. Kupsc37,68, W. Kühn30, J. J. Lane59, J. S. Lange30,
P. Larin14, A. Lavania21, L. Lavezzi67A,67C, Z. H. Lei64,50, H. Leithoff28, M.
Lellmann28, T. Lenz28, C. Li36, C. Li40, C. H. Li32, Cheng Li64,50, D. M.
Li73, F. Li1,50, G. Li1, H. Li44, H. Li64,50, H. B. Li1,55, H. J. Li15, H. N.
Li48,j, J. Q. Li4, J. S. Li51, J. W. Li42, Ke Li1, L. J Li1, L. K. Li1, Lei
Li3, M. H. Li36, P. R. Li31,k,l, S. X. Li9, S. Y. Li53, T. Li42, W. D. Li1,55,
W. G. Li1, X. H. Li64,50, X. L. Li42, Xiaoyu Li1,55, Z. Y. Li51, H. Liang1,55,
H. Liang64,50, H. Liang27, Y. F. Liang46, Y. T. Liang25,55, G. R. Liao12, L.
Z. Liao42, J. Libby21, A. Limphirat52, C. X. Lin51, D. X. Lin25,55, T. Lin1,
B. J. Liu1, C. X. Liu1, D. Liu14,64, F. H. Liu45, Fang Liu1, Feng Liu6, G. M.
Liu48,j, H. Liu31,k,l, H. M. Liu1,55, Huanhuan Liu1, Huihui Liu16, J. B.
Liu64,50, J. L. Liu65, J. Y. Liu1,55, K. Liu1, K. Y. Liu33, Ke Liu17, L.
Liu64,50, M. H. Liu9,g, P. L. Liu1, Q. Liu55, S. B. Liu64,50, T. Liu9,g, W. K.
Liu36, W. M. Liu64,50, X. Liu31,k,l, Y. Liu31,k,l, Y. B. Liu36, Z. A.
Liu1,50,55, Z. Q. Liu42, X. C. Lou1,50,55, F. X. Lu51, H. J. Lu18, J. G.
Lu1,50, X. L. Lu1, Y. Lu1, Y. P. Lu1,50, Z. H. Lu1, C. L. Luo34, M. X. Luo72,
T. Luo9,g, X. L. Luo1,50, X. R. Lyu55, Y. F. Lyu36, F. C. Ma33, H. L. Ma1, L.
L. Ma42, M. M. Ma1,55, Q. M. Ma1, R. Q. Ma1,55, R. T. Ma55, X. Y. Ma1,50, Y.
Ma39,h, F. E. Maas14, M. Maggiora67A,67C, S. Maldaner4, S. Malde62, Q. A.
Malik66, A. Mangoni23B, Y. J. Mao39,h, Z. P. Mao1, S. Marcello67A,67C, Z. X.
Meng58, J. G. Messchendorp56,d, G. Mezzadri24A, H. Miao1, T. J. Min35, R. E.
Mitchell22, X. H. Mo1,50,55, N. Yu. Muchnoi10,b, H. Muramatsu60, Y. Nefedov29,
F. Nerling11,e, I. B. Nikolaev10,b, Z. Ning1,50, S. Nisar8,m, Y. Niu 42, S. L.
Olsen55, Q. Ouyang1,50,55, S. Pacetti23B,23C, X. Pan9,g, Y. Pan59, A. Pathak1,
A. Pathak27, M. Pelizaeus4, H. P. Peng64,50, K. Peters11,e, J. Pettersson68,
J. L. Ping34, R. G. Ping1,55, S. Plura28, S. Pogodin29, R. Poling60, V.
Prasad64,50, F. Z. Qi1, H. Qi64,50, H. R. Qi53, M. Qi35, T. Y. Qi9,g, S.
Qian1,50, W. B. Qian55, Z. Qian51, C. F. Qiao55, J. J. Qin65, L. Q. Qin12, X.
P. Qin9,g, X. S. Qin42, Z. H. Qin1,50, J. F. Qiu1, S. Q. Qu53, S. Q. Qu36, K.
H. Rashid66, C. F. Redmer28, K. J. Ren32, A. Rivetti67C, V. Rodin56, M.
Rolo67C, G. Rong1,55, Ch. Rosner14, S. N. Ruan36, H. S. Sang64, A.
Sarantsev29,c, Y. Schelhaas28, C. Schnier4, K. Schoenning68, M.
Scodeggio24A,24B, K. Y. Shan9,g, W. Shan19, X. Y. Shan64,50, J. F.
Shangguan47, L. G. Shao1,55, M. Shao64,50, C. P. Shen9,g, H. F. Shen1,55, X.
Y. Shen1,55, B.-A. Shi55, H. C. Shi64,50, J. Y. Shi1, R. S. Shi1,55, X.
Shi1,50, X. D Shi64,50, J. J. Song15, W. M. Song27,1, Y. X. Song39,h, S.
Sosio67A,67C, S. Spataro67A,67C, F. Stieler28, K. X. Su69, P. P. Su47, Y.-J.
Su55, G. X. Sun1, H. Sun55, H. K. Sun1, J. F. Sun15, L. Sun69, S. S. Sun1,55,
T. Sun1,55, W. Y. Sun27, X Sun20,i, Y. J. Sun64,50, Y. Z. Sun1, Z. T. Sun42,
Y. H. Tan69, Y. X. Tan64,50, C. J. Tang46, G. Y. Tang1, J. Tang51, L. Y Tao65,
Q. T. Tao20,i, J. X. Teng64,50, V. Thoren68, W. H. Tian44, Y. Tian25,55, I.
Uman54B, B. Wang1, B. L. Wang55, D. Y. Wang39,h, F. Wang65, H. J. Wang31,k,l,
H. P. Wang1,55, K. Wang1,50, L. L. Wang1, M. Wang42, M. Z. Wang39,h, Meng
Wang1,55, S. Wang9,g, T. Wang9,g, T. J. Wang36, W. Wang51, W. H. Wang69, W. P.
Wang64,50, X. Wang39,h, X. F. Wang31,k,l, X. L. Wang9,g, Y. D. Wang38, Y. F.
Wang1,50,55, Y. H. Wang40, Y. Q. Wang1, Ying Wang51, Z. Wang1,50, Z. Y.
Wang1,55, Ziyi Wang55, D. H. Wei12, F. Weidner61, S. P. Wen1, D. J. White59,
U. Wiedner4, G. Wilkinson62, M. Wolke68, L. Wollenberg4, J. F. Wu1,55, L. H.
Wu1, L. J. Wu1,55, X. Wu9,g, X. H. Wu27, Y. Wu64, Z. Wu1,50, L. Xia64,50, T.
Xiang39,h, D. Xiao31,k,l, H. Xiao9,g, S. Y. Xiao1, Y. L. Xiao9,g, Z. J.
Xiao34, X. H. Xie39,h, Y. Xie42, Y. G. Xie1,50, Y. H. Xie6, Z. P. Xie64,50, T.
Y. Xing1,55, C. F. Xu1, C. J. Xu51, G. F. Xu1, Q. J. Xu13, S. Y. Xu63, X. P.
Xu47, Y. C. Xu55, F. Yan9,g, L. Yan9,g, W. B. Yan64,50, W. C. Yan73, H. J.
Yang43,f, H. L. Yang27, H. X. Yang1, L. Yang44, S. L. Yang55, Tao Yang1, Y. X.
Yang1,55, Yifan Yang1,55, M. Ye1,50, M. H. Ye7, J. H. Yin1, Z. Y. You51, B. X.
Yu1,50,55, C. X. Yu36, G. Yu1,55, T. Yu65, C. Z. Yuan1,55, L. Yuan2, S. C.
Yuan1, X. Q. Yuan1, Y. Yuan1,55, Z. Y. Yuan51, C. X. Yue32, A. A. Zafar66, F.
R. Zeng42, X. Zeng6, Y. Zeng20,i, Y. H. Zhan51, A. Q. Zhang1, B. L. Zhang1, B.
X. Zhang1, D. H. Zhang36, G. Y. Zhang15, H. Zhang64, H. H. Zhang51, H. H.
Zhang27, H. Y. Zhang1,50, J. L. Zhang70, J. Q. Zhang34, J. W. Zhang1,50,55, J.
X. Zhang31,k,l, J. Y. Zhang1, J. Z. Zhang1,55, Jianyu Zhang1,55, Jiawei
Zhang1,55, L. M. Zhang53, L. Q. Zhang51, Lei Zhang35, P. Zhang1, Q. Y.
Zhang32,73, Shulei Zhang20,i, X. D. Zhang38, X. M. Zhang1, X. Y. Zhang42, X.
Y. Zhang47, Y. Zhang62, Y. T. Zhang73, Y. H. Zhang1,50, Yan Zhang64,50, Yao
Zhang1, Z. H. Zhang1, Z. Y. Zhang36, Z. Y. Zhang69, G. Zhao1, J. Zhao32, J. Y.
Zhao1,55, J. Z. Zhao1,50, Lei Zhao64,50, Ling Zhao1, M. G. Zhao36, Q. Zhao1,
S. J. Zhao73, Y. B. Zhao1,50, Y. X. Zhao25,55, Z. G. Zhao64,50, A.
Zhemchugov29,a, B. Zheng65, J. P. Zheng1,50, Y. H. Zheng55, B. Zhong34, C.
Zhong65, X. Zhong51, H. Zhou42, L. P. Zhou1,55, X. Zhou69, X. K. Zhou55, X. R.
Zhou64,50, X. Y. Zhou32, Y. Z. Zhou9,g, J. Zhu36, K. Zhu1, K. J. Zhu1,50,55,
L. X. Zhu55, S. H. Zhu63, T. J. Zhu70, W. J. Zhu9,g, Y. C. Zhu64,50, Z. A.
Zhu1,55, B. S. Zou1, J. H. Zou1 (BESIII Collaboration) 1 Institute of High
Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9 Fudan University, Shanghai 200433, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
14 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
15 Henan Normal University, Xinxiang 453007, People’s Republic of China
16 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
17 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 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
24 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
25 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27 Jilin University, Changchun 130012, People’s Republic of China
28 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31 Lanzhou University, Lanzhou 730000, People’s Republic of China
32 Liaoning Normal University, Dalian 116029, People’s Republic of China
33 Liaoning University, Shenyang 110036, People’s Republic of China
34 Nanjing Normal University, Nanjing 210023, People’s Republic of China
35 Nanjing University, Nanjing 210093, People’s Republic of China
36 Nankai University, Tianjin 300071, People’s Republic of China
37 National Centre for Nuclear Research, Warsaw 02-093, Poland
38 North China Electric Power University, Beijing 102206, People’s Republic of
China
39 Peking University, Beijing 100871, People’s Republic of China
40 Qufu Normal University, Qufu 273165, People’s Republic of China
41 Shandong Normal University, Jinan 250014, People’s Republic of China
42 Shandong University, Jinan 250100, People’s Republic of China
43 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
44 Shanxi Normal University, Linfen 041004, People’s Republic of China
45 Shanxi University, Taiyuan 030006, People’s Republic of China
46 Sichuan University, Chengdu 610064, People’s Republic of China
47 Soochow University, Suzhou 215006, People’s Republic of China
48 South China Normal University, Guangzhou 510006, People’s Republic of China
49 Southeast University, Nanjing 211100, People’s Republic of China
50 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
51 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
52 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
53 Tsinghua University, Beijing 100084, People’s Republic of China
54 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
55 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
56 University of Groningen, NL-9747 AA Groningen, The Netherlands
57 University of Hawaii, Honolulu, Hawaii 96822, USA
58 University of Jinan, Jinan 250022, People’s Republic of China
59 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
60 University of Minnesota, Minneapolis, Minnesota 55455, USA
61 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
62 University of Oxford, Keble Rd, Oxford, UK OX13RH
63 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
64 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
65 University of South China, Hengyang 421001, People’s Republic of China
66 University of the Punjab, Lahore-54590, Pakistan
67 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
68 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
69 Wuhan University, Wuhan 430072, People’s Republic of China
70 Xinyang Normal University, Xinyang 464000, People’s Republic of China
71 Yunnan University, Kunming 650500, People’s Republic of China
72 Zhejiang University, Hangzhou 310027, People’s Republic of China
73 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 Currently at Istanbul Arel University, 34295 Istanbul, Turkey
e Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
f 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
g 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
h Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
i Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
k Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
m Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan

###### Abstract

The cross sections of $e^{+}e^{-}\rightarrow K^{+}K^{-}J/\psi$ at center-of-
mass energies from 4.127 to 4.600 GeV are measured based on 15.6 fb-1 data
collected with the BESIII detector operating at the BEPCII storage ring. Two
resonant structures are observed in the line shape of the cross sections. The
mass and width of the first structure are measured to be ($4225.3\pm 2.3\pm
21.5$) MeV and ($72.9\pm 6.1\pm 30.8$) MeV, respectively. They are consistent
with those of the established $Y(4230)$. The second structure is observed for
the first time with a statistical significance greater than 8$\sigma$, denoted
as $Y(4500)$. Its mass and width are determined to be ($4484.7\pm 13.3\pm
24.1$) MeV and ($111.1\pm 30.1\pm 15.2$) MeV, respectively. The first
presented uncertainties are statistical and the second ones are systematic.
The product of the electronic partial width with the decay branching fraction
$\Gamma(Y(4230)\to e^{+}e^{-})\mathcal{B}(Y(4230)\to K^{+}K^{-}J/\psi)$ is
reported.

Y states, charmonium-like states, BESIII

††preprint: APS/123-QED

##  I Introduction

Recent discoveries of charmonium-like states expand our perspective of the
hadron spectrum around the $\tau$-charm energy region, and provide excellent
laboratories to study perturbative and non-perturbative strong interaction.
The dynamic, however, is more complex than the one of conventional mesons due
to possible additional degrees of freedom. Therefore, their nature has not yet
been established despite many different speculations about them being hybrids,
tetra-quarks, molecules, cusp effects, and so on [1].

Among the exotic states, the $Y(4230)$ state, previously called $Y(4260)$, is
the first observed vector charmonium-like state. It was discovered in
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ channel by the BaBar Collaboration using
initial-state-radiation (ISR) technique [2], and confirmed by CLEO [3] and
Belle [4]. In addition, the $Y(4230)$ state was observed in various modes by
BESIII, including $e^{+}e^{-}\to\pi\pi J/\psi$ [5, 6, 7],
$\pi^{+}\pi^{-}h_{c}$ [8], $\pi\pi\psi(2S)$ [9, 10, 11], $\omega\chi_{c0}$
[12, 13], and $\pi^{+}D^{0}D^{*-}+c.c.$ [14]. It was also found to decay into
$X(3872)$ via radiative transition [15] and to $Z_{c}(3900)$ via pion
transition [6]. A better understanding of its internal structure and quark
components is crucial, and it will also be helpful to understand the series of
the tetra-quark candidates [16].

Two recent measurements on the cross sections of $e^{+}e^{-}\to\eta J/\psi$
[17] and $e^{+}e^{-}\to\eta^{\prime}J/\psi$ [18] indicate a considerable
strange quark component in $Y(4230)$. This hypothesis was analyzed in Ref.
[19]. However, a concrete conclusion is still missing due to the large
uncertainty. Thus a measurement of $Y(4230)\to K\bar{K}J/\psi$ is important to
clarify the puzzle by comparison with the results of $Y(4230)\to\pi\pi
J/\psi$. The first evidence for $Y(4230)\to K^{+}K^{-}J/\psi$ was found by
CLEO [20]. Later, the cross sections of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ at
center-of-mass (c.m.) energies between threshold and $6.0$ GeV were measured
for the first time via ISR process [21] and was updated [22] by Belle. No
significant signal was observed in both measurements, and an upper limit
$\mathcal{B}(Y(4230)\to K^{+}K^{-}J/\psi)\Gamma(Y(4230)\to e^{+}e^{-})<1.7$ eV
at 90% confidence level was obtained. Here and following the natural unit
system is adopted, i.e., $\hbar=c=1$. Recently, BESIII has measured the cross
sections of $e^{+}e^{-}\to K\bar{K}J/\psi$ at c.m. energies from $4.189$ to
$4.600$ GeV with an integrated luminosity ($\mathcal{L}_{int}$) of $4.7$ fb-1
[23], and no significant signal of the $Y(4230)$ is observed too. In Refs.
[21, 22, 23], a structure around $\sqrt{s}=4.5$ GeV was seen, though the
statistics is too small to identify its properties. The structure with higher
mass is consistent with the calculation by Ref. [24], which suggests a
conventional charmonium state $\psi(4500)$ with mass of $4489-4529$ MeV by the
5S-4D mixing scheme. It is also consistent with a virtual state predicted in
Ref. [25], where the mass of the heavy-antiheavy hadronic molecule is
calculated to be $4483-4503$ MeV that is just below the $D_{s}\bar{D}_{s1}$
threshold. Meanwhile, Ref. [26] predicts an exotic state with
$c\bar{c}s\bar{s}$ component in quenched lattice quantum chromodynamics with
exact chiral symmetry, and its mass is predicted to be ($4450\pm 100$) MeV.

In this Letter, we present an updated measurement of the Born cross sections
of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ at the c.m. energies from 4.127 to 4.600
GeV, using data samples composed of twenty-eight c.m. energy points [27, 28]
with $\mathcal{L}_{int}$ = 15.6 fb-1 [29, 30], collected at the BESIII
detector operating at the BEPCII storage ring [31]. These samples are about
three times in luminosity compared with that used in Ref. [23], and they
overlap at eleven energy points. The added points are mainly around the
$Y(4230)$ mass region, except two points around $4.5$ GeV that are crucial for
determining the line shape of $Y(4500)$. Furthermore, although previously in
Ref. [23] a full reconstruction method is applied, considering the low
efficiency of kaon reconstruction with low momentum, a new partial
reconstruction method is applied improving significantly the efficiency.

##  II Data Analysis

The BESIII detector is described in detail in Ref. [32, 31]. The geant4-based
[33] Monte Carlo (MC) simulation software framework boost [34], which consists
of detector geometry and its response, is used to produce large simulated
event samples. These samples are used to optimize the event selection
criteria, determine the detection efficiency, evaluate the ISR correction
factor, and estimate background contributions. The signal events are generated
at each c.m. energy, where signal events include $e^{+}e^{-}\to
K^{+}K^{-}J/\psi$ (phase space (PHSP) model ), $f_{0}(980)J/\psi$, and
$f_{2}(1270)J/\psi$. Both $f_{0}(980)$ and $f_{2}(1270)$ decay into
$K^{+}K^{-}$, and $J/\psi$ decays into one lepton pair
($\mu^{+}\mu^{-}/e^{+}e^{-}$). The simulation includes the beam energy spread
and ISR in the $e^{+}e^{-}$ annihilation modelled with the generator kkmc [35,
36] and evtgen [37, 38], where the angular distributions according to spin and
parity are considered by the specific models implemented in the generator. The
final state radiation effect associated with leptons is handled by the photos
package [39]. The potential backgrounds are estimated by the inclusive MC
sample, that includes the production of open charm processes, the ISR
production of vector charmonium(-like) states, and the continuum processes
incorporated in kkmc. All particle decays are modelled with evtgen using
branching fractions either taken from the Particle Data Group [40], when
available, or otherwise estimated with lundcharm [41, 42].

A pair of leptons ($e^{+}e^{-}/\mu^{+}\mu^{-}$) and at least one kaon is
required for a signal candidate. For each track, the polar angle $\theta$,
with respect to the symmetry axis of the multiple drift chamber, must satisfy
$|\cos\theta|<0.93$, and the point of the closest approach to the $e^{+}e^{-}$
interaction point must be less than 10.0 cm in the beam direction and 1.0 cm
in the plane perpendicular to the beam direction. Each charged track with
momentum larger than 1.0 GeV in the laboratory frame is assumed to be a
lepton. The ratio of the energy deposited in the electromagnetic calorimeter
over the momentum of each lepton candidate is required to be greater than 0.8
for electrons and less than 0.8 for muons. For particle identification (PID),
the energy loss in the main drift chamber and the time measured with the time-
of-flight system are combined to calculate the confidence levels ($C.L.$) with
kaon and pion hypotheses, and the confidence level is required to satisfy
$C.L.(K)>0.001$ and $C.L.(K)>C.L.(\pi)$ for each kaon candidate. In order to
improve resolution and suppress backgrounds, a vertex fit and a kinematic fit
with one constraint on the mass of the missing kaon are performed. If there is
more than one kaon track candidate, the selected one has the smallest
$\chi^{2}_{F}$, defined as the sum of the $\chi^{2}$ of vertex and kinematic
fits and required to be less than $20$. To remove radiative Bhabha background,
where the radiative photon would convert into an $e^{+}e^{-}$ pair, all pairs
of oppositely charged tracks must have an opening angle
$\cos(\theta_{open})<0.98$ for the $e^{+}e^{-}$ mode. For the $\mu^{+}\mu^{-}$
mode, the penetration depth for one of the muon candidate in the muon counter
is required to be greater than $40$ cm to remove hadron backgrounds.

After applying the aforementioned event selection, the invariant mass
distributions of $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ are shown in Fig. 1 with
all data samples. The signal region of $J/\psi$ is defined as a region with
2.5 times of the mass resolution, and the side band regions are taken as the
same size as the signal with 0.01 GeV gap from the signal region. The
distributions of $M(\ell^{+}\ell^{-})$ at all the energies are presented in
Appendix .1, where $\ell^{+}\ell^{-}$ is either $e^{+}e^{-}$ or
$\mu^{+}\mu^{-}$. There are no peaking backgrounds based on the study of
inclusive MC samples. The yields of signals ($N^{obs}$) are obtained from
background subtraction and the corresponding uncertainties are estimated by
the profile likelihood method [43].

To explore potential intermediate states, the invariant mass distributions of
$K^{+}K^{-}$ and $K^{+}J/\psi$ are shown in Fig. 2, with all data samples,
PHSP signal MC samples, and the weighted signal MC samples, where PHSP and
weighted signal MC samples are both normalized to data. There is no obvious
structure in the $KJ/\psi$ invariant mass distributions. The distributions of
invariant mass of $K^{+}K^{-}$ show signs of mesons $f_{0}(980)$ and
$f_{2}(1270)$. These distributions roughly match with the theoretical
calculation at $\sqrt{s}$ = 4.23 and 4.26 GeV [44], while the corresponding
predictions are still missing at higher energy regions. We extract the
contributions of different components (PHSP, $f_{0}(980)$, $f_{2}(1270)$) by
fitting to the data samples with large signal yields ($N^{obs}>55$), then
expanding the results to the smaller samples by linear interpolation. The
efficiency of event selection ($\varepsilon$) is calculated by the sum of
weighted efficiencies of these components.

Figure 1: The invariant mass distributions of the lepton pairs from different
$J/\psi$ decay modes. (a) is $e^{+}e^{-}$ mode, (b) is $\mu^{+}\mu^{-}$ mode,
where the dots with error bar indicate data, the blue line histograms indicate
PHSP signal MC sample, the red line histograms indicate backgrounds from
inclusive MC sample. Figure 2: (a) and (b) are distributions of
$M(K^{+}K^{-})$ and $M(K^{+}J/\psi$), respectively. The black dots with error
bars indicate data from the $J/\psi$ signal region with all data samples, the
red dashed curves indicate data from $J/\psi$ side band regions, the blue
solid curves indicate PHSP signal MC sample, and the yank dashed-dot curves
indicated the weighted signal MC sample.

##  III Born Cross Section and Resonant Parameter

The Born cross section of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ at each energy is
obtained by

$\sigma^{B}(\sqrt{s})=\frac{N^{obs}}{\mathcal{L}_{int}\cdot\varepsilon\cdot(1+\delta)_{ISR}\cdot\frac{1}{|1-\Pi|^{2}}\cdot\mathcal{B}(J/\psi\rightarrow\ell^{+}\ell^{-})},$
(1)

where $\frac{1}{|1-\Pi|^{2}}$ is the vacuum polarization factor calculated by
QED [45], $\mathcal{B}(J/\psi\rightarrow\ell^{+}\ell^{-})$ is the branching
fraction that is quoted as $(11.93\pm 0.05)\%$ [40], $(1+\delta)_{ISR}$ is the
radiative correction factor obtained by an iteration method, described in
details in Ref. [46]. The results of Born cross sections are listed in
Appendix .2. Compared with the previous BESIII measurements [23], the Born
cross sections at the same c.m. energies are consistent, and the precision is
slightly improved. The observation of two clear structures in the distribution
of the cross sections is due to more data samples used.

A maximum likelihood fit is applied to the dressed cross sections
($\sigma^{D}(\sqrt{s})$, including vacuum polarization effects) of
$e^{+}e^{-}\rightarrow K^{+}K^{-}J/\psi$ to determine the parameters of the
resonant structures, and the dressed cross sections are shown in Fig. 3. The
likelihood is constructed taking the fluctuations of the number of signal
events into account. Its definition is described in Appendix .3. The fit
function is parameterized as a coherent sum of two relativistic Breit-Wigner
($BW$) functions

$\sigma^{D}(\sqrt{s})=|BW_{1}(\sqrt{s})\cdot
e^{i\varphi}+BW_{2}(\sqrt{s})|^{2},$ (2)

where $\varphi$ is the relative phase angle, and

$BW(\sqrt{s})=\frac{M}{\sqrt{s}}\cdot\frac{\sqrt{12\pi\Gamma_{ee}\Gamma_{tot}\mathcal{B}}}{s-M^{2}+iM\Gamma_{tot}}\cdot\sqrt{\frac{\Phi(\sqrt{s})}{\Phi(M)}},$
(3)

where $M,~{}\Gamma_{tot},~{}\Gamma_{ee}$, and $\mathcal{B}$ are the mass, full
width, electronic partial width (whose definition includes vacuum polarization
effects, that is why the dressed cross sections are fitted to rather than the
Born cross sections), and branching fraction of corresponding resonance,
respectively.
$\Phi(\sqrt{s})=\int\int\frac{1}{(2\pi)^{3}32(\sqrt{s})^{3}}dm_{12}^{2}dm_{23}^{2}$
is the three-body phase space [40], where $m_{ij}$ is the invariant mass of
particles i and j. The fitting curve is shown in Fig. 3, and the fit quality
is estimated to be $\chi^{2}/{\rm n.d.f}$ = 37.45 / 21, where n.d.f is the
number of degrees of freedom. The resonance with lower mass is consistent with
the previously established $Y(4230)$. Its mass and width are determined to be
$M(Y(4230))=(4225.3\pm 2.3)$ MeV and $\Gamma(Y(4230))=(72.9\pm 6.1)$ MeV.
Since there is no observed state corresponding to the resonance with higher
mass, we name it as $Y(4500)$. Its mass and width are determined to be
$M(Y(4500))=(4484.7\pm 13.3)$ MeV, $\Gamma(Y(4500))=(111.1\pm 30.1)$ MeV,
respectively. The statistical significance of $Y(4230)$ and $Y(4500)$ have
been estimated to be 29$\sigma$ and 8$\sigma$, respectively, via the
differences of the likelihood values and the degrees of freedom with and
without considering the corresponding resonance by the Wilk’s theorem [47].
Two solutions with equal goodness-of-fit qualities are found. The masses and
total widths are unchanged in the two solutions, while the amplitudes vary
significantly due to constructive and destructive interferences between
$Y(4230)$ and $Y(4500)$, which are consistent with the mathematical
expectation of multiple solutions [48]. The products of the electron partial
width and branching fraction, of the states $Y(4230)$ and $Y(4500)$, are
listed in Table. 1.

Figure 3: Dressed cross sections of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$,
indicated by error bars with only statistical uncertainties. They are fitted
by a coherent sum of two Breit-Wigner functions indicated by red solid curves,
and the blue and pink dashed curves are the amplitudes describing the
resonances $Y(4230)$ and $Y(4500)$, respectively. (a) corresponds to solution
I, (b) corresponds to solution II. Table 1: Fitted parameters of the two
resonant structures observed in the cross sections of $e^{+}e^{-}\to
K^{+}K^{-}J/\psi$, where the first uncertainty is statistical and the second
one is systematic.

| Parameters | Solution I | Solution II
---|---|---|---
$Y(4230)$ |  $M$(MeV) | $4225.3\pm 2.3\pm 21.5$
$\Gamma_{tot}$(MeV) | $72.9\pm 6.1\pm 30.8$
$\Gamma_{ee}\mathcal{B}$(eV) | $0.42\pm 0.04\pm 0.15$ | $0.29\pm 0.02\pm 0.10$
$Y(4500)$ |  $M$(MeV) | $4484.7\pm 13.3\pm 24.1$
$\Gamma_{tot}$(MeV) | $111.1\pm 30.1\pm 15.2$
$\Gamma_{ee}\mathcal{B}$(eV) | $1.35\pm 0.14\pm 0.07$ | $0.41\pm 0.08\pm 0.13$
Phase angle |  $\varphi$(rad) | $1.72\pm 0.09\pm 0.52$ | $5.49\pm 0.35\pm 0.58$

The main sources of systematic uncertainties of the cross section measurements
are: the integrated luminosity $\mathcal{L}_{int}$, the tracking and PID
efficiency, the branching fraction of $J/\psi\to\ell^{+}\ell^{-}$, kinematic
fit, $(1+\delta)_{ISR}$, the intermediate structures of the $K^{+}K^{-}$
system, the resolution of $J/\psi$, and the requirement on the penetration
depth in the muon counter. The $\mathcal{L}_{int}$ is measured with Bhabha
events and the uncertainty is found to be 1.0% [29, 30]. The differences
between data and MC in the tracking and PID efficiencies are studied by using
the process $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$, and 2.5% per charged
kaon is quoted [23]. For tracking efficiency of charged leptons, the
systematic uncertainty is studied by using the process
$\psi(3686)\to\pi^{+}\pi^{-}J/\psi(J/\psi\to\ell^{+}\ell^{-})$, and 1.0%
uncertainty per lepton is quoted [49]. The uncertainty of the branching
fraction of $J/\psi\to\ell^{+}\ell^{-}$ is quoted as 0.4% [40]. The
uncertainty of kinematic fit is estimated by correcting the corresponding
track parameters, and the difference between the efficiencies with (default)
and without this correction is quoted as the relevant uncertainty [50]. Varied
line shape of the input cross sections is constructed by connecting each
nearby points with a smooth curve. The difference between the final cross
sections with this new line shape and the nominal one is taken as the
uncertainty of radiation correction. The uncertainty associated with the
intermediate structures is estimated by weighting the PHSP MC samples
according to the observed $M(K^{+}K^{-})$ distribution of data, and the
difference between the two efficiencies of the two methods is quoted as the
corresponding uncertainty. To estimate the uncertainty according to the
difference in the resolutions between data and MC samples, the efficiency is
re-obtained by smearing the resolution of the $J/\psi$ signal of the MC
sample. The difference is quoted as the associated uncertainty. To consider
the uncertainty of the criteria on the penetration depth in the muon counter,
the difference between the final results with and without this criterion is
adopted. The total uncertainties are calculated by summing all individual
items in quadrature. They are energy dependent and vary from $8.6\%$ to
$11.1\%$. All the systematic uncertainties, including the individual and
total, are listed in Appendix .4.

The systematic uncertainties for the parameters of resonances mainly come from
c.m. energy measurements, the form and parameterization of the fit function,
and the systematic uncertainties in the cross section measurements that will
be discussed later. The c.m. energies were measured with
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ events and the uncertainties are
determined correspondingly for different data samples [27, 28]. The associated
systematic uncertainty is estimated by varying the c.m. energies during the
fit. A three-body PHSP shape for a non-resonant component is added to the two
coherent $BW$ sum to estimate the uncertainty of the cross section description
in the fit, which turns out to be negligible. The uncertainty of the formalism
of the full width is estimated by replacing the $\Gamma_{tot}$ with
$\Gamma=\Gamma_{tot}\frac{\Phi(\sqrt{s})}{\Phi(M)}$ in the denominator of Eq.
(3), where $\Gamma_{tot}$ is the nominal width of the resonance. The
systematic uncertainties due to cross section measurements can be divided into
two categories. The first one is uncorrelated among the different c.m. energy
points including kinematic fit, radiation correction, intermediate structures,
and resolution of $J/\psi$. The associated uncertainty is estimated by
considering them while doing the fit and comparing to the results obtained
only considering statistical uncertainties of the cross sections. The second
category of the systematic uncertainties is correlated and common for all data
samples (5.1%), therefore only affects the $\Gamma_{ee}\mathcal{B}$. All of
these uncertainties on the parameters of resonances are listed in Appendix .4.

##  IV Summary and Discussion

In summary, the Born cross sections of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ at
c.m. energy from 4.127 to 4.600 GeV are measured with a new partial
reconstruction method and larger data samples compared with Ref [23]. Two
resonances are observed with high significance. One is consistent to the
previous observed $Y(4230)$, and its mass and width are measured to be
$M(Y(4230))=(4225.3\pm 2.3\pm 21.5)$ MeV, $\Gamma(Y(4230))=(72.9\pm 6.1\pm
30.8)$ MeV, where the first uncertainties are statistical and the second are
systematic. The other one can not be assigned into any experimentally observed
resonance, named as $Y(4500)$, and its mass and width are determined to be
$M(Y(4500))=(4484.7\pm 13.3\pm 24.1)$ MeV, $\Gamma(Y(4500))=(111.1\pm 30.1\pm
15.2)$ MeV, respectively. There are also evidences of this new structure in
the measurements of the cross sections of $e^{+}e^{-}\to\pi\pi J/\psi$ [7],
even in the same channel $K^{+}K^{-}J/\psi$ [21, 22, 23]. But only with
additional data samples at BESIII and improved analysis method, this state has
been observed with a significance more than $5\sigma$ for the first time. The
mass of $Y(4500)$ is consistent with the prediction of the 5S-4D mixing scheme
[24], the heavy-antiheavy hadronic molecules model [25] and the lattice
quantum chromodynamics result for a $(cs\bar{c}\bar{s})$ state [26], while the
width is 2$\sigma$ larger than the prediction of Ref. [24]. More experimental
measurements and theoretical studies are needed to reveal its nature.

For the first time, the state $Y(4230)$ has been observed in the
$K\bar{K}J/\psi$ mode with the significance larger than 5$\sigma$. The product
of the electronic partial width and the decay branching fraction is measured
to be $\mathcal{B}(Y(4230)\to K^{+}K^{-}J/\psi)\Gamma(e^{+}e^{-}\to Y(4230))$
= ($0.29\pm 0.02\pm 0.10$) eV or ($0.42\pm 0.04\pm 0.15$) eV, according to
different interferences, respectively. The ratio between the branching
fractions of the $Y(4230)$ decaying into $K\bar{K}J/\psi$ and $\pi\pi J/\psi$
[7] are calculated and shown in Table. 2. Author of Ref. [51] predicts the
$K\bar{K}$ mode should be suppressed if the $Y(4230)$ is $\Lambda_{c}$
baryonium. Even at present, no conclusion can be drawn due to the multiple
solutions. However, once the physics solution is determined as done in Ref.
[19], it will provide very useful information for understanding the nature of
$Y(4230)$.

Table 2: The ratios between the branching fractions of the $Y(4230)$ decaying
into $K\bar{K}J/\psi$ and $\pi\pi J/\psi$, depending on the various
combinations of the multiple solutions.

|  $K\bar{K}J/\psi$ Sol. I |  $K\bar{K}J/\psi$ Sol. II
---|---|---
$\pi\pi J/\psi$ Sol. I | $0.17\pm 0.02$ | $0.25\pm 0.04$
$\pi\pi J/\psi$ Sol. II | $0.097\pm 0.017$ | $0.14\pm 0.03$
$\pi\pi J/\psi$ Sol. III | $0.035\pm 0.004$ | $0.051\pm 0.007$
$\pi\pi J/\psi$ Sol. IV | $0.020\pm 0.002$ | $0.028\pm 0.004$

##  V Acknowledgement

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11625523, 11635010, 11735014, 11822506, 11835012, 11935015, 11935016,
11935018, 11961141012, 12022510, 12025502, 12035009, 12035013, 12061131003;
the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program;
Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under
Contracts Nos. U1732263, U1832207; CAS Key Research Program of Frontier
Sciences under Contract No. QYZDJ-SSW-SLH040; 100 Talents Program of CAS;
INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC
under Contract No. 758462; European Union Horizon 2020 research and innovation
programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790;
German Research Foundation DFG under Contracts Nos. 443159800, Collaborative
Research Center CRC 1044, FOR 2359, GRK 214; Istituto Nazionale di Fisica
Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; National Science and Technology fund; Olle Engkvist
Foundation under Contract No. 200-0605; STFC (United Kingdom); The Knut and
Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal
Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research
Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374,
DE-SC-001206.

## Appendix

###  .1 Distributions of $M(\ell^{+}\ell^{-})$ and $M(K^{+}K^{-})$

Fig. A1 shows the distributions of the invariant mass of lepton pairs,
$M(\ell^{+}\ell^{-})$, for data and PHSP signal MC samples at various c.m.
energies. Here, the $J/\psi$ signal and sideband regions are indicated with
red and blue arrows, respectively. In addition, the distributions of invariant
mass of kaons, $M(K^{+}K^{-})$, for data, PHSP signal MC, and weighted MC
samples at various c.m. energies are shown in Fig. A2.

Figure A1: The distributions of invariant mass of lepton pairs
$M(\ell^{+}\ell^{-})$, where the dots with error bars are data, the blue
histogram is PHSP signal MC sample (normalized to data.), the red and blue
arrows identify the $J/\psi$ signal and sideband regions, respectively.

Figure A2: The distributions of $K^{+}K^{-}$ invariant mass of the data, PHSP
signal MC and weighted MC samples at each c.m. energy, where the black dots
with error bars indicate data from the $J/\psi$ signal region, the red dashed
curves indicate data from $J/\psi$ sideband regions, the blue histograms
indicate PHSP signal MC sample (normalized to data) and the yank dashed-dot
curves indicate the weighted signal MC sample (normalized to data).

###  .2 Born Cross Sections

The Born cross sections of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ and related
quantities such as the c.m. energy, the integrated luminosity, the numbers of
observed signal events, the efficiencies, the radiative correction factors,
the vacuum polarization factors are listed in Table. A1.

Table A1: The Born cross sections of $e^{+}e^{-}\to K^{+}K^{-}J/\psi$ and
related quantities. Here $\sqrt{s}$ is the c.m. energy, $\mathcal{L}_{int}$ is
the integrated luminosity, $N^{obs}$ is the number of observed signal events,
$\varepsilon$ is the efficiency, $(1+\delta)_{ISR}$ is the radiative
correction factor, $\frac{1}{|1-\Pi|^{2}}$ is the vacuum polarization factor,
and the $\sigma^{B}$ is the Born cross section, the first uncertainties are
statistical, and the second systematic.

Data sample |  $\sqrt{s}$(GeV) |  $\mathcal{L}_{int}$ (pb-1) | $N^{obs}$ | $\varepsilon$ | $(1+\delta)_{ISR}$ | $\frac{1}{|1-\Pi|^{2}}$ |  $\sigma^{B}$(pb)
---|---|---|---|---|---|---|---
4130 | 4.127 | 401.50 | $6.5_{-2.4}^{+3.0}$ | 0.141 | 0.896 | 1.052 | $1.02_{-0.38}^{+0.47}\pm 0.09$
4160 | 4.157 | 408.70 | $6.5_{-2.4}^{+3.0}$ | 0.263 | 0.856 | 1.053 | $0.56_{-0.21}^{+0.26}\pm 0.05$
4180 | 4.178 | 3194.50 | $72.0_{-9.1}^{+9.7}$ | 0.315 | 0.820 | 1.054 | $0.69_{-0.09}^{+0.09}\pm 0.06$
4190 | 4.189 | 570.10 | $18.5_{-4.1}^{+4.7}$ | 0.335 | 0.792 | 1.056 | $0.97_{-0.22}^{+0.25}\pm 0.08$
4200 | 4.199 | 526.00 | $25.0_{-4.6}^{+5.2}$ | 0.357 | 0.787 | 1.056 | $1.34_{-0.25}^{+0.28}\pm 0.12$
4210 | 4.209 | 572.10 | $33.5_{-5.8}^{+6.4}$ | 0.358 | 0.762 | 1.057 | $1.70_{-0.29}^{+0.33}\pm 0.15$
4220 | 4.219 | 569.20 | $69.0_{-8.1}^{+8.7}$ | 0.379 | 0.771 | 1.056 | $3.29_{-0.39}^{+0.41}\pm 0.37$
4230 | 4.226 | 1100.90 | $183.0_{-13.3}^{+14.0}$ | 0.395 | 0.771 | 1.056 | $4.33_{-0.31}^{+0.33}\pm 0.38$
4237 | 4.236 | 530.30 | $65.5_{-8.3}^{+8.9}$ | 0.391 | 0.789 | 1.056 | $3.18_{-0.40}^{+0.43}\pm 0.30$
4245 | 4.242 | 55.88 | $8.0_{-2.5}^{+3.2}$ | 0.405 | 0.805 | 1.055 | $3.49_{-1.09}^{+1.39}\pm 0.31$
4246 | 4.244 | 538.10 | $74.5_{-8.5}^{+9.2}$ | 0.393 | 0.812 | 1.056 | $3.44_{-0.39}^{+0.43}\pm 0.30$
4260 | 4.258 | 828.40 | $107.5_{-10.3}^{+10.9}$ | 0.396 | 0.866 | 1.054 | $3.01_{-0.29}^{+0.31}\pm 0.26$
4270 | 4.267 | 531.10 | $45.0_{-6.7}^{+7.4}$ | 0.390 | 0.911 | 1.053 | $1.90_{-0.28}^{+0.31}\pm 0.17$
4280 | 4.278 | 175.70 | $16.5_{-4.0}^{+4.7}$ | 0.372 | 0.935 | 1.053 | $2.15_{-0.52}^{+0.61}\pm 0.19$
4290 | 4.287 | 502.40 | $35.0_{-6.3}^{+6.9}$ | 0.363 | 0.937 | 1.053 | $1.63_{-0.29}^{+0.32}\pm 0.14$
4310 | 4.308 | 45.08 | $3.0_{-1.4}^{+2.1}$ | 0.367 | 0.982 | 1.052 | $1.47_{-0.69}^{+1.03}\pm 0.13$
4315 | 4.311 | 501.20 | $24.0_{-5.6}^{+6.2}$ | 0.358 | 0.985 | 1.052 | $1.08_{-0.25}^{+0.28}\pm 0.10$
4340 | 4.337 | 505.00 | $18.5_{-5.6}^{+6.1}$ | 0.364 | 0.963 | 1.051 | $0.83_{-0.25}^{+0.28}\pm 0.07$
4360 | 4.358 | 544.00 | $26.5_{-5.7}^{+6.3}$ | 0.378 | 0.935 | 1.051 | $1.10_{-0.24}^{+0.26}\pm 0.10$
4380 | 4.377 | 522.70 | $33.0_{-7.1}^{+7.6}$ | 0.379 | 0.891 | 1.051 | $1.49_{-0.32}^{+0.34}\pm 0.13$
4390 | 4.387 | 55.57 | $3.0_{-1.4}^{+2.1}$ | 0.397 | 0.865 | 1.051 | $1.25_{-0.58}^{+0.88}\pm 0.11$
4400 | 4.395 | 507.80 | $16.0_{-6.1}^{+6.6}$ | 0.401 | 0.861 | 1.051 | $0.73_{-0.28}^{+0.30}\pm 0.06$
4420 | 4.416 | 1090.70 | $57.0_{-8.8}^{+9.4}$ | 0.417 | 0.815 | 1.052 | $1.23_{-0.19}^{+0.20}\pm 0.11$
4440 | 4.436 | 569.90 | $55.5_{-8.3}^{+8.9}$ | 0.436 | 0.780 | 1.054 | $2.28_{-0.34}^{+0.37}\pm 0.20$
4470 | 4.467 | 111.09 | $14.0_{-4.0}^{+4.6}$ | 0.450 | 0.738 | 1.055 | $3.02_{-0.86}^{+0.99}\pm 0.26$
4530 | 4.527 | 112.12 | $23.5_{-4.8}^{+5.4}$ | 0.460 | 0.751 | 1.054 | $4.82_{-0.99}^{+1.11}\pm 0.41$
4575 | 4.574 | 48.93 | $0.0_{-0.0}^{+2.3}$ | 0.430 | 0.856 | 1.054 | $0.00_{-0.00}^{+1.01}\pm 0.00$
4600 | 4.600 | 586.90 | $52.0_{-8.2}^{+8.7}$ | 0.400 | 0.899 | 1.055 | $1.96_{-0.31}^{+0.33}\pm 0.17$

###  .3 Definition of likelihood function

In the maximum likelihood fit to the dressed cross sections of
$e^{+}e^{-}\rightarrow K^{+}K^{-}J/\psi$, the likelihood is constructed as:

$L(\mu;\vartheta)=\prod\limits_{i}^{6}L_{i}(\mu_{i};~{}\vartheta_{i})\prod\limits_{j}^{22}L_{j}(\mu_{j};~{}\vartheta_{j}),$
(A1)

where $\mu_{i}$ and $\mu_{j}$ are the numbers of observed signal events,
$\vartheta_{i}$ and $\vartheta_{j}$ are the parameters in the likelihood
functions, and $L_{i}$ and $L_{j}$ are the likelihood functions for the data
samples with $\mu\leq 10$ and $\mu>10$, respectively.

The likelihood functions are defined variously according to the different
numbers of observed events. For the data samples with $\mu\leq 10$, the
likelihood function is described by a Poisson function:

$L_{i}(\mu_{i};\vartheta_{i})=P_{i}(\mu_{i};\vartheta_{i})=\frac{1}{\mu_{i}!}\vartheta_{i}^{\mu_{i}}e^{-\vartheta_{i}},$
(A2)

While for the data samples with $\mu>10$, the likelihood function is described
by an asymmetric Gaussian function:

$L_{j}(\mu_{j},~{}\sigma_{1j},~{}\sigma_{2j};~{}\vartheta_{j})=G(\mu_{j},~{}\sigma_{1j},~{}\sigma_{2j};~{}\vartheta_{j})=\left\\{\begin{array}[]{rcl}\frac{1}{\sqrt{2\pi}~{}(\sigma_{1j}+\sigma_{2j})}~{}e^{-\frac{(\vartheta_{j}-\mu_{j})^{2}}{2\sigma_{1j}^{2}}},&&{\vartheta_{j}>\mu_{j};{}}\\\
{}\hfil\\\
\frac{1}{\sqrt{2\pi}~{}(\sigma_{1j}+\sigma_{2j})}~{}e^{-\frac{(\vartheta_{j}-\mu_{j})^{2}}{2\sigma_{2j}^{2}}},&&{\vartheta_{j}\leq\mu_{j};{}}\\\
{}\hfil\\\ \end{array}\right.$ (A3)

where $\sigma_{1}$ and $\sigma_{2}$ are the upper and lower statistical
uncertainties of $\mu$, respectively.

###  .4 The systematic uncertainties of the Born cross sections and the
resonance parameters

The systematic uncertainties of the Born cross sections are listed in Tab. A2
and the systematic uncertainties of the resonance parameters are listed in
Tab. A3. Fig. A3 shows dressed cross sections fitting results with different
function forms. Generally, there should be four solutions when three coherent
amplitudes are used to describe a lineshape of cross sections. However, only
two have been found by us. We suppose that the other solutions are very close
to the found ones, then cannot be separated by the scanning method. And one
should notice that the fitting results, with the tentatively additional
continuum term, only change slightly.

Table A2: The systematic uncertainties (in units %) in the measurement of Born
cross sections. Here $\sqrt{s}$ is the c.m. energy of data samples,
$\mathcal{L}_{int}$ is the integrated luminosity,
$\mathcal{B}(J/\psi\rightarrow l^{+}l^{-})$ is the branching fraction,
$(1+\delta)_{ISR}$ is the radiation correction factor, $M(K^{+}K^{-})$ is the
intermediate structures in $K^{+}K^{-}$ system, $R(J/\psi)$ is the resolution
of $J/\psi$, MUC is the criteria applied on the penetration depth in the muon
counter.

Data sample |  $\sqrt{s}$(GeV) | $\mathcal{L}_{int}$ | Tracking and PID | $\mathcal{B}(J/\psi\rightarrow l^{+}l^{-})$ | kinematic fit | $(1+\delta)_{ISR}$ | $M(K^{+}K^{-})$ | $R(J/\psi)$ | MUC | Total
---|---|---|---|---|---|---|---|---|---|---
4130 | 4.127 | 1.0 | 4.5 | 0.4 | 0.4 | 1.2 | 6.9 | 0.3 | 2.1 | 8.7
4160 | 4.157 | 1.0 | 4.5 | 0.4 | 0.7 | 0.7 | 6.9 | 0.3 | 2.1 | 8.6
4180 | 4.178 | 1.0 | 4.5 | 0.4 | 0.9 | 2.6 | 6.9 | 0.2 | 2.1 | 9.0
4190 | 4.189 | 1.0 | 4.5 | 0.4 | 1.0 | 1.0 | 6.9 | 0.1 | 2.1 | 8.7
4200 | 4.199 | 1.0 | 4.5 | 0.4 | 1.1 | 0.6 | 6.9 | 0.3 | 2.1 | 8.7
4210 | 4.209 | 1.0 | 4.5 | 0.4 | 1.1 | 1.7 | 6.9 | 0.3 | 2.1 | 8.8
4220 | 4.219 | 1.0 | 4.5 | 0.4 | 1.1 | 6.9 | 6.9 | 0.2 | 2.1 | 11.1
4230 | 4.226 | 1.0 | 4.5 | 0.4 | 1.1 | 0.6 | 6.9 | 0.4 | 2.1 | 8.7
4237 | 4.236 | 1.0 | 4.5 | 0.4 | 1.2 | 3.5 | 6.9 | 0.4 | 2.1 | 9.3
4245 | 4.242 | 1.0 | 4.5 | 0.4 | 1.2 | 2.4 | 6.9 | 0.2 | 2.1 | 9.0
4246 | 4.244 | 1.0 | 4.5 | 0.4 | 1.2 | 1.5 | 6.9 | 0.2 | 2.1 | 8.8
4260 | 4.258 | 1.0 | 4.5 | 0.4 | 1.3 | 1.3 | 6.9 | 0.3 | 2.1 | 8.8
4270 | 4.267 | 1.0 | 4.5 | 0.4 | 1.3 | 2.7 | 6.9 | 0.3 | 2.1 | 9.1
4280 | 4.278 | 1.0 | 4.5 | 0.4 | 1.3 | 1.3 | 6.9 | 0.4 | 2.1 | 8.8
4290 | 4.287 | 1.0 | 4.5 | 0.4 | 1.2 | 1.2 | 6.9 | 0.3 | 2.1 | 8.7
4310 | 4.308 | 1.0 | 4.5 | 0.4 | 1.6 | 1.3 | 6.9 | 0.4 | 2.1 | 8.8
4315 | 4.312 | 1.0 | 4.5 | 0.4 | 1.6 | 0.7 | 6.9 | 0.4 | 2.1 | 8.8
4340 | 4.337 | 1.0 | 4.5 | 0.4 | 1.6 | 1.5 | 6.9 | 0.7 | 2.1 | 8.9
4360 | 4.358 | 1.0 | 4.5 | 0.4 | 1.4 | 0.5 | 6.9 | 0.5 | 2.1 | 8.7
4380 | 4.377 | 1.0 | 4.5 | 0.4 | 1.4 | 0.3 | 6.9 | 0.6 | 2.1 | 8.7
4390 | 4.387 | 1.0 | 4.5 | 0.4 | 1.2 | 1.0 | 6.9 | 0.5 | 2.1 | 8.7
4400 | 4.395 | 1.0 | 4.5 | 0.4 | 1.1 | 0.9 | 6.9 | 0.5 | 2.1 | 8.7
4420 | 4.416 | 1.0 | 4.5 | 0.4 | 0.9 | 1.0 | 6.9 | 0.7 | 2.1 | 8.7
4440 | 4.436 | 1.0 | 4.5 | 0.4 | 0.7 | 1.6 | 6.9 | 0.6 | 2.1 | 8.8
4470 | 4.467 | 1.0 | 4.5 | 0.4 | 0.4 | 0.2 | 6.9 | 0.4 | 2.1 | 8.6
4530 | 4.527 | 1.0 | 4.5 | 0.4 | 0.3 | 0.9 | 6.9 | 0.6 | 2.1 | 8.6
4575 | 4.574 | 1.0 | 4.5 | 0.4 | 0.2 | 0.0 | 6.9 | 0.4 | 2.1 | 8.6
4600 | 4.600 | 1.0 | 4.5 | 0.4 | 0.2 | 1.2 | 6.9 | 0.3 | 2.1 | 8.7

Table A3: The systematic uncertainty in the measurement of the resonance
parameters for solution I (solution II), where Non-Resonant indicated a three-
body PHSP shape is added in the default fit and considering the coherent
between them, the cross section1 is uncorrelated systematic uncertainties from
the measurement of cross sections, while cross section2 is correlated.

Sources Parameters | $Y(4230)$ | $Y(4500)$ | phase angle
---|---|---|---
$M$ (MeV) |  $\Gamma_{tot}$ (MeV) |  $\Gamma_{ee}\mathcal{B}$ (eV) |  $M$ (MeV) |  $\Gamma_{tot}$ (MeV) |  $\Gamma_{ee}\mathcal{B}$ (eV) |  $\varphi$ (rad)
c.m. Energy | 0.7 | 0.1 | – | 1.8 | 1.4 | – | 0.00 (0.02)
Non-Resonant | – | – | – | – | – | – | 0.13 (0.51)
$\Gamma_{tot}$ | 21.5 | 30.8 | 0.15 (0.10) | 24.1 | 15.1 | 0.02 (0.13) | 0.50 (0.28)
cross section1 | – | – | – (–) | 0.1 | 0.1 | – (–) | –
cross section2 | – | – | 0.02 (0.01) | – | – | 0.07 (0.02) | –
Total | 21.5 | 30.8 | 0.15 (0.10) | 24.1 | 15.2 | 0.07 (0.13) | 0.52 (0.58)

\begin{overpic}[scale={0.42}]{fit_2bw_phsp_1.eps} \put(75.0,50.0){(a)}
\end{overpic}\begin{overpic}[scale={0.42}]{fit_2bw_phsp_2.eps}
\put(75.0,50.0){(b)}
\end{overpic}\begin{overpic}[scale={0.42}]{fit_G_tot_1.eps}
\put(75.0,50.0){(c)}
\end{overpic}\begin{overpic}[scale={0.42}]{fit_G_tot_2.eps}
\put(75.0,50.0){(d)} \end{overpic}

Figure A3: The dressed cross sections fitting results with different function
forms. (a), (b) are two sets of solutions to the sum of two coherent Breit-
Wigner functions and 3-body phase space function fitting, respectively. (c),
(d) are two sets of solutions to the two coherent Breit-Wigner functions with
an energy-dependent full width fitting, respectively.

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# Perceiving and Modeling Density is All You Need for Image Dehazing

Tian Ye1∗ Mingchao Jiang2∗ Yunchen Zhang3 Equal contribution.†Corresponding
author. Liang Chen4 Erkang Chen1† Pen Chen1 Zhiyong Lu2
1School of Ocean Information Engineering  Jimei University  Xiamen  China
2JOYY AI GROUP  Guangzhou  China
3China Design Group Co  Ltd. Nanjing  China
4Fujian Provincial Key Laboratory of Photonics Technology  Fujian Normal
University  Fuzhou  China.

###### Abstract

In the real world, the degradation of images taken under haze can be quite
complex, where the spatial distribution of haze is varied from image to image.
Recent methods adopt deep neural networks to recover clean scenes from hazy
images directly. However, due to the paradox caused by the variation of real
captured haze and the fixed degradation parameters of the current networks,
the generalization ability of recent dehazing methods on real-world hazy
images is not ideal. To address the problem of modeling real-world haze
degradation, we propose to solve this problem by perceiving and modeling
density for uneven haze distribution.

We propose a novel Separable Hybrid Attention (SHA) module to encode haze
density by capturing features in the orthogonal directions to achieve this
goal. Moreover, a density map is proposed to model the uneven distribution of
the haze explicitly. The density map generates positional encoding in a semi-
supervised way—such a haze density perceiving and modeling capture the
unevenly distributed degeneration at the feature level effectively. Through a
suitable combination of SHA and density map, we design a novel dehazing
network architecture, which achieves a good complexity-performance trade-off.

The extensive experiments on two large-scale datasets demonstrate that our
method surpasses all state-of-the-art approaches by a large margin both
quantitatively and qualitatively, boosting the best published PSNR metric from
28.53 dB to 33.49 dB on the Haze4k test dataset and from 37.17 dB to 38.41 dB
on the SOTS indoor test dataset.

##  1 Introduction

Single image dehazing aims to generate a haze-free image from a hazy image. It
is a classical image processing problem, which has been an important research
topic in the vision communities within the last decade [22, 27, 2, 1].
Numerous real-world vision tasks (e.g., object detection and auto drive)
require high-quality clean images, and the fog and haze usually lead to
degraded images. Therefore, it is of great interest to develop an effective
algorithm to recover haze-free images.

Figure 1: Examples of our image dehazing. Hazy images are in the top row, of
which (a) and (b) are from synthetic hazy dataset haze4k [20], (c) and (d) are
Real-world hazy images from web. The corresponding dehazed results are in the
bottom row.

Haze is a common atmospheric phenomenon in our daily life. Images with haze
and fog lose details and color fidelity. Mathematically, the image degradation
caused by haze could be formulated by the following model:

$\mathbf{I}(\mathbf{x})=\mathbf{J}(\mathbf{x})t(\mathbf{x})+\mathbf{A}(1-t(\mathbf{x}))$
(1)

where $\mathbf{I}(\mathbf{x})$ is the hazy image, $\mathbf{J}(\mathbf{x})$ is
the clear image, $t(\mathbf{x})$ is the transmission map, $\mathbf{A}$ is the
global atmospheric light.

Methods [10, 8, 3, 13] based on priors first estimate the above two unknown
parameters $t(\mathbf{x})$ and $\mathbf{A}$, then inversely solve the above
infinitive formula. Unfortunately, these handcrafted priors do not always hold
in diverse real-world hazy scenes, resulting in inaccurate estimated
$t(\mathbf{x})$.

Figure 2: The Overview of network architecture. The shallow layers is used to
make the density map joined with the density estimation module, which consists
of a stack of MHAC blocks and a Tail Module. The Deep layers emphasize on
detailed reconstruction by Adaptive Features Fusion (AFF) module and Multi-
branch Hybrid Attention Block (MHA). We give details of the structure and
configurations in Section 3.

Therefore, in recent years, more and more researchers have begun to pay
attention to data-driven learning algorithms [15, 7, 11, 27, 22]. Different
from traditional methods, deep learning methods achieve superior performance
via training on large-scale datasets. However, deep learning based algorithms
still has the following challenges: (1) _Number of parameters: huge._ The
previous methods [18, 7, 28] boost their performance by increasing the
capacity of models which result in a large number of parameters. Lack of
consideration of the characteristics of haze also leads to the application
failure when encountering practical scenes. (2) _Training Strategy: Complex._
Complicated loss functions [11, 27, 28, 26, 20] and fancy training strategies
are design to better optimize the reconstruction process. However, these
designs lead to high training costs and poor convergence. (3) _Speed: slow._
Some recent methods [22] extract the feature maps of hazy images under full
resolution in every model stage in order to achieve effective performance,
thus slowing the speed in the practical test.

To address the above problems, we formulate the problem in terms of a
perceiving and modeling density for uneven haze distribution. From the Eq.1,
we can notice that the haze degradation model is highly associated with
absolute position of image pixels. The key to solve the dehazing problem lies
in correctly encoding the haze intensity with its absolute position.
Therefore, we propose a network to model the density of haze distribution in
an end-to-end manner, which encodes the co-relationship between haze intensity
and its absolute position. And our method has a powerful ability to restore
image details and color fidelity on real domain and synthetic domain images,
as seen in Fig. 1.

We propose the method from three different levels: primary block of the
network, architecture of the network and map of haze density information to
refine features:

$\bullet$ Primary Block: We propose an efficient attention mechanism to
perceive the uneven distribution of degradation of features among channel and
spatial dimensions: Separable Hybrid Attention (SHA), which effectively
samples the input features through a combination of different pooling
operations, and strengthens the interaction of different dimensional
information by channel scaling and channel shuffle. Our SHA based on
horizontal and vertical encoding can obtain sufficient spatial clue from input
features.

$\bullet$ Density Map: Density Map is a coefficient matrix that encodes the
co-relationship between haze intensity and absolute position. The density map
explicitly models the intensity of the haze degradation model at corresponding
spatial locations. The density map is obtained in an end-to-end matter,
semantic information of the scene is also introduced implicitly, which makes
the density map more consistent with the actual distribution.

$\bullet$ Network Architecture: We design a novel network architecture that
restores hazy images with the coarse-to-fine strategy. The architecture of our
network mainly consists of three parts: shallow layers, deep layers and
density map. We build the Shallow Layers and Deep Layers of our method based
on SHA. The shallow layers will generate the coarse haze-free image, which we
call the pseudo-haze-free image. For modeling the uneven degradation of hazy
image to refine features explicitly, we utilize the pseudo-haze-free image and
the input hazy sample to generate the Density Map.

Our main contributions are summarized as follows:

  * •

We propose SHA as a universal attention mechanism perceives efficiently the
degenerated density, which can further improve the performance of various
image restoration networks.

  * •

We propose density map as a novel method to build relationship between
different parts in the network, which enhances the coupling of our model. In
addition, it can be applied in other low-level vision tasks in the future.

  * •

We propose a novel dehazing method based on perceiving and modeling the haze
density by highly efficient SHA and density map. Our method achieves the best
performance compared with the state-of-the approaches.

##  2 Related Works

###  2.1 Single Image Dehazing

Single image dehazing is mainly divided into two categories: a prior-based
defogging method [10, 14] and a data-driven method based on deep learning.
With the introduction of large hazy datasets [16, 20], image dehazing based on
the deep neural network has developed rapidly. MSBDN [7] uses the classic
Encoder-Decoder architecture, but repeated up-sampling and down-sampling
operations result in texture information loss. The number of parameters of
MSBDN [7] is large, and the model is complex. FFA-Net [22] proposes an FA
block based on channel attention and pixel attention, obtains the final haze-
free image by fusing features of different levels. With the help of two
different-scale attention mechanisms, FFA-Net has obtained impressive PSNR and
SSIM, but the entire model is convolutional operation at the resolution of the
original image, resulting in a large amount of calculation and slow speed. It
is unfriendly to restore large-size hazy images on devices with low memory.
AECR-Net [27] reuses the FA block and proposes to use a novel loss function
based on the contrast learning to make full use of hazy samples, and use a
deformable convolution block to improve the expression ability of the model,
pushing the index on SOTS [16] to a new high, but memory consumption of the
loss function is so high. Compared with AECR-Net [27], our model only needs to
utilize a simple Charboonier[5] loss function to achieve higher PSNR and SSIM.

###  2.2 Attention Mechanism

Attention mechanisms have been proven essential in various computer vision
tasks, such as image classification [12], segmentation[9], dehazing [22, 27,
11] and deblurring [29]. One of the classical attention mechanisms is
SENet[12], which is widely used as a comparative baseline of the plugin of the
backbone network. CBAM [25] introduces spatial information encoding via
convolutions with large-size kernels, which sequentially infers attention maps
along the channel and spatial dimension. Later attention mechanism extends the
idea of CBAM [25] by adopting different dimension attention mechanisms to
design the advanced attention module. This shows that the key of the
performance of attention mechanism is to sample the original feature map
fully. There is no effective information exchange between the different
dimension attention encoding in before works, thus limiting the lifting of
networks. In response to the above problems, we utilize two types of pooling
operations to sample the original feature map and cleverly inserted the
channel shuffle block in our attention module. Experiments demonstrate that
the performance of our attention mechanism has been dramatically improved due
to sufficient feature sampling and efficient feature exchange among different
dimensions.

##  3 Method

We propose a novel architecture as shown in Fig.2, which consists of three
parts: shallow layers, deep layers and density map. Shallow layers are
responsible for reconstruct high-level contextual content, and deep layers are
responsible for rebuilding pixel-level detailed features.

###  3.1 Separable Hybrid Attention Module

Figure 3: An illustration of the Separable Hybird Attention Module (SHA). Our
SHA focuses on the directional embedding, which consists of unilateral
directional pooling and convolution. The @k denotes the convolutuion kernel
size.

Attention mechanisms based on pooling operations have been applied in many
visual tasks and have improved indicators, but previous attention mechanisms
such as CBAM [25], and DANet [9] often calculate the attention weights
subsequently in the spatial and channel dimensions. But there is no useful
information exchanging between the attention of different dimensions. Non-
local [24] and Trilinear Attention [30] have interactive feature-capability
but the calculation is very large. So we propose a novel attention mechanism
called Separable Hybrid Attention (SHA), which can effectively generate the
fine-grained attention weights to guidance the degenerated density perception.
Specifically, the combination of average pooling and maximum pooling not only
smoothens the noise in features but also effectively enhances high-frequency
information of features. So we utilize the average pooling
($\mathbf{AvgPool}$) and maximum pooling ($\mathbf{MaxPool}$) calculate in the
feature $F_{in}$ along the horizontal and vertical directions to obtain the
encoding features $v$. The formulas are as follows:

$v_{avg}^{h}=\mathbf{AvgPool_{h}}(F_{in}),v_{avg}^{v}=\mathbf{AvgPool_{v}}(F_{in})$
(2)
$v_{max}^{h}=\mathbf{MaxPool_{h}}(F_{in}),v_{max}^{v}=\mathbf{MaxPool_{v}}(F_{in})$
(3)

We get the direction encode features $v^{h}$ and $v^{v}$ from Eq.2 add Eq.3
correspondingly, so we can make the distribution of features like a normal
distribution, which can better reinforce the important information of input
features:

$v^{h}=v_{avg}^{h}+v_{max}^{h},v^{v}=v_{avg}^{v}+v_{max}^{v}$ (4)

We concatenate the encoded features and utilize the channel shuffle to
interaction channel information, aiming to exchange the encoding information
between different channels. Then utilizing the 1x1 convolution to reduce the
dimension of features, which pass through the nonlinear activation function,
so that the features of different channels are fully interactive. The formulas
are as follows:

$[y^{h}_{c/r},y^{v}_{c/r}]=\delta(\mathbf{Conv}(\mathbf{cat}([v^{h}_{c},v^{v}_{c}])))$
(5)

Wherein Eq.5, the $\delta$ is ReLU6 activation function, $c$ is the dimensions
number and $r$ is the channel scaling factor, $r$ is usually 4. We use the
shared 3x3 convolution to restore the number of channel dimensions for encoded
features of different directions, making the isotropic features get similar
attention weight values. The final weights can be determined by the larger
receptive field of the input features.

$y^{h}_{c}=\mathbf{Conv}(y^{h}_{c/r}),y^{v}_{c}=\mathbf{Conv}(y^{v}_{c/r})$
(6)

Figure 4: An illustration of the pipeline that generates a density map.

Our experiments demonstrate that the 3x3 convolution will help generate a more
accurate attention weight matrix. After restoring the channel of the feature,
multiply the $y_{c}^{h}$ and $y_{c}^{w}$ to obtain the attention weight matrix
with the same size as the input feature. Finally, a $Sigmoid$ function works
to get the attention map $W_{c\times h\times w}$:

$W_{c\times h\times w}=\mathbf{Sigmoid}(y^{h}_{c}\times y^{v}_{c})$ (7)

We multiply the attention weight matrix $W_{c\times h\times w}$ with the input
feature $F_{in}$ to get the output feature $F_{out}$:

$F_{out}=W_{c\times h\times w}\otimes F_{in}$ (8)

The entire SHA module calculates the attention weight matrix and the process
of applying attention weight to the input features, the overview as shown in
Fig.3. Compared with the previous attention mechanism for image dehazing, our
method is less computationally and more effective.

Figure 5: An illustration of the histogram of the diff map and density map.
The diff map is the numerical difference between the pseudo-haze-free image
and haze images. The value of the density map is mapped from [0,1] to [0,255].
The diff map and density map are visualized by ColorJet for observation and
comparison. Note that the histograms of the diff map and density map have
similar intensity distributions. Figure 6: Visual comparisons on real-world
hazy images, the diff map is the difference between the pseudo-haze-free and
hazy images. It’s worth noting that our density map effectively models the
density of the degradation spatially compared with the diff map.

###  3.2 Shallow Layers

The shallow layers is stacked by several Multi-branch Hybrid Attention with
COT Block and the Encoding-Decoding context module. We use the shallow layers
to generate the pseudo-haze-free image, which has high-level semantic
information.

#### Multi-branch Hybrid Attention.

We design the Multi-branch Hybrid Attention (MHA) Block, mainly consisting of
the SHA module with parallel convolution. The multi-branch design will improve
the expressive ability of the network by introducing multi-scale receptive-
filed. The multi-branch block comprises parallel 3x3 convolution, 1x1
convolution, and a residual connection, as shown in Fig.7 (a). The degradation
is often uneven in space in degraded images, such as hazy images and rainy
images, and the spatial structure and color of some areas in the picture are
not affected by the degradation of the scene, so we set the local residual
learning to let the feature pass the current block directly without any
process, which also avoids the disappearance of the gradient.

Figure 7: An illustration of the basic modules of the network. (a) The Multi-
branch Hybrid Attention Block (MHAB). (b) The MHAC block of shallow layers.
(c) The Tail Module.

#### Adaptive Features Fusion Module.

We hope that the network can adjust the proportion of feature fusion
adaptively according to the importance of different information. Different
from the MixUP [27] module in AECR-Net, we utilize the Adaptive Features
Fusion Module to combine two different blocks. The formula is as follows:

$\begin{split}F_{out}&=\mathbf{AFF}(\mathbf{block1},\mathbf{block2})\\\
&=\sigma(\theta)*\mathbf{block1}+\sigma(1-\theta)*\mathbf{block2}\\\
\end{split}$ (9)

Wherein, $block$ denotes the block module, which has the same output size,
$\sigma$ is the $Sigmoid$ activation function, and $\theta$ is a learnable
factor.

#### Multi-branch Hybrid Attention with CoT.

Long-range dependence is essential for feature representation, so we introduce
an improved CoT [17] block combined with the MHA block in the shallow layers
to mine the long-distance dependence of sample features that further expand
the receptive field. Specifically, we design a parallel mix block that uses
the MHA block and the improved CoT block to capture local features and global
dependencies simultaneously. To fusion the attention result, an Adaptive
Features Fusion module is followed back as shown in Fig.7 (b), we call this
MHAC block. The formulas are as follows:

$\begin{split}F_{out}&=\mathbf{AFF}(\mathbf{MHAB}(F_{in}),\mathbf{CoT}(F_{in}))\\\
&=\sigma(\theta)*(\mathbf{MHAB}(F_{in})+\sigma(1-\theta)*\mathbf{CoT}(F_{in})\\\
\end{split}$ (10)

The $F_{in}$ denotes the input features, $F_{out}$ is the adaptive mixing
results from MHA and CoT block. Considering that the BN layer will destroy the
internal features of the sample, we use the IN layer to replace the BN layer
in the improved CoT Block and use the ELU as the activation function. The
basic block of shallow layers.

#### Tail Module.

As shown in Fig.7(c), we design the Tail module, which fusions the extracted
features and restores the hazy image. The $tanh$ activation function is often
used on degradation reconstructions. So we use that as the activation of the
output after a stack of 3x3 convolution.

#### Architecture of Shallow Layers.

As shown in Fig.2, we use the shallow layers to reconstruct the degraded image
context content. In order to effectively reduce the amount of calculation and
expand the receptive field of the convolution of MHAC, we utilize 2
convolutions with a stride of 2 to reduce the resolution of the feature map to
1/4 of the original input firstly; each convolution follows a SHA module. Then
use a stack of 8 MHAC blocks with 256 channels, which have 2 skip connections
to introduce shallow features before up-sampling, and utilize the Tail module
for obtaining the residual of the restored image of the shallow layers:

$S(x)=\mathbf{Shallowlayers}(x)+x$ (11)

Wherein the $S(x)$ denotes the pseudo-haze-free image, the $x$ denotes the
hazy input image.

###  3.3 Density Map

To alleviate the problem of uneven haze distribution in the spatial position,
we adopt the density map to perceive the implicit spatial correlation between
pseudo-haze-free images generated by shallow layers and hazy images.

Like all classical deep learning methods, our approach adopts convolutions to
encode input to high-level feature representations that map the position
information to the activation area onto high-level feature maps. Unlike the
classical deep learning methods that only utilize information from the hazy
image, the proposed density map is based on fully exploring the correlation
between the pseudo-haze-free image and the hazy image.

As is depicted in Fig. 5, it is obvious that shallow convolution network has
the ability to reconstruct a pseudo haze-free image. But it’s weak to model
complex spatial distribution that is disturbed by contextual information
variation, which leads to an inaccurate estimation of the degradation model.

Since hand-crafted prior may not model spatial co-relationship between image
pairs effectively, we adopt a simple convolution network to estimate a density
map that shares the same spatial dimensions as input. As the illustration in
Fig. 4, Density Estimation Module splices the pseudo-haze-free image and the
hazy sample of the shallow layers in the channel dimension. We expand the
features to 64 channels using the 3x3 convolution after utilizing the
Reflected Padding to avoid the detail loss of the feature edge. And then
utilize the SHA module to explore perceiving the uneven degeneration of input
features fully, finally using a convolution operation to compress the shape of
the feature. The sigmoid function is used to get the density map
$M\in\mathbb{R}^{1\times H\times W}$.After getting the density map $M$, we
multiply $M$ by the input feature $F_{in}\in\mathbb{R}^{C\times H\times W}$ to
get the final output $F_{out}\in\mathbb{R}^{C\times H\times W}$:

$F_{out}=F_{in}\otimes M$ (12)

As is depicted in Fig. 5 and Fig. 6, the visualization of density map clearly
demonstrate that our density map effectively model the uneven haze
distribution spatially. The diff map is the numerical difference between the
pseudo-haze-free and haze input images, which can be considered a degenerated
layer of the hazy veil. The intensity of the density map changes with the
depth of scenes and hazy distribution.

###  3.4 Deep layers

A lot of degraded image restoration methods [6, 29, 7] often use the fully
Encoder-Decoder structure to learn a large receptive fields. However, repeated
down-sampling and up-sampling operations can easily cause the loss of texture
details, which will greatly affect the visual perception of the image and the
natural effect, so we utilize the deep layers to repair the pixel-level
texture details with the supervised signal from density map. As shown in Fig.
2, we utilize 10 MHA blocks with 16 channels to extract features at the
resolution of the original input. In order to avoid unnecessary calculations
caused by repeated extraction of features, we use the AFF module to introduce
and mix features refined by our density map from the shallow layers
adaptively.

##  4 Experiments

###  4.1 Datasets and Metrics

We chose the PSNR and SSIM as experimental metrics to measure the performance
of our SHA-Net.We train on two large synthetic datasets, RESIDE [16] and
Haze4k [20], and testing on SOTS [16] and Haze4k [20] testing sets,
respectively. The indoor training set of RESIDE [16] contains 1,399 clean
image and 13,990 hazy images generated by corresponding clean images. The
indoor testing set of SOTS contains 500 indoor images. The training set of
Haze4k [20] contains 3,000 hazy images with ground truth images and the
testing set of Haze4k [20] contains 1,000 hazy images with ground truth
images.

###  4.2 Training Settings

We augment the training dataset with randomly rotated by 90,180,270 degrees
and horizontal flip. The training image patches with the size $256\times 256$
are extracted as input $I_{in}$ of our network. The network is trained for
$7.5\times 10^{5}$, $1.5\times 10^{6}$ steps on Haze4k [20] and RESIDE [16]
respectively.We use Adam optimizer with initial learning rate of $2\times
10^{-4}$, and adopt the CyclicLR to adjust the learning rate, where on the
triangular mode, the value of gamma is 1.0,base momentum is 0.8, max momentum
is 0.9, base learning rate is initial learning rate and max learning rate is
$3\times 10^{-4}$. PyTorch [21] was used to implement our models with 4 RTX
3080 GPU with total batchsize of 40.

###  4.3 Loss Function

We only use Charbonnier loss [5] as our optimization objective:

$\mathcal{L}(\Theta)=\mathcal{L}_{\text{char
}}(S(x),J_{gt}(x))+\mathcal{L}_{\text{char }}(D(x),J_{gt}(x)))$ (13)

Where $\Theta$ denotes the parameters of our network, the $S(x)$ denotes the
pseudo-haze-free image, $D(x)$ denotes the output of deep layers, which is the
final output image, $J_{gt}$ stands for ground truth, and
$\mathcal{L}_{\text{char }}$ is the Charbonnier loss [5]:

$\mathcal{L}_{\text{char
}}=\frac{1}{N}\sum_{i=1}^{N}\sqrt{\left\|X^{i}-Y^{i}\right\|^{2}+\epsilon^{2}}$
(14)

with constant $\epsilon$ emiprically set to $1e^{-}3$ for all experiments.

###  4.4 Ablation Study

Table 1: Comparisons on Haze4k testset for different configurations of Shallow Layers. Model |  |  |  |  |
---|---|---|---|---|---
Base | ✓ | ✓ | ✓ | ✓ | ✓
+FA |  | ✓ |  |  |
+SHA |  |  | ✓ | ✓ | ✓
+CoT |  |  |  | ✓ | ✓
+AFF |  |  |  |  | ✓
PSNR | 24.40 | 25.30 | 26.39 | 26.84 | 27.28

To demonstrate the effectiveness of the proposed method, we conduct ablation
study to analyze different elements, including SHA, CoT, AFF and different
configurations of our network. We first construct our base Network as the
baseline of shallow layers of dehazing network, which lack SHA, CoT, and AFF.
Subsequently, we add the different modules into the baseline as: (1) base+SHA:
Add the Separable Hybrid Attention module into baseline. (2) base+FA: Add the
Feature Attention [22] module into baseline. (3) base+SHA+CoT: Add both SHA
and CoT operation into baseline. (4) base+SHA+CoT+AFF: The complete
combination of our shallow layers.

We employ the Charbonnier loss [5] as training loss function for ablation
study, and utilize Haze4k [20] dataset for both training and testing. The
performance of above models are summarized in Table 1.

Effect of Separable Hybrid Attention. Separable Hybrid Attention module
significantly improves the performance from Base to Base+SHA with an increase
of 1.99 dB PSNR. Therefore, SHA is a significant component due to the high-
performance gain. We also evaluate the performance of the FA [22] module with
the base model as show in Table 1.

We compared the Flops and Parameters of SHA module with other universal and
dehazing attention modules, such as the Feature Attention module [22] used in
dehazing networks [22, 27, 26], and the common CBAM [25],as shown in Table
2.Compared with other attention module used in image dehazing networks, the
Flops of our SHA module is lower.

The result demonstrates that our attention mechanism has better performance
because of the excellent feature coding mechanism.

Table 2: The Flops and Params comparison of universal attention mechanism and dehazing network. The SWRCA is the attention module of KDDN[11]. Attention | Flops(M) | Params
---|---|---
SE[12] | 4.195 | 512
ECA[23] | 4.195 | 3
CBA[25] | 10.619 | 1.122K
FA[22] | 38.864 | 1.625K
SWRCA[11] | 2424.311 | 41.088K
SHA(Ours) | 15.29 | 5.192K
Figure 8: Visual comparisons on dehazed results of various methods on
synthetic hazy images and our density map, the images from haze4k [20] test
dataset. The images are best viewed in the full-screen mode.

Study on Separable Hybrid Attention. We verify the effect of main operations
of Separable Hybrid Attention we proposed. As shown in Table 3, we demonstrate
that the combination of AvgPool and MaxPool is nontrivial for the performance
of attention mechanism we proposed. The Channel Shuffle operation boost the
encoding information exchanging among different channels, which is essential
for an efficient and effective attention mechanism.

Table 3: Comparisons on Haze4k testset for different configurations of Separable Hybrid Attention. AvgPool | ✓ | ✓ | ✓ |
---|---|---|---|---
MaxPool |  | ✓ | ✓ |
Channel Shuffle |  |  | ✓ |
PSNR | 28.28 | 31.16 | 33.49 |
Conv@k1 | ✓ |  |  |
Conv@k3 |  | ✓ |  |
PSNR | 32.21 | 33.49 | - |

We also verify the suitable kernel size of the last convolution of SHA, as
shown in Table 3; we choose the 3x3 convolution as the final processing
operation for parameter-performance trade-off.

Table 4: Comparisons on Haze4k testset for different configurations of our methods. Shallow Layers | ✓ | ✓ | ✓ |
---|---|---|---|---
Deep Layers |  | ✓ | ✓ |
density map |  |  | ✓ |
PSNR | 27.28 | 30.33 | 33.49 |

Effect of Deep Layers and Density Map. We verify the effect of deep layers of
our method. As described in section 3.4, the deep layers effectively relieve
the texture detail loss by repeated sampling. The deep layers significantly
improve the model performance with an increase of 3.05 dB PSNR, as shown in
Table 4. The result demonstrates that the feature extracted processing with
the original image resolution is essential for pixel-level texture detail
restoration. And we also verify the effect of density map we proposed, as
shown in Table 4 and Fig.6. It’s worth noting that our density map effectively
predicts the hazy density of hazy input, which models the uneven distribution
of hazy degradation and improves the performance with an increase of 3.16 dB
PSNR.

##  5 Compare with SOTA Methods

#### Visual Comparison.

To validate the superiority of our method, as shown in Fig.8, firstly we
compare the visual results of our method with previous SOTA methods on
synthetic hazy images from Haze4k [20] dataset. It can be seen that the other
methods are not able to remove the haze in all the cases, while the proposed
method produced results close to the real clean scenes visually. Our method is
superior in the recovery performance of image details and color fidelity.
Please refer to the supplementary materials for more visual comparisons on the
synthetic hazy images and real-world hazy images.

Table 5: Quantitative comparisons of our models with the state-of-the-art
dehazing methods on Haze4k [20] and SOTS [16] datasets (PSNR(dB)/SSIM). Best
results are underlined.

Method | Haze4k [20] | SOTS [16]
---|---|---
PSNR$\uparrow$ | SSIM$\uparrow$ | PSNR$\uparrow$ | SSIM$\uparrow$
DCP [10] | 14.01 | 0.76 | 15.09 | 0.76
NLD[3] | 15.27 | 0.67 | 17.27 | 0.75
DehazeNet [4] | 19.12 | 0.84 | 20.64 | 0.80
AOD-Net [15] | 17.15 | 0.83 | 19.82 | 0.82
GDN [19] | 23.29 | 0.93 | 32.16 | 0.98
MSBDN [7] | 22.99 | 0.85 | 33.79 | 0.98
FFA-Net [22] | 26.96 | 0.95 | 36.39 | 0.98
AECR-Net [27] | - | - | 37.17 | 0.99
DMT-Net [20] | 28.53 | 0.96 | - | -
Ours | 33.49 | 0.98 | 38.41 | 0.99

#### Quantitative Comparison.

We compare quantitatively the dehazing results of our SHA-Net with SOTA single
image dehazing methods on Haze4k [20] and SOTS [16] datasets. As shown in
Table 5, the SHA-Net outperforms all SOTA methods, achieving 33.49dB PSNR and
0.98 SSIM on Haze4k [20]. It increases the PSNR by 4.93dB compared to the
second best method. On SOTS [16] indoor test set, the SHA-Net also outperforms
all SOTA methods, achieving 38.41dB PSNR and 0.99 SSIM. It increases the PSNR
by 1.24dB, compared to the second best method.

##  6 Conclusion

In this paper, we propose a powerful image dehazing method to recover haze-
free images directly. Specifically, the Separable Hybrid Attention is design
to better perceive haze density and a density map is to further refine
extracted features. Although our method is simple, it is superior to all the
previous state-of-art methods with a very large margin on two large-scale hazy
datasets. Our method has a powerful advantage in the restoration of image
detail and color fidelity. We hope to further promote our method to other low-
level vision tasks such as deraining, super-resolution, denosing and
desnowing.

Limitations: The proposed method recovers high-fidelity haze-free images. As
is shown in Fig.8, haze free images are usually in low-light mode, which is
not visually pleasant in real-life scenes. Following the main idea of this
work, future research can be made in various aspects to generate high-quality
images with pleasant visual perceptions.

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