# Observation of a resonant structure in $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$

M. Ablikim1, M. N. Achasov10,e, P. Adlarson63, S. Ahmed15, M. Albrecht4, A.
Amoroso62A,62C, Q. An59,47, Anita21, Y. Bai46, O. Bakina28, R. Baldini
Ferroli23A, I. Balossino24A, Y. Ban37,m, K. Begzsuren26, J. V. Bennett5, N.
Berger27, M. Bertani23A, D. Bettoni24A, F. Bianchi62A,62C, J Biernat63, J.
Bloms56, I. Boyko28, R. A. Briere5, H. Cai64, X. Cai1,47, A. Calcaterra23A, G.
F. Cao1,51, N. Cao1,51, S. A. Cetin50B, J. F. Chang1,47, W. L. Chang1,51, G.
Chelkov28,c,d, D. Y. Chen6, G. Chen1, H. S. Chen1,51, M. L. Chen1,47, S. J.
Chen35, X. R. Chen25, Y. B. Chen1,47, W. Cheng62C, G. Cibinetto24A, F.
Cossio62C, X. F. Cui36, H. L. Dai1,47, J. P. Dai41,i, X. C. Dai1,51, A.
Dbeyssi15, D. Dedovich28, Z. Y. Deng1, A. Denig27, I. Denysenko28, M.
Destefanis62A,62C, F. De Mori62A,62C, Y. Ding33, C. Dong36, J. Dong1,47, L. Y.
Dong1,51, M. Y. Dong1,47,51, S. X. Du67, J. Fang1,47, S. S. Fang1,51, Y.
Fang1, R. Farinelli24A,24B, L. Fava62B,62C, F. Feldbauer4, G. Felici23A, C. Q.
Feng59,47, M. Fritsch4, C. D. Fu1, Y. Fu1, X. L. Gao59,47, Y. Gao37,m, Y.
Gao60, Y. G. Gao6, I. Garzia24A,24B, E. M. Gersabeck54, A. Gilman55, K.
Goetzen11, L. Gong36, W. X. Gong1,47, W. Gradl27, M. Greco62A,62C, L. M. Gu35,
M. H. Gu1,47, S. Gu2, Y. T. Gu13, C. Y Guan1,51, A. Q. Guo22, L. B. Guo34, R.
P. Guo39, Y. P. Guo27, Y. P. Guo9,j, A. Guskov28, S. Han64, T. T. Han40, T. Z.
Han9,j, X. Q. Hao16, F. A. Harris52, K. L. He1,51, F. H. Heinsius4, T. Held4,
Y. K. Heng1,47,51, M. Himmelreich11,h, T. Holtmann4, Y. R. Hou51, Z. L. Hou1,
H. M. Hu1,51, J. F. Hu41,i, T. Hu1,47,51, Y. Hu1, G. S. Huang59,47, L. Q.
Huang60, X. T. Huang40, N. Huesken56, T. Hussain61, W. Ikegami Andersson63, W.
Imoehl22, M. Irshad59,47, S. Jaeger4, S. Janchiv26,l, Q. Ji1, Q. P. Ji16, X.
B. Ji1,51, X. L. Ji1,47, H. B. Jiang40, X. S. Jiang1,47,51, X. Y. Jiang36, J.
B. Jiao40, Z. Jiao18, D. P. Jin1,47,51, S. Jin35, Y. Jin53, T. Johansson63, N.
Kalantar-Nayestanaki30, X. S. Kang33, R. Kappert30, M. Kavatsyuk30, B. C.
Ke42,1, I. K. Keshk4, A. Khoukaz56, P. Kiese27, R. Kiuchi1, R. Kliemt11, L.
Koch29, O. B. Kolcu50B,g, B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc63, M.
G. Kurth1,51, W. Kühn29, J. J. Lane54, J. S. Lange29, P. Larin15, L.
Lavezzi62C, H. Leithoff27, M. Lellmann27, T. Lenz27, C. Li38, C. H. Li32,
Cheng Li59,47, D. M. Li67, F. Li1,47, G. Li1, H. B. Li1,51, H. J. Li9,j, J. C.
Li1, J. L. Li40, Ke Li1, L. K. Li1, Lei Li3, P. L. Li59,47, P. R. Li31, S. Y.
Li49, W. D. Li1,51, W. G. Li1, X. H. Li59,47, X. L. Li40, X. N. Li1,47, Z. B.
Li48, Z. Y. Li48, H. Liang1,51, H. Liang59,47, Y. F. Liang44, Y. T. Liang25,
L. Z. Liao1,51, J. Libby21, C. X. Lin48, D. X. Lin15, B. Liu41,i, B. J. Liu1,
C. X. Liu1, D. Liu59,47, D. Y. Liu41,i, F. H. Liu43, Fang Liu1, Feng Liu6, H.
B. Liu13, H. M. Liu1,51, Huanhuan Liu1, Huihui Liu17, J. B. Liu59,47, J. Y.
Liu1,51, K. Liu1, K. Y. Liu33, Ke Liu6, L. Liu59,47, L. Y. Liu13, Q. Liu51, S.
B. Liu59,47, Shuai Liu45, T. Liu1,51, X. Liu31, X. Y. Liu1,51, Y. B. Liu36, Z.
A. Liu1,47,51, Z. Q. Liu40, Y. F. Long37,m, X. C. Lou1,47,51, H. J. Lu18, J.
D. Lu1,51, J. G. Lu1,47, X. L. Lu1, Y. Lu1, Y. P. Lu1,47, C. L. Luo34, M. X.
Luo66, P. W. Luo48, T. Luo9,j, X. L. Luo1,47, S. Lusso62C, X. R. Lyu51, F. C.
Ma33, H. L. Ma1, L. L. Ma40, M. M. Ma1,51, Q. M. Ma1, R. Q. Ma1,51, R. T.
Ma51, X. N. Ma36, X. X. Ma1,51, X. Y. Ma1,47, Y. M. Ma40, F. E. Maas15, M.
Maggiora62A,62C, S. Maldaner27, S. Malde57, Q. A. Malik61, A. Mangoni23B, Y.
J. Mao37,m, Z. P. Mao1, S. Marcello62A,62C, Z. X. Meng53, J. G.
Messchendorp30, G. Mezzadri24A, J. Min1,47, T. J. Min35, R. E. Mitchell22, X.
H. Mo1,47,51, Y. J. Mo6, C. Morales Morales15, N. Yu. Muchnoi10,e, H.
Muramatsu55, S. Nakhoul11,h, Y. Nefedov28, F. Nerling11,h, I. B. Nikolaev10,e,
Z. Ning1,47, S. Nisar8,k, S. L. Olsen51, Q. Ouyang1,47,51, S. Pacetti23B, X.
Pan45, Y. Pan54, M. Papenbrock63, A. Pathak1, P. Patteri23A, M. Pelizaeus4, H.
P. Peng59,47, K. Peters11,h, J. Pettersson63, J. L. Ping34, R. G. Ping1,51, A.
Pitka4, R. Poling55, V. Prasad59,47, H. Qi59,47, H. R. Qi49, M. Qi35, T. Y.
Qi2, S. Qian1,47, C. F. Qiao51, L. Q. Qin12, X. P. Qin13, X. S. Qin4, Z. H.
Qin1,47, J. F. Qiu1, S. Q. Qu36, K. H. Rashid61, K. Ravindran21, C. F.
Redmer27, M. Richter4, A. Rivetti62C, V. Rodin30, M. Rolo62C, G. Rong1,51, Ch.
Rosner15, M. Rump56, A. Sarantsev28,f, M. Savrié24B, Y. Schelhaas27, C.
Schnier4, K. Schoenning63, D. C. Shan45, W. Shan19, X. Y. Shan59,47, M.
Shao59,47, C. P. Shen2, P. X. Shen36, X. Y. Shen1,51, H. Y. Sheng1, H. C.
Shi59,47, R. S. Shi1,51, X. Shi1,47, X. D Shi59,47, J. J. Song40, Q. Q.
Song59,47, X. Y. Song1, Y. X. Song37,m, S. Sosio62A,62C, C. Sowa4, S.
Spataro62A,62C, F. F. Sui40, G. X. Sun1, J. F. Sun16, L. Sun64, S. S. Sun1,51,
T. Sun1,51, W. Y. Sun34, Y. J. Sun59,47, Y. K Sun59,47, Y. Z. Sun1, Z. J.
Sun1,47, Z. T. Sun1, Y. X. Tan59,47, C. J. Tang44, G. Y. Tang1, J. Tang48, X.
Tang1, V. Thoren63, B. Tsednee26, I. Uman50D, B. Wang1, B. L. Wang51, C. W.
Wang35, D. Y. Wang37,m, H. P. Wang1,51, K. Wang1,47, L. L. Wang1, L. S. Wang1,
M. Wang40, M. Z. Wang37,m, Meng Wang1,51, P. L. Wang1, W. P. Wang59,47, X.
Wang37,m, X. F. Wang31, X. L. Wang9,j, Y. Wang59,47, Y. Wang48, Y. D. Wang15,
Y. F. Wang1,47,51, Y. Q. Wang1, Z. Wang1,47, Z. G. Wang1,47, Z. Y. Wang1, Ziyi
Wang51, Zongyuan Wang1,51, T. Weber4, D. H. Wei12, P. Weidenkaff27, F.
Weidner56, H. W. Wen34,a, S. P. Wen1, D. J. White54, U. Wiedner4, G.
Wilkinson57, M. Wolke63, L. Wollenberg4, J. F. Wu1,51, L. H. Wu1, L. J.
Wu1,51, X. Wu9,j, Z. Wu1,47, L. Xia59,47, H. Xiao9,j, S. Y. Xiao1, Y. J.
Xiao1,51, Z. J. Xiao34, Y. G. Xie1,47, Y. H. Xie6, T. Y. Xing1,51, X. A.
Xiong1,51, G. F. Xu1, J. J. Xu35, Q. J. Xu14, W. Xu1,51, X. P. Xu45, L.
Yan62A,62C, L. Yan9,j, W. B. Yan59,47, W. C. Yan67, Xu Yan45, H. J. Yang41,i,
H. X. Yang1, L. Yang64, R. X. Yang59,47, S. L. Yang1,51, Y. H. Yang35, Y. X.
Yang12, Yifan Yang1,51, Zhi Yang25, M. Ye1,47, M. H. Ye7, J. H. Yin1, Z. Y.
You48, B. X. Yu1,47,51, C. X. Yu36, G. Yu1,51, J. S. Yu20,n, T. Yu60, C. Z.
Yuan1,51, W. Yuan62A,62C, X. Q. Yuan37,m, Y. Yuan1, C. X. Yue32, A.
Yuncu50B,b, A. A. Zafar61, Y. Zeng20,n, B. X. Zhang1, B. Y. Zhang1,47, C. C.
Zhang1, D. H. Zhang1, Guangyi Zhang16, H. H. Zhang48, H. Y. Zhang1,47, J. L.
Zhang65, J. Q. Zhang4, J. W. Zhang1,47,51, J. Y. Zhang1, J. Z. Zhang1,51,
Jianyu Zhang1,51, Jiawei Zhang1,51, L. Zhang1, Lei Zhang35, S. Zhang48, S. F.
Zhang35, T. J. Zhang41,i, X. Y. Zhang40, Y. Zhang57, Y. H. Zhang1,47, Y. T.
Zhang59,47, Yan Zhang59,47, Yao Zhang1, Yi Zhang9,j, Z. H. Zhang6, Z. Y.
Zhang64, G. Zhao1, J. Zhao32, J. W. Zhao1,47, J. Y. Zhao1,51, J. Z. Zhao1,47,
Lei Zhao59,47, Ling Zhao1, M. G. Zhao36, Q. Zhao1, S. J. Zhao67, T. C. Zhao1,
Y. B. Zhao1,47, Z. G. Zhao59,47, A. Zhemchugov28,c, B. Zheng60, J. P.
Zheng1,47, Y. Zheng37,m, Y. H. Zheng51, B. Zhong34, C. Zhong60, L. Zhou1,47,
L. P. Zhou1,51, Q. Zhou1,51, X. Zhou64, X. K. Zhou51, X. R. Zhou59,47, A. N.
Zhu1,51, J. Zhu36, K. Zhu1, K. J. Zhu1,47,51, S. H. Zhu58, W. J. Zhu36, X. L.
Zhu49, Y. C. Zhu59,47, Y. S. Zhu1,51, Z. A. Zhu1,51, J. Zhuang1,47, 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 200443, 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 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, 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 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
and University of Perugia, I-06100, Perugia, Italy
24 (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 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
28 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
29 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
30 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
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 Peking University, Beijing 100871, People’s Republic of China
38 Qufu Normal University, Qufu 273165, People’s Republic of China
39 Shandong Normal University, Jinan 250014, People’s Republic of China
40 Shandong University, Jinan 250100, People’s Republic of China
41 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
42 Shanxi Normal University, Linfen 041004, People’s Republic of China
43 Shanxi University, Taiyuan 030006, People’s Republic of China
44 Sichuan University, Chengdu 610064, People’s Republic of China
45 Soochow University, Suzhou 215006, People’s Republic of China
46 Southeast University, Nanjing 211100, People’s Republic of China
47 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
48 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
49 Tsinghua University, Beijing 100084, People’s Republic of China
50 (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (C)Uludag University, 16059 Bursa,
Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
51 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
52 University of Hawaii, Honolulu, Hawaii 96822, USA
53 University of Jinan, Jinan 250022, People’s Republic of China
54 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
55 University of Minnesota, Minneapolis, Minnesota 55455, USA
56 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
57 University of Oxford, Keble Rd, Oxford, UK OX13RH
58 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
59 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
60 University of South China, Hengyang 421001, People’s Republic of China
61 University of the Punjab, Lahore-54590, Pakistan
62 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
63 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
64 Wuhan University, Wuhan 430072, People’s Republic of China
65 Xinyang Normal University, Xinyang 464000, People’s Republic of China
66 Zhejiang University, Hangzhou 310027, People’s Republic of China
67 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Ankara University,06100 Tandogan, Ankara, Turkey
b Also at Bogazici University, 34342 Istanbul, Turkey
c Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
d Also at the Functional Electronics Laboratory, Tomsk State University,
Tomsk, 634050, Russia
e Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
f Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
g Also at Istanbul Arel University, 34295 Istanbul, Turkey
h Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
i 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
j 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
k Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA
l Currently at: Institute of Physics and Technology, Peace Ave.54B,
Ulaanbaatar 13330, Mongolia
m Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
n School of Physics and Electronics, Hunan University, Changsha 410082, China

###### Abstract

Using $e^{+}e^{-}$ collision data samples with center-of-mass energies ranging
from 2.000 to 2.644 GeV, collected by the BESIII detector at the BEPCII
collider, and with a total integrated luminosity of 300 pb-1, a partial-wave
analysis is performed for the process $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$. The total Born cross sections for the process
$e^{+}e^{-}\to$ $K^{+}K^{-}\pi^{0}\pi^{0}$, as well as the Born cross sections
for the subprocesses $e^{+}e^{-}\to\phi\pi^{0}\pi^{0}$, $K^{+}(1460)K^{-}$,
$K^{+}_{1}(1400)K^{-}$, $K^{+}_{1}(1270)K^{-}$ and $K^{*+}(892)K^{*-}(892)$,
are measured versus the center-of-mass energy. The corresponding results for
$e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ and $\phi\pi^{0}\pi^{0}$ are
consistent with those of BaBar and have much improved precision. By analyzing
the cross sections for the four subprocesses, $K^{+}(1460)K^{-}$,
$K^{+}_{1}(1400)K^{-}$, $K^{+}_{1}(1270)K^{-}$ and $K^{*+}(892)K^{*-}(892)$, a
structure with mass $M$ = (2126.5 $\pm$ 16.8 $\pm$ 12.4) MeV/$c^{2}$ and width
$\Gamma$ = (106.9 $\pm$ 32.1 $\pm$ 28.1) MeV is observed with an overall
statistical significance of 6.3$\sigma$, although with very limited
significance in the subprocesses $e^{+}e^{-}\to K^{+}_{1}(1270)K^{-}$ and
$K^{*+}(892)K^{*-}(892)$. The resonant parameters of the observed structure
suggest it can be identified with the $\phi(2170)$, thus the results provide
valuable input to the internal nature of the $\phi(2170)$.

The vector meson state $Y(2175)$, denoted as the $\phi(2170)$ by the Particle
Data Group (PDG) PDG , is currently one of the most interesting particles in
light hadron spectroscopy. The $\phi(2170)$ was first observed by BaBar
Y2170Babar and subsequently confirmed by several other experiments Y2170Babar1
; Y2170belle ; Y2170Bes ; Y2170Bes3 ; 2017bes32 . The internal constituents of
the $\phi(2170)$ are still unknown, which has stimulated extensive theoretical
discussions. Possible interpretations of the $\phi(2170)$ include a
conventional $3^{3}S_{1}$ or $2^{3}D_{1}$ $s\bar{s}$ state strange ; 2017ding
; 2017wang ; 2017afonin , an $s\bar{s}g$ hybrid 2017ding ; 2017ding2 ; hybrid2
, a tetraquark state 2017wang2 ; 2017chen ; 2019ke ; 2017drenska , a
$\Lambda\bar{\Lambda}(^{3}S_{1})$ bound state 2017zhao ; 2017deng ; 2017dong ,
or a $\phi KK$ resonance state 2017oset , etc., but no interpretation has yet
been established. Each of these theoretical models can accommodate a resonant
state with parameters similar to those of the $\phi(2170)$, but they predict
significantly different partial widths for individual decay modes, especially
the $K^{(*)}K^{(*)}$ decay modes, where the $K^{(*)}$ is the ground or excited
state of a $K$ meson with different spin-parities. Consequently, studying the
decay modes of the $\phi(2170)$, and precisely measuring their partial widths,
plays a key role in determining the internal structure of the $\phi(2170)$.

The BESII collaboration searched for the decay $\phi(2170)\to
K^{*0}(892)\bar{K}^{*0}(892)$ via $J/\psi\to\eta\phi(2170)$ by using 58
million $J/\psi$ events ksksupper . No significant signal was observed. The
BaBar collaboration performed an analysis of $e^{+}e^{-}\to
K^{+}K^{-}\pi^{+}\pi^{-}$ and $K^{+}K^{-}\pi^{0}\pi^{0}$ using 454 fb-1 data
via the initial state radiation (ISR) process Y2170Babar . Beside clearly
observing the process $e^{+}e^{-}\to\phi\pi\pi$, abundant $K^{*}$ structures
were observed in the $K\pi(\pi)$ invariant mass spectrum, such as the
$K^{*}(892)$ and $K^{*}_{2}(1430)$, as well as the $K_{1}(1270)$ and
$K_{1}(1400)$. It is worth noting that only about 1% of the $e^{+}e^{-}\to
K^{+}K^{-}\pi^{+}\pi^{-}$ events were from the subprocess $e^{+}e^{-}\to
K^{*0}(892)\bar{K}^{*0}(892)$, while roughly 30% of the $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$ events were from $e^{+}e^{-}\to
K^{*+}(892)K^{*-}(892)$. A comprehensive analysis, _e.g._ a partial-wave
analysis (PWA), is desired to resolve the contribution of individual
components in these decays.

Besides an excited $\phi$ state, the quark model also predicts excited $\rho$
and $\omega$ states in the 2 GeV/$c^{2}$ mass range Godfrey:1985xj . Finding
this set of excited vector mesons would help establish the corresponding
$\rho$, $\omega$, and $\phi$ meson families and would set a baseline for
theoretical models. Since these excited vector mesons can each decay into
$K^{(*)}K^{(*)}$ final states, analyzing the $K^{(*)}K^{(*)}$ invariant mass
spectra in $e^{+}e^{-}$ annihilation becomes an effective means to discover
them.

In this Letter, we present a PWA of the process $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$ using data collected with the BESIII detector. The
ten data samples used in this analysis have center-of-mass (c.m.) energies
ranging from 2.000 to 2.644 GeV and have a total integrated luminosity of 300
pb-1. The c.m. energy values and integrated luminosities of each data set are
presented in Table I in the supplemental material supply . Charge-conjugated
processes are always included by default.

Detailed descriptions of the design and performance of the BESIII detector can
be found in Ref. besint . A Monte Carlo (MC) simulation based on Geant4 geant4
, including the geometric description of the BESIII detector and its response,
is used to optimize the event selection criteria, estimate backgrounds, and
determine the detection efficiency. The signal MC samples are generated using
the package ConExc conexc , which incorporates a higher-order ISR correction.
Background samples of the processes $e^{+}e^{-}\to e^{+}e^{-}$,
$\mu^{+}\mu^{-}$ and $\gamma\gamma$ are generated with the Babayaga babayaga
generator, while $e^{+}e^{-}\to$ hadrons and two photon events are generated
by the Luarlw lumarlw and Bestwogam bestwogam generators, respectively.

The selection criteria for charged tracks, particle identification (PID), and
photon candidates are the same as those in Ref. selec .

The process $e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ results in the final
state $K^{+}K^{-}\gamma\gamma\gamma\gamma$. Thus, candidate events with only
two oppositely-charged kaons and at least four photons are selected. To
improve the kinematic resolution and suppress background, a six-constraint
(6C) kinematic fit imposing energy-momentum conservation, as well as two
additional $\pi^{0}$ mass constraints, is carried out under the hypothesis
$e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$. The combination with minimum
$\chi^{2}_{6C}$ is retained for further analysis. The candidate events are
required to satisfy $\chi^{2}_{6C}$ $<$ $80$. After the above selection
criteria, detailed studies indicate that the backgrounds are negligible.

Using the GPUPWA framework PWAframe , a PWA is performed on the surviving
candidate events to disentangle the intermediate processes present in
$e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$. The quasi two-body decay amplitudes
in the sequential decays are constructed using covariant tensor amplitudes
PWAtensor . The intermediate states are parameterized with relativistic Breit-
Wigner (BW) functions, except for the $f_{0}(980)$, which is described with a
Flatté formula PWA:sigma1 . The resonance parameters of the $f_{0}(980)$ and
the wide resonance $\sigma$ in the fit are fixed to those in Ref. PWA:sigma1
and Ref. PWA:sigma1 ; PWA:sigma2 , respectively, and those of other
intermediate states are fixed to PDG values, or measured in the analysis. To
include the resolution for the narrow $\phi(1020)$ resonance, a Gaussian
function is convolved with the BW function, but this is not done for the other
resonances. The relative magnitudes and phases of the individual intermediate
processes are determined by performing an unbinned maximum likelihood fit
using MINUIT PWA:minuit1 .

We start the fit procedure by including all possible intermediate states in
the PDG that conserve $\rm J^{PC}$, where these intermediate states can decay
into $K^{+}K^{-}$, $\pi^{0}\pi^{0}$, $K^{\pm}\pi^{0}$, $K^{+}K^{-}\pi^{0}$ or
$K^{\pm}\pi^{0}\pi^{0}$ final states. Then we examine the statistical
significance of the individual amplitudes, and drop the ones with statistical
significance less than 5$\sigma$. The process is repeated until no amplitude
remains with a statistical significance less than 5$\sigma$. After that, all
the removed processes are reintroduced individually to make sure that they are
not needed in the fit. In the above approach, the statistical significance of
each individual amplitude is determined by the changes in the negative log
likelihood (NLL) value and the number of free parameters in the fit with and
without the corresponding amplitude included.

The above strategy is performed individually on the data sets at
$\sqrt{s}=2.125$ and 2.396 GeV, which have the largest luminosities among the
ten data sets. The nominal solution for data at $\sqrt{s}=2.125$ GeV includes
the two-body decay processes $K^{+}(1460)K^{-}$, $K^{+}_{1}(1270)K^{-}$,
$K^{+}_{1}(1400)K^{-}$, $K^{*+}(892)K^{*-}(892)$,
$K^{*+}_{0}(1430)K^{*-}(892)$, $\phi(1020)\sigma$, $\phi(1020)f_{0}(980)$,
$\phi(1020)f_{2}(1270)$ and $\omega(1420)\pi^{0}$, as well as the three-body
decay processes $K^{+}K^{-}\sigma$, $K^{+}K^{-}f_{0}(980)$ and
$K^{+}K^{-}f_{0}(1370)$. For the data at $\sqrt{s}=2.396$ GeV, the additional
intermediate processes $K^{*+}_{2}(1430)K^{*-}(892)$,
$K^{*+}(892)K^{-}\pi^{0}$ and $\phi(1020)f_{0}(1370)$ are included, but
without the $\phi(1020)\sigma$ and $\phi(1020)f_{2}(1270)$ processes. An
interesting decay mode $K^{*+}(1410)K^{-}$, which is expected to have a
sizeable decay rate for a conventional $3^{3}S_{1}$ $s\bar{s}$ state 2017ding
, is found to be less than 3$\sigma$ in both data samples. In the above, the
three-body decays are treated as consecutive quasi two-body decays with a very
broad resonance decaying into $K^{+}K^{-}$ or $K^{+}\pi^{0}$ and modeled as a
1- phase space distribution. The intermediate states $K^{+}(1460)$,
$K^{+}_{1}(1270)$, $K^{+}_{1}(1400)$ decay into $K^{*+}(892)\pi^{0}$, and
$\omega(1420)$ decays into $K^{*\pm}(892)K^{\mp}$, followed by $K^{*+}(892)\to
K^{+}\pi^{0}$. The state $K^{*+}_{0}(1430)$ decays into $K^{+}\pi^{0}$. The
state $\phi(1020)$ decays into $K^{+}K^{-}$ and $\sigma$, $f_{0}(980)$,
$f_{2}(1270)$, $f_{0}(1370)$ decay into $\pi^{0}\pi^{0}$. The masses and
widths of the $K(1460)$, $K_{1}(1400)$, $K_{1}(1270)$ and $\omega(1420)$ in
the fit are determined by scanning the likelihood value, and the results are
consistent with the parameters in the PDG. The masses and widths of other
intermediate states are fixed to PDG values. The statistical significance of
all intermediate processes are summarized in sections II and III of the
supplemental material supply , respectively. The corresponding comparison of
invariant mass spectra and angular distributions between data and MC
projections are shown in section IV of the supplemental material.

For the other eight data samples, due to limited statistics, we do not perform
the above optimization strategy to determine which intermediate processes to
include. Instead, we use the same intermediate processes as the data sets with
nearby c.m. energy. The data sets with $\sqrt{s}=$ 2.000, 2.100, 2.175, 2.200
and 2.232 GeV (referred to as group I data), use the same processes as
$\sqrt{s}=2.125~{}\mbox{GeV}$, while the other three points (group II data)
use the same processes as $\sqrt{s}=2.396~{}\mbox{GeV}$.

The total Born cross sections for $e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ and
the Born cross sections for the intermediate processes are obtained at each
c.m. energy using:

$\sigma^{B}=\frac{N^{sig}}{\mathcal{L}_{int}~{}\frac{1}{|1-\Pi|^{2}}~{}(1+\delta)^{r}~{}\mathcal{B}r~{}\epsilon},$
(1)

where $N^{sig}$ is the corresponding signal yield, and is determined by
calculating the fraction according to the PWA results for the individual
intermediate process; $\mathcal{L}_{int}$ is the integrated luminosity;
$(1+\delta)^{r}$ is the ISR correction factor obtained from a QED calculation
VR ; conexc and incorporating the input cross section from this analysis
iteratively; $\frac{1}{|1-\Pi|^{2}}$ is the vacuum polarization factor taken
from a QED calculation VP ; $\epsilon$ is the detection efficiency obtained
from a PWA-weighted MC sample; and $\mathcal{B}r$ is the product of branching
ratios of the intermediate states as quoted in the PDG PDG . In the decay
$e^{+}e^{-}\to K^{+}(1460)K^{-}$, the branching fraction of $K(1460)\to
K^{*}(892)\pi$ is included in the measured cross section since it has never
been measured.

Two categories of systematic uncertainties are considered in the measurement
of the Born cross sections. The first category includes uncertainties
associated with the luminosity, track detection, PID, kinematic fit, ISR
correction, and the branching fractions of intermediate states. The
uncertainty associated with the integrated luminosity is 1% at each energy
point lum . The uncertainty of the detection efficiency is 1% for each charged
track trackerror and photon photonerror . The PID efficiency uncertainty is
1.0% for each charged track trackerror . The uncertainty related to the
kinematic fit is estimated by correcting the helix parameters of the simulated
charged tracks to match the resolution helixsys . The uncertainty associated
with the ISR correction factor is estimated to be the difference of
$(1+\delta^{r})\epsilon$ between the last two iterations in the cross section
measurement. The systematic uncertainties from the branching ratios of
intermediate states in the subsequent decays are taken from the PDG PDG . The
second category of uncertainties are from the PWA fit procedure. Fits with
alternative scenarios are performed, and the changes of signal yields are
taken as systematic uncertainties. Uncertainties from the BW parameterization
are estimated by replacing the constant-width BW with the mass-dependent
width. Uncertainties associated with the resonance parameters, which are taken
from the PDG and fixed in the fit, are estimated by alternative fits
superposing additional constraints on these resonance parameters, where the
superposed constraints follow Gaussian distributions with widths equal to
their uncertainties. One thousand fits are performed, and the resultant
standard deviations of the signal yields are taken as systematic
uncertainties. Uncertainties associated with the additional resonances are
estimated by alternative fits including the components $K^{*}(1410)K$ or the
$K^{*}_{2}(1430)K^{*}$, which are most significant, but less than 5$\sigma$.
Uncertainties due to the barrier factor are estimated by varying the radius of
the centrifugal barrier from 0.7 to 1.0 fm. To estimate the uncertainties on
the detection efficiency related to the fit parameters in the PWA, one hundred
MC samples are generated with five hundred groups of parameters of PWA
amplitudes which is sampled from a multi-variable Gaussian function according
to their mean values and their covariance error matrix from the nominal fit.
The standard deviations of the resultant detection efficiencies are considered
as the uncertainties.

In the above procedure, the uncertainties associated with the barrier factor,
resonance parameterization and additional resonances are strongly affected by
the statistics. Thus, those uncertainties of data with $\sqrt{s}$=2.125 GeV
are assigned to the group I data, while those of data with $\sqrt{s}$=2.396
GeV are assigned to the group II data. Assuming all sources of systematic
uncertainties are independent, the total uncertainties are the quadratic sums
of the individual values, shown in section V of the supplemental material
supply , where the sources of the uncertainties tagged with ‘*’ are assumed to
be 100% correlated among each energy points.

The measured total Born cross sections for $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$ and the Born cross sections for the subprocess
$e^{+}e^{-}\to\phi\pi^{0}\pi^{0}$, summing over all the $\pi^{0}\pi^{0}$
intermediate processes and their interferences, are shown in Fig. 1. Good
agreement is found with the previous results from BaBar. In order to study the
properties of $1^{--}$ states, the cross sections for the processes
$e^{+}e^{-}\to K^{+}(1460)K^{-}$, $K^{+}_{1}(1400)K^{-}$,
$K^{+}_{1}(1270)K^{-}$ and $K^{*+}(892)K^{*-}(892)$, referred to as the $KK$
processes, are shown in Fig. 2. A clear peak between 2.1 and 2.2 GeV is
present in the process $e^{+}e^{-}\to K^{+}(1460)K^{-}$, and dips are observed
for the processes $e^{+}e^{-}\to K^{+}_{1}(1400)K^{-}$ and
$K^{+}_{1}(1270)K^{-}$ in almost the same energy region. This may be due to
destructive interference between different components. No obvious structure or
dip is present in the process $e^{+}e^{-}\to K^{*+}(892)K^{*-}(892)$. All the
various numbers used in the cross section calculation are summarized in
section I of the supplemental material supply .

Figure 1: The Born cross sections for (a) the process $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$ and (b) the subprocess
$e^{+}e^{-}\to\phi\pi^{0}\pi^{0}$. The red squares are from this analysis; the
blue dots are from the BaBar experiment.

Figure 2: Fit to the cross sections for $e^{+}e^{-}$ to the final states (a)
$K^{+}(1460)K^{-}$, (b) $K^{+}_{1}(1400)K^{-}$, (c) $K^{+}_{1}(1270)K^{-}$ and
(d) $K^{*+}(892)K^{*-}(892)$, where black dots with errors are data, the black
solid curves are the overall fit results, the red long-dashed curves are from
the intermediate state, the green short-dashed curves are from the continuum
component, and the blue dash-dotted curves are the interference contribution
for solution 1.

To further examine the structure, a binned $\chi^{2}$ fit, incorporating the
correlated and uncorrelated uncertainties among different energy points, is
performed to the cross sections for the $K^{+}(1460)K^{-}$,
$K^{+}_{1}(1400)K^{-}$, $K^{+}_{1}(1270)K^{-}$ and $K^{*+}(892)K^{*-}(892)$
processes. The fit probability density function (PDF) for the individual
processes is the coherent sum of a continuum component $f_{1}$ and a resonant
component $f_{2}$ :

$\mathcal{A}=f_{1}+e^{i\phi}f_{2},$ (2)

where $\phi$ is the relative phase between the two components. By considering
phase space $\Phi(\sqrt{s})$, the energy-dependent cross section of the QED
process, and the relative orbital angular momentum $L$ in the two-body decay,
the amplitude $f_{1}$ is described as

$f_{1}=q^{L}\frac{\sqrt{\Phi(\sqrt{s})}}{s^{n}},$ (3)

where $q$ is the momentum of the daughter particle. The resonant amplitude
$f_{2}$ is described with a BW function,

$f_{2}=\frac{M_{R}}{\sqrt{s}}\frac{\sqrt{12\pi\mathcal{B}r\Gamma^{e^{+}e^{-}}_{R}\Gamma_{R}}}{s-M_{R}^{2}+iM_{R}\Gamma_{R}}(\frac{q}{q_{0}})^{L}\sqrt{\frac{\Phi(\sqrt{s})}{\Phi(M_{R})}},$
(4)

where $M_{R}$ is the mass of the structure, $\Gamma_{R}$ is the total width,
$\Gamma^{e^{+}e^{-}}_{R}$ is its partial width to $e^{+}e^{-}$, $\mathcal{B}r$
is the decay branching fraction to a given final state, and $q_{0}$ is the
momenta of the daughter particle in the rest frame of the parent particle
($M_{R}$).

A simultaneous fit, assuming the same structure among the $K^{+}(1460)K^{-}$,
$K^{+}_{1}(1400)K^{-}$, $K^{+}_{1}(1270)K^{-}$ and $K^{*+}(892)K^{*-}(892)$
processes, is performed to the measured cross sections, as shown in Fig. 2. In
the fit, $M_{R}$ and $\Gamma_{R}$ are shared parameters between the four
processes and are floated, while $\it n$, the production
$\mathcal{B}r\Gamma^{e^{+}e^{-}}_{R}$, and the relative phase angle $\phi$ are
floated and final state dependent. For $e^{+}e^{-}\to K^{+}_{1}(1270)K^{-}$
and $K^{+}_{1}(1400)K^{-}$, $L=0$, while $L=1$ for the other two modes. The
fit results have two solutions with equal fit quality, identical
$M_{R}=(2126.5\pm 16.8)~{}\mbox{MeV/$c^{2}$}$ and $\Gamma_{R}=(106.9\pm
32.1)~{}\mbox{MeV}$, but different $\mathcal{B}r\Gamma^{e^{+}e^{-}}_{R}$ and
$\phi$ for the processes $e^{+}e^{-}\to K^{+}_{1}(1400)K^{-}$ and
$K^{+}_{1}(1270)K^{-}$, as summarized in Table 1. The statistical significance
of the structure is estimated with the change of $\chi^{2}$ ($\Delta\chi^{2}$)
and the number of degrees of freedom ($\Delta$ndof) between the scenarios with
and without the structure included in the fit. The overall statistical
significance is 6.3$\sigma$, obtained with $\Delta\chi^{2}$=63.8 and
$\Delta$ndof=10. The significance of the resonant state for each $KK$ process
is also estimated and summarized in Table 1. The significances of the resonant
state in the processes $e^{+}e^{-}\to K^{+}(1460)K^{-}$ and
$K_{1}^{+}(1400)K^{-}$ are greater than 4.5$\sigma$, while no significant
signal is found in the other two processes. We also estimate the upper limit
at the 90% confidence level on the production
$\mathcal{B}r\Gamma^{e^{+}e^{-}}_{R}$ to be 1.9 eV for $e^{+}e^{-}\to
K^{*+}(892)K^{*-}(892)$ and 12.5(297.6) eV for $e^{+}e^{-}\to
K^{+}_{1}(1270)K^{-}$.

Table 1: A summary of fit results. Channel |  |  $\mathcal{B}_{r}\Gamma^{e^{+}e^{-}}_{R}$ (eV) |  $\phi$ (rad) |  Sig. ($\sigma$)
---|---|---|---|---
$K^{+}(1460)K^{-}$ |  |  3.0 $\pm$ 3.8 |  5.6 $\pm$ 1.5 | 4.4
$K^{+}_{1}(1400)K^{-}$ | solution 1 |  4.7 $\pm$ 3.3 |  3.7 $\pm$ 0.4 | 4.8
solution 2 |  98.8 $\pm$ 7.8 |  4.5 $\pm$ 0.3
$K^{+}_{1}(1270)K^{-}$ | solution 1 |  7.6 $\pm$ 3.7 |  4.0 $\pm$ 0.2 | 1.4
solution 2 |  152.6 $\pm$ 14.2 |  4.5 $\pm$ 0.1
$K^{*+}(892)K^{*-}(892)$ |  |  0.04 $\pm$ 0.2 |  5.8 $\pm$ 1.9 | 1.2

The systematic uncertainties on the resonant parameters come from the absolute
c.m. energy measurement, the measured cross section, and the fit procedure.
The uncertainty of the c.m. energy from BEPCII is small, and is ignored in the
determination of the parameters of the structure. The statistical and
systematic uncertainties of the measured cross section are incorporated in the
fit, thus no further uncertainty is necessary. The uncertainties associated
with the fit procedure include those from the fit range and signal model. The
uncertainty from the fit range is investigated by excluding the last energy
point $\sqrt{s}=2.644$ GeV in the fit. The resultant changes, 5.1 MeV/$c^{2}$
for mass and 9.1 MeV for width, are taken as the systematic uncertainties. To
assess the systematic uncertainty associated with the signal model, an
alternative BW function with energy-dependent width is implemented in the fit,
and results in differences of 11.3 MeV/$c^{2}$ and 26.5 MeV for mass and
width, respectively, which are taken as the systematic uncertainties. The
overall systematic uncertainties are the quadratic sum of the individual ones,
12.4 MeV/$c^{2}$ and 28.1 MeV for the mass and width, respectively.

In summary, a PWA of the process $e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ is
performed for ten data samples with c.m. energies from 2.000 to 2.644 GeV and
with an integrated luminosity of 300 pb-1. The Born cross sections for
$e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ and $\phi\pi^{0}\pi^{0}$ are obtained
and are consistent with those from the BaBar experiment. We also measure the
cross sections for the processes $e^{+}e^{-}\to K^{+}(1460)K^{-}$,
$K^{+}_{1}(1400)K^{-}$, $K^{+}_{1}(1270)K^{-}$, and $K^{*+}(892)K^{*-}(892)$,
individually, and perform a simultaneous fit on the obtained results. The fit
results in a structure with mass $M$ = (2126.5 $\pm$ 16.8 $\pm$ 12.4)
MeV/$c^{2}$, width $\Gamma$ = (106.9 $\pm$ 32.1 $\pm$ 28.1) MeV, and
statistical significance 6.3 $\sigma$, where the uncertainties are statistical
and systematic, respectively. The structure is directly produced in
$e^{+}e^{-}$ collisions, thus has $J^{PC}=1^{--}$. This structure has a mass
close to the masses of the vector particles $\phi(2170)$, $\rho(2150)$ and
$\omega(2290)$ listed in the PDG PDG . Its width is only consistent with the
$\phi(21770)$ and is different from the others by more than 3$\sigma$.

Assuming the observed structure is $\phi(2170)$, our measurement implies that
the $\phi(2170)$ has a sizable partial width to $K^{+}(1460)K^{-}$,
$K_{1}^{+}(1400)K^{-}$, and $K_{1}^{+}(1270)K^{-}$, but a much smaller partial
width to $K^{*+}(892)K^{*-}(892)$ and $K^{*+}(1410)K^{-}$. According to Ref.
2017ding , the $3^{3}S_{1}$ $s\bar{s}$ state mainly decays to
$K^{*+}(892)K^{*-}(892)$ and $K^{*+}(1410)K^{-}$, but has a much smaller
partial width to $K_{1}^{+}(1400)K^{-}$ and $K^{+}(1460)K^{-}$. On the other
hand, the $2^{3}D_{1}$ $s\bar{s}$ state has an expected partial width to
$K_{1}^{+}(1400)K^{-}$ smaller than that to $K^{*+}(1410)K^{-}$ by a factor of
2-5 2017ding ; 2017wang . A hybrid state is expected to decay dominantly into
$K_{1}^{+}(1270)K^{-}$ and $K_{1}^{+}(1400)K^{-}$, while it should be highly
suppressed in the modes $K^{*+}(892)K^{*-}(892)$ and $K^{+}(1460)K^{-}$
2017ding2 . None of the above theoretical expectations are in good agreement
with our experimental results.

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 Basic Research Program of China under Contract No. 2015CB856700; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 11625523,
11635010, 11735014, 11822506, 11835012; 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. U1532257, U1532258, U1732263,
U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos.
QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and
Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract
No. 758462; German Research Foundation DFG under Contracts Nos. Collaborative
Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare,
Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470;
National Science and Technology fund; 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-0010118, DE-SC-0012069; University of Groningen (RuG) and the
Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt.

## References

  * (1) K. A. Olive et al. [Particle Data Group], Chin. Phys. C, 38, 090001 (2018).
  * (2) B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 74, 091103(R) (2006).
  * (3) B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 76, 012008 (2007).
  * (4) C. P. Shen et al. (Belle Collaboration), Phys. Rev. D 80, 031101(R) (2009)
  * (5) M. Ablikim et al. (BES Collaboration), Phys. Rev. Lett. 100, 102003 (2008).
  * (6) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 91, 052017 (2015).
  * (7) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 99, 012014 (2019).
  * (8) T. Barnes, et al., Phys. Rev. D 68 054014 (2003).
  * (9) G. J. Ding and M. L. Yan, Phys. Lett. B 657, 49 (2007).
  * (10) X. Wang et al., Phys. Rev. D 85, 074024 (2012).
  * (11) S. S. Afonin and I. V. Pusenkov, Phys. Rev. D 90, 094020 (2014).
  * (12) G. J. Ding and M. L. Yan, Phys. Lett. B 650, 390 (2007).
  * (13) P. R. Page, E. S. Swanson, and A. P. Szczepaniak, Phys. Rev. D 59, 034016 (1999).
  * (14) Z. G. Wang, Nucl. Phys. A 791, 106 (2007).
  * (15) H. X. Chen et al., Phys. Rev. D 78, 034012 (2008).
  * (16) H. W. Ke and X. Q. Li, Phys. Rev. D 99, 036014 (2019).
  * (17) N. V. Drenska, R. Faccini and A. D. Polosa, Phys. Lett. B 669, 160 (2008).
  * (18) L. Zhao et al., Phys. Rev. D 87, 054034 (2013).
  * (19) C. Deng et al., Phys. Rev. D 88, 074007 (2013).
  * (20) Yubing Dong et al., Phys. Rev. D 96, 074027 (2017).
  * (21) A. Martinez Torres et al., Phys. Rev. D 78, 074031 (2008); S. Gomez-Avila, M. Napsuciale and E. Oset, Phys. Rev. D 79, 034018 (2009).
  * (22) M. Ablikim et al. (BES Collaboration), Phys. Lett. B 685 27 (2010).
  * (23) S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * (24) See Supplemental Material at [URL to be inserted by publisher] for a summary of the number of the number of signal events, luminosity, cross section and systematic uncertainty at each energy point, the comparison of invariant mass spectra and angular distribution between data and fit results at $\sqrt{s}$ =2.125 and 2.396 GeV.
  * (25) M. Ablikim et al. (BESIII Collaboration), Nucl. Instrum. Meth. A 614, 3 (2010).
  * (26) S. Agostinelli et al. (Geant4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003).
  * (27) R. G. Ping, Chin. Phys. C 38, 083001 (2014).
  * (28) G. Balossini et al., Nucl. Phys. B758, 227 (2006).
  * (29) B. Andersson and H. Hu, arXiv:hep-ph/9910285.
  * (30) M. Ablikim, et al. (BESIII Collaboration), Chin. Phys. C 41, 063001 (2017).
  * (31) M. Ablikim, et al. (BESIII Collaboration), Phys. Rev. D 83, 032003 (2011).
  * (32) N. Berger, B. J. Liu, and J. K.Wang, J. Phys. Conf. Ser. 219, 042031 (2010).
  * (33) B. S. Zou and D. V. Bugg, Eur. Phys. J. A 16, 537 (2003).
  * (34) M. Ablikim, et al. (BESII Collaboration), Phys. Lett. B 598, 149 (2004).
  * (35) M. Ablikim, et al. (BESII Collaboration), Phys. Lett. B 645, 19 (2007).
  * (36) F. James and M. Roos, Comput. Phys. Commun. 10, 343 (1975).
  * (37) E. A. Kuraev and V. S. Fadin, Sov. J. Nucl. Phys. 41, 466 (1985).
  * (38) S. Actis et al., Eur. Phys. J. C 66, 585 (2010).
  * (39) M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 41, 063001 (2017).
  * (40) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 99, 032001 (2019).
  * (41) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 81, 052005 (2010).
  * (42) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 87, 012002 (2013).