# Study of the resonance structures in
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ process

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, 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. F. 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, 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, 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, G. R. Liao12, L. Z.
Liao42, J. Libby21, A. Limphirat52, C. X. Lin51, D. X. Lin25, 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.
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, 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. T. Tian25, 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, 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
Author list

###### Abstract

Using about 23 $\mathrm{fb^{-1}}$ of data collected with the BESIII detector
operating at the BEPCII storage ring, a precise measurement of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ Born cross section is performed at
center-of-mass energies from 3.7730 to 4.7008 GeV. Two structures, identified
as the $Y(4220)$ and the $Y(4320)$ states, are observed in the energy-
dependent cross section with a significance larger than $10\sigma$. The masses
and widths of the two structures are determined to be ($M,~{}\Gamma$) =
($4221.4\pm 1.5\pm 2.0$ MeV/$c^{2}$, $41.8\pm 2.9\pm 2.7$ MeV) and
($M,~{}\Gamma$) = ($4298\pm 12\pm 26$ MeV/$c^{2}$, $127\pm 17\pm 10$ MeV),
respectively. A small enhancement around 4.5 GeV with a significance about
$3\sigma$, compatible with the $\psi(4415)$, might also indicate the presence
of an additional resonance in the spectrum. The inclusion of this additional
contribution in the fit to the cross section affects the resonance parameters
of the $Y(4320)$ state.

##  I Introduction

A series of charmonium-like states, commonly referred to as $XYZ$ states, has
been observed in the last two decades. The $Y(4260)$ was first observed by
BaBar using an initial state radiation (ISR) technique in the process
$e^{+}e^{-}\rightarrow\gamma_{\rm{ISR}}\pi^{+}\pi^{-}J/\psi$ [1] and was soon
confirmed by Belle in the same process [2]. In 2017, the BESIII collaboration
was able to resolve the $Y(4260)$ structure into a combination of two
resonances, the $Y(4220)$ and the $Y(4320)$, using the world’s largest sample
of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ events [3]. Moreover, similar
structures have been observed in the processes
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\psi(2S)$ [4], $\pi^{+}\pi^{-}h_{c}$ [5],
$\pi^{+}D^{0}D^{\ast-}$ [6], $\eta J/\psi$ [7] and $\omega\chi_{cJ}$ $(J=0,\
1,\ 2)$ [8, 9] with the BESIII data. The parameters of the two resonances are
to a large extent consistent among reactions. However, to understand whether
the structures observed in different final states are indeed the same, more
investigations are needed.

Conventional charmonium states, such as the $\psi(4040)$, $\psi(4160)$, and
$\psi(4415)$, mainly decay into open charm final states
($D^{(\ast)}\bar{D}^{(\ast)}$), while the $Y$ states show strong coupling to
the hidden-charm final states [10]. The number of observed vector states in
this energy region exceeds that of the predicted vector charmonium states
[11]. These features suggest that some of these supernumerary vector states
are candidates of an exotic nature, such as hybrid, tetraquark states, or
mesonic molecules [12, 13, 14, 15, 16, 17]. To clarify the nature of these
states and to distinguish between the different theoretical models, precise
measurements of the production cross section and of the resonance parameters
are essential.

In this paper, an improved measurement of the energy-dependent
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ cross section at center-of-mass
energies ($\sqrt{s}$) between 3.7730 to 4.7008 GeV with a total integrated
luminosity of about 23 fb-1 is presented. The $XYZ$ data and the $R$-scan data
(between $\sqrt{s}=4.410$ and $4.590$ GeV) analyzed in Ref. [3] are also
analyzed here, and thus these measurements are correlated and can not be used
in combination. Improvements in this paper include the use of additional data
in the $Y(4220)$/$Y(4320)$ mass region, allowing us to study these two states
in more detail, as well as the inclusion of more detailed background studies
and systematic studies of the event selection.

##  II The BESIII detector and data sets

The BESIII detector [18] records $e^{+}e^{-}$ collisions provided by the
BEPCII storage ring [19] in a center-of-mass energy range from 2.0 to 4.9 GeV.
The cylindrical core of the detector covers 93% of the full solid angle and
consists of a helium-based multilayer drift chamber (MDC), a plastic
scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic
calorimeter (EMC), which are all enclosed in a superconducting solenoidal
magnet providing a 1.0 T magnetic field. The solenoid is supported by an
octagonal flux-return yoke with resistive plate counter muon identification
modules interleaved with steel. The charged-particle momentum resolution at
$1~{}{\rm GeV}/c$ is $0.5\%$, and the $\mathrm{d}E/\mathrm{d}x$ resolution is
$6\%$ for electrons from Bhabha scattering. The EMC measures photon energies
with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel (end cap)
region. The time resolution in the TOF barrel region is 68 ps, while that in
the end cap region is 110 ps. The end cap TOF system was upgraded in 2015
using multi-gap resistive plate chamber technology, providing a time
resolution of 60 ps [20].

The data samples analyzed in this article are listed in appendix. They include
40 energy points (referred to as the $XYZ$ data sample) from 3.7730 to 4.7008
GeV with an integrated luminosity of more than 40 $\rm{pb^{-1}}$ at each
$\sqrt{s}$, and 13 energy points (the $R$-scan data sample) from 4.410 to
4.590 GeV with a luminosity of $7-9$ $\rm{pb^{-1}}$ at each $\sqrt{s}$ to
study possible structure around 4.5 GeV. The integrated luminosities and
$\sqrt{s}$ are measured using Bhabha scattering, radiative di-muon events, and
$\Lambda_{c}$ pairs [21, 23, 22].

The geant4-based [24] Monte Carlo (MC) simulation software packages boost [25]
and evtgen [26] are used to determine detection efficiencies and to estimate
the background contributions. The generator kkmc [27] is used to model the
beam energy spread and the ISR emission in $e^{+}e^{-}$ annihilations. Final
state radiation (FSR) from charged (final state) particles is incorporated
using the photos package [28]. The signal MC samples of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$, with
$J/\psi\rightarrow\ell^{+}\ell^{-}~{}(\ell=e,~{}\mu)$, are generated using a
phase-space (PHSP) model and weighted following the results of a partial wave
analysis (PWA) of the data. The amplitude model and the PWA framework are the
same as in Refs. [29, 30, 31]. Potential background contaminations are studied
via inclusive MC samples described in Refs. [3, 32]. Bhabha, di-muon, and
$e^{+}e^{-}\rightarrow 2(\pi^{+}\pi^{-})$ events are chosen as control samples
to study the tracking efficiencies of $e^{\pm}$, $\mu^{\pm}$, and $\pi^{\pm}$,
respectively. An exclusive MC sample for $e^{+}e^{-}\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}$ (with
$\gamma^{\ast}\gamma^{\ast}\rightarrow\mu^{+}\mu^{-}$, two-photon process)
[33, 34] is generated for Boosted Decision Tree (BDT) training [35] used for
background suppression.

##  III Event selection

Events with exactly four charged tracks and zero net charge are selected. In
order to be considered, a charged track reconstructed in the MDC is required
to satisfy $|\rm{cos}\theta|<0.93$, with $\theta$ being the polar angle, and
the distance of closest approach to the $e^{+}e^{-}$ interaction point must be
within $\pm 10$ cm in the beam direction and within 1 cm in the plane
perpendicular to the beam direction. If there are only two tracks with
momentum greater than 1.0 GeV$/c$, then the two tracks are judged as leptons;
if there are three tracks with momentum greater than 1.0 GeV$/c$, then the two
tracks with opposite charges and with an invariant mass closer to the $J/\psi$
mass [36] are regarded as leptons. The other two tracks are regarded as pions.
The energy deposited in the EMC is used to distinguish between electrons and
muons. For muon candidates, the deposited energy is required to be less than
0.4 GeV, while for electrons, it is required to be larger than 1.1 GeV. In
order to suppress the background contribution and to improve the energy and
momentum resolution, a four-constraint (4C) kinematic fit is applied to the
event with the hypothesis
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\ell^{+}\ell^{-}$, which constrains the
total four-momentum of the final state particles to that of the initial
colliding beams. The $\chi^{2}$ of the kinematic fit is required to be less
than 60. The cosine of the opening angle of the pion-pair ($\rm
cos(\pi^{+}\pi^{-})$) and, only in the $J/\psi\rightarrow e^{+}e^{-}$ decay
channel, of the pion-electron pairs ($\rm cos(\pi^{\pm}e^{\mp})$) are required
to be less than 0.98 to suppress gamma conversion background events of the
radiative Bhabha and di-muon background contributions [3].

In order to reduce the background contributions from low momentum electrons,
pions are identified using the $\mathrm{d}E/\mathrm{d}x$ information recorded
by the MDC. A discriminator $\chi_{\pi^{+/-}}\equiv(\mu_{\rm m}-\mu_{\rm
exp})/\sigma_{\rm m}$ is defined by combining the measured
$\mathrm{d}E/\mathrm{d}x$ value ($\mu_{\rm m}$), the measurement uncertainty
($\sigma_{\rm m}$), and the expected value under a pion hypothesis ($\mu_{\rm
exp}$). The conditions $\chi_{\pi^{+}}<2.8~{}(3.2)$ and
$\chi_{\pi^{-}}<3.0~{}(4.10)$ for the $e^{+}e^{-}$ mode ($\mu^{+}\mu^{-}$
mode) are found to provide an optimal balance between signal efficiency loss
and background rejection power. The polar angle distributions of $e^{+}$ and
$e^{-}$ in the $J/\psi\rightarrow e^{+}e^{-}$ mode (Fig. 1) clearly show a
significant contribution from the $e^{+}e^{-}\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}$ process (a two-photon process corresponding to the
yellow filled area in Fig. 1). A boosted decision tree method, implemented
within the TMVA framework [35], is trained to efficiently suppress this
background.

The energy deposited in the EMC, the time of flight from TOF,
$\mathrm{d}E/\mathrm{d}x$, and the opening angle of the pion and electron
candidates (Fig. 2) are used as input variables to the BDT, as they are mostly
uncorrelated, as can be seen from Figs. 3(a) and (b). The BDT is trained using
the MC simulation of the process $e^{+}e^{-}\rightarrow
e^{+}e^{-}\mu^{+}\mu^{-}$. The signal MC and background MC simulation of
multiple energy points are combined to train the BDT model. Figure 3(c) shows
that the training samples and test samples in the BDT model are in good
agreement, and the model can effectively distinguish between signal and
background. The response of the BDT is required to maximize the $S/\sqrt{S+B}$
distribution (where the $S$ and $B$ indicate the number of signal and
background events passing the condition) shown in Fig. 3(d).

Figure 1: Cosine of the polar angle of the $e^{+}$
($\rm{cos}(\theta_{e^{+}})$) and $e^{-}$ ($\rm{cos}(\theta_{e^{-}})$) from the
sample at $\sqrt{s}=4.1780$ GeV. The blue dots with error bars are data from
the signal region ($3.08<M(\ell^{+}\ell^{-})<3.12$ GeV/$c^{2}$). The black
line is the signal MC, which is normalized to the number of signal events. The
green line comes from the sideband region ($3.02<M(\ell^{+}\ell^{-})<3.06$
GeV/$c^{2}$ or $3.14<M(\ell^{+}\ell^{-})<3.18$ GeV/$c^{2}$ , with an event
weight of 0.5) of $M(\ell^{+}\ell^{-})$. The yellow-filled area is the
background MC (two-photon process MC), and it is normalized to the number of
events in the sideband area. The red line is the sum of signal MC plus
background MC simulation.

(a)(b)(c)(d)(e)(f)(g)(h)

Figure 2: The variables used in the BDT. (a,b) Distribution of the difference
between the measured value and the expected value from the TOF of the two
pions. (c,d) The energy deposited in the EMC by the two pions. (e,f) The
cosine of the polar angle of $e^{+}$ ($\rm{cos}(\theta_{e^{+}})$) and
$\pi^{+}$ ($\rm{cos}(\theta_{\pi^{+}})$). (g,h) Distributions of
$\chi_{\pi^{+}}$ and $\chi_{\pi^{-}}$. The blue line is for the signal and the
red line for the background. In (a,b,c,d), the peak at $-1.0$ corresponds to
the tracks without TOF or EMC information. The background and the signal have
the same magnitude. All the distributions come from the signal MC and
background MC simulation.

(a)(b)

(c)(d)

Figure 3: (a,b) The linear correlation coefficients of the variables in the
BDT training: (a) is for signal MC and (b) is for background MC simulation
samples. (c) The overtraining check of the classifier: the training samples
and test samples are in good agreement. (d) The optimized results of the
sample at $\sqrt{s}=4.1780$ GeV. The numbers of signal and background events
estimated from the data are used to maximize $S/\sqrt{S+B}$ and thus optimize
the event selection criteria.

(a)(b)$\sqrt{s}=4.1780$GeV$\sqrt{s}=4.1780$GeV

(c)(d)

Figure 4: The (a) and (b) plots show the fit results of the invariant mass of
the $e^{+}e^{-}$ mode and $\mu^{+}\mu^{-}$ mode, respectively. The (c) and (d)
are the two-dimensional distributions of the squared invariant mass of the
$\pi^{+}\pi^{-}$ and $\pi^{+}J/\psi$ pairs. Here, (c) is the data, (d) is the
MC simulation from PWA. The contents shown come from the sample at
$\sqrt{s}=4.1780$ GeV.

##  IV Measurement of the dressed Born cross section

After applying the event selection criteria mentioned above, a clear $J/\psi$
signal peak is observed in the invariant mass distribution of the lepton pairs
($M(\ell^{+}\ell^{-})$), as shown in Figs. 4(a)($e^{+}e^{-}$ mode) and
(b)($\mu^{+}\mu^{-}$ mode), Figs. 4(c)(data) and (d)(MC) show the two-
dimensional distribution of the invariant mass of the $\pi^{+}\pi^{-}$ and
$\pi^{+}J/\psi$ pairs. To determine the signal yields, an unbinned maximum-
likelihood fit to $M(\ell^{+}\ell^{-})$ is performed. The signal probability
density function (PDF) is defined as the convolution between the MC signal and
a Gaussian function, while the background contribution is parameterized with a
linear term.

The dressed Born cross section of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$
at a given $\sqrt{s}$ is calculated with

$\sigma(\sqrt{s})=\frac{N^{\rm obs}}{\mathcal{L}_{\rm
int}(1+\delta)\epsilon\mathcal{B}},$ (1)

where $N^{\rm obs}$ is the number of signal events, $\mathcal{L}_{\rm int}$ is
the integrated luminosity, $\epsilon$ is the selection efficiency,
$\mathcal{B}$ is the branching fraction of $J/\psi\rightarrow\ell^{+}\ell^{-}$
[36] and ($1+\delta$) is the ISR correction factor. The final cross section is
given by the average of the $e^{+}e^{-}$ and the $\mu^{+}\mu^{-}$ modes,
weighted with the inverse of the respective statistical uncertainties. The
parameters of Eq. 1 and the measured cross sections at the different
$\sqrt{s}$ are summarized in the appendix.

The ISR correction factor is evaluated using an iterative procedure, in which
the $\sigma(\sqrt{s})$ is initially assumed to be simply flat and iteratively
recomputed until the difference between the final two iterations is less than
0.1%. The $(i+1)^{\rm th}$ iteration $(1+\delta)^{i+1}$ is given by [37]:

$(1+\delta)^{i+1}=(1+\delta)^{0}\sum^{N}_{j}{W_{j}^{i}}/N,~{}W_{j}^{i}=\frac{\sigma^{i}(m_{j})}{\sigma^{i}(\sqrt{s})},$
(2)

where $W_{j}^{i}$ is the weighting factor for the $j^{\rm th}$ event. The ISR
correction factor $(1+\delta)^{0}$ corresponds to the $0^{\rm th}$ iteration,
i.e. to an assumed flat shape of the cross section; $\sigma^{i}$ is the line-
shape given by the $i^{\rm th}$ iteration and $\sigma^{0}$ is calculated with
the correction factor $(1+\delta)^{0}$, which is calculated using the kkmc
[27] program; $m_{j}$ is the invariant mass of the final state of the event
$j$; and $N$ is the total number of generated MC events. Considering that the
sizes of the $R$-scan data samples are small, only the $XYZ$ data samples are
used in the iteration. The ISR correction factors for the $R$-scan data
samples are calculated from the model describing the cross section in Sect.
VI.

##  V Systematic uncertainties of the cross section measurements

The main contributions to the systematic uncertainty of the cross section
measurement are related to the measurement of the luminosity, the MC model,
the tracking efficiency, the ISR correction, the branching fractions of
$J/\psi$ decays, the kinematic fit, the fit to $M(\ell^{+}\ell^{-})$, and the
BDT method. The integrated luminosities of all data sets are measured using
large angle Bhabha scattering events with an uncertainty of 0.66% [21]. The
branching fraction of $J/\psi\rightarrow\ell^{+}\ell^{-}$ is taken from PDG
[36], and the uncertainties are 0.6% for each mode. The uncertainty related to
the kinematic fit is estimated by adjusting the helix parameters of the
charged tracks in MC simulation to match the $\chi^{2}$ distribution of data
and MC [38]. The uncertainty is determined as the difference between the
results before and after the helix parameters correction, resulting in 0.76%
(1.34%) for the $\mu^{+}\mu^{-}$ ($e^{+}e^{-}$) mode. The uncertainty on the
signal yield arising from the fitting of the $M(\ell^{+}\ell^{-})$
distribution is obtained by varying the fit range and by changing the
background modeling from a first to a second order polynomial, which leads to
a difference of 2.44% (1.16%) that is taken as the systematic uncertainty
introduced by the fit method.

In the evaluation of the systematic uncertainties due to the tracking
efficiency, the MC simulation model, the BDT method, and the ISR correction,
300 sets of Gaussian samplings according to the related central value
($\mu_{0}$) and statistical uncertainties, obtained from the control samples,
the PWA fitted parameters, the data-MC simulation ratios of the BDT training
variables, and the ISR correction factors, are generated. A Gaussian function
($\mu_{1},~{}\sigma_{1}$) is used to fit the resulting distributions from the
300 MC simulation samples. The final uncertainty of each source is given by
$(|\mu_{1}-\mu_{0}|+\sigma_{1})/\mu_{0}\times 100\%$.

The uncertainty of the tracking efficiency comes from the uncertainties of the
correction factors obtained from control samples, which are the Bhabha, di-
muon and $e^{+}e^{-}\rightarrow 2(\pi^{+}\pi^{-})$ processes. The correction
factors are given by the ratio of data to MC simulation in the two dimensions
of transverse momentum and the cosine of the polar angle. Therefore, the
Gaussian sampling is based on the correction value and its uncertainty, and
obtains the efficiencies distributions with respect to the new correction
factors. In the MC simulation model, the uncertainties come from the
uncertainties of the amplitude parameters given by the PWA fit to the data. To
estimate the uncertainties of the parameters, toy MC samples are generated
according to the fitted parameters and the error matrix to obtain the
efficiency distribution for each energy point. The difference in the
efficiency determined with pseudoexperiments generated according to the
amplitude model parameters with and without the three-body PHSP process is
regarded as the uncertainty of the partial wave model.

The differences in line shape caused by statistical uncertainties lead to
different ISR correction factors. Therefore, for each energy point, the cross
section and error serve as the parameters of the Gaussian function of
sampling, and after a new round of iteration, the cross section distribution
is obtained for each point. Apart from the systematic uncertainty of the ISR
correction, the systematic uncertainties of the other contributions are
approximately the same for every energy point, and thus the average of each is
taken, as listed in Table 1. The ISR correction uncertainty of $R$-scan data
samples are estimated by the value of closest $XYZ$ data point, and the
uncertainty of each point is listed in the appendix. Assuming all the sources
to be independent, the total systematic uncertainties are obtained by adding
them in quadrature.

Table 1: Systematic uncertainties (%) of the cross section in $\mu^{+}\mu^{-}$ mode and $e^{+}e^{-}$ mode. Source | Uncertainty (%)
---|---
$\mu^{+}\mu^{-}$ | $e^{+}e^{-}$
Luminosities | 0.7 | 0.7
Tracking efficiency | 0.8 | 0.6
MC simulation model | 1.9 | 1.9
Branching fractions | 0.6 | 0.6
Kinematic fit | 0.8 | 1.3
Fit to $M(\ell^{+}\ell^{-})$ | 2.4 | 1.2
BDT method |  | 0.9
Total | 2.7 | 3.1

##  VI Fit to the cross section

To study the possible resonant structures in the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ process, a maximum likelihood fit
is preformed to the measured cross section. The likelihood is constructed
assuming the number of events satisfies a Gaussian distribution in $XYZ$ data
and Poisson distribution in $R$-scan data. The cross section is parameterized
with a coherent sum of Breit-Wigner (BW) functions. Due to the lack of data
near the $\psi(3770)$ resonance, it is not feasible to determine the relative
phase between the $\psi(3770)$ amplitude and other amplitudes. The cross
section line shape is described by

$\footnotesize{\sigma_{\mathrm{fit}}(\sqrt{s})=|R_{\psi(3770)}(\sqrt{s})|^{2}+\left|\sum_{j=0}^{n}R_{j}(\sqrt{s})e^{i\phi_{j}}\right|^{2},}$
(3)

where $R_{\psi(3770)}$ is used to describe the $\psi(3770)$ resonance and its
mass and width are fixed to the world average values [36]. The $i$ is the
imaginary unit. $R_{j}$ represents the amplitude to describe a given resonant
structure and $\phi_{j}$ is the corresponding phase. The phase $\phi_{0}$ is
set to zero and the other phases are given relative to the $R_{0}$. For the
structure near 4.0 GeV, two different parameterization methods are applied,
Model I: a BW function, and Model II: an exponential function (Exp) of the
form
$R_{0}(\sqrt{s})=PS(\sqrt{s})e^{-p_{0}(\sqrt{s}-M_{\rm{threshold}})}p_{1}$
[39], with $M_{\rm{threshold}}=m_{\pi^{+}}+m_{\pi^{-}}+m_{J/\psi}$,
$PS(\sqrt{s})$ is the PHSP factor of the three-body decay
$R_{j}\rightarrow\pi^{+}\pi^{-}J/\psi$ [36], and $p_{0}$ and $p_{1}$ are free
parameters determined by the fit. The number of resonances is denoted by $n$,
comprising the known $Y(4220)$ and $Y(4320)$ as well as further possible
structures. The amplitude $R_{j}$ is defined as

$\footnotesize{R_{j}(\sqrt{s})=\frac{M_{j}}{\sqrt{s}}\frac{\sqrt{12\pi\Gamma^{\rm
ee}_{j}\Gamma^{\rm
tot}_{j}\mathcal{B}(R_{j})}}{s-M^{2}_{j}+iM_{j}\Gamma^{\rm{tot}}_{j}}\times\sqrt{\frac{PS(\sqrt{s})}{PS(M_{j})}},}$
(4)

where $M_{j}$, $\Gamma^{\rm tot}_{j}$ and $\Gamma^{\rm ee}_{j}$ are the mass,
full width and electronic width of resonance $R_{j}$, respectively, and
$\mathcal{B}(R_{j})$ is the branching fraction for
$R_{j}\rightarrow\pi^{+}\pi^{-}J/\psi$.

In case of considering the states $Y(4220)$ and $Y(4320)$ ($n=2$), multiple
sets of solutions are obtained given by the two models (Model I: BW, Model II:
Exp) at 4.0 GeV. The fit results are shown in Fig. 5, and the fit parameters
are summarized in Table 2. Sizable differences between the fit results of
Model I and Model II appear mainly in the energy region between 3.7730 and
4.1574 GeV. The difference of $\chi^{2}/\rm{ndf}$ is 3.72, where $\rm{ndf}$ is
the number of degrees of freedom. Therefore, Model I is chosen to be the
default model for the final cross section fit result. The cross section fit
shows larger fluctuations at $\sqrt{s}=$ 3.8713 GeV. These might be due to the
influence of the $X(3872)$ [36] resonance which was not included in the model
since the $X(3872)$ is very narrow and there are not sufficient data points
around its nominal mass.

(a)(b)

Figure 5: Fit to the energy-dependent cross section of
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ using two different fit models:
Model I (a) and Model II (b). The upper panels show the data points with error
bars overlaid with the fit result represented by the solid (blue) line. The
lower panels show the corresponding fit quality for each data point in terms
of $\chi$ in units of $\sigma$. The point of $\sqrt{s}=3.8713$ GeV is not
included in the fit. For more details of the fit models, see the text. Table
2: The values of $\Gamma^{\rm
e^{+}e^{-}}\mathcal{B}(R\rightarrow\pi^{+}\pi^{-}J/\psi)$ from the fit to the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ cross section. The parameters
$M(R_{n})$, $\Gamma^{\rm tot}_{n}(R_{n})$ and $\Gamma^{\rm
ee}_{n}\mathcal{B}(R_{n})$, ($n=0,1,2$) represent the mass (in MeV/$c^{2}$),
total width (in MeV) and the product of the $e^{+}e^{-}$ partial width (in eV)
with the branching ratio of the resonance decay into $\pi^{+}\pi^{-}J/\psi$
for the resonances, respectively. Here, $p_{0}$ ($c^{2}$/MeV) and $p_{1}$ are
the free parameters of the exponential function. The parameters $\phi_{1}$ and
$\phi_{2}$ (in degrees) are phases of the resonance $R_{1}$ and $R_{2}$, the
phase of resonance $R_{0}$ is set to 0. The numbers in the brackets correspond
to the fit by replacing the resonance $R_{0}$ (BW: Model I) with an
exponential function (Exp: Model II) to describe the structure near 4.0 GeV.
The uncertainties are statistical only. “…” represents a null value.

Parameter | Solution I | Solution II | Solution III | Solution IV
---|---|---|---|---
$\Gamma^{\rm ee}_{3770}\mathcal{B}(R_{3770})$ | $0.6\pm 0.1~{}(0.3\pm 0.1)$
$M(R_{0})$ ($p_{0}$) |  $3905.5\pm 30.1$ ($4.4\pm 0.3$)
$\Gamma^{\rm tot}_{0}(R_{0})$ ($p_{1}$) |  $346.0\pm 48.5$ ($(2.7\pm 0.6)\times 10^{-3}$)
$\Gamma^{\rm ee}_{0}\mathcal{B}(R_{0})$ |  $5.5\pm 0.5$ (…) |  $6.9\pm 0.7$ (…) |  $8.3\pm 0.6$ (…) |  $10.5\pm 0.9$ (…)
$M(R_{1})$ |  $4221.4\pm 1.5$ ($4220.1\pm 1.2$)
$\Gamma^{\rm tot}_{1}(R_{1})$ |  $41.8\pm 2.9$ ($43.6\pm 2.6$)
$\Gamma^{\rm ee}_{1}\mathcal{B}(R_{1})$ | $1.7\pm 0.2~{}(1.7\pm 0.2)$ | $8.2\pm 0.9~{}(8.6\pm 0.5)$ | $3.0\pm 0.5~{}(2.5\pm 0.3)$ | $14.6\pm 1.2~{}(12.7\pm 0.8)$
$M(R_{2})$ |  $4297.5\pm 12.1$ ($4316.2\pm 12.4$)
$\Gamma^{\rm tot}_{2}(R_{2})$ |  $126.6\pm 16.7$ ($124.3\pm 18.0$)
$\Gamma^{\rm ee}_{2}\mathcal{B}(R_{2})$ | $1.2\pm 0.3~{}(0.7\pm 0.2)$ | $2.3\pm 0.8~{}(1.1\pm 0.3)$ | $15.6\pm 2.1~{}(15.0\pm 1.2)$ | $30.2\pm 3.3~{}(23.6\pm 2.9)$
$\phi_{1}$ | $-3.7\pm 5.4~{}(24.3\pm 3.0)$ | $-124.6\pm 11.7~{}(-78.8\pm 5.1)$ | $87.7\pm 21.9~{}(88.0\pm 12.1)$ | $-33.5\pm 11.2~{}(-15.1\pm 7.7)$
$\phi_{2}$ | $79.6\pm 18.5~{}(130.7\pm 15.8)$ | $35.8\pm 27.2~{}(96.6\pm 19.7)$ | $-104.7\pm 26.9~{}(-92.5\pm 6.0)$ | $-148.7\pm 4.5~{}(-127.6\pm 2.3)$
$\chi^{2}/\rm{ndf}$ | 54.0/40 (57.3/41)

Considering the distribution of the pull ($\chi$) values, the above two models
do not describe the interval from 4.4 to 4.6 GeV very well. Therefore, a third
BW function ($n=3$) is added to study whether this deviation is caused by
possible additional structures. When the (fit) parameters of the third BW
function are floated, two possible solutions are obtained, one with a mass
close to the $\psi(4415)$ and the other one close to 4.5 GeV. Compared with
the mode of $n=2$, the significance of these two solutions are $4.0\sigma$
($3.6\sigma$) and $2.1\sigma$ ($2.7\sigma$), respectively. The alternative
fits using the parameters of the $\psi(4415)$ [36] and the newly discovered
$Y(4500)$ structure [40] have also been attempted to describe the structure at
4.5 GeV, and led to results with significance of $2.6\sigma$ ($3.1\sigma$) and
$3.3~{}\sigma$ ($3.3~{}\sigma$), respectively. The numbers in the brackets
correspond to an alternative fit, in which the BW function(Model I) is
replaced by an exponential function(Model II) to describe the structure near
4.0 GeV, as also shown in Table 2 and Table 3.

Due to the limited data samples around 4.4 GeV, the mass and width of the new
additional BW function are fixed to $\psi(4415)$ or $Y(4500)$, the fit result
in Table 3. With larger/new data samples in that region, the structure of the
$\psi(4415)$ or $Y(4500)$ can be studied further. The fit results are shown in
Fig. 6, and the parameters are summarized in Table 3. In conclusion, the
parameters of the $Y(4220)$ state are stable for the different models, while
the ones of $Y(4360)$ heavily depend on the presence of an additional
structure close to 4.5 GeV.

(a)(b)

(c)(d)

Figure 6: Fit to the energy-dependent cross section of the process
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$, where an additional BW function
is added based on Model I. The upper panels show the data points with error
bars overlaid with the fit result represented by the solid (blue) line. The
lower panels show the corresponding fit quality for each data point in terms
of $\chi$ in units of $\sigma$, and the fit results are listed in Table 3. (a)
and (b): The new BW function with free parameters. (c) and (d): The new BW
function with parameters fixed to the $\psi(4415)$ and $Y(4500)$ resonance
parameters. The point of $\sqrt{s}=3.8713$ GeV is not included in the fit.
Table 3: The fit results of the test for the additional structure near 4.5
GeV. The values of $\Gamma^{\rm
e^{+}e^{-}}\mathcal{B}(R\rightarrow\pi^{+}\pi^{-}J/\psi)$ from the fit to the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ cross section. The parameters
$M(R_{n})$, and $\Gamma^{\rm tot}_{n}(R_{n})$ ($n=0,1,2,3$) represent the mass
(in MeV/$c^{2}$), total width (in MeV) of the resonance decay into
$\pi^{+}\pi^{-}J/\psi$ for the resonances, respectively. Finally, the
significance of the additional BW is given. The uncertainties are statistical
only. The numbers in the brackets correspond to the fit by replacing the
BW:Model I with an Exp: Model II to describe the structure near 4.0 GeV.

Parameter | Result I | Result II | Result III | Result IV
---|---|---|---|---
$M(R_{1})$ |  $4221.0\pm 1.6$ ($4220.3\pm 1.6$) |  $4219.8\pm 1.3$ ($4219.1\pm 1.2$) |  $4223.9\pm 1.4$ ($4219.6\pm 1.3$) |  $4220.2\pm 1.3$ ($4219.4\pm 1.1$)
$\Gamma^{\rm tot}_{1}(R_{1})$ |  $41.0\pm 3.0$ ($42.3\pm 3.0$) |  $45.4\pm 2.8$ ($46.3\pm 2.5$) |  $42.2\pm 3.2$ ($44.3\pm 2.7$) |  $44.5\pm 2.9$ ($45.3\pm 2.5$)
$M(R_{2})$ |  $4293.7\pm 13.1$ ($4304.8\pm 18.8$) |  $4345.8\pm 28.4$ ($4357.9\pm 20.2$) |  $4308.5\pm 17.6$ ($4333.2\pm 23.2$) |  $4328.58\pm 18.9$ ($4347.1\pm 14.5$)
$\Gamma^{\rm tot}_{2}(R_{2})$ |  $152.4\pm 23.9$ ($144.3\pm 31.5$) |  $130.1\pm 20.7$ ($107.9\pm 25.6$) |  $161.4\pm 24.6$ ($153.2\pm 26.2$) |  $133.8\pm 20.2$ ($127.5\pm 22.2$)
$M(R_{3})$ |  $4405.6\pm 4.5$ ($4405.0\pm 6.7$) |  $4471.1\pm 36.2$ ($4550.9\pm 16.9$) | 4421 (fixed) | 4485 (fixed)
$\Gamma^{\rm tot}_{3}(R_{3})$ |  $9.1\pm 2.5$ ($8.7\pm 4.9$) |  $159.7\pm 97.0$ ($211.8\pm 132.8$) | 62 (fixed) | 111 (fixed)
$\chi^{2}/\rm{ndf}$ | 40.1/36 (44.8/37) | 47.6/36 (48.7/37) | 45.4/38 (48.7/39) | 48.1/38 (51.3/39)
Significance |  $4.0\sigma$ ($3.6\sigma$) |  $2.1\sigma$ ($2.7\sigma$) |  $3.3\sigma$ ($3.3\sigma$) |  $2.6\sigma$ ($3.1\sigma$)

##  VII Systematic uncertainties of the resonance parameters

Relevant systematic uncertainties of the fit parameters are caused by the fit
model, the $\sqrt{s}$ energy measurement and its spread, and the PHSP factor.
The systematic uncertainty related to the fit model is evaluated as the
difference of the mass and width results of Model I and Model II. The
$\sqrt{s}$ of all data sets have been measured with di-muon events with an
uncertainty of 0.6 MeV that propagates directly to the uncertainty of the mass
of the resonances. The uncertainties included by the $\sqrt{s}$ spread are
obtained by convolving the resonant PDF with a Gaussian function whose width
is taken to be 1.6 MeV, equal to the spread obtained from the Beam Energy
Measurement System [41]. The uncertainty of the PHSP factor, due to the
existence of intermediate states, is estimated by considering the PHSP of
cascade two-body decays of $e^{+}e^{-}\rightarrow RJ/\psi$ (with
$R=\sigma,~{}f_{0}(980),~{}f_{0}(1370)$) and
$e^{+}e^{-}\rightarrow\pi^{\pm}Z_{c}(3900)^{\mp}$ , and the maximum value of
the difference with respect to the result obtained when using the three-body
PHSP factor is taken as the systematic uncertainty. The deviation of the
resonant parameters introduced by the uncertainties of the $\psi(3770)$
resonance parameters are less than 0.1 MeV, and thus can be neglected.
Assuming all of the systematic uncertainties are independent, adding them in
quadrature delivers the total error as listed in Table 4.

Table 4: Summary of the uncertainties of the resonance parameters.

Source | Uncertainty
---|---
$Y(4220)$ | $Y(4320)$
M (MeV/$c^{2}$) |  $\Gamma$ (MeV) | M (MeV/$c^{2}$) |  $\Gamma$ (MeV)
$\sqrt{s}$ | 0.6 |  | 0.6 |
Beam spread | 0.3 | 0.4 | 5.0 | 2.1
Fit model | 1.4 | 1.0 | 15.8 | 6.8
PHSP factor | 1.3 | 2.5 | 19.9 | 7.8
Total | 2.0 | 2.7 | 25.9 | 10.3

##  VIII Summary

In summary, a precision measurement of the energy-dependent cross section for
the process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ from $\sqrt{s}=3.7730$
GeV to $\sqrt{s}=4.7008$ GeV at BESIII is performed. This measurement improves
upon the precision of the previous results from the BESIII collaboration in
the same channel [3] by about 60% at the points of same statistics
($\sqrt{s}=4.2263$ and 4.2580 GeV). This is achieved by the improvements in
the optimized MC simulation model and the enhanced tracking efficiencies.

The energy-dependent cross section is fitted with different fit models for the
cross section line shape, allowing for the search of resonances and the
evaluation of their parameters. It was found that the structure close to 4.0
GeV is better described when using the BW function as compared to an Exp
function. The $Y(4220)$ and $Y(4320)$ resonances were observed with
significances larger than $10\sigma$ and their resonance parameters were
estimated to be consistent with those reported in Ref. [3]. However, the
presence of an additional structure around 4.5 GeV, possibly identifiable with
the $\psi(4415)$, influences the evaluation of the $Y(4320)$ parameters, which
are ($M,~{}\Gamma$) = ($4298\pm 12\pm 26$ MeV/$c^{2}$, $127\pm 17\pm 10$ MeV),
therefore, reported with a large uncertainty.

###### Acknowledgements.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11625523, 11635010, 11735014, 11822506, 11835012, 11935015, 11975141,
11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013,
12061131003, 11875262; 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; The Fundamental Research Funds of Shandong University;
German Research Foundation DFG under Contracts Nos. 443159800, Collaborative
Research Center CRC 1044, FOR 2359, GRK 2149; 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-0012069.

## References

  * [1] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 95, 142001 (2005).
  * [2] C. Z. Yuan et al. [Belle], Phys. Rev. Lett. 99, 182004 (2007).
  * [3] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092001 (2017).
  * [4] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 96, 032004 (2017).
  * [5] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 118, 092002 (2017).
  * [6] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 122, 102002 (2019).
  * [7] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 102, 031101 (2020).
  * [8] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 114, 092003 (2015).
  * [9] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 93, 011102 (2016).
  * [10] X. H. Mo et al., Phys. Lett. B 640, 182 (2006).
  * [11] Eichten et al., Phys. Rev. D 17, 3090 (1978); Phys. Rev. D 21, 203 (1980); S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
  * [12] M. Berwein, N. Brambilla, J. Tarrús Castellà and A. Vairo, Phys. Rev. D 92, 114019 (2015).
  * [13] E. Epelbaum, A. M. Gasparyan, H. Krebs and C. Schat, Eur. Phys. J. A 51, 26 (2015).
  * [14] F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang and Q. Zhao, Phys. Lett. B 725, 127 (2013).
  * [15] M. Cleven et al., Phys. Rev. D 90, 074039 (2014).
  * [16] C. Ayala et al., Eur. Phys. J. C 78, 1002 (2018).
  * [17] J. F. Giron and R. F. Lebed, Phys. Rev. D 101, 074032 (2020).
  * [18] M. Ablikim et al. (BESIII Collaboration), Nucl. Instrum. Meth. A 614, 345 (2010).
  * [19] C. H. Yu et al., Proceedings of IPAC2016, Busan, Korea, 2016, JACoW IPAC2016-TUYA01.
  * [20] X. Li et al., Radiat. Detect. Technol. Methods 1, 13 (2017); Y. X. Guo et al., Radiat. Detect. Technol. Methods 1, 15 (2017); P. Cao et al.,  Nucl. Instrum. Meth. A 953, 163053 (2020).
  * [21] M. Ablikim et al. (BESIII Collaboration), [arXiv:2203.03133 [hep-ex]].
  * [22] M. Ablikim et al. (BESIII Collaboration), [arXiv:2205.04809 [hep-ex]].
  * [23] M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C 45 103001 (2021).
  * [24] S. Agostinelli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003).
  * [25] Z. Y. Deng et al., Chin. Phys. C 30, 371 (2006).
  * [26] D. J. Lange, Nucl. Instrum. Meth. A 462, 152 (2001); R. G. Ping, Chin. Phys. C 32, 599 (2008).
  * [27] S. Jadach, B. F. L. Ward and Z. Was, Phys. Rev. D 63, 113009 (2001).
  * [28] E. Richter-Was, Phys. Lett. B 303, 163 (1993).
  * [29] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 102, 012009 (2020).
  * [30] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 119, 072001 (2017).
  * [31] https://github.com/mashephe/AmpTools.
  * [32] X. Y. Zhou, S. X. Du, G. Li and C. P. Shen, Comput. Phys. Commun. 258, 107540 (2021).
  * [33] F. A. Berends, P. H. Daverveldt and R. Kleiss, Comput. Phys. Commun. 40, 271 (1986).
  * [34] F. A. Berends, P. H. Daverveldt and R. Kleiss, Comput. Phys. Commun. 40, 285 (1986).
  * [35] A. Hocker et al. [arXiv:physics/0703039].
  * [36] P. A. Zyla et al. (Particle Data Group), Theor. Exp. Phys. 2020, 083C01 (2020).
  * [37] W. Sun et al., Front. Phys. (Beijing) 16, 64501 (2021).
  * [38] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 87, 012002 (2013).
  * [39] J. Adam et al. (STAR Collaboration), Phys. Rev. D 99, 051102 (2019).
  * [40] M. Ablikim et al. (BESIII Collaboration), [arXiv:2204.07800 [hep-ex]].
  * [41] E. V. Abakumova et al., Nucl. Instrum. Meth. A 659, 21 (2011).

## Appendix

Table 5 lists the parameters used to calculate the cross section of the
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$ process with the Eq. 1.

Table 5: The cross section of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}J/\psi$.
Here, $\mathcal{L}$ (pb-1) is the integrated luminosity, $1+\delta$ is the
radiation correction factor, “Rsys” is the systematic uncertainties (%) of the
radiation correction factor, Nobs is the number of observed signal events from
$\ell^{+}\ell^{-}$ invariant mass distributions, $\epsilon$ is the event
selection efficiency determined from signal MC, $\sigma$ (pb) is the cross
section. “average” is the weighted average cross section of $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ modes, and the weight is the inverse of the statistical
uncertainty. Samples marked with “$r$” are $R$-scan data, and the results are
given by combined $e^{+}e^{-}$ mode and $\mu^{+}\mu^{-}$ mode. The first term
is statistical uncertainty, and the second term is systematic uncertainty. The
uncertainty of $\mathcal{L}$ consists of statistical and systematic
uncertainties. The uncertainty for Nobs is statistical only.

$\sqrt{s}$ (GeV) | $\mathcal{L}$ (pb-1) | $1+\delta$ | Rsys(%) | Nobs |  $\epsilon$ (%) |  $\sigma$ (pb)
---|---|---|---|---|---|---
$e^{+}e^{-}$ |  $\mu^{+}\mu^{-}$ |  $e^{+}e^{-}$ |  $\mu^{+}\mu^{-}$ |  $e^{+}e^{-}$ |  $\mu^{+}\mu^{-}$ |  average
3.7730 | $2874.72\pm 19.40$ | 0.73 | 2.04 | $1169\pm 36$ | $1756\pm 46$ | 32.41 | 51.34 | $28.74\pm 0.89\pm 1.06$ | $27.30\pm 0.72\pm 0.92$ | $27.94\pm 0.56\pm 0.69$
3.8077 | $50.54\pm 0.49$ | 0.86 | 10.46 | $12\pm 4$ | $17\pm 5$ | 29.60 | 46.21 | $15.67\pm 5.22\pm 1.71$ | $14.26\pm 4.19\pm 1.54$ | $14.89\pm 3.29\pm 1.14$
3.8674 | $108.90\pm 0.72$ | 0.86 | 2.30 | $24\pm 5$ | $40\pm 8$ | 27.09 | 44.39 | $15.77\pm 3.28\pm 0.61$ | $16.05\pm 3.21\pm 0.57$ | $15.91\pm 2.30\pm 0.41$
3.8713 | $110.30\pm 0.73$ | 1.19 | 14.29 | $17\pm 5$ | $30\pm 6$ | 24.96 | 41.99 | $8.67\pm 2.55\pm 1.27$ | $9.10\pm 1.82\pm 1.32$ | $8.92\pm 1.50\pm 0.93$
3.8962 | $52.61\pm 0.51$ | 0.86 | 5.00 | $14\pm 4$ | $19\pm 5$ | 27.39 | 45.41 | $18.93\pm 5.41\pm 1.11$ | $15.52\pm 4.08\pm 0.88$ | $16.98\pm 3.29\pm 0.69$
4.0076 | $482.00\pm 3.18$ | 0.90 | 1.15 | $120\pm 12$ | $229\pm 18$ | 29.63 | 46.71 | $15.62\pm 1.56\pm 0.51$ | $18.94\pm 1.49\pm 0.55$ | $17.32\pm 1.08\pm 0.38$
4.0855 | $52.86\pm 0.35$ | 0.92 | 3.28 | $15\pm 4$ | $20\pm 5$ | 29.65 | 47.00 | $17.43\pm 4.65\pm 0.78$ | $14.68\pm 3.67\pm 0.62$ | $15.89\pm 2.90\pm 0.49$
4.1271 | $397.99\pm 2.63$ | 0.97 | 3.19 | $86\pm 10$ | $127\pm 14$ | 29.66 | 47.37 | $12.52\pm 1.46\pm 0.55$ | $11.60\pm 1.28\pm 0.48$ | $12.03\pm 0.96\pm 0.36$
4.1567 | $409.88\pm 2.70$ | 0.96 | 2.15 | $83\pm 10$ | $118\pm 13$ | 28.08 | 44.98 | $12.61\pm 1.52\pm 0.47$ | $11.21\pm 1.24\pm 0.38$ | $11.84\pm 0.96\pm 0.30$
4.1780 | $3194.50\pm 31.90$ | 0.94 | 2.61 | $612\pm 27$ | $1032\pm 38$ | 26.48 | 42.84 | $12.87\pm 0.57\pm 0.52$ | $13.44\pm 0.49\pm 0.50$ | $13.17\pm 0.37\pm 0.36$
4.1888 | $570.05\pm 2.16$ | 0.90 | 2.83 | $150\pm 13$ | $231\pm 19$ | 28.85 | 46.23 | $16.94\pm 1.47\pm 0.71$ | $16.31\pm 1.34\pm 0.64$ | $16.61\pm 0.99\pm 0.47$
4.1989 | $524.60\pm 2.05$ | 0.81 | 2.49 | $242\pm 17$ | $354\pm 21$ | 31.93 | 50.14 | $29.72\pm 2.09\pm 1.17$ | $27.77\pm 1.65\pm 1.02$ | $28.63\pm 1.30\pm 0.77$
4.2091 | $572.05\pm 1.81$ | 0.79 | 2.26 | $389\pm 21$ | $634\pm 27$ | 32.24 | 49.92 | $44.56\pm 2.41\pm 1.70$ | $47.07\pm 2.00\pm 1.65$ | $45.93\pm 1.55\pm 1.19$
4.2187 | $569.20\pm 1.80$ | 0.77 | 1.94 | $627\pm 26$ | $915\pm 32$ | 32.95 | 50.88 | $72.90\pm 3.02\pm 2.65$ | $69.06\pm 2.42\pm 2.29$ | $70.77\pm 1.90\pm 1.73$
4.2263 | $1100.91\pm 7.00$ | 0.76 | 1.15 | $1517\pm 40$ | $2315\pm 51$ | 34.37 | 52.00 | $88.68\pm 2.34\pm 2.88$ | $89.81\pm 1.98\pm 2.59$ | $89.29\pm 1.52\pm 1.93$
4.2357 | $530.60\pm 2.39$ | 0.82 | 1.54 | $696\pm 27$ | $1103\pm 35$ | 34.05 | 51.69 | $78.50\pm 3.05\pm 2.70$ | $82.20\pm 2.61\pm 2.55$ | $80.50\pm 1.99\pm 1.85$
4.2438 | $593.98\pm 2.69$ | 0.84 | 1.42 | $744\pm 28$ | $1133\pm 36$ | 33.22 | 51.12 | $74.99\pm 2.82\pm 2.54$ | $74.37\pm 2.36\pm 2.26$ | $74.65\pm 1.82\pm 1.69$
4.2580 | $828.40\pm 5.47$ | 0.89 | 1.15 | $888\pm 31$ | $1327\pm 39$ | 32.77 | 50.17 | $61.25\pm 2.14\pm 2.01$ | $59.89\pm 1.76\pm 1.75$ | $60.50\pm 1.37\pm 1.32$
4.2667 | $529.70\pm 3.13$ | 0.91 | 1.43 | $509\pm 23$ | $781\pm 30$ | 31.73 | 49.14 | $56.03\pm 2.53\pm 1.90$ | $55.56\pm 2.13\pm 1.69$ | $55.78\pm 1.64\pm 1.26$
4.2776 | $175.70\pm 0.96$ | 0.91 | 2.87 | $152\pm 13$ | $253\pm 17$ | 31.01 | 48.02 | $51.29\pm 4.39\pm 2.16$ | $55.20\pm 3.71\pm 2.17$ | $53.41\pm 2.84\pm 1.54$
4.2866 | $498.52\pm 3.29$ | 0.90 | 2.32 | $451\pm 22$ | $684\pm 28$ | 31.34 | 47.82 | $53.53\pm 2.61\pm 2.06$ | $53.27\pm 2.18\pm 1.89$ | $53.40\pm 1.68\pm 1.39$
4.3079 | $45.08\pm 0.30$ | 0.87 | 4.04 | $35\pm 6$ | $59\pm 9$ | 30.94 | 47.07 | $48.41\pm 8.30\pm 2.46$ | $53.74\pm 7.29\pm 2.61$ | $51.25\pm 5.49\pm 1.79$
4.3115 | $499.19\pm 3.30$ | 1.05 | 5.74 | $345\pm 19$ | $573\pm 26$ | 29.99 | 45.90 | $36.85\pm 2.03\pm 2.40$ | $40.03\pm 1.82\pm 2.54$ | $38.53\pm 1.36\pm 1.75$
4.3370 | $511.48\pm 3.38$ | 0.98 | 1.86 | $313\pm 19$ | $453\pm 24$ | 28.85 | 44.30 | $36.08\pm 2.19\pm 1.30$ | $34.06\pm 1.80\pm 1.11$ | $34.97\pm 1.40\pm 0.85$
4.3583 | $543.94\pm 3.59$ | 1.06 | 2.98 | $267\pm 17$ | $342\pm 21$ | 27.01 | 42.17 | $28.55\pm 1.82\pm 1.22$ | $23.47\pm 1.44\pm 0.94$ | $25.72\pm 1.14\pm 0.75$
4.3768 | $531.41\pm 3.51$ | 1.14 | 3.41 | $195\pm 15$ | $297\pm 20$ | 26.46 | 41.38 | $20.32\pm 1.56\pm 0.93$ | $19.83\pm 1.34\pm 0.86$ | $20.06\pm 1.02\pm 0.63$
4.3874 | $55.57\pm 0.37$ | 0.98 | 4.93 | $22\pm 5$ | $32\pm 6$ | 27.14 | 41.77 | $24.82\pm 5.64\pm 1.44$ | $23.51\pm 4.41\pm 1.32$ | $24.09\pm 3.50\pm 0.97$
4.3954 | $515.95\pm 3.41$ | 1.38 | 7.05 | $140\pm 13$ | $194\pm 17$ | 23.54 | 37.26 | $14.02\pm 1.30\pm 1.08$ | $12.29\pm 1.08\pm 0.93$ | $13.07\pm 0.83\pm 0.70$
$4.410^{r}$ | $7.19\pm 0.01$ | 1.14 | 7.05 | $3.3^{+2.6}_{-1.8}$ | 31.75 | $10.00^{+8.39}_{-5.81}\pm 0.82$
4.4156 | $1090.66\pm 6.89$ | 1.17 | 2.68 | $233\pm 16$ | $401\pm 24$ | 22.19 | 34.64 | $13.76\pm 0.94\pm 0.56$ | $15.18\pm 0.91\pm 0.57$ | $14.49\pm 0.66\pm 0.40$
$4.430^{r}$ | $6.96\pm 0.01$ | 1.17 | 2.83 | $4.0^{+2.2}_{-1.9}$ | 29.82 | $13.50^{+7.62}_{-6.58}\pm 0.54$
4.4359 | $582.34\pm 3.84$ | 1.16 | 2.83 | $110\pm 12$ | $219\pm 18$ | 22.23 | 35.27 | $12.17\pm 1.33\pm 0.51$ | $15.36\pm 1.26\pm 0.60$ | $13.81\pm 0.92\pm 0.39$
$4.450^{r}$ | $7.86\pm 0.01$ | 1.19 | 2.83 | $3.4^{+2.6}_{-1.9}$ | 29.32 | $10.42^{+7.97}_{-5.82}\pm 0.41$
$4.460^{r}$ | $8.96\pm 0.01$ | 1.19 | 10.22 | $6.0^{+3.1}_{-2.4}$ | 29.14 | $16.11^{+8.32}_{-6.44}\pm 1.70$
4.4671 | $111.09\pm 0.73$ | 1.17 | 10.22 | $25\pm 6$ | $26\pm 7$ | 22.05 | 35.61 | $15.10\pm 3.48\pm 1.61$ | $9.38\pm 2.52\pm 0.99$ | $11.78\pm 2.07\pm 0.89$
$4.480^{r}$ | $8.39\pm 0.01$ | 1.20 | 10.22 | $2.6^{+2.9}_{-1.6}$ | 28.97 | $10.30^{+8.30}_{-4.58}\pm 0.79$
$4.500^{r}$ | $8.24\pm 0.01$ | 1.22 | 10.22 | $0.4^{+1.7}_{-0.9}$ | 28.40 | $3.82^{+4.99}_{-2.64}\pm 0.12$
$4.520^{r}$ | $8.95\pm 0.01$ | 1.22 | 10.22 | $2.8^{+2.2}_{-1.5}$ | 28.37 | $7.82^{+5.93}_{-4.04}\pm 0.80$
4.5271 | $112.12\pm 0.74$ | 1.12 | 2.84 | $25\pm 6$ | $27\pm 6$ | 23.59 | 38.18 | $14.08\pm 3.38\pm 0.59$ | $9.76\pm 2.44\pm 0.38$ | $11.57\pm 2.00\pm 0.33$
$4.540^{r}$ | $9.66\pm 0.01$ | 1.23 | 2.84 | $1.5^{+2.6}_{-1.9}$ | 28.20 | $11.98^{+6.49}_{-4.74}\pm 0.15$
$4.550^{r}$ | $9.05\pm 0.01$ | 1.23 | 2.84 | $3.5^{+2.6}_{-1.9}$ | 28.13 | $11.74^{+6.93}_{-5.07}\pm 0.36$
$4.560^{r}$ | $8.59\pm 0.01$ | 1.24 | 2.84 | $2.6^{+2.4}_{-1.7}$ | 28.01 | $8.99^{+6.74}_{-4.78}\pm 0.28$
$4.570^{r}$ | $8.69\pm 0.01$ | 1.24 | 2.84 | $1.9^{+2.2}_{-1.5}$ | 28.71 | $5.93^{+5.93}_{-4.05}\pm 0.20$
4.5745 | $48.93\pm 0.32$ | 1.06 | 3.93 | $7\pm 3$ | $21\pm 5$ | 23.50 | 39.88 | $9.63\pm 4.13\pm 0.48$ | $16.21\pm 4.05\pm 0.77$ | $12.95\pm 2.89\pm 0.45$
$4.580^{r}$ | $8.84\pm 0.01$ | 1.25 | 12.26 | $3.9^{+2.8}_{-2.0}$ | 28.51 | $13.87^{+7.47}_{-5.34}\pm 1.31$
$4.590^{r}$ | $8.50\pm 0.01$ | 1.25 | 12.26 | $0.9^{+1.8}_{-1.1}$ | 28.67 | $3.86^{+4.96}_{-3.03}\pm 0.31$
4.5995 | $586.89\pm 3.87$ | 1.49 | 12.26 | $64\pm 9$ | $99\pm 14$ | 19.37 | 31.79 | $6.35\pm 0.89\pm 0.80$ | $5.98\pm 0.85\pm 0.75$ | $6.16\pm 0.61\pm 0.55$
4.6119 | $103.83\pm 0.68$ | 2.35 | 55.62 | $7\pm 3$ | $5\pm 4$ | 14.22 | 23.18 | $3.86\pm 1.89\pm 2.18$ | $1.80\pm 1.50\pm 1.00$ | $2.72\pm 1.20\pm 1.13$
4.6280 | $521.52\pm 3.37$ | 1.22 | 7.34 | $43\pm 8$ | $91\pm 13$ | 18.55 | 31.20 | $6.09\pm 1.13\pm 0.49$ | $7.68\pm 1.10\pm 0.61$ | $6.90\pm 0.80\pm 0.39$
4.6409 | $552.41\pm 3.57$ | 1.43 | 8.72 | $54\pm 8$ | $75\pm 12$ | 19.77 | 32.22 | $5.79\pm 0.86\pm 0.54$ | $4.95\pm 0.79\pm 0.46$ | $5.35\pm 0.59\pm 0.35$
4.6612 | $529.63\pm 3.45$ | 1.48 | 3.75 | $42\pm 7$ | $69\pm 12$ | 18.87 | 31.09 | $4.73\pm 0.79\pm 0.23$ | $4.74\pm 0.82\pm 0.22$ | $4.73\pm 0.58\pm 0.16$
4.6819 | $1669.31\pm 10.77$ | 1.35 | 2.59 | $134\pm 14$ | $210\pm 20$ | 18.41 | 30.66 | $5.40\pm 0.56\pm 0.22$ | $5.10\pm 0.49\pm 0.19$ | $5.24\pm 0.38\pm 0.15$
4.6988 | $536.20\pm 3.47$ | 1.33 | 3.97 | $46\pm 8$ | $72\pm 12$ | 18.65 | 31.97 | $5.77\pm 1.00\pm 0.29$ | $5.22\pm 0.88\pm 0.25$ | $5.48\pm 0.68\pm 0.19$


# Matsumoto theorem for skeleta

M. Gorelik  Weizmann Institute of Science<EMAIL_ADDRESS>,  V.
Hinich  University of Haifa<EMAIL_ADDRESS>and  V. Serganova  UC Berkeley
<EMAIL_ADDRESS>
###### Abstract.

We present a proof of a generalization of the theorem of H. Matsumoto on
Coxeter groups. Our generalized version is applicable to “graphs admitting
geometric realization”. The original version of the theorem for Coxeter groups
is a special case when applied to the Cayley graph and the geometric
representation of a Coxeter group. Our version of Matsumoto theorem is also
applicable to skeleta, graphs that were defined in the recent paper [GHS] on
root Lie superalgebras.

##  1\. Introduction

### 1.1.

Let $(G,S)$ be a Coxeter group. Recall that this means that

$G=\mathit{Free}_{S}/((st)^{m_{st}}),$

the factor of the free group generated by $S$, where $M=(m_{st})$ is a
symmetric matrix over $\mathbb{Z}$ satisfying the conditions $m_{ss}=1,\
m_{st}>1$ for $s\neq t$. A reduced decomposition of $g\in G$ is a shortest
expression

$g=s_{1}\ldots s_{n},\ s_{i}\in S$

and we denote by $\ell(g)$ the length of a reduced decomposition of $g$.

It is convenient to rewrite the relations as follows

$\begin{cases}s^{2}=1,&s\in S\\\ sts\ldots=tst\ldots\text{ (both expressions
of length }m_{st}),&s\neq t\in S.\end{cases}$

The second equality is called the generalized braid (also Artin-Tits)
relation. We define an equivalence relation (called the braid relation) on the
set of words in the alphabet $S$ as the minimal relation closed under
concatenation and containing the Artin-Tits relation.

The following result is proven by Matsumoto [M] in 1964.

######  1.1.1 Theorem.

Let $g=s_{1}\ldots s_{n}=t_{1}\ldots t_{n}$ be two reduced decompositions of
$g$, with $s_{i}$ and $t_{j}$ in $S$. Then the sequences
$(s_{1},\ldots,s_{n})$ and $(t_{1},\ldots,t_{n})$ are braid equivalent.

The original proof of Matsumoto theorem is based on the following replacement
property.

######  Lemma.

Let $g=s_{1}\ldots,s_{n}$ be a reduced decomposition and let $s\in S$ satisfy
the condition

$\ell(sg)<n.$

Then there exists $j$ such that

$ss_{1}\ldots s_{n}=s_{1}\ldots s_{j-1}s_{j+1}\ldots s_{n}.$

### 1.2.

In a recent paper [GHS] a family of graphs (skeleta) $\mathtt{Sk}(v)$ having a
certain property of Coxeter type was described. One can formulate a version of
Matsumoto theorem for these graphs but the replacement property does not seem
to hold for skeleta. This led us to an attempt to generalize Matsumoto theorem
so that it would cover the case of skeleta.

Our solution turns out to be very simple: it gives a proof that is, in our
opinion, easier even for the Coxeter groups. Note that our proof does not use
replacement property. Our approach is somewhat similar (though not completely
equivalent) to that of Heckenberger-Yamane [HY]. It seems that validity of the
Matsumoto theorem for skeleta in the fully reflectable case could have been
deduced from [HY] and the Coxeter property of skeleta proven in [GHS], Sect.
6.

##  2\. Matsumoto graphs

###  2.1. Rays and cones

Our basic geometric object will consist of closed rays in a real vector space
$\mathbb{V}$.

For a collection of rays $\underline{\alpha}=(\alpha_{1},\ldots,\alpha_{k})$
their sum $C(\underline{\alpha})=C(\alpha_{1},\ldots,\alpha_{k})$ is the
convex cone generated by $\alpha_{i}$. For a collection
$\underline{\alpha}=(\alpha_{i},\ i=1,\ldots,\dim\mathbb{V})$ of linearly
independent rays the cone $C(\underline{\alpha})$ will be called the
simplicial cone spanned by $\alpha_{i}$.

Note that the rays spanning a simplicial cone $C$ are uniquely determined by
$C$.

###  2.2. Graphs and their realization

Our graphs $\Gamma=(V,E)$ are connected. Their edges may be compact (having
two ends) or noncompact (infinite, connected to only one vertex), and we write
$E=E_{c}\sqcup E_{\infty}$.

We think of compact edges as the ones allowing both orientations, whereas
infinite edges are always incoming. We define the set of oriented edges
$\stackrel{{\scriptstyle\to}}{{E}}=\stackrel{{\scriptstyle\to}}{{E_{c}}}\sqcup
E_{\infty}$ where $\stackrel{{\scriptstyle\to}}{{E_{c}}}$ is a two-fold
covering of $E_{c}$ that we identify with $E_{c}\times\\{\pm 1\\}$. Thus,

$\stackrel{{\scriptstyle\to}}{{E}}=\\{(e,\epsilon)\in E\times\\{\pm 1\\}|\
\epsilon=1\mathrm{\ if\ }e\in E_{\infty}\\}.$

A geometric realization of a graph $\Gamma$ is an assignment of a ray
$\alpha_{e,\epsilon}$ to each oriented edge $(e,\epsilon)$ so that
$\alpha_{e,-\epsilon}=-\alpha_{e,\epsilon}$ for $e\in E_{c}$. If $e\in
E_{\infty}$, we will write $\alpha_{e}$ instead of $\alpha_{e,1}$.

We define the set of roots $R=\\{\alpha_{e,\epsilon}\\}$ and we require the
following axioms.

  * 1.

If $e\in E_{\infty}$ then $-\alpha_{e}\not\in R$.

  * 2.

For any $v\in V$ the set of roots $\alpha_{1},\ldots,\alpha_{d}$ assigned to
the edges ingoing to $v$, form a basis of $\mathbb{V}$. In particular, the
graph $\Gamma$ is regular of degree $d=\dim\mathbb{V}$. We denote
$R^{+}_{v}=\\{\alpha\in R|\ \alpha\in C(\alpha_{1},\ldots,\alpha_{d})\\}$.

  * 3.

For each $(e,\epsilon):v\to v^{\prime}$ one has

$R^{+}_{v^{\prime}}\setminus R^{+}_{v}=\\{\alpha_{e,\epsilon}\\}.$

Note that this condition implies that a graph having a geometric realization
has no loops $e:v\to v$ and no multiple edges.

  * 4.

If $R^{+}_{v}=R^{+}_{v^{\prime}}$ then $v=v^{\prime}$.

######  2.2.1 Definition.

A Matsumoto graph is a connected graph endowed with a geometric realization.

Note that a root $\alpha$ corresponds to a compact edge iff $-\alpha\not\in
R$. Having this in mind, the roots corresponding to compact edges will be
called invertible and the roots of infinite edges will be called non-
invertible.

Our generalization of Matsumoto theorem is formulated in terms of Matsumoto
graphs. It claims that any two shortest paths from one vertex to another in a
Matsumoto graph are braid equivalent. The result is formulated only in 2.5.2
as the braid equivalence requires a certain effort to define.

###  2.3. Basic properties

The property below immediately follows from the axioms.

######  2.3.1 Lemma.

  * 1.

For any $e\in E_{\infty}$ and any $v\in V$ one has $\alpha_{e}\in R^{+}_{v}$.

  * 2.

$R\subset R^{+}_{v}\sqcup(-R^{+}_{v})$.

∎

We define $R^{-}_{v}:=R\setminus R^{+}_{v}\subset-R^{+}_{v}$.

######  2.3.2 Lemma.

If $R^{+}_{v}\subset R^{+}_{v^{\prime}}$ then $v=v^{\prime}$.

###### Proof.

Since the graph is connected, the difference between any pair of sets
$R^{+}_{v}$ and $R^{+}_{v^{\prime}}$ is finite. Therefore, we can talk about
relative cardinality of these sets. A passage from a vertex to its neighbor
along an edge does not change the cardinality. Therefore, the relative
cardinality is always zero. Thus, by the assumption
$R^{+}_{v}=R^{+}_{v^{\prime}}$ and by Axiom 4 $v=v^{\prime}$. ∎

For vertices $v,v^{\prime}$ we define the distance $d(v,v^{\prime})$ as the
length of a shortest path connecting $v$ with $v^{\prime}$.

######  2.3.3 Lemma.

Any path from $v$ to $v^{\prime}$ has length equal to $d(v,v^{\prime})$ modulo
2. $d(v,v^{\prime})$ is the cardinality of $R^{+}_{v^{\prime}}\setminus
R^{+}_{v}$. In particular, any closed path has an even cardinality; that is,
any graph admitting a geometric realization, is bipartite.

###### Proof.

The first claim follows from Axiom 3. The inequality
$d(v,v^{\prime})\geq|R^{+}_{v^{\prime}}\setminus R^{+}_{v}|$ is obvious. The
converse is deduced by induction, as by 2.3.2 for $v^{\prime}\neq v$ there
exists an incoming edge $(e,\epsilon)$ to $v$ which does not belong to
$R^{+}_{v^{\prime}}$. ∎

#### 2.3.4.

Matsumoto graphs are regular of degree $d=\dim\mathbb{V}$. The lemma below
shows that any edge $e$ connecting two vertices $v$ and $w$ establishes a one-
to-one correspondence between the edges adjacent to $v$ and to $w$ so that $e$
corresponds to itself. In Proposition 5.1.1 we will show that this
correspondence leads to a consistent coloring of the set of edges $E$ with the
elements of a set $X$ of cardinality $d$, so that the correspondence above
assigns to an edge adjacent to $v$ the edge adjacent to $w$ of the same color.

######  2.3.5 Lemma.

Let $(e,\epsilon):v\to v^{\prime}$, $\alpha=\alpha_{e,\epsilon}$ and
$R^{+}_{v}=C(\alpha_{1},\ldots,\alpha_{d})$ with $\alpha_{1}=-\alpha$. Then
there is a unique numbering of the basic roots of
$R^{+}_{v^{\prime}}=C(\alpha^{\prime}_{1},\ldots,\alpha^{\prime}_{d})$ such
that $\alpha^{\prime}_{1}=-\alpha_{1}=\alpha$ and for each $i>1$
$\alpha^{\prime}_{i}$ lies in the cone spanned by $\alpha_{i}$ and
$\alpha_{1}$.

###### Proof.

Let $\mathbb{V}^{\prime}=\mathbb{V}/\mathbb{R}\alpha$ and
$\pi:\mathbb{V}\to\mathbb{V}^{\prime}$ denote the natural projection. Then
$C^{\prime}=\pi(C(\alpha_{2},\dots,\alpha_{d}))=\pi(C(\alpha_{2}^{\prime},\dots\alpha_{d}^{\prime}))$
coincides with the image under $\pi$ of the convex hull of the set
$R^{+}_{v}\cap R^{+}_{v^{\prime}}$. On the other hand, $C^{\prime}$ is a
simplicial cone in $\mathbb{V}^{\prime}$ with generators
$\pi(\alpha_{2}),\dots,\pi(\alpha_{d})$ and similarly with generators
$\pi(\alpha^{\prime}_{2}),\dots,\pi(\alpha^{\prime}_{d})$. We choose the
enumeration so that $\pi(\alpha_{i})=\pi(\alpha_{i}^{\prime})$. ∎

######  2.3.6 Lemma.

Let $v$ be a vertex of a Matsumoto graph with basis
$\\{\alpha_{1},\dots,\alpha_{d}\\}$. Let $\\{\alpha_{1},\dots,\alpha_{k}\\}$
be a subset and $\mathbb{W}$ be the span of
$\\{\alpha_{1},\dots,\alpha_{k}\\}$. Then $R\cap\mathbb{W}$ gives a geometric
realization of a full subgraph $\Gamma^{\prime}$ of degree $k$.

###### Proof.

We define $\Gamma^{\prime}$ as the connected component of $v$ of the subgraph
$\Gamma^{\prime\prime}$ of $\Gamma$ having the same vertices as $\Gamma$ and
the edges $e\in E$ such that $\alpha_{e,1}\in\mathbb{W}$.

The vertex $v\in\Gamma^{\prime}$ has $k$ incoming edges marked by
$\\{\alpha_{1},\dots,\alpha_{k}\\}$. Lemma 2.3.5 implies that
$\Gamma^{\prime}$ is the Matsumoto graph of degree $k$ with the set of roots
$R^{\prime}\subset R\cap\mathbb{W}$. We will show that
$R^{\prime}=R\cap\mathbb{W}$.

For any vertex $v^{\prime}\in\Gamma^{\prime}$ we denote by
$\alpha_{1}^{\prime},\dots,\alpha^{\prime}_{k}$ the basis at $v^{\prime}$.
Note first that for any $v^{\prime}\in\Gamma^{\prime}$ we have
$R^{+}_{v^{\prime}}\cap\mathbb{W}\subset
C(\alpha_{1}^{\prime},\dots,\alpha_{k}^{\prime})$ again by Lemma 2.3.5. Next
by induction on the cardinality of
$(R^{+}_{v^{\prime}}\cap\mathbb{W})\setminus(R^{+}_{u}\cap\mathbb{W})$ we can
prove that for any $u\in\Gamma$ there exists $v^{\prime}\in\Gamma^{\prime}$
such that
$(R^{+}_{v^{\prime}}\cap\mathbb{W})\subset(R^{+}_{u}\cap\mathbb{W})$.

Now let $\beta\in R\cap\mathbb{W}$. There exists $u\in\Gamma$ such that
$\beta$ is an element of the basis at $u$. Then either $\beta$ or $-\beta$
lies in $R^{+}_{v^{\prime}}\cap\mathbb{W}\subset
C(\alpha^{\prime}_{1},\dots,\alpha_{k}^{\prime})$ where
$v^{\prime}\in\Gamma^{\prime}$ is as above. Suppose that $\beta\in
R^{+}_{v^{\prime}}\cap\mathbb{W}$. We claim that $\beta=\alpha^{\prime}_{i}$
for some $i\leq k$. Indeed, consider a linear function $\psi$ such that
$\psi(\beta)=0$ and $\psi(\gamma)>0$ for all other $\gamma\in R^{+}_{u}$. If
$\beta\neq\alpha^{\prime}_{i}$ for all $i\leq k$ then $\psi(\beta)>0$ and we
get a contradiction. Similarly, if $-\beta\in
R^{+}_{v^{\prime}}\cap\mathbb{W}$ we can prove that
$-\beta=\alpha^{\prime}_{i}$ by the same method. The proof is complete. ∎

######  2.3.7 Notation.

The graph $\Gamma^{\prime}$ in the above construction will be denoted
$\Gamma(v,\alpha_{1},\ldots,\alpha_{k})$.

###  2.4. The case $d=2$

#### 2.4.1.

Here are all connected graphs of degree two.

  * 1.

A polygon. Here all edges are compact. If an $n$-gon admits a geometric
realization, $n$ has to be even by Lemma 2.3.3. A polygon with an even number
of edges is the Cayley graph of a dihedral group, so it admits a realization,
see 3.1.

  * 2.

A graph with an infinite number of compact edges. This is the Cayley graph of
the infinite dihedral group, so it admits a realization, see 3.1.

  * 3.

A graph with one infinite edge and an infinite number of compact edges.

  * 4.

A graph with two infinite edges and a finite number of compact edges.

A geometric realization for the cases 3,4 can be easily found. Note that in
all Matsumoto cases there is a canonical coloring of the edges so that any two
adjacent edges have different colors.

###  2.5. Braid relation and Matsumoto theorem

####  2.5.1. Braid relation

Fix a vertex $v\in\Gamma$. Choose two edges adjacent to $v$, one incoming and
one outcoming. Let $\alpha$ and $\beta$ be the corresponding roots. There is a
unique continuation of the path $(\alpha,\beta)$, potentially in both
directions, so that the roots assigned to the new edges belong to
$\operatorname{Span}_{\mathbb{R}}(\alpha,\beta)$. This yields a rank 2
Matsumoto subgraph $\Gamma(v,\alpha,\beta)$ of $\Gamma$ spanned by $\alpha$
and $\beta$.

We denote by $\mathcal{P}=\mathcal{P}(\Gamma)$ the set of paths in $\Gamma$.
We define the braid relation on $\mathcal{P}$ as the minimal equivalence
relation satisfying the following two properties.

  * 1.

For any rank two Matsumoto subgraph $\Gamma(v,\alpha,\beta)$ presented by a
$2m$-gon and colored by $X=\\{x,y\\}$, the length $m$ paths

$xyx\ldots\mathrm{and}\ yxy\ldots$

starting at $v$ (and having the same end) are equivalent.

  * 2.

The equivalence relation is closed under the concatenation of paths.

Note that equivalent paths have the same ends and the same length.

The main result of this note is the following generalization of Matsumoto
theorem [M].

######  2.5.2 Theorem.

Let $\Gamma$ be a Matsumoto graph. Then any two shortest paths in $\Gamma$
from $v$ to $v^{\prime}$ are braid equivalent.

The proof is given in Section 4.

##  3\. Examples

In this section we present two examples of Matsumoto graphs: the Cayley graphs
of Coxeter groups and the skeleta of admissible components of the root
groupoid, see [GHS], Section 5.

###  3.1. Coxeter groups

The Matsumoto structure on the Cayley graph of a general Coxeter group
$(G,X)$, based on its geometric representation [H], 5.3, is described below in
3.1.2. We present here an even more classical construction that makes sense
for Weyl groups.

####  3.1.1. Weyl group

Let $\mathfrak{g}$ be a semisimple Lie algebra with a Cartan subalgebra
$\mathfrak{h}$, the root system $\Delta$ and the set of simple roots
$\alpha_{1},\ldots,\alpha_{n}$. The Weyl group $W$ is generated by the simple
reflections $s_{i}=s_{\alpha_{i}}$.

We define $\Gamma$ as the Cayley graph of $W$ with respect to the set $S$ of
simple reflections, so that the edges (they are all compact) are of the form
$w\to ws$, $s\in S$. The arrow $w\to ws_{i}$ is marked with the root
$\mathbb{R}_{\geq 0}\cdot w(\alpha_{i})$.

The same construction of Matsumoto structure generalizes to Weyl groups of
Kac-Moody Lie algebras, as well as for root Lie superalgebras [GHS].

Note that the Matsumoto structures corresponding to the classical root systems
$B_{n}$ and $C_{n}$ are isomorphic, as in our formalism roots are rays and not
vectors.

####  3.1.2. General Coxeter group

Let $(G,X)$ be a Coxeter group with the generating set $X$. We define $\Gamma$
as the Cayley graph of $G$ with respect to the set $X$ of generators. All
edges in $\Gamma$ are compact and are colored by the set of generators $X$.
The geometric representation $\sigma:G\to GL(\mathbb{V})$, [H], 5.3, where
$\mathbb{V}=\operatorname{Span}_{\mathbb{R}}\\{\alpha_{x},x\in X\\}$, yields a
geometric realization of $\Gamma$ as follows. It assigns to an arrow $x:g\to
gx$ the root $\mathbb{R}_{\geq 0}\cdot\sigma_{g}(\alpha_{x})\in\mathbb{V}$.
The set $R^{+}_{g}$ is spanned by $-\sigma_{g}(\alpha_{x})$, $x\in X$. The
axioms 1–3 of the geometric realization are verified immediately. The Axiom 4
follows from [H], 5.6.

###  3.2. Skeleta

The application to skeleta is the raison-d’être of the present note. Let $X$
be a finite set and let
$v=(\mathfrak{h},a:X\to\mathfrak{h},b:X\to\mathfrak{h}^{*},p:X\to\mathbb{Z}_{2})$
be an admissible root datum, see [GHS], Section 2. Let $\mathtt{Sk}(v)$ be the
connected component of $v$ in the skeleton of the root groupoid $\mathcal{R}$,
see [GHS], Sect. 3, 4. The graph $\Gamma$ assigned to it is a regular graph
with the edges colored by $X$ having the same vertices as $\mathtt{Sk}(v)$.
The compact edges of $\Gamma$ are the edges of $\mathtt{Sk}(v)$. The infinite
edges of $\Gamma$ are described by the nonreflectable pairs $(v^{\prime},x)$,
with $v^{\prime}\in\mathtt{Sk}(v)$ and $x\in X$. The geometric realization of
$\Gamma$ assigns to each arrow $r_{x}:v\to v^{\prime}$ the real root
$b_{v^{\prime}}(x)$ and to an infinite edge defined by the nonreflectable pair
$(v^{\prime},x)$ the real nonreflectable root $b_{v^{\prime}}(x)$. In this
realization the roots of $\Gamma$ are precisely the real roots of the
component of $v\in\mathcal{R}$.

It is proven in [GHS] that $\mathtt{Sk}(v)$ is a Coxeter graph. Now Theorem
2.5.2 implies that the analog of Matsumoto theorem holds of $\mathtt{Sk}(v)$.
Note that Theorem 2.5.2 gives another proof of coxeterity of $\mathtt{Sk}(v)$.

##  4\. Proof of Theorem 2.5.2

We will prove the theorem by induction in the length $n$ of the shortest
paths.

### 4.1.

The cases $n=0$ are obvious as a Matsumoto graph does not have loops and
multiple edges.

### 4.2.

Assume the result is proven for shortest paths of length $\leq n-1$. Let

$\underline{\alpha}:v=v_{0}\xrightarrow{\alpha_{1}}\ldots\xrightarrow{\alpha_{n}}v_{n}=v^{\prime}$

and

$\underline{\beta}:v=v^{\prime}_{0}\xrightarrow{\beta_{1}}\ldots\xrightarrow{\beta_{n}}v^{\prime}_{n}=v^{\prime}$

be a pair of length $n$ shortest paths leading from $v$ to $v^{\prime}$. By
Lemma 2.3.3 the sets of $\alpha_{i}$ and of $\beta_{j}$ coincide. Thus,
$\beta_{1}=\alpha_{i}$ for some $i$.

### 4.3.

Assume that $i<n$. Consider the path

$v^{\prime}_{1}\xrightarrow{-\beta_{1}}v^{\prime}_{0}=v\xrightarrow{\alpha_{1}}\ldots\xrightarrow{\alpha_{i}}v_{i}.$

Obviously $d(v^{\prime}_{1},v_{i})=i-1$. Choose a path $\gamma$ of length
$i-1$ connecting $v^{\prime}_{1}$ with $v_{i}$. Then, by the inductive
hypothesis, the path $\alpha$ is braid equivalent to the concatenation of
$\beta_{1}$, $\gamma$, and the segment of $\alpha$ connecting $v_{i}$ with
$v_{n}=v^{\prime}$. In the same manner the above concatenation is equivalent
to the path $\beta$. This proves that in this case the paths $\alpha$ and
$\beta$ are braid equivalent. Therefore, we may assume that $i=n$, that is
$\beta_{1}=\alpha_{n}$.

### 4.4.

In the same way, we can assume that $\alpha_{1}=\beta_{n}$.

### 4.5.

It remains to deal with the case when $\alpha_{1}=\beta_{n}$ and
$\alpha_{n}=\beta_{1}$. One has $\alpha_{1},\beta_{1}\in
R^{+}_{v^{\prime}}\cap R^{-}_{v}$. This means that

$\Gamma(v,-\beta_{1},\alpha_{1})=C(\alpha_{1},\beta_{1})\cap R\subset
R^{+}_{v^{\prime}}\cap R^{-}_{v}.$

The roots assigned to infinite edges are everywhere positive, the intersection
$R^{+}_{v^{\prime}}\cap R^{-}_{v}$ is finite, so
$\Gamma(v,-\beta_{1},\alpha_{1})$ is a polygon with an even number of edges.
Denote by $v^{\prime\prime}$ the vertex opposite to $v$. One has

$d(v,v^{\prime})=d(v,v^{\prime\prime})+d(v^{\prime\prime},v^{\prime}).$

Denote by $\gamma$ a shortest path connecting $v^{\prime\prime}$ with
$v^{\prime}$. Denote by $\gamma^{\prime}$, $\gamma^{\prime\prime}$ two
equivalent halves of $\Gamma(v,-\beta_{1},\alpha_{1})$ connecting $v$ to
$v^{\prime\prime}$. Then the concatenation of $\gamma^{\prime}$ with $\gamma$
is equivalent to $\alpha$ by the inductive hypothesis, as they start with the
same root $\alpha_{1}$, and similarly the concatenation of
$\gamma^{\prime\prime}$ with $\gamma$ is equivalent to $\beta$.

This proves the theorem. ∎

##  5\. Complements

###  5.1. Edge coloring

We will now prove the existence of a consistent coloring of the set of edges
$E$ of a Matsumoto graph $\Gamma$.

######  5.1.1 Proposition.

Let $\Gamma$ be a graph of degree $d$ endowed with a geometric realization.
Let $|X|=d$. There exists a coloring $E\to X$ of the edges of $\Gamma$
consistent with the isomorphisms defined by Lemma 2.3.5 for each arrow $v\to
v^{\prime}$. The coloring is unique up to automorphism of $X$.

###### Proof.

Let us identify $X$ with the set of generators of $R^{+}_{v}$. Any closed path
$\gamma:v\to v$ in $\Gamma$ defines an automorphism $\theta_{\gamma}:X\to X$.
It is enough to prove that $\theta_{\gamma}=\mathrm{id}_{X}$ for any $\gamma$.
Theorem 2.5.2 implies that if $\gamma^{\prime},\gamma^{\prime\prime}:v\to
v^{\prime}$ are two shortest paths,
$\theta_{\gamma^{\prime}\circ\gamma^{\prime\prime-1}}=\mathrm{id}_{X}$.

Let us assume, to the contrary, that there exists a closed path $\gamma:v\to
v$ for which $\theta_{\gamma}\neq\mathrm{id}_{X}$. Choose a shortest such
$\gamma$, of length $2m$. For any root $\alpha$ in $\gamma$ the root $-\alpha$
should also appear. Let us choose a closest pair of edges is $\gamma$ with
assigned roots $\alpha$ and $-\alpha$.

It is easy to see that if $\alpha$ and $-\alpha$ are not precisely opposite to
each other, one can produce a shorter counterexample.

In the remaining case when all pairs $\alpha$ and $-\alpha$ are opposite to
each other, the loop $\gamma$ can necessarily be presented as
$\gamma^{\prime}\circ\gamma^{\prime\prime-1}$ where $\gamma^{\prime}$ and
$\gamma^{\prime\prime}$ are two shortest paths with the same ends. This proves
the claim. ∎

######  5.1.2 Remark.

Note that in the examples of graphs with a geometric realization presented in
Section 3, the coloring provided by Proposition 5.1.1 coincides with the
apriori coloring given, in the first case by the set $S$ of generator, and in
the case of skeleta — by the set $X$.

###  5.2. Exchange condition

In this subsection we discuss a partial exchange condition that holds in the
context of skeleta [GHS], 4.2.5. The edges of a skeleton $\mathtt{Sk}(v)$ are
reflexions that can be anisotropic or isotropic, see definitions in [GHS],
4.1.1. The following result shows that, in the context of skeleta, the
exchange condition holds for anisotropic reflexions.

######  5.2.1 Proposition.

Let

$v_{0}\stackrel{{\scriptstyle r_{x_{1}}}}{{\to}}\ldots\stackrel{{\scriptstyle
r_{x_{n}}}}{{\to}}v_{n}$

be a sequence of reflexions that is a shortest path from $v_{0}$ to $v_{n}$
and let $r_{y}:v^{\prime}_{0}\to v_{0}$ be an anisotropic reflexion such that
the path $v_{0}^{\prime}\to v_{0}\to\ldots\to v_{n}$ is not shortest. Then
there exists $j$ so that the composition

$v^{\prime}_{0}\stackrel{{\scriptstyle
r_{y}}}{{\to}}v_{0}\to\ldots\stackrel{{\scriptstyle r_{x_{i}}}}{{\to}}v_{j}$

coincides with the composition

$v^{\prime}_{0}\stackrel{{\scriptstyle
r_{x_{1}}}}{{\to}}v^{\prime}_{1}\to\ldots\stackrel{{\scriptstyle
r_{x_{j-1}}}}{{\to}}v^{\prime}_{j-1}=v_{j}.$

###### Proof.

By the assumptions, $\alpha=b_{v_{0}}(y)=b_{v_{j}}(x_{j})$ for some $j$. The
path $v^{\prime}_{0}\to\ldots\to v^{\prime}_{j-1}$ exists as the namesake of
the path $v_{0}\to\ldots\to v_{j-1}$, see [GHS], 4.3.6, 4.3.7. The equality
$v^{\prime}_{j-1}=v_{j}$ then follows from 4.3.7. ∎

###  5.3. Geometric properties of roots

######  5.3.1 Lemma.

Let $\Gamma$ be a Matsumoto graph with set of roots $R$. Consider $R$ as a set
of points on the unit sphere in $\mathbb{V}$. If $\alpha,-\alpha\in R$ (that
is, if $\alpha$ is the root of a compact edge) then $\alpha$ is an isolated
pointed of $R$ in the usual topology on the unit sphere.

###### Proof.

Let $\alpha$ mark an arrow $v^{\prime}\to v$, so that
$\alpha=\alpha_{1},\ldots,\alpha_{d}$ form the basis of roots at $v$. If there
is a sequence of roots $\beta_{k}\in R^{+}_{v}$ converging to $\alpha$, then
$\alpha$ belongs to the closure of $R^{+}_{v}\setminus\alpha=R^{+}_{v}\cap
R^{+}_{v^{\prime}}\subset C(\alpha_{2},\ldots,\alpha_{d})$ which is
impossible. For the same reason a sequence of $\beta_{k}\in R^{-}_{v}$ cannot
possibly converge to $\alpha$: then $-\beta_{k}\in
C(\alpha_{2},\ldots,\alpha_{d})$ would converge to $-\alpha$. This proves the
result. ∎

Note that a noninvertible root $\alpha$ may be a limit point in $R$ as we will
see in 5.4.4 below.

###  5.4. Dual picture

The notion of Matsumoto graph can be equivalently expressed in the dual
picture, in terms of convex geometry. We present below this equivalent
language.

#### 5.4.1.

Let $\Gamma$ be a Matsumoto graph with the set of roots $R$ in a vector space
$\mathbb{V}$. For every $\alpha\in R$ define a hyperplane $H_{\alpha}$ and a
half-space $U_{\alpha}$ in the dual space $\mathbb{V}^{*}$ by

$H_{\alpha}=\\{\xi\in\mathbb{V}^{*}\mid\langle\xi,\alpha\rangle=0\\},\quad
U_{\alpha}=\\{\xi\in\mathbb{V}^{*}\mid\langle\xi,\alpha\rangle\geq 0\\}.$

For a vertex $v\in\Gamma$ with assigned basis $\alpha_{1},\dots,\alpha_{d}$ we
define a chamber $C^{\vee}_{v}$ to be the intersection $\cap U_{\alpha_{i}}$.
Then $C^{\vee}_{v}$ is a simplicial cone in $\mathbb{V}^{*}$.

######  5.4.2 Lemma.

Let $D=\cup_{v\in\Gamma}C^{\vee}_{v}$ and $D^{\prime}$ be the set of all
$\xi\in\mathbb{V}^{*}$ such that $\langle\xi,\alpha\rangle\geq 0$ for all but
finitely many invertible $\alpha\in R_{v}^{+}$. Then $D=D^{\prime}$. In
particular, $D$ is conical convex set.

###### Proof.

Obviously $D\subset D^{\prime}$. Let us show that $D=D^{\prime}$. Suppose that
$\xi\in D^{\prime}$ and $M$ be the set of all invertible $\alpha\in R_{v}^{+}$
such that $\langle\xi,\alpha\rangle<0$. If $M\neq\emptyset$ then there is an
invertible root $\alpha$ in the basis at $v$ such that $\alpha\in M$. Consider
the arrow $v^{\prime}\to v$ marked by $\alpha$. Then
$\langle\xi,\beta\rangle\geq 0$ for all $\beta\in R_{v^{\prime}}^{+}\setminus
M^{\prime}$ with $M^{\prime}=M\setminus\\{\alpha\\}$. Using induction on $|M|$
we can prove existence of a vertex $u$ such that $\xi$ is nonnegative on all
roots of $R^{+}_{u}$. This shows that $\xi\in C_{u}\subset D$. ∎

#### 5.4.3.

Vice versa, here is a way to reconstruct a Matsumoto graph from a convex
conical set in the dual space $\mathbb{V}^{*}$.

Let $D$ be a conical convex set in $\mathbb{V}^{*}$ and $R$ a collection of
rays in $\mathbb{V}$ so that each $H_{\alpha}$ intersects $D$ nontrivially. We
assume that all connected components of $D\setminus(\cup H_{\alpha})$ are
interiors of simplicial cones $C^{\vee}_{v}$ and $D=\cup C^{\vee}_{v}$. One
can now construct a Matsumoto graph $\Gamma$ whose vertices correspond to the
chambers and edges correspond to $d-1$-dimensional faces of chambers with
adjacency of chambers defined in the obvious way. This construction gives a
dual geometric realization of a Matsumoto graph.

######  5.4.4 Example.

The convex conic set $D$ is uniquely determined by its intersection with the
unit sphere in $\mathbb{V}^{*}$. Moreover, if $D\setminus\\{0\\}$ lies in some
open half-space

$\\{\xi\in\mathbb{V}^{*}|\ \langle\xi,\alpha\rangle>0\\},$

one can use instead the intersection of $D$ with the affine hyperplane
$H=\\{\xi\in\mathbb{V}^{*}|\ \langle\xi,\alpha\rangle=1\\}$.

Let us give one example in the case $\dim H=2$. Consider a triangle $A,B,C$ in
$H$, let $M_{0}$ be the midpoint $BC$, $M_{1}$ the midpoint of $AC$, $M_{2}$
the midpoint of $M_{1}C$, and then inductively $M_{n}$ is defined to be the
midpoint of $M_{n-1}C$. This configuration gives a dual realization of a
Matsumoto graph with chambers $ABM_{0},AM_{0}M_{1},\dots,AM_{n}M_{n+1}\
(n>0),\dots$. The roots are dual to the lines $AB$, $BM_{0}$, $M_{0}C$,
$M_{0}M_{i}$ for all $i>0$, $AM_{1}$, $M_{n}M_{n+1}$. The noninvertible roots
are $AB$, $BM_{0}$, $M_{0}C$, $AM_{1}$ and $M_{n}M_{n+1}$ for $n>0$. Note that
$M_{0}C$ is the limit of the set $\\{M_{0}M_{n}\\}_{n\geq 1}$.

## References

  * [B] N. Bourbaki, Groupes et algèbres de Lie., Springer 1968.
  * [GHS] M. Gorelik, V. Hinich, V. Serganova, Root groupoid and related Lie superalgebras, arXiv:2209.06253.
  * [HY] I. Heckenberger, H. Yamane, A generalization of Coxeter groups, roots systems and Matsumoto’s theorem, Math. Z., 259 (2008), 255–276.
  * [H] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge UP, 1990.
  * [M] H. Matsumoto, Générateurs et relations des groupes de Weyl généralisés, C. R. Acad. Sci. Paris, 258 (1964), 3419-–3422.