# Measurement of the Born cross sections for
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ at center-of-mass energies between
$2.00$ and $3.08$ GeV

M. Ablikim1, M. N. Achasov10,c, P. Adlarson67, S. Ahmed15, M. Albrecht4, R.
Aliberti28, A. Amoroso66a,66c, Q. An63,50, Anita21, X. H. Bai57, Y. Bai49, O.
Bakina29, R. Baldini Ferroli23a, I. Balossino24, Y. Ban39,k, K. Begzsuren26,
N. Berger28, M. Bertani23a, D. Bettoni24a, F. Bianchi66a,66c, J Biernat67, J.
Bloms60, A. Bortone66a,66c, I. Boyko29, R. A. Briere5, H. Cai68, 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,b, D. Y. Chen6, G. Chen1, H. S. Chen1,55, M. L.
Chen1,50, S. J. Chen36, X. R. Chen25, Y. B. Chen1,50, Z. J Chen20,l, W. S.
Cheng66c, G. Cibinetto24a, F. Cossio66c, X. F. Cui37, H. L. Dai1,50, X. C.
Dai1,55, A. Dbeyssi15, R. E. de Boer4, D. Dedovich29, Z. Y. Deng1, A. Denig28,
I. Denysenko29, M. Destefanis66a,66c, F. De Mori66a,66c, Y. Ding34, C. Dong37,
J. Dong1,50, L. Y. Dong1,55, M. Y. Dong1,50,55, X. Dong68, S. X. Du71, J.
Fang1,50, S. S. Fang1,55, Y. Fang1, R. Farinelli24a, L. Fava66b,66c, F.
Feldbauer4, G. Felici23a, C. Q. Feng63,50, M. Fritsch4, C. D. Fu1, Y. Fu1, Y.
Gao39,k, Y. Gao64, Y. Gao63,50, Y. G. Gao6, I. Garzia24a,24b, E. M.
Gersabeck58, A. Gilman59, K. Goetzen11, L. Gong34, W. X. Gong1,50, W. Gradl28,
M. Greco66a,66c, L. M. Gu36, M. H. Gu1,50, S. Gu2, Y. T. Gu13, C. Y Guan1,55,
A. Q. Guo22, L. B. Guo35, R. P. Guo41, Y. P. Guo28, Y. P. Guo9,h, A. Guskov29,
T. T. Han42, X. Q. Hao16, F. A. Harris56, K. L. He1,55, F. H. Heinsius4, C. H.
Heinz28, T. Held4, Y. K. Heng1,50,55, C. Herold52, M. Himmelreich11,f, T.
Holtmann4, Y. R. Hou55, Z. L. Hou1, H. M. Hu1,55, J. F. Hu48,m, T. Hu1,50,55,
Y. Hu1, G. S. Huang63,50, L. Q. Huang64, X. T. Huang42, Y. P. Huang1, Z.
Huang39,k, N. Huesken60, T. Hussain65, W. Ikegami Andersson67, W. Imoehl22, M.
Irshad63,50, S. Jaeger4, S. Janchiv26,j, Q. Ji1, Q. P. Ji16, X. B. Ji1,55, X.
L. Ji1,50, H. B. Jiang42, X. S. Jiang1,50,55, X. Y. Jiang37, J. B. Jiao42, Z.
Jiao18, S. Jin36, Y. Jin57, T. Johansson67, N. Kalantar-Nayestanaki31, X. S.
Kang34, R. Kappert31, M. Kavatsyuk31, B. C. Ke44,1, I. K. Keshk4, A.
Khoukaz60, P. Kiese28, R. Kiuchi1, R. Kliemt11, L. Koch30, O. B. Kolcu54a,e,
B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc67, M. G. Kurth1,55, W. Kühn30,
J. J. Lane58, J. S. Lange30, P. Larin15, L. Lavezzi66a,66c, Z. H. Lei63,50, H.
Leithoff28, M. Lellmann28, T. Lenz28, C. Li40, C. H. Li33, Cheng Li63,50, D.
M. Li71, F. Li1,50, G. Li1, H. Li44, H. Li63,50, H. B. Li1,55, H. J. Li9,h, J.
L. Li42, J. Q. Li4, Ke Li1, L. K. Li1, Lei Li3, P. L. Li63,50, P. R. Li32, S.
Y. Li53, W. D. Li1,55, W. G. Li1, X. H. Li63,50, X. L. Li42, Z. Y. Li51, H.
Liang63,50, H. Liang1,55, Y. F. Liang46, Y. T. Liang25, L. Z. Liao1,55, J.
Libby21, C. X. Lin51, B. J. Liu1, C. X. Liu1, D. Liu63,50, F. H. Liu45, Fang
Liu1, Feng Liu6, H. B. Liu13, H. M. Liu1,55, Huanhuan Liu1, Huihui Liu17, J.
B. Liu63,50, J. Y. Liu1,55, K. Liu1, K. Y. Liu34, Ke Liu6, L. Liu63,50, M. H.
Liu9,h, Q. Liu55, S. B. Liu63,50, Shuai Liu47, T. Liu1,55, W. M. Liu63,50, X.
Liu32, Y. B. Liu37, Z. A. Liu1,50,55, Z. Q. Liu42, X. C. Lou1,50,55, F. X.
Lu16, H. J. Lu18, J. D. Lu1,55, J. G. Lu1,50, X. L. Lu1, Y. Lu1, Y. P. Lu1,50,
C. L. Luo35, M. X. Luo70, P. W. Luo51, T. Luo9,h, X. L. Luo1,50, S. Lusso66c,
X. R. Lyu55, F. C. Ma34, H. L. Ma1, L. L. Ma42, M. M. Ma1,55, Q. M. Ma1, R. Q.
Ma1,55, R. T. Ma55, X. N. Ma37, X. X. Ma1,55, X. Y. Ma1,50, F. E. Maas15, M.
Maggiora66a,66c, S. Maldaner28, S. Malde61, Q. A. Malik65, A. Mangoni23b, Y.
J. Mao39,k, Z. P. Mao1, S. Marcello66a,66c, Z. X. Meng57, J. G.
Messchendorp31, G. Mezzadri24a, T. J. Min36, R. E. Mitchell22, X. H.
Mo1,50,55, Y. J. Mo6, N. Yu. Muchnoi10,c, H. Muramatsu59, S. Nakhoul11,f, Y.
Nefedov29, F. Nerling11,f, I. B. Nikolaev10,c, Z. Ning1,50, S. Nisar8,i, S. L.
Olsen55, Q. Ouyang1,50,55, S. Pacetti23b,23c, X. Pan9,h, Y. Pan58, A. Pathak1,
P. Patteri23a, M. Pelizaeus4, H. P. Peng63,50, K. Peters11,f, J. Pettersson67,
J. L. Ping35, R. G. Ping1,55, A. Pitka4, R. Poling59, V. Prasad63,50, H.
Qi63,50, H. R. Qi53, K. H. Qi25, M. Qi36, T. Y. Qi2, T. Y. Qi9, S. Qian1,50,
W.-B. Qian55, Z. Qian51, C. F. Qiao55, L. Q. Qin12, X. S. Qin4, Z. H. Qin1,50,
J. F. Qiu1, S. Q. Qu37, K. H. Rashid65, K. Ravindran21, C. F. Redmer28, A.
Rivetti66c, V. Rodin31, M. Rolo66c, G. Rong1,55, Ch. Rosner15, M. Rump60, H.
S. Sang63, A. Sarantsev29,d, Y. Schelhaas28, C. Schnier4, K. Schoenning67, M.
Scodeggio24a, D. C. Shan47, W. Shan19, X. Y. Shan63,50, M. Shao63,50, C. P.
Shen9, P. X. Shen37, X. Y. Shen1,55, H. C. Shi63,50, R. S. Shi1,55, X.
Shi1,50, X. D Shi63,50, W. M. Song27,1, Y. X. Song39,k, S. Sosio66a,66c, S.
Spataro66a,66c, K. X. Su68, F. F. Sui42, G. X. Sun1, H. K. Sun1, J. F. Sun16,
L. Sun68, S. S. Sun1,55, T. Sun1,55, W. Y. Sun35, X Sun20,l, Y. J. Sun63,50,
Y. K. Sun63,50, Y. Z. Sun1, Z. T. Sun1, Y. H. Tan68, Y. X. Tan63,50, C. J.
Tang46, G. Y. Tang1, J. Tang51, J. X. Teng63,50, V. Thoren67, I. Uman54b, C.
W. Wang36, D. Y. Wang39,k, H. P. Wang1,55, K. Wang1,50, L. L. Wang1, M.
Wang42, M. Z. Wang39,k, Meng Wang1,55, W. H. Wang68, W. P. Wang63,50, X.
Wang39,k, X. F. Wang32, X. L. Wang9,h, Y. Wang51, Y. Wang63,50, Y. D. Wang38,
Y. F. Wang1,50,55, Y. Q. Wang1, Z. Wang1,50, Z. Y. Wang1, Ziyi Wang55,
Zongyuan Wang1,55, D. H. Wei12, P. Weidenkaff28, F. Weidner60, S. P. Wen1, D.
J. White58, U. Wiedner4, G. Wilkinson61, M. Wolke67, L. Wollenberg4, J. F.
Wu1,55, L. H. Wu1, L. J. Wu1,55, X. Wu9,h, Z. Wu1,50, L. Xia63,50, H. Xiao9,h,
S. Y. Xiao1, Y. J. Xiao1,55, Z. J. Xiao35, X. H. Xie39,k, Y. G. Xie1,50, Y. H.
Xie6, T. Y. Xing1,55, G. F. Xu1, J. J. Xu36, Q. J. Xu14, W. Xu1,55, X. P.
Xu47, F. Yan9,h, L. Yan66a,66c, L. Yan9,h, W. B. Yan63,50, W. C. Yan71, Xu
Yan47, H. J. Yang43,g, H. X. Yang1, L. Yang44, R. X. Yang63,50, S. L. Yang55,
S. L. Yang1,55, Y. H. Yang36, Y. X. Yang12, Yifan Yang1,55, Zhi Yang25, M.
Ye1,50, M. H. Ye7, J. H. Yin1, Z. Y. You51, B. X. Yu1,50,55, C. X. Yu37, G.
Yu1,55, J. S. Yu20,l, T. Yu64, C. Z. Yuan1,55, L. Yuan2, W. Yuan66a,66c, X. Q.
Yuan39,k, Y. Yuan1, Z. Y. Yuan51, C. X. Yue33, A. Yuncu54a,a, A. A. Zafar65,
Y. Zeng20,l, B. X. Zhang1, Guangyi Zhang16, H. Zhang63, H. H. Zhang51, H. Y.
Zhang1,50, J. J. Zhang44, J. L. Zhang69, J. Q. Zhang4, J. Q. Zhang35, J. W.
Zhang1,50,55, J. Y. Zhang1, J. Z. Zhang1,55, Jianyu Zhang1,55, Jiawei
Zhang1,55, Lei Zhang36, S. Zhang51, S. F. Zhang36, Shulei Zhang20,l, X. D.
Zhang38, X. Y. Zhang42, Y. Zhang61, Y. H. Zhang1,50, Y. T. Zhang63,50, Yan
Zhang63,50, Yao Zhang1, Yi Zhang9,h, Z. H. Zhang6, Z. Y. Zhang68, G. Zhao1, J.
Zhao33, J. Y. Zhao1,55, J. Z. Zhao1,50, Lei Zhao63,50, Ling Zhao1, M. G.
Zhao37, Q. Zhao1, S. J. Zhao71, Y. B. Zhao1,50, Y. X. Zhao25, Z. G. Zhao63,50,
A. Zhemchugov29,b, B. Zheng64, J. P. Zheng1,50, Y. Zheng39,k, Y. H. Zheng55,
B. Zhong35, C. Zhong64, L. P. Zhou1,55, Q. Zhou1,55, X. Zhou68, X. K. Zhou55,
X. R. Zhou63,50, A. N. Zhu1,55, J. Zhu37, K. Zhu1, K. J. Zhu1,50,55, S. H.
Zhu62, T. J. Zhu69, W. J. Zhu37, X. L. Zhu53, Y. C. Zhu63,50, Z. A. Zhu1,55,
B. S. Zou1, J. H. Zou1
(BESIII Collaboration)
1Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2Beihang University, Beijing 100191, People’s Republic of China
3Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4Bochum Ruhr-University, D-44780 Bochum, Germany
5Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6Central China Normal University, Wuhan 430079, People’s Republic of China
7China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9Fudan University, Shanghai 200443, People’s Republic of China
10G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12Guangxi Normal University, Guilin 541004, People’s Republic of China
13Guangxi University, Nanning 530004, People’s Republic of China
14Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16Henan Normal University, Xinxiang 453007, People’s Republic of China
17Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18Huangshan College, Huangshan 245000, People’s Republic of China
19Hunan Normal University, Changsha 410081, People’s Republic of China
20Hunan University, Changsha 410082, People’s Republic of China
21Indian Institute of Technology Madras, Chennai 600036, India
22Indiana University, Bloomington, Indiana 47405, USA
23aINFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
23bINFN Sezione di Perugia, I-06100 Perugia, Italy
23cUniversity of Perugia, I-06100, Perugia, Italy
24aNFN Sezione di Ferrara, I-44122 Ferrara, Italy
24bUniversity of Ferrara, I-44122, Ferrara, Italy
25Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27Jilin University, Changchun 130012, People’s Republic of China
28Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
32Lanzhou University, Lanzhou 730000, People’s Republic of China
33Liaoning Normal University, Dalian 116029, People’s Republic of China
34Liaoning University, Shenyang 110036, People’s Republic of China
35Nanjing Normal University, Nanjing 210023, People’s Republic of China
36Nanjing University, Nanjing 210093, People’s Republic of China
37Nankai University, Tianjin 300071, People’s Republic of China
38North China Electric Power University, Beijing 102206, People’s Republic of
China
39Peking University, Beijing 100871, People’s Republic of China
40Qufu Normal University, Qufu 273165, People’s Republic of China
41Shandong Normal University, Jinan 250014, People’s Republic of China
42Shandong University, Jinan 250100, People’s Republic of China
43Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
44Shanxi Normal University, Linfen 041004, People’s Republic of China
45Shanxi University, Taiyuan 030006, People’s Republic of China
46Sichuan University, Chengdu 610064, People’s Republic of China
47Soochow University, Suzhou 215006, People’s Republic of China
48South China Normal University, Guangzhou 510006, People’s Republic of China
49Southeast University, Nanjing 211100, People’s Republic of China
50State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
51Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
52Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima
30000, Thailand
53Tsinghua University, Beijing 100084, People’s Republic of China
54aTurkish Accelerator Center Particle Factory Group, Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey
54bNear East University, Nicosia, North Cyprus, Mersin 10, Turkey
55University of Chinese Academy of Sciences, Beijing 100049, People’s Republic
of China
56University of Hawaii, Honolulu, Hawaii 96822, USA
57University of Jinan, Jinan 250022, People’s Republic of China
58University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
59University of Minnesota, Minneapolis, Minnesota 55455, USA
60University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
61University of Oxford, Keble Rd, Oxford, UK OX13RH
62University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
63University of Science and Technology of China, Hefei 230026, People’s
Republic of China
64University of South China, Hengyang 421001, People’s Republic of China
65University of the Punjab, Lahore-54590, Pakistan
66aUniversity of Turin, I-10125, Turin, Italy
66bUniversity of Eastern Piedmont, I-15121, Alessandria, Italy
66cINFN, I-10125, Turin, Italy
67Uppsala University, Box 516, SE-75120 Uppsala, Sweden
68Wuhan University, Wuhan 430072, People’s Republic of China
69Xinyang Normal University, Xinyang 464000, People’s Republic of China
70Zhejiang University, Hangzhou 310027, People’s Republic of China
71Zhengzhou University, Zhengzhou 450001, People’s Republic of China
aAlso at Bogazici University, 34342 Istanbul, Turkey
bAlso at the Moscow Institute of Physics and Technology, Moscow 141700, Russia
cAlso at the Novosibirsk State University, Novosibirsk, 630090, Russia
dAlso at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
eAlso at Istanbul Arel University, 34295 Istanbul, Turkey
fAlso at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
gAlso 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
hAlso 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
iAlso at Harvard University, Department of Physics, Cambridge, MA, 02138, USA
jCurrently at: Institute of Physics and Technology, Peace Ave.54B, Ulaanbaatar
13330, Mongolia
kAlso at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
lSchool of Physics and Electronics, Hunan University, Changsha 410082, China
mAlso at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China

(August 27, 2024)

###### Abstract

The Born cross sections for the process
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ at different center-of-mass
energies between $2.00$ and $3.08$ GeV are reported with improved precision
from an analysis of data samples collected with the BESIII detector operating
at the BEPCII storage ring. An obvious structure is observed in the Born cross
section line shape. Fit as a Breit-Wigner resonance, it has a statistical
significance of $6.3\sigma$ and a mass and width of $M=(2111\pm 43\pm 25)$
MeV/$c^{2}$ and $\Gamma=(135\pm 34\pm 30)$ MeV, where the uncertainties are
statistical and systematic, respectively. These measured resonance parameters
agree with the measurements of BABAR in
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ and BESIII in
$e^{+}e^{-}\to\omega\pi^{0}$ within two standard deviations.

##  I Introduction

Low energy $e^{+}e^{-}$ collision experiments, where $\rho$, $\omega$ and
$\phi$ resonances as well as their excited states are produced copiously,
offer an ideal test-bed to thoroughly investigate the properties of these
resonances. Many experimental results regarding these states have been
summarized in the Particle Data Group (PDG) review PDG . Still, the properties
of some states are still ambiguous. Notably, the status of the $\rho(2000)$,
$\rho(2150)$, and $\rho(2270)$ states is unclear, due to insufficient
experimental information. The $\rho(2000)$ was found in $p\bar{p}$ collisions
Hasan1994 ; Anisovich491 ; Anisovich508 ; Anisovich513 ; Anisovich2002 , and
it was explained as a radial excitation of the $\rho(1700)$ Anisovich2002  or
a mixed state with a significant ${}^{3}D_{1}$ component Bugg2004 . The
$\rho(2150)$ was initially regarded as a $2^{3}D_{1}$ state Godfrey1985 , but
later was considered to be a $4^{3}S_{1}$ state Bugg2013 ; Anisovich051502 ;
Masjuan2013 ; He2013 ; arXiv:2102.05356 . The $\rho(2270)$ was first observed
in photo-production Atkinson1985  and categorized as a $3^{3}D_{1}$ state
He2013 . There are no published results on $\rho(2000)$ and $\rho(2270)$ from
$e^{+}e^{-}$ collision experiments. The $\rho(2150)$ has been widely studied
in $e^{+}e^{-}$, $p\bar{p}$, $s$-channel $N\bar{N}$ and $\pi p$ collision
experiments, but inconsistencies in the measured masses and widths make the
$\rho(2150)$ more controversial.

According to the vector meson dominance model VMD , the isovector part of the
electromagnetic current in the positive G-parity process
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ allows direct production of
$\rho-$like states. Therefore this process can be used to extract the
resonance parameters of the $\rho-$like states. The BABAR Collaboration has
measured the Born cross section line shape of
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ at center-of-mass (c.m.) energies
($\sqrt{s}$) from 1.58 to 3.42 GeV with the initial state radiation (ISR)
technique. A resonance-like structure around 2.1 GeV/$c^{2}$ reported by BABAR
could be interpreted as the $\rho(2150)$ BarBarCS .

By using 19 data sets taken at $\sqrt{s}$ between $2.00$ and $3.08$ GeV, the
BESIII Collaboration recently reported a $J^{\rm PC}=1^{--}$ vector state,
$Y(2040)$, in $e^{+}e^{-}\to\omega\pi^{0}$ Dong2020  with a mass and width of
($2034\pm 13\pm 9$) MeV/$c^{2}$ and ($234\pm 30\pm 25$) MeV, respectively.
Here, we report the Born cross sections for
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ based on the same data sets. By
fitting to these cross sections, we measure the parameters of the possible
$\rho-$like resonances.

##  II BESIII AND MONTE CARLO SIMULATION

BESIII is a general-purpose detector located at the Beijing Electron Positron
Collider (BEPCII) BEPCII  and is designed for studies of hadron spectroscopy
and $\tau$-charm physics besphysics ; besphysicsv . The cylindrical detector
has a geometrical acceptance of 93% of $4\pi$ solid angle and consists of four
main components:

(i) A small cell, helium-based main drift chamber (MDC) with 43 layers
immersed in the 1.0 T magnetic field of a super conducting solenoid. The
average single-hit resolution is 135 $\upmu$m, and the charged particle
momentum resolution is 0.5% at 1.0 GeV/$c$.

(ii) A time-of-flight system (TOF) made from two layers of plastic
scintillator, with 88 counters 5 cm thick and 2.4 m long in each layer for the
barrel, and 96 fan-shaped counters in each end cap. It provides timing
information with a resolution of 68 ps in the barrel and 110 ps in the end
caps, which yields 2$\sigma$ $K/\pi$ separation at 1.0 GeV/$c$.

(iii) An electromagnetic calorimeter (EMC) consisting of 6240 CsI (Tl)
crystals in a cylindrical barrel and two end caps to measure shower energies.
The photon energy resolution at 1.0 GeV is 2.5% in the barrel and 5% in the
end caps, while the position resolution is 6 mm and 9 mm for barrel and end
caps, respectively.

(iv) A resistive plate chamber (RPC)-based muon chamber (MUC) with nine layers
in the barrel and eight layers in the end caps providing 2 cm position
resolution.

Monte Carlo (MC) simulations of the full detector, based on geant4 Geant4
simulation software, and the BESIII Object Oriented Simulation Tool (BOOST)
Boost , are used to optimize the event selection criteria, understand
potential backgrounds, and determine the detection efficiency. The BOOST
package contains the detector geometry and material description, the detector
response and signal digitization models, as well as records of the detector
running conditions and performance. Large inclusive MC samples at
$\sqrt{s}=2.1250$ and 2.3960 GeV are generated to estimate potential
backgrounds. The processes $e^{+}e^{-}\to e^{+}e^{-},\mu^{+}\mu^{-}$ and
$\gamma\gamma$ are generated with the Babayaga generator Babayaga , while
$e^{+}e^{-}\to$ hadrons and two photon processes are simulated by the luarlw
LUARLW  and bestwogam BESTWOGAM  generators, respectively. Due to the
dominance of $\rho\eta^{\prime}$ in $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$
process BarBarCS , the signal MC is generated with conexc ConExc  as
$e^{+}e^{-}\to\rho\eta^{\prime}$. The cross sections measured by the BABAR
experiment BarBarCS  are used as initial input. The ISR, vacuum polarization
and the angular distributions of the final state have been taken into account
in the generator. The wide $\rho$ resonance in the intermediate state is
described by the Gounaris-Sakurai (GS) model GSi . The
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$ decay is simulated with a model based
on the results of the amplitude analysis in Ref. Gammapipi , and the
$\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ decay is produced using a phase-space
(PHSP) model.

##  III  Data Analysis

In this analysis, the $\eta^{\prime}$ is reconstructed via the two decay modes
$\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ and
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$, which will be referred to as mode I
and mode II, respectively. For mode I, the $\eta$ is reconstructed in the
decay $\eta\to\gamma\gamma$.

Charged tracks are reconstructed using hits in the MDC. Each track is required
to be within the polar angle ($\theta$) region $|\cos\theta|<0.93$ and have a
distance of closest approach to the interaction point within $\pm$10 cm along
the beam direction (z-axis) and within 1 cm in the transverse plane. For both
modes, it is required that there are exactly four charged tracks with net zero
charge. Particle identification (PID) for charged tracks combines the
measurements of the specific ionization energy, $dE/dx$, in the MDC and the
flight time in the TOF. The charged track is identified as a pion if the
confidence level for the pion hypothesis is greater than those for both the
kaon and proton hypotheses.

Showers in the EMC are chosen as photon candidates if they satisfy the
following requirements: the deposited energy must be larger than 25 MeV in the
barrel region ($|\cos\theta|<0.80$) and 50 MeV in the end cap
($0.86<|\cos\theta|<0.92$). To suppress electronic noise and showers unrelated
to the event, the EMC time deviation from the event start time is required to
be within (0,700) ns. For mode I (mode II), it is required that there are at
least two (one) photons.

A vertex fit is imposed on the selected charged tracks to ensure that they
originate from the same interaction point. To improve momentum resolution and
to suppress background, a four-constraint (4C) kinematic fit imposing energy-
momentum conservation under the hypotheses of
$e^{+}e^{-}\to\gamma\gamma\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and
$e^{+}e^{-}\to\gamma\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ is employed on the selected
candidates for mode I and mode II, respectively. For events with additional
photon(s), the combination with the smallest $\chi^{2}_{\rm{4C}}$ is retained.
Based on an optimization of $s/\sqrt{s^{\prime}+b}$ for the requirement on
$\chi^{2}_{\rm{4C}}$, where $s$ and $s^{\prime}+b$ is the number of events in
the $\eta^{\prime}$ signal region
(($M(\eta\pi^{+}\pi^{-},\gamma\pi^{+}\pi^{-})\in(0.943,0.973)$ GeV/$c^{2}$))
in signal MC and data, respectively, candidate events with
$\chi^{2}_{\rm{4C}}<100$ ($\chi^{2}_{\rm{4C}}<50$) for mode I (mode II) are
accepted for further analysis. For mode II, to suppress contaminations from
$e^{+}e^{-}\to 2(\pi^{+}\pi^{-})$ and $e^{+}e^{-}\to 2(\gamma\pi^{+}\pi^{-})$,
two additional 4C kinematic fits under each of these two hypotheses are
performed. Events are discarded if the $\chi^{2}_{\rm{4C}}$ for either of
these fits is less than the signal mode 4C kinematic fit. To further suppress
background from $e^{+}e^{-}\to 2(\pi^{+}\pi^{-})$, all photon energies are
required to be greater than 0.1 GeV for mode II. For convenience, we take the
data set at $\sqrt{s}=$2.1250 GeV, which has the the largest statistics, as an
example in this section. Figure 1 shows the $M(\gamma\gamma)$ distribution of
the accepted candidates for $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ using
mode I. The events in the $\eta$ mass signal region,
$M(\gamma\gamma)\in(0.523,0.573)$ GeV/$c^{2}$ (the region between two solid
blue arrows), are kept for further analysis.

Figure 2 presents the distributions of $M(\eta\pi^{+}\pi^{-})$ for mode I and
$M(\gamma\pi^{+}\pi^{-})$ for mode II of the accepted candidates. Clear
$\eta^{\prime}$ signals are observed. For mode I, the non-$\eta$ backgrounds
in the $\eta$ mass signal region are examined by the events in the $\eta$ mass
sideband region, which is defined as
$M(\gamma\gamma)\in(0.488,0.513)\cup(0.583,0.608)$ GeV/$c^{2}$ (the region
between two neighboring dashed green arrows in Figure. 1). The resulting
$M(\eta\pi^{+}\pi^{-})$ distribution of the $\eta$ sideband events is shown as
the green shaded histogram in Figure. 2 (a). Further studies based on the
inclusive MC samples show that the dominant backgrounds are $e^{+}e^{-}\to\eta
2(\pi^{+}\pi^{-})|_{\rm non{\text{-}}\eta^{\prime}}$ for mode I and
$e^{+}e^{-}\to 2(\pi^{+}\pi^{-})$ for mode II. The resulting
$M(\eta\pi^{+}\pi^{-},\gamma\pi^{+}\pi^{-})$ distribution of the accepted
background events from the inclusive MC samples for mode I and mode II are
shown as the magenta histograms of Figure. 2 (a) and (b), respectively. No
peaking backgrounds are seen near the $\eta^{\prime}$ mass. Therefore, we will
fit the $M(\eta\pi^{+}\pi^{-})$ ($M(\gamma\pi^{+}\pi^{-})$) spectra using a
smooth background shape to account for the remaining backgrounds in the
analysis.

\begin{overpic}[width=213.39566pt]{Figure1_Meta0_draft_new.eps} \end{overpic}
Figure 1:  The $M(\gamma\gamma)$ spectrum of the
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ candidate events for mode I in data
(dots with error bars) and signal MC (histogram) at $\sqrt{s}=$2.1250 GeV. The
region between the two solid blue arrows is the $\eta$ signal region, while
the regions between the pairs of dashed green arrows are the $\eta$ sideband
regions.

Possible intermediate states in $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ are
examined by the corresponding Dalitz plots selecting the $\eta^{\prime}$
signal region as $M(\eta\pi^{+}\pi^{-},\gamma\pi^{+}\pi^{-})\in(0.943,0.973)$
GeV/$c^{2}$, shown in Figure 3. After subtracting the non-$\eta^{\prime}$
background in the $\eta^{\prime}$ sideband region,
$M(\eta\pi^{+}\pi^{-},\gamma\pi^{+}\pi^{-})\in(0.918,0.938)\cup(0.978,0.998)$
GeV/$c^{2}$, with a weight factor of 0.8. Figure 4 presents the projections of
the corresponding Dalitz plots on the $M^{2}(\pi^{+}\pi^{-})$ axis for two
modes. As expected, the dominant component is
$e^{+}e^{-}\to\rho\eta^{\prime}$, and the non-$\rho$ contribution is less than
10%. A fit to the $M(\pi^{+}\pi^{-})$ in the high-statistics bin at c.m.
energy of 2.125 GeV shows that the apparent shift between MC and data of the
$\rho(770)$ peak can be explained by interference between $\rho(770)$ and
non-$\rho(770)$ process.

\begin{overpic}[width=213.39566pt]{Figure2a_Fit_etapipi_update.eps}
\par\put(25.0,55.0){(a)}
\end{overpic}\begin{overpic}[width=213.39566pt]{Figure2b_Fit_gmapipi_update.eps}
\par\put(25.0,55.0){(b)} \end{overpic}

Figure 2: Fits to (a) $M(\eta\pi^{+}\pi^{-})$ and (b)
$M(\gamma\pi^{+}\pi^{-})$ for the accepted candidate events in data at
$\sqrt{s}=$2.1250 GeV. Dots with error bars show the data, black lines give
the total fit results, blue dotted lines are the signal components, red lines
are the smooth backgrounds, and magenta histograms are the inclusive MC
samples. The inclusive MC samples are normalized to the data luminosity. The
green shaded histogram in (a) shows the events from the $\eta$ sidebands in
data.

\begin{overpic}[width=213.39566pt]{Figure3a_Dalitz_etapipi_update.eps}
\put(70.0,57.0){(a)}
\end{overpic}\begin{overpic}[width=213.39566pt]{Figure3b_Dalitz_gammapipi_update.eps}
\put(70.0,57.0){(b)} \end{overpic}

Figure 3: Dalitz plots of the accepted
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ candidate events for (a) mode I and
(b) mode II in data at $\sqrt{s}=$2.1250 GeV. The requirement
$M(\eta\pi^{+}\pi^{-},\gamma\pi^{+}\pi^{-})\in(0.943,0.973)$ GeV/$c^{2}$ has
been imposed.

\begin{overpic}[width=213.39566pt]{Figure4a_Mpipi_etapipi.eps}
\put(25.0,55.0){(a)}
\end{overpic}\begin{overpic}[width=213.39566pt]{Figure4b_Mpipi_gammapipi.eps}
\put(25.0,55.0){(b)} \end{overpic}

Figure 4: Projections of the corresponding Dalitz plots on
$M^{2}(\pi^{+}\pi^{-})$ axis for (a) mode I and (b) mode II in data (dots with
error bars) and signal MC (histograms) at $\sqrt{s}=$2.1250 GeV.

##  IV Born Cross section Measurement

The Born cross section, $\sigma^{\rm B}$, at each c.m. energy is determined as

$\sigma^{\rm B}=\frac{N^{\rm
obs}}{\mathcal{L}\cdot\epsilon\cdot{\mathcal{B}}\cdot(1+\delta^{\mathit{r}})\cdot\frac{1}{|1-\Pi|^{2}}},$
(1)

where $N^{\rm{obs}}$ is the signal yield, $\mathcal{L}$ is the integrated
luminosity of the data set, $\epsilon$ is the detection efficiency,
${\mathcal{B}}$ is the product of the relevant daughter branching fractions,
i.e.,
${\mathcal{B}}={\mathcal{B}}(\eta^{\prime}\to\eta\pi^{+}\pi^{-})\cdot{\mathcal{B}}(\eta\to\gamma\gamma)=16.8\%$
for mode I and
${\mathcal{B}}={\mathcal{B}}(\eta^{\prime}\to\gamma\pi^{+}\pi^{-})=28.9\%$ for
mode II PDG . The factor $(1+\delta^{\mathit{r}})$ is the ISR correction
factor and $\frac{1}{|1-\Pi|^{2}}$ is the vacuum polarization factor. Both
$(1+\delta^{\mathit{r}})$ and $\frac{1}{|1-\Pi|^{2}}$ are obtained from MC
simulations ISR ; VP . To obtain reliable detection efficiencies and ISR
factors, the Born cross sections used as input in the generator have been
iterated until the product $(1+\delta^{\mathit{r}})\cdot\epsilon$ has
converged, defined as an iteration leading to a relative change of less than
1.0%.

The signal yields are obtained from a simultaneous unbinned maximum-likelihood
fit to the $M(\eta\pi^{+}\pi^{-})$ and $M(\gamma\pi^{+}\pi^{-})$ spectra at
each c.m. energy. The signal is described by a MC-simulated shape convolved
with a Gaussian function. The parameters of the Gaussian function are free.
Among the different data sets, a common Gaussian convolution is used to
compensate for potential differences in calibration and resolution between
data and MC simulation. A second-order Chebychev polynomial is used to
describe the combinatorial background shape. In the fit, the two modes share
the same Born cross section, and the expected signal yields are
$N^{obs}=\sigma^{B}\cdot\mathcal{L}\cdot\epsilon\cdot{\mathcal{B}}\cdot(1+\delta^{\mathit{r}})\cdot\frac{1}{|1-\Pi|^{2}}$.
Figure 2 shows the fit result for data taken at $\sqrt{s}=2.1250$ GeV. Similar
combined fits to the two final states are performed for each c.m. energy; the
resulting cross sections and related variables are listed in Table 1. They are
consistent with those from the BABAR experiment BarBarCS , as seen in Figure.
5. Independent fits to the $M(\eta\pi^{+}\pi^{-})$ and
$M(\gamma\pi^{+}\pi^{-})$ spectra are also performed and the individual signal
yields are also summarized in Table 1.

.

Table 1: Summary of the integrated luminosities ($\mathcal{L}$)
LuminosityFinal , observed event yields from independent fit ($N^{obs}$),
detection efficiencies ($\epsilon$), radiative correction factors
$(1+\delta^{\gamma})$, vacuum polarization factors $\frac{1}{|1-\Pi|^{2}}$,
and the obtained Born cross section ($\sigma^{\rm B}$) at different c.m.
energies ($\sqrt{s}$). The first uncertainties for $\sigma^{\rm B}$ are
statistical and the second are systematic; those for $N^{\rm
obs\leavevmode\nobreak\ I}$ and $N^{\rm obs\leavevmode\nobreak\ II}$ are
statistical only. The symbols of I and II represent the
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ processes reconstructed via
$\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ and
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$, respectively.

$\sqrt{s}$ (GeV) |  $\mathcal{L}$ (pb-1) | $N^{\rm obs\leavevmode\nobreak\ I}$ | $N^{\rm obs\leavevmode\nobreak\ II}$ | $\epsilon^{\rm I}$ | $\epsilon^{\rm II}$ | $1+\delta^{\mathit{r}}$ | $\frac{1}{|1-\Pi|^{2}}$ |  $\sigma^{\rm B}$ (pb)
---|---|---|---|---|---|---|---|---
2.0000 | 10.1 | $35.4\pm 7.5$ | $119.2\pm 16.8$ | 0.157 | 0.265 | 0.983 | 1.037 | $144.0\pm 17.0\pm 9.7$
2.0500 | 3.34 | $23.9\pm 5.5$ | $55.6\pm 10.6$ | 0.167 | 0.275 | 0.941 | 1.038 | $229.3\pm 33.2\pm 15.7$
2.1000 | 12.2 | $58.6\pm 9.1$ | $207.9\pm 19.6$ | 0.166 | 0.269 | 0.979 | 1.039 | $200.1\pm 16.3\pm 14.1$
2.1250 | 108. | $555.4\pm 27.4$ | $1684.6\pm 58.2$ | 0.160 | 0.258 | 1.016 | 1.039 | $191.0\pm 5.4\pm 12.7$
2.1500 | 2.84 | $8.7\pm 3.7$ | $53.0\pm 9.3$ | 0.154 | 0.252 | 1.045 | 1.040 | $184.8\pm 31.9\pm 12.5$
2.1750 | 10.6 | $43.7\pm 7.7$ | $116.4\pm 15.7$ | 0.159 | 0.251 | 1.059 | 1.040 | $137.6\pm 14.7\pm 9.1$
2.2000 | 13.7 | $36.0\pm 7.6$ | $130.1\pm 16.6$ | 0.155 | 0.247 | 1.075 | 1.040 | $108.8\pm 12.0\pm 9.3$
2.2324 | 11.9 | $35.1\pm 7.4$ | $127.0\pm 15.8$ | 0.156 | 0.244 | 1.089 | 1.041 | $122.0\pm 13.2\pm 9.9$
2.3094 | 21.1 | $45.8\pm 8.2$ | $149.1\pm 17.9$ | 0.153 | 0.237 | 1.109 | 1.041 | $83.3\pm 8.4\pm 5.6$
2.3864 | 22.5 | $57.5\pm 8.8$ | $158.1\pm 17.4$ | 0.152 | 0.232 | 1.120 | 1.041 | $87.9\pm 7.9\pm 6.3$
2.3960 | 66.9 | $167.9\pm 15.2$ | $496.1\pm 31.7$ | 0.156 | 0.235 | 1.119 | 1.041 | $89.3\pm 4.7\pm 5.9$
2.6444 | 33.7 | $46.8\pm 7.9$ | $164.1\pm 16.7$ | 0.160 | 0.223 | 1.154 | 1.039 | $55.4\pm 4.9\pm 3.6$
2.6464 | 34.0 | $51.8\pm 8.3$ | $148.2\pm 17.0$ | 0.160 | 0.222 | 1.157 | 1.039 | $52.7\pm 5.0\pm 3.8$
2.9000 | 105. | $137.4\pm 13.0$ | $288.7\pm 22.4$ | 0.163 | 0.211 | 1.192 | 1.033 | $37.3\pm 2.2\pm 2.8$
2.9500 | 15.9 | $19.6\pm 4.9$ | $43.2\pm 8.3$ | 0.161 | 0.203 | 1.197 | 1.029 | $37.4\pm 5.7\pm 2.9$
2.9810 | 16.1 | $13.5\pm 4.3$ | $35.5\pm 8.3$ | 0.163 | 0.206 | 1.199 | 1.025 | $28.1\pm 5.2\pm 2.0$
3.0000 | 15.9 | $16.1\pm 4.5$ | $33.8\pm 7.8$ | 0.163 | 0.204 | 1.197 | 1.021 | $29.7\pm 5.3\pm 2.0$
3.0200 | 17.3 | $13.6\pm 4.8$ | $33.9\pm 7.6$ | 0.163 | 0.204 | 1.199 | 1.014 | $26.1\pm 5.0\pm 2.3$
3.0800 | 126. | $88.0\pm 10.2$ | $218.3\pm 20.1$ | 0.157 | 0.191 | 1.135 | 0.915 | $28.0\pm 2.0\pm 1.9$

By implementing the same strategy described in prd0320012019 ; Yankun ; Yateng
; Yankun2020 , several sources of systematic uncertainty on the measured cross
section are considered. These uncertainties for all datasets are summarized in
Table 2.

The uncertainty in the measurement of the integrated luminosity of the data
set has been determined to be $1.0\%$ LuminosityFinal . The uncertainties of
the track reconstruction and PID efficiencies of charged pions have been
studied by using a control sample of $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$
prd0320012019 , resulting in $1.0$% per pion for tracking and $1.0$% per pion
for PID. The uncertainty on the photon efficiency is estimated as $1.0$% per
photon by using a control sample of $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$
photonerror . Since the number of photon for mode I and mode II are two and
one respectively, the larger change in the cross section due to shifting the
detection efficiency by $\pm 2.0$% for mode I and $\pm 1.0$% for mode II in
the simultaneous fit (with both shifts having the same sign), is taken as the
systematic uncertainty.

To estimate the uncertainty of the MC modeling, we examine the efficiency at
2.1250 GeV, which has the largest statistics among all the c.m. energies. The
signal MC samples are weighted according to the the Dalitz distribution of
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ in data after background
subtraction. The difference between the weighted efficiency and the nominal
efficiency, 1.3%, is taken as the systematic uncertainty. Due to the limited
statistics at other c.m. energies, this systematic uncertainty is taken to be
the same as that for 2.1250 GeV at all energies.

The track helix parameters for the MC simulation are corrected before the 4C
kinematic fit to account for observed data-MC differences Aixiaocong . The
differences in detection efficiencies with and without corrections,
($0.8-1.9$)%,are assigned as the systematic uncertainty from the 4C kinematic
fit.

The uncertainty originating from the ISR correction factor is taken as the
relative difference of the values of $(1+\delta^{\mathit{r}})\cdot\epsilon$
between the last two iterations of the cross section measurement.

Three uncertainties associated with the fits to the mass spectra are examined.
The background shape is replaced by a third-order Chebychev polynomial
function. The signal shape is replaced with a Gaussian function with fixed
resolution, obtained by shifting the resolution from the nominal fit by one
standard deviation in each direction. Finally, the fit range is varied by $\pm
5$ MeV. The quadrature sum of the changes in the fitted yields is taken as the
uncertainty.

The impact of uncertainties on the branching fractions of the intermediate
states is examined by changing the branching fractions of $\eta$ and
$\eta^{\prime}$ by $\pm 1\sigma$ in the simultaneous fit, where $\sigma$ is
the uncertainty of the individual branching fractions. The difference on the
cross section, $1.8$%, is taken as the systematic uncertainty.

Adding the systematic uncertainties in quadrature yields the total systematic
uncertainties of the measured Born cross sections, which are summarized in
Table 2.

Table 2: Systematic uncertainties (in %) from luminosity ($\mathcal{L}$), photon reconstruction (photon), tracking, PID, MC modeling (MC), kinematic fit (KF), radiation correction (RC), fitting and quoted branching fraction in the cross section measurements. The sources with ∗ superscript are common systematic uncertainties for different c.m. energies. $\sqrt{s}$ (GeV) | $\mathcal{L}^{*}$ | photon∗ | tracking∗ | PID∗ | MC∗ | KF | RC | Fitting | ${\mathcal{B}}^{*}$ | Sum
---|---|---|---|---|---|---|---|---|---|---
2.0000 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.6 | 0.5 | 1.7 | 1.8 | 6.7
2.0500 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.5 | 0.1 | 2.3 | 1.8 | 6.9
2.1000 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.9 | 0.5 | 2.5 | 1.8 | 7.1
2.1250 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.8 | 0.4 | 1.2 | 1.8 | 6.7
2.1500 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.7 | 0.4 | 1.7 | 1.8 | 6.8
2.1750 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.7 | 0.1 | 1.3 | 1.8 | 6.7
2.2000 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.8 | 0.8 | 5.4 | 1.8 | 8.5
2.2324 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.7 | 0.1 | 4.9 | 1.8 | 8.2
2.3094 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.7 | 0.2 | 1.8 | 1.8 | 6.8
2.3864 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.5 | 0.6 | 3.2 | 1.8 | 7.2
2.3960 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.4 | 0.3 | 1.3 | 1.8 | 6.6
2.6444 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.4 | 0.5 | 0.7 | 1.8 | 6.5
2.6464 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.3 | 0.4 | 3.3 | 1.8 | 7.2
2.9000 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.3 | 0.1 | 3.9 | 1.8 | 7.5
2.9500 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 0.8 | 0.1 | 4.2 | 1.8 | 7.6
2.9810 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.9 | 0.1 | 3.0 | 1.8 | 7.2
3.0000 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.3 | 0.1 | 1.9 | 1.8 | 6.7
3.0200 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.2 | 0.4 | 5.8 | 1.8 | 8.7
3.0800 | 1.0 | 1.3 | 4.0 | 4.0 | 1.3 | 1.4 | 0.7 | 2.3 | 1.8 | 6.9

##  V FIT TO THE BORN CROSS SECTION

The obtained Born cross sections are shown in Figure. 5 in which a clear
structure around 2.05 GeV is observed. To determine the mass and width of the
possible resonance, a $\chi^{2}$ fit is performed to these cross sections. The
cross section is parameterized as the coherent sum of a resonant amplitude
described by a Breit-Wigner function and an $s$-dependent continuum amplitude
s_depend :

$\sigma(s)=\left|\frac{C_{0}}{s^{n}}\sqrt{\Phi(\sqrt{s})}+C_{1}\cdot
BW(\sqrt{s})\times e^{i\phi}\right|^{2},$ (2)

where $C_{0}$ and $n$ are the continuum parameters, $C_{1}=3.894\times
10^{5}\;\textrm{nb}\cdot\text{GeV}^{2}$ is a unit conversion factor, and
$\phi$ is the phase angle between the amplitudes. The relativistic Breit-
Wigner amplitude is given by

$BW(\sqrt{s})=\frac{\sqrt{12\pi\Gamma^{ee}_{R}\mathcal{B}_{R}\Gamma^{\rm
tot}_{R}}}{s-M_{R}^{2}+iM_{R}\Gamma^{\rm
tot}_{R}}\sqrt{\frac{\Phi(\sqrt{s})}{\Phi(M_{R})}},$ (3)

where $M_{R}$, $\Gamma_{R}^{ee}$ and $\Gamma^{\rm tot}_{R}$ are the mass,
partial width to $e^{+}e^{-}$ and total width of the assumed resonance $R$.
$\mathcal{B}_{R}$ is the branching fraction for
$R\to\pi^{+}\pi^{-}\eta^{\prime}$, and $\Phi(\sqrt{s})$ is the two-body PHSP
factor of $R\to\rho\eta^{\prime}$ PDG .

In the fit, the correlated and uncorrelated uncertainties are incorporated and
the $\chi^{2}$ is constructed as

$\chi^{2}=\Delta X^{T}\mathcal{M}^{-1}\Delta X,$ (4)

where $\Delta X$ is the difference between the measured and predicted cross
sections; the uncertainty of the measured value includes the uncorrelated
statistical and systematic components. $\mathcal{M}$ is the covariance matrix;
its diagonal elements represent the total uncertainty and off-diagonal
elements are correlated systematic uncertainties.

The systematic uncertainties marked with ∗ in Table 2 are treated as fully
correlated uncertainties, while the other systematic uncertainties are
considered independent for the various c.m. energies. $\mathcal{M}$ is defined
as

$\mathcal{M}_{i,j}=\sum_{k}x_{i}\;\epsilon_{i,j,k}\;x_{j}\;\epsilon_{j,i,k},$
(5)

where $x_{i}$ is the measured value at c.m. energy $i$,
$\epsilon_{i,j,k}=\epsilon_{j,i,k}$ is the common relative systematic
uncertainty of $x_{i}$ and $x_{j}$ from correlated source $k$.

Figure 5 shows the result of the fit to the Born cross sections. There are two
solutions with equal fit quality and very similar mass and width for the
resonance, while the product $\mathcal{B}_{R}\Gamma^{ee}_{R}$ and phases are
different in the two solutions. The goodness of fit is $\chi^{2}$/n.d.f. =
9.4/13 = 0.72, where n.d.f. is the number of degrees of freedom. The fit
parameters are summarized in Table 3. Since the mass, width, $n(n^{\prime})$
and $C_{0}(C_{0}^{\prime})$ values of the two solutions are consistent within
$0.2$ standard deviation, we present the average values of the two solutions
in Table 3. The statistical significance of this resonance is estimated to be
$6.3\sigma$, by comparing the change of $\chi^{2}$ ($\Delta\chi^{2}=50.5$),
with and without the $R$ amplitude in the fit and taking the change of degrees
of freedom ($\Delta$n.d.f.=4) into account.

\begin{overpic}[width=213.39566pt]{Figure5a_Lineshapefit_V1_update.eps}
\put(35.0,63.0){(a)}
\end{overpic}\begin{overpic}[width=213.39566pt]{Figure5b_Lineshapefit_V2_update.eps}
\put(35.0,63.0){(b)} \end{overpic}

Figure 5: Fit to the $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ Born cross sections (only BESIII results). (a) Solution I, constructive interference. (b) Solution II, destructive interference. Red solid dots with error bars are BESIII data, and hollow dots with error bars are BABAR data. The black solid curve is the total fit result, the blue dashed line is the resonant component, the green dashed line is the continuum contribution, and the magenta dot-dashed line represents the interference between the resonance and the continuum contribution. The systematic uncertainties are included. Table 3: Results of the fit to $e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ Born cross section. The first uncertainties are statistical and the second systematic. $n^{\prime}$ and $C_{0}^{\prime}$ are parameters of the alternative parameterization of the continuum contribution. Parameter | Solution 1 | Solution 2
---|---|---
$M_{R}$ (MeV/$c^{2}$) | $2111\pm 43\pm 25$
$\Gamma_{R}^{\rm tot}$ (MeV) | $135\pm 34\pm 30$
$\mathcal{B}_{R}\Gamma^{ee}_{R}$ (eV) | $0.64\pm 0.49\pm 0.42$ | $23.3\pm 5.3\pm 3.3$
$\phi$ (rad) | $2.24\pm 0.73\pm 0.48$ | $4.46\pm 0.06\pm 0.10$
$n(n^{\prime})$ | $4.42\pm 0.22\pm 0.20\leavevmode\nobreak\ (1.66\pm 0.12\pm 0.07)$
$C_{0}(C_{0}^{\prime})$ | $921\pm 240\pm 114\leavevmode\nobreak\ (53.0\pm 13.2\pm 0.1)$

The systematic uncertainties of the resonance parameters come from the c.m.
energy calibration, the resonance model, the parameterization of the continuum
and the type of PHSP factor. The uncertainties of the measured Born cross
sections have been included in the fit.

The systematic uncertainty of the c.m. energy is found to be negligible.

To estimate the uncertainty related to the fit model, a modified Breit-Wigner
function, in which the width is energy dependent, is employed in the fit. The
width of the modified Breit-Wigner function is written as:

$\Gamma(\sqrt{s})=\Gamma_{R}\left|\frac{p}{p_{R}}\right|^{2L+1}\frac{M_{R}}{\sqrt{s}}\frac{B(p)}{B\left(p_{R}\right)},$
(6)

where $\Gamma_{R}$ is the nominal width. $p$ and $p_{R}$ are the daughter
momenta in the rest frame of $P$, when $P$ taken as $\sqrt{s}$ or $M_{R}$,
respectively. $L$ is the angular momentum of the decay specified in its
subscript. $B(p)$ is the Blatt-Weisskopf form factor Blatte . The shifts of
the mass and width, which are $14$ MeV$/c^{2}$ and $17$ MeV, respectively, are
taken as the systematic uncertainties.

The uncertainty of the parameterization of the continuum contribution is
estimated by replacing $C_{0}/s^{n}$ with an exponential function of the form
$C_{0}^{\prime}\cdot e^{-n^{\prime}(\sqrt{s}-M_{th})}$, where
$M_{th}=m_{\rho}+m_{\eta^{\prime}}$. The differences of the obtained mass and
width, which are $21$ MeV$/c^{2}$ and $24$ MeV, respectively, are assigned as
the corresponding systematic uncertainties.

To assess the uncertainty regarding the PHSP factor, we replace the two-body
PHSP factor with an alternative PHSP factor consist of 90% two-body and 10%
three-body PHSP factor. The resulting changes in the fit of $0.4$ MeV$/c^{2}$
for the mass and $4.1$ MeV for the width are taken as the systematic
uncertainties.

A quadrature sum of all contributions yields total systematic uncertainties
for the mass and width of 25 MeV$/c^{2}$ and 30 MeV, respectively.

##  VI Summary and Discussion

We present measurements of the Born cross sections for
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ using the data samples collected by
the BESIII detector at c.m. energies between 2.00 and 3.08 GeV. The measured
Born cross sections are consistent with those of BABAR but have much improved
precision. The Born cross section line shape fit has two solutions with equal
fit quality and identical mass and width of the resonance, while the product
$\mathcal{B}_{R}\Gamma^{ee}_{R}$ and phase are different in the two solutions.
The statistical significance of the observed resonant structure is
$6.3\sigma$, and its mass, width and $\mathcal{B}_{R}\Gamma^{ee}_{R}$ are
determined to be $M=2111\pm 43\pm 25$ MeV/$c^{2}$,$\Gamma=135\pm 34\pm 30$ MeV
and $\mathcal{B}_{R}\Gamma^{ee}_{R}=(0.64\pm 0.49\pm 0.42)$ eV or $(23.3\pm
5.3\pm 3.3)$ eV, respectively, where the first uncertainties are statistical
and the second systematic. The mass and width measured in this work agree with
those of the $Y(2040)$ resonance found in $e^{+}e^{-}\to\omega\pi^{0}$ by
BESIII ($M=2034\pm 13\pm 9$ MeV/$c^{2}$, $\Gamma=234\pm 30\pm 25$ MeV)
Dong2020  and with those of the $\rho(2150)$ resonance found in
$e^{+}e^{-}\to\eta^{\prime}\pi^{+}\pi^{-}$ by BABAR ($M=1990\pm 80$
MeV/$c^{2}$, $\Gamma=310\pm 140$ MeV) BarBarCS  within two standard deviation.

The $e^{+}e^{-}\to\pi^{+}\pi^{-}\eta^{\prime}(\eta)$ processes are also
studied in the Resonance Chiral Theory framework and the extended Nambu-Jona-
Lasinio model Gomez2012 ; Dai2013 ; Volkov2014 ; GammapipiDai ;
arXiv:2011.09618 . However, most of the comparisons of experimental data with
those theory predictions are performed in the energy region below 2.0 GeV.
With more resonances being included and precise experimental measurements
available, these theory models could be tested above 2 GeV in the future.

##  VII acknowledgments

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
key R & D program of China under Contracts No. 2020YFA0406400 and No.
2020YFA0406300; National Natural Science Foundation of China (NSFC) under
Contracts No. 11975118, No. 11625523, No. 11635010, No. 11735014, No.
11822506, No. 11835012, No. 11935015, No. 11935016, No. 11935018 and No.
11961141012, No. 11605196, No. 11605198, No. 11950410506, No. 12061131003, No.
11705192, and No. 12035013; Natural Science Foundation of Hunan Province under
Contract No. 2019JJ30019; The Chinese Academy of Sciences (CAS) Large-Scale
Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of
the NSFC and CAS under Contracts No. U1732263, No. U1832103, No. U1832207, No.
U2032111, No. U2032105; 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.
443159800, Collaborative Research Center CRC 1044, FOR 2359, FOR 2359, GRK
214; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of
Turkey under Contract No. DPT2006K-120470; National Science and Technology
fund; Olle Engkvist Foundation under Contract No. 200-0605; STFC (United
Kingdom); The Knut and Alice Wallenberg Foundation (Sweden) under Contract No.
2016.0157; The Royal Society, UK under Contracts Nos. DH140054, DH160214; The
Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-
FG02-05ER41374, DE-SC-0012069

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