# Measurement of $e^{+}e^{-}\to\phi\eta^{\prime}$ cross sections at center-of-
mass energies from 3.508 to 4.951 GeV and search for the decay
$\psi(3770)\to\phi\eta^{\prime}$

M. Ablikim1, M. N. Achasov5,b, P. Adlarson74, X. C. Ai80, R. Aliberti35, A.
Amoroso73A,73C, M. R. An39, Q. An70,57, Y. Bai56, O. Bakina36, I.
Balossino29A, Y. Ban46,g, V. Batozskaya1,44, K. Begzsuren32, N. Berger35, M.
Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi73A,73C, E.
Bianco73A,73C, A. Bortone73A,73C, I. Boyko36, R. A. Briere6, A. Brueggemann67,
H. Cai75, X. Cai1,57, A. Calcaterra28A, G. F. Cao1,62, N. Cao1,62, S. A.
Cetin61A, J. F. Chang1,57, T. T. Chang76, W. L. Chang1,62, G. R. Che43, G.
Chelkov36,a, C. Chen43, Chao Chen54, G. Chen1, H. S. Chen1,62, M. L.
Chen1,57,62, S. J. Chen42, S. M. Chen60, T. Chen1,62, X. R. Chen31,62, X. T.
Chen1,62, Y. B. Chen1,57, Y. Q. Chen34, Z. J. Chen25,h, W. S. Cheng73C, S. K.
Choi11A, X. Chu43, G. Cibinetto29A, S. C. Coen4, F. Cossio73C, J. J. Cui49, H.
L. Dai1,57, J. P. Dai78, A. Dbeyssi18, R. E. de Boer4, D. Dedovich36, Z. Y.
Deng1, A. Denig35, I. Denysenko36, M. Destefanis73A,73C, F. De Mori73A,73C, B.
Ding65,1, X. X. Ding46,g, Y. Ding34, Y. Ding40, J. Dong1,57, L. Y. Dong1,62,
M. Y. Dong1,57,62, X. Dong75, M. C. Du1, S. X. Du80, Z. H. Duan42, P.
Egorov36,a, Y.H. Y. Fan45, Y. L. Fan75, J. Fang1,57, S. S. Fang1,62, W. X.
Fang1, Y. Fang1, R. Farinelli29A, L. Fava73B,73C, F. Feldbauer4, G. Felici28A,
C. Q. Feng70,57, J. H. Feng58, K Fischer68, M. Fritsch4, C. Fritzsch67, C. D.
Fu1, J. L. Fu62, Y. W. Fu1, H. Gao62, Y. N. Gao46,g, Yang Gao70,57, S.
Garbolino73C, I. Garzia29A,29B, P. T. Ge75, Z. W. Ge42, C. Geng58, E. M.
Gersabeck66, A Gilman68, K. Goetzen14, L. Gong40, W. X. Gong1,57, W. Gradl35,
S. Gramigna29A,29B, M. Greco73A,73C, M. H. Gu1,57, C. Y Guan1,62, Z. L.
Guan22, A. Q. Guo31,62, L. B. Guo41, M. J. Guo49, R. P. Guo48, Y. P. Guo13,f,
A. Guskov36,a, T. T. Han49, W. Y. Han39, X. Q. Hao19, F. A. Harris64, K. K.
He54, K. L. He1,62, F. H H.. Heinsius4, C. H. Heinz35, Y. K. Heng1,57,62, C.
Herold59, T. Holtmann4, P. C. Hong13,f, G. Y. Hou1,62, X. T. Hou1,62, Y. R.
Hou62, Z. L. Hou1, H. M. Hu1,62, J. F. Hu55,i, T. Hu1,57,62, Y. Hu1, G. S.
Huang70,57, K. X. Huang58, L. Q. Huang31,62, X. T. Huang49, Y. P. Huang1, T.
Hussain72, N Hüsken27,35, W. Imoehl27, N. in der Wiesche67, J. Jackson27, S.
Jaeger4, S. Janchiv32, J. H. Jeong11A, Q. Ji1, Q. P. Ji19, X. B. Ji1,62, X. L.
Ji1,57, Y. Y. Ji49, X. Q. Jia49, Z. K. Jia70,57, H. J. Jiang75, P. C.
Jiang46,g, S. S. Jiang39, T. J. Jiang16, X. S. Jiang1,57,62, Y. Jiang62, J. B.
Jiao49, Z. Jiao23, S. Jin42, Y. Jin65, M. Q. Jing1,62, T. Johansson74, X. K.1,
S. Kabana33, N. Kalantar-Nayestanaki63, X. L. Kang10, X. S. Kang40, M.
Kavatsyuk63, B. C. Ke80, A. Khoukaz67, R. Kiuchi1, R. Kliemt14, O. B.
Kolcu61A, B. Kopf4, M. Kuessner4, A. Kupsc44,74, W. Kühn37, J. J. Lane66, P.
Larin18, A. Lavania26, L. Lavezzi73A,73C, T. T. Lei70,57, Z. H. Lei70,57, H.
Leithoff35, M. Lellmann35, T. Lenz35, C. Li43, C. Li47, C. H. Li39, Cheng
Li70,57, D. M. Li80, F. Li1,57, G. Li1, H. Li70,57, H. B. Li1,62, H. J. Li19,
H. N. Li55,i, Hui Li43, J. R. Li60, J. S. Li58, J. W. Li49, K. L. Li19, Ke
Li1, L. J Li1,62, L. K. Li1, Lei Li3, M. H. Li43, P. R. Li38,j,k, Q. X. Li49,
S. X. Li13, T. Li49, W. D. Li1,62, W. G. Li1, X. H. Li70,57, X. L. Li49,
Xiaoyu Li1,62, Y. G. Li46,g, Z. J. Li58, C. Liang42, H. Liang1,62, H.
Liang70,57, H. Liang34, Y. F. Liang53, Y. T. Liang31,62, G. R. Liao15, L. Z.
Liao49, Y. P. Liao1,62, J. Libby26, A. Limphirat59, D. X. Lin31,62, T. Lin1,
B. J. Liu1, B. X. Liu75, C. Liu34, C. X. Liu1, F. H. Liu52, Fang Liu1, Feng
Liu7, G. M. Liu55,i, H. Liu38,j,k, H. M. Liu1,62, Huanhuan Liu1, Huihui Liu21,
J. B. Liu70,57, J. L. Liu71, J. Y. Liu1,62, K. Liu1, K. Y. Liu40, Ke Liu22, L.
Liu70,57, L. C. Liu43, Lu Liu43, M. H. Liu13,f, P. L. Liu1, Q. Liu62, S. B.
Liu70,57, T. Liu13,f, W. K. Liu43, W. M. Liu70,57, X. Liu38,j,k, Y. Liu38,j,k,
Y. Liu80, Y. B. Liu43, Z. A. Liu1,57,62, Z. Q. Liu49, X. C. Lou1,57,62, F. X.
Lu58, H. J. Lu23, J. G. Lu1,57, X. L. Lu1, Y. Lu8, Y. P. Lu1,57, Z. H. Lu1,62,
C. L. Luo41, M. X. Luo79, T. Luo13,f, X. L. Luo1,57, X. R. Lyu62, Y. F. Lyu43,
F. C. Ma40, H. L. Ma1, J. L. Ma1,62, L. L. Ma49, M. M. Ma1,62, Q. M. Ma1, R.
Q. Ma1,62, R. T. Ma62, X. Y. Ma1,57, Y. Ma46,g, Y. M. Ma31, F. E. Maas18, M.
Maggiora73A,73C, S. Malde68, Q. A. Malik72, A. Mangoni28B, Y. J. Mao46,g, Z.
P. Mao1, S. Marcello73A,73C, Z. X. Meng65, J. G. Messchendorp14,63, G.
Mezzadri29A, H. Miao1,62, T. J. Min42, R. E. Mitchell27, X. H. Mo1,57,62, N.
Yu. Muchnoi5,b, J. Muskalla35, Y. Nefedov36, F. Nerling18,d, I. B.
Nikolaev5,b, Z. Ning1,57, S. Nisar12,l, W. D. Niu54, Y. Niu 49, S. L. Olsen62,
Q. Ouyang1,57,62, S. Pacetti28B,28C, X. Pan54, Y. Pan56, A. Pathak34, P.
Patteri28A, Y. P. Pei70,57, M. Pelizaeus4, H. P. Peng70,57, K. Peters14,d, J.
L. Ping41, R. G. Ping1,62, S. Plura35, S. Pogodin36, V. Prasad33, F. Z. Qi1,
H. Qi70,57, H. R. Qi60, M. Qi42, T. Y. Qi13,f, S. Qian1,57, W. B. Qian62, C.
F. Qiao62, J. J. Qin71, L. Q. Qin15, X. P. Qin13,f, X. S. Qin49, Z. H.
Qin1,57, J. F. Qiu1, S. Q. Qu60, C. F. Redmer35, K. J. Ren39, A. Rivetti73C,
M. Rolo73C, G. Rong1,62, Ch. Rosner18, S. N. Ruan43, N. Salone44, A.
Sarantsev36,c, Y. Schelhaas35, K. Schoenning74, M. Scodeggio29A,29B, K. Y.
Shan13,f, W. Shan24, X. Y. Shan70,57, J. F. Shangguan54, L. G. Shao1,62, M.
Shao70,57, C. P. Shen13,f, H. F. Shen1,62, W. H. Shen62, X. Y. Shen1,62, B. A.
Shi62, H. C. Shi70,57, J. L. Shi13, J. Y. Shi1, Q. Q. Shi54, R. S. Shi1,62, X.
Shi1,57, J. J. Song19, T. Z. Song58, W. M. Song34,1, Y. J. Song13, Y. X.
Song46,g, S. Sosio73A,73C, S. Spataro73A,73C, F. Stieler35, Y. J. Su62, G. B.
Sun75, G. X. Sun1, H. Sun62, H. K. Sun1, J. F. Sun19, K. Sun60, L. Sun75, S.
S. Sun1,62, T. Sun1,62, W. Y. Sun34, Y. Sun10, Y. J. Sun70,57, Y. Z. Sun1, Z.
T. Sun49, Y. X. Tan70,57, C. J. Tang53, G. Y. Tang1, J. Tang58, Y. A. Tang75,
L. Y Tao71, Q. T. Tao25,h, M. Tat68, J. X. Teng70,57, V. Thoren74, W. H.
Tian51, W. H. Tian58, Y. Tian31,62, Z. F. Tian75, I. Uman61B, S. J. Wang 49,
B. Wang1, B. L. Wang62, Bo Wang70,57, C. W. Wang42, D. Y. Wang46,g, F. Wang71,
H. J. Wang38,j,k, H. P. Wang1,62, J. P. Wang 49, K. Wang1,57, L. L. Wang1, M.
Wang49, Meng Wang1,62, S. Wang38,j,k, S. Wang13,f, T. Wang13,f, T. J. Wang43,
W. Wang71, W. Wang58, W. P. Wang70,57, X. Wang46,g, X. F. Wang38,j,k, X. J.
Wang39, X. L. Wang13,f, Y. Wang60, Y. D. Wang45, Y. F. Wang1,57,62, Y. H.
Wang47, Y. N. Wang45, Y. Q. Wang1, Yaqian Wang17,1, Yi Wang60, Z. Wang1,57, Z.
L. Wang71, Z. Y. Wang1,62, Ziyi Wang62, D. Wei69, D. H. Wei15, F. Weidner67,
S. P. Wen1, C. W. Wenzel4, U. Wiedner4, G. Wilkinson68, M. Wolke74, L.
Wollenberg4, C. Wu39, J. F. Wu1,62, L. H. Wu1, L. J. Wu1,62, X. Wu13,f, X. H.
Wu34, Y. Wu70, Y. H. Wu54, Y. J. Wu31, Z. Wu1,57, L. Xia70,57, X. M. Xian39,
T. Xiang46,g, D. Xiao38,j,k, G. Y. Xiao42, S. Y. Xiao1, Y. L. Xiao13,f, Z. J.
Xiao41, C. Xie42, X. H. Xie46,g, Y. Xie49, Y. G. Xie1,57, Y. H. Xie7, Z. P.
Xie70,57, T. Y. Xing1,62, C. F. Xu1,62, C. J. Xu58, G. F. Xu1, H. Y. Xu65, Q.
J. Xu16, Q. N. Xu30, W. Xu1,62, W. L. Xu65, X. P. Xu54, Y. C. Xu77, Z. P.
Xu42, Z. S. Xu62, F. Yan13,f, L. Yan13,f, W. B. Yan70,57, W. C. Yan80, X. Q.
Yan1, H. J. Yang50,e, H. L. Yang34, H. X. Yang1, Tao Yang1, Y. Yang13,f, Y. F.
Yang43, Y. X. Yang1,62, Yifan Yang1,62, Z. W. Yang38,j,k, Z. P. Yao49, M.
Ye1,57, M. H. Ye9, J. H. Yin1, Z. Y. You58, B. X. Yu1,57,62, C. X. Yu43, G.
Yu1,62, J. S. Yu25,h, T. Yu71, X. D. Yu46,g, C. Z. Yuan1,62, L. Yuan2, S. C.
Yuan1, X. Q. Yuan1, Y. Yuan1,62, Z. Y. Yuan58, C. X. Yue39, A. A. Zafar72, F.
R. Zeng49, X. Zeng13,f, Y. Zeng25,h, Y. J. Zeng1,62, X. Y. Zhai34, Y. C.
Zhai49, Y. H. Zhan58, A. Q. Zhang1,62, B. L. Zhang1,62, B. X. Zhang1, D. H.
Zhang43, G. Y. Zhang19, H. Zhang70, H. H. Zhang58, H. H. Zhang34, H. Q.
Zhang1,57,62, H. Y. Zhang1,57, J. Zhang80, J. J. Zhang51, J. L. Zhang20, J. Q.
Zhang41, J. W. Zhang1,57,62, J. X. Zhang38,j,k, J. Y. Zhang1, J. Z. Zhang1,62,
Jianyu Zhang62, Jiawei Zhang1,62, L. M. Zhang60, L. Q. Zhang58, Lei Zhang42,
P. Zhang1,62, Q. Y. Zhang39,80, Shuihan Zhang1,62, Shulei Zhang25,h, X. D.
Zhang45, X. M. Zhang1, X. Y. Zhang49, Xuyan Zhang54, Y. Zhang68, Y. Zhang71,
Y. T. Zhang80, Y. H. Zhang1,57, Yan Zhang70,57, Yao Zhang1, Z. H. Zhang1, Z.
L. Zhang34, Z. Y. Zhang75, Z. Y. Zhang43, G. Zhao1, J. Zhao39, J. Y. Zhao1,62,
J. Z. Zhao1,57, Lei Zhao70,57, Ling Zhao1, M. G. Zhao43, S. J. Zhao80, Y. B.
Zhao1,57, Y. X. Zhao31,62, Z. G. Zhao70,57, A. Zhemchugov36,a, B. Zheng71, J.
P. Zheng1,57, W. J. Zheng1,62, Y. H. Zheng62, B. Zhong41, X. Zhong58, H.
Zhou49, L. P. Zhou1,62, X. Zhou75, X. K. Zhou7, X. R. Zhou70,57, X. Y. Zhou39,
Y. Z. Zhou13,f, J. Zhu43, K. Zhu1, K. J. Zhu1,57,62, L. Zhu34, L. X. Zhu62, S.
H. Zhu69, S. Q. Zhu42, T. J. Zhu13,f, W. J. Zhu13,f, Y. C. Zhu70,57, Z. A.
Zhu1,62, J. H. Zou1, J. Zu70,57
(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 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
6 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
7 Central China Normal University, Wuhan 430079, People’s Republic of China
8 Central South University, Changsha 410083, People’s Republic of China
9 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
10 China University of Geosciences, Wuhan 430074, People’s Republic of China
11 Chung-Ang University, Seoul, 06974, Republic of Korea
12 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
13 Fudan University, Shanghai 200433, People’s Republic of China
14 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
15 Guangxi Normal University, Guilin 541004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University, Kaifeng 475004, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 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
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Inner Mongolia University, Hohhot 010021, People’s Republic of China
31 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica
1000000, Chile
34 Jilin University, Changchun 130012, People’s Republic of China
35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
38 Lanzhou University, Lanzhou 730000, People’s Republic of China
39 Liaoning Normal University, Dalian 116029, People’s Republic of China
40 Liaoning University, Shenyang 110036, People’s Republic of China
41 Nanjing Normal University, Nanjing 210023, People’s Republic of China
42 Nanjing University, Nanjing 210093, People’s Republic of China
43 Nankai University, Tianjin 300071, People’s Republic of China
44 National Centre for Nuclear Research, Warsaw 02-093, Poland
45 North China Electric Power University, Beijing 102206, People’s Republic of
China
46 Peking University, Beijing 100871, People’s Republic of China
47 Qufu Normal University, Qufu 273165, People’s Republic of China
48 Shandong Normal University, Jinan 250014, People’s Republic of China
49 Shandong University, Jinan 250100, People’s Republic of China
50 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
51 Shanxi Normal University, Linfen 041004, People’s Republic of China
52 Shanxi University, Taiyuan 030006, People’s Republic of China
53 Sichuan University, Chengdu 610064, People’s Republic of China
54 Soochow University, Suzhou 215006, People’s Republic of China
55 South China Normal University, Guangzhou 510006, People’s Republic of China
56 Southeast University, Nanjing 211100, People’s Republic of China
57 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
58 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
59 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
60 Tsinghua University, Beijing 100084, People’s Republic of China
61 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
62 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
63 University of Groningen, NL-9747 AA Groningen, The Netherlands
64 University of Hawaii, Honolulu, Hawaii 96822, USA
65 University of Jinan, Jinan 250022, People’s Republic of China
66 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
67 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
68 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
69 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
70 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
71 University of South China, Hengyang 421001, People’s Republic of China
72 University of the Punjab, Lahore-54590, Pakistan
73 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
74 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
75 Wuhan University, Wuhan 430072, People’s Republic of China
76 Xinyang Normal University, Xinyang 464000, People’s Republic of China
77 Yantai University, Yantai 264005, People’s Republic of China
78 Yunnan University, Kunming 650500, People’s Republic of China
79 Zhejiang University, Hangzhou 310027, People’s Republic of China
80 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan

###### Abstract

The cross sections of the $e^{+}e^{-}\to\phi\eta^{\prime}$ process at center-
of-mass energies from 3.508 to 4.951 GeV are measured with high precision
using 26.1 fb-1 data collected with the BESIII detector operating at the
BEPCII storage ring. The cross sections are of the order of a few picobarn,
and decrease as the center-of-mass energy increases as $s^{-n/2}$ with
$n=4.35\pm 0.14$. This result is in agreement with the Nambu-Jona-Lasinio
model prediction of $n=3.5\pm 0.9$. In addition, the charmless decay
$\psi(3770)\to\phi\eta^{\prime}$ is searched for by fitting the measured cross
sections, yet no significant signal is observed. The upper limit of ${\cal
B}(\psi(3770)\to\phi\eta^{\prime})$ at the 90% confidence level is determined
to be $2.3\times 10^{-5}$.

###### pacs:

Valid PACS appear here

##  I Introduction

The study of $e^{+}e^{-}$ annihilation into light hadrons is essential to
understand the production mechanism of light hadrons and reveal fundamental
aspects of the strong interactions of light quarks Davier:2013vna . Within the
chiral perturbation framework, the Nambu-Jona-Lasinio (NJL) model has been
successful in describing the dynamics of strong interactions between
elementary particles and predicting the cross sections of the $e^{+}e^{-}\to$
vector meson + pseudoscalar meson (VP) process over a range of center-of-mass
(c.m.) energies ($\sqrt{s}$) from 2.3 to 5.5 GeV Nambu:1961tp ; Volkov:1986zb
; Volkov:2005kw ; Prades:1993ys , here we only refer to the light hadrons
final states. Within large experimental uncertainties, the cross sections of
the $e^{+}e^{-}\to\rho\eta^{\prime}$ process measured by the BABAR
Collaboration BaBar:2007qju  are consistent with the NJL model predictions
Bystritskiy:2008pr . The NJL model also predicts that the cross sections of
the $e^{+}e^{-}\to\phi\eta^{\prime}$ process from $\sqrt{s}$=3.5 to 5.0 GeV
are expected to to be $\sim 60$ picobarn and follow an asymptotic power-law
behavior of $\sigma\propto s^{-n/2}$ with $n=3.5\pm 0.9$ as can be derived
from the result in Bystritskiy:2008pr . However, these predictions have never
been tested with high-precision experimental data.

The $\psi(3770)$, observed in the 1970s int30 , is the first charmonium
resonance above the $D\bar{D}$ production threshold. It is expected to decay
almost entirely to Okubo-Zweig-Iizuka-allowed $D\bar{D}$ final states while
the fractions of hadronic and radiative transitions to lower lying charmonium
states and decays into light hadrons (states with $u$, $d$, $s$ quarks) are
predicted to be small Eichten:1974af ; Eichten:1979ms . Unexpectedly, the BES
experiment measured a branching fraction for $\psi(3770)\to{\rm
non}$-$D\bar{D}$ to be $(14.7\pm 3.2)\%$ int37 ; int38 ; int39 ; int40 , while
the CLEO result is $(-3.3\pm 1.4^{+6.6}_{-4.8})\%$ int41 . Taking the
charmonium transition into account int31 ; int32 ; int33 ; int34 ; int35 , a
few percent of $\psi(3770)$ decays into light hadrons are still allowed due to
the large width of $\psi(3770)$. Although BES, CLEO-c, and BESIII searched for
exclusive $\psi(3770)$ decays into light hadrons, no significant signal for
any decay was reported int36 ; CLEO:2005tkm . Until now, the sum of the
branching fractions of the known ${\rm non}$-$D\bar{D}$ decays of the
$\psi(3770)$ remains to be less than 2% int1 . Recently, BESIII measured the
branching fraction for the inclusive decay $\psi(3770)\to J/\psi X$, which are
consistent within error with the sum of published branching fractions
$(0.47\pm 0.06)\%$ int1 , of
$\psi(3770)\to\pi^{+}\pi^{-}J/\psi,\pi^{0}\pi^{0}J/\psi,\eta J/\psi$, and
$\gamma\chi_{cJ}$ with $J=0,1,2$ BESIII:2020xfc .

Although the $\psi(3770)$ is believed to be primarily the $1^{3}D_{1}$ state
of the $c\bar{c}$ system, its large leptonic width of $(0.262\pm 0.018)$ keV
indicates substantial mixing with the $S$-wave state Eichten:1978tg ;
Kuang:1989ub . The $S$\- and $D$-wave mixing scheme has been used to explain
the “$\rho\pi$ puzzle" in $\psi(2S)$ and $J/\psi$ decays int42 ; int421  and
may suggest that the small branching fractions for $\psi(2S)\to$ VP decays can
be enhanced in $\psi(3770)\to$ VP decays, including $\rho\pi$, $\phi\eta$,
$\phi\eta^{\prime}$, and so on. Analogous to the observed charmless decay of
$\psi(3770)\to\phi\eta$ int36 , we search for the related decay
$\psi(3770)\to\phi\eta^{\prime}$. By using the known branching fractions of
$J/\psi\to\phi\eta^{\prime}$ and $\psi(2S)\to\phi\eta^{\prime}$ int1 , and
taking into account the mixing angle $\theta$ between $S$\- and $D$-wave, the
$\psi(3770)\to\phi\eta^{\prime}$ decay branching fraction (${\cal
B}_{\phi\eta^{\prime}}$) is predicted to be $(0.46\pm 0.18)\times
10^{-5}<{\cal B}_{\phi\eta^{\prime}}<(3.5\pm 1.3)\times 10^{-5}$ int45 ; int46
. This decay can be searched for with the BESIII data at the $\psi(3770)$ peak
and in its vicinity.

A recent BESIII paper BESIII:2022tjc  reported a study of the
$e^{+}e^{-}\to\phi\eta^{\prime}$ process and a search for the charmless decay
$Y(4230)\to\phi\eta^{\prime}$ with part of the data used in this analysis. In
this article, we report a measurement of the cross sections of the
$e^{+}e^{-}\to\phi\eta^{\prime}$ process with 26.1 fb-1 data collected at
$\sqrt{s}=3.508\sim 4.951$ GeV dec1  to test the NJL model prediction and to
search for the charmless decay $\psi(3770)\to\phi\eta^{\prime}$. Due to the
low integrated luminosity of the $\psi(3770)$ scan data samples, we do not use
the $\psi(3770)$ scan presented in the BESIII work BESIII:2020xfc .

##  II Detector and data samples

The BESIII detector dec1  records symmetric $e^{+}e^{-}$ collisions provided
by the BEPCII storage ring dec2  in the $\sqrt{s}$ range from 2.0 to 4.95 GeV,
with a peak luminosity of $1.05\times 10^{33}~{}\text{cm}^{-2}\text{s}^{-1}$
achieved at $\sqrt{s}=3.77~{}\text{GeV}$. The cylindrical core of BESIII
detector 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 detvis . The solenoid is supported by
an octagonal flux-return yoke with resistive plate counter muon identifier
modules interleaved with steel. The acceptance of charged particles and
photons is 93% over 4$\pi$ solid angle. The charged particle momentum
resolution at 1 GeV/$c$ is 0.5%, and the specific ionization energy loss
$dE/dx$ resolution is 6% for the 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 of the TOF barrel part is 68 ps, while
that of the end cap part is 110 ps. The end cap TOF system was upgraded in
2015 with multi-gap resistive plate chamber technology, providing a time
resolution of 60 ps dec3 ; dec4 .

Details of the data samples used in this analysis are listed in Table 1. The
optimization of event selection criteria and efficiency determination are
based on Monte Carlo (MC) simulations. The geant4-based geant4  simulation
software, BESIII Object Oriented Simulation Tool (boost) boost , includes the
geometric description of the BESIII detectors. The beam energy spread and
initial state radiation (ISR) in the $e^{+}e^{-}$ annihilation are modeled
with the generator kkmc kkmc1 ; kkmc2 .

Samples of MC events for the signal process $e^{+}e^{-}\to\phi\eta^{\prime}$
are generated using the HELAMP model within the evtgen framework evtgen1 ;
evtgen2 , which allow simulations of any two-body decay by specifying the
helicity amplitudes for the final state particles. The subsequent decay
$\phi\to K^{+}K^{-}$ is generated with the Vector-Scalar-Scalar (VSS) model in
which a vector particle decays into two scalars. The angular distribution of
$K$ is proportional to sin${}^{2}\theta_{K}$ in the $\phi$ helicity frame
evtgen1 , where $\theta_{K}$ is the helicity angle of $K$ in $\phi$ helicity
frame. The decay $\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$ is generated with the
DIY generator taking into account both $\rho-\omega$ interference and the box
anomaly Qin:2017vkw . The decays $\eta^{\prime}\to\eta\pi^{+}\pi^{-}$and
$\eta\to\gamma\gamma$ are both generated uniformly in phase space. Final state
radiation associated with charged final state particles is handled by photos
photos .

The inclusive MC samples generated at $\sqrt{s}$=3.650, 3.773, 4.178, 4.416,
and 4.843 GeV are used to estimate the possible background. The continuum
processes are incorporated in kkmc, and the QED processes such as Bhabha
scattering, $\mu^{+}\mu^{-}$, $\tau^{+}\tau^{-}$, and $\gamma\gamma$ events
are generated with kkmc and babayaga Babayaga . All particle decays are
modelled with evtgen using branching fractions either taken from the Particle
Data Group (PDG) int1 , when available, or otherwise estimated with lundcharm
for $J/\psi$ and $\psi(2S)$ decays lund1 ; lund2 .

##  III Event Selection

Charged tracks detected in the MDC are required to be within a polar angle
($\theta$) range of $|\cos\theta|<0.93$, where $\theta$ is defined with
respect to the $z$-axis, the symmetry axis of the MDC. For charged tracks, the
distances of the closest approach to the interaction point along the $z$-axis
and in the transverse plane must be less than 10 cm and 1 cm, respectively.

Fig. 1: The two-dimensional distribution of $M(K^{+}K^{-})$ versus
$M(\gamma\pi^{+}\pi^{-})$ for mode I (a) and versus $M(\eta\pi^{+}\pi^{-})$
for mode II (b) from data at $\sqrt{s}=$3.773 GeV. The red rectangle indicates
the signal region.

Photon candidates are identified using showers in the EMC. The deposited
energy of each shower must be more than 25 MeV in the barrel region
($|\cos\theta|<0.80$) and more than 50 MeV in the end cap region
($0.86<|\cos\theta|<0.92$). To significantly reduce showers that originate
from charged tracks, the angle subtended by the EMC shower and the position of
the closest charged track at the EMC must be greater than 10 degrees as
measured from the interaction point. To suppress electronic noise and showers
unrelated to the event, the difference between the EMC time and the event
start time is required to be within [0, 700] ns.

Fig. 2: The unbinned maximum likelihood fits on the distributions of
$M(\gamma\pi^{+}\pi^{-})$ for mode I (a) and $M(\eta\pi^{+}\pi^{-})$ for mode
II (b) at $\sqrt{s}=$3.773 GeV. The black dots with error bars represent the
data sample, the red dot-dashed lines represent the signal MC sample, the
green dashed lines are the fitted background and the blue solid lines
represent the total fit. The signal region lies between the two pink solid
arrows.

To reconstruct $e^{+}e^{-}\to\phi\eta^{\prime}$, the $\phi$ candidate is
reconstructed via the $K^{+}K^{-}$ decays, and the $\eta^{\prime}$ candidate
is reconstructed via $\gamma\pi^{+}\pi^{-}$ (mode I) and $\eta\pi^{+}\pi^{-}$
with $\eta\to\gamma\gamma$ (mode II). We require exactly four charged tracks
with zero net-charge and at least one (two) photon(s) candidates in an event
for mode I (II). A vertex fit is performed on the four charged tracks to
ensure they originate from the same vertex. In order to select $\phi$
candidates, for both modes I and II, we test all possible combinations of
oppositely charged particles which are assumed to be kaons. A $\phi$ candidate
is selected if it satisfies $|M(K^{+}K^{-})-m_{\phi}|<0.04$ GeV/$c^{2}$
($M(K^{+}K^{-})$ is the invariant mass of $K^{+}K^{-}$ and $m_{\phi}$ is the
nominal mass of $\phi$ from PDG int1 ). To remove a combinatorial background,
events with more than one selected $\phi$ candidate are rejected. According to
the MC simulation, this reduces the signal statistics by about 1%. A four-
constraint (4C) kinematic fit, which constrains the sum of four momentum of
the final-state particles to the four momentum of the initial colliding beams,
is performed for the final state particles of mode I in each event to improve
resolutions and suppress the background. If there is more than one combination
due to multiple photons in an event, the one with the smallest $\chi^{2}_{\rm
4C}$ is selected. The $\chi^{2}_{\rm 4C}$ is required to be less than 100 for
mode I. For mode II, a five-constraint (5C) kinematic fit is performed with an
additional constraint to the invariant mass of the $\eta$ candidate to the
$\eta$ nominal mass from PDG int1 . We loop over all possible combinations of
$\eta$ candidates and select the one with the least $\chi^{2}_{\rm 5C}$ of the
kinematic fit. The $\chi^{2}_{\rm 5C}$ is required to be less than 200 for
mode II. All the selection criteria have been optimized by maximizing the
figure of merit $S/\sqrt{S+B}$, where $S(B)$ is the number of signal
(background) events in the signal region, determined by MC simulation. The
signal candidates for $\phi$ and $\eta^{\prime}$ mesons are required to be
within the mass ranges $1.004\leq M(K^{+}K^{-})\leq 1.034$ GeV/$c^{2}$ for
both modes, and $0.943\leq M(\gamma(\eta)\pi^{+}\pi^{-})\leq 0.973$
GeV/$c^{2}$ for mode I (II).

After applying the mentioned selection criteria, Fig. 1 shows the scatter
plots of $M(K^{+}K^{-})$ versus $M(\gamma\pi^{+}\pi^{-})$ and
$M(\eta\pi^{+}\pi^{-})$ in data at $\sqrt{s}=$3.773 GeV for modes I and II.
Two clear accumulations of signal are observed in Fig. 1 around the
intersection of the $\phi$ and $\eta^{\prime}$ signal regions. We have used
the sidebands of $\phi$ and $\eta^{\prime}$ in data to check the possible
peaking backgrounds from $K^{+}K^{-}\eta^{\prime}$ and
$\gamma(\eta)\pi^{+}\pi^{-}\phi$, and we do not observe obvious peaking
backgrounds.

To extract the signal yield, an unbinned maximum likelihood fit is performed
on the $M(\gamma\pi^{+}\pi^{-})$ and $M(\eta\pi^{+}\pi^{-})$ distributions for
modes I and II, respectively. In the fit, the signal is described by the line
shape of the $\eta^{\prime}$ from MC simulation convolved with a Gaussian
function, which accounts for the difference in resolutions between data and MC
simulation, and the background is described by a linear function. The
parameters of the Gaussian function and the linear function are free in the
fit. Due to the low statistics at $\sqrt{s}=$3.670, 3.867, 3.871, 4.278,
4.308, 4.467, 4.527, 4.699, 4.740, 4.918, and 4.951 GeV, the parameters of the
Gaussian function are fixed to the values obtained from the nearby data
samples with large statistics for mode I. For mode II, the background level is
very low and the parameters of the Gaussian function are free in the fit. The
fit results at $\sqrt{s}=$3.773 GeV for modes I and II are shown in Fig. 2.
The results for the number of events within the signal region, $N^{\rm
signal}$, are obtained from the fits for all data samples and listed in Table
1. The uncertainty of $N^{\rm signal}$ obtained from the fit may be smaller
than $\sqrt{N^{\rm signal}}$, because the fit range is larger than the signal
region. The uncertainty of $N^{\rm signal}$ is estimated with MINOS in TMINUIT
sta .

Table 1: Signal yields and cross sections for the $e^{+}e^{-}\to\phi\eta^{\prime}$ process at different c.m. energies ($\sqrt{s}$) BESIII:2015zbz ; BESIII:2015qfd ; BESIII:2022xyz . Here, ${\cal L}$, ${N^{\rm signal}}$, $\epsilon$, $(1+\delta)$, and $\frac{1}{{|1-\Pi|}^{2}}$ are the integrated luminosity BESIII:2015qfd ; BESIII:2022xyz ; BESIII:2022xyz2 , the number of signal events in the signal region, the detection efficiency, the radiative correction factor, and the vacuum polarization factor, respectively. $\sigma^{\rm dressed}_{\rm com}$ and $\sigma^{\rm Born}_{\rm com}$ are the combined dressed cross section and Born cross section of the $e^{+}e^{-}\to\phi\eta^{\prime}$ process, where the first uncertainties are statistical, and the second uncertainties are systematic. The subscripts I and II represent modes I and II, respectively. $\sqrt{s}$ (GeV) |  ${\cal L}$ $({\rm pb^{-1}})$ | ${N_{\rm I}^{\rm signal}}$ | ${N_{\rm II}^{\rm signal}}$ |  $\epsilon_{\rm I}$ (%) |  $\epsilon_{\rm II}$ (%) | $(1+\delta)$ | $\frac{1}{{|1-\Pi|}^{2}}$ | $\sigma^{\rm dressed}_{\rm com}({\rm pb})$ | $\sigma^{\rm Born}_{\rm com}({\rm pb})$
---|---|---|---|---|---|---|---|---|---
3.508 | 181. | 8 | 46. | 1${}^{+7.3}_{-7.3}$ | 15. | 6${}^{+4.3}_{-3.6}$ | 32.52% | 25.79% | 0.734 | 1.044 | 6.75${}^{+0.93}_{-0.89}\pm$0.34 | 6.47${}^{+0.89}_{-0.85}\pm$0.32
3.511 | 184. | 6 | 50. | 3${}^{+7.5}_{-7.5}$ | 22. | 4${}^{+5.4}_{-4.7}$ | 32.47% | 26.02% | 0.737 | 1.044 | 7.79${}^{+0.99}_{-0.95}\pm$0.39 | 7.46${}^{+0.95}_{-0.91}\pm$0.37
3.582 | 85. | 7 | 25. | 0${}^{+5.2}_{-5.2}$ | 7. | 5${}^{+3.1}_{-2.5}$ | 31.09% | 25.02% | 0.792 | 1.039 | 7.28${}^{+1.36}_{-1.29}\pm$0.36 | 7.01${}^{+1.31}_{-1.24}\pm$0.35
3.650 | 453. | 9 | 94. | 0${}^{+10.4}_{-10.4}$ | 37. | 5${}^{+6.7}_{-6.0}$ | 30.48% | 24.23% | 0.915 | 1.021 | 4.93${}^{+0.46}_{-0.45}\pm$0.25 | 4.83${}^{+0.45}_{-0.44}\pm$0.24
3.670 | 84. | 7 | 13. | 2${}^{+3.6}_{-3.6}$ | 6. | 8${}^{+2.9}_{-2.2}$ | 30.12% | 24.20% | 0.839 | 0.994 | 4.42${}^{+1.02}_{-0.93}\pm$0.22 | 4.45${}^{+1.03}_{-0.94}\pm$0.24
3.773 | 2931. | 8 | 538. | 8${}^{+24.9}_{-24.9}$ | 237. | 0${}^{+15.6}_{-15.6}$ | 28.33% | 22.28% | 0.960 | 1.056 | 4.63${}^{+0.18}_{-0.18}\pm$0.23 | 4.39${}^{+0.17}_{-0.17}\pm$0.22
3.808 | 50. | 5 | 7. | 4${}^{+3.2}_{-2.6}$ | 2. | 9${}^{+2.0}_{-1.4}$ | 30.09% | 23.88% | 0.896 | 1.056 | 3.59${}^{+1.32}_{-1.03}\pm$0.18 | 3.40${}^{+1.25}_{-0.98}\pm$0.17
3.867 | 108. | 9 | 8. | 8${}^{+3.7}_{-3.1}$ | 6. | 8${}^{+3.1}_{-2.5}$ | 29.26% | 23.45% | 0.918 | 1.051 | 2.53${}^{+0.78}_{-0.64}\pm$0.13 | 2.40${}^{+0.74}_{-0.61}\pm$0.12
3.871 | 110. | 3 | 19. | 9${}^{+4.6}_{-4.6}$ | 5. | 8${}^{+2.7}_{-2.1}$ | 29.29% | 23.17% | 0.919 | 1.051 | 4.12${}^{+0.85}_{-0.81}\pm$0.21 | 3.92${}^{+0.81}_{-0.77}\pm$0.20
3.896 | 52. | 6 | 11. | 0${}^{+3.8}_{-3.2}$ | 3. | 9${}^{+2.3}_{-1.6}$ | 29.44% | 23.61% | 0.927 | 1.049 | 4.91${}^{+1.46}_{-1.18}\pm$0.25 | 4.68${}^{+1.40}_{-1.12}\pm$0.23
4.008 | 482. | 0 | 52. | 5${}^{+7.9}_{-7.9}$ | 30. | 4${}^{+6.0}_{-5.4}$ | 28.52% | 22.79% | 0.964 | 1.044 | 2.96${}^{+0.35}_{-0.34}\pm$0.15 | 2.84${}^{+0.34}_{-0.33}\pm$0.14
4.085 | 52. | 9 | 7. | 1${}^{+3.1}_{-2.5}$ | 2. | 0${}^{+1.7}_{-1.1}$ | 28.34% | 22.61% | 0.988 | 1.051 | 2.91${}^{+1.13}_{-0.87}\pm$0.15 | 2.77${}^{+1.08}_{-0.83}\pm$0.14
4.128 | 401. | 5 | 37. | 4${}^{+6.5}_{-6.5}$ | 21. | 4${}^{+4.9}_{-4.2}$ | 27.66% | 21.95% | 1.047 | 1.052 | 2.40${}^{+0.33}_{-0.32}\pm$0.12 | 2.28${}^{+0.32}_{-0.30}\pm$0.11
4.157 | 408. | 7 | 41. | 3${}^{+6.8}_{-6.8}$ | 9. | 2${}^{+3.6}_{-2.9}$ | 27.75% | 22.26% | 1.052 | 1.053 | 2.00${}^{+0.31}_{-0.29}\pm$0.10 | 1.90${}^{+0.29}_{-0.28}\pm$0.10
4.178 | 3189. | 0 | 299. | 1${}^{+17.9}_{-17.9}$ | 113. | 6${}^{+11.1}_{-10.4}$ | 27.82% | 21.80% | 1.014 | 1.054 | 2.19${}^{+0.11}_{-0.11}\pm$0.11 | 2.08${}^{+0.11}_{-0.10}\pm$0.10
4.189 | 570. | 0 | 63. | 5${}^{+8.3}_{-8.3}$ | 21. | 3${}^{+5.2}_{-4.5}$ | 27.61% | 21.93% | 1.017 | 1.056 | 2.52${}^{+0.29}_{-0.28}\pm$0.13 | 2.38${}^{+0.28}_{-0.27}\pm$0.12
4.199 | 526. | 0 | 36. | 0${}^{+6.4}_{-6.4}$ | 17. | 6${}^{+4.9}_{-4.1}$ | 27.92% | 21.99% | 1.020 | 1.056 | 1.70${}^{+0.26}_{-0.24}\pm$0.09 | 1.61${}^{+0.24}_{-0.23}\pm$0.08
4.209 | 572. | 0 | 46. | 5${}^{+7.3}_{-7.3}$ | 10. | 7${}^{+3.6}_{-2.9}$ | 27.45% | 21.63% | 1.021 | 1.057 | 1.70${}^{+0.24}_{-0.23}\pm$0.08 | 1.61${}^{+0.23}_{-0.22}\pm$0.08
4.219 | 569. | 2 | 48. | 0${}^{+7.1}_{-7.1}$ | 18. | 7${}^{+4.8}_{-4.1}$ | 27.43% | 21.65% | 1.024 | 1.056 | 1.98${}^{+0.26}_{-0.24}\pm$0.10 | 1.88${}^{+0.24}_{-0.23}\pm$0.09
4.226 | 1100. | 9 | 106. | 2${}^{+10.9}_{-10.9}$ | 53. | 6${}^{+7.6}_{-6.9}$ | 27.90% | 22.14% | 1.026 | 1.056 | 2.41${}^{+0.20}_{-0.19}\pm$0.12 | 2.28${}^{+0.19}_{-0.18}\pm$0.11
4.236 | 530. | 3 | 44. | 1${}^{+6.9}_{-6.9}$ | 17. | 0${}^{+4.5}_{-3.9}$ | 27.57% | 21.99% | 1.029 | 1.056 | 1.93${}^{+0.26}_{-0.25}\pm$0.10 | 1.82${}^{+0.25}_{-0.24}\pm$0.09
4.244 | 538. | 1 | 45. | 3${}^{+7.1}_{-7.1}$ | 18. | 5${}^{+4.6}_{-3.9}$ | 27.78% | 22.04% | 1.031 | 1.056 | 1.97${}^{+0.26}_{-0.25}\pm$0.10 | 1.86${}^{+0.25}_{-0.24}\pm$0.09
4.258 | 828. | 4 | 73. | 5${}^{+9.0}_{-9.0}$ | 25. | 1${}^{+6.1}_{-4.8}$ | 27.82% | 22.04% | 1.035 | 1.054 | 1.96${}^{+0.22}_{-0.20}\pm$0.10 | 1.86${}^{+0.21}_{-0.19}\pm$0.09
4.267 | 531. | 1 | 45. | 2${}^{+6.9}_{-6.9}$ | 23. | 4${}^{+5.5}_{-4.8}$ | 27.76% | 21.95% | 1.038 | 1.053 | 2.13${}^{+0.27}_{-0.26}\pm$0.11 | 2.02${}^{+0.26}_{-0.25}\pm$0.10
4.278 | 175. | 7 | 7. | 9${}^{+3.2}_{-2.6}$ | 4. | 9${}^{+2.5}_{-1.9}$ | 27.19% | 21.48% | 1.042 | 1.053 | 1.22${}^{+0.39}_{-0.31}\pm$0.06 | 1.16${}^{+0.37}_{-0.29}\pm$0.06
4.288 | 502. | 4 | 32. | 0${}^{+6.3}_{-6.3}$ | 15. | 6${}^{+4.2}_{-3.6}$ | 26.99% | 21.29% | 1.083 | 1.053 | 1.54${}^{+0.25}_{-0.24}\pm$0.08 | 1.46${}^{+0.23}_{-0.22}\pm$0.07
4.308 | 45. | 1 | 1. | 9${}^{+1.7}_{-1.0}$ | 1. | 0${}^{+1.3}_{-0.0}$ | 27.53% | 21.83% | 1.051 | 1.052 | 1.06${}^{+0.78}_{-0.36}\pm$0.05 | 1.00${}^{+0.74}_{-0.35}\pm$0.05
4.312 | 501. | 2 | 34. | 0${}^{+6.1}_{-6.1}$ | 12. | 7${}^{+4.0}_{-3.4}$ | 27.08% | 21.06% | 1.087 | 1.052 | 1.51${}^{+0.24}_{-0.23}\pm$0.08 | 1.44${}^{+0.22}_{-0.22}\pm$0.07
4.337 | 505. | 0 | 27. | 1${}^{+5.6}_{-5.6}$ | 12. | 7${}^{+3.8}_{-3.2}$ | 26.98% | 21.27% | 1.093 | 1.051 | 1.27${}^{+0.22}_{-0.21}\pm$0.06 | 1.21${}^{+0.21}_{-0.20}\pm$0.06
4.358 | 543. | 9 | 52. | 7${}^{+7.4}_{-7.4}$ | 12. | 5${}^{+4.1}_{-3.4}$ | 27.69% | 21.65% | 1.065 | 1.051 | 1.94${}^{+0.25}_{-0.24}\pm$0.10 | 1.85${}^{+0.24}_{-0.23}\pm$0.09
4.377 | 522. | 7 | 24. | 7${}^{+5.3}_{-5.3}$ | 17. | 6${}^{+4.4}_{-3.9}$ | 26.92% | 21.13% | 1.102 | 1.051 | 1.30${}^{+0.21}_{-0.20}\pm$0.07 | 1.24${}^{+0.20}_{-0.19}\pm$0.06
4.396 | 507. | 8 | 37. | 5${}^{+6.3}_{-6.3}$ | 10. | 6${}^{+3.8}_{-3.2}$ | 26.80% | 20.86% | 1.106 | 1.051 | 1.53${}^{+0.23}_{-0.22}\pm$0.08 | 1.45${}^{+0.22}_{-0.21}\pm$0.07
4.416 | 1090. | 7 | 83. | 3${}^{+9.4}_{-9.4}$ | 24. | 4${}^{+5.2}_{-4.6}$ | 27.14% | 21.32% | 1.082 | 1.052 | 1.60${}^{+0.16}_{-0.16}\pm$0.08 | 1.52${}^{+0.15}_{-0.15}\pm$0.08
4.436 | 569. | 9 | 47. | 0${}^{+7.5}_{-7.5}$ | 14. | 4${}^{+4.3}_{-3.6}$ | 26.65% | 21.00% | 1.116 | 1.054 | 1.72${}^{+0.24}_{-0.23}\pm$0.09 | 1.64${}^{+0.23}_{-0.22}\pm$0.08
4.467 | 111. | 1 | 4. | 8${}^{+2.5}_{-1.9}$ | 1. | 0${}^{+1.3}_{-0.7}$ | 26.94% | 20.93% | 1.096 | 1.055 | 0.85${}^{+0.41}_{-0.30}\pm$0.04 | 0.80${}^{+0.39}_{-0.28}\pm$0.04
4.527 | 112. | 1 | 6. | 6${}^{+2.8}_{-2.2}$ | 2. | 9${}^{+2.0}_{-1.4}$ | 26.64% | 20.61% | 1.112 | 1.054 | 1.37${}^{+0.50}_{-0.38}\pm$0.07 | 1.30${}^{+0.47}_{-0.36}\pm$0.06
4.600 | 586. | 9 | 24. | 0${}^{+4.9}_{-4.9}$ | 14. | 6${}^{+4.1}_{-3.5}$ | 26.40% | 20.54% | 1.132 | 1.055 | 1.05${}^{+0.17}_{-0.16}\pm$0.05 | 1.00${}^{+0.17}_{-0.16}\pm$0.05
4.612 | 103. | 8 | 4. | 9${}^{+2.5}_{-1.9}$ | 0. | 0${}^{+0.5}_{-0.0}$ | 26.01% | 19.95% | 1.158 | 1.055 | 0.75${}^{+0.39}_{-0.29}\pm$0.04 | 0.71${}^{+0.37}_{-0.28}\pm$0.04
4.628 | 521. | 5 | 28. | 5${}^{+5.6}_{-5.6}$ | 8. | 7${}^{+3.5}_{-2.8}$ | 25.85% | 19.76% | 1.162 | 1.054 | 1.14${}^{+0.20}_{-0.19}\pm$0.06 | 1.08${}^{+0.19}_{-0.18}\pm$0.05
4.641 | 552. | 4 | 27. | 5${}^{+5.4}_{-5.4}$ | 8. | 8${}^{+3.6}_{-2.9}$ | 25.90% | 19.74% | 1.164 | 1.054 | 1.05${}^{+0.19}_{-0.18}\pm$0.05 | 0.99${}^{+0.18}_{-0.17}\pm$0.05
4.661 | 529. | 6 | 27. | 6${}^{+5.1}_{-5.1}$ | 17. | 5${}^{+4.4}_{-3.8}$ | 25.59% | 19.59% | 1.169 | 1.054 | 1.37${}^{+0.20}_{-0.19}\pm$0.07 | 1.30${}^{+0.19}_{-0.18}\pm$0.06
4.682 | 1669. | 3 | 66. | 4${}^{+8.5}_{-8.5}$ | 21. | 2${}^{+5.3}_{-4.4}$ | 25.69% | 19.53% | 1.174 | 1.054 | 0.84${}^{+0.10}_{-0.09}\pm$0.04 | 0.79${}^{+0.09}_{-0.09}\pm$0.04
4.699 | 536. | 5 | 16. | 7${}^{+4.3}_{-4.3}$ | 8. | 5${}^{+3.3}_{-2.7}$ | 25.76% | 19.25% | 1.178 | 1.055 | 0.75${}^{+0.16}_{-0.15}\pm$0.04 | 0.71${}^{+0.15}_{-0.14}\pm$0.04
4.740 | 164. | 3 | 11. | 3${}^{+3.4}_{-3.4}$ | 2. | 9${}^{+2.0}_{-1.4}$ | 26.18% | 19.52% | 1.188 | 1.055 | 1.35${}^{+0.37}_{-0.35}\pm$0.07 | 1.28${}^{+0.35}_{-0.33}\pm$0.06
4.750 | 367. | 2 | 8. | 9${}^{+3.6}_{-3.0}$ | 8. | 8${}^{+3.3}_{-2.6}$ | 26.23% | 19.61% | 1.192 | 1.055 | 0.75${}^{+0.21}_{-0.17}\pm$0.04 | 0.71${}^{+0.20}_{-0.16}\pm$0.04
4.781 | 512. | 8 | 20. | 2${}^{+4.7}_{-4.7}$ | 11. | 7${}^{+3.7}_{-3.1}$ | 25.82% | 19.57% | 1.199 | 1.055 | 0.97${}^{+0.18}_{-0.17}\pm$0.05 | 0.92${}^{+0.17}_{-0.16}\pm$0.05
4.843 | 527. | 3 | 25. | 7${}^{+5.3}_{-5.3}$ | 7. | 5${}^{+3.2}_{-2.5}$ | 25.63% | 19.13% | 1.213 | 1.056 | 0.98${}^{+0.18}_{-0.17}\pm$0.05 | 0.93${}^{+0.17}_{-0.16}\pm$0.05
4.918 | 208. | 1 | 9. | 4${}^{+3.3}_{-2.7}$ | 1. | 9${}^{+1.7}_{-1.1}$ | 25.04% | 18.94% | 1.234 | 1.056 | 0.85${}^{+0.28}_{-0.22}\pm$0.04 | 0.80${}^{+0.26}_{-0.21}\pm$0.04
4.951 | 160. | 4 | 6. | 4${}^{+2.8}_{-2.2}$ | 4. | 9${}^{+2.8}_{-2.2}$ | 24.78% | 18.49% | 1.244 | 1.056 | 1.11${}^{+0.39}_{-0.30}\pm$0.06 | 1.05${}^{+0.37}_{-0.29}\pm$0.05

##  IV CROSS SECTION MEASUREMENT OF THE $e^{+}e^{-}\to\phi\eta^{\prime}$
PROCESS

The dressed cross section of the $e^{+}e^{-}\to\phi\eta^{\prime}$ process at
each energy point is calculated with

$\sigma^{\rm dressed}_{i}=\frac{\sigma^{\rm
Born}_{i}}{{|1-\Pi|}^{2}}=\frac{N^{\rm signal}_{i}}{{{\cal
L}\cdot(1+\delta)\cdot\epsilon_{i}\cdot{\cal B}_{i}}},$ (1)

where $\sigma^{\rm Born}_{i}$ is the Born cross section, $N^{\rm signal}_{i}$
is the number of signal events extracted from the fit to the
$M(\gamma\pi^{+}\pi^{-})$ or $M(\eta\pi^{+}\pi^{-})$ distribution as described
in Sec. III, ${\cal L}$ is the integrated luminosity, $\epsilon_{i}$ is the
selection efficiency obtained from the MC simulation, and ${\cal B}_{i}$ is
the product of the branching fractions ${\cal B}(\phi\to K^{+}K^{-})\cdot{\cal
B}(\eta^{\prime}\to\gamma\pi^{+}\pi^{-})$ or ${\cal B}(\phi\to
K^{+}K^{-})\cdot{\cal B}(\eta^{\prime}\to\eta\pi^{+}\pi^{-})\cdot{\cal
B}$($\eta\to\gamma\gamma$), taken from PDG int1 . The subscript $i$ represents
mode I or mode II. The vacuum polarization factor $\frac{1}{{|1-\Pi|}^{2}}$ is
taken from Ref. WorkingGrouponRadiativeCorrections:2010bjp , and the radiative
correction factor $(1+\delta)$ is defined as

${(1+\delta)=\frac{\sigma^{\rm obs}_{i}}{\sigma^{\rm
dressed}_{i}}=\frac{\int\sigma^{\rm dressed}_{i}(s(1-x))F(x,s)dx}{\sigma^{\rm
dressed}_{i}(s)}},$ (2)

where $F(x,s)$ is the QED radiator function with an accuracy of 0.1% isr ,
$x=2E_{\gamma}/\sqrt{s}$ the scaled ISR photon energy, and ${\sigma^{\rm
obs}_{i}}={N^{\rm signal}_{i}}/({{{\cal L}\cdot\epsilon_{i}\cdot{\cal
B}_{i}}})$ the observed cross section.

Fig. 3: Dressed cross sections of the $e^{+}e^{-}\to\phi\eta^{\prime}$process
and the fits under different assumptions: (a) continuum amplitude only; (b)
coherent sum of continuum and $\psi(3770)$ amplitudes, the solution with
$\Phi=2.0$ rad, ${\cal B}_{\phi\eta^{\prime}}=4.6\times 10^{-6}$, and the
significance of the $\psi(3770)\to\phi\eta^{\prime}$ decay is $1.5\sigma$; (c)
coherent sum of continuum and $\psi(3770)$ amplitudes, the solution with
$\Phi=4.7$ rad, ${\cal B}_{\phi\eta^{\prime}}=1.9\times 10^{-3}$, and the
significance of the $\psi(3770)\to\phi\eta^{\prime}$ decay is $1.6\sigma$. The
black dots with error bars are data. The red solid lines are the total fits,
the magenta dot dashed lines represent the $\psi(3770)$ component, and the
blue dashed lines the continuum process.

The cross sections we get from each of the modes are compatible with each
other, and the combined dressed cross section of these two modes is calculated
with

${\sigma^{\rm dressed}_{\rm com}=\frac{N^{\rm signal}_{\rm com}}{{\cal
L}\cdot(1+\delta)\cdot\sum\limits_{i}{\epsilon_{i}\cdot{\cal B}_{i}}}},$ (3)

where $N^{\rm signal}_{\rm com}$ is the total number of signal events in the
two modes. To get more accurate ISR factors, an iterative procedure is used
iteration . The results of the cross sections are summarized in Table 1 and
shown in Fig. 3. We can see that the cross sections range between 0.7 and 7.5
pb and decrease gradually as $\sqrt{s}$ increases. There is no visible
structure from the contribution of a vector charmonium or a charmonium-like
state in this energy region.

Least-chi-square fits to the measured dressed cross sections are performed to
describe the continuum line shape and to search for the
$\psi(3770)\to\phi\eta^{\prime}$ decay. The $\chi^{2}$ is given by

$\chi^{2}=\sum_{j}{\frac{(\sigma_{j}-\sigma_{j}^{\rm
fit})^{2}}{\delta_{j}^{2}}},$ (4)

where $\sigma_{j}$, $\sigma_{j}^{\rm fit}$, and $\delta_{j}$ are the measured
dressed cross section, the fitted cross section, and the statistical
uncertainty of the measured cross section of the $j$th energy point,
respectively.

The cross sections are described with a coherent sum of the continuum and the
$\psi(3770)$ amplitudes

$\sigma^{\rm dressed}(\sqrt{s})=\left|\frac{a}{({\sqrt{s}})^{n}}\sqrt{{\rm
PS}(\sqrt{s})}+e^{i\Phi}{\rm BW}(\sqrt{s})\right|^{2},$ (5)

where the constants $a$ and $n$ describe the magnitude and slope of the
continuum process, $\Phi$ is the relative phase between the continuum and the
$\psi(3770)$ amplitudes, ${\rm PS}(\sqrt{s})=q^{3}(\sqrt{s})/s$ is the P-wave
phase space factor int1 , and $q(\sqrt{s})$ is the momentum of $\phi$ and
$\eta^{\prime}$ in the $e^{+}e^{-}$ c.m. frame.

The $\psi(3770)$ resonance is parameterized with a Breit-Wigner (${\rm BW}$)
function

${\rm BW(\sqrt{s})}=\frac{\sqrt{12\pi\Gamma_{e^{+}e^{-}}\Gamma_{\rm tot}{\cal
B}_{\phi\eta^{\prime}}}}{s-M^{2}+iM\Gamma_{\rm tot}}\sqrt{\frac{{\rm
PS}(\sqrt{s})}{{\rm PS}(M)}},$ (6)

where $M$, $\Gamma_{\rm tot}$, and $\Gamma_{e^{+}e^{-}}$ are the mass, width,
and electronic partial width of $\psi(3770)$, and all of them are set to their
PDG values int1 . ${\cal B}_{\phi\eta^{\prime}}$ is the branching fraction of
$\psi(3770)\to\phi\eta^{\prime}$.

For the data samples far from $\sqrt{s}=3.773$ GeV, the interference between
the $\psi(3770)$ and continuum amplitudes can be ignored. Without the data
sample at $\sqrt{s}=3.773$ GeV, the pure continuum process ($\sigma^{\rm
dressed}(\sqrt{s})=|\frac{a}{(\sqrt{s})^{n}}\sqrt{\rm PS(\sqrt{s})}|^{2})$ is
used to fit the dressed cross sections. From this fit, $a=1.97\pm 0.40$ (${\rm
GeV}^{n-0.5}{\rm pb}^{0.5}$) and $n=4.35\pm 0.14$ are obtained, where the
uncertainties are statistical only. The systematic uncertainties of the cross
section measurements (see below) have negligible effect on $n$. The fit result
is shown in Fig. 3(a).

Due to the lack of high integrated luminosity data in the vicinity of the
$\psi(3770)$, the fit including the $\sqrt{s}=3.773$ GeV data and the
resonance contribution is not stable. A two-dimensional parameter scan of
${\cal B}_{\phi\eta^{\prime}}$ and $\Phi$ is performed under all possible line
shape assumptions. In the scan, the parameters $a$ and $n$ describing the
continuum amplitudes are fixed to the central values obtained from the
continuum-only fit. The resonance parameters of $\psi(3770)$, $M$,
$\Gamma_{\rm tot}$, and $\Gamma_{e^{+}e^{-}}$ are fixed to the corresponding
PDG values int1 . We scan ${\cal B}_{\phi\eta^{\prime}}$ and $\Phi$ from 0 to
$2.3\times 10^{-3}$ and from 0 to $2\pi$ in steps of $7.6\times 10^{-6}$ and
0.02, respectively. Figure 4 shows the scanning result, where the color scale
represents the value of $\chi^{2}$ defined in Eq. (4). The $i$th $\sigma$
contour is defined by $\chi^{2}=\chi^{2}_{\rm min}+i^{2}$ sta , where
$\chi^{2}_{\rm min}$ is the minimal $\chi^{2}$ value. The probability that
both parameters simultaneously take values within the one standard deviation
($1\sigma$) contour is 39.3%. The fit result corresponding to the minimum in
the scan (shown in Fig. 4) is shown in Fig. 3(c). The parameters ${\cal
B}_{\phi\eta^{\prime}}$ and $\Phi$ corresponding to the minimum $\chi^{2}$
value is ($1.9\times 10^{-3}$, 4.7 rad), and the corresponding $\chi^{2}_{\rm
min}$ is 48.0, which is slightly smaller than the $\chi^{2}$ of Fig. 3(b)
($\chi^{2}$ = 48.2). Therefore, these two solutions correspond to different
line shapes and do not originate from the multiple-solution problems discussed
in Ref. Zhu:2011ha . The significance of the $\psi(3770)\to\phi\eta^{\prime}$
decay is less than $2\sigma$ by comparing the fit $\chi^{2}$s
($\Delta~{}{\chi^{2}}=4.4$) with and without including the $\psi(3770)$
amplitude and taking the change of the number of degrees of freedom
($\Delta{\rm ndf}=2$) into account.

Fig. 4: The two-dimensional scan result of the parameters ${\cal
B}_{\phi\eta^{\prime}}$ and $\Phi$. In (a), the $y$-axis is from 0 to
$2.3\times 10^{-3}$, and the upper right contour corresponds to Fig. 3(c) and
the bottom contour corresponds to Fig. 3(b), respectively; in (b), the
$y$-axis is from 0 to $2.3\times 10^{-4}$ (Plot (b) is the subset of plot
(a)). The color in the figure represents the value of $\chi^{2}$ in different
$\sigma$ intervals defined in Eq. (4).

For the fitted parameter of ${\cal B}_{\phi\eta^{\prime}}=1.9\times 10^{-3}$,
the cross section of $\psi(3770)\to\phi\eta^{\prime}$ is of the same order of
magnitude as that of the continuum process, as shown in Fig. 3(c). Due to
destructive interference, a small structure appears above the expectation of
the continuum process, which has also been observed in
$e^{+}e^{-}\to\Lambda\bar{\Lambda},~{}\omega\pi^{0}$ and $\omega\eta$
processes BESIII:2022tjc ; BESIII:2021ccp ; BESIII:2022zxr . Considering the
fact that the cross section of $\psi(3770)\to$ light hadrons is much smaller
than the continuum production of $e^{+}e^{-}\to$ light hadrons, we exclude the
non-physical solution where the cross section is compatible to the continuum
production. The physical solution, which lies within the $1\sigma$ contour
range of ${\cal B}_{\phi\eta^{\prime}}\in[0.3,1.9]\times 10^{-5}$ and
$\Phi\in[0.6,2.8]$ rad, is shown in Fig. 4(b).

##  V SYSTEMATIC UNCERTAINTY

The systematic uncertainties of the cross section measurements are classified
into two categories: correlated terms and uncorrelated terms. Correlated terms
are common to the two $\eta^{\prime}$ decay modes, including the luminosity
measurements, the differences between data and MC simulation for the tracking
efficiency, photon reconstruction efficiency, the branching fraction of the
$\phi$ decay, and the mass window for $\phi$ candidate. Uncorrelated terms are
different in each $\eta^{\prime}$ decay mode, including the branching
fractions of $\eta^{\prime}$ and $\eta$ decays, the $\eta$ reconstruction
efficiency, the 4C/5C kinematic fits, the mass window for $\eta^{\prime}$
candidate, the signal shape and the background shape. These two categories of
uncertainties are discussed in detail below.

The integrated luminosities of the data samples used in this study are
measured using large angle Bhabha scattering events, with an uncertainty
ranging within (0.5-1)% BESIII:2015qfd ; BESIII:2022xyz ; BESIII:2022xyz2 . We
conservatively take 1% as the corresponding systematic uncertainty. The
uncertainty of the difference in the tracking efficiencies between data and MC
simulations is estimated to be 1% per track track1 ; track2 . The uncertainty
of the photon reconstruction efficiency is determined through dedicated
studies sys74 ; sys75  and is found to be 1% per photon. The uncertainties
related to the branching fractions of $\phi\to K^{+}K^{-}$,
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$, $\eta^{\prime}\to\eta\pi^{+}\pi^{-}$,
and $\eta\to\gamma\gamma$ are taken from PDG int1 . The uncertainty caused by
the $\eta$ reconstruction is determined by using a high purity control sample
of $J/\psi\to\eta p\bar{p}$ decays BESIII:2010tfr . The study utilized the
$\eta$ yields revealed in the $\gamma\gamma$ mass spectrum with or without
$\eta$ selection requirements to determine the $\eta$ reconstruction
efficiency. Given the negligible statistical error of the test sample, the
observed 1% difference of the $\eta$ reconstruction efficiencies between data
and MC simulations is taken as the systematic uncertainty.

To address the uncertainty caused by the 4C/5C kinematic fit, we adjust the
track helix parameters of charged tracks in the MC simulation so that it
better describes the momentum spectra of the data sys7 . In this study, we
utilize the efficiency after the helix correction to obtain the nominal
results. The difference in the MC efficiencies before and after the correction
is taken as the systematic uncertainty.

The efficiencies obtained from MC simulation are corrected according to the
data-MC difference in efficiencies obtained with the control sample of
$\psi(2S)\to\phi\eta^{\prime}$ decays. The correction factor $f^{v}$ is
defined as

$f^{v}=\varepsilon_{\rm MC}^{v}/\varepsilon_{\rm data}^{v},$ (7)

with

$\varepsilon^{v}_{\rm data(MC)}=N^{v}_{\rm data(MC)}/M^{v}_{\rm data(MC)},$
(8)

where the subscripts “MC" and “data" represent MC and data samples,
respectively, $N^{v}_{\rm data(MC)}$ is the number of events in the signal
region of a selection criterion $v$, and $M^{v}_{\rm data(MC)}$ is the number
of events in the full range of $v$. The relative uncertainty of the correction
factor $f^{v}$ is calculated as a quadratic sum of the relative statistical
errors of $\varepsilon^{v}_{\rm data}$ and $\varepsilon^{v}_{\rm MC}$.

A correction will not be applied if the correction factor $f^{v}$ differs from
unity by less than its uncertainty; otherwise the MC efficiency will be
corrected as $\varepsilon=\varepsilon/f^{v}$, and $\sigma_{f^{v}}$ will be
taken as the systematic uncertainty. The systematic uncertainties due to the
$\phi$ and $\eta^{\prime}$ mass windows are estimated using the control sample
of $\psi(2S)\to\phi\eta^{\prime}$(with
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$and $\eta^{\prime}\to\eta\pi^{+}\pi^{-}$
($\eta\to\gamma\gamma$) for modes I and II, respectively). The correction
factors $f^{v}$ are $1.0072\pm 0.0019$ for the $\phi$ mass window, and
$1.0065\pm 0.0032$ (mode I) and $1.0104\pm 0.0045$ (mode II) for the
$\eta^{\prime}$ mass window.

For mode I, the parameters of the Gaussian functions convolved with the signal
MC shape to describe the data are fixed at those obtained at $\sqrt{s}$ =
3.670, 3.867, 3.871, 4.278, 4.308, 4.467, 4.527, 4.699, 4.740, 4.918 and 4.951
GeV. We vary the nominal mean and standard deviation of the Gaussian function
by $\pm 1\sigma$, and the maximum difference in the signal yields resulting
from the various Gaussian function parameters is used to estimate the
systematic uncertainty caused by the signal shape. We weight the systematic
uncertainty of the signal shape by the luminosity of the data samples and
obtain an estimated uncertainty of 1.6%. For mode II, because the background
is relatively small, the Gaussian function parameters in all data samples are
free in the fit, and there is no systematic uncertainty caused by the signal
shape. The uncertainty associated with the background shape is estimated by
changing from a first-order Chebyshev polynomial to a second-order one. To
avoid any influence from statistical uncertainty, high statistical data
samples at $\sqrt{s}$ = 3.773, 4.178, 4.420, and 4.680 GeV are used to
evaluate the systematic uncertainty caused by the parameterization of the
background shape. The largest differences in the signal yields between
different parameterizations of the background shape are taken as the
systematic uncertainties, which are 1.4% and 2.7% for modes I and II,
respectively.

Due to correlations between the two modes, the combined $i$th item of the
systematic uncertainty $\xi^{i}$ is calculated with

$\xi^{i}=\frac{\sqrt{(w_{\rm I}\xi_{\rm I}^{i})^{2}+(w_{\rm II}\xi_{\rm
II}^{i})^{2}+2w_{\rm I}w_{\rm II}\rho_{\rm I,II}\xi_{\rm I}^{i}\xi_{\rm
II}^{i}}}{w_{\rm I}+w_{\rm II}},$ (9)

where $w_{\rm I}={\cal
B}$($\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$)$\cdot\epsilon_{1}$, and $w_{\rm
II}={\cal B}$($\eta^{\prime}\to\eta\pi^{+}\pi^{-}$)$\cdot{\cal
B}$($\eta\to\gamma\gamma$)$\cdot\epsilon_{2}$. Here, $\xi_{\rm I}^{i}$ and
$\xi_{\rm II}^{i}$ are the systematic uncertainties for modes I and II listed
in Table 2, respectively, and $\rho_{\rm I,II}$ is the correlation coefficient
between modes I and II. For the systematic uncertainties caused by the
luminosity measurements, the tracking efficiency, the photon reconstruction
efficiency, the branching fraction of the $\phi$ decay, and the mass window
for $\phi$ candidate, $\rho_{\rm I,II}$ is taken as 1, for the other sources
as 0.

Table 2: The systematic uncertainties for the cross section measurements. Source | Systematic uncertainty (%)
---|---
$\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$ | $\eta^{\prime}\to\eta\pi^{+}\pi^{-}$ | Combination
Luminosity | 1.0 | 1.0 | 1.0
Tracking | 4.0 | 4.0 | 4.0
Photon | 1.0 | 2.0 | 1.3
${\cal B}$($\phi\to K^{+}K^{-}$) | 1.0 | 1.0 | 1.0
${\cal B}$($\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$) | 1.4 | - | 1.0
${\cal B}$($\eta^{\prime}\to\eta\pi^{+}\pi^{-}$) | - | 1.2 | 0.4
${\cal B}$($\eta\to\gamma\gamma$) | - | 0.5 | 0.2
$\eta$ reconstruction | - | 1.0 | 0.3
Kinematic fit | 1.2 | 0.7 | 0.9
Mass window of $\phi$ | 0.2 | 0.2 | 0.2
Mass window of $\eta^{\prime}$ | 0.3 | 0.5 | 0.3
Signal shape | 1.6 | - | 1.1
Background shape | 1.4 | 2.7 | 1.3
Total | 5.2 | 5.7 | 5.0

Table 2 summarizes all the systematic uncertainties related to the cross
section measurements for the individual decay modes and the combined one. The
overall multiplicative systematic uncertainties are obtained by adding all
systematic uncertainties in quadrature assuming they are independent.

##  VI UPPER LIMIT ON THE BRANCHING FRACTION of
$\psi(3770)\to\phi\eta^{\prime}$

There is no obvious structure in the dressed cross sections of
$e^{+}e^{-}\to\phi\eta^{\prime}$, as shown in Fig. 3. Based on the Bayesian
method upp1 , the upper limit of the ${\cal B}_{\phi\eta^{\prime}}$ at the 90%
confidence level (C.L.) is calculated with the systematic uncertainty taken
into account. The least-chi-square fits are performed on the dressed cross
sections of $e^{+}e^{-}\to\phi\eta^{\prime}$, and the fitting estimator
$Q^{2}$ is constructed as

$\displaystyle Q^{2}=\sum_{j}{\frac{(\sigma_{j}-\sigma_{j}^{\rm
fit})^{2}}{\zeta_{j}^{2}}},$ (10)

where $\sigma_{j}$, and $\sigma_{j}^{\rm fit}$ are defined according to Eq.
(4), and $\zeta_{j}$ includes the statistical uncertainty and uncorrelated
systematic uncertainty of the measured cross sections, which are summed in
quadrature assuming they are independent.

We vary the parameters ${\cal B}_{\phi\eta^{\prime}}$ to get the distribution
of $Q^{2}$. Then the likelihood distribution (${\rm L^{\prime}}$) as a
function of the parameters ${\cal B}_{\phi\eta^{\prime}}$ is constructed as
${{\rm L^{\prime}}({\cal B}_{\phi\eta^{\prime}})=e^{-0.5Q^{2}}}$. The
normalized likelihood distribution is smeared with a Gaussian function, with a
mean $\varepsilon_{0}$ and a standard deviation
$\delta_{\varepsilon}=\varepsilon\times\delta^{cor}_{\rm sys}$, where
$\delta_{\varepsilon}$ and $\delta^{cor}_{\rm sys}$ are the absolute and
relative correlated systematic uncertainties, respectively. The normalized
likelihood is defined as

$\displaystyle{{\rm L}({\cal B}_{\phi\eta^{\prime}})=\int_{0}^{1}{\rm
L^{\prime}}({\frac{\varepsilon}{\varepsilon_{0}}\cdot{\cal
B}_{\phi\eta^{\prime}}})\cdot\frac{1}{\sqrt{2\pi}\delta_{\varepsilon}}e^{-\frac{{(\varepsilon-\varepsilon_{0})}^{2}}{2\delta_{\varepsilon}^{2}}}}{\rm
d\varepsilon},$ (11)

where $\varepsilon_{0}$ is the efficiency obtained from the MC simulation.

The parameters ${\cal B}_{\phi\eta^{\prime}}$ at the 90% C.L. (${\cal
B}_{\phi\eta^{\prime}}^{\rm up}$) are determined as

$\displaystyle\int_{0}^{{\cal B}_{\phi\eta^{\prime}}^{\rm up}}{\rm L}({\cal
B}_{\phi\eta^{\prime}}){\rm d{\cal
B}_{\phi\eta^{\prime}}}=0.9\int_{0}^{\infty}{\rm L}({\cal
B}_{\phi\eta^{\prime}}){\rm d{\cal B}_{\phi\eta^{\prime}}}.$ (12)

Considering the effect on the upper limit from the uncertainties of the mass
and width measurements, we vary the nominal mass and width by $\pm 1\sigma$
and choose the largest upper limit of ${\cal B}_{\phi\eta^{\prime}}$ among
these combinations. The maximum of ${\cal B}_{\phi\eta^{\prime}}$ is near
$\Phi=2.81$ rad within the $1\sigma$ contour shown in Fig. 4. In order to
obtain more conservative upper limits, the phase $\Phi$ is fixed to 2.81 rad
when determining the upper limit of ${\cal B}_{\phi\eta^{\prime}}$. The
normalized likelihood distribution is displayed in Fig. 5, and an upper limit
of $2.3\times 10^{-5}$ is obtained for ${\cal B}_{\phi\eta^{\prime}}$ for the
charmless decay of $\psi(3770)\to\phi\eta^{\prime}$.

Fig. 5: The distribution of the likelihood versus ${\cal
B}_{\phi\eta^{\prime}}$, with the phase $\Phi$ fixed to 2.81 rad. The black
arrow shows the result corresponding to the 90% C.L.

##  VII SUMMARY and discussions

The measurement of the $e^{+}e^{-}\to\phi\eta^{\prime}$ cross sections at
center-of-mass energies from 3.508 to 4.951 GeV have been reported. The
results in Table 1 show that the cross sections range from 0.71$\pm 0.20$ to
7.46$\pm 0.95$ pb, which is not consistent with $\sim 60$ pb predicted by the
NJL model Bystritskiy:2008pr . The power parameter, $n=4.35\pm 0.14$, is
obtained from the fit to the cross sections. This value is in good agreement
with the prediction of the NJL model ($n=3.5\pm 0.9$) Bystritskiy:2008pr ,
where the uncertainty is solely systematic. However, it should be noted that
the accuracy of the NJL model is (20-30)% for the calculation of the cross
sections of the $e^{+}e^{-}\to$ VP process. Our present study can therefore
offer valuable insights to advance the development of the NJL model and
enhance its precision in calculating the cross sections of the VP process.

By fitting the measured cross sections, we have also searched for the
charmless decay $\psi(3770)\to\phi\eta^{\prime}$ and found that its
statistical significance is less than $2\sigma$. The upper limit for the
parameters ${\cal B}_{\phi\eta^{\prime}}$ of the $\psi(3770)$ line shape has
been determined to be $2.3\times 10^{-5}$, which is in agreement with the
model prediction of $(0.5-3.5)\times 10^{-5}$ int45 ; int46 . This suggests
that the S- and D-wave charmonium states mixing scheme may play a role in
explaining the “$\rho\pi$ puzzle” in $J/\psi$, $\psi(2S)$, and $\psi(3770)$
charmless decays. This study, together with previous searches for exclusive
$\psi(3770)$ charmless decays, provides valuable insights into the nature of
the $\psi(3770)$, but the large $\psi(3770)$ ${\rm non}$-$D\bar{D}$ decay rate
remains a puzzle. In the future, a fine energy scan of the $\psi(3770)$
resonance would be necessary to determine the presence and magnitude of its
charmless decays.

##  VIII Acknowledgment

The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012,
12022510, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the
CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207; CAS
Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-
SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; The Institute of Nuclear
and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics
and Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020
research and innovation programme under Marie Sklodowska-Curie grant agreement
under Contract No. 894790; German Research Foundation DFG under Contracts Nos.
455635585, Collaborative Research Center CRC 1044, FOR5327, GRK 2149; Istituto
Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; National Research Foundation of Korea under
Contract No. NRF-2022R1A2C1092335; National Science and Technology fund of
Mongolia; National Science Research and Innovation Fund (NSRF) via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation of Thailand under Contract No. B16F640076; Polish National Science
Centre under Contract No. 2019/35/O/ST2/02907; The Swedish Research Council;
U. S. Department of Energy under Contract No. DE-FG02-05ER41374.

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