# Search for the Rare Decays $D_{s}^{+}\to h^{+}(h^{0})e^{+}e^{-}$

M. Ablikim1, M. N. Achasov4,b, P. Adlarson75, O. Afedulidis3, X. C. Ai80, R.
Aliberti35, A. Amoroso74A,74C, Q. An71,58, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,g, H.-R. Bao63, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi74A,74C, E.
Bianco74A,74C, A. Bortone74A,74C, I. Boyko36, R. A. Briere5, A. Brueggemann68,
H. Cai76, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,63, N. Cao1,63, S. A.
Cetin62A, J. F. Chang1,58, W. L. Chang1,63, G. R. Che43, G. Chelkov36,a, C.
Chen43, C. H. Chen9, Chao Chen55, G. Chen1, H. S. Chen1,63, M. L. Chen1,58,63,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,63, X. R. Chen31,63, X. T.
Chen1,63, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,h, Z. Y. Chen1,63, S. K.
Choi10A, X. Chu43, G. Cibinetto29A, F. Cossio74C, J. J. Cui50, H. L. Dai1,58,
J. P. Dai78, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng72, Z. Y.
Deng1, A. Denig35, I. Denysenko36, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding46,g, Y. Ding34, Y. Ding40, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, M. C. Du1, S. X. Du80, Z. H. Duan42, P.
Egorov36,a, Y. H. Fan45, J. Fang59, J. Fang1,58, S. S. Fang1,63, W. X. Fang1,
Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava74B,74C, F. Feldbauer3, G.
Felici28A, C. Q. Feng71,58, J. H. Feng59, Y. T. Feng71,58, K. Fischer69, M.
Fritsch3, C. D. Fu1, J. L. Fu63, Y. W. Fu1, H. Gao63, Y. N. Gao46,g, Yang
Gao71,58, S. Garbolino74C, I. Garzia29A,29B, P. T. Ge76, Z. W. Ge42, C.
Geng59, E. M. Gersabeck67, A. Gilman69, K. Goetzen13, L. Gong40, W. X.
Gong1,58, W. Gradl35, S. Gramigna29A,29B, M. Greco74A,74C, M. H. Gu1,58, Y. T.
Gu15, C. Y. Guan1,63, Z. L. Guan22, A. Q. Guo31,63, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,f, A. Guskov36,a, J. Gutierrez27, K. L. Han63, T. T.
Han1, X. Q. Hao19, F. A. Harris65, K. K. He55, K. L. He1,63, F. H. Heinsius3,
C. H. Heinz35, Y. K. Heng1,58,63, C. Herold60, T. Holtmann3, P. C. Hong12,f,
G. Y. Hou1,63, X. T. Hou1,63, Y. R. Hou63, Z. L. Hou1, B. Y. Hu59, H. M.
Hu1,63, J. F. Hu56,i, T. Hu1,58,63, Y. Hu1, G. S. Huang71,58, K. X. Huang59,
L. Q. Huang31,63, X. T. Huang50, Y. P. Huang1, T. Hussain73, F. Hölzken3, N
Hüsken27,35, N. in der Wiesche68, M. Irshad71,58, J. Jackson27, S. Janchiv32,
J. H. Jeong10A, Q. Ji1, Q. P. Ji19, W. Ji1,63, X. B. Ji1,63, X. L. Ji1,58, Y.
Y. Ji50, X. Q. Jia50, Z. K. Jia71,58, D. Jiang1,63, H. B. Jiang76, P. C.
Jiang46,g, S. S. Jiang39, T. J. Jiang16, X. S. Jiang1,58,63, Y. Jiang63, J. B.
Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y. Jin66, M. Q. Jing1,63, X. M.
Jing63, T. Johansson75, S. Kabana33, N. Kalantar-Nayestanaki64, X. L. Kang9,
X. S. Kang40, M. Kavatsyuk64, B. C. Ke80, V. Khachatryan27, A. Khoukaz68, R.
Kiuchi1, O. B. Kolcu62A, B. Kopf3, M. Kuessner3, X. Kui1,63, A. Kupsc44,75, W.
Kühn37, J. J. Lane67, P. Larin18, L. Lavezzi74A,74C, T. T. Lei71,58, Z. H.
Lei71,58, H. Leithoff35, M. Lellmann35, T. Lenz35, C. Li43, C. Li47, C. H.
Li39, Cheng Li71,58, D. M. Li80, F. Li1,58, G. Li1, H. Li71,58, H. B. Li1,63,
H. J. Li19, H. N. Li56,i, Hui Li43, J. R. Li61, J. S. Li59, Ke Li1, L. J
Li1,63, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k, Q. M. Li1,63, Q. X.
Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,63, W. G. Li1, X. Li1,63, X.
H. Li71,58, X. L. Li50, Xiaoyu Li1,63, Y. G. Li46,g, Z. J. Li59, Z. X. Li15,
C. Liang42, H. Liang1,63, H. Liang71,58, Y. F. Liang54, Y. T. Liang31,63, G.
R. Liao14, L. Z. Liao50, Y. P. Liao1,63, J. Libby26, A. Limphirat60, D. X.
Lin31,63, T. Lin1, B. J. Liu1, B. X. Liu76, C. Liu34, C. X. Liu1, F. H. Liu53,
Fang Liu1, Feng Liu6, G. M. Liu56,i, H. Liu38,j,k, H. B. Liu15, H. M. Liu1,63,
Huanhuan Liu1, Huihui Liu21, J. B. Liu71,58, J. Y. Liu1,63, K. Liu38,j,k, K.
Y. Liu40, Ke Liu22, L. Liu71,58, L. C. Liu43, Lu Liu43, M. H. Liu12,f, P. L.
Liu1, Q. Liu63, S. B. Liu71,58, T. Liu12,f, W. K. Liu43, W. M. Liu71,58, X.
Liu38,j,k, X. Liu39, Y. Liu80, Y. Liu38,j,k, Y. B. Liu43, Z. A. Liu1,58,63, Z.
D. Liu9, Z. Q. Liu50, X. C. Lou1,58,63, F. X. Lu59, H. J. Lu23, J. G. Lu1,58,
X. L. Lu1, Y. Lu7, Y. P. Lu1,58, Z. H. Lu1,63, C. L. Luo41, M. X. Luo79, T.
Luo12,f, X. L. Luo1,58, X. R. Lyu63, Y. F. Lyu43, F. C. Ma40, H. Ma78, H. L.
Ma1, J. L. Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M. Ma1, R. Q. Ma1,63, X. T.
Ma1,63, X. Y. Ma1,58, Y. Ma46,g, Y. M. Ma31, F. E. Maas18, M. Maggiora74A,74C,
S. Malde69, A. Mangoni28B, Y. J. Mao46,g, Z. P. Mao1, S. Marcello74A,74C, Z.
X. Meng66, J. G. Messchendorp13,64, G. Mezzadri29A, H. Miao1,63, T. J. Min42,
R. E. Mitchell27, X. H. Mo1,58,63, B. Moses27, N. Yu. Muchnoi4,b, J.
Muskalla35, Y. Nefedov36, F. Nerling18,d, I. B. Nikolaev4,b, Z. Ning1,58, S.
Nisar11,l, Q. L. Niu38,j,k, W. D. Niu55, Y. Niu 50, S. L. Olsen63, Q.
Ouyang1,58,63, S. Pacetti28B,28C, X. Pan55, Y. Pan57, A. Pathak34, P.
Patteri28A, Y. P. Pei71,58, M. Pelizaeus3, H. P. Peng71,58, Y. Y. Peng38,j,k,
K. Peters13,d, J. L. Ping41, R. G. Ping1,63, S. Plura35, V. Prasad33, F. Z.
Qi1, H. Qi71,58, H. R. Qi61, M. Qi42, T. Y. Qi12,f, S. Qian1,58, W. B. Qian63,
C. F. Qiao63, J. J. Qin72, L. Q. Qin14, X. S. Qin50, Z. H. Qin1,58, J. F.
Qiu1, S. Q. Qu61, Z. H. Qu72, C. F. Redmer35, K. J. Ren39, A. Rivetti74C, M.
Rolo74C, G. Rong1,63, Ch. Rosner18, S. N. Ruan43, N. Salone44, A.
Sarantsev36,c, Y. Schelhaas35, K. Schoenning75, M. Scodeggio29A, K. Y.
Shan12,f, W. Shan24, X. Y. Shan71,58, J. F. Shangguan55, L. G. Shao1,63, M.
Shao71,58, C. P. Shen12,f, H. F. Shen1,63, W. H. Shen63, X. Y. Shen1,63, B. A.
Shi63, H. C. Shi71,58, J. L. Shi12, J. Y. Shi1, Q. Q. Shi55, R. S. Shi1,63, S.
Y. Shi72, X. Shi1,58, J. J. Song19, T. Z. Song59, W. M. Song34,1, Y. J.
Song12, S. Sosio74A,74C, S. Spataro74A,74C, F. Stieler35, Y. J. Su63, G. B.
Sun76, G. X. Sun1, H. Sun63, H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun76, S.
S. Sun1,63, T. Sun51,e, W. Y. Sun34, Y. Sun9, Y. J. Sun71,58, Y. Z. Sun1, Z.
Q. Sun1,63, Z. T. Sun50, C. J. Tang54, G. Y. Tang1, J. Tang59, Y. A. Tang76,
L. Y. Tao72, Q. T. Tao25,h, M. Tat69, J. X. Teng71,58, V. Thoren75, W. H.
Tian59, Y. Tian31,63, Z. F. Tian76, I. Uman62B, Y. Wan55, S. J. Wang 50, B.
Wang1, B. L. Wang63, Bo Wang71,58, D. Y. Wang46,g, F. Wang72, H. J.
Wang38,j,k, J. P. Wang 50, K. Wang1,58, L. L. Wang1, M. Wang50, Meng Wang1,63,
N. Y. Wang63, S. Wang12,f, S. Wang38,j,k, T. Wang12,f, T. J. Wang43, W.
Wang72, W. Wang59, W. P. Wang71,58, X. Wang46,g, X. F. Wang38,j,k, X. J.
Wang39, X. L. Wang12,f, X. N. Wang1, Y. Wang61, Y. D. Wang45, Y. F.
Wang1,58,63, Y. L. Wang19, Y. N. Wang45, Y. Q. Wang1, Yaqian Wang17, Yi
Wang61, Z. Wang1,58, Z. L. Wang72, Z. Y. Wang1,63, Ziyi Wang63, D. Wei70, D.
H. Wei14, F. Weidner68, S. P. Wen1, Y. R. Wen39, U. Wiedner3, G. Wilkinson69,
M. Wolke75, L. Wollenberg3, C. Wu39, J. F. Wu1,8, L. H. Wu1, L. J. Wu1,63, X.
Wu12,f, X. H. Wu34, Y. Wu71, Y. H. Wu55, Y. J. Wu31, Z. Wu1,58, L. Xia71,58,
X. M. Xian39, B. H. Xiang1,63, T. Xiang46,g, D. Xiao38,j,k, G. Y. Xiao42, S.
Y. Xiao1, Y. L. Xiao12,f, Z. J. Xiao41, C. Xie42, X. H. Xie46,g, Y. Xie50, Y.
G. Xie1,58, Y. H. Xie6, Z. P. Xie71,58, T. Y. Xing1,63, C. F. Xu1,63, C. J.
Xu59, G. F. Xu1, H. Y. Xu66, Q. J. Xu16, Q. N. Xu30, W. Xu1, W. L. Xu66, X. P.
Xu55, Y. C. Xu77, Z. P. Xu42, Z. S. Xu63, F. Yan12,f, L. Yan12,f, W. B.
Yan71,58, W. C. Yan80, X. Q. Yan1, H. J. Yang51,e, H. L. Yang34, H. X. Yang1,
Tao Yang1, Y. Yang12,f, Y. F. Yang43, Y. X. Yang1,63, Yifan Yang1,63, Z. W.
Yang38,j,k, Z. P. Yao50, M. Ye1,58, M. H. Ye8, J. H. Yin1, Z. Y. You59, B. X.
Yu1,58,63, C. X. Yu43, G. Yu1,63, J. S. Yu25,h, T. Yu72, X. D. Yu46,g, C. Z.
Yuan1,63, J. Yuan34, L. Yuan2, S. C. Yuan1, Y. Yuan1,63, Z. Y. Yuan59, C. X.
Yue39, A. A. Zafar73, F. R. Zeng50, S. H. Zeng72, X. Zeng12,f, Y. Zeng25,h, Y.
J. Zeng59, Y. J. Zeng1,63, X. Y. Zhai34, Y. C. Zhai50, Y. H. Zhan59, A. Q.
Zhang1,63, B. L. Zhang1,63, B. X. Zhang1, D. H. Zhang43, G. Y. Zhang19, H.
Zhang71, H. C. Zhang1,58,63, H. H. Zhang59, H. H. Zhang34, H. Q. Zhang1,58,63,
H. Y. Zhang1,58, J. Zhang59, J. Zhang80, J. J. Zhang52, J. L. Zhang20, J. Q.
Zhang41, J. W. Zhang1,58,63, J. X. Zhang38,j,k, J. Y. Zhang1, J. Z. Zhang1,63,
Jianyu Zhang63, L. M. Zhang61, Lei Zhang42, P. Zhang1,63, Q. Y. Zhang39,80,
Shuihan Zhang1,63, Shulei Zhang25,h, X. D. Zhang45, X. M. Zhang1, X. Y.
Zhang50, Y. Zhang72, Y. T. Zhang80, Y. H. Zhang1,58, Y. M. Zhang39, Yan
Zhang71,58, Yao Zhang1, Z. D. Zhang1, Z. H. Zhang1, Z. L. Zhang34, Z. Y.
Zhang76, Z. Y. Zhang43, G. Zhao1, J. Y. Zhao1,63, J. Z. Zhao1,58, Lei
Zhao71,58, Ling Zhao1, M. G. Zhao43, R. P. Zhao63, S. J. Zhao80, Y. B.
Zhao1,58, Y. X. Zhao31,63, Z. G. Zhao71,58, A. Zhemchugov36,a, B. Zheng72, J.
P. Zheng1,58, W. J. Zheng1,63, Y. H. Zheng63, B. Zhong41, X. Zhong59, H.
Zhou50, J. Y. Zhou34, L. P. Zhou1,63, X. Zhou76, X. K. Zhou6, X. R. Zhou71,58,
X. Y. Zhou39, Y. Z. Zhou12,f, J. Zhu43, K. Zhu1, K. J. Zhu1,58,63, L. Zhu34,
L. X. Zhu63, S. H. Zhu70, S. Q. Zhu42, T. J. Zhu12,f, W. J. Zhu12,f, Y. C.
Zhu71,58, Z. A. Zhu1,63, J. H. Zou1, J. Zu71,58
(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 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, 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 Renmin University of China, Beijing 100872, People’s Republic of China
49 Shandong Normal University, Jinan 250014, People’s Republic of China
50 Shandong University, Jinan 250100, People’s Republic of China
51 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
52 Shanxi Normal University, Linfen 041004, People’s Republic of China
53 Shanxi University, Taiyuan 030006, People’s Republic of China
54 Sichuan University, Chengdu 610064, People’s Republic of China
55 Soochow University, Suzhou 215006, People’s Republic of China
56 South China Normal University, Guangzhou 510006, People’s Republic of China
57 Southeast University, Nanjing 211100, People’s Republic of China
58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
59 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
60 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
61 Tsinghua University, Beijing 100084, People’s Republic of China
62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
63 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
64 University of Groningen, NL-9747 AA Groningen, The Netherlands
65 University of Hawaii, Honolulu, Hawaii 96822, USA
66 University of Jinan, Jinan 250022, People’s Republic of China
67 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
68 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
69 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
70 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
71 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
72 University of South China, Hengyang 421001, People’s Republic of China
73 University of the Punjab, Lahore-54590, Pakistan
74 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
75 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
76 Wuhan University, Wuhan 430072, 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 MOE 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

Using 7.33 fb-1 of $e^{+}e^{-}$ collision data collected by the BESIII
detector at center-of-mass energies in the range of $\sqrt{s}=4.128-4.226$
GeV, we search for the rare decays $D_{s}^{+}\to h^{+}(h^{0})e^{+}e^{-}$,
where $h$ represents a kaon or pion. By requiring the $e^{+}e^{-}$ invariant
mass to be consistent with a $\phi(1020)$, $0.98<M(e^{+}e^{-})<1.04$
GeV/$c^{2}$, the decay $D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$ is
observed with a statistical significance of 7.8$\sigma$, and evidence for the
decay $D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ is found for the first
time with a statistical significance of 4.4$\sigma$. The decay branching
fractions are measured to be $\mathcal{B}(D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-})=(1.17^{+0.23}_{-0.21}\pm 0.03)\times 10^{-5}$, and
$\mathcal{B}(D_{s}^{+}\to\rho^{+}\phi,\phi\to
e^{+}e^{-})=(2.44^{+0.67}_{-0.62}\pm 0.16)\times 10^{-5}$, where the first
uncertainties are statistical and the second systematic. No significant signal
for the three four-body decays of $D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-},\
D_{s}^{+}\to K^{+}\pi^{0}e^{+}e^{-}$, and $D_{s}^{+}\to
K_{S}^{0}\pi^{+}e^{+}e^{-}$ is observed. For
$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, the $\phi$ mass region is vetoed to
minimize the long-distance effects. The 90$\%$ confidence level upper limits
set on the branching fractions of these decays are in the range of
$(7.0-8.1)\times 10^{-5}$.

In flavor physics, rare decays play an important role in precision tests of
the Standard Model (SM) and as probes of physics beyond the SM. In the SM, the
rare decays $D_{s}^{+}\to h^{+}(h^{0})e^{+}e^{-}$, where $h$ denotes a pion or
kaon, involve both short-distance (SD) and long-distance (LD) contributions.
The SD contributions proceed via the $c\to ul^{+}l^{-}$ flavor-changing
neutral-current (FCNC) transition, which can only occur at the loop level and
are highly suppressed by the Glashow$-$Iliopoulos$-$Maiani (GIM) mechanism
[1]. The GIM suppression is more effective in the charm sector compared to the
bottom and strange sectors, leading to naive SD-only branching fractions (BFs)
as low as $10^{-9}$ [2, 3, 4, 2]. The tiny SM contribution makes the FCNC
transitions in the charm sector particularly sensitive to new physics, which
may significantly enhance the BFs through the presence of new particles and
interactions [4, 5, 2, 3]. The LD contributions, occurring through a radiated
photon or an intermediate meson decaying to dileptons, dominate the decays of
$D_{s}^{+}\to h^{+}(h^{0})e^{+}e^{-}$ and can enhance the BFs to the order of
$10^{-6}$ [8, 6, 7]. Furthermore, as $D_{s}^{+}\to Ve^{+}e^{-}$ decays ($V$ is
a light vector meson) receive considerable contributions from virtual photons,
and $D_{s}^{+}\to V\gamma$ decays are predicted to have BFs as high as ${\cal
O}(10^{-3})$ [9], one would expect the BFs of $D_{s}^{+}\to Ve^{+}e^{-}$ to
reach $10^{-5}$. Therefore, the FCNC processes are often overshadowed by the
LD effects. The SD effects can be accessed through measurements in the
dilepton mass regions away from those of the intermediate mesons, such as
$\eta,\rho,\omega$ and $\phi$. Moreover, measurements of angular dependence
and the charge-parity-conjugation asymmetry in regions away from or dominated
by the resonances as SM null tests are also helpful for exploring the SD
effects [4, 5, 2, 3, 8, 6, 7, 10].

Recently, experimental results for the tests of lepton universality (LU) in
the beauty-quark FCNC decays $b\to sl^{+}l^{-}$ have received significant
attention [11]. The latest measurement from LHCb shows the tests of muon-
electron universality using $B^{+}\to K^{+}l^{+}l^{-}$ and $B^{0}\to
K^{*}l^{+}l^{-}$ decays are in agreement with the SM predictions [12]. An
analogous measurement of $c\to ul^{+}l^{-}$ provides an important
complementary test of lepton universality in the up-quark sector [13]. However
due to limited experimental results on di-electron modes, LU tests in $c\to
ul^{+}l^{-}$ are largely unexplored [10]. These modes also provide sensitivity
to a broader class of new physics, which could affect angular observables and
CP asymmetries at a few percent level in $D^{0}\to h^{+}h^{-}l^{+}l^{-}$
decays [14].

Experimentally, the decays $D^{0}\to K^{-}\pi^{+}\mu^{+}\mu^{-}$,
$D^{0}\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$, $D^{0}\to K^{-}K^{+}\mu^{+}\mu^{-}$,
and $D^{0}\to K^{-}\pi^{+}e^{+}e^{-}$ have been observed with BFs of ${\cal
O}(10^{-6}-10^{-7})$ in the LHCb and BaBar experiments [15, 16, 17], where the
decays are dominated by the LD contributions with lepton pairs originating
from $\rho$ and $\omega$ resonances. Lately, LHCb experiment reports the
measurement of the branching fraction ratio
$\mathcal{B}(\phi\to\mu^{+}\mu^{-})/\mathcal{B}(\phi\to e^{+}e^{-})$ with
$D_{(s)}^{+}\to\pi^{+}\phi$ decays, and the result is compatible with the SM
predictions [18]. Additionally, the FCNC decays of $D^{0,\pm}$ mesons have
been searched for extensively [19, 20], but the present experimental upper
bounds are still much higher than the SM predictions [21]. For the rare decays
of $D_{s}^{+}$ meson, the upper limits on the three-body decays $D_{s}^{+}\to
h^{+}l^{+}l^{-}$, where the dilepton mass is away from the $\phi$ mass, are in
the range ${\cal O}(10^{-8}-10^{-6})$ as reported in the BaBar [22] and LHCb
experiments [20].

In this Letter, we measure the LD contributions of $D_{s}^{+}\to\pi^{+}\phi$
with $\phi\to e^{+}e^{-}$, $D_{s}^{+}\to\rho^{+}\phi$ with
$\rho^{+}\to\pi^{+}\pi^{0}$ and $\phi\to e^{+}e^{-}$, and search for the four-
body decays of $D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, $D_{s}^{+}\to
K^{+}\pi^{0}e^{+}e^{-}$, $D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-}$ using data
samples corresponding to an integrated luminosity of $7.33$ fb-1 accumulated
with the BESIII detector at $e^{+}e^{-}$ center-of-mass (c.m.) energies in the
range $\sqrt{s}=4.128-4.226$ GeV [23, 24]. In these data samples, the
$D_{s}^{\pm}$ mesons are dominantly produced in the process $e^{+}e^{-}\to
D_{s}^{*\pm}D_{s}^{\mp}$ with $D_{s}^{*\pm}$ predominantly decaying via
$D_{s}^{*\pm}\to D_{s}^{\pm}\gamma$ [25]. The cross section of $e^{+}e^{-}\to
D_{s}^{*\pm}D_{s}^{\mp}$ is about 20 times larger than that of $e^{+}e^{-}\to
D_{s}^{+}D_{s}^{-}$. Consequently, the $D_{s}^{+}$ candidates are selected
from the process of $e^{+}e^{-}\to D_{s}^{*\pm}(\to
D_{s}^{\pm}\gamma)D_{s}^{\mp}$ in this work. Throughout this Letter, charged
conjugate modes are included, and $\rho$ denotes the $\rho(770)$.

Details about the BESIII detector are described in Refs. [26, 27]. Simulated
Monte-Carlo (MC) event samples produced with a geant4-based [28] package,
which includes the geometric description of the BESIII detector and the
detector response, are used to determine detection efficiencies and to
estimate backgrounds. The simulation models the beam energy spread and initial
state radiation (ISR) in the $e^{+}e^{-}$ annihilations with the generator
kkmc [29]. The input cross-section line shape of $e^{+}e^{-}\to
D_{s}^{*\pm}D_{s}^{\mp}$ is modelled according to the measurement in Ref.
[30]. The signal $D_{s}^{+}\to\pi^{+}\phi$ ($D_{s}^{+}\to\rho^{+}\phi$) is
generated using scalar-to-vector-scalar (scalar-to-two-vectors) model in
evtgen [31]. For the $\phi\to e^{+}e^{-}$ ($\rho^{+}\to\pi^{+}\pi^{0}$) decay,
the vector-to-a-lepton-pair (vector-to-two-scalars) model is employed. The
signal processes of $D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$,
$K^{+}\pi^{0}e^{+}e^{-}$, and $K_{s}^{0}\pi^{+}e^{+}e^{-}$ are generated
uniformly in phase space. The inclusive MC sample, corresponding to 40 times
the integrated luminosity of data, includes the production of open charm
processes, the ISR production of vector charmonium(-like) states, and the
continuum processes incorporated in kkmc. All particle decays are modelled
with evtgen using BFs either taken from the Particle Data Group (PDG) [32]
when available, or otherwise estimated with lundcharm [33]. Final state
radiation from charged final state particles is incorporated using the photos
package [34].

We apply a single tag (ST) method to search for $D_{s}^{+}$ candidates. The ST
method requires one $D_{s}^{+}$ meson to be fully reconstructed in the signal
mode in each event. The BF of $D_{s}^{+}\to h^{+}(h^{0})e^{+}e^{-}$ is given
by

$\mathcal{B}(D_{s}^{+}\to h^{+}(h^{0})e^{+}e^{-})=\frac{N_{\rm{sig}}}{2\cdot
N_{D_{s}^{*\pm}D_{s}^{\mp}}\cdot\epsilon\cdot\mathcal{B}_{\rm{inter}}},$ (1)

where $N_{\rm{sig}}$ is the signal yield and
$N_{D_{s}^{*\pm}D_{s}^{\mp}}=(64.72\pm 0.28)\times 10^{5}$ is the total number
of $D_{s}^{*\pm}D_{s}^{\mp}$ pairs in the data samples [40]. The signal
efficiency $\epsilon$ weighted over eight energy points is given by
$\epsilon=\sum^{i}\epsilon^{i}N_{D_{s}^{*\pm}D_{s}^{\mp}}^{i}/N_{D_{s}^{*\pm}D_{s}^{\mp}}$,
where $\epsilon^{i}$ and $N_{D_{s}^{*\pm}D_{s}^{\mp}}^{i}$ are the detection
efficiency and the number of $D_{s}^{*\pm}D_{s}^{\mp}$ pairs at the $i$-th
energy point, respectively, and $\mathcal{B}_{\rm{inter}}$ is the product BF
of intermediate state decays.

The final states include $\pi^{+}$, $\pi^{0}$, $K^{+}$, $K_{S}^{0}$, and
$e^{\pm}$ particles. We require that charged tracks detected in the main drift
chamber (MDC) satisfy $|\rm{cos\theta}|<0.93$, where the polar angle $\theta$
is defined with respect to the $z$-axis, which is the symmetry axis of the
MDC. For charged tracks not used for reconstruction of $K_{S}^{0}$, the
distance of the closest approach to the interaction point (IP) must be less
than 10 cm along the $z$-axis, $|V_{z}|$, and less than 1 cm in the transverse
plane, $|V_{xy}|$. Particle identification (PID) is applied for charged pion
and kaon tracks by combining the measurements of the specific ionization
energy loss in the MDC (d$E$/d$x$) and the flight time in the time-of-flight
system (TOF) to form likelihoods $\mathcal{L}(h)~{}(h=K,\pi)$ for each hadron
hypothesis. Kaons are identified by requiring
$\mathcal{L}(K)>\mathcal{L}(\pi)$, while pions are identified by requiring
$\mathcal{L}(K)<\mathcal{L}(\pi)$. To suppress the contamination from
$K_{S}^{0}\to\pi^{+}\pi^{-}$ decays, the vertices of all $\pi^{+}\pi^{-}$
combinations are required to be less than three times the vertex resolution
from the IP. Electron PID uses the measured information in the MDC, TOF, and
electromagnetic calorimeter (EMC). Electron candidates are required to satisfy
$\mathcal{L}^{\prime}(e)>0.001$ and
$\mathcal{L}^{\prime}(e)/(\mathcal{L}^{\prime}(e)+\mathcal{L}^{\prime}(\pi)+\mathcal{L}^{\prime}(K))>0.8$.
To reduce background from hadrons and muons, the electron and positron
candidates are required to have deposited energy in the EMC greater than 0.8
(0.7) times its momentum for those with momentum larger (less) than 0.4
GeV$/c$ [35]. To effectively suppress electron-positron pairs originating from
$\gamma$-conversions, events with the distance from the reconstructed vertex
point of the $e^{+}e^{-}$ pair to the IP in the range (2.0, 8.0) cm are
discarded. Furthermore, any photons within a cone of $5^{\circ}$ around the
electron direction is recovered to the electron momentum to improve the
electron momentum resolution.

The $K_{S}^{0}$ candidates are reconstructed from two oppositely charged
tracks satisfying $|V_{z}|<$ 20 cm. The two charged tracks are assigned as
$\pi^{+}\pi^{-}$ without imposing further PID criteria. They are constrained
to originate from a common vertex and are required to have an invariant mass
within $|M(\pi^{+}\pi^{-})-m_{K_{S}^{0}}|<$ 12 MeV$/c^{2}$, where
$m_{K_{S}^{0}}$ is the $K^{0}_{S}$ nominal mass [32]. The decay length of the
$K^{0}_{S}$ candidate must be greater than twice the vertex resolution. The
$\pi^{0}$ candidate is reconstructed from a pair of photon candidates, which
are reconstructed using showers in the EMC detector. The showers are required
to have energy deposited greater than 25 MeV in the barrel region
($|\cos\theta|<0.80$) and 50 MeV in the end cap region
($0.86<|\cos\theta|<0.92$). The difference between the EMC time and the event
start time for each photon is required to be within [0, 700] ns to suppress
electronic noise and showers unrelated to the event. The angle between the
vectors from the IP to the EMC cluster position and the projection of the
closest charged track at the EMC for each photon is required to be greater
than $10^{\circ}$. The $\pi^{0}$ candidates must have a $\gamma\gamma$
invariant mass in the region $[0.115,0.150]$ GeV/c2. The backgrounds
associated with mis-paired photons are effectively suppressed by requiring the
energy of $\pi^{0}$ candidate to be greater than 0.17 GeV. A kinematic fit
constraining the $\gamma\gamma$ invariant mass to the $\pi^{0}$ nominal mass
is performed to improve the $\pi^{0}$ four-vector for use in later kinematic
calculations.

Candidate $D_{s}^{+}$ mesons are formed from combinations of identified
hadrons and lepton pairs. The $h^{+}(h^{0})e^{+}e^{-}$ invariant mass is
required to be within $[1.88,2.02]$ GeV$/c^{2}$. To further suppress
backgrounds and identify the $D_{s}^{+}$ candidates from $e^{+}e^{-}\to
D_{s}^{*\pm}D_{s}^{\mp}$ process, we introduce two variables, the recoil mass
of $D_{s}^{+}$, $M_{\rm rec}$, and the mass difference, $\Delta M$:

$\begin{split}&M_{\rm rec}=\sqrt{\left(E_{\rm
cm}-\sqrt{|\vec{P}_{D_{s}^{+}}|^{2}+m_{D_{s}^{+}}^{2}}\right)^{2}-|P_{D_{s}^{+}}|^{2}},\\\
&\Delta M=M(D_{s}^{+}\gamma)-M(D_{s}^{+}),\\\ \end{split}$ (2)

where $E_{\rm cm}$ is the c.m. energy of $e^{+}e^{-}$ system,
$\vec{P}_{D_{s}^{+}}$ is the three-momentum of the $D_{s}^{+}$ in the
$e^{+}e^{-}$ c.m. frame, and $m_{D_{s}^{+}}$ is the $D_{s}^{+}$ nominal mass
[32]. $M(D_{s}^{+}\gamma)$ is the invariant mass of the $D_{s}^{+}$ candidate
and the photon candidate for the $D_{s}^{*+}\to\gamma D_{s}^{+}$ process. If
there are multiple photon candidates, the one with the recoil mass of the
$D_{s}^{+}\gamma$ closest to $m_{D_{s}^{+}}$ is chosen. $M(D_{s}^{+})$ is the
invariant mass of the $D_{s}^{+}$ candidate. The signals peak in the $M_{\rm
rec}$ spectrum around $m_{D_{s}^{*-}}$ for a $D_{s}^{+}$ from the
$e^{+}e^{-}\to D_{s}^{+}D_{s}^{*-}$ process or in the $\Delta M$ spectrum
around $m_{D_{s}^{*+}}-m_{D_{s}^{+}}$ for a $D_{s}^{+}$ from $e^{+}e^{-}\to
D_{s}^{-}D_{s}^{*+}(D_{s}^{*+}\to\gamma D_{s}^{+})$ process [36]. Signal
candidates are required to lie within the signal windows in the two-
dimensional plane of $M_{\rm rec}$ versus $\Delta M$ listed in the
Supplemental Material [37]. These windows are determined independently for
each energy point and decay mode based on the figure-of-merit of
$S/\sqrt{S+B}$ for the two $\phi$ decay modes or $\epsilon/(1.5+\sqrt{B})$
[38] for the three four-body decays. Two different figure-of-merit are
utilized due to the significantly different contributions of the SM in the two
cases. Here, $S$ represents the expected yield from the SM contribution, while
$B$ denotes the background yield from the inclusive MC sample.

For the two $\phi$ decay modes, the invariant mass of the electron pair,
$M(e^{+}e^{-})$ (as shown in the Supplemental Material [37]), must lie within
the $\phi$ mass window of $[0.98,1.04]$ GeV$/c^{2}$, determined from the
signal MC simulation. The signal candidates for the
$D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ decay are selected by further
requiring the invariant mass $M(\pi^{+}\pi^{0})$ to be within $[0.60,0.95]$
GeV$/c^{2}$. For the non-resonant $D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$
decay, we exclude events with $M(e^{+}e^{-})$ in the range of $[0.96,1.05]$
GeV$/c^{2}$ to reject potential LD effects from
$D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$. For the $D_{s}^{+}\to
K^{+}\pi^{0}e^{+}e^{-}$ and $D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-}$ decays,
the contributions from the resonant $\phi\to e^{+}e^{-}$ decay are expected to
be insignificant with BFs of the order of $10^{-8}$ after taking the BF of
$\mathcal{B}(\phi\to e^{+}e^{-})=(2.979\pm 0.033)\times 10^{-4}$ [32] into
account. Since this value is already beyond the sensitivity of BESIII, no
further requirement on $M(e^{+}e^{-})$ is required for these two decays. After
the application of all selection criteria, the signal efficiencies range from
5.3$\%$ to 25.1$\%$ for different decays, as outlined in Table. 1. The
variation in signal efficiencies is mainly caused by the different $M_{\rm
rec}$ and $\Delta M$ requirements, and different tracking and PID efficiencies
in the signal decays.

The signal yields of different $D_{s}^{+}$ decays in the data sample are
determined independently using unbinned maximum likelihood fits to the
invariant mass distributions of each final state in the range of 1.88 to 2.02
GeV$/c^{2}$. In these fits, the signal shapes are modelled by a double-sided
Crystal Ball function [39] plus a bifurcated Gaussian function (i.e., using
asymmetric widths), with parameters determined from MC simulation. The
residual backgrounds primarily originate from random combinations of the final
states from $D_{(s)}$ meson decays and a small fraction of the continuum
process $e^{+}e^{-}\to q\bar{q}$, which vary smoothly across the fit range.
Consequently, the background shape is modelled by a first-order Chebychev
polynomial function for $D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$ mode and
$D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ mode and a second-order
Chebychev polynomial function for the others. The data distributions and fit
results for two resonant decay modes are shown in Fig. 1. The obtained signal
yields are $38.2_{-6.8}^{+7.8}$ for $D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-}$ and $37.8_{-9.6}^{+10.3}$ for $D_{s}^{+}\to\rho^{+}\phi,\phi\to
e^{+}e^{-}$, where the uncertainties are statistical only. The signal from
$D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$ is observed with a statistical
significance of 7.8$\sigma$. The statistical significance is calculated by
$\sqrt{-2\Delta\ln L}$, where $\Delta\ln L$ is the difference of the log-
likelihoods with and without the signal component in the fit. Additionally,
evidence for $D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ is found for the
first time with a statistical significance of 4.4$\sigma$. Figure 2 shows the
data distributions and fit results for the three four-body decay modes. No
significant signal is observed, and the corresponding statistical significance
is found to be less than 2$\sigma$ in each case.

$D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$

$D_{s}^{+}\to\rho^{+}\phi,\rho^{+}\to\pi^{+}\pi^{0},\phi\to e^{+}e^{-}$

Figure 1: Fits to the $M(D_{s}^{+})$ distributions for
$D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$ and
$D_{s}^{+}\to\rho^{+}\phi,\rho^{+}\to\pi^{+}\pi^{0},\phi\to e^{+}e^{-}$. The
signals are shown as the magenta dashed curves. The blue long-dashed curves
are the combinatorial background components, and the dots with error bars are
data. The $\chi^{2}/\rm{d.o.f}$ is displayed on each figure as an indication
of the goodness of fit, where $\rm{d.o.f}$ is the number of degrees of freedom
in each fit. The low $\chi^{2}/\rm{d.o.f}$ values are the result of the low
statistics.

$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$

$D_{s}^{+}\to K^{+}\pi^{0}e^{+}e^{-}$

$D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-}$

Figure 2: Fits to the $M(D_{s}^{+})$ distributions for
$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, $D_{s}^{+}\to K^{+}\pi^{0}e^{+}e^{-}$,
and $D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-}$. For
$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, the $M(e^{+}e^{-})$ is required to be
outside the $\phi$ mass window. The signals are shown as the magenta dashed
curves. The blue long-dashed curves are the combinatorial background
components, and the dots with error bars are data. The $\chi^{2}/\rm{d.o.f}$
is displayed on each figure to demonstrate the goodness of fit.

The main sources of systematic uncertainties in the BF measurements include
those associated with the total number of $D_{s}^{+}D_{s}^{*-}$ pairs, the BFs
of $\pi^{0}\to\gamma\gamma$, $K_{S}^{0}\to\pi^{+}\pi^{-}$ and
$\rho^{+}\to\pi^{+}\pi^{0}$, the signal detection efficiency, and the signal
extraction. The systematic uncertainty from the total number of
$D_{s}^{+}D_{s}^{*-}$ pairs is 0.4$\%$ [40]. An additional effect of possible
contamination from $e^{+}e^{-}\to D_{s}^{+}D_{s}^{-}$ and $e^{+}e^{-}\to
D_{s}^{*+}D_{s}^{*-}$ is estimated to be 0.3$\%$. The uncertainties from the
BFs of $\pi^{0}\to\gamma\gamma$, $K_{S}^{0}\to\pi^{+}\pi^{-}$, and
$\rho^{+}\to\pi^{+}\pi^{0}$ are negligible [32]. The uncertainties from the
tracking (PID) efficiencies of $K^{\pm}$ and $\pi^{\pm}$ are estimated to be
0.8% and 0.3% (0.8% and 0.5%) per track, respectively [41]. The uncertainties
associated with the tracking and PID efficiencies of $e^{\pm}$ are studied
with a radiative Bhabha ($e^{+}e^{-}\to\gamma e^{+}e^{-}$) control sample,
which results in a systematic uncertainty of 1.0$\%$ for each of tracking and
PID [42]. The uncertainties related to the $\pi^{0}$ and $K_{S}^{0}$
reconstruction are assigned as 2.0$\%$ and 1.5$\%$, respectively [41, 43]. The
uncertainty due to the $\gamma$-conversion background veto is quoted as
1.8$\%$ [44].

The uncertainty from the requirements of $\Delta M$ and $M_{\rm rec}$ is
studied with a control sample of $D_{s}^{+}\to K^{+}K^{-}\pi^{+}$ candidates,
giving a difference of 1.5$\%$ between data and MC simulation. For the
$D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ decay, the uncertainty
associated with the $\rho^{+}$ mass window requirement is estimated using the
$D_{s}^{+}\to\rho^{+}\phi,\phi\to K^{+}K^{-}$ control sample. The resulting
efficiency difference between data and MC simulation, 3.6$\%$, is assigned as
the corresponding systematic uncertainty. The signal MC samples of
$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, $D_{s}^{+}\to K^{+}\pi^{0}e^{+}e^{-}$,
and $D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-}$ are generated uniformly in phase
space. To estimate the uncertainty from the MC model, we generate a series of
alternative MC samples by requiring the $e^{+}e^{-}$ pairs to originate from a
vector meson whose mass varied across the $M(e^{+}e^{-})$ spectrum. The
difference in efficiency between the two MC samples at each $M(e^{+}e^{-})$
bin are used to correct the phase space MC sample. The corresponding
efficiency changes are assigned as mode-dependent systematic uncertainties:
3.7$\%$ for $D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, 1.0$\%$ for $D_{s}^{+}\to
K^{+}\pi^{0}e^{+}e^{-}$, and 2.0$\%$ for $D_{s}^{+}\to
K_{S}^{+}\pi^{+}e^{+}e^{-}$. Moreover, additional MC samples are generated
with the $\pi^{+}\pi^{0},K\pi$ coming from $\rho,K^{*}$ decays and the
resulting efficiency changes, compared to the phase-space MC samples, are
assigned as systematic uncertainties: 9.3$\%$ for
$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$, 15.5$\%$ for $D_{s}^{+}\to
K^{+}\pi^{0}e^{+}e^{-}$, and 18.0$\%$ for $D_{s}^{+}\to
K_{S}^{0}\pi^{+}e^{+}e^{-}$.

The resolution difference between data and MC simulation for the signal shape
is investigated using the signal from $D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-}$. An alternative fit is performed with the signal modelled by the
signal MC shape convolved with a Gaussian function, representing the
resolution difference between data and MC simulation. The resulting signal
yield changes by 0.8$\%$, which is assigned as the systematic uncertainty. The
systematic uncertainties associated with the background model are estimated by
repeating the fit with the background MC simulated shape; the relative changes
of the signal yields are 0.8$\%$ for $D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-}$, and 4.7$\%$ for $D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$
decays. For the upper limit measurements, the variation giving the largest
upper limit on the signal yield is taken as the final result.

Assuming that all the sources of systematic uncertainties (listed in the
Supplemental Material [37]) are independent, the total systematic
uncertainties are obtained to be 2.4$\%$ for $D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-}$, 6.6$\%$ for $D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$,
9.9$\%$ for $D_{s}^{+}\to\pi^{+}\pi^{-}e^{+}e^{-}$, 15.9$\%$ for $D_{s}^{+}\to
K^{+}\pi^{0}e^{+}e^{-}$, and 18.3$\%$ for $D_{s}^{+}\to
K_{S}^{0}\pi^{+}e^{+}e^{-}$, by adding all sources in quadrature.

Using Eq. 1, the measured BFs of the two LD measurements are determined and
presented in Table 1, where the first uncertainty is statistical and the
second systematic. These two values are consistent with the direct
calculations of $\mathcal{B}(D_{s}^{+}\to\pi^{+}\phi)\cdot\mathcal{B}(\phi\to
e^{+}e^{-})=(1.34\pm 0.12)\times 10^{-5}$ and
$\mathcal{B}(D_{s}^{+}\to\rho^{+}\phi)\cdot\mathcal{B}(\phi\to
e^{+}e^{-})=(1.67\pm 0.11)\times 10^{-5}$ from the PDG [32]. For the three
four-body decay modes, no significant signal is found. Therefore, we set upper
limits on the BFs at the 90$\%$ confidence level. Applying a Bayesian method
[45], a likelihood scan is performed by fixing the signal yield at various
values. The effects of the systematic uncertainty are included by convolving
the likelihood curve with a Gaussian function, where the standard deviation is
set to the total systematic uncertainty. The corresponding normalized
likelihood curves can be found in Ref. [37]. These 90$\%$ upper limits for the
four body decays are also listed in Table 1.

Table 1: The signal yields ($N_{\rm{sig}}$), signal efficiencies, the BFs, and the 90$\%$ confidence level upper limits. Decay | $N_{\rm{sig}}$ |  $\epsilon$ ($\%$) |  $\mathcal{B}~{}$($\times 10^{-5}$)
---|---|---|---
$D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$ | $38.2_{-6.8}^{+7.8}$ | 25.1 | $1.17^{+0.23}_{-0.21}\pm 0.03$
$D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ | $37.8_{-9.6}^{+10.3}$ | 12.1 | $2.44^{+0.67}_{-0.62}\pm 0.16$
$D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-}$ | … | 7.4 | $<7.0$
$D_{s}^{+}\to K^{+}\pi^{0}e^{+}e^{-}$ | … | 5.3 | $<7.1$
$D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-}$ | … | 6.7 | $<8.1$

In conclusion, we search for the rare decays $D_{s}^{+}\to
h^{+}(h^{0})e^{+}e^{-}$ using a data sample corresponding to an integrated
luminosity of $7.33$ fb-1 [23] taken at $e^{+}e^{-}$ c.m. energies in the
range $\sqrt{s}=4.128-4.226$ GeV. The $D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-}$ decay is observed with a statistical significance of 7.8$\sigma$.
Evidence of the $D_{s}^{+}\to\rho^{+}\phi,\phi\to e^{+}e^{-}$ decay is found
for the first time with a statistical significance of 4.4$\sigma$. The BFs of
these two decays are measured to be
$\mathcal{B}(D_{s}^{+}\to\pi^{+}\phi,\phi\to
e^{+}e^{-})=(1.17^{+0.23}_{-0.21}\pm 0.03)\times 10^{-5}$ and
$\mathcal{B}(D_{s}^{+}\to\rho^{+}\phi,\phi\to
e^{+}e^{-})=(2.44^{+0.67}_{-0.62}\pm 0.16)\times 10^{-5}$, where the first
uncertainties are statistical and the second systematic. The BF of
$D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-}$ is in agreement with the CLEO
[46] result quoted by the PDG [32], with a precision improved by a factor of
three. Our result is also in agreement with the BaBar result,
$\mathcal{B}(D_{s}^{+}\to\pi^{+}\phi,\phi\to e^{+}e^{-})=(0.97\pm 0.18)\times
10^{-5}$, obtained indirectly (based on the yields and efficiencies listed in
Ref. [22]) and thus not used by the PDG. Both BFs are consistent with the
products of the PDG values,
$\mathcal{B}(D_{s}^{+}\to\pi^{+}\phi)\cdot\mathcal{B}(\phi\to e^{+}e^{-})$ and
$\mathcal{B}(D_{s}^{+}\to\rho^{+}\phi)\cdot\mathcal{B}(\phi\to e^{+}e^{-})$
[32] within uncertainties. No significant signal of the four-body rare decays
is observed, and the upper limits on the BFs of these decays are set to be
$\mathcal{B}(D_{s}^{+}\to\pi^{+}\pi^{0}e^{+}e^{-})<7.0\times 10^{-5}$,
$\mathcal{B}(D_{s}^{+}\to K^{+}\pi^{0}e^{+}e^{-})<7.1\times 10^{-5}$, and
$\mathcal{B}(D_{s}^{+}\to K_{S}^{0}\pi^{+}e^{+}e^{-})<8.1\times 10^{-5}$, at
the 90$\%$ confidence level. These results represent the first upper limits on
the BFs of these decays. Our measurements improve our knowledge of LU in $c\to
ul^{+}l^{-}$ and $D_{s}^{+}\to V\gamma$ decays.

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. 2020YFA0406400,
2020YFA0406300; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11635010, 11735014, 11835012, 11935015, 11935016, 11935018,
11961141012, 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; 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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