# Measurement of the absolute branching fractions of $J/\psi\to\gamma\eta$ and
$\eta$ decay modes

M. Ablikim1, M. N. Achasov10,b, P. Adlarson67, S. Ahmed15, M. Albrecht4, R.
Aliberti28, A. Amoroso66A,66C, M. R. An32, Q. An63,49, X. H. Bai57, Y. Bai48,
O. Bakina29, R. Baldini Ferroli23A, I. Balossino24A, Y. Ban38,h, K.
Begzsuren26, N. Berger28, M. Bertani23A, D. Bettoni24A, F. Bianchi66A,66C, J.
Bloms60, A. Bortone66A,66C, I. Boyko29, R. A. Briere5, H. Cai68, X. Cai1,49,
A. Calcaterra23A, G. F. Cao1,54, N. Cao1,54, S. A. Cetin53A, J. F. Chang1,49,
W. L. Chang1,54, G. Chelkov29,a, G. Chen1, H. S. Chen1,54, M. L. Chen1,49, S.
J. Chen35, X. R. Chen25, Y. B. Chen1,49, Z. J. Chen20,i, W. S. Cheng66C, G.
Cibinetto24A, F. Cossio66C, X. F. Cui36, H. L. Dai1,49, J. P. Dai42,f, X. C.
Dai1,54, A. Dbeyssi15, R. E. de Boer4, D. Dedovich29, Z. Y. Deng1, A. Denig28,
I. Denysenko29, M. Destefanis66A,66C, F. De Mori66A,66C, Y. Ding33, C. Dong36,
J. Dong1,49, L. Y. Dong1,54, M. Y. Dong1,49,54, X. Dong68, S. X. Du71, Y. L.
Fan68, J. Fang1,49, S. S. Fang1,54, Y. Fang1, R. Farinelli24A, L. Fava66B,66C,
F. Feldbauer4, G. Felici23A, C. Q. Feng63,49, J. H. Feng50, M. Fritsch4, C. D.
Fu1, Y. N. Gao38,h, Ya Gao64, Yang Gao63,49, I. Garzia24A,24B, P. T. Ge68, C.
Geng50, E. M. Gersabeck58, A Gilman61, K. Goetzen11, L. Gong33, W. X.
Gong1,49, W. Gradl28, M. Greco66A,66C, L. M. Gu35, M. H. Gu1,49, Y. T. Gu13,
C. Y Guan1,54, A. Q. Guo22, L. B. Guo34, R. P. Guo40, Y. P. Guo9,g, A.
Guskov29,a, T. T. Han41, W. Y. Han32, X. Q. Hao16, F. A. Harris56, K. L.
He1,54, F. H. Heinsius4, C. H. Heinz28, Y. K. Heng1,49,54, C. Herold51, M.
Himmelreich11,e, T. Holtmann4, G. Y. Hou1,54, Y. R. Hou54, Z. L. Hou1, H. M.
Hu1,54, J. F. Hu47,j, T. Hu1,49,54, Y. Hu1, G. S. Huang63,49, L. Q. Huang64,
X. T. Huang41, Y. P. Huang1, Z. Huang38,h, T. Hussain65, N Hüsken22,28, W.
Ikegami Andersson67, W. Imoehl22, M. Irshad63,49, S. Jaeger4, S. Janchiv26, Q.
Ji1, Q. P. Ji16, X. B. Ji1,54, X. L. Ji1,49, Y. Y. Ji41, H. B. Jiang41, X. S.
Jiang1,49,54, J. B. Jiao41, Z. Jiao18, S. Jin35, Y. Jin57, M. Q. Jing1,54, T.
Johansson67, N. Kalantar-Nayestanaki55, X. S. Kang33, R. Kappert55, M.
Kavatsyuk55, B. C. Ke71, I. K. Keshk4, A. Khoukaz60, P. Kiese28, R. Kiuchi1,
R. Kliemt11, L. Koch30, O. B. Kolcu53A, B. Kopf4, M. Kuemmel4, M. Kuessner4,
A. Kupsc67, M. G. Kurth1,54, W. Kühn30, J. J. Lane58, J. S. Lange30, P.
Larin15, A. Lavania21, L. Lavezzi66A,66C, Z. H. Lei63,49, H. Leithoff28, M.
Lellmann28, T. Lenz28, C. Li39, C. H. Li32, Cheng Li63,49, D. M. Li71, F.
Li1,49, G. Li1, H. Li63,49, H. Li43, H. B. Li1,54, H. J. Li16, J. L. Li41, J.
Q. Li4, J. S. Li50, Ke Li1, L. K. Li1, Lei Li3, P. R. Li31,k,l, S. Y. Li52, W.
D. Li1,54, W. G. Li1, X. H. Li63,49, X. L. Li41, Xiaoyu Li1,54, Z. Y. Li50, H.
Liang1,54, H. Liang27, H. Liang63,49, Y. F. Liang45, Y. T. Liang25, G. R.
Liao12, L. Z. Liao1,54, J. Libby21, A. Limphirat51, C. X. Lin50, T. Lin1, B.
J. Liu1, C. X. Liu1, D. Liu15,63, F. H. Liu44, Fang Liu1, Feng Liu6, H. B.
Liu13, H. M. Liu1,54, Huanhuan Liu1, Huihui Liu17, J. B. Liu63,49, J. L.
Liu64, J. Y. Liu1,54, K. Liu1, K. Y. Liu33, L. Liu63,49, M. H. Liu9,g, P. L.
Liu1, Q. Liu68, Q. Liu54, S. B. Liu63,49, Shuai Liu46, T. Liu1,54, W. M.
Liu63,49, X. Liu31,k,l, Y. Liu31,k,l, Y. B. Liu36, Z. A. Liu1,49,54, Z. Q.
Liu41, X. C. Lou1,49,54, F. X. Lu50, H. J. Lu18, J. D. Lu1,54, J. G. Lu1,49,
X. L. Lu1, Y. Lu1, Y. P. Lu1,49, C. L. Luo34, M. X. Luo70, P. W. Luo50, T.
Luo9,g, X. L. Luo1,49, X. R. Lyu54, F. C. Ma33, H. L. Ma1, L. L. Ma41, M. M.
Ma1,54, Q. M. Ma1, R. Q. Ma1,54, R. T. Ma54, X. X. Ma1,54, X. Y. Ma1,49, Y.
Ma38,h, F. E. Maas15, M. Maggiora66A,66C, S. Maldaner4, S. Malde61, Q. A.
Malik65, A. Mangoni23B, Y. J. Mao38,h, Z. P. Mao1, S. Marcello66A,66C, Z. X.
Meng57, J. G. Messchendorp55,d, G. Mezzadri24A, T. J. Min35, R. E. Mitchell22,
X. H. Mo1,49,54, N. Yu. Muchnoi10,b, H. Muramatsu59, S. Nakhoul11,e, Y.
Nefedov29, F. Nerling11,e, I. B. Nikolaev10,b, Z. Ning1,49, S. Nisar8,m, S. L.
Olsen54, Q. Ouyang1,49,54, S. Pacetti23B,23C, X. Pan9,g, Y. Pan58, A. Pathak1,
A. Pathak27, P. Patteri23A, M. Pelizaeus4, H. P. Peng63,49, K. Peters11,e, J.
Pettersson67, J. L. Ping34, R. G. Ping1,54, S. Pogodin29, R. Poling59, V.
Prasad63,49, H. Qi63,49, H. R. Qi52, K. H. Qi25, M. Qi35, T. Y. Qi9,g, S.
Qian1,49, W. B. Qian54, Z. Qian50, C. F. Qiao54, L. Q. Qin12, X. P. Qin9,g, X.
S. Qin41, Z. H. Qin1,49, J. F. Qiu1, S. Q. Qu36, K. H. Rashid65, K.
Ravindran21, C. F. Redmer28, A. Rivetti66C, V. Rodin55, M. Rolo66C, G.
Rong1,54, Ch. Rosner15, M. Rump60, H. S. Sang63, A. Sarantsev29,c, Y.
Schelhaas28, C. Schnier4, K. Schoenning67, M. Scodeggio24A,24B, D. C. Shan46,
W. Shan19, X. Y. Shan63,49, J. F. Shangguan46, M. Shao63,49, C. P. Shen9,g, H.
F. Shen1,54, P. X. Shen36, X. Y. Shen1,54, H. C. Shi63,49, R. S. Shi1,54, X.
Shi1,49, X. D Shi63,49, W. M. Song27,1, Y. X. Song38,h, S. Sosio66A,66C, S.
Spataro66A,66C, K. X. Su68, P. P. Su46, G. X. Sun1, H. K. Sun1, J. F. Sun16,
L. Sun68, S. S. Sun1,54, T. Sun1,54, W. Y. Sun27, W. Y. Sun34, X Sun20,i, Y.
J. Sun63,49, Y. Z. Sun1, Z. T. Sun41, Y. H. Tan68, Y. X. Tan63,49, C. J.
Tang45, G. Y. Tang1, J. Tang50, J. X. Teng63,49, V. Thoren67, W. H. Tian43, Y.
T. Tian25, I. Uman53B, B. Wang1, C. W. Wang35, D. Y. Wang38,h, H. J.
Wang31,k,l, H. P. Wang1,54, K. Wang1,49, L. L. Wang1, M. Wang41, M. Z.
Wang38,h, Meng Wang1,54, S. Wang9,g, W. Wang50, W. H. Wang68, W. P. Wang63,49,
X. Wang38,h, X. F. Wang31,k,l, X. L. Wang9,g, Y. Wang63,49, Y. D. Wang37, Y.
F. Wang1,49,54, Y. Q. Wang1, Y. Y. Wang31,k,l, Ying Wang50, Z. Wang1,49, Z. Y.
Wang1, Ziyi Wang54, Zongyuan Wang1,54, D. H. Wei12, F. Weidner60, S. P. Wen1,
D. J. White58, U. Wiedner4, G. Wilkinson61, M. Wolke67, L. Wollenberg4, J. F.
Wu1,54, L. H. Wu1, L. J. Wu1,54, X. Wu9,g, Z. Wu1,49, L. Xia63,49, T.
Xiang38,h, H. Xiao9,g, S. Y. Xiao1, Z. J. Xiao34, X. H. Xie38,h, Y. G.
Xie1,49, Y. H. Xie6, T. Y. Xing1,54, C. J. Xu50, G. F. Xu1, Q. J. Xu14, W.
Xu1,54, X. P. Xu46, Y. C. Xu54, F. Yan9,g, L. Yan9,g, W. B. Yan63,49, W. C.
Yan71, Xu Yan46, H. J. Yang42,f, H. X. Yang1, L. Yang43, S. L. Yang54, Y. X.
Yang12, Yifan Yang1,54, Zhi Yang25, M. Ye1,49, M. H. Ye7, J. H. Yin1, Z. Y.
You50, B. X. Yu1,49,54, C. X. Yu36, G. Yu1,54, J. S. Yu20,i, T. Yu64, C. Z.
Yuan1,54, L. Yuan2, X. Q. Yuan38,h, Y. Yuan1, Z. Y. Yuan50, C. X. Yue32, A. A.
Zafar65, X. Zeng Zeng6, Y. Zeng20,i, A. Q. Zhang1, B. X. Zhang1, G. Y.
Zhang16, H. Zhang63, H. H. Zhang27, H. H. Zhang50, H. Y. Zhang1,49, J. L.
Zhang69, J. Q. Zhang34, J. W. Zhang1,49,54, J. Y. Zhang1, J. Z. Zhang1,54,
Jianyu Zhang1,54, Jiawei Zhang1,54, L. M. Zhang52, L. Q. Zhang50, Lei Zhang35,
S. Zhang50, S. F. Zhang35, Shulei Zhang20,i, X. D. Zhang37, X. Y. Zhang41, Y.
Zhang61, Y. T. Zhang71, Y. H. Zhang1,49, Yan Zhang63,49, Yao Zhang1, Z. Y.
Zhang68, G. Zhao1, J. Zhao32, J. Y. Zhao1,54, J. Z. Zhao1,49, Lei Zhao63,49,
Ling Zhao1, M. G. Zhao36, Q. Zhao1, S. J. Zhao71, Y. B. Zhao1,49, Y. X.
Zhao25, Z. G. Zhao63,49, A. Zhemchugov29,a, B. Zheng64, J. P. Zheng1,49, Y. H.
Zheng54, B. Zhong34, C. Zhong64, L. P. Zhou1,54, Q. Zhou1,54, X. Zhou68, X. K.
Zhou54, X. R. Zhou63,49, X. Y. Zhou32, A. N. Zhu1,54, J. Zhu36, K. Zhu1, K. J.
Zhu1,49,54, S. H. Zhu62, T. J. Zhu69, W. J. Zhu36, W. J. Zhu9,g, Y. C.
Zhu63,49, Z. A. Zhu1,54, B. S. Zou1, J. H. Zou1 (BESIII Collaboration) 1
Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9 Fudan University, Shanghai 200443, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
24 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
25 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27 Jilin University, Changchun 130012, People’s Republic of China
28 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31 Lanzhou University, Lanzhou 730000, People’s Republic of China
32 Liaoning Normal University, Dalian 116029, People’s Republic of China
33 Liaoning University, Shenyang 110036, People’s Republic of China
34 Nanjing Normal University, Nanjing 210023, People’s Republic of China
35 Nanjing University, Nanjing 210093, People’s Republic of China
36 Nankai University, Tianjin 300071, People’s Republic of China
37 North China Electric Power University, Beijing 102206, People’s Republic of
China
38 Peking University, Beijing 100871, People’s Republic of China
39 Qufu Normal University, Qufu 273165, People’s Republic of China
40 Shandong Normal University, Jinan 250014, People’s Republic of China
41 Shandong University, Jinan 250100, People’s Republic of China
42 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
43 Shanxi Normal University, Linfen 041004, People’s Republic of China
44 Shanxi University, Taiyuan 030006, People’s Republic of China
45 Sichuan University, Chengdu 610064, People’s Republic of China
46 Soochow University, Suzhou 215006, People’s Republic of China
47 South China Normal University, Guangzhou 510006, People’s Republic of China
48 Southeast University, Nanjing 211100, People’s Republic of China
49 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
50 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
51 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
52 Tsinghua University, Beijing 100084, People’s Republic of China
53 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
54 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
55 University of Groningen, NL-9747 AA Groningen, The Netherlands
56 University of Hawaii, Honolulu, Hawaii 96822, USA
57 University of Jinan, Jinan 250022, People’s Republic of China
58 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
59 University of Minnesota, Minneapolis, Minnesota 55455, USA
60 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
61 University of Oxford, Keble Rd, Oxford, UK OX13RH
62 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
63 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
64 University of South China, Hengyang 421001, People’s Republic of China
65 University of the Punjab, Lahore-54590, Pakistan
66 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
67 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
68 Wuhan University, Wuhan 430072, People’s Republic of China
69 Xinyang Normal University, Xinyang 464000, People’s Republic of China
70 Zhejiang University, Hangzhou 310027, People’s Republic of China
71 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Currently at Istanbul Arel University, 34295 Istanbul, Turkey
e Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
f Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
g Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
h Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
i Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
k Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
m Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan

###### Abstract

Based on a data sample of $(1.0087\pm 0.0044)\times 10^{10}$ $J/\psi$ events
collected by the BESIII detector at the BEPCII accelerator, the absolute
branching fraction (BF) of the decay $J/\psi\to\gamma\eta$ is measured with
high precision using events in which the radiative photon converts to
$e^{+}e^{-}$. Using the measured absolute BF of $J/\psi\to\gamma\eta$, the
absolute BFs of four dominant $\eta$ decay modes are measured for the first
time. The results are $\mathcal{B}(J/\psi\to\gamma\eta)=(1.067\pm 0.005\pm
0.023)\times 10^{-3}$, $\mathcal{B}(\eta\to\gamma\gamma)=(39.86\pm 0.04\pm
0.99)\%$, $\mathcal{B}(\eta\to\pi^{0}\pi^{0}\pi^{0})=(31.96\pm 0.07\pm
0.84)\%$, $\mathcal{B}(\eta\to\pi^{+}\pi^{-}\pi^{0})=(23.04\pm 0.03\pm
0.54)\%$, and $\mathcal{B}(\eta\to\pi^{+}\pi^{-}\gamma)=(4.38\pm 0.02\pm
0.10)\%$, where the first and second uncertainties are statistical and
systematic, respectively. The results are consistent with the world average
values within two standard deviations.

##  I Introduction

As two members of the ground-state nonet of pseudo-scalar mesons, the $\eta$
and $\eta^{\prime}$ mesons play an important part in understanding low energy
Quantum Chromodynamics (QCD) [1, 2]. Precise measurements of their branching
fractions (BFs) are important for a wide variety of physics topics. For
example, the decay widths of $\eta,~{}\eta^{\prime}\to\gamma\gamma$ are
related to the quark content of the two mesons [3], the BFs of
$\eta,~{}\eta^{\prime}\to 3\pi$ decays can provide valuable information on
light quark masses [4], the BFs of
$\eta,~{}\eta^{\prime}\to\pi^{+}\pi^{-}\gamma$ decays are related to details
of chiral dynamics [5, 6], and the BFs of some rare decays of the $\eta$ and
$\eta^{\prime}$ can test fundamental QCD symmetries [7] and probe for physics
beyond the standard model [8]. As the BFs of the rare decays are obtained via
normalization to the dominant decay modes, a precise determination of the BFs
of the dominant decay modes of the $\eta$ and $\eta^{\prime}$ is essential.
While the absolute BFs of dominant $\eta^{\prime}$ decays have been measured
with high precision by the BESIII experiment [9], no absolute BFs of $\eta$
decays have yet been measured. The exclusive BFs of the $\eta$ summarized by
the Particle Data Group (PDG) [10] are all relative measurements. This is due
to the difficulty of tagging inclusive decays of the $\eta$. The most precise
measurements so far are from the CLEO experiment [11], where the BFs were
presented under the assumption that the five dominant decay modes measured in
their work account for 99.9% of all $\eta$ decays.

In the previous work by BESIII on $\eta^{\prime}$ decays [9], absolute BFs
were measured using a specially developed method which allows tagging
inclusive decays of the $\eta^{\prime}$. In this work, using a similar but
optimized method and a much larger $J/\psi$ sample, inclusive decays of the
$\eta$ are tagged and the absolute BFs of dominant $\eta$ decay modes are
measured for the first time.

##  II BESIII Detector

The BESIII detector [13] records symmetric $e^{+}e^{-}$ collisions provided by
the BEPCII storage ring [14], which operates with a design luminosity of
$1\times 10^{33}$ cm-2s-1 in the center-of-mass energy range from 2.0 to 4.9
GeV. BESIII has collected large data samples in this energy region [15]. The
cylindrical core of the BESIII detector covers 93% of the full solid angle and
consists of a beam pipe, 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 (0.9 T in 2012) magnetic field. Around
10.8% of $J/\psi$ events were collected in 2012. The beam pipe has two layers
with 2 mm gaps between them. The inner layer diameter is 63 mm with a
thickness of 0.8 mm, while the thickness of the outer layer is 0.5 mm. The
inner diameter of the MDC is 118 mm. The solenoid is supported by an octagonal
flux-return yoke with resistive plate counter muon identification modules
interleaved with steel. The charged-particle momentum resolution at $1~{}{\rm
GeV}/c$ is $0.5\%$, and the $dE/dx$ resolution is $6\%$ for electrons from
Bhabha scattering. The EMC measures photon energies with a resolution of
$2.5\%$ ($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution
in the TOF barrel region is 68 ps, while that in the end cap region is 110 ps.
The end cap TOF system was upgraded in 2015 using multi-gap resistive plate
chamber technology, providing a time resolution of 60 ps [16].

##  III Dataset and MC Simulation

A sample of $(1.0087\pm 0.0044)\times 10^{10}$ $J/\psi$ events collected by
BESIII is used for this analysis. The total number of $J/\psi$ events
collected in the years of 2009, 2012, 2018 and 2019 at BESIII is determined
using inclusive $J/\psi$ decays with the method described in Ref. [17]. For
the selected inclusive $J/\psi$ events, the background due to QED processes,
beam-gas interactions, and cosmic rays is estimated using the continuum data
samples at $\sqrt{s}=3.08$ GeV. The detection efficiency for the inclusive
$J/\psi$ decays is obtained using the data sample of
$\psi(3686)\rightarrow\pi^{+}\pi^{-}J/\psi$. The efficiency difference between
the $J/\psi$ produced at rest and the $J/\psi$ from the decay
$\psi(3686)\rightarrow\pi^{+}\pi^{-}J/\psi$ is estimated by comparing the
corresponding efficiencies in MC simulation. The uncertainties related to the
signal MC model, track reconstruction efficiency, fit to the $J/\psi$ mass
peak, background estimation, noise mixing, and reconstruction efficiency for
the pions recoiling against the $J/\psi$ are studied. Finally, the number of
$J/\psi$ events collected at BESIII is determined to be $N_{J/\psi}=(10087\pm
44)\times 10^{6}$.

Simulated data samples are produced with a geant4-based [18] Monte Carlo (MC)
package [19], which includes the geometric description of the BESIII detector
and the detector response [20, 21]. They are used to determine the detection
efficiency and estimate the backgrounds. The simulation includes the beam
energy spread and initial state radiation (ISR) in the $e^{+}e^{-}$
annihilations modeled with the generator kkmc [22]. A sample of $1.0011\times
10^{10}$ simulated inclusive $J/\psi$ events is used to estimate the
background events. This inclusive MC sample includes both the production of
the $J/\psi$ resonance and the continuum processes incorporated in kkmc. The
known decay modes are modeled with evtgen [23] using BFs taken from the PDG
[10], and the remaining unknown charmonium decays are modeled with lundcharm
[24]. Final state radiation (FSR) from charged final state particles is
incorporated using photos [25].

In addition, a sample of $1\times 10^{8}$ $J/\psi\to\gamma\eta$ simulated
events is generated to determine the detection efficiency. In the simulation,
the $\eta$ decay BFs from the PDG [10] are used, and the decay modes are
described with theoretical models that have been validated in previous works,
as listed in Table 1. To study background distributions, exclusive MC samples
for specific background processes, such as $e^{+}e^{-}\to\gamma\gamma$, are
generated. The simulated processes and the corresponding theoretical models
are listed in Table 1.

Table 1: Generator models used for MC simulations. Decay mode | Generator model
---|---
$J/\psi\to\gamma\eta$ | Helicity amplitude [26]
$\eta\to\gamma\gamma$ | Phase space [26]
$\eta\to\pi^{0}\pi^{0}\pi^{0}$ | Dalitz plot analyses [27]
$\eta\to\pi^{+}\pi^{-}\pi^{0}$ | Dalitz plot analyses [27]
$\eta\to\pi^{+}\pi^{-}\gamma$ | Box anomaly proceed [28]
$\eta\to\gamma e^{+}e^{-}$ | Electromagnetic Dalitz decays [28]
$\eta\to\gamma\mu^{+}\mu^{-}$ | Electromagnetic Dalitz decays [28]
Other $\eta$ decays | Phase space [26]
$e^{+}e^{-}\to\gamma\gamma$ | BABAYAGA [29, 30, 31]
$e^{+}e^{-}\to e^{+}e^{-}$ | BABAYAGA [29, 30, 31]
$J/\psi\to e^{+}e^{-}\eta$ | Electromagnetic Dalitz decays [32]
$J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ | Dalitz plot analyses [33]
$J/\psi\to\omega\eta$ | Helicity amplitude [26]
$J/\psi\to\omega\pi^{0}$ | Helicity amplitude [26]
$J/\psi\to\gamma\eta^{\prime}$ | Helicity amplitude [26]
$\eta^{\prime}$ decays | Same as in [9]

##  IV Event Selection and Background Analysis

To tag inclusive decays of the $\eta$, $J/\psi\to\gamma\eta$ events in which
the radiative photon converts to an $e^{+}e^{-}$ pair are selected using the
Photon Conversion Finder (PCF) package [12]. Reconstructed photon conversion
events have an energy resolution twice as good as photons reconstructed in the
EMC. The signal of the $\eta$ meson is extracted from the recoil mass spectrum
of the $e^{+}e^{-}$ conversion pair, $M_{\rm{recoil}}(e^{+}e^{-})$. The BF of
$J/\psi\to\gamma\eta$ is calculated with

$\mathcal{B}(J/\psi\to\gamma\eta)=\frac{{{N_{\gamma\eta}^{\rm{obs}}}}}{N_{J/\psi}\cdot\varepsilon_{\gamma\eta}\cdot
f}.$ (1)

Here, $N_{\gamma\eta}^{\rm{obs}}$ is the number of observed
$J/\psi\to\gamma\eta$, $\gamma\to e^{+}e^{-}$ events, $N_{J/\psi}$ is the
total number of $J/\psi$ decays, $\varepsilon_{\gamma\eta}$ is the detection
efficiency obtained from MC simulation, and $f$ is a factor used to correct
for the difference in photon conversion efficiencies between data and MC
simulation.

After that, $J/\psi\to\gamma\eta,~{}\eta\to X$ events are reconstructed to
study the $\eta$ decay BFs. Here, $X$ stands for one of the four dominant
$\eta$ decay modes: $\gamma\gamma$, $\pi^{0}\pi^{0}\pi^{0}$,
$\pi^{+}\pi^{-}\pi^{0}$, and $\pi^{+}\pi^{-}\gamma$. To improve statistics,
the radiative photons are required to be detected in the EMC istead of
converting to $e^{+}e^{-}$. The absolute BFs of $\eta\to X$ are then obtained
with

$\mathcal{B}(\eta\\!\to\\!X)\\!=\\!\frac{N_{X}^{\rm{obs}}}{\varepsilon_{X}\\!\cdot\\!N_{J/\psi}\\!\cdot\\!\mathcal{B}(J/\psi\\!\to\\!\gamma\eta)}\\!=\\!\frac{N_{X}^{\rm{obs}}}{\varepsilon_{X}}\cdot\frac{\varepsilon_{\gamma\eta}\\!\cdot\\!f}{N_{\gamma\eta}^{\rm{obs}}}.$
(2)

Here $N_{X}^{\rm{obs}}$ denotes the number of observed $J/\psi\to\gamma\eta$,
$\eta\to X$ events, and $\varepsilon_{X}$ the MC-determined reconstruction
efficiency.

###  IV.1 Inclusive channel

To select $J/\psi\to\gamma\eta$ events where the radiative photon converts to
$e^{+}e^{-}$, candidate events are required to have at least two oppositely
charged tracks. The charged tracks are reconstructed using information from
the MDC and are required to pass within $\pm 30$ cm of the run-by-run
determined interaction point (IP) along the beam direction. They must also
have a polar angle ($\theta$) within the range $|\\!\cos\theta|<0.93$, where
$\theta$ is defined with respect to the MDC axis. In the next step, particle
identification (PID) requirements are applied. The combined information from
the specific energy loss in the MDC ($dE/dx$), TOF, and EMC is used to
calculate the probability that the track originates from an electron or
positron. This probability is then compared to the corresponding probability
that the track originates from a muon, pion, kaon, or proton. A track is
assumed to be an electron or positron if its probability is larger than the
other particle hypotheses. The event is kept for further analysis if there is
at least one positron and one electron candidate.

The radiative photon is reconstructed from the $e^{+}e^{-}$ pair using the
PCF. At BESIII, the helix parameters of charged tracks are determined assuming
that the IP is the origin, which is not true in our case since the conversion
point (CP) is generally displaced from the IP. The conversion point of the
photon is estimated using the track projections of $e^{+}e^{-}$ in the $x$-$y$
plane, perpendicular to the beam direction. The midpoint of the centers of the
two track projections is taken as the CP, as shown in Fig. 1. As most photon
conversions occur at the beam pipe and the inner wall of the MDC, the
distances from the CP to IP in the $x$-$y$ plane, denoted by $R_{xy}$, is
usually greater than 2 cm. Hence, $R_{xy}>2$ cm is required to suppress non-
conversion $e^{+}e^{-}$ pairs. Moreover, as the radiative photon has high
energy, the opening angle between the conversion $e^{+}$ and $e^{-}$ tracks is
close to zero. Based on this, several selection criteria are applied to
suppress the non-conversion $e^{+}e^{-}$ tracks: (i) The sum of the minimum
distances from the CP to the two track projections, denoted by $|\Delta xy|$,
has to be less than 0.2 cm, see Fig. 1. (ii) The minimum distance between the
tracks of $e^{+}e^{-}$ in the beam direction, denoted by $\Delta z$, has to be
less than 1.5 cm. (iii) The angle between the $x$-$y$ plane and the plane
determined by the momentum vectors of $e^{+}$ and $e^{-}$, denoted by
$\Psi_{\text{pair}}$, has to be within $[-0.5,0.5]$ radians, see Fig. 1. The
$\Psi_{\text{pair}}$ of converted $e^{+}e^{-}$ concentrate around zero because
their polar angles are essentially the same, but their azimuth angles are
slightly different as the track parameters are extrapolated to the IP rather
than the CP. (iv) The angle between the momentum vector of the radiative
photon and the direction from IP to CP, denoted by $\theta_{eg}$, has to
satisfy $\\!\cos\theta_{eg}>0.8$, see Fig. 1.

(a)(b)

Figure 1: (a): Projections of $e^{+}$ and $e^{-}$ tracks in the $x$-$y$ plane.
The $z$-axis shows the direction of the magnetic field. The reconstructed
vertex of the tracks is near the IP (since the IP is used in the determination
of the track helix parameters). Point CP is the conversion point obtained with
the PCF, and is supposed to be the true vertex of the tracks. The distance
from the IP to CP is $R_{xy}$. The points $O_{1}$, $O_{2}$ are the centers of
the two track projections. The points A, B are the intersection points of
$O_{1}O_{2}$ and the two track projections, and the distance between A and B
is $|\Delta xy|$. The arrow $P$γ represents the momentum of the converted
photon, and the angle between the arrow and the IP-CP is $\theta_{eg}$. (b):
Illustration of $\Psi_{\rm{pair}}$. The arrow $P$${}_{e^{+}}$
($P$${}_{e^{-}}$) represents the momentum of the $e^{+}$ ($e^{-}$).

After the reconstruction of the radiative photon, further selection criteria
are applied to suppress background events. Since almost all known $\eta$
decays contain at least one photon, we require that at least one photon is
detected in the EMC to suppress fully charged background contributions. The
photon candidate must have a deposited energy greater than 25 MeV when
detected in the barrel region ($|\\!\cos\theta|<0.80$) and greater than 50 MeV
when detected in the end cap region ($0.86<|\\!\cos\theta|<0.92$). The angle
between the detected position of the photon candidate and the closest
extrapolated charged track must be larger than 10 degrees to exclude photons
that originate from charged tracks. The difference between the EMC time of the
photon candidate and the event start time is required to be within [0, 700] ns
to suppress electronic noise and photons unrelated to the event. Furthermore,
to suppress $e^{+}e^{-}\to\gamma\gamma$ and
$J/\psi\to\gamma_{1}X,~{}X\to\gamma_{2}\gamma_{3}$, where $\gamma_{2}$ or
$\gamma_{3}$ converts to $e^{+}e^{-}$, three selection criteria are applied.
(i) The energy of all photons (except for the radiative one) are required to
be less than 1.4 GeV. (ii) For events that have fewer than five photons,
$-0.998<\\!\cos\theta_{\gamma\gamma}<0$ is required, where
$\theta_{\gamma\gamma}$ is the angle between the radiative photon and the most
energetic one of the other photons. (iii) For events that have only two
charged tracks and fewer than four photons, $|\\!\cos\theta_{\rm{miss}}|<0.98$
is required, where $\theta_{\rm{miss}}$ is the polar angle of the missing
momentum of the event. Finally, to suppress $J/\psi\to
e^{+}e^{-}(\gamma),~{}\gamma\to e^{+}e^{-}$ events, all events that have more
than two charged tracks are required to satisfy $2P_{\rm{trk}}-P_{\gamma}<0.8$
GeV, where $P_{\gamma}$ and $P_{\rm{trk}}$ are the magnitude of the momentum
of the radiative photon and the most energetic charged track excluding the
converted $e^{+}e^{-}$, respectively.

According to a study with the MC sample, only 0.06% of the
$J/\psi\to\gamma\eta$ events that passed all of the above selection criteria
have more than one $e^{+}e^{-}$ combination. All combinations are retained for
further analysis.

Figure 2: Recoil mass spectrum of $e^{+}e^{-}$ for the final event sample. The
black dots with error bars represent data and the solid blue line represents
the inclusive MC sample. The five shaded (or solid) histograms represent five
different background components from the inclusive MC sample and are drawn
stacked upon each other. The histogram filled with light green oblique lines
represents the $J/\psi\to\gamma\eta^{\prime}$ events, the green filled solid
histogram $J/\psi\to\omega\pi^{0}$ events, the histogram filled with light
blue grids $J/\psi\to\omega\eta$ events, the histogram filled with dark blue
horizontal lines $J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ events, and the histogram
filled with pink vertical lines the other background events.

The recoil mass spectra of $e^{+}e^{-}$ for data and the corresponding
inclusive MC sample, after all the selection criteria have been applied, are
shown in Fig. 2. There are large differences between the two samples. The
reason is that some background processes are not included in the inclusive MC
sample. These missing backgrounds are from the processes
$e^{+}e^{-}\to\gamma\gamma$, $e^{+}e^{-}\to e^{+}e^{-}$ and $J/\psi\to
e^{+}e^{-}\eta$. These processes are instead studied with exclusive MC
samples. In addition, to describe the background events more accurately, some
processes already included in the inclusive MC sample are simulated
exclusively. These processes are $J/\psi\to\gamma\eta^{\prime}$,
$J/\psi\to\pi^{+}\pi^{-}\pi^{0}$, $J/\psi\to\omega\eta$, and
$J/\psi\to\omega\pi^{0}$. The remaining backgrounds are studied with the
inclusive MC sample. Different background components from the inclusive MC
sample are shown in 2. The background caused by wrong $e^{+}e^{-}$
combinations of $J/\psi\to\gamma\eta$ events is ignored.

###  IV.2 Exclusive channels

Candidate events for the processes $J/\psi\\!\to\\!\gamma\eta$,
$\eta\\!\to\\!X$ ($X\\!=\\!\gamma\gamma$, $\pi^{0}\pi^{0}\pi^{0}$,
$\pi^{+}\pi^{-}\pi^{0}$ or $\pi^{+}\pi^{-}\gamma$) are reconstructed with the
following common selection criteria. (i) Charged tracks detected in the MDC
are required to have a polar angle $|\\!\cos\theta|<0.93$, and the distance of
closest approach to the IP must be less than 10 cm along the beam direction
and less than 1 cm in the transverse plane. (ii) Photons are reconstructed
with the same selection criteria as described in section IV.1, except that
only photons detected in the barrel region ($|\\!\cos\theta|<0.80$) of the EMC
are used, where the photon detection efficiency of data is in good agreement
with that of MC simulation. In addition, for the neutral decays of
$\eta\to\gamma\gamma$ and $\eta\to\pi^{0}\pi^{0}\pi^{0}$, instead of being
within [0, 700] ns of the event start time, the EMC times of the photons are
required to be within $[-500,500]$ ns of the EMC time of the most energetic
photon. (iii) The events must have the correct number of charged tracks, and
at least the minimum number of photons associated with the given final state.
(iv) A kinematic fit on the final state particle candidates is performed. The
kinematic fit adjusts the track energy and momentum within the measured
uncertainties so as to satisfy energy and momentum conservation for the given
final state hypothesis. This improves the momentum resolution and reduces the
background. (v) To maximize the figure of merit, defined as $S/\sqrt{S+B}$,
the maximum value of the kinematic fit quality of the candidate events,
$\chi^{2}$, is restricted. Here, $S$ is the number of corresponding signal
events estimated by MC simulation, and $S+B$ is the number of data events.
(vi) If there are multiple possible photon combinations, the combination with
the minimum $\chi^{2}$ is chosen for further analysis. (vii) For the channels
$X=\pi^{0}\pi^{0}\pi^{0}$, $\pi^{+}\pi^{-}\pi^{0}$, and
$\pi^{+}\pi^{-}\gamma$, the energy of the radiative photon is much larger than
that of the other photons. Therefore, the most energetic photon is taken as
the radiative photon.

In the case of $J/\psi\to\gamma\eta,~{}\eta\to\gamma\gamma$, a four-constraint
(4C) kinematic fit imposing energy-momentum conservation is performed, and the
fit quality $\chi^{2}_{\rm{4C}}$ is required to be less than 80. To suppress
the $e^{+}e^{-}\to\gamma\gamma(\gamma)$ process, the energy of the photons is
required to be greater than 0.07 GeV. The $\eta$ is reconstructed using
$\gamma\gamma$ pairs, shown in Fig. 3. As it is impossible to separate the
radiative photons from the $\eta$-decay photons, all $\gamma\gamma$
combinations are kept. MC simulations of the signal show that the mass
spectrum of the wrong $\gamma\gamma$ combinations is flat. In addition, the
background distributions have been investigated with the inclusive MC sample.
Except for events from the processes
$J/\psi\to\omega\eta,~{}\omega\to\gamma\pi^{0},~{}\eta\to\gamma\gamma$, and
$J/\psi\to\gamma
f_{0}(2100),~{}f_{0}(2100)\to\eta\eta,~{}\eta\to\gamma\gamma$, which form a
small peak in the signal region, the distribution of the other background
contributions is smooth.

For the decay $J/\psi\to\gamma\eta$, $\eta\to\pi^{0}\pi^{0}\pi^{0}$, a seven-
constraint (7C) kinematic fit imposing energy-momentum conservation and
constraints on the three $\pi^{0}$ masses is performed. The fit
$\chi^{2}_{\rm{7C}}$ is required to be less than 100. The three-$\pi^{0}$
combination with the least $\chi^{2}_{\rm{7C}}$ is used to reconstruct the
$\eta$, as displayed in Fig. 3. A very clean $\eta$ peak is observed. Using
the inclusive MC sample of $J/\psi$ decays, the background study indicates
that only the decays $J/\psi\to\omega\eta,~{}\omega\to\gamma\pi^{0},~{}\eta\to
3\pi^{0}$, and $J/\psi\to\gamma
f_{0}(2100),~{}f_{0}(2100)\to\eta\eta,~{}\eta\to 3\pi^{0}$ may contribute to a
very small peak in the signal region, as indicated by the dashed line in Fig.
3.

The $J/\psi\to\gamma\eta$, $\eta\to\pi^{+}\pi^{-}\pi^{0}$ candidates are
selected with a five-constraint (5C) kinematic fit imposing energy-momentum
conservation and a constraint on the mass of the $\pi^{0}$, and the
$\chi^{2}_{\rm{5C}}$ is required to be less than 100. After the above
requirements, the $\pi^{+}\pi^{-}\pi^{0}$ invariant mass is illustrated in
Fig. 3, where a prominent $\eta$ peak is seen. We also perform a background
study with the inclusive MC sample, and the result indicates that no peaking
background is seen in the $\eta$ mass region.

The $J/\psi\to\gamma\eta$, $\eta\to\pi^{+}\pi^{-}\gamma$ candidates are
selected using a 4C kinematic fit, and the $\chi^{2}_{\rm{4C}}$ is required to
be less than 60. If more than two good photons are found, a 5C kinematic fit
under the $J/\psi\to\gamma\eta$, $\eta\to\pi^{+}\pi^{-}\pi^{0}$ hypothesis is
performed. After requiring that the kinematic fit probability of
$\gamma\pi^{+}\pi^{-}\gamma$ is greater than that under the hypothesis of
$\gamma\pi^{+}\pi^{-}\pi^{0}$, the mass spectrum of $\pi^{+}\pi^{-}\gamma$ is
shown in Fig. 3. The MC simulation shows that the background events from
$\eta\to\pi^{+}\pi^{-}\pi^{0}$ may contribute to a broad bump on the left side
of the $\eta$ peak, while no peaking background events are found in the signal
region.

(a)(b)
(c)(d)

Figure 3: Unbinned maximum likelihood fit to the mass spectra of the $\eta$
decay modes. The black dots with error bars represent data, the red solid
lines represent the fit results, the light blue dashed lines represent the
peaking backgrounds, and the dark blue dotted lines represent the other
backgrounds. The bottom panel shows the corresponding pull distribution. (a)
The $\eta\to\gamma\gamma$ channel. (b) The $\eta\to\pi^{0}\pi^{0}\pi^{0}$
channel. (c) The $\eta\to\pi^{+}\pi^{-}\pi^{0}$ channel. (d) The
$\eta\to\pi^{+}\pi^{-}\gamma$ channel.

##  V Measurement of $\boldmath\mathcal{B}(J/\psi\to\gamma\eta)$

To measure the BF of the $J/\psi\to\gamma\eta$ decay, an unbinned maximum
likelihood fit is performed to the recoil mass spectrum of the $e^{+}e^{-}$
pair. The fit includes a signal component and background contributions
estimated from exclusive and inclusive simulations.

The distribution of the signal events is described by a modified double-tailed
Crystal Ball function:

$X\\!=\\!\frac{x\\!-\\!\mu}{\sigma},~{}f(x)\\!=\\!\begin{cases}\frac{\rm{exp}\left(-\frac{1}{2}|k|^{1+\frac{1}{1+0.5|k|}}\right)}{(1-\frac{1}{n}-\frac{|k|}{n}X)^{n}},&X\\!<\\!-|k|\\\
\\\
\rm{exp}\left(-\frac{1}{2}|X|^{1+\frac{1}{1+0.5|X|}}\right),&\\!-|k|\\!\leq\\!X\\!\leq\\!|K|\\\
\\\
\frac{\left(-\frac{1}{2}|K|^{1+\frac{1}{1+0.5|K|}}\right)}{(1-\frac{1}{N}+\frac{|K|}{N}X)^{N}},&X>|K|\end{cases}$
(3)

Compared to the standard Crystal Ball function [34], it uses a modified
Gaussian function as the core region. The parameters $n,k,N$, and $K$ describe
the two tails, while $\mu$ and $\sigma$ are parameters of the modified
Gaussian function. We first fit the signal MC sample with Eq. (3), where all
the parameters are included in the fit. In the second step, a fit to data is
performed, in which the values of $n,k,N$, and $K$ are fixed to the results
obtained by fitting the signal MC sample. The parameters $\mu$ and $\sigma$
are determined by the fit to data.

Background events from the processes $e^{+}e^{-}\to\gamma\gamma$,
$e^{+}e^{-}\to e^{+}e^{-}$, $J/\psi\to e^{+}e^{-}\eta$,
$J/\psi\to\pi^{+}\pi^{-}\pi^{0}$, $J/\psi\to\omega\eta$,
$J/\psi\to\omega\pi^{0}$, and $J/\psi\to\gamma\eta^{\prime}$ are described
with shapes extracted from their corresponding exclusive MC samples. The
number of $J/\psi\to\gamma\eta^{\prime}$ events is left free in the fit, while
the numbers of events from the other six processes are fixed according to
$N=L\sigma\varepsilon$ or $N=N_{J/\psi}\mathcal{B}\varepsilon$. Here, $L$ is
the integrated luminosity of the data sample, $N_{J/\psi}$ is the number of
$J/\psi$ events, $\varepsilon$ stands for the efficiency estimated with the
corresponding MC sample, and $\sigma$ (or $\mathcal{B}$) stands for the
corresponding cross section (or BF). The BFs of the $J/\psi\to e^{+}e^{-}\eta$
and $J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ decays are obtained from Ref. [35] and
[36]. The BFs of $J/\psi\to\omega\eta$ and $J/\psi\to\omega\pi^{0}$ are
obtained from Ref. [37].

The remaining background events are described using the shape extracted from
the inclusive MC sample. The normalization of this component is determined by
the fit. The result of the fit is shown in Fig. 4.

Figure 4: Unbinned maximum likelihood fit to the $e^{+}e^{-}$ recoil mass
spectrum. In the top panel, the black dots with error bars are data. The black
solid curve gives the fit result. The red dashed line represents the
distribution of all backgrounds. The blue solid, purple dotted, pink solid,
dark blue dashed, brown solid, green dotted, light blue dashed, and yellow
dashed lines represent the backgrounds from $e^{+}e^{-}\to\gamma\gamma$,
$e^{+}e^{-}\to e^{+}e^{-}$, $J/\psi\to e^{+}e^{-}\eta$,
$J/\psi\to\gamma\eta^{\prime}$, $J/\psi\to\omega\eta$,
$J/\psi\to\omega\pi^{0}$, $J/\psi\to\pi^{+}\pi^{-}\pi^{0}$, and the other
backgrounds, respectively. The bottom panel shows the corresponding pull
distribution.

In the last step, a correction factor $f$, which accounts for the difference
in the photon conversion efficiency between data and MC simulation, is
implemented in the BF calculation. Using $e^{+}e^{-}\to\gamma\gamma$ events
collected at $\sqrt{s}=3.08$ GeV, just below the $J/\psi$ resonance, the
factor $f=1.010\pm 0.004\pm 0.005$ is estimated with

$f\equiv\frac{N^{\rm{data}}_{\rm{conv}}/N^{\rm{data}}_{\gamma\gamma}}{N^{\rm{mc}}_{\rm{conv}}/N^{\rm{mc}}_{\gamma\gamma}}\approx\frac{\varepsilon^{\rm{data}}_{\rm{conv}}}{\varepsilon^{\rm{mc}}_{\rm{conv}}},$
(4)

where $N_{\rm{conv}}$ and $N_{\gamma\gamma}$ are the observed numbers of
$e^{+}e^{-}\to\gamma\gamma$ events with and without one $\gamma\to e^{+}e^{-}$
conversion, respectively. The energy of the radiative photons in the
$e^{+}e^{-}\to\gamma\gamma$ sample and the $J/\psi\to\gamma\eta$ sample are
very close.

Both the reconstruction of the converted and non-converted photons are the
same as described above. The systematic uncertainty of $f$ is conservatively
estimated to be $(f-1)/2$. The yields of $J/\psi\to\gamma\eta$, $\gamma\to
e^{+}e^{-}$ events is $N_{\gamma\eta}^{\rm{obs}}=87887\pm 373$. The detection
efficiency obtained from MC simulation is $\varepsilon_{\gamma\eta}=(8.090\pm
0.009)\times 10^{-3}$, and $\mathcal{B}(J/\psi\to\gamma\eta)$ is determined to
be $(1.067\pm 0.005)\\!\times\\!{10^{-3}}$, where the uncertainty is
statistical.

##  VI Measurement of $\boldmath\mathcal{B}(\eta\to X)$

The signal yields of the exclusive channels are obtained by performing
unbinned maximum likelihood fits to the mass spectra of $\gamma\gamma$,
$\pi^{0}\pi^{0}\pi^{0}$, $\pi^{+}\pi^{-}\pi^{0}$, and $\pi^{+}\pi^{-}\gamma$
candidates, respectively. In the fits, the signal component is modeled by the
MC-simulated shape convolved with a Gaussian function to account for the
difference in the mass resolution between MC simulation and data. The
parameters of the Gaussian function are free in the fit. When simulating the
signal shape of the $\eta\to\gamma\gamma$ channel, only the right
$\gamma\gamma$ combination of the MC events is used. The wrong $\gamma\gamma$
combinations are removed according to the simulation information.

The backgrounds of the $\eta\to\gamma\gamma$ and
$\eta\to\pi^{0}\pi^{0}\pi^{0}$ channels are modeled with two components: (i) a
second-order Chebyshev polynomial function which describes the non-peaking
background; (ii) a peaking background shape which is determined with MC
simulation, and the number of the peaking background events is fixed according
to the corresponding BF from PDG [10]. The background of the
$\eta\to\pi^{+}\pi^{-}\pi^{0}$ channel is modeled by the shape obtained from
the inclusive MC sample. The background of the $\eta\to\pi^{+}\pi^{-}\gamma$
channel is modeled by the sum of a second-order Chebyshev polynomial function
and the shape obtained from the $J/\psi\to\gamma\eta$,
$\eta\to\pi^{+}\pi^{-}\pi^{0}$ MC sample. Here the magnitudes of different
components are left free in the fit. The fit results of the four channels are
shown in Fig. 3.

The signal yields obtained from the fits, the detection efficiencies estimated
with MC simulations, and the BFs of the four dominant $\eta$ decay modes are
listed in Table 2. Note that the BFs from CLEO and the PDG are all relative
measurements.

Table 2: Summary of BFs and comparison with previous results. The first error is statistical and the second systematic. |  |  | $\mathcal{B}(\eta\to X)~{}(\%)$
---|---|---|---
$X$ |  $N^{\rm{obs}}_{\eta\to X}$($\times 10^{5}$) |  $\varepsilon_{\gamma\eta}$(%) | This Work | CLEO | PDG
$\gamma\gamma$ | 20.78$\pm$0.02 | $48.46\pm 0.01$ | $39.86\pm 0.04\pm 0.99$ | 38.45$\pm$0.40$\pm$0.36 | 39.41$\pm$0.20
$\pi^{0}\pi^{0}\pi^{0}$ | 2.831$\pm$0.006 | $8.230\pm 0.004$ | $31.96\pm 0.07\pm 0.84$ | 34.03$\pm$0.56$\pm$0.49 | 32.68$\pm$0.23
$\pi^{+}\pi^{-}\pi^{0}$ | 6.131$\pm$0.008 | $24.73\pm 0.01$ | $23.04\pm 0.03\pm 0.54$ | 22.60$\pm$0.35$\pm$0.29 | 22.92$\pm$0.28
$\pi^{+}\pi^{-}\gamma$ | 2.018$\pm$0.005 | $42.86\pm 0.01$ | $4.38\pm 0.02\pm 0.10$ | 3.96$\pm$0.14$\pm$0.14 | 4.22$\pm$0.08

##  VII Systematic Uncertainties

The systematic uncertainties of $\mathcal{B}(J/\psi\to\gamma\eta)$ have been
evaluated for the fit procedure, the reconstruction efficiency of the
converted photon, the efficiencies of the background suppression criteria, and
the number of $J/\psi$ candidates. The systematic uncertainties of
$\mathcal{B}(\eta\to X)$ have been evaluated for the fit procedure, the photon
detection efficiency, the tracking efficiencies of charged pions, the
kinematic fit efficiency, the efficiencies of the background suppression
criteria, and the BFs of the decays $\pi^{0}\to\gamma\gamma$ and
$J/\psi\to\gamma\eta$. As the number of $J/\psi$ canceles when calculating
$\mathcal{B}(\eta\to X)$, the systematic uncertainty of $\mathcal{B}(\eta\to
X)$ does not contain the uncertainty of the number of $J/\psi$. The
reconstruction efficiencies cannot be cancelled as the radiative photons are
reconstructed in different ways.

The fit uncertainty comes from three sources: the fit range, the signal shape,
and the background shape. The uncertainty arising from the fit range is
estimated by varying the range. The change of the efficiency caused by the
change of the fit range is considered. The change in the BF is taken as the
systematic uncertainty. To estimate the uncertainty arising from the signal
shape, we use the Bukin function [38] instead of the MC-simulated shape to
describe the signal component. The Bukin function is an asymmetric function
with five parameters: $\mu,~{}\sigma,~{}\xi,~{}\rho_{1}$, and $\rho_{2}$. The
parameters $\mu$ and $\sigma$ are the position and width of the peak,
respectively, $\xi$ describes the asymmetry of the peak, and $\rho_{1}$,
$\rho_{2}$ describe the tails of the peak. In the fit to the exclusive
channels, all the parameters of the Bukin function are free. However, in the
fit to the inclusive channel, we first fit the signal MC sample with all the
parameters of the Bukin function free. Then we fit the data with values of
$\rho_{1}$, $\rho_{2}$ fixed to the results of the fit to the signal MC
sample, and $\mu,~{}\sigma,~{}\xi$ are free. The differences between the
nominal results and the results from the alternative method are taken as the
systematic uncertainties.

The systematic effect arising from the background shapes is estimated with
different methods for different channels. For the inclusive channel, two
uncertainty sources are considered: the peaking background and the other
background shape uncertainty. For the peaking background, the number of events
is fixed during the fit. We vary the number by one standard deviation and the
difference with respect to the nominal result is taken as the uncertainty. For
the uncertainty of the remaining background, a 2nd-order Chebyshev function is
added to the fit, and the induced change of the BF is taken as the
uncertainty. The square root of the summed squares of the two uncertainties is
taken as the background uncertainty. For the $\eta\to\pi^{+}\pi^{-}\pi^{0}$
channel, we use a 2nd-order Chebyshev function instead of the MC-simulated
shape in the fit and take the change of the BF as the uncertainty. For the
$\eta\to\gamma\gamma$, $\eta\to\pi^{0}\pi^{0}\pi^{0}$, and
$\eta\to\pi^{+}\pi^{-}\gamma$ channels, the order of the Chebyshev polynomial
functions used in the fit is changed and the induced change of the BF is
chosen as the uncertainty. For the $\eta\to\gamma\gamma$ and
$\eta\to\pi^{0}\pi^{0}\pi^{0}$ channels, the number of the peaking background
events, which is fixed during the fit, is varied by one standard deviation,
and the induced change of the BF is taken as one source of the uncertainty.
The uncertainties from different sources are added in quadrature.

As the reconstruction efficiency of the converted photon is corrected with the
factor $f$, the uncertainty of $f$ is taken as the associated systematic
uncertainty.

For photons directly detected by the EMC, the uncertainty of the detection
efficiency has been studied using a control sample of
$e^{+}e^{-}\to\gamma\mu^{+}\mu^{-}$ events. The four-momentum of the initial-
state-radiation photon is predicted using only the four-momentum of the
$\mu^{+}\mu^{-}$ pair. The photon detection efficiency is defined as the
fraction of predicted photons with four-momentum matching that of the actual
photons in the EMC. The systematic uncertainty is defined as the relative
difference in efficiency between data and MC simulation. It is found that the
photon detection efficiency of data is consistent with the MC simulation
within 0.5%. The effect of the discrepancy between data and the MC simulation
is estimated by using a reweighing technique. The weighted relative
uncertainties for the $\eta\to\gamma\gamma$, $\eta\to\pi^{0}\pi^{0}\pi^{0}$,
$\eta\to\pi^{+}\pi^{-}\pi^{0}$, and $\eta\to\pi^{+}\pi^{-}\gamma$ channels are
determined to be $0.21$%, $0.18$%, $0.18$%, and $0.17$% per photon,
respectively.

The tracking efficiency uncertainty of charged pions has been studied with the
control sample $J/\psi\to\pi^{+}\pi^{-}\pi^{0}$. The momentum of the $\pi^{+}$
is predicted using the four-momentum of $\pi^{-}\pi^{0}$, and the tracking
efficiency of the $\pi^{+}$ is defined as the fraction of the number of events
in which $\pi^{+}\pi^{-}\pi^{0}$ are reconstructed and the number of events in
which $\pi^{-}\pi^{0}$ are reconstructed. The systematic uncertainty is
defined as the relative difference in efficiency between data and MC
simulation. The weighted average uncertainties for the tracks are obtained
using bins of transverse momentum. The weighted average relative uncertainties
for the $\eta\to\pi^{+}\pi^{-}\pi^{0}$ and $\eta\to\pi^{+}\pi^{-}\gamma$
channels are 0.13% and 0.14% per track, respectively.

The uncertainty associated with the kinematic fit arises from the
inconsistency of the $\chi^{2}$ distribution between data and the MC
simulation. The reconstructed energy and angle of the photons, the helix
parameters of the charged tracks, and their errors of the MC simulation are
corrected to make their distributions more consistent with data. This makes
the $\chi^{2}$ distributions of data and the MC simulation more consistent as
well. The corrected MC simulation is used for the nominal results. The
difference of the kinematic fit efficiencies before and after the correction
is taken as the uncertainty. The relative uncertainties for the four exclusive
channels are $0.11$%, $0.29$%, $0.09$%, and $0.22$%, respectively.

There are also efficiency uncertainties caused by the selection criteria used
to suppress the background. For selection criteria that cause only a small
efficiency loss, the corresponding uncertainties are conservatively estimated
as half of the efficiency loss. Such selection criteria include
$\chi^{2}_{\rm{4C}}<\chi^{2}_{\rm{5C}}$ for the $\eta\to\pi^{+}\pi^{-}\gamma$
channel, and the photon energy requirement $E_{\gamma}>0.07$ GeV for the
$\eta\to\gamma\gamma$ channel. For the selection that requires at least one
photon detected in addition to the radiative photon, which is applied on the
inclusive channel, the change of the BF obtained with or without the selection
is taken as the corresponding uncertainty. For other selection criteria, we
vary the value of the selection criteria, and take the change of the BF as the
corresponding uncertainty. Such selection criteria including $E_{\gamma}<1.4$
GeV, $-0.998<\\!\cos\theta_{\gamma\gamma}<0$,
$|\\!\cos\theta_{\rm{miss}}|<0.98$, and $2P_{\rm{trk}}-P_{\gamma_{ee}}<0.8$
GeV, which are all applied on the inclusive channel.

The uncertainty of the number of $J/\psi$ events is described in Sec. III.
Finally, the BF uncertainty of the $\pi^{0}\to\gamma\gamma$ decay is taken
from the PDG [10].

The systematic uncertainties are summarized in Table 3 and Table 4. In Table
3, the uncertainty of the number of $J/\psi$ events only contributed to the
total uncertainty in the measurement of $\mathcal{B}(J/\psi\to\gamma\eta)$, as
it can be canceled when calculating $\mathcal{B}(\eta\to X)$. The total
systematic uncertainty is given by the quadratic sum of the individual
contributions.

Table 3: Relative systematic uncertainties for the inclusive channel. Source | Relative Uncertainty (%)
---|---
Fit range | $0.87$
Signal shape | 0.57
Background shape | $1.09$
Converted photon | $0.64$
At least one good photon | 0.61
$E_{\gamma}<1.4$ GeV | 0.81
$-0.998<\\!\cos\theta_{\gamma\gamma}<0$ | 0.77
$|\\!\cos\theta_{\rm{miss}}|<0.98$ | 0.61
$(2P_{\rm{trk}}-P_{\gamma})<0.8$ GeV | 0.41
Number of $J/\psi$ events | 0.44
Total (for $\mathcal{B}(J/\psi\to\gamma\eta)$) | 2.24
Total (for $\mathcal{B}(\eta\to X)$) | $2.20$
Table 4: Relative systematic uncertainties for the exclusive channels.

| Relative Uncertainty (%)
---|---
Sourcewhite1 | $\gamma\gamma$ | $\pi^{0}\pi^{0}\pi^{0}$ | $\pi^{+}\pi^{-}\pi^{0}$ | $\pi^{+}\pi^{-}\gamma$
$\mathcal{B}(J/\psi\to\gamma\eta)$* | $2.24$ | $2.24$ | $2.24$ | $2.24$
Fit range | $0.08$ | $0.11$ | $0.08$ | $0.32$
Signal shape | 0.62 | 0.19 | 0.37 | 0.66
Background shape | 0.39 | 0.28 | 0.04 | 0.13
Photon detection efficiency | $0.64$ | $1.26$ | $0.53$ | $0.34$
Tracking efficiency | – | – | $0.26$ | $0.28$
Kinematic fit efficiency | $0.11$ | $0.29$ | $0.09$ | $0.22$
Referenced BF | – |  white00.059 |  white00.034 | –
Other | 0.45 | – | – | 0.07
Total | $2.49$ | $2.62$ | $2.35$ | $2.39$

  * *

Contains both the systematic and statistical uncertainties.

##  VIII Summary

Based on $(1.0087\pm 0.0044)\times 10^{10}$ $J/\psi$ events collected by
BESIII at BEPCII, the BF of the decay $J/\psi\to\gamma\eta$ is measured with
high precision and the absolute BFs of four dominant $\eta$ decays are
measured for the first time. The measured BF of $J/\psi\to\gamma\eta$ is
$(1.067\pm 0.005\pm 0.023)\times 10^{-3}$, which is in agreement with the
world average value, $(1.108\pm 0.027)\times 10^{-3}$ [10], within two
standard deviations, but with improved precision.

The measured BFs of $\eta$ decays are summarized in Table 2. The value of
$\mathcal{B}(\eta\to\pi^{+}\pi^{-}\gamma)$ is consistent with the world
average values [10] within two standard deviations, and the measured BFs of
the other $\eta$ decays are within one standard deviation. Compared with the
BFs measured by CLEO [11], only the
$\mathcal{B}(\eta\to\pi^{+}\pi^{-}\pi^{0})$ is in agreement within one
standard deviation. The ratios of the measured BFs of $\eta$ are summarized in
Table 5, which are in agreement with CLEO’s result within two standard
deviations. The sum of the four BFs, which provides a first constraint on the
unknown decay modes of $\eta$, is $(99.24\pm 0.09\pm 2.31)$%, where the first
error is statistical and the second systematic.

Table 5: Comparison of ratios of BFs. The first error is statistical and the second systematic. |  $\mathcal{B}(\eta\to X)$/$\mathcal{B}(\eta\to\gamma\gamma)$
---|---
$X$ | This Work | CLEO
$\pi^{0}\pi^{0}\pi^{0}$white1 | $0.802\pm 0.002\pm 0.014$ | $0.884\pm 0.022\pm 0.019$
$\pi^{+}\pi^{-}\pi^{0}$ | $0.578\pm 0.001\pm 0.008$ | $0.587\pm 0.011\pm 0.009$
$\pi^{+}\pi^{-}\gamma$ | $0.110\pm 0.001\pm 0.002$ | $0.103\pm 0.004\pm 0.004$

###### Acknowledgements.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key Research and Development Program of China under Contracts Nos.
2020YFA0406300, 2020YFA0406400; National Natural Science Foundation of China
(NSFC) under Contracts Nos. 11625523, 11635010, 11735014, 11822506, 11835012,
11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009,
12035013, 12061131003; the Chinese Academy of Sciences (CAS) Large-Scale
Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of
the NSFC and CAS under Contracts Nos. U1732263, U1832207; CAS Key Research
Program of Frontier Sciences under Contract No. QYZDJ-SSW-SLH040; 100 Talents
Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and
Cosmology; ERC under Contract No. 758462; European Union Horizon 2020 research
and innovation programme under Contract No. Marie Sklodowska-Curie grant
agreement No 894790; German Research Foundation DFG under Contracts Nos.
443159800, Collaborative Research Center CRC 1044, FOR 2359, GRK 214; Istituto
Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; National Science and Technology fund; Olle
Engkvist Foundation under Contract No. 200-0605; STFC (United Kingdom); The
Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157;
The Royal Society, UK under Contracts Nos. DH140054, DH160214; The Swedish
Research Council; U. S. Department of Energy under Contracts Nos. DE-
FG02-05ER41374, DE-SC-0012069

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