# Observation of $\eta_{c}$(1S, 2S) and $\chi_{cJ}$ decays to
2($\pi^{+}$$\pi^{-}$)$\eta$ via $\psi$(3686) radiative transitions

M. Ablikim1, M. N. Achasov13,b, P. Adlarson75, R. Aliberti36, A.
Amoroso74A,74C, M. R. An40, Q. An71,58, Y. Bai57, O. Bakina37, I.
Balossino30A, Y. Ban47,g, V. Batozskaya1,45, K. Begzsuren33, N. Berger36, M.
Berlowski45, M. Bertani29A, D. Bettoni30A, F. Bianchi74A,74C, E.
Bianco74A,74C, J. Bloms68, A. Bortone74A,74C, I. Boyko37, R. A. Briere5, A.
Brueggemann68, H. Cai76, X. Cai1,58, A. Calcaterra29A, G. F. Cao1,63, N.
Cao1,63, S. A. Cetin62A, J. F. Chang1,58, T. T. Chang77, W. L. Chang1,63, G.
R. Che44, G. Chelkov37,a, C. Chen44, Chao Chen55, G. Chen1, H. S. Chen1,63, M.
L. Chen1,58,63, S. J. Chen43, S. M. Chen61, T. Chen1,63, X. R. Chen32,63, X.
T. Chen1,63, Y. B. Chen1,58, Y. Q. Chen35, Z. J. Chen26,h, W. S. Cheng74C, S.
K. Choi10A, X. Chu44, G. Cibinetto30A, S. C. Coen4, F. Cossio74C, J. J. Cui50,
H. L. Dai1,58, J. P. Dai79, A. Dbeyssi19, R. E. de Boer4, D. Dedovich37, Z. Y.
Deng1, A. Denig36, I. Denysenko37, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding47,g, Y. Ding35, Y. Ding41, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, S. X. Du81, Z. H. Duan43, P. Egorov37,a, Y. L.
Fan76, J. Fang1,58, S. S. Fang1,63, W. X. Fang1, Y. Fang1, R. Farinelli30A, L.
Fava74B,74C, F. Feldbauer4, G. Felici29A, C. Q. Feng71,58, J. H. Feng59, K
Fischer69, M. Fritsch4, C. Fritzsch68, C. D. Fu1, J. L. Fu63, Y. W. Fu1, H.
Gao63, Y. N. Gao47,g, Yang Gao71,58, S. Garbolino74C, I. Garzia30A,30B, P. T.
Ge76, Z. W. Ge43, C. Geng59, E. M. Gersabeck67, A Gilman69, K. Goetzen14, L.
Gong41, W. X. Gong1,58, W. Gradl36, S. Gramigna30A,30B, M. Greco74A,74C, M. H.
Gu1,58, Y. T. Gu16, C. Y Guan1,63, Z. L. Guan23, A. Q. Guo32,63, L. B. Guo42,
R. P. Guo49, Y. P. Guo12,f, A. Guskov37,a, X. T. H.1,63, T. T. Han50, W. Y.
Han40, X. Q. Hao20, F. A. Harris65, K. K. He55, K. L. He1,63, F. H H..
Heinsius4, C. H. Heinz36, Y. K. Heng1,58,63, C. Herold60, T. Holtmann4, P. C.
Hong12,f, G. Y. Hou1,63, Y. R. Hou63, Z. L. Hou1, H. M. Hu1,63, J. F. Hu56,i,
T. Hu1,58,63, Y. Hu1, G. S. Huang71,58, K. X. Huang59, L. Q. Huang32,63, X. T.
Huang50, Y. P. Huang1, T. Hussain73, N Hüsken28,36, W. Imoehl28, M.
Irshad71,58, J. Jackson28, S. Jaeger4, S. Janchiv33, J. H. Jeong10A, Q. Ji1,
Q. P. Ji20, X. B. Ji1,63, X. L. Ji1,58, Y. Y. Ji50, Z. K. Jia71,58, P. C.
Jiang47,g, S. S. Jiang40, T. J. Jiang17, X. S. Jiang1,58,63, Y. Jiang63, J. B.
Jiao50, Z. Jiao24, S. Jin43, Y. Jin66, M. Q. Jing1,63, T. Johansson75, X. K.1,
S. Kabana34, N. Kalantar-Nayestanaki64, X. L. Kang9, X. S. Kang41, R.
Kappert64, M. Kavatsyuk64, B. C. Ke81, A. Khoukaz68, R. Kiuchi1, R. Kliemt14,
L. Koch38, O. B. Kolcu62A, B. Kopf4, M. K. Kuessner4, A. Kupsc45,75, W.
Kühn38, J. J. Lane67, J. S. Lange38, P. Larin19, A. Lavania27, L.
Lavezzi74A,74C, T. T. Lei71,k, Z. H. Lei71,58, H. Leithoff36, M. Lellmann36,
T. Lenz36, C. Li44, C. Li48, C. H. Li40, Cheng Li71,58, D. M. Li81, F. Li1,58,
G. Li1, H. Li71,58, H. B. Li1,63, H. J. Li20, H. N. Li56,i, Hui Li44, J. R.
Li61, J. S. Li59, J. W. Li50, Ke Li1, L. J Li1,63, L. K. Li1, Lei Li3, M. H.
Li44, P. R. Li39,j,k, S. X. Li12, T. Li50, W. D. Li1,63, W. G. Li1, X. H.
Li71,58, X. L. Li50, Xiaoyu Li1,63, Y. G. Li47,g, Z. J. Li59, Z. X. Li16, Z.
Y. Li59, C. Liang43, H. Liang71,58, H. Liang1,63, H. Liang35, Y. F. Liang54,
Y. T. Liang32,63, G. R. Liao15, L. Z. Liao50, J. Libby27, A. Limphirat60, D.
X. Lin32,63, T. Lin1, B. J. Liu1, B. X. Liu76, C. Liu35, C. X. Liu1, D.
Liu19,71, F. H. Liu53, Fang Liu1, Feng Liu6, G. M. Liu56,i, H. Liu39,j,k, H.
B. Liu16, H. M. Liu1,63, Huanhuan Liu1, Huihui Liu22, J. B. Liu71,58, J. L.
Liu72, J. Y. Liu1,63, K. Liu1, K. Y. Liu41, Ke Liu23, L. Liu71,58, L. C.
Liu44, Lu Liu44, M. H. Liu12,f, P. L. Liu1, Q. Liu63, S. B. Liu71,58, T.
Liu12,f, W. K. Liu44, W. M. Liu71,58, X. Liu39,j,k, Y. Liu39,j,k, Y. B. Liu44,
Z. A. Liu1,58,63, Z. Q. Liu50, X. C. Lou1,58,63, F. X. Lu59, H. J. Lu24, J. G.
Lu1,58, X. L. Lu1, Y. Lu7, Y. P. Lu1,58, Z. H. Lu1,63, C. L. Luo42, M. X.
Luo80, T. Luo12,f, X. L. Luo1,58, X. R. Lyu63, Y. F. Lyu44, F. C. Ma41, H. L.
Ma1, J. L. Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M. Ma1, R. Q. Ma1,63, R. T.
Ma63, X. Y. Ma1,58, Y. Ma47,g, Y. M. Ma32, F. E. Maas19, M. Maggiora74A,74C,
S. Maldaner4, S. Malde69, A. Mangoni29B, Y. J. Mao47,g, Z. P. Mao1, S.
Marcello74A,74C, Z. X. Meng66, J. G. Messchendorp14,64, G. Mezzadri30A, H.
Miao1,63, T. J. Min43, R. E. Mitchell28, X. H. Mo1,58,63, N. Yu. Muchnoi13,b,
Y. Nefedov37, F. Nerling19,d, I. B. Nikolaev13,b, Z. Ning1,58, S. Nisar11,l,
Y. Niu 50, S. L. Olsen63, Q. Ouyang1,58,63, S. Pacetti29B,29C, X. Pan55, Y.
Pan57, A. Pathak35, P. Patteri29A, Y. P. Pei71,58, M. Pelizaeus4, H. P.
Peng71,58, K. Peters14,d, J. L. Ping42, R. G. Ping1,63, S. Plura36, S.
Pogodin37, V. Prasad34, F. Z. Qi1, H. Qi71,58, H. R. Qi61, M. Qi43, T. Y.
Qi12,f, S. Qian1,58, W. B. Qian63, C. F. Qiao63, J. J. Qin72, L. Q. Qin15, X.
P. Qin12,f, X. S. Qin50, Z. H. Qin1,58, J. F. Qiu1, S. Q. Qu61, C. F.
Redmer36, K. J. Ren40, A. Rivetti74C, V. Rodin64, M. Rolo74C, G. Rong1,63, Ch.
Rosner19, S. N. Ruan44, N. Salone45, A. Sarantsev37,c, Y. Schelhaas36, K.
Schoenning75, M. Scodeggio30A,30B, K. Y. Shan12,f, W. Shan25, 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, X. Shi1,58, J. J. Song20, T. Z.
Song59, W. M. Song35,1, Y. J. Song12, Y. X. Song47,g, S. Sosio74A,74C, S.
Spataro74A,74C, F. Stieler36, Y. J. Su63, G. B. Sun76, G. X. Sun1, H. Sun63,
H. K. Sun1, J. F. Sun20, K. Sun61, L. Sun76, S. S. Sun1,63, T. Sun1,63, W. Y.
Sun35, Y. Sun9, Y. J. Sun71,58, Y. Z. Sun1, Z. T. Sun50, Y. X. Tan71,58, C. J.
Tang54, G. Y. Tang1, J. Tang59, Y. A. Tang76, L. Y Tao72, Q. T. Tao26,h, M.
Tat69, J. X. Teng71,58, V. Thoren75, W. H. Tian59, W. H. Tian52, Y. Tian32,63,
Z. F. Tian76, I. Uman62B, B. Wang1, B. L. Wang63, Bo Wang71,58, C. W. Wang43,
D. Y. Wang47,g, F. Wang72, H. J. Wang39,j,k, H. P. Wang1,63, K. Wang1,58, L.
L. Wang1, M. Wang50, Meng Wang1,63, S. Wang12,f, S. Wang39,j,k, T. Wang12,f,
T. J. Wang44, W. Wang59, W. Wang72, W. H. Wang76, W. P. Wang71,58, X.
Wang47,g, X. F. Wang39,j,k, X. J. Wang40, X. L. Wang12,f, Y. Wang61, Y. D.
Wang46, Y. F. Wang1,58,63, Y. H. Wang48, Y. N. Wang46, Y. Q. Wang1, Yaqian
Wang18,1, Yi Wang61, Z. Wang1,58, Z. L. Wang72, Z. Y. Wang1,63, Ziyi Wang63,
D. Wei70, D. H. Wei15, F. Weidner68, S. P. Wen1, C. W. Wenzel4, U. W.
Wiedner4, G. Wilkinson69, M. Wolke75, L. Wollenberg4, C. Wu40, J. F. Wu1,63,
L. H. Wu1, L. J. Wu1,63, X. Wu12,f, X. H. Wu35, Y. Wu71, Y. J. Wu32, Z.
Wu1,58, L. Xia71,58, X. M. Xian40, T. Xiang47,g, D. Xiao39,j,k, G. Y. Xiao43,
H. Xiao12,f, S. Y. Xiao1, Y. L. Xiao12,f, Z. J. Xiao42, C. Xie43, X. H.
Xie47,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. Xu17, Q. N. Xu31, W.
Xu1,63, W. L. Xu66, X. P. Xu55, Y. C. Xu78, Z. P. Xu43, Z. S. Xu63, F.
Yan12,f, L. Yan12,f, W. B. Yan71,58, W. C. Yan81, X. Q Yan1,63, H. J.
Yang51,e, H. L. Yang35, H. X. Yang1, Tao Yang1, Y. Yang12,f, Y. F. Yang44, Y.
X. Yang1,63, Yifan Yang1,63, Z. W. Yang39,j,k, M. Ye1,58, M. H. Ye8, J. H.
Yin1, Z. Y. You59, B. X. Yu1,58,63, C. X. Yu44, G. Yu1,63, T. Yu72, X. D.
Yu47,g, C. Z. Yuan1,63, L. Yuan2, S. C. Yuan1, X. Q. Yuan1, Y. Yuan1,63, Z. Y.
Yuan59, C. X. Yue40, A. A. Zafar73, F. R. Zeng50, X. Zeng12,f, Y. Zeng26,h, Y.
J. Zeng1,63, X. Y. Zhai35, Y. H. Zhan59, A. Q. Zhang1,63, B. L. Zhang1,63, B.
X. Zhang1, D. H. Zhang44, G. Y. Zhang20, H. Zhang71, H. H. Zhang35, H. H.
Zhang59, H. Q. Zhang1,58,63, H. Y. Zhang1,58, J. J. Zhang52, J. L. Zhang21, J.
Q. Zhang42, J. W. Zhang1,58,63, J. X. Zhang39,j,k, J. Y. Zhang1, J. Z.
Zhang1,63, Jianyu Zhang63, Jiawei Zhang1,63, L. M. Zhang61, L. Q. Zhang59, Lei
Zhang43, P. Zhang1, Q. Y. Zhang40,81, Shuihan Zhang1,63, Shulei Zhang26,h, X.
D. Zhang46, X. M. Zhang1, X. Y. Zhang50, X. Y. Zhang55, Y. Zhang69, Y.
Zhang72, Y. T. Zhang81, Y. H. Zhang1,58, Yan Zhang71,58, Yao Zhang1, Z. H.
Zhang1, Z. L. Zhang35, Z. Y. Zhang44, Z. Y. Zhang76, G. Zhao1, J. Zhao40, J.
Y. Zhao1,63, J. Z. Zhao1,58, Lei Zhao71,58, Ling Zhao1, M. G. Zhao44, S. J.
Zhao81, Y. B. Zhao1,58, Y. X. Zhao32,63, Z. G. Zhao71,58, A. Zhemchugov37,a,
B. Zheng72, J. P. Zheng1,58, W. J. Zheng1,63, Y. H. Zheng63, B. Zhong42, X.
Zhong59, H. Zhou50, L. P. Zhou1,63, X. Zhou76, X. K. Zhou6, X. R. Zhou71,58,
X. Y. Zhou40, Y. Z. Zhou12,f, J. Zhu44, K. Zhu1, K. J. Zhu1,58,63, L. Zhu35,
L. X. Zhu63, S. H. Zhu70, S. Q. Zhu43, 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 Deceased
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
d Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
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 MOE 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 75270,
Pakistan
n Also at Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland
o Also at Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
p Also at School of Physics, Beihang University, Beijing 100191 , China

###### Abstract

Based on $2.7\times 10^{9}~{}\psi(3686)$ decays collected with the BESIII
detector, the radiative decay $\psi(3686)\to\gamma 2(\pi^{+}\pi^{-})\eta$ is
investigated to measure properties of S- and P-wave charmonium states. The
branching fraction of the decay $\eta_{c}(1S)\to 2(\pi^{+}\pi^{-})\eta$, which
is found to have a strong dependence on the interference pattern between
$\eta_{c}(1S)$ and non-$\eta_{c}(1S)$ processes, is measured in both
destructive and constructive interference scenarios for the first time. The
mass and width of the $\eta_{c}(1S)$ are measured to be $M=(2984.14\pm 0.13\pm
0.38)$ MeV/$c^{2}$ and $\Gamma=(28.82\pm 0.11\pm 0.82)$ MeV, respectively.
Clear signals for the decays of the $\chi_{cJ}(J=0,1,2)$ and the
$\eta_{c}(2S)$ to $2(\pi^{+}\pi^{-})\eta$ are also observed for the first
time, and the corresponding branching fractions are measured. The ratio of the
branching fractions between the $\eta_{c}(2S)$ and $\eta_{c}(1S)$ decays is
significantly lower than the theoretical prediction, which might suggest
different dynamics in their decays.

Our understanding of charmonium states with masses below the open-charm
production threshold is far from satisfactory. This is especially true for the
spin singlet states Ref_a , which include the S-wave ground state $\eta_{c}$
($\eta_{c}\equiv\eta_{c}(1S)$) and its first radial excitation $\eta_{c}(2S)$.
The $\eta_{c}(2S)$ state was first observed by the Belle experiment Ref_b  in
$B$ meson decay and later confirmed by the BaBar Ref_c  and CLEO Ref_d
experiments. BESIII was the first to observe the radiative transition
$\psi(3686)\to\gamma\eta_{c}(2S)$ Ref_g . The sum of the measured branching
fractions (BFs) of the $\eta_{c}(2S)$ is currently only about 6$\%$. Many
decay modes of the $\eta_{c}$ also remain unknown and some measured BFs have
large uncertainties.

The expected ratio of BFs for decays of the $\eta_{c}$ and $\eta_{c}(2S)$ into
the same hadronic final state was initially anticipated to be similar to the
ratio (12$\%$) observed for their spin-triplet partners $J/\psi$ and
$\psi(3686)$ Ref_09 ; Ref_06 . However, a more in-depth analysis suggested
that this ratio should actually be close to unity, as the dominant decay
dynamics of the $\eta_{c}$ and $\eta_{c}(2S)$ involve the annihilation of
charmed quark pairs into two gluons Ref_10 . Experimental measurements have
revealed inconsistencies with both of these theoretical predictions Ref_11 .
Further measurements of additional decay modes of the $\eta_{c}$ and
$\eta_{c}(2S)$, in addition to testing these predictions, may provide insight
into the internal structure of these states, for example by testing for the
presence of non-charmonium components in their wave functions.

Previous measurements of the $\eta_{c}$ mass and width, obtained through the
magnetic dipole (M1) transition $\psi(3686)\to\gamma\eta_{c}$, $p\bar{p}$
collisions, or $\gamma\gamma$ collisions, have shown significant discrepancies
between different experiments Ref_02 . However, interference between
$\eta_{c}$ decays and non-resonant processes could have a large impact on
these measurements as well as on $\eta_{c}$ BFs. The CLEO Ref_04 , BESIII
Ref_05 , and KEDR Ref_03  collaborations have studied $\eta_{c}$ decays using
the M1 transition $\psi(3686)\to\gamma\eta_{c}$, which is considered to be a
golden channel for $\eta_{c}$ production due to its low background level and
the few unknown background channels. BESIII successfully described the
$\eta_{c}$ line shape by using the energy dependence of the hindered-M1
transition matrix element and by fully considering interference with non-
resonant $\psi(3686)$ radiative decays, leading to a significant improvement
in the precision of the $\eta_{c}$ mass and width measurements Ref_05 .
However, further improvements are still beneficial since, for example, the
determination of the hyperfine mass splitting of S-wave charmonium can test
theoretical calculations Ref_hf1s1 ; Ref_hf1s2 , and the width directly
relates to the theory of $\eta_{c}$-glueball mixing cheny .

As for the $P$-wave spin-triplet states $\chi_{c1,2}$, the dominant decay
modes are the transitions $\chi_{c1,2}\to\gamma J/\psi$. However, the sum of
all measured $\chi_{c0,1,2}$ BFs are each still far less than 1. Intensive
studies of their multi-body decays are lacking relative to their few-body
decays. The search for more new decay modes of the $\chi_{c0,1,2}$ is useful
in understanding their properties.

In this Letter, we analyze $2.7\times 10^{9}~{}\psi(3686)$ decays to study the
radiative decay $\psi(3686)\to\gamma 2(\pi^{+}\pi^{-})\eta$. We report the
first observations of $\eta_{c}(2S)$ and $\chi_{cJ}$ decays to
$2(\pi^{+}\pi^{-})\eta$. We also present improved measurements of the
$\eta_{c}$ mass and width and the BF of $\eta_{c}\to 2(\pi^{+}\pi^{-})\eta$.
We find that the BF strongly depends on the interference between the
$\eta_{c}$ and non-$\eta_{c}$ components.

The design and performance of the BESIII detector are described in Refs besiii
; besiii2 . The corresponding simulation, analysis framework, and software are
presented in Refs evtgen1 ; evtgen2 . Simulated Monte Carlo (MC) samples are
produced with geant4-based G4  software, which models the experimental
conditions, including the electron-positron collision, the decays of the
particles, and the response of the detector. Final-state radiation (FSR) from
charged final-state particles is incorporated using the photos package fsr .
The exclusive decays of $\psi(3686)\to\gamma X$ are generated following the
angular distribution of (1 + $\lambda\cos^{2}\theta_{\gamma}$), where $X$
refers to $\eta_{c}(1S,2S)$ or $\chi_{cJ}$, $\theta_{\gamma}$ is the polar
angle of the radiative photon in the rest frame of the $\psi(3686)$, and the
value of $\lambda$ is set to 1 for $\eta_{c}(1S,2S)$ and to 1, -1/3, 1/13 for
$\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$ respectively Ref_12 ; Ref_05 . The
$X\to 2(\pi^{+}\pi^{-})\eta$ decays are generated uniformly in phase space
(PHSP).

Charged tracks detected in the multilayer drift chamber (MDC) are required to
be within the polar angle ($\theta$) range of $|\cos\theta|<0.93$. Here
$\theta$ is defined with respect to the $z$-axis, which is the symmetry axis
of the MDC. For each charged track, the distance of closest approach to the
interaction point must be less than 10 cm along the $z$-axis and less than 1
cm in the transverse plane. Photon candidates are reconstructed from isolated
showers in the electromagnetic calorimeter (EMC). The deposited energy of each
shower is required to be at least 25 MeV in the barrel region
$(|\cos\theta|<0.80)$ and 50 MeV in the end cap region
$(0.86<|\cos\theta|<0.92)$. For the $\eta_{c}(2S)/\chi_{cJ}$ modes, the
deposited energy is required to be larger than 40 MeV in the end cap region
due to the low energy of the M1 transition photon. To exclude showers
originating from charged tracks, the angle between the shower direction and
the charged tracks extrapolated to the EMC must be greater than $10^{\circ}$.
To suppress electronic noise and energy depositions unrelated to the event,
the EMC cluster timing from the reconstructed event start time is required to
be within [0, 700] ns.

Candidate events are required to have four charged tracks with zero net charge
and at least three photons. All the charged tracks are taken as pions. To
reconstruct the $\eta\to\gamma\gamma$ decay, the invariant mass of a pair of
photons is required to satisfy $M_{\gamma\gamma}\in$[0.51, 0.57] GeV/$c^{2}$.
Then a one-constraint (1C) kinematic fit is performed for each combination
with $\chi^{2}_{\rm{1C}}(\gamma\gamma)<20$, where the photon pair is
constrained to the $\eta$ mass, and at least one $\eta$ candidate is required.
To suppress background and improve the mass resolution, a five-constraint (5C)
kinematic fit is performed to the total initial four momentum of the colliding
beams. The extra 1C is used to constrain the $M_{\gamma\gamma}$ to the $\eta$
nominal mass. If there is more than one combination in an event, the one with
the minimum $\chi^{2}$ value of the 5C kinematic fit ($\chi^{2}_{\rm{5C}}$) is
chosen. Furthermore, the criterion $\chi^{2}_{\rm{5C}}<15$ is required, which
is determined by optimizing the figure of merit, defined as $S/\sqrt{S+B}$,
where $S(B)$ is the number of signal (background) events in the fit region. In
addition, to suppress backgrounds from processes with missing or additional
photons, a four-constraint (4C) kinematic fit is performed for both signal and
background channels. For the $\gamma\eta_{c}$ mode, the
$\chi^{2}_{\rm{4C}}(3\gamma 2(\pi^{+}\pi^{-}))$ is required to be less than
$\chi^{2}_{\rm{4C}}(4\gamma 2(\pi^{+}\pi^{-}))$ and
$\chi^{2}_{\rm{4C}}(2\gamma 2(\pi^{+}\pi^{-}))$. For the
$\gamma\eta_{c}(2S)/\chi_{cJ}$ modes, only $\chi^{2}_{\rm{4C}}(3\gamma
2(\pi^{+}\pi^{-}))<\chi^{2}_{\rm{4C}}(4\gamma 2(\pi^{+}\pi^{-}))$ is required
since the events with one missing photon mainly come from FSR, which will be
discussed later. Backgrounds from the continuum process $e^{+}e^{-}\to
q\bar{q}$ are investigated using the off-resonance data sample taken at the
center-of-mass (CM) energy of 3.65 GeV. It is found that its shape is
generally smooth in the $\eta_{c}(1S,2S)$ and $\chi_{cJ}$ mass regions.

Background events from the $\psi(3686)\to PJ/\psi(P=\pi^{+}\pi^{-},\eta)$
process are suppressed by requiring the recoil mass of $P$ to not fall in the
$J/\psi$ signal region. In case of multiple $\pi^{+}\pi^{-}$ combinations in
an event, we select the one with $M^{\rm recoil}_{P}$ closest to the $J/\psi$
mass. Background events from the $\psi(3686)\to\pi^{0}H$ process ($H$ denotes
hadronic states) are suppressed by requiring the invariant mass of all
possible photon pair combinations to be outside the $\pi^{0}$ signal region.
For $\gamma\eta_{c}(2S)/\chi_{cJ}$ decay modes, the number of background
events from the $\psi(3686)\to\gamma\chi_{cJ}$; $\chi_{cJ}\to\gamma J/\psi$
process are suppressed by requiring the recoil invariant mass of all possible
photon pair combinations to not fall in the $J/\psi$ signal region.

The process $\psi(3686)\to 2(\pi^{+}\pi^{-})\eta$ can contaminate our signal
channel through FSR from one of the final-state pions. This effect is
particularly notable in the $\eta_{c}(2S)$ mass region, where the FSR photon
has a similar energy to the radiative photon. To separate this background from
the $\eta_{c}(2S)$ signal, we perform a modified kinematic fit where the
energy of the radiative photon is floating in the 5C fit (m5C). Although the
resolution of the $\eta_{c}(2S)$ signal itself becomes slightly worse, the FSR
background shape becomes smoother. This results in a widened gap between the
$\eta_{c}(2S)$ peak and FSR background Ref_12 . Furthermore, the background
shape is sensitive to the fraction of the number of background events between
FSR and non-FSR modes. The discrepancy between data and MC simulation is
corrected using a control sample $\psi(3686)\to\gamma\chi_{c0},\chi_{c0}\to
3(\pi^{+}\pi^{-})(\gamma_{\rm{FSR}})$ using the same method as in Ref. Ref_12
.

The process $\psi(3686)\to\pi^{0}2(\pi^{+}\pi^{-})\eta$, where a photon from
the $\pi^{0}$ decay is lost, is the dominant remaining background.
Particularly in the $\eta_{c}$ mass region, a data-driven method is employed
to determine its contribution by comparing the detection efficiencies between
the signal and background channels, and combining with the actual number of
events in data Ref_05 .

After a topology analysis with the $\psi(3686)$ inclusive MC events Ref_topo ,
we find that there are several potential peaking backgrounds for the
$\chi_{cJ}$ modes. The normalized contributions for these backgrounds using
branching fractions from the PDG Ref_02  are listed in Table 1, and we will
subtract them from the signals directly when calculating the BFs.

Table 1: Estimates of potential peaking backgrounds. Backgrounds | $\chi_{c0}$ | $\chi_{c1}$ | $\chi_{c2}$
---|---|---|---
$\chi_{c0}\to\eta^{\prime}\eta^{\prime};\eta^{\prime}\to\gamma\pi^{+}\pi^{-}$ | 287$\pm$17 | 0 | 0
$\chi_{cJ}\to\gamma J/\psi;J/\psi\to 2(\pi^{+}\pi^{-})\pi^{0}$ | 0 | 210$\pm$21 | 85$\pm$9
$\chi_{cJ}\to\gamma J/\psi;J/\psi\to\gamma 2(\pi^{+}\pi^{-})$ | 0 | 55$\pm$10 | 2.0$\pm$0.4

The signal yields for $\chi_{cJ}/\eta_{c}(2S)$ are obtained by performing an
unbinned maximum likelihood fit to the $M_{2(\pi^{+}\pi^{-})\eta}^{m5C}$
distribution as shown in Fig. 1. The fit function includes the
$\chi_{cJ}/\eta_{c}(2S)$ signals, continuum background, FSR background from
$\psi(3686)\to 2(\pi^{+}\pi^{-})\eta$, and other background contributions from
$\psi(3686)$ decays. The $\chi_{cJ}$ signal functions are described by MC-
simulated shapes convolved with Gaussian functions to account for the
difference in detector resolution between data and MC simulation, and the
parameters of the smeared Gaussian functions are free. The $\eta_{c}(2S)$
signal function is described by

$\textstyle\footnotesize{\epsilon(m)(E_{\gamma}^{3}\times{BW(m)}\times
f_{d}(E_{\gamma})\otimes DG)\otimes G(\delta m,\sigma)},$ (1)

where $\epsilon(m)$ is the energy-dependent detection efficiency determined by
MC simulation with a PHSP model, $E_{\gamma}$ is the energy of the M1
transition photon, and $BW(m)$ is the non-relativisic Breit-Wigner ($BW$)
function, in which the $\eta_{c}(2S)$ mass and width are fixed to those in the
PDG Ref_02 . The function $f_{d}(E_{\gamma})$ is used to damp the diverging
tail caused by the $E_{\gamma}^{3}$ term, and is given by
(${E_{0}^{2}}/{E_{\gamma}E_{0}+(E_{\gamma}-E_{0})^{2}}$), as introduced by the
KEDR experiment Ref_13 . Here, $E_{0}$ =
$({m^{2}_{\psi(3686)}-m^{2}_{\eta_{c}(2S)}})/{2m_{\psi(3686)}}$ denotes the
peaking energy of the transition photon; $\otimes$ is the convolution
operator; $DG$ is a double Gaussian function accounting for the detector
resolution, with two parameters obtained by comparing the difference in
$M_{2(\pi^{+}\pi^{-})\eta}$ before and after going through the detector; and
$G(\delta m,\sigma)$ is a Gaussian function accounting for the difference in
detector resolution between data and MC simulation, where $\sigma$ in the
$\eta_{c}(2S)$ mass region is fixed to the value obtained from linear
extrapolation from the $\chi_{cJ}$ mass region, while $\delta m$ is allowed to
float.

The shape of the contribution from FSR background is fixed to that obtained
from the corrected MC-simulated shape, and the number of events is allowed to
float. The contribution from continuum background is fixed to that obtained
from off-resonance data and is normalized according to the luminosity and
shifted according to the CM energy. The remaining background distribution from
$\psi(3686)$ decay is found to be smooth and described by an ARGUS function
Ref_arg .

Figure 1: Fit to the $M^{m5C}_{2(\pi^{+}\pi^{-})\eta}$ spectrum. The different
components are expressed with different line styles or colors as given in the
legend.

Figure 1 shows the fit result with a goodness of fit $\chi^{2}/ndf=537/360$,
where $ndf$ is the number of degrees of freedom. The insert shows the
$\eta_{c}(2S)$ mass region on a linear scale. The numerical results are
summarized in Table 2. The statistical significance of the $\eta_{c}(2S)$
signal is determined to be 8.4$\sigma$ by comparing the difference in
likelihoods with and without including the $\eta_{c}(2S)$ signal component,
taking into account the difference in $ndf$ ($\Delta ndf$=2).

The BFs for our studied channels are calculated via:

$\textstyle{\displaystyle\mathcal{B}(X\to 2(\pi^{+}\pi^{-})\eta)={\frac{N^{\rm
obs}}{N^{\rm
tot}_{\rm\psi(3686)\times\displaystyle\mathcal{B}_{1}\times\displaystyle\mathcal{B}_{2}\times\epsilon}}}},$
(2)

where $N^{\rm obs}$ is the number of signal events, $N^{\rm tot}_{\psi(3686)}$
is the total number of $\psi(3686)$ events, $\displaystyle\mathcal{B}_{1}$ and
$\displaystyle\mathcal{B}_{2}$ are the BFs of $\psi(3686)\to\gamma X$ and
$\eta\to\gamma\gamma$ Ref_02 , respectively, and $\epsilon$ is the detection
efficiency after correcting the helix parameters of charged tracks Ref_18 .
Table 2 lists the numerical results, and table 4 lists the product BFs of
$\mathcal{B}(\psi(3686)\to\gamma X)\times\mathcal{B}(X\to
2(\pi^{+}\pi^{-})\eta)$.

Table 2: The signal yields, detection efficiencies, and the obtained BFs for $\chi_{cJ}/\eta_{c}(2S)\to 2(\pi^{+}\pi^{-})\eta$ decays. The first and second uncertainties are statistical and systematic, respectively. Decay | $N^{\rm obs}$ | $\epsilon~{}(\%)$ |  $\displaystyle\mathcal{B}$ ($\times 10^{-3}$)
---|---|---|---
$\eta_{c}(2S)\to 2(\pi^{+}\pi^{-})\eta$ | 565 $\pm$ $30$ | 6.27 | 12.1 $\pm$0.6$\pm$6.2
$\chi_{c0}\to 2(\pi^{+}\pi^{-})\eta$ | $42530\pm 240$ | 8.90 | 4.54$\pm$0.03$\pm$0.38
$\chi_{c1}\to 2(\pi^{+}\pi^{-})\eta$ | $78440\pm 290$ | 9.31 | 8.07$\pm$0.03$\pm$0.76
$\chi_{c2}\to 2(\pi^{+}\pi^{-})\eta$ | $50980\pm 240$ | 8.62 | 5.81$\pm$0.03$\pm$0.56

The $\eta_{c}$ signal yield is obtained by performing an unbinned maximum
likelihood fit to the $M_{2(\pi^{+}\pi^{-})\eta}^{\rm 5C}$ distribution Ref_05
. The fit function includes the $\eta_{c}$ signal function and four background
components: the background from $\psi(3686)\to\pi^{0}2(\pi^{+}\pi^{-})\eta$,
continuum background, non-resonant background ($\psi(3686)\to\gamma
2(\pi^{+}\pi^{-})\eta$), and other potential backgrounds from $\psi(3686)$
decays estimated using the $\psi(3686)$ inclusive MC sample. Interference
between the $\eta_{c}$ and the non-resonant process is considered. The fit
function is expressed as

${\textstyle{{(\epsilon(m)\lvert
e^{i\phi}E_{\gamma}^{2}BW(m)+\alpha\mathcal{N}\rvert^{2}E_{\gamma}^{3})}\otimes
G+f_{\rm BKG}}},$ (3)

where the first term is the $\eta_{c}$ signal shape, which is described by the
sum of a modified $BW$ function and a coherent amplitude squared, convolved
with the resolution function. Here, $\epsilon(m)$ is the mass-dependent
efficiency; $\phi$ is the interference phase angle; $E_{\gamma}$ =
$({m_{\psi(3686)}^{2}-m^{2}})/2m_{\psi(3686)}$ is the energy of the radiative
photon; $BW(m)$ is a nonrelativistic $BW$ function defined as
$\textstyle{\frac{\Gamma/2}{{(m-m_{0})}+i\Gamma/2}}$, where $m_{0}$ and
$\Gamma$ are the mass and width of the $\eta_{c}$; and $\alpha$ is the
strength of the non-resonant component. The non-resonant PDF, denoted as
$\mathcal{N}$, is described by a second order Chebychev function. The Gaussian
function $G$ accounts for the detector resolution, and $f_{\rm BKG}$ denotes
the sum of the other three background components mentioned earlier, excluding
the non-resonant background. In the fit, $\phi$ and $\alpha$ are free
parameters.

The mass-dependent efficiency is determined from MC simulations of the PHSP
decay $\eta_{c}\to 2(\pi^{+}\pi^{-})\eta$. We use a Gaussian function to
describe the discrepancy between data and MC simulation, and the parameters
are determined by fitting the $J/\psi$ peak in the control sample
$\psi(3686)\to\gamma\gamma J/\psi,J/\psi\to 2(\pi^{+}\pi^{-})\eta$.

Figure 2: The fit results for the $\eta_{c}$ signal with destructive (a) and
constructive (b) interference solutions. The different components are
expressed with different line styles or colors.

Figure 2 shows the fit result, where the goodness of fit is
$\chi^{2}/ndf=164/118$. The total number of $\eta_{c}$ signal is determined to
be $N^{\rm obs}=(2.60\pm 0.24)\times 10^{4}$ (destructive), or $N^{\rm
obs}=(1.24\pm 0.20)\times 10^{4}$ (constructive). Using the $\eta_{c}$ fitted
signal yield, we determine the BF of $\eta_{c}\to 2(\pi^{+}\pi^{-})\eta$,
listed in Table 3, as well as the mass and width of the $\eta_{c}$ under the
two different interference scenarios.

Table 3: The obtained mass, width, and BF of the $\eta_{c}$ for the two
interference solutions. The first and second uncertainties are statistical and
systematic, respectively.

Solution | Destructive | Constructive
---|---|---
Mass (MeV/$c^{2}$) | $2984.14\pm 0.12\pm 0.34$ | $2984.13\pm 0.13\pm 0.38$
Width (MeV) | $28.82\pm 0.11\pm 0.81$ | $28.81\pm 0.11\pm 0.82$
$\mathcal{B}$ ($\%$) | $5.9\pm 0.5\pm 2.1$ | $2.8\pm 0.5\pm 1.3$
$\alpha(\%)$ | $2.541\pm 0.023$ | $3.50\pm 0.18$
$\phi$(rad) | $-1.949\pm 0.014$ | $2.270\pm 0.030$

Table 4: The product BFs of $\mathcal{B}(\psi(3686)\to\gamma
X)\times\mathcal{B}(X\to 2(\pi^{+}\pi^{-})\eta)$. The first and second
uncertainties are statistical and systematic, respectively.

Decay | $\displaystyle\mathcal{B}(\times 10^{-5})$
---|---
$\psi(3686)\to\gamma\eta_{c}(2S);\eta_{c}(2S)\to 2(\pi^{+}\pi^{-})\eta$ | $0.85\pm 0.04\pm 0.13$
$\psi(3686)\to\gamma\chi_{c0};\chi_{c0}\to 2(\pi^{+}\pi^{-})\eta$ | $44.4\pm 0.3\pm 3.7$
$\psi(3686)\to\gamma\chi_{c1};\chi_{c1}\to 2(\pi^{+}\pi^{-})\eta$ | $78.7\pm 0.3\pm 7.4$
$\psi(3686)\to\gamma\chi_{c2};\chi_{c2}\to 2(\pi^{+}\pi^{-})\eta$ | $55.3\pm 0.3\pm 5.3$
$\psi(3686)\to\gamma\eta_{c};\eta_{c}\to 2(\pi^{+}\pi^{-})\eta$ (destructive) | $2.01\pm 0.17\pm 0.64$
$\psi(3686)\to\gamma\eta_{c};\eta_{c}\to 2(\pi^{+}\pi^{-})\eta$ (constructive) | $0.95\pm 0.17\pm 0.44$

The systematic uncertainties related to the measurements for ${\cal
B}(\eta_{c}(2S)/\chi_{cJ}\to 2(\pi^{+}\pi^{-})\eta)$ include the following
sources. The tracking and PID efficiencies are all assigned to be 1.0$\%$ per
charged track Ref_14 , and the photon efficiency is also assigned to be
1.0$\%$ per photon Ref_15 . The uncertainties caused by the kinematic fit are
determined to be (5.5-6.9)$\%$ by comparing the difference in detection
efficiency with and without correcting the helix parameters of charged tracks
in MC simulation Ref_18 . The uncertainties caused by the background veto are
determined to be (0.3-6.1)$\%$ by randomly sampling multiple mass windows
Ref_19 . The uncertainties caused by the number of continuum background events
are determined to be 0.23$\%$ for the $\eta_{c}(2S)$ signal and negligible for
$\chi_{cJ}$ signals. They are estimated by varying the number of events in the
fit by one standard deviation assuming they follow either a Poisson or
binomial distribution. The uncertainties caused by MC imperfections are
estimated to be (0.1-1.3)% by comparing the differences with and without
correcting the momentum distributions in MC simulation based on data. The
uncertainty related to $\mathcal{B}(\eta\to\gamma\gamma)$ is taken from the
PDG Ref_02 , and the uncertainty related to
$\mathcal{B}(\psi(3686)\to\gamma\eta_{c}(2S))$ is taken from Ref. Ref_11 . The
uncertainty due to the total number of $\psi(3686)$ events is determined to be
0.5$\%$ Ref_16 . The uncertainties due to MC statistics are determined to be
0.3% by the same method as for the continuum background events number. The
uncertainties caused by the signal yields include the following components:
continuum background shape, FSR factor, mass and width of the $\eta_{c}(2S)$,
efficiency curve, damping function, detector resolution, and other potential
background from $\psi(3686)$ decays. Among them, the uncertainty due to the
efficiency curve is estimated by comparing the difference in the fit results
with and without including the curve. The uncertainties caused by the
continuum and the potential background shapes are estimated by alternatively
employing different functions. The uncertainties from other components in
signal yields are estimated by varying the corresponding parameters by one
standard deviation. The resulting uncertainties due to signal yields are
determined to be (0.2-11.0)$\%$. We assume all these sources are independent
and take their sum in quadrature as the total systematic uncertainties, which
are determined to be 15.7$\%$ (not including the uncertainty from ${\cal
B}(\psi(3686)\to\gamma\eta_{c}(2S))$ measurement), 9.1$\%$, 9.7$\%$, and
9.8$\%$ for $\eta_{c}(2S)/\chi_{cJ}$, respectively.

The uncertainties related to the measurement of ${\cal B}(\eta_{c}\to
2(\pi^{+}\pi^{-})\eta)$ have the same or similar sources as above, and are
estimated with the same or similar methods. However, the same source may
introduce two different uncertainties, corresponding to the constructive and
destructive solutions, due to interference between the $\eta_{c}(1S)$ decay
and the non-resonant process. Additionally, the dominant uncertainties come
from the non-resonant process with different interference fraction
assumptions, which are estimated to be 30.5% (constructive) or 44.1%
(destructive) by mixing different fractions within one standard deviation, and
the largest differences relative to the nominal results are taken as the
uncertainties. Finally, the total systematic uncertainty is determined to be
35.2% (constructive) or 49.1% (destructive).

The uncertainties related to the measurements of the mass and width of the
$\eta_{c}$ are the same as those mentioned above for the $\eta_{c}$ branching
fraction measurement and are estimated with the same methods, except for the
mass scale, which is estimated using fits to the $\chi_{cJ}$ peaks in the same
final state. The relative uncertainties are listed in Table 3.

In summary, the radiative decay $\psi(3686)\to\gamma 2(\pi^{+}\pi^{-})\eta$ is
investigated using a sample of 2.7 billion $\psi(3686)$ decays. The BF of
$\eta_{c}\to 2(\pi^{+}\pi^{-})\eta$ is determined for the first time under
different interference scenarios. Notably, there is a significant difference
between the constructive and destructive solutions. The
$\chi_{cJ}/\eta_{c}(2S)\to 2(\pi^{+}\pi^{-})\eta$ decays are observed for the
first time with statistical significance greater than 8$\sigma$, and the
corresponding BFs are measured. We present the most precise single measurement
of the $\eta_{c}$ mass and width by assuming all radiative non-resonant events
interfere with the $\eta_{c}$. The relative phase for constructive or
destructive interference is consistent with the earlier BESIII result Ref_05
within $1\sigma$, which may suggest a common phase in all modes. The physics
behind this possible common phase needs to be understood. With the measured
$\eta_{c}$ mass, we obtain the hyperfine mass splitting to be $\Delta
M_{hf}(1S)_{c\bar{c}}=M(J/\psi)-M(\eta_{c})=112.8\pm 0.4$ MeV/$c^{2}$, which
is not in conflict with theoretical calculations Ref_hf1s1 ; Ref_hf1s2 .

Furthermore, using the BFs measured in this analysis, the ratio of BFs is

$\scriptstyle{{\frac{\scriptstyle\mathcal{B}(\eta_{c}(2S)\to
2(\pi^{+}\pi^{-})\eta)}{\scriptstyle\mathcal{B}(\eta_{c}\to
2(\pi^{+}\pi^{-})\eta)}}=\left\\{\begin{aligned} 0.21\pm
0.13~{}(destructive)\\\ 0.43\pm 0.31~{}(constructive)\end{aligned}\right.}.$
(4)

The results are consistent with most experimental measurements Ref_11 , and
obviously disagree with the prediction in Ref. Ref_10 . The inconsistency
between the experimental measurements and the theoretical prediction on the
ratio ${\cal B}(\eta_{c}(2S)\to hadrons)/{\cal B}(\eta_{c}\to hadrons)$
implies that mixing of the pseudoscalar glueballs to $\eta_{c}$ or
$\eta_{c}(2S)$ may play an important role in charmonium decays cheny ;
Ref_mixing1 ; Ref_mixing2 . Additionally, the distinct contributions of the
open-charm loop to the decays of the $\eta_{c}$ and $\eta_{c}(2S)$ cannot be
ignored loop .

The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400,
2023YFA1606000; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11635010, 11735014, 11935015, 11935016, 11935018, 12025502,
12035009, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263,
12192264, 12192265, 12221005, 12225509, 12235017, 12361141819, 12375070; 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. U2032108, No.
U1832207; 100 Talents Program of CAS; The Institute of Nuclear and Particle
Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and
Cosmology; German Research Foundation DFG under Contracts Nos. 455635585,
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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# On the need of an ultramassive black hole in OJ 287

Mauri J. Valtonen,1,2 Staszek Zola,3 A. Gopakumar,4 Anne Lähteenmäki,5,6 Merja
Tornikoski,5 Lankeswar Dey,4 Alok C. Gupta,7,20 Tapio Pursimo,8 Emil
Knudstrup,8 Jose L. Gomez,9 Rene Hudec,10,11 Martin Jelínek,11 Jan Štrobl,11
Andrei V. Berdyugin,2 Stefano Ciprini,12,13 Daniel E. Reichart,14 Vladimir V.
Kouprianov,14 Katsura Matsumoto,15 Marek Drozdz,16 Markus Mugrauer,17 Alberto
Sadun,18 Michal Zejmo,19 Aimo Sillanpää,2 Harry J. Lehto,2 Kari Nilsson,1 Ryo
Imazawa,21, Makoto Uemura 22 and James W. Davidson Jr. 23
1 FINCA, University of Turku, FI-20014 Turku, Finland
2 Tuorla Observatory, Department of Physics and Astronomy, University of
Turku, FI-20014 Turku, Finland
3 Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244
Krakow, Poland
4 Department of Astronomy and Astrophysics, Tata Institute of Fundamental
Research, 4000005 Mumbai, India
5 Aalto University, Metsähovi Radio Observatory, Metsähovintie 114, 02540
Kylmälä, Finland
6 Aalto University, Department of Electronics and Nanoengineering, P.O. Box
15500, FI-00076 AALTO, Finland
7 Aryabhatta Research Institute of Observational Sciences (ARIES), Manora
Park, Nainital 263001, India
8 Nordic Optical Telescope, Apartado 474, E-38700 Santa Cruz de La Palma,
Spain
9 Instituto de Astrofisica de Andalucia - CSIC, Glorieta de la Astronomia s/n,
18008 Granada, Spain
10 Czech Technical University, Faculty of Electrical Engineering, 16000
Prague, Czech Republic
11 Astronomical Institute (ASU CAS), 251 65 Ondřejov, Czech Republic
12 Instituto Nazionale di Fisica Nucleare (INFN) Sezione di Roma Tor Vergata,
Via della Ricerca Scientifica 1,
00133, Roma, Italy
13 ASI Space Science Data Center (SSDC), Via del Politecnico, 00133, Roma,
Italy
14 University of North Carolina at Chapel Hill, Chapel Hill, North Carolina,
NC 27599, USA
15 Astronomical Institute, Osaka Kyoiku University, 4-698 Asahigaoka,
Kashiwara, Osaka, 582-8582, Japan
16 Mt. Suhora Observatory, Pedagogical University, ul. Podchorazych 2, 30-084
Krakow, Poland
17 Astrophysikalisches Institut und Universitäts-Sternwarte, Schillergässchen
2, D-07745 Jena, Germany
18 Department of Physics, University of Colorado, Denver, CO 80217, USA
19 Kepler Institute of Astronomy, University of Zielona Gora, Lubuska 2,
65-265 Zielona Gora, Poland
20 Key Laboratory for Research in Galaxies and Cosmology, Shanghai
Astronomical Observatory,
Chinese Academy of Sciences, Shanghai 200030, China
21 Department of Physics, Graduate School of Advanced Science and Engineering,
Hiroshima University,
1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
22 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1
Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
23 Department of Astronomy, University of Virginia, 530 McCormick Rd.,
Charlottesville, VA 22904, USA E-mail<EMAIL_ADDRESS>(MJV)

(Accepted … Received … in original form …)

###### Abstract

The highly variable blazar OJ 287 is commonly discussed as an example of a
binary black hole system. The 130 year long optical light curve is well
explained by a model where the central body is a massive black hole of
18.35$\times$109 solar mass that supports a thin accretion disc. The secondary
black hole of 0.15$\times$109 solar mass impacts the disc twice during its 12
year orbit, and causes observable flares. Recently, it has been argued that an
accretion disc with a typical AGN accretion rate and above mentioned central
body mass should be at least six magnitudes brighter than OJ 287’s host galaxy
and would therefore be observationally excluded. Based on the observations of
OJ 287’s radio jet, detailed in Marscher and Jorstad (2011), and up-to-date
accretion disc models of Azadi et al. (2022), we show that the V-band
magnitude of the accretion disc is unlikely to exceed the host galaxy
brightness by more than one magnitude, and could well be fainter than the
host. This is because accretion power is necessary to launch the jet as well
as to create electromagnetic radiation, distributed across many wavelengths,
and not concentrated especially on the optical V-band. Further, we note that
the claimed V-band concentration of accretion power leads to serious problems
while interpreting observations of other Active Galactic Nuclei. Therefore, we
infer that the mass of the primary black hole and its accretion rate do not
need to be smaller than what is determined in the standard model for OJ 287.

###### keywords:

BL Lacertae objects: individual: OJ 287 – quasars: supermassive black holes –
accretion, accretion discs – gravitational waves – galaxies: jets

††pubyear: 2022††pagerange: On the need of an ultramassive black hole in OJ
287–On the need of an ultramassive black hole in OJ 287

##  1 Introduction

OJ 287 has the longest and best covered optical light curve among all AGN
(Active Galactic Nuclei). It has been photographed as a part of other projects
since 1888, and for the last 50 years it is among the most studied
extragalactic objects. The number of optical observations in the light curve
now exceeds 100,000. The light curve is shown in Figure 1. A binary black hole
(BH) model for OJ 287 was proposed in 1988 (Sillanpää et al., 1988). The model
and its subsequent elaborations (Lehto & Valtonen, 1996; Valtonen, 2007; Dey
et al., 2018) have made increasingly accurate predictions about future light
curve events, in 1994, 1995, 2005, 2007, 2015, and 2019 which have been
subsequently verified (Sillanpää et al., 1996a, b; Valtonen et al., 2006,
2008, 2016; Laine et al., 2020; Valtonen et al., 2021). The multi-wavelength
observations at these times as well as the study of the historical light curve
as a whole, has produced an increasingly accurate picture of the central
engine of OJ 287 (Figure 1). At the heart of this model is a massive BH that
weighs more than $10^{10}$ solar mass. It is customary to refer such massive
black holes as ultramassive black holes, even though they simply represent a
continuum distribution which extends well beyond the mass of the BH in OJ 287
(Saglia et al., 2016).

Recently, Komossa et al. (2023) (K23 hereafter) claimed that the brightness of
the accretion disc in OJ 287 should be at least V $\sim$ 13.5 in the V-band.
Looking at Figure 1, we obviously do not see such a constant contribution from
the accretion disc. Therefore K23 conclude that the standard model has a mass
which is at least two orders of magnitude too high, or its accretion rate is
similarly two orders of magnitude too high. In the latter case the disc should
not be geometrically thin which in their opinion invalidates the standard
model.

Figure 1: The historical light curve of OJ 287 in the optical V-band (on the
left) and the binary black hole picture of its central engine (on the right).
The model has a 18.35$\times$109 solar mass central BH, with an accretion disc
with the accretion rate of 8$\pm$4% of the Eddington rate. According to
Komossa et al. (2023), an accretion disc of these parameters should be as
bright as 15 mJy in the V-band, which is clearly not seen, as it would provide
the bottom level of flux in this graph. In the standard accretion disc models
the disc brightness is orders of magnitude lower, and therefore the disc
cannot be observed in the light curve.

K23 base their arguments on calculating the total accretion power of OJ 287
and channelling a major part of it to the V-band. They assume (1) that 100
percent of the accretion power appears as accretion disc radiation, (2) that
1/9 of the disc radiation goes to the optical wavebands (so called bolometric
correction, (Kaspi et al., 2000)), and (3) that 100 percent of the optical
radiation is concentrated in the V-band. In this way they calculate the total
emission $L_{V}\sim 2\times 10^{46}$ ergs sec-1. Then they convert this
emission rate to the V band magnitude corresponding to the distance of OJ 287
(Nilsson et al., 2010), and find that it is at least 10 times higher than the
recent minimum brightness observed in 2022. This means that the disc should at
least 6 magnitudes brighter than the host galaxy (Nilsson et al., 2020).

It is straightforward to note that the above estimate provides an upper limit
for OJ 287’s disc radiation, and thus gives a lower limit for V band
magnitude. We note that K23’s assumption (2) is consistent with the detailed
studies on various accretion disc models by Azadi et al. (2022) though their
assumptions (1) and (3) can lead to a huge overestimate for OJ 287’s accretion
disc energy flux.

In this paper, we provide realistic estimates for the upper limit for OJ 287’s
disc luminosity, influenced by the fact that there is no observational
evidence for the accretion disc in OJ 287.

For this purpose, we employ recent detailed models of accretion discs by Azadi
et al. (2022) and take into account the observed radio jet in our energy
budget estimates (Marscher & Jorstad, 2011), while considering the fact that
power can also be dissipated into accretion disc winds (Rodriguez-Ramirez et
al., 2020).

Thereafter, we explore consequences of the K23-type estimates while dealing
with AGNs in the nearby Universe where we have much more detailed
observational inputs compared to OJ 287.

##  2  Estimating Accretion Disc Luminosities in AGNs

The calculation of the accretion disc structure and its bolometric corrections
is not a simple task and has required lots of work in modelling as well as in
comparing with observations (Shakura & Sunyaev, 1973; Mineshige & Wood, 1990;
Chen et al., 1995; Zdziarski, 1998; DeKool & Wickramasinghe, 1999; Collin &
Hure, 2001; Nemmen & Brotherton, 2010; Azadi et al., 2022). The recent models
of Kubota & Done (2018) and Azadi et al. (2022) give us the luminosity of the
disc as a function of the main parameters of the system, the mass of the
central BH $M_{BH}$, the accretion rate $\lambda_{Edd}$ with respect to the
the rate that produces the Eddington luminosity, and the spin of the black
hole. For the OJ 287 primary we have found the central mass
$M_{BH}=18.35\times 10^{9}$ solar mass, accretion rate $0.08\pm 0.04$ of the
Eddington limit and the normalised spin $\sim$ 0.38 (Dey et al., 2018;
Valtonen et al., 2019).

The Eddington ratio is defined as the mass accretion rate, $\dot{m}$,
normalised by the rate that produces the Eddington luminosity,
$\dot{m}_{Edd}$, as:

$\lambda_{Edd}=\dot{m}/\dot{m}_{Edd},$ (1)

where

$\dot{m}_{Edd}=L_{Edd}/(\eta_{r}c^{2}),$ (2)

and $\eta_{r}$ is the radiation efficiency, usually taken as $\eta_{r}=0.057$.
The models of Kubota & Done (2018) and Azadi et al. (2022) as well as our
standard model Dey et al. (2018) use this definition (Stella & Rosner, 1984;
Valtonen et al., 2019; Jiang et al., 2019).

In magnetic disc models a large fraction of the accretion power goes to
maintaining the hot corona (about $50\%$ in the model AGN0.07 of Jiang et al.
(2019), representing $\lambda_{Edd,corona}\sim 0.035$), mechanical power of
the disc wind (about $40\%$ in the model of Rodriguez-Ramirez et al. (2020),
representing $\lambda_{Edd,wind}\sim 0.048$), and the mechanical power of the
twin jets ($\sim 5\times 10^{46}$ erg sec-1 from the observed megaparsec jet
(Marscher & Jorstad, 2011), representing $\lambda_{Edd,jet}\sim 0.033$ for the
twin jets). Added together, these magnetic and mechanical contributions could
in principle soak nearly all the accretion energy in our model, where
$\lambda_{Edd,model}=0.08\pm 0.04$.

In the Azadi et al. (2022) grid of models this puts us in the area slightly
beyond $M_{BH}=10^{10}$ solar mass, log$(\lambda_{Edd})$ smaller than -1.5,
and the spin closer to 0 than 1. Figure 5 (top left panel) of Azadi et al.
(2022) gives log$(\nu L_{\nu})\sim 45.8$ where $\nu L_{\nu}$ is in units of
ergs sec-1. Extrapolation from Figure 7 (top right panel) of Azadi et al.
(2022) to log$(\lambda_{Edd})\sim-2.5$ gives $\nu L_{\nu}\sim(0.5-2)\times
10^{45}$ ergs sec-1. Considering the large fraction of the accretion power
likely to go to the halo, winds and jets, this may be a realistic estimate for
the power that may appear as disc luminosity.

Out of the disc luminosity (1 - 4)$\times$1044 ergs sec-1 is shared into the
V-band. This corresponds roughly to V$\sim 19\pm 1$. It is close to the
magnitude of the host galaxy measured directly (Nilsson et al., 2020), and
agrees with the V$\sim$17.5 observational upper bound at the historical
brightness minimum when the disc was definitely not detected, based on the
colours, unless the disc mimics the colours of the host galaxy (Takalo et al.,
1990; Valtonen et al., 2022).

OJ 287 is not the optimal system for testing accretion disc theories since in
this system the disc is overpowered by the jet. Therefore we may ask where
does the K23-type calculation lead when applied to other AGN.

We first use empirical relations to estimate the AGN contribution in the
V-band lumininosity of OJ 287. Jalan et al. (2023) find that for an AGN around
redshift 0.3 and Eddington ratio around 8 $\%$, the AGN and host galaxy
luminosities are approximately equal, with a wide variation in individual
cases. This agrees with what we just found out for OJ 287.

If a central BH of $M_{BH}\sim 1.835\times 10^{10}$ solar mass and a typical
accretion rate makes the accretion disc 6 magnitudes brighter than the host
galaxy, what would be the implications to other accretions discs with lower BH
mass? The BH mass has a nearly linear correlation with the host luminosity
(Saglia et al., 2016), and since the accretion disc power also correlates
directly with the BH-mass (Azadi et al., 2022), in close to linear manner, we
have to conclude that also in general the accretion discs should be brighter
than their host galaxies.

In Figure 2 we plot the BH mass - host galaxy mass diagram for the local AGN.
It is the usual diagram (Saglia et al., 2016; Terrazas et al., 2017) for well
determined BH masses, but including only those galaxies which have been
reported to possess an AGN. Then we draw a line which runs two orders of
magnitude below the observed point for OJ287 (Nilsson et al., 2020), and has a
slope from the above mentioned correlations. Therefore in the K23-calculation
it represents the line of equality between the accretion disc and the host
galaxy. The observational points are not far from this line which means that
all AGN in the nearby Universe should be about equally bright as their hosts,
and many of them much brighter than their hosts.

Figure 2: The correlation between black hole mass and the host mass in nearby
AGN (crosses) (Saglia et al., 2016; Terrazas et al., 2017). We start from the
K23 estimate that the brightness of a 1.835$\times 10^{10}$ solar mass BH
accretion disc is V = 13.5 at the redshift 0.306 of OJ 287, given the
accretion rate of 8$\%$ of the Eddington rate. Then we use the disc luminosity
- BH mass correlation from Azadi et al. (2022) to transfer the estimate to the
AGN of the local Universe, and draw a line representing systems where the
accretion disc luminosity should be equal to the host galaxy luminosity in the
K23 accretion disc calculation. We see that the observational points
concentrate around the line, suggesting that the accretions discs should be
similar in brightness to the host galaxies. Also a large number of local AGN
should overpower their host galaxies by more than an order of magnitude.
However, observationally we know that this is not the case, and thus the
K23-calculations are oversimplified.

From observations we know that this is not the case: the nearby AGN do not
dominate their host galaxies. Typically it is difficult to see the accretion
disc, and even if we see it, it is definitely very much fainter than host
galaxy (Sun & Malkan, 1989). A way to get around this problem in the K23 model
is to reduce the accretion rate from $\sim 8\%$ to a very much lower value,
for all AGN, but that does not seem to be possible (Kynoch et al., 2023).

In quasars it has been found that accretion discs may be up to ten times
brighter than their host galaxies in the optical wavebands (Sun & Malkan,
1989). The discs are thought to be responsible for the "Big Blue Bump", which
dominates the optical luminosity in these objects (Collin & Hure, 2001). Even
here the K23-type calculation gives a one magnitude excess of the disc
luminosity, and considering that quasars often accrete close to their
Eddington rates, much higher than the $8\%$ in the K23 calculation, the
problem becomes even worse. It is possible that also all quasars contradict
the K23-type calculation.

##  3 Discussion and conclusions

One may wonder what is the origin of the difference of 5.5 magnitudes of the
V-band luminosity of OJ 287 between the K23 calculation and our calculation
based on recent disc models. About one half of the difference (in magnitude
scale) appears to come from the fact that the K23 model concentrates the
accretion power very strongly on the V-band. The other half has to do with the
jet: the mechanical power required by the jet is very large (Marscher &
Jorstad, 2011), and this energy cannot appear as disc luminosity. In addition,
the mechanical luminosity of the AGN wind in OJ 287 may carry away up to 50
$\%$ of the mass accretion power (Rodriguez-Ramirez et al., 2020). Therefore a
calculation based on the theoretical accretion power, and assuming that the
disc has to radiate all of this, provides an upper limit on the disc
luminosity at best, and as we have seen, this upper limit is not very useful.

Also we have to remember that the accretion rate in the standard model of Dey
et al. (2018) is not solved directly, but it comes via a further calculation
from the parameters of the orbit solution (Valtonen et al., 2019). Therefore
it has wide error bounds which have to be taken into consideration if one
attempts to estimate the upper bound on the disc luminosity.

Our estimate for the accretion disc brightness in OJ 287 is V$=19\pm 1$. The
upper limit of brightness, corresponding to V=18, is consistent not having
detected the disc during the faintest state ever seen in OJ 287 (Takalo et
al., 1990). Or possibly, if the accretion disc has very similar colours as the
host host galaxy, it could have been detected at the level of V=18 (Valtonen
et al., 2022).

To resolve this problem, K23 proposed to lower the central BH mass to $\sim
10^{8}$ solar mass. However, if there is really a problem with the disc
brightness, changing the BH mass is not a solution. The power of the jet must
still be supported, and it depends on the same product of mass and mass
accretion rate as the disc brightness. If mass is changed, the accretion rate
must be adjusted equally, but in opposite direction, to keep the product
constant. The product is fixed by the jet power (Marscher & Jorstad, 2011),
and so is the disc brightness also.

The suggestion of lowering the BH mass to $\sim 10^{8}$ solar mass rather
brings problems of its own. It would imply that the accretion rate of OJ 287
would have to be highly super-Eddington even to produce the mechanical power
of the jet, and from the great length of its jet we may deduce that it would
have been in a super-Eddington state for a long time (Marscher & Jorstad,
2011). This is rather unlikely (Jalan et al., 2023). This is a problem of the
low BH mass scenario both in binary and single BH models of OJ 287 (Britzen et
al., 2018).

Further problems arise, if OJ 287 is interpreted as a low BH mass binary
system. The usual interpretation is based on the double peaked structure of
the optical light curve. The simplest way to obtain the double peaks in a
binary system is to associate the observed flares with disc impacts twice
during each orbit. But lowering the mass scale by two orders of magnitude
lowers also the brightness of the flares by the same amount, and makes them
invisible against the background of the general jet emission. This applies
equally to direct impact flares and flares generated by tidally induced
accretion flow variations (Valtonen et al., 2012). This of course does not
exclude the possibility that a new way of producing double peaked structure is
invented in future.

The main unanswered question in the low BH mass scenario is why OJ 287 behaves
in a highly predictable manner in the standard model of Dey et al. (2018). The
standard model relies heavily on General Relativity, and the latter is not a
scale-free theory in the same sense as the Newtonian theory is. Therefore the
predictability, which has now been witnessed for 40 years (Valtonen et al.,
2023), is lost if the mass scale is lowered.

There were two major flares predicted to take place in OJ 287 also in 2022,
the first one in January/February and the second one in July/August (Valtonen
et al., 2021). Because OJ 287 is close to the sun during the latter period,
ground based optical observations are not possible, and thus the 2022
observing campaign carried out by us was concentrated on the first half of the
year. The first flare is caused directly by the impact of the secondary BH on
the disk. A cloud of plasma is pulled out of the disc (Ivanov et al., 1998),
and its radiation may be described by the standard van der Laan (1971)
expanding cloud model. The model provides an explanation for the peculiar
behaviour of the flare at the time when the source becomes optically thin to
synchrotron radiation.

The flare was observed at the predicted time, with properties that have been
observed once before in OJ 287, in 2005 (Ciprini & Rizzi, 2008). In 2005 and
in 2022 the impacts took place at practically the same distance from the
primary BH which should make the impact flares comparable to each other. The
time difference between the 2005 and 2022 flares was expected to be 6161$\pm$3
days from the Dey et al. (2018) orbit model, while the observed distance
between the flat spectrum peaks was within the same time interval. In 2005, we
know from the orbit model that the flare peaked 35$\pm 3$ days after the disc
impact. In 2022 we also have strong evidence for the association with the disc
impact, with the same time delay from the impact to the flat spectrum peak.
The light curves of both flares are displayed in Valtonen et al. (2023). The
configuration of the emitting cloud at the brightness peak is illustrated by
Ivanov et al. (1998) (Figure 4, top right hand panel).

As to the July/August flare, it arises when the plasma cloud has expanded
further by about a factor of ten, and the cloud becomes optically thin to
bremsstrahlung radiation. At the start of our observing campaign, the impact
configuration had not been calculated yet, and therefore there was a real
possibility that the flare could have shifted outside the "summer gap", and
this possibility was communicated also to wider circles (a preprint posted in
arXiv).

However, once the configuration was determined, it became clear that the
second flare was unobservable, and indeed, was not observed. Since we could
not observe the second flare, we can only say that its absence in our light
curve is consistent with our model (Valtonen, 2007; Dey et al., 2018; Valtonen
et al., 2021, 2023).

## Acknowledgements

This work was partly funded by NCN grant No. 2018/29/B/ST9/01793 (SZ) and JSPS
KAKENHI grant No. 19K03930 (KM). SC acknowledges support by ASI through
contract ASI-INFN 2021-43-HH.0 for SSDC, and Instituto Nazionale di Fisica
Nucleare (INFN). RH acknowledges the EU project H2020 AHEAD2020, grant
agreement 871158, and internal CTU grant SGS21/120/OHK3/2T/13. ACG is
partially supported by Chinese Academy of Sciences (CAS) President’s
International Fellowship Initiative (PIFI) (grant no. 2016VMB073). MJV
acknowledges a grant from the Finnish Society for Sciences and Letters.

## Data Availability

We do not present previously unpublished data in this work.

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