#  Observation of $D\to a_{0}(980)\pi$ in the decays
$D^{0}\rightarrow\pi^{+}\pi^{-}\eta$ and $D^{+}\rightarrow\pi^{+}\pi^{0}\eta$

M. Ablikim1, M. N. Achasov4,c, P. Adlarson75, O. Afedulidis3, X. C. Ai80, R.
Aliberti35, A. Amoroso74A,74C, Q. An71,58,a, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,h, H.-R. Bao63, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi74A,74C, E.
Bianco74A,74C, A. Bortone74A,74C, I. Boyko36, R. A. Briere5, A. Brueggemann68,
H. Cai76, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,63, N. Cao1,63, S. A.
Cetin62A, J. F. Chang1,58, G. R. Che43, G. Chelkov36,b, C. Chen43, C. H.
Chen9, Chao Chen55, G. Chen1, H. S. Chen1,63, H. Y. Chen20, M. L. Chen1,58,63,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,63, X. R. Chen31,63, X. T.
Chen1,63, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,i, Z. Y. Chen1,63, S. K.
Choi10A, G. Cibinetto29A, F. Cossio74C, J. J. Cui50, H. L. Dai1,58, J. P.
Dai78, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng72, Z. Y. Deng1,
A. Denig35, I. Denysenko36, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding46,h, Y. Ding34, Y. Ding40, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, M. C. Du1, S. X. Du80, Y. Y. Duan55, Z. H.
Duan42, P. Egorov36,b, Y. H. Fan45, J. Fang59, J. Fang1,58, S. S. Fang1,63, W.
X. Fang1, Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava74B,74C, F.
Feldbauer3, G. Felici28A, C. Q. Feng71,58, J. H. Feng59, Y. T. Feng71,58, M.
Fritsch3, C. D. Fu1, J. L. Fu63, Y. W. Fu1,63, H. Gao63, X. B. Gao41, Y. N.
Gao46,h, Yang Gao71,58, S. Garbolino74C, I. Garzia29A,29B, L. Ge80, P. T.
Ge76, Z. W. Ge42, C. Geng59, E. M. Gersabeck67, A. Gilman69, K. Goetzen13, L.
Gong40, W. X. Gong1,58, W. Gradl35, S. Gramigna29A,29B, M. Greco74A,74C, M. H.
Gu1,58, Y. T. Gu15, C. Y. Guan1,63, A. Q. Guo31,63, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,g, A. Guskov36,b, J. Gutierrez27, K. L. Han63, T. T.
Han1, F. Hanisch3, X. Q. Hao19, F. A. Harris65, K. K. He55, K. L. He1,63, F.
H. Heinsius3, C. H. Heinz35, Y. K. Heng1,58,63, C. Herold60, T. Holtmann3, P.
C. Hong34, G. Y. Hou1,63, X. T. Hou1,63, Y. R. Hou63, Z. L. Hou1, B. Y. Hu59,
H. M. Hu1,63, J. F. Hu56,j, S. L. Hu12,g, T. Hu1,58,63, Y. Hu1, G. S.
Huang71,58, K. X. Huang59, L. Q. Huang31,63, X. T. Huang50, Y. P. Huang1, Y.
S. Huang59, T. Hussain73, F. Hölzken3, N. Hüsken35, N. in der Wiesche68, J.
Jackson27, S. Janchiv32, J. H. Jeong10A, Q. Ji1, Q. P. Ji19, W. Ji1,63, X. B.
Ji1,63, X. L. Ji1,58, Y. Y. Ji50, X. Q. Jia50, Z. K. Jia71,58, D. Jiang1,63,
H. B. Jiang76, P. C. Jiang46,h, S. S. Jiang39, T. J. Jiang16, X. S.
Jiang1,58,63, Y. Jiang63, J. B. Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y.
Jin66, M. Q. Jing1,63, X. M. Jing63, T. Johansson75, S. Kabana33, N. Kalantar-
Nayestanaki64, X. L. Kang9, X. S. Kang40, M. Kavatsyuk64, B. C. Ke80, V.
Khachatryan27, A. Khoukaz68, R. Kiuchi1, O. B. Kolcu62A, B. Kopf3, M.
Kuessner3, X. Kui1,63, N. Kumar26, A. Kupsc44,75, W. Kühn37, J. J. Lane67, L.
Lavezzi74A,74C, T. T. Lei71,58, Z. H. Lei71,58, M. Lellmann35, T. Lenz35, C.
Li47, C. Li43, C. H. Li39, Cheng Li71,58, D. M. Li80, F. Li1,58, G. Li1, H. B.
Li1,63, H. J. Li19, H. N. Li56,j, Hui Li43, J. R. Li61, J. S. Li59, K. Li1, L.
J. Li1,63, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k,l, Q. M. Li1,63, Q.
X. Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,63, W. G. Li1,a, X.
Li1,63, X. H. Li71,58, X. L. Li50, X. Y. Li1,63, X. Z. Li59, Y. G. Li46,h, Z.
J. Li59, Z. Y. Li78, C. Liang42, H. Liang1,63, H. Liang71,58, Y. F. Liang54,
Y. T. Liang31,63, G. R. Liao14, Y. P. Liao1,63, J. Libby26, A. Limphirat60, C.
C. Lin55, D. X. Lin31,63, T. Lin1, B. J. Liu1, B. X. Liu76, C. Liu34, C. X.
Liu1, F. Liu1, F. H. Liu53, Feng Liu6, G. M. Liu56,j, H. Liu38,k,l, H. B.
Liu15, H. H. Liu1, H. M. Liu1,63, Huihui Liu21, J. B. Liu71,58, J. Y. Liu1,63,
K. Liu38,k,l, K. Y. Liu40, Ke Liu22, L. Liu71,58, L. C. Liu43, Lu Liu43, M. H.
Liu12,g, P. L. Liu1, Q. Liu63, S. B. Liu71,58, T. Liu12,g, W. K. Liu43, W. M.
Liu71,58, X. Liu38,k,l, X. Liu39, Y. Liu80, Y. Liu38,k,l, Y. B. Liu43, Z. A.
Liu1,58,63, Z. D. Liu9, Z. Q. Liu50, X. C. Lou1,58,63, F. X. Lu59, H. J. Lu23,
J. G. Lu1,58, X. L. Lu1, Y. Lu7, Y. P. Lu1,58, Z. H. Lu1,63, C. L. Luo41, J.
R. Luo59, M. X. Luo79, T. Luo12,g, X. L. Luo1,58, X. R. Lyu63, Y. F. Lyu43, F.
C. Ma40, H. Ma78, H. L. Ma1, J. L. Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M.
Ma1, R. Q. Ma1,63, T. Ma71,58, X. T. Ma1,63, X. Y. Ma1,58, Y. Ma46,h, Y. M.
Ma31, F. E. Maas18, M. Maggiora74A,74C, S. Malde69, Y. J. Mao46,h, Z. P. Mao1,
S. Marcello74A,74C, Z. X. Meng66, J. G. Messchendorp13,64, G. Mezzadri29A, H.
Miao1,63, T. J. Min42, R. E. Mitchell27, X. H. Mo1,58,63, B. Moses27, N. Yu.
Muchnoi4,c, J. Muskalla35, Y. Nefedov36, F. Nerling18,e, L. S. Nie20, I. B.
Nikolaev4,c, Z. Ning1,58, S. Nisar11,m, Q. L. Niu38,k,l, W. D. Niu55, Y. Niu
50, S. L. Olsen63, Q. Ouyang1,58,63, S. Pacetti28B,28C, X. Pan55, Y. Pan57, A.
Pathak34, Y. P. Pei71,58, M. Pelizaeus3, H. P. Peng71,58, Y. Y. Peng38,k,l, K.
Peters13,e, J. L. Ping41, R. G. Ping1,63, S. Plura35, V. Prasad33, F. Z. Qi1,
H. Qi71,58, H. R. Qi61, M. Qi42, T. Y. Qi12,g, S. Qian1,58, W. B. Qian63, C.
F. Qiao63, X. K. Qiao80, J. J. Qin72, L. Q. Qin14, L. Y. Qin71,58, X. P.
Qin12,g, X. S. Qin50, Z. H. Qin1,58, J. F. Qiu1, Z. H. Qu72, C. F. Redmer35,
K. J. Ren39, A. Rivetti74C, M. Rolo74C, G. Rong1,63, Ch. Rosner18, S. N.
Ruan43, N. Salone44, A. Sarantsev36,d, Y. Schelhaas35, K. Schoenning75, M.
Scodeggio29A, K. Y. Shan12,g, W. Shan24, X. Y. Shan71,58, Z. J. Shang38,k,l,
J. F. Shangguan16, L. G. Shao1,63, M. Shao71,58, C. P. Shen12,g, H. F.
Shen1,8, W. H. Shen63, X. Y. Shen1,63, B. A. Shi63, H. Shi71,58, H. C.
Shi71,58, J. L. Shi12,g, J. Y. Shi1, Q. Q. Shi55, S. Y. Shi72, X. Shi1,58, J.
J. Song19, T. Z. Song59, W. M. Song34,1, Y. J. Song12,g, Y. X. Song46,h,n, S.
Sosio74A,74C, S. Spataro74A,74C, F. Stieler35, Y. J. Su63, G. B. Sun76, G. X.
Sun1, H. Sun63, H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun76, S. S. Sun1,63, T.
Sun51,f, W. Y. Sun34, Y. Sun9, Y. J. Sun71,58, Y. Z. Sun1, Z. Q. Sun1,63, Z.
T. Sun50, C. J. Tang54, G. Y. Tang1, J. Tang59, M. Tang71,58, Y. A. Tang76, L.
Y. Tao72, Q. T. Tao25,i, M. Tat69, J. X. Teng71,58, V. Thoren75, W. H. Tian59,
Y. Tian31,63, Z. F. Tian76, I. Uman62B, Y. Wan55, S. J. Wang 50, B. Wang1, B.
L. Wang63, Bo Wang71,58, D. Y. Wang46,h, F. Wang72, H. J. Wang38,k,l, J. J.
Wang76, J. P. Wang 50, K. Wang1,58, L. L. Wang1, M. Wang50, N. Y. Wang63, S.
Wang12,g, S. Wang38,k,l, T. Wang12,g, T. J. Wang43, W. Wang59, W. Wang72, W.
P. Wang35,71,o, X. Wang46,h, X. F. Wang38,k,l, X. J. Wang39, X. L. Wang12,g,
X. N. Wang1, Y. Wang61, Y. D. Wang45, Y. F. Wang1,58,63, Y. L. Wang19, Y. N.
Wang45, Y. Q. Wang1, Yaqian Wang17, Yi Wang61, Z. Wang1,58, Z. L. Wang72, Z.
Y. Wang1,63, Ziyi Wang63, D. H. Wei14, F. Weidner68, S. P. Wen1, Y. R. Wen39,
U. Wiedner3, G. Wilkinson69, M. Wolke75, L. Wollenberg3, C. Wu39, J. F. Wu1,8,
L. H. Wu1, L. J. Wu1,63, X. Wu12,g, X. H. Wu34, Y. Wu71,58, Y. H. Wu55, Y. J.
Wu31, Z. Wu1,58, L. Xia71,58, X. M. Xian39, B. H. Xiang1,63, T. Xiang46,h, D.
Xiao38,k,l, G. Y. Xiao42, S. Y. Xiao1, Y. L. Xiao12,g, Z. J. Xiao41, C. Xie42,
X. H. Xie46,h, 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. Xu66,2,p, M. Xu71,58, Q.
J. Xu16, Q. N. Xu30, W. Xu1, W. L. Xu66, X. P. Xu55, Y. C. Xu77, Z. S. Xu63,
F. Yan12,g, L. Yan12,g, W. B. Yan71,58, W. C. Yan80, X. Q. Yan1, H. J.
Yang51,f, H. L. Yang34, H. X. Yang1, T. Yang1, Y. Yang12,g, Y. F. Yang1,63, Y.
F. Yang43, Y. X. Yang1,63, Z. W. Yang38,k,l, Z. P. Yao50, M. Ye1,58, M. H.
Ye8, J. H. Yin1, Z. Y. You59, B. X. Yu1,58,63, C. X. Yu43, G. Yu1,63, J. S.
Yu25,i, T. Yu72, X. D. Yu46,h, Y. C. Yu80, C. Z. Yuan1,63, J. Yuan34, J.
Yuan45, L. Yuan2, S. C. Yuan1,63, Y. Yuan1,63, Z. Y. Yuan59, C. X. Yue39, A.
A. Zafar73, F. R. Zeng50, S. H. Zeng72, X. Zeng12,g, Y. Zeng25,i, Y. J.
Zeng59, Y. J. Zeng1,63, X. Y. Zhai34, Y. C. Zhai50, Y. H. Zhan59, A. Q.
Zhang1,63, B. L. Zhang1,63, B. X. Zhang1, D. H. Zhang43, G. Y. Zhang19, H.
Zhang80, H. Zhang71,58, H. C. Zhang1,58,63, H. H. Zhang34, H. H. Zhang59, H.
Q. Zhang1,58,63, H. R. Zhang71,58, H. Y. Zhang1,58, J. Zhang80, J. Zhang59, J.
J. Zhang52, J. L. Zhang20, J. Q. Zhang41, J. S. Zhang12,g, J. W. Zhang1,58,63,
J. X. Zhang38,k,l, J. Y. Zhang1, J. Z. Zhang1,63, Jianyu Zhang63, L. M.
Zhang61, Lei Zhang42, P. Zhang1,63, Q. Y. Zhang34, R. Y. Zhang38,k,l, S. H.
Zhang1,63, Shulei Zhang25,i, X. D. Zhang45, X. M. Zhang1, X. Y. Zhang50, Y.
Zhang72, Y. Zhang1, Y. T. Zhang80, Y. H. Zhang1,58, Y. M. Zhang39, Yan
Zhang71,58, Z. D. Zhang1, Z. H. Zhang1, Z. L. Zhang34, Z. Y. Zhang76, Z. Y.
Zhang43, Z. Z. Zhang45, G. Zhao1, J. Y. Zhao1,63, J. Z. Zhao1,58, L. Zhao1,
Lei Zhao71,58, M. G. Zhao43, N. Zhao78, R. P. Zhao63, S. J. Zhao80, Y. B.
Zhao1,58, Y. X. Zhao31,63, Z. G. Zhao71,58, A. Zhemchugov36,b, B. Zheng72, B.
M. Zheng34, J. P. Zheng1,58, W. J. Zheng1,63, Y. H. Zheng63, B. Zhong41, X.
Zhong59, H. Zhou50, J. Y. Zhou34, L. P. Zhou1,63, S. Zhou6, X. Zhou76, X. K.
Zhou6, X. R. Zhou71,58, X. Y. Zhou39, Y. Z. Zhou12,g, J. Zhu43, K. Zhu1, K. J.
Zhu1,58,63, K. S. Zhu12,g, L. Zhu34, L. X. Zhu63, S. H. Zhu70, T. J. Zhu12,g,
W. D. Zhu41, 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

We report the first amplitude analysis of the decays
$D^{0}\to\pi^{+}\pi^{-}\eta$ and $D^{+}\rightarrow\pi^{+}\pi^{0}\eta$ using a
data sample taken with the BESIII detector at the center-of-mass energy of
3.773 GeV, corresponding to an integrated luminosity of 7.9 ${\rm fb}^{-1}$.
The contribution from the process $D^{0(+)}\to a_{0}(980)^{+}\pi^{-(0)}$ is
significantly larger than the $D^{0(+)}\to a_{0}(980)^{-(0)}\pi^{+}$
contribution. The ratios $\mathcal{B}(D^{0}\rightarrow
a_{0}(980)^{+}\pi^{-})/\mathcal{B}(D^{0}\rightarrow a_{0}(980)^{-}\pi^{+})$
and $\mathcal{B}(D^{+}\rightarrow
a_{0}(980)^{+}\pi^{0})/\mathcal{B}(D^{+}\rightarrow a_{0}(980)^{0}\pi^{+})$
are measured to be $7.5^{+2.5}_{-0.8\,\mathrm{stat.}}\pm 1.7_{\mathrm{syst.}}$
and $2.6\pm 0.6_{\mathrm{stat.}}\pm 0.3_{\mathrm{syst.}}$, respectively. The
measured $D^{0}$ ratio disagrees with the theoretical predictions by orders of
magnitudes, thus implying a substantial contribution from final-state
interactions.

Theoretical predictions of the strong interaction in the charm sector are
challenging, since quantum chromodynamics calculations involve non-
perturbative contributions. The W-annihilation (WA) and W-exchange (WE)
processes, which are strictly suppressed in $B$-meson decays, can occur in $D$
decays as a result of final-state interactions (FSI). They are expected to be
dominated by non-perturbative effects, which leads to significant
uncertainties when making theoretical predictions, since these effects depend
strongly on the cutoff values and the unknown phases between different
processes Cheng:2010ry ; Qin:2013tje . Therefore, the study of decays with a
significant contribution from WA or WE processes represents a promising method
for investigating the dynamics of charm decays.

The BESIII Collaboration has observed the pure WA decays $D_{s}^{+}\to
a_{0}(980)^{+(0)}\pi^{0(+)}$ BESIII:2019jjr , which indicates a sizeable
contribution from FSI in the WA processes for the $D\to SP$ sector (where $S$
and $P$ denote scalar and pseudo-scalar particles, respectively). Theorists
have explained the observed large WA amplitude, as well as the amplitude
symmetry $A(D_{s}^{+}\to a_{0}(980)^{+}\pi^{0})=-A(D_{s}^{+}\to
a_{0}(980)^{0}\pi^{+})$, taking into account the contribution from the
$D_{s}^{+}\to\rho^{+}\eta$ and
$D_{s}^{+}\to\bar{K}^{*}(892)^{0}K^{+}(K^{*}(892)^{+}\bar{K}^{0})$ decays,
since they exhibit large branching fractions (BF) and involve the WA amplitude
process at the quark level Hsiao:2019ait ; Ling:2021qzl . This behavior
supports the interpretations of $a_{0}(980)$ as a tetraquark or a molecular
state Ling:2021qzl . More recently, further measurements involving $D_{s}^{+}$
decays have been performed, in particular those that have led to the
observation of the $a_{0}(1817)^{+(0)}$ resonance BESIII:2021anf ;
BESIII:2022npc , which is expected as an excited state of the
$a_{0}(980)^{+(0)}$ Guo:2022xqu , support the interpretation of these two
resonances as $K^{(*)}\bar{K}^{(*)}$ molecules Wang:2021jub ; Oset:2023hyt .

In $D^{0}$ decays, the relative ratio $r_{+/-}=\mathcal{B}(D^{0}\to
a_{0}(980)^{+}\pi^{-})/\mathcal{B}(D^{0}\to a_{0}(980)^{-}\pi^{+})$ is
expected to be less than 0.05 Cheng:2022vbw , when ignoring the WE process.
Until now, attempts to measure this ratio have not been conclusive; both the
CLEO and LHCb Collaborations have studied the $a_{0}(980)^{\pm}\pi^{\mp}$
contributions to the decays $D^{0}\to K_{S}^{0}K^{\pm}\pi^{\mp}$ CLEO:2012obf
; LHCb:2015lnk , but with large uncertainties; the Belle Collaboration has
studied $D^{0}\to\pi^{+}\pi^{-}\eta$ decays Belle:2021dfa , and has only
observed the $a_{0}(980)$ peak in the $M(\pi^{+}\eta)$ projection.

In analogy to what is observed in $D_{s}^{+}\to a_{0}(980)^{+(0)}\pi^{0(+)}$
decays BESIII:2019jjr , sizeable contributions from FSI are expected to
enhance the WA process in the corresponding $D^{+}$ decays. However, in this
case the symmetry observed in $D_{s}^{+}$ decays is expected to be violated,
since further short-distance contributions must be considered, which can be
expressed as a color-allowed external $W$-emission tree (T) diagram and a
color-suppressed external $W$-emission tree diagram in the topological diagram
approach Cheng:2022vbw . The measurement of the BFs for $D^{0}$ ($D^{+}$)
decays to $a_{0}(980)\pi$ and of the corresponding relative ratios $r_{+/-}$ (
$r_{+/0}=\mathcal{B}(D^{+}\to a_{0}(980)^{+}\pi^{0})/\mathcal{B}(D^{+}\to
a_{0}(980)^{0}\pi^{+})$) can constrain the size and phase of the amplitude of
the WE (WA) process and improve the knowledge about the role the $a_{0}(980)$
plays in charm decays Ikeno:2021kzf .

In this Letter, we perform amplitude analyses of
$D^{0(+)}\to\pi^{+}\pi^{-(0)}\eta$ decays, to study the contributions from the
intermediate processes $D^{0(+)}\to a_{0}(980)^{+}\pi^{-(0)}$, $D^{0(+)}\to
a_{0}(980)^{-(0)}\pi^{+}$ and $D^{0(+)}\to\rho^{0(+)}\eta$. The analyses are
based on $e^{+}e^{-}$ collision data recorded with the BESIII detector at the
center-of-mass energy of 3.773 GeV, corresponding to an integrated luminosity
of 7.9 ${\rm fb}^{-1}$. Charge conjugation is implied throughout this Letter,
as well as the equivalences $a_{0}(980)^{\pm(0)}\to\pi^{\pm(0)}\eta$ and
$\mathcal{B}(a_{0}(980)^{+}\to\pi^{+}\eta)=\mathcal{B}(a_{0}(980)^{0}\to\pi^{0}\eta)$.

A detailed description of the BESIII detector design and performance can be
found in Ref. detector . About 63$\%$ of the data analyzed in this Letter
profits from an upgrade of the end-cap time-of-flight system with multi-gap
resistive plate chambers with a time resolution of 60 ps MRPC . Simulated data
samples produced with a geant4-based sim  Monte Carlo (MC) package, which
includes the geometric description of the BESIII detector and the detector
response, are used to determine detection efficiencies and to estimate
backgrounds. The simulation models the beam-energy spread and initial-state
radiation (ISR) in $e^{+}e^{-}$ annihilations with the generator kkmc ref:kkmc
. The inclusive MC sample includes the production of $D\bar{D}$ pairs
(including quantum coherence for the neutral $D$ channels), the non-$D\bar{D}$
decays of the $\psi(3770)$, the ISR production of the $J/\psi$ and
$\psi(3686)$ states, and the continuum processes incorporated in kkmc ref:kkmc
.

The charged-track selection, particle identification (PID), $K_{S}^{0}$,
$\pi^{0}$ and $\eta$ reconstruction use the same criteria described in Ref.
BESIII:2019jjr , except for the invariant-mass window around the $\eta$, which
is set to $0.45<M(\gamma\gamma)_{\eta}<0.55$ GeV$/c^{2}$. The $D$ mesons are
identified using the beam-constrained mass
$M_{\mathrm{BC}}=\sqrt{E_{\mathrm{beam}}^{2}-|\vec{P}_{D}|^{2}}$ and the
deviation of the reconstructed energy from the expected energy $\Delta
E=E_{D}-E_{\mathrm{beam}}$, where $(E_{D},\vec{P}_{D})$ is the four-momentum
of the $D$ meson, and $E_{\mathrm{beam}}$ is the beam energy. A double-tag
(DT) technique MARK-III:1985hbd  is employed to suppress the background. On
both the tag and the signal side, any candidate with $M_{\mathrm{BC}}<1.83$
GeV$/c^{2}$ or $|\Delta E|>0.1$ GeV is rejected; if multiple combinations
survive in an event, the one with the $M_{\mathrm{BC}}$ closest to the known
$D$ meson mass from Particle Data Group (PDG) Workman:2022ynf  is retained.

For the $D^{0}$ channel, four tag modes ($\bar{D}^{0}\to K^{+}\pi^{-}$,
$\bar{D}^{0}\to K^{+}\pi^{-}\pi^{0}(\pi^{0})$ and $\bar{D}^{0}\to
K^{+}\pi^{-}\pi^{-}\pi^{+}$) are used, while for the $D^{+}$ channel we use
six tag modes ($D^{-}\to K^{+}\pi^{-}\pi^{-}(\pi^{0})$, $D^{-}\to
K^{0}_{S}\pi^{-}(\pi^{0})$, $D^{-}\to K^{0}_{S}\pi^{-}\pi^{-}\pi^{+}$ and
$D^{-}\to K^{+}K^{-}\pi^{-}$). Signal MC samples with $\psi(3770)\to
D\bar{D}$, $\bar{D}\to tag~{}modes$ and $D\to signal~{}modes$ are produced, in
which the signal decays $D^{0(+)}\to\pi^{+}\pi^{-(0)}\eta$ are generated with
the amplitude models that result from the studies presented in this Letter.
For the tag channels, the $M_{\mathrm{BC}}$ signal windows are set to be $\pm
6$ MeV$/c^{2}$ around the known $D$ mass Workman:2022ynf , while the $\Delta
E$ signal windows are set to be 3.5 times the $\Delta E$ resolution around the
fitted peak.

On the signal side, we select $D^{0}$ ($D^{+}$) candidates with
$M_{\mathrm{BC}}$ within $[1.858,~{}1.874]$ GeV$/c^{2}$ ($[1.860,~{}1.880]$
GeV$/c^{2}$). Furthermore, for the $D^{0}$ channel, the requirement
$|M(\pi^{+}\pi^{-})-m(K_{S}^{0})|>0.03$ GeV$/c^{2}$ is imposed to remove the
peaking background from the $D^{0}\to K_{S}^{0}\eta$ decays, where
$m(K_{S}^{0})$ is the known $K_{S}^{0}$ mass Workman:2022ynf . Since the
dominant background originates from wrong $\eta\to\gamma\gamma$ candidates, a
multivariate analysis (MVA) Hocker:2007ht  is performed to select events for
use in the amplitude analysis. This MVA involves the development of a Gradient
Boosted Decision Tree (BDTG) classifier based on the inclusive MC sample,
which operates on three discriminating variables: the $\gamma\gamma$ invariant
mass $M(\gamma\gamma)_{\eta}$, the goodness of the kinematic fit constraining
the $\gamma\gamma$ invariant mass to the known $\eta$ mass $\chi^{2}(\eta)$,
and the helicity angle of the higher energy photon from the $\eta$ decay. A
requirement on the BDTG output is imposed, which retains 83% (77%) of the
signal and rejects 78% (84%) of the background for the $D^{0}$ ($D^{+}$)
channel according to studies performed with MC simulation. Additionally, the
selection $|\Delta E|<0.045$ GeV ($|\Delta E|<0.040$ GeV) for the $D^{0}$
($D^{+}$) channel is applied. The final sample contains 1678 (1226)
$D^{0}(D^{+})\to\pi^{+}\pi^{-}(\pi^{0})\eta$ candidates with a purity of
$(74.1\pm 1.2)\%$ ($(65.7\pm 1.7)\%$).

The amplitude analysis is performed on the accepted candidate events with an
unbinned maximum-likelihood fit. The logarithm of the likelihood is
constructed as

$\displaystyle\begin{aligned} \ln
L=\ln(f_{s}\tilde{S}(p)+(1-f_{s})\tilde{B}(p)),\end{aligned}$ (1)

where $f_{s}$ is the signal purity, $p$ is the four-momenta of final
particles, $\tilde{S}(p)$ and $\tilde{B}(p)$ are the signal and background
probability density function (PDF), expressed as

$\displaystyle\begin{aligned}
\tilde{S}(p)&=\frac{\epsilon(p)|\mathcal{M}(p)|^{2}R_{3}(p)}{\int{\epsilon(p)|\mathcal{M}(p)|^{2}R_{3}(p)\mathrm{d}p}},\\\
\tilde{B}(p)&=\frac{\epsilon(p)B_{\epsilon}(p)R_{3}(p)}{\int{\epsilon(p)B_{\epsilon}(p)R_{3}(p)\mathrm{d}p}}.\end{aligned}$
(2)

Here, $R_{3}(p)$ is the three-body phase-space factor, $\epsilon(p)$ is the
efficiency function, $\mathcal{M}(p)$ is the signal amplitude, $B(p)$ is the
background shape, and $B_{\epsilon}(p)=B(p)/\epsilon(p)$ is the efficiency-
corrected background shape. For the $D^{0}$ channel, the $B(p)$ term is
extracted from the $\Delta E$ sideband region ($0.05<|\Delta E|<0.10$ GeV);
for the $D^{+}$ channel, since the resolution is much wider than for the
$D^{0}$, the inclusive MC sample is used. The total signal amplitude
$\mathcal{M}(p)=\sum_{\alpha}{c_{\alpha}e^{i\phi_{\alpha}}A_{\alpha}}$ is
modeled as the coherent sum of the amplitudes of all the intermediate
processes, where $c_{\alpha}$ and $\phi_{\alpha}$ are the magnitude and phase
of the $\alpha^{\mathrm{th}}$ amplitude, respectively, and the
$\alpha^{\mathrm{th}}$ amplitude is given by
$A_{\alpha}=P_{\alpha}S_{\alpha}F^{r}_{\alpha}F^{D}_{\alpha}$. Here,
$P_{\alpha}$ is the propagator, generally following the relativistic Breit-
Wigner formula except for the $\rho$ and $a_{0}(980)$ states. For the
$\rho^{+}$ we use the GS formula GS , and for the $\rho^{0}$ description
$\rho-\omega$ mixing is additionally considered BESIII:2019ymv . When modeling
the $a_{0}(980)$ the two channel-coupled Flatté formula
$P_{a_{0}(980)}=1/[(m_{0}^{2}-s_{a})-i(g_{\eta\pi}^{2}\rho_{\eta\pi}+g_{K\bar{K}}^{2}\rho_{K\bar{K}})]$
is used, with $g_{\pi\eta}$ ($g_{K\bar{K}}$) and $\rho_{\pi\eta}$
($\rho_{K\bar{K}}$) representing the coupling constant from Ref.
BESIII:2016tqo  and the phase-space factor $q/\sqrt{s_{a}}$, respectively,
where $s_{a}$ is the invariant-mass squared of the $a_{0}(980)$ candidate and
$q$ is the total momentum of the daughter particles in the $a_{0}(980)$ rest
frame. For $\pi^{+}\pi^{-}$ $S$-wave scattering, the K-matrix formalism
Anisovich:2002ij  is used, with parameters taken from Ref. LHCb:2015klp . The
$S_{\alpha}$ term is the spin factor and is constructed with the covariant-
tensor formalism Zou:2002ar . The $F^{r}_{\alpha}$ ($F^{D}_{\alpha}$) term is
the barrier factor for the intermediate state (the $D$ meson) BESIII:2022npc .
The relative contribution of the $\alpha^{\mathrm{th}}$ amplitude to the
$\beta^{\mathrm{th}}$ amplitude is quantified by the ratio
$r_{\alpha/\beta}=\int{|c_{\alpha}A_{\alpha}(p)|^{2}R_{3}(p)\mathrm{d}p}/\int{|c_{\beta}A_{\beta}(p)|^{2}R_{3}(p)\mathrm{d}p}$.
When substituting $c_{\beta}A_{\beta}$ with $\mathcal{M}(p)$, the ratio
becomes the fit fraction of the $\alpha^{\mathrm{th}}$ amplitude
($\mathrm{FF}_{\alpha}$).

In the data projections, a significant $\rho^{0}$ peak appears for the $D^{0}$
channel, while for the $D^{+}$ channel there is no evident $\rho^{+}$ peak but
a significant $a_{0}(980)^{+}$ peak is observed. Therefore, the decays
$D^{0}\to\rho^{0}\eta$ and $D^{+}\to a_{0}(980)^{+}\pi^{0}$ are chosen as the
reference amplitudes for the $D^{0}$ and $D^{+}$ channels, respectively. All
contributions with significance larger than $3\sigma$ are retained for further
analysis. Here, the significance is calculated using the changes of $\ln L$
and the number of degrees of freedom (NDF) when the fit is performed with and
without the corresponding amplitude included. Following this criterion, six
intermediate states are retained in the fit model for both channels. The decay
amplitudes and the corresponding $\phi_{\alpha}$, $\mathrm{FF}_{\alpha}$,
significance, BF, and $r_{+/-(0)}$ values are listed in Table 1.

Table 1: The phases, FFs, statistical significances and BFs for various
amplitudes. The first and second uncertainties are statistical and systematic,
respectively. The intermediate states are reconstructed in the decays
$\rho\to\pi\pi$, $a_{0}\to\pi\eta$ and $a_{2}\to\pi\eta$.

Amplitude | Phase (in unit rad) | FF (%) | Significance ($\sigma$) | BF ($\times 10^{-3}$)
---|---|---|---|---
$D^{0}\to\rho^{0}\eta$ |  $0$ (fixed) | $15.2\pm 1.7\pm 1.0$ | $>10$ | $0.19\pm 0.02\pm 0.01$
$D^{0}\to a_{0}(980)^{-}\pi^{+}$ |  $0.06\pm 0.16\pm 0.12$ |  $5.9\pm 1.3\pm 1.0$ |  8.9 | $0.07\pm 0.02\pm 0.01$
$D^{0}\to a_{0}(980)^{+}\pi^{-}$ | $-1.06\pm 0.12\pm 0.10$ | $44.0\pm 4.0\pm 5.3$ | $>10$ | $0.55\pm 0.05\pm 0.07$
$D^{0}\to a_{2}(1320)^{+}\pi^{-}$ | $-1.16\pm 0.25\pm 0.23$ |  $2.1\pm 0.9\pm 0.8$ |  4.5 | $0.03\pm 0.01\pm 0.01$
$D^{0}\to a_{2}(1700)^{+}\pi^{-}$ |  $0.08\pm 0.17\pm 0.23$ |  $5.5\pm 1.8\pm 2.7$ |  6.1 | $0.07\pm 0.02\pm 0.03$
$D^{0}\to(\pi^{+}\pi^{-})_{S-\mathrm{wave}}\eta$ | $-0.92\pm 0.29\pm 0.14$ |  $3.9\pm 1.8\pm 2.1$ |  5.3 | $0.05\pm 0.02\pm 0.03$
$r_{+/-}$ |  | $7.5^{+2.5}_{-0.8}\pm 1.7$ |  7.7* | -
$D^{+}\to\rho^{+}\eta$ | $-4.03\pm 0.19\pm 0.13$ |  $9.3\pm 3.0\pm 2.1$ |  6.0 | $0.20\pm 0.07\pm 0.05$
$D^{+}\to(\pi^{+}\pi^{0})_{V}\eta$ | $-0.64\pm 0.22\pm 0.19$ | $15.8\pm 4.8\pm 5.2$ |  4.7 | $0.34\pm 0.11\pm 0.11$
$D^{+}\to a_{0}(980)^{+}\pi^{0}$ |  $0$ (fixed) | $43.7\pm 5.6\pm 1.9$ |  9.1 | $0.95\pm 0.12\pm 0.05$
$D^{+}\to a_{0}(980)^{0}\pi^{+}$ | $~{}~{}\,2.44\pm 0.20\pm 0.10$ | $17.0\pm 4.4\pm 1.7$ |  7.9 | $0.37\pm 0.10\pm 0.04$
$D^{+}\to a_{2}(1700)^{+}\pi^{0}$ | $~{}~{}\,0.92\pm 0.20\pm 0.14$ |  $4.2\pm 2.1\pm 0.7$ |  3.6 | $0.09\pm 0.05\pm 0.02$
$D^{+}\to a_{0}(1450)^{+}\pi^{0}$ | $~{}~{}\,0.63\pm 0.41\pm 0.30$ |  $7.0\pm 2.8\pm 0.7$ |  4.7 | $0.15\pm 0.06\pm 0.02$
$r_{+/0}$ |  |  $2.6\pm 0.6\pm 0.3$ |  4.0* | -

  * *

The significance is for the test hypothesis $r=1.0$.

The Dalitz plots and the projections are shown in Fig. 1. The fit quality is
determined by calculating the $\chi^{2}$ of the fit using an adaptive binning
of the $M^{2}(\pi^{+}\eta)$ versus $M^{2}(\pi^{-}\eta)$ ($M^{2}(\pi^{0}\eta)$)
Dalitz plot with each bin containing at least 10 events. The resulting
$\chi^{2}/\mathrm{NDF}$ is $136.6/138$ ($131.6/99$) for the $D^{0}(D^{+})$
channel.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1: The Dalitz plot (a), the projections on $M(\pi^{+}\eta)$ (b),
$M(\pi^{-}\eta)$ (c), $M(\pi^{+}\pi^{-})$ (d) the for
$D^{0}\to\pi^{+}\pi^{-}\eta$ channel and the Dalitz plot (e), the projections
on $M(\pi^{+}\eta)$ (f), $M(\pi^{0}\eta)$(g), $M(\pi^{+}\pi^{0})$ (h) for the
$D^{+}\to\pi^{+}\pi^{0}\eta$ channel. In the projections, the dots with error
bars are data, the blue lines are the fit curves and the green histograms are
the backgrounds; the cyan solid, pink dashed and red dashed lines are the
contributions from the intermediate states $\rho^{0(+)}$, $a_{0}(980)^{-(0)}$
and $a_{0}(980)^{+}$, respectively.

In Table 1, the second uncertainties are systematic and arise from the
following sources: (I) the coupling with the $\pi\eta^{\prime}$ channel in the
$a_{0}(980)$ line shape BESIII:2016tqo ; (II) the $a_{0}(980)^{\pm}$ mass and
coupling constants, changed within the uncertainties given by Ref.
BESIII:2016tqo ; (III) the masses and widths of the $\rho^{0}$,
$a_{2}(1320)^{+}$ and $a_{2}(1700)^{+}$ states, changed within the
uncertainties given by PDG Workman:2022ynf ; (IV) the parameters in the
K-matrix formalism, changed within the statistical uncertainties given in Ref.
LHCb:2015klp  (only for the $D^{0}$ channel); (V) the effective radii for the
intermediate resonances and for the $D^{0(+)}$ state, estimated by varying the
effective radii by $\pm 1~{}\mathrm{GeV}^{-1}$; (VI) the background level,
estimated from the uncertainty of the signal ratio in data; (VII) the
background shape, estimated by checking the effect on the fit results with
background shapes extracted from the different sideband regions; (VIII) the
fitter performance, estimated by fitting three hundred signal MC samples with
the same size as the data sample. The fits show good agreement between the
fitted and the input values for the parameters in the amplitude model. These
systematic uncertainties are estimated separately by taking the difference
between the values of $\phi_{\alpha}$, $\mathrm{FF}_{\alpha}$, $r_{+/-}$ and
$r_{+/0}$ obtained by the alternative and the baseline fits. Since varying the
propagators causes different normalization factors, only the effect on the
$\mathrm{FF}_{\alpha}$ is considered BESIII:2019jjr  from source (I) for the
corresponding amplitudes, which is also similar for source (IV). The total
systematic uncertainties are obtained by adding each term in quadrature.

The BFs of each sub-process in Table 1 are calculated with
$\mathcal{B}_{\alpha}=\mathrm{FF}_{\alpha}\times\mathcal{B}(D^{0(+)}\to\pi^{+}\pi^{-(0)}\eta)$.
In the BF measurements, tighter windows than those used for the MVA
selections, $0.505<M(\gamma\gamma)_{\eta}<0.570$ GeV$/c^{2}$ and
$\chi^{2}(\eta)<50$, are used. The total decay BFs are measured with the DT
method as
$\mathcal{B}=Y_{\mathrm{DT}}/Y_{\mathrm{ST}}\epsilon_{\mathrm{sig}}\mathcal{B}_{\mathrm{sub}}$;
here, $Y_{\mathrm{ST}}$ is the total single tag (ST) yield, which is
$(6897.1\pm 8.2)\times 10^{3}$ [$(4176.9\pm 2.8)\times 10^{3}$] for the
$D^{0}$ ($D^{+}$) channel; $\mathcal{B}_{\mathrm{sub}}$ is the BF of
$\pi^{0}(\eta)\to\gamma\gamma$; $\epsilon_{\mathrm{sig}}$ is the weighted
signal efficiency
$\epsilon_{\mathrm{sig}}=\sum_{i}{\frac{Y^{(i)}_{\mathrm{ST}}}{\epsilon^{(i)}_{\mathrm{ST}}}\epsilon^{(i)}_{\mathrm{DT}}}/Y_{\mathrm{ST}}$,
where the $Y^{(i)}_{\mathrm{ST}}$ and $\epsilon^{(i)}_{\mathrm{ST~{}(DT)}}$
are the ST yield and ST (DT) efficiencies for the $i^{\mathrm{th}}$ tag
channels, respectively. The DT yields $Y_{\mathrm{DT}}$ are extracted using a
fit to the $\Delta E$ distributions without applying the signal window, as
shown in Fig. 2. In the fit, the signal PDF is parameterized as the sum of a
bifurcated Gaussian BifurcatedGauss  and a double Gaussian function, where the
two functions have the same mean value. All the parameters except for the mean
value are determined by the fit to the signal MC sample. The background
function is described by a second-order Chebychev polynomial, validated by
using the inclusive MC sample. From the fits, we obtain
$Y_{\mathrm{DT}}(D^{0})=1369\pm 48$ and $Y_{\mathrm{DT}}(D^{+})=949\pm 54$.

(a)

(b)

Figure 2: Fits to the $\Delta E$ distributions for (a)
$D^{0}\to\pi^{+}\pi^{-}\eta$ and (b) $D^{+}\to\pi^{+}\pi^{0}\eta$. The dots
with error bars are data. The blue solid, red dashed and green dashed lines
are the total fit, the signal and the background contribution, respectively.

The total BFs of the $D^{0}\to\pi^{+}\pi^{-}\eta$ and
$D^{+}\to\pi^{+}\pi^{0}\eta$ channels are measured to be $(1.24\pm
0.04_{\mathrm{stat.}}\pm 0.03_{\mathrm{syst.}})\times 10^{-3}$ and $(2.18\pm
0.12_{\mathrm{stat.}}\pm 0.05_{\mathrm{syst.}})\times 10^{-3}$, respectively.
Here, the systematic uncertainties for the $D^{0}$ and $D^{+}$ channels
include: PID (1.0% and 0.5%); tracking efficiency (1.0% and 0.5%);
$\eta/\pi^{0}$ reconstruction (0.8% and 1.6%), determined from hadronic DT
$D\bar{D}$ events; signal shape (0.3% and 0.1%), estimated by the change of
$Y_{\mathrm{\mathrm{DT}}}$ by altering the parameters in the signal shape
within the uncertainties; background shape (0.6% and 1.4%), estimated by using
a third-order Chebychev polynomial instead of the second-order one;
$M_{\mathrm{BC}}$ window (0.3% and 0.0%), determined with the $D^{0}\to
K^{-}\pi^{+}\eta$ control sample for the $D^{0}$ channel and negligible for
the $D^{+}$ channel due to the loose window requirement; MC sample size (0.1%
and 0.1%); quantum correlations (0.9%, only for the $D^{0}$ channel), quoted
from Ref. BESIII:2019xhl ; fitter performance (0.3% and 0.6%), estimated from
the inclusive MC sample; MC generator (0.2% and 0.4%), estimated by varying
the input parameters in the generator according to the error matrix obtained
from the fit to data; uncertainties from
$\mathcal{B}(\eta/\pi^{0}\to\gamma\gamma)$ (0.5% and 0.5%), quoted from the
PDG Workman:2022ynf .

From the amplitude model we calculate the ratios
$r_{+/-}=7.5^{+2.5}_{-0.8\,{\rm stat.}}\pm 1.7_{\rm syst.}$ and
$r_{+/0}=2.6\pm 0.6_{\rm stat.}\pm 0.3_{\rm syst.}$, which are both
significantly higher than unity. Especially for the $D^{0}$ channel, the
$r_{+/-}$ higher than the results of naive calculations that do not allow for
enhancements to the WE contributions by two orders of magnitude Cheng:2022vbw
.

In summary, we present the first amplitude analysis of the
$D^{0(+)}\to\pi^{+}\pi^{-(0)}\eta$ channel. The decays $D^{0(+)}\to
a_{0}(980)^{+}\pi^{-(0)}$ and $D^{0(+)}\to a_{0}(980)^{-(0)}\pi^{+}$ are
observed for the first time with statistical significances $>10\sigma$
($9.1\sigma$) and $8.9\sigma$ ($7.9\sigma$), respectively. The
$a_{0}(980)^{+}$ is identified as the dominant intermediate resonance in both
channels, and its contribution is found to be significantly larger than that
of the $a_{0}(980)^{-(0)}$ state.

For the $D^{+}$ channel, the measured value of $r_{+/0}$ indicates that the
symmetry observed in $D_{s}^{+}$ decays is here violated. Furthermore, in
contrast with the large BF of $D_{s}^{+}\to\rho^{+}\eta$, the low BF of
$D^{+}\to\rho^{+}\eta$ shows the importance of the $K^{*}K\to a_{0}(980)\pi$
re-scattering process. For the $D^{0}$ channel, the measured value of
$r_{+/-}$ disagrees with theoretical predictions that ignore non-perturbative
effects by two orders of magnitude. Estimations suggest that the size for the
amplitude of WE process is even larger than that of the T diagram
Cheng:2022vbw ; npcontribution . This dominance of the WE process in $D\to SP$
decays is to be contrasted with the situation in $D\to SP$, $D\to PP$ and
$D\to VP$ ($V$ denotes vector particle) decays Cheng:2022vbw ; Cheng:2024hdo ,
indicating the very important role that FSI plays in the $D\to SP$ sector. In
analogy to theoretical interpretations for WA process, the re-scattering
contributions also suggest that we would expect $r_{+/-}<1$ c2is0P3 , in
contradiction to our measurement. In addition, the resonance $a_{0}(1817)$ is
not observed in both channels. The measured BFs and ratios are highly valuable
for improving the understanding of the role that the $a_{0}(980)$ plays in
charm-meson decays and the theoretical interpretations of the nature of the
$a_{0}(980)$ state.

The authors greatly thank Professor H.Y. Cheng from Institute of Physics,
Academia Sinica for the useful discussions. 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. 2023YFA1606000, 2020YFA0406300, 2020YFA0406400; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 11635010,
11735014, 11935015, 11935016, 11935018, 11961141012, 12025502, 12035009,
12035013, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264,
12192265, 12221005, 12225509, 12235017, 12205384, 12361141819; the Chinese
Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS
Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207,
U2032104; 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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  * (32) The diagram for $D^{0}\to a_{0}(980)^{\pm}\pi^{\mp}$ consists of a T diagram and a WE diagram. To estimate the relative size of the WE amplitude over the T amplitude, we set the amplitude of the T diagram in $D^{0}\to a_{0}(980)^{-}\pi^{+}$ to be $1.0e^{i0.0}$, and denote that for the T diagram in $D^{0}\to a_{0}(980)^{+}\pi^{-}$ to be $be^{i\phi_{b}}$, those amplitudes for the WE diagrams for $a_{0}(980)^{-}$ and $a_{0}(980)^{+}$ involved sub-modes to be $xe^{i\phi_{-}}$ and $cxe^{i\phi_{+}}$, respectively, and write $r_{+/-}$ as $r$ for brevity. Here, the parameters $b$, $c$ and $x$ are real numbers. According to how the relative BF is determined, we obtain $\frac{|cxe^{i\phi_{+}}+be^{\phi_{b}}|^{2}}{|xe^{i\phi_{-}}+1|^{2}}=r$. Then $(r-c^{2})x^{2}+2[r\cos(\phi_{-}-\phi_{b})-cb\cos(\phi_{+}-\phi_{b})]x+(r-b^{2})=0$; therefore, the minimum of $|x|$ is $\sqrt{\frac{r-b^{2}}{r-c^{2}}}$. The parameter $b^{2}$ is expected to be less than 0.05 Cheng:2022vbw . The parameter $c$, the relative magnitude of the amplitude for the WE diagram in $D^{0}\to a_{0}(980)^{+}\pi^{-}$ over that in $D^{0}\to a_{0}(980)^{-}\pi^{+}$, can be estimated from the largest re-scattering contribution, which is $\frac{\mathcal{B}(D^{0}\to K^{+}K^{*-})}{\mathcal{B}(D^{0}\to K^{-}K^{*+})}$ c2is0P3 , therefore $c^{2}\sim 0.3$. It follows that we can expect that $b^{2}<c^{2}$, and so $|x|>1$.
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  * (34) In analogy to theoretical interpretations for WA process, re-scattering through $\rho\eta\to a_{0}(980)\pi$ and $K^{*}K\to a_{0}(980)\pi$ triangle diagrams can also be invoked to explain the large FSI in the WE process. However, since the BFs of $D^{0}\to\rho^{0}\eta$, $D^{0}\to K^{*0}\bar{K}^{0}$ and $D^{0}\to\bar{K}^{*0}K^{0}$ Workman:2022ynf  are small, the large FSI contribution is then expected from the re-scattering of $K^{*\pm}K^{\mp}\to a_{0}(980)^{\mp}\pi^{\pm}$. While through the triangle diagram, only the re-scatterings of $K^{-}K^{*+}\to\pi^{+}a_{0}(980)^{-}$ and $K^{+}K^{*-}\to\pi^{-}a_{0}(980)^{+}$ are allowed. Therefore if the WE contribution is dominant, $r_{+/-}\sim\frac{\mathcal{B}(D^{0}\to K^{+}K^{*-})}{\mathcal{B}(D^{0}\to K^{-}K^{*+})}$, which is only about 0.3.