# First observation of the decay $\Lambda^{+}_{c}\to nK^{0}_{S}\pi^{+}\pi^{0}$

M. Ablikim1, M. N. Achasov4,b, P. Adlarson75, O. Afedulidis3, X. C. Ai80, R.
Aliberti35, A. Amoroso74A,74C, Q. An71,58, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,g, H.-R. Bao63, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi74A,74C, E.
Bianco74A,74C, A. Bortone74A,74C, I. Boyko36, R. A. Briere5, A. Brueggemann68,
H. Cai76, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,63, N. Cao1,63, S. A.
Cetin62A, J. F. Chang1,58, W. L. Chang1,63, G. R. Che43, G. Chelkov36,a, C.
Chen43, C. H. Chen9, Chao Chen55, G. Chen1, H. S. Chen1,63, M. L. Chen1,58,63,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,63, X. R. Chen31,63, X. T.
Chen1,63, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,h, Z. Y. Chen1,63, S. K.
Choi10A, X. Chu43, G. Cibinetto29A, F. Cossio74C, J. J. Cui50, H. L. Dai1,58,
J. P. Dai78, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng72, Z. Y.
Deng1, A. Denig35, I. Denysenko36, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding46,g, Y. Ding34, Y. Ding40, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, M. C. Du1, S. X. Du80, Z. H. Duan42, P.
Egorov36,a, Y. H. Fan45, J. Fang1,58, J. Fang59, S. S. Fang1,63, W. X. Fang1,
Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava74B,74C, F. Feldbauer3, G.
Felici28A, C. Q. Feng71,58, J. H. Feng59, Y. T. Feng71,58, K. Fischer69, M.
Fritsch3, C. D. Fu1, J. L. Fu63, Y. W. Fu1, H. Gao63, Y. N. Gao46,g, Yang
Gao71,58, S. Garbolino74C, I. Garzia29A,29B, 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, Z. L. Guan22, A. Q. Guo31,63, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,f, A. Guskov36,a, J. Gutierrez27, K. L. Han63, T. T.
Han1, X. Q. Hao19, F. A. Harris65, K. K. He55, K. L. He1,63, F. H. Heinsius3,
C. H. Heinz35, Y. K. Heng1,58,63, C. Herold60, T. Holtmann3, P. C. Hong12,f,
G. Y. Hou1,63, X. T. Hou1,63, Y. R. Hou63, Z. L. Hou1, B. Y. Hu59, H. M.
Hu1,63, J. F. Hu56,i, T. Hu1,58,63, Y. Hu1, G. S. Huang71,58, K. X. Huang59,
L. Q. Huang31,63, X. T. Huang50, Y. P. Huang1, T. Hussain73, F. Hölzken3, N
Hüsken27,35, N. in der Wiesche68, M. Irshad71,58, J. Jackson27, S. Janchiv32,
J. H. Jeong10A, Q. Ji1, Q. P. Ji19, W. Ji1,63, X. B. Ji1,63, X. L. Ji1,58, Y.
Y. Ji50, X. Q. Jia50, Z. K. Jia71,58, D. Jiang1,63, H. B. Jiang76, P. C.
Jiang46,g, S. S. Jiang39, T. J. Jiang16, X. S. Jiang1,58,63, Y. Jiang63, J. B.
Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y. Jin66, M. Q. Jing1,63, X. M.
Jing63, T. Johansson75, S. Kabana33, N. Kalantar-Nayestanaki64, X. L. Kang9,
X. S. Kang40, M. Kavatsyuk64, B. C. Ke80, V. Khachatryan27, A. Khoukaz68, R.
Kiuchi1, O. B. Kolcu62A, B. Kopf3, M. Kuessner3, X. Kui1,63, N. Kumar26, A.
Kupsc44,75, W. Kühn37, J. J. Lane67, P. Larin18, L. Lavezzi74A,74C, T. T.
Lei71,58, Z. H. Lei71,58, H. Leithoff35, M. Lellmann35, T. Lenz35, C. Li47, C.
Li43, C. H. Li39, Cheng Li71,58, D. M. Li80, F. Li1,58, G. Li1, H. Li71,58, H.
B. Li1,63, H. J. Li19, H. N. Li56,i, Hui Li43, J. R. Li61, J. S. Li59, Ke Li1,
L. J Li1,63, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k, Q. M. Li1,63, Q.
X. Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,63, W. G. Li1, X. Li1,63,
X. H. Li71,58, X. L. Li50, Xiaoyu Li1,63, Y. G. Li46,g, Z. J. Li59, Z. X.
Li15, C. Liang42, H. Liang71,58, H. Liang1,63, Y. F. Liang54, Y. T.
Liang31,63, G. R. Liao14, L. Z. Liao50, Y. P. Liao1,63, J. Libby26, A.
Limphirat60, D. X. Lin31,63, T. Lin1, B. J. Liu1, B. X. Liu76, C. Liu34, C. X.
Liu1, F. H. Liu53, Fang Liu1, Feng Liu6, G. M. Liu56,i, H. Liu38,j,k, H. B.
Liu15, H. M. Liu1,63, Huanhuan Liu1, Huihui Liu21, J. B. Liu71,58, J. Y.
Liu1,63, K. Liu38,j,k, K. Y. Liu40, Ke Liu22, L. Liu71,58, L. C. Liu43, Lu
Liu43, M. H. Liu12,f, P. L. Liu1, Q. Liu63, S. B. Liu71,58, T. Liu12,f, W. K.
Liu43, W. M. Liu71,58, X. Liu38,j,k, X. Liu39, Y. Liu38,j,k, Y. Liu80, Y. B.
Liu43, Z. A. Liu1,58,63, Z. D. Liu9, Z. Q. Liu50, X. C. Lou1,58,63, F. X.
Lu59, H. J. Lu23, J. G. Lu1,58, X. L. Lu1, Y. Lu7, Y. P. Lu1,58, Z. H. Lu1,63,
C. L. Luo41, M. X. Luo79, T. Luo12,f, X. L. Luo1,58, X. R. Lyu63, Y. F. Lyu43,
F. C. Ma40, H. Ma78, H. L. Ma1, J. L. Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M.
Ma1, R. Q. Ma1,63, X. T. Ma1,63, X. Y. Ma1,58, Y. Ma46,g, Y. M. Ma31, F. E.
Maas18, M. Maggiora74A,74C, S. Malde69, A. Mangoni28B, Y. J. Mao46,g, Z. P.
Mao1, S. Marcello74A,74C, Z. X. Meng66, J. G. Messchendorp13,64, G.
Mezzadri29A, H. Miao1,63, T. J. Min42, R. E. Mitchell27, X. H. Mo1,58,63, B.
Moses27, N. Yu. Muchnoi4,b, J. Muskalla35, Y. Nefedov36, F. Nerling18,d, I. B.
Nikolaev4,b, Z. Ning1,58, S. Nisar11,l, Q. L. Niu38,j,k, W. D. Niu55, Y. Niu
50, S. L. Olsen63, Q. Ouyang1,58,63, S. Pacetti28B,28C, X. Pan55, Y. Pan57, A.
Pathak34, P. Patteri28A, Y. P. Pei71,58, M. Pelizaeus3, H. P. Peng71,58, Y. Y.
Peng38,j,k, K. Peters13,d, J. L. Ping41, R. G. Ping1,63, S. Plura35, V.
Prasad33, F. Z. Qi1, H. Qi71,58, H. R. Qi61, M. Qi42, T. Y. Qi12,f, S.
Qian1,58, W. B. Qian63, C. F. Qiao63, X. K. Qiao80, J. J. Qin72, L. Q. Qin14,
X. S. Qin50, Z. H. Qin1,58, J. F. Qiu1, S. Q. Qu61, Z. H. Qu72, C. F.
Redmer35, K. J. Ren39, A. Rivetti74C, M. Rolo74C, G. Rong1,63, Ch. Rosner18,
S. N. Ruan43, N. Salone44, A. Sarantsev36,c, Y. Schelhaas35, K. Schoenning75,
M. Scodeggio29A, K. Y. Shan12,f, W. Shan24, X. Y. Shan71,58, Z. J Shang38,j,k,
J. F. Shangguan55, L. G. Shao1,63, M. Shao71,58, C. P. Shen12,f, H. F.
Shen1,8, W. H. Shen63, X. Y. Shen1,63, B. A. Shi63, H. C. Shi71,58, J. L.
Shi12, J. Y. Shi1, Q. Q. Shi55, R. S. Shi1,63, S. Y. Shi72, X. Shi1,58, J. J.
Song19, T. Z. Song59, W. M. Song34,1, Y. J. Song12, Y. X. Song46,g,m, S.
Sosio74A,74C, S. Spataro74A,74C, F. Stieler35, Y. J. Su63, G. B. Sun76, G. X.
Sun1, H. Sun63, H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun76, S. S. Sun1,63, T.
Sun51,e, W. Y. Sun34, Y. Sun9, Y. J. Sun71,58, Y. Z. Sun1, Z. Q. Sun1,63, Z.
T. Sun50, C. J. Tang54, G. Y. Tang1, J. Tang59, Y. A. Tang76, L. Y. Tao72, Q.
T. Tao25,h, M. Tat69, J. X. Teng71,58, V. Thoren75, W. H. Tian59, Y.
Tian31,63, Z. F. Tian76, I. Uman62B, Y. Wan55, S. J. Wang 50, B. Wang1, B. L.
Wang63, Bo Wang71,58, D. Y. Wang46,g, F. Wang72, H. J. Wang38,j,k, J. P. Wang
50, K. Wang1,58, L. L. Wang1, M. Wang50, Meng Wang1,63, N. Y. Wang63, S.
Wang38,j,k, S. Wang12,f, T. Wang12,f, T. J. Wang43, W. Wang59, W. Wang72, W.
P. Wang71,58, X. Wang46,g, X. F. Wang38,j,k, X. J. Wang39, X. L. Wang12,f, X.
N. Wang1, Y. Wang61, Y. D. Wang45, Y. F. Wang1,58,63, Y. L. Wang19, Y. N.
Wang45, Y. Q. Wang1, Yaqian Wang17, Yi Wang61, Z. Wang1,58, Z. L. Wang72, Z.
Y. Wang1,63, Ziyi Wang63, D. Wei70, D. H. Wei14, F. Weidner68, S. P. Wen1, Y.
R. Wen39, U. Wiedner3, G. Wilkinson69, M. Wolke75, L. Wollenberg3, C. Wu39, J.
F. Wu1,8, L. H. Wu1, L. J. Wu1,63, X. Wu12,f, X. H. Wu34, Y. Wu71, Y. H. Wu55,
Y. J. Wu31, Z. Wu1,58, L. Xia71,58, X. M. Xian39, B. H. Xiang1,63, T.
Xiang46,g, D. Xiao38,j,k, G. Y. Xiao42, S. Y. Xiao1, Y. L. Xiao12,f, Z. J.
Xiao41, C. Xie42, X. H. Xie46,g, Y. Xie50, Y. G. Xie1,58, Y. H. Xie6, Z. P.
Xie71,58, T. Y. Xing1,63, C. F. Xu1,63, C. J. Xu59, G. F. Xu1, H. Y. Xu66, Q.
J. Xu16, Q. N. Xu30, W. Xu1, W. L. Xu66, X. P. Xu55, Y. C. Xu77, Z. P. Xu42,
Z. S. Xu63, F. Yan12,f, L. Yan12,f, W. B. Yan71,58, W. C. Yan80, X. Q. Yan1,
H. J. Yang51,e, H. L. Yang34, H. X. Yang1, Tao Yang1, Y. Yang12,f, Y. F.
Yang43, Y. X. Yang1,63, Yifan Yang1,63, Z. W. Yang38,j,k, Z. P. Yao50, M.
Ye1,58, M. H. Ye8, J. H. Yin1, Z. Y. You59, B. X. Yu1,58,63, C. X. Yu43, G.
Yu1,63, J. S. Yu25,h, T. Yu72, X. D. Yu46,g, Y. C. Yu80, C. Z. Yuan1,63, J.
Yuan34, L. Yuan2, S. C. Yuan1, Y. Yuan1,63, Z. Y. Yuan59, C. X. Yue39, A. A.
Zafar73, F. R. Zeng50, S. H. Zeng72, X. Zeng12,f, Y. Zeng25,h, Y. J. Zeng59,
Y. J. Zeng1,63, X. Y. Zhai34, Y. C. Zhai50, Y. H. Zhan59, A. Q. Zhang1,63, B.
L. Zhang1,63, B. X. Zhang1, D. H. Zhang43, G. Y. Zhang19, H. Zhang71, H. C.
Zhang1,58,63, H. H. Zhang59, H. H. Zhang34, H. Q. Zhang1,58,63, H. Y.
Zhang1,58, J. Zhang80, J. Zhang59, J. J. Zhang52, J. L. Zhang20, J. Q.
Zhang41, J. W. Zhang1,58,63, J. X. Zhang38,j,k, J. Y. Zhang1, J. Z. Zhang1,63,
Jianyu Zhang63, L. M. Zhang61, Lei Zhang42, P. Zhang1,63, Q. Y. Zhang39,80, R.
Y Zhang38,j,k, Shuihan Zhang1,63, Shulei Zhang25,h, X. D. Zhang45, X. M.
Zhang1, X. Y. Zhang50, Y. Zhang72, Y. T. Zhang80, Y. H. Zhang1,58, Y. M.
Zhang39, Yan Zhang71,58, Yao Zhang1, Z. D. Zhang1, Z. H. Zhang1, Z. L.
Zhang34, Z. Y. Zhang76, Z. Y. Zhang43, G. Zhao1, J. Y. Zhao1,63, J. Z.
Zhao1,58, Lei Zhao71,58, Ling Zhao1, M. G. Zhao43, R. P. Zhao63, S. J. Zhao80,
Y. B. Zhao1,58, Y. X. Zhao31,63, Z. G. Zhao71,58, A. Zhemchugov36,a, B.
Zheng72, J. P. Zheng1,58, W. J. Zheng1,63, Y. H. Zheng63, B. Zhong41, X.
Zhong59, H. Zhou50, J. Y. Zhou34, L. P. Zhou1,63, X. Zhou76, X. K. Zhou6, X.
R. Zhou71,58, X. Y. Zhou39, Y. Z. Zhou12,f, J. Zhu43, K. Zhu1, K. J.
Zhu1,58,63, L. Zhu34, L. X. Zhu63, S. H. Zhu70, S. Q. Zhu42, T. J. Zhu12,f, W.
J. Zhu12,f, Y. C. Zhu71,58, Z. A. Zhu1,63, J. H. Zou1, J. Zu71,58 (BESIII
Collaboration) 1 Institute of High Energy Physics, Beijing 100049, People’s
Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University, Kaifeng 475004, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Inner Mongolia University, Hohhot 010021, People’s Republic of China
31 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica
1000000, Chile
34 Jilin University, Changchun 130012, People’s Republic of China
35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
38 Lanzhou University, Lanzhou 730000, People’s Republic of China
39 Liaoning Normal University, Dalian 116029, People’s Republic of China
40 Liaoning University, Shenyang 110036, People’s Republic of China
41 Nanjing Normal University, Nanjing 210023, People’s Republic of China
42 Nanjing University, Nanjing 210093, People’s Republic of China
43 Nankai University, Tianjin 300071, People’s Republic of China
44 National Centre for Nuclear Research, Warsaw 02-093, Poland
45 North China Electric Power University, Beijing 102206, People’s Republic of
China
46 Peking University, Beijing 100871, People’s Republic of China
47 Qufu Normal University, Qufu 273165, People’s Republic of China
48 Renmin University of China, Beijing 100872, People’s Republic of China
49 Shandong Normal University, Jinan 250014, People’s Republic of China
50 Shandong University, Jinan 250100, People’s Republic of China
51 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
52 Shanxi Normal University, Linfen 041004, People’s Republic of China
53 Shanxi University, Taiyuan 030006, People’s Republic of China
54 Sichuan University, Chengdu 610064, People’s Republic of China
55 Soochow University, Suzhou 215006, People’s Republic of China
56 South China Normal University, Guangzhou 510006, People’s Republic of China
57 Southeast University, Nanjing 211100, People’s Republic of China
58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
59 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
60 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
61 Tsinghua University, Beijing 100084, People’s Republic of China
62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
63 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
64 University of Groningen, NL-9747 AA Groningen, The Netherlands
65 University of Hawaii, Honolulu, Hawaii 96822, USA
66 University of Jinan, Jinan 250022, People’s Republic of China
67 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
68 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
69 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
70 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
71 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
72 University of South China, Hengyang 421001, People’s Republic of China
73 University of the Punjab, Lahore-54590, Pakistan
74 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
75 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
76 Wuhan University, Wuhan 430072, People’s Republic of China
77 Yantai University, Yantai 264005, People’s Republic of China
78 Yunnan University, Kunming 650500, People’s Republic of China
79 Zhejiang University, Hangzhou 310027, People’s Republic of China
80 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan
m Also at Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne,
Switzerland

###### Abstract

Based on 4.5 fb-1 of $e^{+}e^{-}$ collision data accumulated at center-of-mass
energies between $4599.53\,\mbox{MeV}$ and $4698.82\,\mbox{MeV}$ with the
BESIII detector, the decay $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ is
observed for the first time with a significance of $9.2\sigma$. The branching
fraction is measured to be $(0.85\pm 0.13\pm 0.03)\%$, where the first
uncertainty is statistical and the second systematic, which differs from the
theoretical prediction based on isospin by 4.4$\sigma$. This indicates that
there may be resonant contributions or some unknown dynamics in this decay.

##  I Introduction

Experimental studies of charmed baryon decays provide important information on
the strong and weak interactions in the environment of heavy quarks. The
lightest charmed baryon, $\Lambda_{c}^{+}$, was first observed in $e^{+}e^{-}$
annihilation at the Mark II experiment Abrams:1979iu . So far, 70% of the
$\Lambda_{c}^{+}$ decays have been observed ParticleDataGroup:2020ssz ;
Li:2021iwf , of which the total branching fraction (BF) of the decays
involving a neutron is $(32.4\pm 1.7)\%$ BESIII:liqingzhang_work . To deeply
understand the properties of non-perturbative quantum chromodynamics, studies
of multi-body hadronic decays which include potential intermediate processes
are needed. Recently, the BFs of the Cabibbo-favored four body decays
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}\pi^{0}$ BESIII:2015bjk , $\Lambda_{c}^{+}\to
nK^{-}\pi^{+}\pi^{+}$ BESIII:yinghao_work , and $\Lambda_{c}^{+}\to
pK_{S}^{0}\pi^{+}\pi^{-}$ BESIII:2015bjk  were measured to be $(4.53\pm
0.38)\%$, $(1.90\pm 0.12)\%$, and $(1.53\pm 0.14)\%$, respectively. However,
the isospin-related decay $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ has
not been observed yet.

Theoretically, the decay amplitude of $\Lambda^{+}_{c}$ consists of
factorizable and non-factorizable contributions Chau:1986jb ; Chau:1995gk . It
is known that the non-factorizable contribution is negligible compared to the
factorizable one in describing the non-leptonic weak decays of charmed mesons
Bauer:1986bm . However, the contributions are different in $\Lambda^{+}_{c}$
decays, where the $W$-exchange diagram manifesting a pole diagram is no longer
subject to helicity and color suppression Cheng:1993gf . As shown in Fig. 1,
the $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ decay proceeds through
external and internal $W$-emission processes, where the dynamics includes both
factorizable and non-factorizable contributions. There has been much progress
in the theoretical and experimental studies of the two-body decays of the
$\Lambda_{c}^{+}$ CHENG2022324 . However, due to possible intermediate
resonances, the dynamics of multi-body decays of $\Lambda^{+}_{c}$ is more
complex, and theoretical calculations are not reliable yet. The decay
$\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ is dominated by the weak
interaction process $c\to su\bar{d}$. A phenomenological model based on
isospin Gronau:2018vei  predicts a BF for this decay of $(1.54\pm 0.08)\%$ and
further relates the BFs for all $N\bar{K}\pi\pi$ final states.

Figure 1: Feynman diagrams for $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ :
(a),(c),(e) External $W$-emission; (b),(d),(f) Internal $W$-emission.

In this paper, we report the first observation of $\Lambda_{c}^{+}\to
nK_{S}^{0}\pi^{+}\pi^{0}$ based on the data samples accumulated at center-of-
mass (c.m.) energies between $4599.53\,\mbox{MeV}$ and $4698.82\,\mbox{MeV}$
with the BESIII detector. These data samples correspond to an integrated
luminosity of $4.5~{}\rm fb^{-1}$ BESIII:2015zbz ; BESIII:2022dxl ;
BESIII:2022ulv ; Ke:2023qzc , as listed in Table 1. Due to the energy being
just above the threshold of $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pair
production, the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs are generated
without other accompanied hadrons, which makes it feasible to apply the
double-tag (DT) method MARK-III:1985hbd  and reconstruct the neutron with
missing neutron mass technique. The $\Lambda_{c}^{+}$ is reconstructed by
recoiling against the single-tag (ST) candidate $\bar{\Lambda}_{c}^{-}$ at
these c.m. energies, and an event containing an ST $\bar{\Lambda}^{-}_{c}$ and
a signal $\Lambda^{+}_{c}$ is referred to as a DT candidate. Charge-conjugated
decays are implied throughout this paper.

Table 1: The c.m. energies and integrated luminosities for each data sample, where the first uncertainties are statistical, and the second ones are systematic. $\sqrt{s}$ (MeV) | Luminosity ($\rm{pb}^{-1}$)
---|---
$4599.53\pm 0.07\pm 0.74$ | $586.90\pm 0.10\pm 3.90$
$4611.86\pm 0.12\pm 0.32$ | $103.65\pm 0.05\pm 0.55$
$4628.00\pm 0.06\pm 0.32$ | $521.53\pm 0.11\pm 2.76$
$4640.91\pm 0.06\pm 0.38$ | $551.65\pm 0.12\pm 2.92$
$4661.24\pm 0.06\pm 0.29$ | $529.43\pm 0.12\pm 2.81$
$4681.92\pm 0.08\pm 0.29$ | $1667.39\pm 0.21\pm 8.84$
$4698.82\pm 0.10\pm 0.39$ | $535.54\pm 0.12\pm 2.84$

##  II BESIII Experiment and Monte Carlo Simulation

The BESIII detector Ablikim:2009aa  records symmetric $e^{+}e^{-}$ collisions
provided by the BEPCII storage ring Yu:2016cof  in the c.m. energy ranging
from 2.0 to 4.95 GeV, with a peak luminosity of $1\times 10^{33}$ cm-2s-1
achieved at $\sqrt{s}=3.773$ GeV. BESIII has collected large data samples in
this energy region Ablikim:2019hff . The cylindrical core of the BESIII
detector covers 93% of the full solid angle and consists of a helium-based
multilayer drift chamber (MDC), a plastic scintillator time-of-flight system
(TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed
in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The
solenoid is supported by an octagonal flux-return yoke with resistive plate
counter muon identification modules interleaved with steel. The charged-
particle momentum resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the
resolution of energy deposited (d$E$/d$x$) is $6\%$ for electrons from Bhabha
scattering. The EMC measures photon energies with a resolution of $2.5\%$
($5\%$) at $1$ GeV in the barrel (end-cap) region. The time resolution in the
TOF barrel region is 68 ps, while that in the end-cap region is 110 ps. The
end cap TOF system was upgraded in 2015 using multigap resistive plate chamber
technology, providing a time resolution of 60 ps. About 87% of the data used
in this analysis benefits from this upgrade. More detailed descriptions can be
found in Refs. Ablikim:2009aa ; Yu:2016cof .

Monte-Carlo (MC) simulated samples are used to determine the detection
efficiencies, optimize selection criteria, and study backgrounds. The
simulation is carried out with a geant4-based geant4  package including the
geometric description of the BESIII detector and the detector response detvis
, and models the beam-energy spread and initial-state radiation (ISR) in the
$e^{+}e^{-}$ annihilation with the generator kkmc KKMC . Final-state radiation
from charged final-state particles is incorporated using the photos package
photos . The inclusive MC samples include the production of
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs, open-charmed mesons, ISR
production of vector charmonium(-like) states, and continuum processes, which
are used to determine ST efficiencies, analyze backgrounds, and extract
background shapes. Known decay modes are modeled with evtgen Lange:2001uf ;
Ping:2008zz  using BFs taken from the Particle Data Group (PDG)
ParticleDataGroup:2020ssz . The remaining unknown charmonium decays are
modeled with lundcharm lundcharm ; Yang:2014vra . The angular distribution of
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ is generated as
$1+\alpha\cos^{2}\theta_{\Lambda_{c}^{+}}$, where $\theta_{\Lambda_{c}^{+}}$
is the polar angle between the $\Lambda_{c}^{+}$ and the positron beam in the
c.m. frame, and $\alpha$ is the angular parameter of $\Lambda_{c}^{+}$
production, which is different at the seven c.m. energy points BESIII:2023rwv
. The signal MC sample consists of the exclusive process where the
$\bar{\Lambda}_{c}^{-}$ decays to eleven ST tag modes and the
$\Lambda_{c}^{+}$ decays to $nK_{S}^{0}\pi^{+}\pi^{0}$, with
$K_{S}^{0}\to\pi^{+}\pi^{-}$ and $\pi^{0}\to\gamma\gamma$, which is used to
determine DT efficiencies and extract signal shapes. The $\Lambda_{c}^{+}\to
nK_{S}^{0}\pi^{+}\pi^{0}$ signal MC sample is simulated using a phase space
(PHSP) model which then has the four daughter ($K_{S}^{0},n,\pi^{+},\pi^{0}$)
momentum and the $n\pi^{+}\pi^{0}$ invariant mass distributions weighted to
match those of data.

##  III Event Selection

The selection criteria of the ST candidates
$\bar{\Lambda}_{c}^{-}\to\bar{p}K_{S}^{0}$, $\bar{p}K^{+}\pi^{-}$,
$\bar{p}K^{+}\pi^{-}\pi^{0}$, $\bar{\Lambda}\pi^{-}$,
$\bar{\Lambda}\pi^{-}\pi^{0}$, $\bar{\Sigma}^{-}\pi^{-}\pi^{+}$,
$\bar{p}K_{S}^{0}\pi^{0}$, $\bar{\Sigma}^{-}\pi^{0}$,
$\bar{\Sigma}^{0}\pi^{-}$, $\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$, and
$\bar{p}K_{S}^{0}\pi^{-}\pi^{+}$ are the same as Ref. BESIII:yinghao_work .
The ST $\bar{\Lambda}_{c}^{-}$ is identified with the beam-constrained mass
$M_{\rm BC}\equiv\sqrt{E_{\rm beam}^{2}/c^{4}-p_{\Lambda_{c}}^{2}/c^{2}}$,
where $E_{\rm beam}$ is the beam energy and $p_{\Lambda_{c}}$ is the measured
momentum of the $\bar{\Lambda}_{c}^{-}$ in the c.m. system of $e^{+}e^{-}$
collision. Figure 2 shows the $M_{\rm BC}$ distributions of various ST modes
for the data sample at $\sqrt{s}=4681.92$ MeV; clear $\bar{\Lambda}_{c}^{-}$
signals are observed in each mode. The signal and sideband regions for the ST
candidates are defined as $(2.280,2.296)\,\mbox{GeV/$c^{2}$}$ and
$(2.250,2.270)\,\mbox{GeV/$c^{2}$}$, respectively. Candidates in the signal
region are used for the further DT reconstruction, and those in the sideband
region are used to estimate the background contribution.

\begin{overpic}[width=208.13574pt,angle={0}]{figure/fit_data_ST_v2.eps}
\end{overpic} Figure 2: The $M_{\rm BC}$ distributions of the ST modes for the
data sample at $\sqrt{s}=4681.92$ MeV. The points with error bars represent
data. The blue solid curves indicate the fit results, the red dashed curves
describe the background shapes, and the green lines are defined as signal
region.

The $K_{S}^{0}(\pi^{+}\pi^{-}$), $\pi^{+}$, $\pi^{0}(\gamma\gamma)$ from the
signal side decay $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ are
reconstructed with the following criteria. Charged tracks detected in the MDC
are required to be within a polar angle ($\theta$) range of
$|\cos\,\theta|<0.93$, where $\theta$ is defined with respect to the $z$-axis
which is the symmetry axis of the MDC. The $K_{S}^{0}$ candidate is
reconstructed from two oppositely charged tracks satisfying
$|V_{z}|<20\,\mathrm{cm}$, where $V_{z}$ denotes the distance to the
interaction point (IP) along $z$-axis. The two charged tracks are assigned as
$\pi^{+}\pi^{-}$ without imposing further particle identification (PID)
criteria to improve the $K_{S}^{0}$ reconstruction efficiency. They are
required to originate from a common vertex. The decay length ($L$) of the
$K_{S}^{0}$ candidate is required to be greater than twice the vertex
resolution ($\sigma_{L}$) away from IP, $i.e.$, $L/\sigma_{L}>2$. If there are
multiple $K_{S}^{0}$ candidates, the one with the largest $L/\sigma_{L}$ is
retained.

Apart from the $K_{S}^{0}$ candidate, there is one other charged track: the
$\pi^{+}$ from the $\Lambda_{c}^{+}$ decay. The distance of closest approach
to the IP for this track must be less than 10 cm along the z-axis,
$|V_{z}|<10\,$cm, and less than 1 cm in the transverse plane,
$|V_{xy}|<1\,$cm. The PID for this track combines the measurements of
d$E$/d$x$ in the MDC and the flight time in the TOF to form likelihoods
$\mathcal{L}(h)~{}(h=p,K,\pi)$ for each hadron ($h$) hypothesis. The track
needs to satisfy $\mathcal{L}(\pi)>\mathcal{L}(K)$.

The photon candidates are identified as showers in the EMC. The deposited
energy of each shower must be more than 25 MeV in the barrel region
($|\\!\cos\theta|<0.80$) and more than 50 MeV in the end cap region
($0.86<|\\!\cos\theta|<0.92$) of the EMC. To exclude showers that originate
from the charged track radiation, the angle subtended by the EMC shower and
the position of the closest charged track at the EMC must be greater than 10
degrees as measured from the IP. To suppress electronic noise and showers
unrelated to the event, the difference between the EMC time and the event
start time is required to be within [0, 700] ns. The invariant mass
$m_{\gamma\gamma}$ of the two photons from the $\pi^{0}$ decay has to satisfy
$0.115<m_{\gamma\gamma}<0.150$ GeV$/c^{2}$. In addition, a one-constraint (1C)
kinematic fit is performed to constrain $m_{\gamma\gamma}$ to the $\pi^{0}$
known mass ParticleDataGroup:2020ssz . The fit chi-squared, $\chi^{2}_{1\rm
C}$, is required to be less than 200. If there are more than one $\pi^{0}$
candidates, the one with the smallest $\chi^{2}_{1\rm C}$ is retained.

Considering that the neutron is difficult to detect, it is reconstructed with
the missing-mass technique, using the kinematic variable $M(n)=\sqrt{E_{\rm
miss}^{2}/c^{4}-\,|\vec{p}_{\rm miss}\,|^{2}/c^{2}}$. Here, the $E_{\rm miss}$
and $\vec{p}_{\rm miss}$ are calculated by $E_{\rm miss}=E_{\rm beam}-E_{\rm
rec}$ and $\vec{p}_{\rm miss}=\vec{p}_{\Lambda_{c}^{+}}-\vec{p}_{\rm rec}$,
where $E_{\rm rec}~{}(\vec{p}_{\rm rec})$ is the sum of the energies (vector
momenta) of the reconstructed $K_{S}^{0}$, $\pi^{+}$ and $\pi^{0}$ in the
$e^{+}e^{-}$ c.m. system. The $\Lambda_{c}^{+}$ momentum
$\vec{p}_{\Lambda_{c}^{+}}$ is derived by
$\vec{p}_{\Lambda_{c}^{+}}=-\hat{p}_{\rm tag}\sqrt{E_{\rm
beam}^{2}/c^{2}-m_{\Lambda_{c}^{+}}^{2}c^{2}}$, where $\hat{p}_{\rm tag}$ is
the unit vector of the $\bar{\Lambda}_{c}^{-}$ momentum direction and
$m_{\Lambda_{c}^{+}}$ is the $\Lambda_{c}^{+}$ nominal mass
ParticleDataGroup:2020ssz .

A study of the inclusive MC sample shows that the peaking backgrounds are
mainly from $\Lambda_{c}^{+}\to\Sigma^{-}(n\pi^{-})\,\pi^{+}\pi^{+}\pi^{0}$,
$\Lambda(n\pi^{0})\,\pi^{+}\pi^{-}\pi^{+}$, and
$\Sigma^{+}(n\pi^{+})\,\omega(\pi^{+}\pi^{-}\pi^{0})$. In order to remove
these peaking backgrounds, the difference of the invariant masses
$M(n\pi^{-})-M(n)$ is required to be outside the interval
$(0.22,0.27)\,\mbox{GeV/$c^{2}$}$ and $M(n\pi^{0})-M(n)$ must be larger than
$0.20\,\mbox{GeV/$c^{2}$}$. Also, both values of $M(n\pi^{+})-M(n)$ are
required to lie outside of the interval of $(0.23,0.28)\,\mbox{GeV/$c^{2}$}$,
where the two $M(n\pi^{+})$ combinations include the $\pi^{+}$ from either the
$\Lambda_{c}^{+}$ or the $K_{S}^{0}$. Here, $M(n\pi^{+})$, $M(n\pi^{-})$, and
$M(n\pi^{0})$ denote the invariant masses of the missing neutron and
$\pi^{+}$, $\pi^{-}$, and $\pi^{0}$, respectively.

##  IV Absolute BF measurement

The signal yield of $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ is obtained
by performing a two-dimensional (2D) unbinned maximum likelihood fit to the
$M(n)$ and $M(\pi^{+}\pi^{-})$ spectra of the candidates combined from the
data sets at seven c.m. energies shown in Fig. 3. The signal shapes ($f_{\rm
sig}$) are determined from signal MC samples convolved with a 2D Gaussian
function, which accounts for the difference in mass resolution between data
and MC simulation. This 2D Gaussian function is extracted by using two
Gaussian functions to fit the 2D $M(n)$ and $M(\pi^{+}\pi^{-})$ spectra in
data with the correlation between them taken into account.

\begin{overpic}[width=390.25534pt]{figure/simulFit_legend_v8_v3.eps}
\end{overpic} Figure 3: The 2D simultaneous fit on the accepted candidates in
the signal (top) and sideband (bottom) regions. The black dots with error bars
represent data. The green solid lines represent the total fit results. The
blue lines, purple lines, and red dashed lines represent the signal,
non-$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ background, and
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pair decay background, respectively.

In the fit, the probability density functions of the signal and sideband
regions are constructed as:

$\displaystyle f_{\rm signal}=$ $\displaystyle N_{\rm sig}\cdot f_{\rm
sig}+N_{\rm\Lambda_{c}^{+}bkg}\cdot f_{\rm\Lambda_{c}^{+}bkg}$ (1)
$\displaystyle+N_{\operatorname{\rm non-\Lambda_{c}^{+}}}\cdot
f_{\operatorname{\rm non-\Lambda_{c}^{+}}},$

$\displaystyle f_{\rm sideband}$
$\displaystyle=N^{{}^{\prime}}_{\operatorname{\rm non-\Lambda_{c}^{+}}}\cdot
f_{\operatorname{\rm non-\Lambda_{c}^{+}}}$ (2) $\displaystyle=h\cdot
N_{\operatorname{\rm non-\Lambda_{c}^{+}}}\cdot f_{\operatorname{\rm
non-\Lambda_{c}^{+}}},$

where $N_{\rm sig}$, $N_{\rm\Lambda_{c}^{+}bkg}$, and $N_{\operatorname{\rm
non-\Lambda_{c}^{+}}}$ denote the signal yield, the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ background yield, and the
non$\operatorname{-}\Lambda_{c}^{+}$ background yield in the signal region,
$N^{{}^{\prime}}_{\rm non\operatorname{-}\Lambda_{c}^{+}}$ denotes the
non$\operatorname{-}\Lambda_{c}^{+}$ yield in the sideband region. The ratio
$h$ between $N^{{}^{\prime}}_{\rm non\operatorname{-}\Lambda_{c}^{+}}$ and
$N_{\rm non\operatorname{-}\Lambda_{c}^{+}}$ is fixed to 1.262 by fitting the
$M_{\rm BC}$ distribution of the tagged samples. This method ensures that the
non$\operatorname{-}\Lambda_{c}^{+}$ (or $q\bar{q}$) background is well-
estimated and as done in the previous BESIII measurement of
$\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}$BESIII:2016yrc . The background from the
$\Lambda_{c}^{+}$ decays, $f_{\rm\Lambda_{c}^{+}bkg}$, is described by a shape
extracted from the inclusive MC sample. The background from the
non-$\Lambda_{c}^{+}$ decay is modeled as:

$\displaystyle f_{\rm non\operatorname{-}\Lambda_{c}^{+}}=\,((1-k)\cdot
f_{1}+k\cdot f_{K_{S}^{0}}\,)\cdot f_{2},$ (3)

where $f_{1}$ and $f_{2}$ represent the first-order Chebychev polynomials, $k$
is the fraction of the $K_{S}^{0}$ component and free in the fit, and
$f_{K_{S}^{0}}$ represents the shape of the $K_{S}^{0}$ signal extracted from
the signal MC samples which has the same $K_{S}^{0}$ shape as the background
process.

Fig. 3 shows the fit results. The total signal yield of $\Lambda_{c}^{+}\to
nK_{S}^{0}\pi^{+}\pi^{0}$ summing over eleven ST modes and seven c.m. energies
is determined to be $N^{\rm DT}=98\pm 15$. The BF of $\Lambda_{c}^{+}\to
nK_{S}^{0}\pi^{+}\pi^{0}$ is calculated by

$\mathcal{B}=\frac{N^{\rm DT}}{\sum_{ij}N_{ij}^{\rm
ST}\cdot(\varepsilon_{ij}^{\rm DT}/\varepsilon_{ij}^{\rm
ST})\cdot\mathcal{B}_{\rm int}},$ (4)

where the indices $i$ and $j$ are the eleven ST modes and seven c.m. energies,
respectively, and $\mathcal{B}_{\rm int}$ are the BFs of
$K_{S}^{0}\to\pi^{+}\pi^{-}$ and $\pi^{0}\to\gamma\gamma$
ParticleDataGroup:2020ssz . The DT efficiencies $\varepsilon_{ij}^{\rm DT}$,
ST yields $N^{\rm ST}_{ij}$, and ST efficiencies $\varepsilon_{ij}^{\rm ST}$
are listed in Tables 2, 3, and 4, respectively. The significance considering
systematic uncertainty (see below) is evaluated to be $9.2\sigma$ via
$\sqrt{-2\times\Delta\ln{\mathcal{L}}}$, where $\Delta{\ln\mathcal{L}}$ is the
variation in $\ln{\mathcal{L}}$ of the likelihood fit with and without the
signal component included. Here, the method of considering systematic
uncertainty is the same as the systematic uncertainty from fitting model
described in the next section, and the minimum significance among all the
fitting models is taken as the final significance.

Table 2: The ST yields, $N_{ij}^{\rm ST}$, at seven c.m. energies. The uncertainties are statistical only. Tag mode | 4599.53 MeV | 4611.86 MeV | 4628.00 MeV | 4640.91 MeV | 4661.24 MeV | 4681.92 MeV | 4698.82 MeV
---|---|---|---|---|---|---|---
$\bar{p}K_{S}^{0}$ | $1243\pm 35$ | $~{}\,226\pm 15$ | $~{}\,994\pm 33$ | $1048\pm 34$ | $1044\pm 33$ | $~{}\,3141\pm 57$ | $~{}\,889\pm 30$
$\bar{p}K^{+}\pi^{-}$ | $6607\pm 89$ | $1094\pm 37$ | $5513\pm 37$ | $5842\pm 83$ | $5447\pm 79$ | $~{}\,15919\pm 134$ | $4680\pm 73$
$\bar{p}K_{S}^{0}\pi^{0}$ | $~{}\,587\pm 33$ | $~{}\,119\pm 16$ | $~{}\,569\pm 33$ | $~{}\,552\pm 33$ | $~{}\,527\pm 32$ | $~{}\,1591\pm 56$ | $~{}\,414\pm 30$
$\bar{p}K_{S}^{0}\pi^{-}\pi^{+}$ | $~{}\,594\pm 33$ | $~{}\,100\pm 15$ | $~{}\,475\pm 30$ | $~{}\,484\pm 30$ | $~{}\,487\pm 21$ | $~{}\,1365\pm 51$ | $~{}\,414\pm 28$
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | $1965\pm 71$ | $~{}\,331\pm 30$ | $1453\pm 75$ | $1458\pm 63$ | $1460\pm 63$ | $~{}~{}~{}4361\pm 109$ | $1172\pm 62$
$\bar{\Lambda}\pi^{-}$ | $~{}\,738\pm 27$ | $~{}\,116\pm 11$ | $~{}\,636\pm 27$ | $~{}\,664\pm 27$ | $~{}\,624\pm 26$ | $~{}\,1916\pm 45$ | $~{}\,495\pm 23$
$\bar{\Lambda}\pi^{-}\pi^{0}$ | $1681\pm 54$ | $~{}\,281\pm 22$ | $1342\pm 50$ | $1483\pm 50$ | $1338\pm 46$ | $~{}\,3900\pm 78$ | $1145\pm 43$
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | $~{}\,744\pm 35$ | $~{}\,130\pm 14$ | $~{}\,547\pm 31$ | $~{}\,690\pm 34$ | $~{}\,703\pm 33$ | $~{}\,1847\pm 55$ | $~{}\,569\pm 31$
$\bar{\Sigma}^{0}\pi^{-}$ | $~{}\,502\pm 25$ | $~{}~{}~{}95\pm 12$ | $~{}\,384\pm 22$ | $~{}\,413\pm 23$ | $~{}\,414\pm 22$ | $~{}\,1267\pm 38$ | $~{}\,334\pm 20$
$\bar{\Sigma}^{-}\pi^{0}$ | $~{}\,309\pm 24$ | $~{}~{}~{}68\pm 10$ | $~{}\,242\pm 21$ | $~{}\,271\pm 22$ | $~{}\,264\pm 22$ | $~{}\,~{}\,770\pm 38$ | $~{}\,216\pm 21$
$\bar{\Sigma}^{-}\pi^{-}\pi^{+}$ | $1146\pm 47$ | $~{}\,204\pm 21$ | $~{}\,922\pm 19$ | $~{}\,995\pm 46$ | $~{}\,949\pm 44$ | $~{}\,2729\pm 79$ | $~{}\,848\pm 42$
Table 3: The ST efficiencies, $\varepsilon_{ij}^{\rm ST}$%, at seven c.m. energies. The uncertainties are statistical only. The quoted efficiencies do not include any sub-decay BFs. Tag mode | 4599.53 MeV | 4611.86 MeV | 4628.00 MeV | 4640.91 MeV | 4661.24 MeV | 4681.92 MeV | 4698.82 MeV
---|---|---|---|---|---|---|---
$\bar{p}K_{S}^{0}$ | $54.6\pm 0.2$ | $50.8\pm 0.6$ | $48.9\pm 0.2$ | $47.9\pm 0.2$ | $46.4\pm 0.2$ | $45.2\pm 0.1$ | $44.1\pm 0.2$
$\bar{p}K^{+}\pi^{-}$ | $49.9\pm 0.1$ | $47.8\pm 0.2$ | $46.1\pm 0.1$ | $45.3\pm 0.1$ | $44.3\pm 0.1$ | $42.8\pm 0.1$ | $41.9\pm 0.1$
$\bar{p}K_{S}^{0}\pi^{0}$ | $22.2\pm 0.2$ | $20.8\pm 0.4$ | $19.2\pm 0.2$ | $19.1\pm 0.2$ | $18.2\pm 0.2$ | $17.6\pm 0.1$ | $16.7\pm 0.2$
$\bar{p}K_{S}^{0}\pi^{-}\pi^{+}$ | $22.8\pm 0.2$ | $20.4\pm 0.4$ | $19.2\pm 0.2$ | $19.3\pm 0.2$ | $18.3\pm 0.2$ | $18.7\pm 0.1$ | $17.4\pm 0.2$
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | $19.4\pm 0.1$ | $18.1\pm 0.2$ | $16.8\pm 0.1$ | $16.2\pm 0.1$ | $15.7\pm 0.1$ | $15.4\pm 0.0$ | $14.9\pm 0.1$
$\bar{\Lambda}\pi^{-}$ | $47.1\pm 0.3$ | $44.2\pm 0.6$ | $40.7\pm 0.3$ | $40.2\pm 0.3$ | $38.8\pm 0.3$ | $38.2\pm 0.2$ | $36.2\pm 0.3$
$\bar{\Lambda}\pi^{-}\pi^{0}$ | $20.8\pm 0.1$ | $18.4\pm 0.2$ | $17.6\pm 0.1$ | $17.5\pm 0.1$ | $16.9\pm 0.1$ | $16.1\pm 0.1$ | $15.7\pm 0.1$
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | $15.1\pm 0.1$ | $12.7\pm 0.3$ | $12.7\pm 0.1$ | $13.2\pm 0.1$ | $12.7\pm 0.1$ | $12.5\pm 0.1$ | $13.0\pm 0.1$
$\bar{\Sigma}^{0}\pi^{-}$ | $28.4\pm 0.2$ | $24.8\pm 0.5$ | $25.3\pm 0.2$ | $24.2\pm 0.2$ | $24.0\pm 0.2$ | $23.2\pm 0.1$ | $21.9\pm 0.2$
$\bar{\Sigma}^{-}\pi^{0}$ | $22.8\pm 0.3$ | $21.0\pm 0.6$ | $21.5\pm 0.3$ | $22.3\pm 0.3$ | $20.5\pm 0.3$ | $19.6\pm 0.1$ | $18.3\pm 0.3$
$\bar{\Sigma}^{-}\pi^{-}\pi^{+}$ | $24.5\pm 0.1$ | $23.8\pm 0.3$ | $21.9\pm 0.1$ | $21.6\pm 0.1$ | $20.9\pm 0.1$ | $20.0\pm 0.1$ | $19.9\pm 0.1$
Table 4: The DT efficiencies of $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$, $\varepsilon_{ij}^{\rm DT}$%, at seven c.m. energies. The uncertainties are statistical only. The quoted efficiencies do not include any sub-decay BFs. Tag mode | 4599.53 MeV | 4611.86 MeV | 4628.00 MeV | 4640.91 MeV | 4661.24 MeV | 4681.92 MeV | 4698.82 MeV
---|---|---|---|---|---|---|---
$\bar{p}K_{S}^{0}$ | $8.3\pm 0.1$ | $8.0\pm 0.1$ | $7.5\pm 0.1$ | $7.0\pm 0.1$ | $6.8\pm 0.1$ | $6.7\pm 0.1$ | $6.5\pm 0.1$
$\bar{p}K^{+}\pi^{-}$ | $8.1\pm 0.1$ | $7.3\pm 0.1$ | $7.1\pm 0.1$ | $7.0\pm 0.1$ | $7.0\pm 0.1$ | $6.6\pm 0.2$ | $6.4\pm 0.1$
$\bar{p}K_{S}^{0}\pi^{0}$ | $3.1\pm 0.1$ | $2.9\pm 0.1$ | $2.8\pm 0.1$ | $2.6\pm 0.1$ | $2.7\pm 0.1$ | $2.6\pm 0.1$ | $2.4\pm 0.1$
$\bar{p}K_{S}^{0}\pi^{-}\pi^{+}$ | $3.0\pm 0.1$ | $2.5\pm 0.1$ | $2.4\pm 0.1$ | $2.4\pm 0.1$ | $2.4\pm 0.1$ | $2.3\pm 0.1$ | $2.3\pm 0.1$
$\bar{p}K^{+}\pi^{-}\pi^{0}$ | $3.1\pm 0.1$ | $2.8\pm 0.1$ | $2.6\pm 0.1$ | $2.6\pm 0.1$ | $2.4\pm 0.1$ | $2.3\pm 0.1$ | $2.3\pm 0.1$
$\bar{\Lambda}\pi^{-}$ | $7.2\pm 0.1$ | $6.3\pm 0.1$ | $5.9\pm 0.1$ | $5.9\pm 0.1$ | $5.9\pm 0.1$ | $5.5\pm 0.2$ | $5.5\pm 0.1$
$\bar{\Lambda}\pi^{-}\pi^{0}$ | $3.0\pm 0.1$ | $2.7\pm 0.1$ | $2.5\pm 0.1$ | $2.3\pm 0.1$ | $2.3\pm 0.1$ | $2.3\pm 0.1$ | $2.3\pm 0.1$
$\bar{\Lambda}\pi^{-}\pi^{+}\pi^{-}$ | $2.0\pm 0.1$ | $1.8\pm 0.1$ | $1.7\pm 0.1$ | $1.7\pm 0.1$ | $1.7\pm 0.1$ | $1.7\pm 0.1$ | $1.7\pm 0.1$
$\bar{\Sigma}^{0}\pi^{-}$ | $4.3\pm 0.1$ | $3.8\pm 0.1$ | $3.6\pm 0.1$ | $3.5\pm 0.1$ | $3.5\pm 0.1$ | $3.3\pm 0.1$ | $3.3\pm 0.1$
$\bar{\Sigma}^{-}\pi^{0}$ | $3.5\pm 0.1$ | $3.4\pm 0.1$ | $3.2\pm 0.1$ | $3.1\pm 0.1$ | $2.8\pm 0.1$ | $2.7\pm 0.1$ | $2.6\pm 0.1$
$\bar{\Sigma}^{-}\pi^{-}\pi^{+}$ | $3.8\pm 0.1$ | $3.6\pm 0.1$ | $3.4\pm 0.1$ | $3.3\pm 0.1$ | $3.2\pm 0.1$ | $3.0\pm 0.1$ | $2.9\pm 0.1$

In order to consider potential intermediate resonant states in the decay of
$\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$, the distributions of the
momenta of the four daughter particles $p(K_{S})$, $p(n)$, $p(\pi^{0})$,
$p(\pi^{+})$, and the invariant mass $M(n\pi^{+}\pi^{0})$ are re-weighted
according to the data to obtain the DT efficiency. The derivations of the ST
yields and ST efficiencies are the same as Ref. BESIII:yinghao_work . The BF
of $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ is calculated to be $(0.85\pm
0.13)\%$, where the uncertainty is statistical only.

##  V Systematic uncertainties

The systematic uncertainties include the $\pi^{+}$ tracking and PID, $\pi^{0}$
and $K_{S}^{0}$ reconstruction, fitting models of tag and signal sides, MC
statistics, mass window of peaking backgrounds, and MC model, as summarized in
Table 5.

The systematic uncertainty from the $\pi^{+}$ tracking and PID is studied by
using the control sample $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$. An
alternative efficiency is calculated by re-weighting events with momentum-
dependent efficiency correction factors extracted from the control sample. The
difference between the nominal and alternative efficiencies, 0.3%, is taken as
systematic uncertainty.

The systematic uncertainties due to the reconstruction of $\pi^{0}$ and
$K_{S}^{0}$ candidates are determined using the control samples $e^{+}e^{-}\to
K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$ and $J/\psi\to K^{*}(892)^{+}K^{-}\to
K_{S}^{0}\pi^{+}K^{-}$. The difference of efficiencies between data and MC
simulation is estimated with the same method used for $\pi^{+}$ tracking and
PID. The systematic uncertainty is estimated to be 0.9% for $K_{S}^{0}$
reconstruction and 0.2% for $\pi^{0}$ reconstruction. The uncertainties of the
BFs of $\pi^{0}\to\gamma\gamma$ and $K_{S}^{0}\to\pi^{+}\pi^{-}$, which are
quoted from the PDG ParticleDataGroup:2020ssz , are 0.03% and 0.1%,
respectively.

Uncertainties on the ST yields, DT efficiencies, and ST efficiencies all
contribute to systematic uncertainties. The propagated uncertainties, as
described by Eq. (4), lead to a total uncertainty of 0.4%.

The uncertainty of 0.2% on the fitting model of the ST side is quoted from
Ref. BESIII:yinghao_work . To estimate the uncertainty of the 2D signal shape
in the fit, we use two Gaussian functions to describe the signal contribution
and take the difference of the fitted signal yields, 0.6%, as the systematic
uncertainty. In the nominal fit, the ratio between $N^{{}^{\prime}}_{\rm
non\operatorname{-}\Lambda_{c}^{+}}$ and $N_{\rm
non\operatorname{-}\Lambda_{c}^{+}}$, is fixed at $1.262\pm 0.005$. To
evaluate the systematic uncertainty from the ratio, it is varied by one
standard deviation and the change of the signal yield is less than 0.1%, which
is negligible. In the nominal fit, the background shape is exacted from the
inclusive MC sample by RooKeysPdf Cranmer:2000du . The uncertainty of the
shape of $\Lambda_{c}^{+}$ background is considered by changing the smoothness
factor of the RooKeysPdf from 0 to 1, and the difference of the signal yield,
1.8%, is taken as the systematic uncertainty. Hence, the total systematic
uncertainty associated with the fitting model of signal side is 1.9%.

The systematic uncertainty of the peaking background window is estimated by
using the control samples of
$\Lambda_{c}^{+}\to\Sigma^{+}\,(n\pi^{+})\,\pi^{+}\pi^{-}$,
$\Sigma^{-}\,(n\pi^{-})\,\pi^{+}\pi^{+}$, and $\Lambda\,(n\pi^{0})\,\pi^{+}$.
Gaussian functions are used to describe the difference between data and MC
simulation. Using the simulated shapes convolved with Gaussian functions to
fit the distributions of $M(n\pi^{+})$, $M(n\pi^{-})$, and $M(n\pi^{0})$, the
parameters of the Gaussian functions are determined. The signal MC samples are
smeared based on the widths and means of these Gaussian functions. The
difference of the fitted signal yield, 0.3%, is taken as the systematic
uncertainty.

The nominal DT efficiencies are calculated by weighting the signal MC samples.
The efficiency difference between the weighted and PHSP signal MC samples,
3.0%, is taken as the systematic uncertainty.

The total systematic uncertainty, 3.7%, is obtained via the quadrature sum of
the individual components.

Table 5: Systematic uncertainties in the BF measurement.  Source |  $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ (%)
---|---
$\pi^{+}$ tracking and PID | 0.3
$\pi^{0}$ reconstruction | 0.2
$K_{S}^{0}$ reconstruction | 0.9
${\mathcal{B}}(K_{S}^{0}\to\pi^{+}\pi^{-})$ | 0.1
MC statistics | 0.4
Fitting model of tag side | 0.2
Fitting model of signal side | 1.9
Peaking background window | 0.3
MC model | 3.0
Total | 3.7

##  VI Summary

In summary, based on 4.5 fb-1 of $e^{+}e^{-}$ collision data samples taken at
c.m. energies between $4599.53\,\mbox{MeV}$ and $4698.82\,\mbox{MeV}$ with the
BESIII detector, the BF of $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ is
measured to be $(0.85\pm 0.13\pm 0.03)$%, with a significance of 9.2$\sigma$,
where the first uncertainty is statistical and the second systematic. Table 6
summarizes the BFs of $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ and its
isospin partners. The measured BF differs with the prediction of isospin
statistical model Gronau:2018vei , $(1.54\pm 0.08)$% by $4.4\sigma$. This
indicates that there may be resonant contributions or some unknown dynamics in
this decay. The total BF of the four body decay $\Lambda_{c}^{+}\to
N\bar{K}\pi\pi$ is predicted to be $(12.88\pm 0.69)$% Gronau:2018vei . Our BF,
together with the BFs of its isospin partners, offer important constraint on
the theoretical prediction.

Table 6: The BFs of $\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ and its isospin partners. Decay mode |  BF $(\times 10^{-2})$
---|---
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}\pi^{0}$ |  $4.53\pm 0.38$ BESIII:2015bjk
$\Lambda_{c}^{+}\to nK^{-}\pi^{+}\pi^{+}$ |  $1.90\pm 0.12$ BESIII:yinghao_work
$\Lambda_{c}^{+}\to pK_{S}^{0}\pi^{+}\pi^{-}$ |  $1.53\pm 0.14$ BESIII:2015bjk
$\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ (Theoretical prediction) |  $1.54\pm 0.08$ Gronau:2018vei
$\Lambda_{c}^{+}\to nK_{S}^{0}\pi^{+}\pi^{0}$ (This work) | $0.85\pm 0.13\pm 0.03$

##  VII ACKNOWLEDGMENTS

The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
12375070, 11635010, 11735014, 11835012, 11935015, 11935016, 11935018,
11961141012, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the
CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U2032108, No.
U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos.
QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; Fundamental
Research Funds for the Central Universities, Lanzhou University, University of
Chinese Academy of Sciences; The Institute of Nuclear and Particle Physics
(INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology;
European Union’s Horizon 2020 research and innovation programme under Marie
Sklodowska-Curie grant agreement under Contract No. 894790; German Research
Foundation DFG under Contracts Nos. 455635585, Collaborative Research Center
CRC 1044, FOR5327, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy;
Ministry of Development of Turkey under Contract No. DPT2006K-120470; National
Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National
Science and Technology fund of Mongolia; National Science Research and
Innovation Fund (NSRF) via the Program Management Unit for Human Resources &
Institutional Development, Research and Innovation of Thailand under Contract
No. B16F640076; Polish National Science Centre under Contract No.
2019/35/O/ST2/02907; The Swedish Research Council; U. S. Department of Energy
under Contract No. DE-FG02-05ER41374.

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