# First Measurement of the Decay Asymmetry of pure W-exchange Decay
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$

M. Ablikim1, M. N. Achasov4,b, P. Adlarson75, O. Afedulidis3, X. C. Ai80, R.
Aliberti35, A. Amoroso74A,74C, Q. An71,58, Y. Bai57, O. Bakina36, I.
Balossino29A, Y. Ban46,g, H.-R. Bao63, V. Batozskaya1,44, K. Begzsuren32, N.
Berger35, M. Berlowski44, M. Bertani28A, D. Bettoni29A, F. Bianchi74A,74C, E.
Bianco74A,74C, A. Bortone74A,74C, I. Boyko36, R. A. Briere5, A. Brueggemann68,
H. Cai76, X. Cai1,58, A. Calcaterra28A, G. F. Cao1,63, N. Cao1,63, S. A.
Cetin62A, J. F. Chang1,58, W. L. Chang1,63, G. R. Che43, G. Chelkov36,a, C.
Chen43, C. H. Chen9, Chao Chen55, G. Chen1, H. S. Chen1,63, M. L. Chen1,58,63,
S. J. Chen42, S. L. Chen45, S. M. Chen61, T. Chen1,63, X. R. Chen31,63, X. T.
Chen1,63, Y. B. Chen1,58, Y. Q. Chen34, Z. J. Chen25,h, Z. Y. Chen1,63, S. K.
Choi10A, X. Chu43, G. Cibinetto29A, F. Cossio74C, J. J. Cui50, H. L. Dai1,58,
J. P. Dai78, A. Dbeyssi18, R. E. de Boer3, D. Dedovich36, C. Q. Deng72, Z. Y.
Deng1, A. Denig35, I. Denysenko36, M. Destefanis74A,74C, F. De Mori74A,74C, B.
Ding66,1, X. X. Ding46,g, Y. Ding34, Y. Ding40, J. Dong1,58, L. Y. Dong1,63,
M. Y. Dong1,58,63, X. Dong76, M. C. Du1, S. X. Du80, Z. H. Duan42, P.
Egorov36,a, Y. H. Fan45, J. Fang59, J. Fang1,58, S. S. Fang1,63, W. X. Fang1,
Y. Fang1, Y. Q. Fang1,58, R. Farinelli29A, L. Fava74B,74C, F. Feldbauer3, G.
Felici28A, C. Q. Feng71,58, J. H. Feng59, Y. T. Feng71,58, K. Fischer69, M.
Fritsch3, C. D. Fu1, J. L. Fu63, Y. W. Fu1, H. Gao63, Y. N. Gao46,g, Yang
Gao71,58, S. Garbolino74C, I. Garzia29A,29B, P. T. Ge76, Z. W. Ge42, C.
Geng59, E. M. Gersabeck67, A. Gilman69, K. Goetzen13, L. Gong40, W. X.
Gong1,58, W. Gradl35, S. Gramigna29A,29B, M. Greco74A,74C, M. H. Gu1,58, Y. T.
Gu15, C. Y. Guan1,63, Z. L. Guan22, A. Q. Guo31,63, L. B. Guo41, M. J. Guo50,
R. P. Guo49, Y. P. Guo12,f, A. Guskov36,a, J. Gutierrez27, K. L. Han63, T. T.
Han1, X. Q. Hao19, F. A. Harris65, K. K. He55, K. L. He1,63, F. H. Heinsius3,
C. H. Heinz35, Y. K. Heng1,58,63, C. Herold60, T. Holtmann3, P. C. Hong12,f,
G. Y. Hou1,63, X. T. Hou1,63, Y. R. Hou63, Z. L. Hou1, B. Y. Hu59, H. M.
Hu1,63, J. F. Hu56,i, T. Hu1,58,63, Y. Hu1, G. S. Huang71,58, K. X. Huang59,
L. Q. Huang31,63, X. T. Huang50, Y. P. Huang1, T. Hussain73, F. Hölzken3, N
Hüsken27,35, N. in der Wiesche68, M. Irshad71,58, J. Jackson27, S. Janchiv32,
J. H. Jeong10A, Q. Ji1, Q. P. Ji19, W. Ji1,63, X. B. Ji1,63, X. L. Ji1,58, Y.
Y. Ji50, X. Q. Jia50, Z. K. Jia71,58, D. Jiang1,63, H. B. Jiang76, P. C.
Jiang46,g, S. S. Jiang39, T. J. Jiang16, X. S. Jiang1,58,63, Y. Jiang63, J. B.
Jiao50, J. K. Jiao34, Z. Jiao23, S. Jin42, Y. Jin66, M. Q. Jing1,63, X. M.
Jing63, T. Johansson75, S. Kabana33, N. Kalantar-Nayestanaki64, X. L. Kang9,
X. S. Kang40, M. Kavatsyuk64, B. C. Ke80, V. Khachatryan27, A. Khoukaz68, R.
Kiuchi1, O. B. Kolcu62A, B. Kopf3, M. Kuessner3, X. Kui1,63, A. Kupsc44,75, W.
Kühn37, J. J. Lane67, P. Larin18, L. Lavezzi74A,74C, T. T. Lei71,58, Z. H.
Lei71,58, H. Leithoff35, M. Lellmann35, T. Lenz35, C. Li43, C. Li47, C. H.
Li39, Cheng Li71,58, D. M. Li80, F. Li1,58, G. Li1, H. Li71,58, H. B. Li1,63,
H. J. Li19, H. N. Li56,i, Hui Li43, J. R. Li61, J. S. Li59, Ke Li1, L. J
Li1,63, L. K. Li1, Lei Li48, M. H. Li43, P. R. Li38,k, Q. M. Li1,63, Q. X.
Li50, R. Li17,31, S. X. Li12, T. Li50, W. D. Li1,63, W. G. Li1, X. Li1,63, X.
H. Li71,58, X. L. Li50, Xiaoyu Li1,63, Y. G. Li46,g, Z. J. Li59, Z. X. Li15,
C. Liang42, H. Liang1,63, H. Liang71,58, Y. F. Liang54, Y. T. Liang31,63, G.
R. Liao14, L. Z. Liao50, Y. P. Liao1,63, J. Libby26, A. Limphirat60, D. X.
Lin31,63, T. Lin1, B. J. Liu1, B. X. Liu76, C. Liu34, C. X. Liu1, F. H. Liu53,
Fang Liu1, Feng Liu6, G. M. Liu56,i, H. Liu38,j,k, H. B. Liu15, H. M. Liu1,63,
Huanhuan Liu1, Huihui Liu21, J. B. Liu71,58, J. Y. Liu1,63, K. Liu38,j,k, K.
Y. Liu40, Ke Liu22, L. Liu71,58, L. C. Liu43, Lu Liu43, M. H. Liu12,f, P. L.
Liu1, Q. Liu63, S. B. Liu71,58, T. Liu12,f, W. K. Liu43, W. M. Liu71,58, X.
Liu38,j,k, X. Liu39, Y. Liu80, Y. Liu38,j,k, Y. B. Liu43, Z. A. Liu1,58,63, Z.
D. Liu9, Z. Q. Liu50, X. C. Lou1,58,63, F. X. Lu59, H. J. Lu23, J. G. Lu1,58,
X. L. Lu1, Y. Lu7, Y. P. Lu1,58, Z. H. Lu1,63, C. L. Luo41, M. X. Luo79, T.
Luo12,f, X. L. Luo1,58, X. R. Lyu63, Y. F. Lyu43, F. C. Ma40, H. Ma78, H. L.
Ma1, J. L. Ma1,63, L. L. Ma50, M. M. Ma1,63, Q. M. Ma1, R. Q. Ma1,63, X. T.
Ma1,63, X. Y. Ma1,58, Y. Ma46,g, Y. M. Ma31, F. E. Maas18, M. Maggiora74A,74C,
S. Malde69, A. Mangoni28B, Y. J. Mao46,g, Z. P. Mao1, S. Marcello74A,74C, Z.
X. Meng66, J. G. Messchendorp13,64, G. Mezzadri29A, H. Miao1,63, T. J. Min42,
R. E. Mitchell27, X. H. Mo1,58,63, B. Moses27, N. Yu. Muchnoi4,b, J.
Muskalla35, Y. Nefedov36, F. Nerling18,d, I. B. Nikolaev4,b, Z. Ning1,58, S.
Nisar11,l, Q. L. Niu38,j,k, W. D. Niu55, Y. Niu 50, S. L. Olsen63, Q.
Ouyang1,58,63, S. Pacetti28B,28C, X. Pan55, Y. Pan57, A. Pathak34, P.
Patteri28A, Y. P. Pei71,58, M. Pelizaeus3, H. P. Peng71,58, Y. Y. Peng38,j,k,
K. Peters13,d, J. L. Ping41, R. G. Ping1,63, S. Plura35, V. Prasad33, F. Z.
Qi1, H. Qi71,58, H. R. Qi61, M. Qi42, T. Y. Qi12,f, S. Qian1,58, W. B. Qian63,
C. F. Qiao63, J. J. Qin72, L. Q. Qin14, X. S. Qin50, Z. H. Qin1,58, J. F.
Qiu1, S. Q. Qu61, Z. H. Qu72, C. F. Redmer35, K. J. Ren39, A. Rivetti74C, M.
Rolo74C, G. Rong1,63, Ch. Rosner18, S. N. Ruan43, N. Salone44, A.
Sarantsev36,c, Y. Schelhaas35, K. Schoenning75, M. Scodeggio29A, K. Y.
Shan12,f, W. Shan24, X. Y. Shan71,58, J. F. Shangguan55, L. G. Shao1,63, M.
Shao71,58, C. P. Shen12,f, H. F. Shen1,63, W. H. Shen63, X. Y. Shen1,63, B. A.
Shi63, H. C. Shi71,58, J. L. Shi12, J. Y. Shi1, Q. Q. Shi55, R. S. Shi1,63, S.
Y. Shi72, X. Shi1,58, J. J. Song19, T. Z. Song59, W. M. Song34,1, Y. J.
Song12, S. Sosio74A,74C, S. Spataro74A,74C, F. Stieler35, Y. J. Su63, G. B.
Sun76, G. X. Sun1, H. Sun63, H. K. Sun1, J. F. Sun19, K. Sun61, L. Sun76, S.
S. Sun1,63, T. Sun51,e, W. Y. Sun34, Y. Sun9, Y. J. Sun71,58, Y. Z. Sun1, Z.
Q. Sun1,63, Z. T. Sun50, C. J. Tang54, G. Y. Tang1, J. Tang59, Y. A. Tang76,
L. Y. Tao72, Q. T. Tao25,h, M. Tat69, J. X. Teng71,58, V. Thoren75, W. H.
Tian59, Y. Tian31,63, Z. F. Tian76, I. Uman62B, Y. Wan55, S. J. Wang 50, B.
Wang1, B. L. Wang63, Bo Wang71,58, D. Y. Wang46,g, F. Wang72, H. J.
Wang38,j,k, J. P. Wang 50, K. Wang1,58, L. L. Wang1, M. Wang50, Meng Wang1,63,
N. Y. Wang63, S. Wang12,f, S. Wang38,j,k, T. Wang12,f, T. J. Wang43, W.
Wang72, W. Wang59, W. P. Wang71,58, X. Wang46,g, X. F. Wang38,j,k, X. J.
Wang39, X. L. Wang12,f, X. N. Wang1, Y. Wang61, Y. D. Wang45, Y. F.
Wang1,58,63, Y. L. Wang19, Y. N. Wang45, Y. Q. Wang1, Yaqian Wang17, Yi
Wang61, Z. Wang1,58, Z. L. Wang72, Z. Y. Wang1,63, Ziyi Wang63, D. Wei70, D.
H. Wei14, F. Weidner68, S. P. Wen1, Y. R. Wen39, U. Wiedner3, G. Wilkinson69,
M. Wolke75, L. Wollenberg3, C. Wu39, J. F. Wu1,8, L. H. Wu1, L. J. Wu1,63, X.
Wu12,f, X. H. Wu34, Y. Wu71, Y. H. Wu55, Y. J. Wu31, Z. Wu1,58, L. Xia71,58,
X. M. Xian39, B. H. Xiang1,63, T. Xiang46,g, D. Xiao38,j,k, G. Y. Xiao42, S.
Y. Xiao1, Y. L. Xiao12,f, Z. J. Xiao41, C. Xie42, X. H. Xie46,g, Y. Xie50, Y.
G. Xie1,58, Y. H. Xie6, Z. P. Xie71,58, T. Y. Xing1,63, C. F. Xu1,63, C. J.
Xu59, G. F. Xu1, H. Y. Xu66, Q. J. Xu16, Q. N. Xu30, W. Xu1, W. L. Xu66, X. P.
Xu55, Y. C. Xu77, Z. P. Xu42, Z. S. Xu63, F. Yan12,f, L. Yan12,f, W. B.
Yan71,58, W. C. Yan80, X. Q. Yan1, H. J. Yang51,e, H. L. Yang34, H. X. Yang1,
Tao Yang1, Y. Yang12,f, Y. F. Yang43, Y. X. Yang1,63, Yifan Yang1,63, Z. W.
Yang38,j,k, Z. P. Yao50, M. Ye1,58, M. H. Ye8, J. H. Yin1, Z. Y. You59, B. X.
Yu1,58,63, C. X. Yu43, G. Yu1,63, J. S. Yu25,h, T. Yu72, X. D. Yu46,g, C. Z.
Yuan1,63, J. Yuan34, L. Yuan2, S. C. Yuan1, Y. Yuan1,63, Z. Y. Yuan59, C. X.
Yue39, A. A. Zafar73, F. R. Zeng50, S. H. Zeng72, X. Zeng12,f, Y. Zeng25,h, Y.
J. Zeng59, Y. J. Zeng1,63, X. Y. Zhai34, Y. C. Zhai50, Y. H. Zhan59, A. Q.
Zhang1,63, B. L. Zhang1,63, B. X. Zhang1, D. H. Zhang43, G. Y. Zhang19, H.
Zhang71, H. C. Zhang1,58,63, H. H. Zhang59, H. H. Zhang34, H. Q. Zhang1,58,63,
H. Y. Zhang1,58, J. Zhang59, J. Zhang80, J. J. Zhang52, J. L. Zhang20, J. Q.
Zhang41, J. W. Zhang1,58,63, J. X. Zhang38,j,k, J. Y. Zhang1, J. Z. Zhang1,63,
Jianyu Zhang63, L. M. Zhang61, Lei Zhang42, P. Zhang1,63, Q. Y. Zhang39,80,
Shuihan Zhang1,63, Shulei Zhang25,h, X. D. Zhang45, X. M. Zhang1, X. Y.
Zhang50, Y. Zhang72, Y. T. Zhang80, Y. H. Zhang1,58, Y. M. Zhang39, Yan
Zhang71,58, Yao Zhang1, Z. D. Zhang1, Z. H. Zhang1, Z. L. Zhang34, Z. Y.
Zhang76, Z. Y. Zhang43, G. Zhao1, J. Y. Zhao1,63, J. Z. Zhao1,58, Lei
Zhao71,58, Ling Zhao1, M. G. Zhao43, R. P. Zhao63, S. J. Zhao80, Y. B.
Zhao1,58, Y. X. Zhao31,63, Z. G. Zhao71,58, A. Zhemchugov36,a, B. Zheng72, J.
P. Zheng1,58, W. J. Zheng1,63, Y. H. Zheng63, B. Zhong41, X. Zhong59, H.
Zhou50, J. Y. Zhou34, L. P. Zhou1,63, X. Zhou76, X. K. Zhou6, X. R. Zhou71,58,
X. Y. Zhou39, Y. Z. Zhou12,f, J. Zhu43, K. Zhu1, K. J. Zhu1,58,63, L. Zhu34,
L. X. Zhu63, S. H. Zhu70, S. Q. Zhu42, T. J. Zhu12,f, W. J. Zhu12,f, Y. C.
Zhu71,58, Z. A. Zhu1,63, J. H. Zou1, J. Zu71,58 (BESIII Collaboration) 1
Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University, Kaifeng 475004, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Inner Mongolia University, Hohhot 010021, People’s Republic of China
31 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
32 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
33 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica
1000000, Chile
34 Jilin University, Changchun 130012, People’s Republic of China
35 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
36 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
37 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
38 Lanzhou University, Lanzhou 730000, People’s Republic of China
39 Liaoning Normal University, Dalian 116029, People’s Republic of China
40 Liaoning University, Shenyang 110036, People’s Republic of China
41 Nanjing Normal University, Nanjing 210023, People’s Republic of China
42 Nanjing University, Nanjing 210093, People’s Republic of China
43 Nankai University, Tianjin 300071, People’s Republic of China
44 National Centre for Nuclear Research, Warsaw 02-093, Poland
45 North China Electric Power University, Beijing 102206, People’s Republic of
China
46 Peking University, Beijing 100871, People’s Republic of China
47 Qufu Normal University, Qufu 273165, People’s Republic of China
48 Renmin University of China, Beijing 100872, People’s Republic of China
49 Shandong Normal University, Jinan 250014, People’s Republic of China
50 Shandong University, Jinan 250100, People’s Republic of China
51 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
52 Shanxi Normal University, Linfen 041004, People’s Republic of China
53 Shanxi University, Taiyuan 030006, People’s Republic of China
54 Sichuan University, Chengdu 610064, People’s Republic of China
55 Soochow University, Suzhou 215006, People’s Republic of China
56 South China Normal University, Guangzhou 510006, People’s Republic of China
57 Southeast University, Nanjing 211100, People’s Republic of China
58 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
59 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
60 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
61 Tsinghua University, Beijing 100084, People’s Republic of China
62 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
63 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
64 University of Groningen, NL-9747 AA Groningen, The Netherlands
65 University of Hawaii, Honolulu, Hawaii 96822, USA
66 University of Jinan, Jinan 250022, People’s Republic of China
67 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
68 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
69 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
70 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
71 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
72 University of South China, Hengyang 421001, People’s Republic of China
73 University of the Punjab, Lahore-54590, Pakistan
74 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
75 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
76 Wuhan University, Wuhan 430072, People’s Republic of China
77 Yantai University, Yantai 264005, People’s Republic of China
78 Yunnan University, Kunming 650500, People’s Republic of China
79 Zhejiang University, Hangzhou 310027, People’s Republic of China
80 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan

###### Abstract

Based on $4.4~{}\text{fb}^{-1}$ of $e^{+}e^{-}$ annihilation data collected at
the center-of-mass energies between $4.60$ and $4.70~{}\text{GeV}$ with the
BESIII detector at the BEPCII collider, the pure $W$-exchange decay
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ is studied with a full angular analysis. The
corresponding decay asymmetry is measured for the first time to be
$\alpha_{\Xi^{0}K^{+}}=0.01\pm 0.16({\rm stat.})\pm 0.03({\rm syst.})$. This
result reflects the interference between the $S$\- and $P$-wave amplitudes.
The phase shift between $S$\- and $P$-wave amplitudes is
$\delta_{p}-\delta_{s}=-1.55\pm 0.25({\rm stat.})\pm 0.05({\rm
syst.})~{}\text{rad}$.

Investigations of charmed baryon decay dynamics are essential to explore the
weak and strong interactions in the Standard Model (SM) of particle physics.
The ground state of the singly-charmed baryons, $\Lambda_{c}^{+}$, was
discovered in 1979 [1]. Many studies have since been made of the properties of
charmed baryons, such as the decay branching fractions (BFs) and decay
asymmetries. But experimental results of the decay asymmetries, which are
sensitive to the different amplitudes in the decay dynamics, have been
limited. Since 2014, there has been some progress on the weak hadronic decays
of $\Lambda_{c}^{+}$, $\Xi_{c}^{+,0}$, and $\Omega_{c}^{0}$, both
experimentally and theoretically [3, 3, 4]. This provides crucial information
about the properties of all the singly-charmed baryons and the searches for
doubly-charmed baryons ($\Xi_{cc}$ and $\Omega_{cc}$) [5]. Nonetheless, the
understanding of the decay dynamics of charmed baryons is still limited, due
to the lack of precision experimental measurements and the difficulties in the
theoretical treatment of strong interaction effects.

Compared to heavy meson decays, charmed baryon decays have a significant
dependence on nonfactorizable contributions from $W$-exchange diagrams.
However, these contributions cannot currently be calculated using theoretical
approaches. Additionally, no experimental measurements exist for the decay
asymmetries of $W$-exchange hadronic decays. An example of such a process is
the decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, which can only be produced via a
$W$-exchange process as depicted in Fig. 1. Experimental measurements of the
asymmetry parameters of the decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ can aid
understanding of the internal dynamics and can also explore charge-parity
($CP$) violation in baryons [5].

Figure 1: Feynman diagrams for $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$.

Table 1 lists the theoretical calculations of the BF and asymmetry parameters
of $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, as well as the experimental measurements
of the BF. Various predictions of the BFs based on the covariant confined
quark model (CCQM), the pole model, and current algebra (CA) in the 1990s are
all smaller than the experimental results [3]. This is explained as a strong
cancellation in the $S$\- and $P$-wave amplitudes. Moreover, the decay
asymmetry parameter was predicted to be zero in these models owing to the
vanishing $S$-wave amplitude [7, 8, 9, 10, 11]. This long-standing puzzle has
recently experienced a renewed interest in the theoretical community [5, 12,
13], especially after the report of new BF measured by BESIII in 2018 [14]. To
reproduce the relatively large experimental branching fraction of this decay,
Ref. [5] adopted a variant of the CA approach and obtained a larger
$\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})\simeq 0.71\%$. This modification
introduces a large positive decay asymmetry of $0.90$, which is quite close to
the calculations based on SU(3) symmetry [12, 13]. However, regarding the
significant enhancement of $\alpha_{\Xi^{0}K^{+}}$ from 0 to about 0.9, the
authors of Ref. [15] pointed out that the particular construction of the
$S$-wave amplitude in Ref. [5] is not well-justified. So experimental
measurement of the asymmetry parameter of the decay
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ will be crucial to test these calculations
and confirm the vanishing $S$-wave contribution [15].

Table 1: Theoretical calculations and experimental measurements of the BF, $\alpha_{\Xi^{0}K^{+}}$, $|A|$, $|B|$ and $\delta_{p}-\delta_{s}$ of $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$. $G_{F}$ is the Fermi constant. The superscript $a$ denotes a model with SU(3) symmetry, while model $b$ includes SU(3) symmetry-breaking effects. The PDG Fit BF also includes a CLEO result on $\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})/\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ [3]. Theory or experiment | $\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ | $\alpha_{\Xi^{0}K^{+}}$ | $|A|$ | $|B|$ | $\delta_{p}-\delta_{s}$
---|---|---|---|---|---
| ($\times 10^{-3}$) |  | $(\times 10^{-2}G_{F}~{}\text{GeV}^{2})$ | $(\times 10^{-2}G_{F}~{}\text{GeV}^{2})$ | $(\text{rad})$
Körner (1992), CCQM [7] | 2.6 | 0 | - | - | -
Xu (1992), Pole [8] | 1.0 | 0 | $0$ | $7.94$ | -
Źencaykowski (1994), Pole [9] | 3.6 | 0 | - | - | -
Ivanov (1998), CCQM [10] | 3.1 | 0 | - | - | -
Sharma (1999), CA [11] | 1.3 | 0 | - | - | -
Geng (2019), SU(3) [12] | $5.7\pm 0.9$ | $0.94^{+0.06}_{-0.11}$ | $2.7\pm 0.6$ | $16.1\pm 2.6$ | -
Zou (2020), CA [5] | $7.1$ | $0.90$ | $4.48$ | $12.10$ | -
Zhong (2022), SU(3)a [13] | $3.8^{+0.4}_{-0.5}$ | $0.91^{+0.03}_{-0.04}$ | $3.2\pm 0.2$ | $8.7^{+0.6}_{-0.8}$ | -
Zhong (2022), SU(3)b [13] | $5.0^{+0.6}_{-0.9}$ | $0.99\pm 0.01$ | $3.3^{+0.5}_{-0.7}$ | $12.3^{+1.2}_{-1.8}$ | -
BESIII (2018) [14] | $5.90\pm 0.86\pm 0.39$ | - | - | - | -
PDG Fit (2022) [3] | $5.5\pm 0.7$ | - | - | - | -

In the SM, the amplitude for a spin-$1/2$ baryon decaying into a spin-$1/2$
baryon and a spin-$0$ meson can be written as
$M=i\bar{u}_{f}(A-B\gamma_{5})u_{i}$, where $A$ and $B$ are constants, $u_{i}$
and $\bar{u}_{f}$ are spinors describing the initial and final baryons [8].
For the decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, the decay asymmetry is defined
by $\alpha_{\Xi^{0}K^{+}}=2\text{Re}(s^{*}p)/(|s|^{2}+|p|^{2})$, where $s=A$
and $p=|\vec{p}_{\Xi^{0}}|B/(E_{\Xi^{0}}+m_{\Xi^{0}})$; here $E_{\Xi^{0}}$ and
$\vec{p}_{\Xi^{0}}$ are the energy and momentum of the $\Xi^{0}$ in the
$\Lambda_{c}^{+}$ rest frame [3]. The effect of the $S$\- and $P$-wave phase
shift difference, $\delta_{p}-\delta_{s}$, is not well accounted for in the
theoretical calculations of decay asymmetries. It can be extracted from
experiments combined with the BF cited from the Particle Data Group (PDG) [3]
and provides an important experimental parameter for the theoretical
prediction of $CP$ violation [16].

In this Letter, we present the first measurement of the decay asymmetry of
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ and its decay dynamic parameters ($|A|$,
$|B|$, and $\delta_{p}-\delta_{s}$). A multidimensional angular analysis of
the cascade-decay $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$,
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, $\Xi^{0}\to\Lambda\pi^{0}$, and $\Lambda\to
p\pi^{-}$ is performed using a technique similar to that used to measure the
asymmetry parameters of $\Lambda_{c}^{+}\to pK_{S}^{0}$, $\Lambda\pi^{+}$,
$\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$ [17]. The data samples used in
this analysis, with an integrated luminosity of $4.4~{}\text{fb}^{-1}$ [18],
were collected at center-of-mass (CM) energies of $4.60$, $4.63$, $4.64$,
$4.66$, $4.68$, and $4.70~{}{\rm GeV}$ with the BESIII detector at the BEPCII
collider. Details about BEPCII as well as BESIII and its sub-detectors – a
helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-
flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC) – can be
found in Refs. [19, 20, 21, 22, 23]. Large data samples taken around the
$\Lambda^{+}_{c}\bar{\Lambda}_{c}^{-}$ production threshold allow to measure
decay asymmetry of low BF decays with the single-tag technique. The low-
background environment is more favorable to measure $\alpha_{\Xi^{0}K^{+}}$
accurately. Charge-conjugate modes are always implied unless explicitly stated
otherwise.

Simulated event samples produced with a geant4-based [24] Monte Carlo (MC)
package, which provides the geometric description of the BESIII detector and
the detector response, are used to determine detection efficiencies and to
estimate backgrounds. The simulation models the beam energy spread and initial
state radiation (ISR) in the $e^{+}e^{-}$ annihilations with the generator
kkmc [26, 27]. The inclusive MC sample includes the production of open charm
processes, the ISR production of vector charmonium(-like) states, and the
continuum processes incorporated in kkmc. All particle decays are modelled
with evtgen [28, 29] using BFs either taken from the PDG [3], when available,
or otherwise estimated with lundcharm [30, 31]. Final state radiation from
charged final state particles is incorporated using the photos package [32].
The phase space MC sample is generated uniformly over phase space, which is
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ followed by
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$, $\Xi^{0}\to\Lambda(\to
p\pi^{-})\pi^{0}(\to\gamma\gamma)$ and $\bar{\Lambda}_{c}^{-}$ decaying
inclusively. For the signal MC sample, the signal process is generated by the
helicity formalism using the decay asymmetry parameters measured in this work
or cited from the PDG [3].

The tracks of $K^{+}$ candidates detected in the MDC are required to be within
a polar angle ($\theta$) range of $|\rm{cos\theta}|<0.93$, where $\theta$ is
defined with respect to the $z$ axis, which is the symmetry axis of the MDC.
Besides, the distance of the closest approach to the interaction point (IP)
must be less than 10 cm along the $z$ axis, $|V_{z}|$, and less than 1 cm in
the transverse plane, $|V_{xy}|$. The particle identification (PID) combining
the $\text{d}E/\text{d}x$ information from the MDC and the flight time from
the TOF is implemented to evaluate the likelihoods
$\mathcal{L}(h)~{}(h=\pi,K,p)$ for different hadron hypotheses. Charged tracks
satisfying $\mathcal{L}(K)>\mathcal{L}(\pi)$, and $\mathcal{L}(K)>0$ are
assigned as kaon candidates.

The $\pi^{0}$ candidates are formed by photon pairs with an invariant mass in
the interval $0.115<M_{\gamma\gamma}<0.150~{}\text{GeV}/c^{2}$. The $\gamma$
candidates are reconstructed from showers in the EMC that are not associated
with any charged track. The deposited energy must be larger than 25 $\rm MeV$
in the barrel region $(|\text{cos}\theta|<0.80)$ or 50 $\rm MeV$ in the end
cap region $(0.86<|\text{cos}\theta|<0.92)$. 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. A mass-constrained fit
to the known $\pi^{0}$ mass [3] is applied to the photon pair and the
resulting energy and momentum of the $\pi^{0}$ is used for further analysis.

The $\Lambda$ candidates are obtained from two oppositely charged tracks in
the final states $p\pi^{-}$. Their distance of closest approaches to the IP
must be within $\pm 20~{}\text{cm}$ along the beam direction. In order to
improve signal purity, the proton PID is required to satisfy
$\mathcal{L}(p)>\mathcal{L}(K)$, $\mathcal{L}(p)>\mathcal{L}(\pi)$ and
$\mathcal{L}(p)>0$. The two daughter tracks are constrained to originate from
a common decay vertex by requiring the vertex fit $\chi^{2}$ to be less than
$20$. The momentum of the $\Lambda$ corrected by the vertex fit is used in
subsequent analysis. Furthermore, the distance of the $\Lambda$ decay vertex
from the IP is required to be greater than twice the fitted vertex resolution.
The $p\pi^{-}$ invariant mass is required to lie in the $\Lambda$ mass region
$(1.111,1.121)~{}\text{GeV}/c^{2}$. The $\Xi^{0}$ candidates are formed by
$\Lambda\pi^{0}$ combinations and the invariant mass $M_{\Lambda\pi^{0}}$ must
be within the mass region $(1.30,1.33)~{}\text{GeV}/c^{2}$. These mass regions
are selected to be about three times the resolution.

Two kinematic variables, the energy difference $\Delta E\equiv
E_{\Lambda_{c}^{+}}-E_{\text{beam}}$ and the beam-constrained mass
$M_{\text{BC}}\equiv\sqrt{E_{\text{beam}}^{2}/c^{4}-|\vec{p}_{\Lambda_{c}^{+}}|^{2}/c^{2}}$,
are defined to identify $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ candidates. Here,
$E_{\Lambda_{c}^{+}}$ and $\vec{p}_{\Lambda_{c}^{+}}$ are the reconstructed
energy and momentum of the $\Lambda_{c}^{+}$ candidates calculated in the
$e^{+}e^{-}$ rest frame, and $E_{\text{beam}}$ is the average energy of the
$e^{+}$ and $e^{-}$ beams. All candidates are required to satisfy
$-0.05<\Delta E<0.05~{}\text{GeV}$ and
$2.25~{}\text{GeV}/c^{2}<M_{\text{BC}}<E_{\text{beam}}$. If more than one
candidate satisfies the above requirements, the one with minimal $|\Delta E|$
is kept. After applying these conditions, the $M_{\text{BC}}$ distribution in
data collected at $4.60~{}{\rm GeV}$ is shown in the Fig. 2. In the fit to
this data, the correctly and mis-reconstructed signal shapes are modeled with
the MC-simulated shape convolved with a Gaussian function representing the
resolution difference between data and MC simulation, and the background shape
is described by an ARGUS function [33]. Finally, $378\pm 21$ signal events are
obtained by combining the six energy points.

Figure 2: Distribution of the $M_{\text{BC}}$ fitting result at $4.60~{}{\rm
GeV}$. Black points with error bars are data; the blue shaded region indicates
the combinatorial background events and pink shaded region is the mis-
reconstructed signal events.

The decay asymmetry parameters are determined by analyzing the multi-
dimensional angular distributions, where the full cascade-decay chain is
considered. The joint angular formula is obtained using helicity basis [34].
Figure 3 illustrates the definitions of the helicity angles for the three-
level cascade decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$,
$\Xi^{0}\to\Lambda\pi^{0}$, and $\Lambda\to p\pi^{-}$ following the process of
$e^{+}e^{-}\to\gamma^{*}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$. In the
helicity frame of the $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$,
$\theta_{0}$ is the polar angle of the $\Lambda_{c}^{+}$ with respect to the
$e^{+}$ beam axis in the $e^{+}e^{-}$ CM system. For the
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ decay, $\phi_{1}$ is the angle between the
$e^{+}\Lambda_{c}^{+}$ and $\Xi^{0}K^{+}$ planes, and $\theta_{1}$ is the
polar angle of the $\Xi^{0}$ momentum in the $\Lambda_{c}^{+}$ rest frame with
respect to the $\Lambda_{c}^{+}$ momentum in the CM frame. For the
$\Xi^{0}\to\Lambda\pi^{0}$ decay, $\phi_{2}$ is the angle between the
$\Xi^{0}K^{+}$ and $\Lambda\pi^{0}$ planes, and $\theta_{2}$ is the polar
angle of the $\Lambda$ momentum with respect to the $\Xi^{0}$ momentum in the
rest frame of $\Lambda_{c}^{+}$. For the helicity angles describing the
$\Lambda\to p\pi^{-}$ decay, $\phi_{3}$ is the angle between the
$\Lambda\pi^{0}$ and $p\pi^{-}$ planes, and $\theta_{3}$ is the polar angle of
the proton with respect to the $\Lambda$ momentum in the $\Xi^{0}$ rest frame.

Figure 3: Definitions of the helicity frames and related angles for
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-},\Lambda_{c}^{+}\to\Xi^{0}K^{+},\Xi^{0}\to\Lambda\pi^{0}$,
and $\Lambda\to p\pi^{-}$.

In Ref. [34], $\Delta_{0}$ is defined as the phase shift between two
individual helicity amplitudes, $\mathcal{H}_{\lambda_{1},\lambda_{2}}$, for
the $\Lambda_{c}^{+}$ production process
$\gamma^{*}\to\Lambda_{c}^{+}(\lambda_{1})\bar{\Lambda}_{c}^{-}(\lambda_{2})$
with total helicities $|\lambda_{1}-\lambda_{2}|=0$ and $1$, respectively. In
the case where one-photon exchange dominates the production process,
$\Delta_{0}$ is also the phase between the electric and magnetic form factors
of $\Lambda_{c}^{+}$ [35, 36], and $\alpha_{0}$ is the angular distribution
parameter of $\Lambda_{c}^{+}$ defined by the helicity amplitude
$\alpha_{0}=(|\mathcal{H}_{1/2,-1/2}|^{2}-2\,|\mathcal{H}_{1/2,1/2}|^{2})/(|\mathcal{H}_{1/2,-1/2}|^{2}+2\,|\mathcal{H}_{1/2,1/2}|^{2})$.
Similarly, the $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ decay is described by two
parameters, $\alpha_{\Xi^{0}K^{+}}$ and $\Delta_{\Xi^{0}K^{+}}$, where the
latter one is the phase shift between the two helicity amplitudes. The Lee-
Yang parameters [34, 1] can be obtained with the relations

$\begin{split}\beta_{\Xi^{0}K^{+}}&=\sqrt{1-(\alpha_{\Xi^{0}K^{+}})^{2}}~{}\text{sin}\Delta_{\Xi^{0}K^{+}},\\\
\gamma_{\Xi^{0}K^{+}}&=\sqrt{1-(\alpha_{\Xi^{0}K^{+}})^{2}}~{}\text{cos}\Delta_{\Xi^{0}K^{+}}.\\\
\end{split}$ (1)

In this analysis, the common free parameters ($\alpha_{\Xi^{0}K^{+}}$ and
$\Delta_{\Xi^{0}K^{+}}$) describing the angular distributions for the six data
sets are determined by a simultaneous unbinned maximum likelihood fit. The
likelihood function is constructed from the joint probability density function
(PDF) by

$\mathcal{L}_{\text{total}}=\sum^{\text{energy}}\mathcal{L}_{\text{data}}=\prod_{i=1}^{N_{\text{data}}}f_{s}(\vec{\xi}_{i}).$
(2)

Here, $f_{s}(\vec{\xi}_{i})$ is the PDF of the signal process,
$N_{\text{data}}$ is the number of events in data and $i$ is the event index.
The signal PDF $f_{s}(\vec{\xi}_{i})$ is formulated as

$f_{s}(\vec{\xi}_{i})=\frac{\epsilon(\vec{\xi}_{i})|M(\vec{\xi}_{i};\vec{\eta})|^{2}}{\int\epsilon(\vec{\xi}_{i})|M(\vec{\xi}_{i};\vec{\eta})|^{2}\text{d}\vec{\xi}_{i}}~{},$
(3)

where $\vec{\xi}_{i}$ denotes the kinematic angular observables
($\theta_{0,1,2,3}$ and $\phi_{1,2,3}$) and $\vec{\eta}$ denotes the free
parameters ($\alpha_{\Xi^{0}K^{+}}$ and $\Delta_{\Xi^{0}K^{+}}$) to be
determined. The $M(\vec{\xi}_{i};\vec{\eta})$ is the total decay amplitude
[34] and $\epsilon(\vec{\xi}_{i})$ is the detection efficiency parameterized
in terms of the kinematic variables $\vec{\xi}_{i}$. The background
contribution to the joint likelihood is subtracted according to the calculated
likelihoods for the combinatorial background based on the inclusive MC
simulation and for the mis-reconstructed signal events based on the signal MC
simulation. The integration of the normalization factor is calculated with a
large phase space MC sample as

$\int\epsilon(\vec{\xi}_{i})|M(\vec{\xi}_{i};\vec{\eta})|^{2}\text{d}\vec{\xi}_{i}=\frac{1}{N_{\text{gen}}}\sum_{k_{\text{MC}}}^{N_{\text{MC}}}|M(\vec{\xi}_{k_{\text{MC}}};\vec{\eta})|^{2}~{},$
(4)

where $N_{\text{gen}}$ is the total number of the simulated phase space MC
events, $N_{\text{MC}}$ is the number of the phase space MC events surviving
all selection criteria and $k_{\text{MC}}$ is the event index.

Using the MINUIT package [38], we minimize the negative logarithmic likelihood
with background subtraction over the six data samples. Here, $\alpha_{0}$,
$\Delta_{0}$, $\alpha_{\Lambda\pi^{0}}$, $\alpha_{\bar{\Lambda}\pi^{0}}$,
$\Delta_{\Lambda\pi^{0}}$, and $\Delta_{\bar{\Lambda}\pi^{0}}$ are fixed to
individual values measured by BESIII [39, 40], $\alpha_{p\pi^{-}}$ and
$\alpha_{\bar{p}\pi^{+}}$ are fixed to the values from the PDG [3]. In the
fit, $\alpha_{\Xi^{0}K^{+}}$ and $\Delta_{\Xi^{0}K^{+}}$ are free parameters,
and $\alpha_{\Xi^{0}K^{+}}=-\alpha_{\bar{\Xi}^{0}K^{-}}$ and
$\Delta_{\Xi^{0}K^{+}}=-\Delta_{\bar{\Xi}^{0}K^{-}}$ as required under the
$CP$ invariance assumption. The projections of the best fit onto several
variables are shown in Fig. 4. The data are compared with the MC events
weighted by the nominal fitting result. From this fit, we obtain
$\alpha_{\Xi^{0}K^{+}}=0.01\pm 0.16$ and $\Delta_{\Xi^{0}K^{+}}=3.84\pm
0.90~{}\text{rad}$. Hence, the other two Lee-Yang parameters are calculated to
be $\beta_{\Xi^{0}K^{+}}=-0.64\pm 0.69$ and $\gamma_{\Xi^{0}K^{+}}=-0.77\pm
0.58$, where the uncertainties are statistical only.

The systematic uncertainties arise mainly from the reconstruction of final
state particles, which is studied with $J/\psi\rightarrow
K^{0}_{s}K^{\pm}\pi^{\mp}$ for kaon, $\Lambda^{+}_{c}\rightarrow\Lambda X$ for
$\Lambda$, $\psi(3686)\rightarrow J/\psi\,\pi^{0}\pi^{0}$ and
$e^{+}e^{-}\rightarrow\omega\pi^{0}$ for $\pi^{0}$. For $\Delta E$ requirement
and $M_{\text{BC}}$ signal regions, which are estimated by smearing the phase
space MC samples using resolution parameters. For the background subtraction,
which is taking into account the background shape and size. The uncertainties
from the quoted values of $\alpha_{0}$, $\Delta_{0}$,
$\alpha_{\Lambda\pi^{0}}$, $\alpha_{\bar{\Lambda}\pi^{0}}$,
$\Delta_{\Lambda\pi^{0}}$, $\Delta_{\bar{\Lambda}\pi^{0}}$,
$\alpha_{p\pi^{-}}$ and $\alpha_{\bar{p}\pi^{+}}$ are estimated by Gaussian
sampling considered their uncertainties and refit the angular distribution,
and take the values in one time uncertainty of Gaussian fit as the
uncertainties of this part. The fit bias, which is the difference before and
after the correction from pull distribution check. Systematic uncertainties
from all sources are combined in quadrature to calculate the total systematic
uncertainties. All systematic uncertainties are summarized in Table 2 and the
relevant details can be found in Ref. [34].

\begin{overpic}[width=433.62pt]{figures_data_topping.pdf} \end{overpic} Figure 4: Projections of the best fit onto various variables. Black points with error bars are data; red solid lines are phase space MC events re-weighted by angular distribution formula, and represent the fitting result; the blue shaded region denotes the combinatorial background events and the pink shaded region is the mis-reconstructed signal events. Table 2: Systematic uncertainties in $\alpha_{\Xi^{0}K^{+}}$ and $\Delta_{\Xi^{0}K^{+}}$. Source | $\alpha_{\Xi^{0}K^{+}}$ | $\Delta_{\Xi^{0}K^{+}}[\text{rad}]$
---|---|---
Tracking and PID efficiencies | negligible | negligible
$\pi^{0}$ reconstruction | negligible | negligible
$\Lambda$ reconstruction | $0.01$ | $0.01$
$\Delta E$ and $M_{\text{BC}}$ signal regions | negligible | negligible
Background subtraction | 0.03 | $0.08$
Input parameters | $0.01$ | $0.14$
Fit bias | negligible | $0.05$
Total | $0.03$ | $0.17$
Figure 5: The comparison between this work and theoretical predictions, where
the branching fraction is taken from PDG (2022) [3]. The blue symbols are
theoretical predictions and the red star is result from this work. The
definitions of the superscripts $a$ and $b$ can be found in Table 1 and the
theory acronyms are explained in the text.

The $\delta_{p}-\delta_{s}$ can be expressed by
$\delta_{p}-\delta_{s}=\text{arctan}(\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}\text{sin}\Delta_{\Xi^{0}K^{+}}/\alpha_{\Xi^{0}K^{+}})$
and the derivation of $|A|$ and $|B|$ can be found in Ref. [34]. The study has
uncovered two distinct physical solutions, with one group characterized by
$|A|=1.6^{+1.9}_{-1.6}({\rm stat.})\pm 0.4({\rm syst.})$ and $|B|=18.3\pm
2.8({\rm stat.})\pm 0.7({\rm syst.})$, and the other by $|A|=4.3\pm 0.7({\rm
stat.})\pm 0.2({\rm syst.})$ and $|B|=6.7^{+8.3}_{-6.7}({\rm stat.})\pm
1.6({\rm syst.})$.

In summary, by analyzing $4.4~{}\text{fb}^{-1}$ of $e^{+}e^{-}$ annihilation
data collected at the CM energies between $4.60$ and $4.70~{}\text{GeV}$ with
the BESIII detector, the pure $W$-exchange decay
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ from the
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ production has been
studied. The decay asymmetry parameters are measured for the first time as
$\alpha_{\Xi^{0}K^{+}}=0.01\pm 0.16({\rm stat.})\pm 0.03({\rm syst.})$ and
$\Delta_{\Xi^{0}K^{+}}=3.84\pm 0.90({\rm stat.})\pm 0.17({\rm
syst.})~{}\text{rad}$. The other two Lee-Yang parameters are calculated to be
$\beta_{\Xi^{0}K^{+}}=-0.64\pm 0.69({\rm stat.})\pm 0.13({\rm syst.})$ and
$\gamma_{\Xi^{0}K^{+}}=-0.77\pm 0.58({\rm stat.})\pm 0.11({\rm syst.})$. The
comparison between this work and theoretical predictions is shown in Fig. 5.
Our measurement of $\alpha_{\Xi^{0}K^{+}}$ is in good agreement with zero,
which is consistent with the theoretical predictions from the 1990s. The decay
dynamics parameters $|A|$, $|B|$, and $\delta_{p}-\delta_{s}$ are derived. The
value of $\delta_{p}-\delta_{s}$ is observed to be $-1.55\pm 0.25({\rm
stat.})\pm 0.05({\rm syst.})$. This is of great significance for decay
asymmetries, as $\text{cos}(\delta_{p}-\delta_{s})$ measured in this study is
close to zero, an effect that had not been anticipated in previous literature.
This measurement fills the long-standing puzzle and deepens our understanding
of the strong dynamics in charmed baryon sector.

The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012,
12022510, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261,
12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the
CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207; CAS
Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-
SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; 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; ERC under Contract
No. 758462; 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. 443159800, 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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Supplemental Material for “First Measurement of the Decay Asymmetry of pure
W-exchange Decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$”

Supplemental Material for “First Measurement of the Decay Asymmetry of pure
W-exchange Decay $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$”

##  Appendix A i. DECAY ASYMMETRY PARAMETERS

For the process
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$,$\Lambda_{c}^{+}\to BP$
and $\bar{\Lambda}_{c}^{-}$ decaying to anything, where $B$ and $P$ denote a
$J^{P}=\frac{1}{2}^{+}$ baryon and a $J^{P}=0^{-}$ pseudoscalar meson,
respectively, the amplitude can be constructed using the helicity basis. For
the weak non-leptonic decay $\Lambda_{c}^{+}\to BP$, the Lee-Yang variables
[1] $\alpha_{BP}$, $\beta_{BP}$ and $\gamma_{BP}$ are defined with respect to
the $s$-wave and $p$-wave amplitudes, such as

$\alpha_{BP}=\frac{2\text{Re}(s^{*}p)}{|s|^{2}+|p|^{2}},\quad\beta_{BP}=\frac{2\text{Im}(s^{*}p)}{|s|^{2}+|p|^{2}},\quad\gamma_{BP}=\frac{|s|^{2}-|p|^{2}}{|s|^{2}+|p|^{2}},$
(i.1)

where $s$ and $p$ are the parity-odd and parity-even decay amplitudes. In a
non-relativistic picture, they correspond to the L = 0 ($S$-wave) and L = 1
($P$-wave) orbital angular momenta of the baryon-meson system, respectively.

The parameters $\alpha_{BP}$, $\beta_{BP}$, and $\gamma_{BP}$ satisfy

$\alpha_{BP}^{2}+\beta_{BP}^{2}+\gamma_{BP}^{2}=1.$ (i.2)

We work with helicity amplitudes. For $\Lambda_{c}^{+}\to
B(\frac{1}{2}^{+})\,P(0^{-})$ decays, we have two helicity amplitudes,
$\mathcal{H}_{\frac{1}{2}}$ and $\mathcal{H}_{-\frac{1}{2}}$. Using the
relations
$s=\frac{1}{\sqrt{2}}(\mathcal{H}_{\frac{1}{2}}+\mathcal{H}_{-\frac{1}{2}})$,
$p=\frac{1}{\sqrt{2}}(\mathcal{H}_{\frac{1}{2}}-\mathcal{H}_{-\frac{1}{2}})$,
the asymmetry parameters defined with helicity amplitudes are

$\begin{split}\alpha_{BP}&=|\mathcal{H}_{\frac{1}{2}}|^{2}-|\mathcal{H}_{-\frac{1}{2}}|^{2},\\\
\beta_{BP}&=\sqrt{1-\alpha_{BP}^{2}}\text{sin}\Delta_{BP},\\\
\gamma_{BP}&=\sqrt{1-\alpha_{BP}^{2}}\text{cos}\Delta_{BP},\\\ \end{split}$
(i.3)

where we take the normalization
$|\mathcal{H}_{\frac{1}{2}}|^{2}+|\mathcal{H}_{-\frac{1}{2}}|^{2}=1$, and
$\Delta_{BP}$ is the phase angle difference between two helicity amplitudes
$\mathcal{H}_{\frac{1}{2}}$ and $\mathcal{H}_{-\frac{1}{2}}$.

If the $\Lambda_{c}^{+}$ and $\bar{\Lambda}_{c}^{-}$ decays respect CP
symmetry, we have relations between the $\Lambda_{c}^{+}$ and
$\bar{\Lambda}_{c}^{-}$ asymmetry parameters

$\alpha_{\bar{B}\bar{P}}=-\alpha_{BP},~{}\beta_{\bar{B}\bar{P}}=-\beta_{BP},~{}\gamma_{\bar{B}\bar{P}}=\gamma_{BP}.$
(i.4)

##  Appendix B ii. JOINT ANGULAR DISTRIBUTION FORMULA FOR THE DECAY
$\Lambda_{c}^{+}\to\Xi^{0}K^{+}$

In the helicity frame of the
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ system, $\theta_{0}$ is
the polar angle of the $\Lambda_{c}^{+}$ with respect to the $e^{+}$ beam axis
in the $e^{+}e^{-}$ CM system. For the $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ decay,
$\phi_{1}$ is the angle between the $e^{+}\Lambda_{c}^{+}$ and $\Xi^{0}K^{+}$
planes and $\theta_{1}$ is the polar angle of the $\Xi^{0}$ momentum in the
$\Lambda_{c}^{+}$ rest frame with respect to the $\Lambda_{c}^{+}$ momentum in
the CM frame. For the helicity system describing $\Xi^{0}\to\Lambda\pi^{0}$
decay, $\phi_{2}$ is the angle between the $\Xi^{0}K^{+}$ and $\Lambda\pi^{0}$
planes and $\theta_{2}$ is the polar angle of the $\Lambda$ momentum with
respect to the $\Xi^{0}$ momentum in the rest frame of $\Lambda_{c}^{+}$. For
the helicity angles describing the $\Lambda\to p\pi^{-}$ decay, $\phi_{3}$ is
the angle between the $\Lambda\pi^{0}$ and $p\pi^{-}$ planes and $\theta_{3}$
is the polar angle of the proton with respect to the $\Lambda$ momentum in the
$\Xi^{0}$ rest frame. In the $\Lambda_{c}^{+}\to\Xi^{0}K^{+}$ process, as
shown in Table 1, $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$
and $\lambda_{5}$ indicate the helicity of $\Lambda_{c}^{+}$,
$\bar{\Lambda}_{c}^{-}$, $\Xi^{0}$, $\Lambda$ and $p$.
$A_{\lambda_{1},\lambda_{2}}$, $B_{\lambda_{3}}$, $C_{\lambda_{4}}$ and
$D_{\lambda_{5}}$ are the helicity amplitudes.

Table 1: Definition of decays, helicity angles and amplitudes, where $\lambda_{i}$ indicates the helicity values for the corresponding hadron. Level | Decay | Helicity angle | Helicity amplitude
---|---|---|---
0 | $e^{+}e^{-}\to\Lambda_{c}^{+}(\lambda_{1})\,\bar{\Lambda}_{c}^{-}(\lambda_{2})$ |  ($\theta_{0}$) | $A_{\lambda_{1},\lambda_{2}}$
1 | $\Lambda_{c}^{+}\to\Xi^{0}(\lambda_{3})\,K^{+}$ |  ($\theta_{1}$,$\phi_{1}$) | $B_{\lambda_{3}}$
2 | $\Xi^{0}\to\Lambda(\lambda_{4})\,\pi^{0}$ |  ($\theta_{2}$,$\phi_{2}$) | $C_{\lambda_{4}}$
3 | $\Lambda\to p(\lambda_{5})\,\pi^{-}$ |  ($\theta_{3}$,$\phi_{3}$) | $D_{\lambda_{5}}$

The joint angular distribution is defined as

$\small\begin{split}&\frac{d\varGamma}{dcos\theta_{0}\,dcos\theta_{1}\,dcos\theta_{2}\,dcos\theta_{3}\,d\phi_{1}\,d\phi_{2}\,d\phi_{3}}\\\
\propto&\sum_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}}\rho^{\lambda_{1}-\lambda_{2},\lambda^{{}^{\prime}}_{1}-\lambda_{2}}(\theta_{0})\,D^{\frac{1}{2}*}_{\lambda_{1},\lambda_{3}}(\theta_{1},\phi_{1},0)\,D^{\frac{1}{2}}_{\lambda^{{}^{\prime}}_{1},\lambda^{{}^{\prime}}_{3}}(\theta_{1},\phi_{1},0)\,B_{\lambda_{3}}\,B^{*}_{\lambda^{{}^{\prime}}_{3}}\\\
&D^{\frac{1}{2}*}_{\lambda_{3},\lambda_{4}}(\theta_{2},\phi_{2},0)\,D^{\frac{1}{2}}_{\lambda^{{}^{\prime}}_{3},\lambda^{{}^{\prime}}_{4}}(\theta_{2},\phi_{2},0)\,C_{\lambda_{4}}\,C^{*}_{\lambda^{{}^{\prime}}_{4}}\,D^{\frac{1}{2}*}_{\lambda_{4},\lambda_{5}}(\theta_{3},\phi_{3},0)\,D^{\frac{1}{2}}_{\lambda^{{}^{\prime}}_{4},\lambda_{5}}(\theta_{3},\phi_{3},0)\,|D_{\lambda_{5}}|^{2},\end{split}$
(ii.1)

where
$\rho^{\lambda_{1}-\lambda_{2},\lambda^{{}^{\prime}}_{1}-\lambda_{2}}(\theta_{0})=\sum_{M=\pm
1}\,d^{1}_{M,\lambda_{1}-\lambda_{2}}(\theta_{0})\,d^{1}_{M,\lambda^{{}^{\prime}}_{1}-\lambda_{2}}(\theta_{0})\,A_{\lambda_{1},\lambda_{2}}\,A^{*}_{\lambda^{{}^{\prime}}_{1},\lambda_{2}}$
corresponds to the $\Lambda_{c}^{+}$ spin density matrix and
$D^{J}_{m,n}(\phi,\theta,\gamma)=e^{-im\phi}\,d^{J}_{m,n}(\theta)\,e^{-in\gamma}$
is Wigner-D function [2]. The helicity amplitudes
$A_{\lambda_{1},\lambda_{2}}$ are related to the asymmetry parameters
$\alpha_{0}=\frac{\left|A_{\frac{1}{2},-\frac{1}{2}}\right|^{2}-\,2\,\left|A_{\frac{1}{2},\frac{1}{2}}\right|^{2}}{\left|A_{\frac{1}{2},-\frac{1}{2}}\right|^{2}+\,2\,\left|A_{\frac{1}{2},\frac{1}{2}}\right|^{2}}$,
and helicity $B_{\lambda_{3}}$ is related to the asymmetry parameter
$\alpha_{\Xi^{0}K^{+}}=\frac{\left|B_{\frac{1}{2}}\right|^{2}-\,\left|B_{-\frac{1}{2}}\right|^{2}}{\left|B_{\frac{1}{2}}\right|^{2}+\,\left|B_{-\frac{1}{2}}\right|^{2}}$,
helicity $C_{\lambda_{4}}$ is related to the asymmetry parameter
$\alpha_{\Lambda\pi^{0}}=\frac{\left|C_{\frac{1}{2}}\right|^{2}-\,\left|C_{-\frac{1}{2}}\right|^{2}}{\left|C_{\frac{1}{2}}\right|^{2}+\,\left|C_{-\frac{1}{2}}\right|^{2}}$
and helicity $D_{\lambda_{5}}$ is related to the asymmetry parameter
$\alpha_{p\pi^{-}}=\frac{\left|D_{\frac{1}{2}}\right|^{2}-\,\left|D_{-\frac{1}{2}}\right|^{2}}{\left|D_{\frac{1}{2}}\right|^{2}+\,\left|D_{-\frac{1}{2}}\right|^{2}}$.
Then the joint angular distribution becomes

$\scriptsize\begin{split}&\frac{d\Gamma}{d\text{cos}\theta_{0}~{}d\text{cos}\theta_{1}~{}d\text{cos}\theta_{2}~{}d\text{cos}\theta_{3}~{}d\phi_{1}~{}d\phi_{2}~{}d\phi_{3}}\\\
&\propto 1+\alpha_{0}\text{cos}^{2}\theta_{0}\\\
&+(1+\alpha_{0}\text{cos}^{2}\theta_{0})~{}\alpha_{\Xi^{0}K^{+}}\alpha_{\Lambda\pi^{0}}\text{cos}\theta_{2}\\\
&+(1+\alpha_{0}\text{cos}^{2}\theta_{0})~{}\alpha_{\Xi^{0}K^{+}}\alpha_{p\pi^{-}}\text{cos}\theta_{2}\text{cos}\theta_{3}\\\
&+(1+\alpha_{0}\text{cos}^{2}\theta_{0})~{}\alpha_{\Lambda\pi^{0}}\alpha_{p\pi^{-}}\text{cos}\theta_{3}\\\
&-(1+\alpha_{0}\text{cos}^{2}\theta_{0})~{}\alpha_{\Xi^{0}K^{+}}\sqrt{1-\alpha^{2}_{\Lambda\pi^{0}}}~{}\alpha_{p\pi^{-}}\text{sin}\theta_{2}\text{sin}\theta_{3}\text{cos}(\Delta_{\Lambda\pi^{0}}+\phi_{3})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\alpha_{\Xi^{0}K^{+}}\text{sin}\theta_{1}\text{sin}\phi_{1}\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\alpha_{\Lambda\pi^{0}}\text{sin}\theta_{1}\text{sin}\phi_{1}\text{cos}\theta_{2}\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\alpha_{\Xi^{0}K^{+}}\alpha_{\Lambda\pi^{0}}\alpha_{p\pi^{-}}\text{sin}\theta_{1}\text{sin}\phi_{1}\text{cos}\theta_{3}\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\alpha_{p\pi^{-}}\text{sin}\theta_{1}\text{sin}\phi_{1}\text{cos}\theta_{2}\text{cos}\theta_{3}\\\
&-\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Lambda\pi^{0}}}~{}\alpha_{p\pi^{-}}\text{sin}\theta_{1}\text{sin}\phi_{1}\text{sin}\theta_{2}\text{sin}\theta_{3}\text{cos}(\Delta_{\Lambda\pi^{0}}+\phi_{3})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}~{}\alpha_{\Lambda\pi^{0}}\text{cos}\phi_{1}\text{sin}\theta_{2}\text{sin}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}~{}\alpha_{\Lambda\pi^{0}}\text{cos}\theta_{1}\text{sin}\phi_{1}\text{sin}\theta_{2}\text{cos}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}~{}\alpha_{p\pi^{-}}\text{cos}\theta_{1}\text{sin}\phi_{1}\text{sin}\theta_{2}\text{cos}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\text{cos}\theta_{3}\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}~{}\alpha_{p\pi^{-}}\text{cos}\phi_{1}\text{sin}\theta_{2}\text{sin}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\text{cos}\theta_{3}\\\
&-\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}\sqrt{1-\alpha^{2}_{\Lambda\pi^{0}}}~{}\alpha_{p\pi^{-}}\text{cos}\theta_{1}\text{sin}\phi_{1}\text{sin}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\text{sin}\theta_{3}\text{sin}(\Delta_{\Lambda\pi^{0}}+\phi_{3})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}~{}\sqrt{1-\alpha^{2}_{\Lambda\pi^{0}}}~{}\alpha_{p\pi^{-}}\text{cos}\theta_{1}\text{sin}\phi_{1}\text{cos}\theta_{2}\text{cos}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\text{sin}\theta_{3}\text{cos}(\Delta_{\Lambda\pi^{0}}+\phi_{3})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}\sqrt{1-\alpha^{2}_{\Lambda\pi^{0}}}~{}\alpha_{p\pi^{-}}\text{cos}\phi_{1}\text{cos}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\text{sin}\theta_{3}\text{sin}(\Delta_{\Lambda\pi^{0}}+\phi_{3})\\\
&+\sqrt{1-\alpha^{2}_{0}}~{}\text{sin}\Delta_{0}\text{sin}\theta_{0}\text{cos}\theta_{0}\sqrt{1-\alpha^{2}_{\Xi^{0}K^{+}}}\sqrt{1-\alpha^{2}_{\Lambda\pi^{0}}}~{}\alpha_{p\pi^{-}}\text{cos}\phi_{1}\text{cos}\theta_{2}\text{sin}(\Delta_{\Xi^{0}K^{+}}+\phi_{2})\text{sin}\theta_{3}\text{cos}(\Delta_{\Lambda\pi^{0}}+\phi_{3})\\\
\end{split}$ (ii.2)

where the $\alpha_{0}$ is the angular distribution parameter of
$e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ and $\Delta_{0}$ is the
transverse polarization for $\Lambda_{c}^{+}$. The decay asymmetry parameters
$\alpha_{\Lambda\pi^{0}}$ and $\alpha_{p\pi^{-}}$ are taken from PDG [3]. The
$\Delta_{\Xi^{0}K^{+}}=\delta^{B}_{\frac{1}{2}}-\delta^{B}_{-\frac{1}{2}}$ and
$\Delta_{\Lambda\pi^{0}}=\delta^{C}_{\frac{1}{2}}-\delta^{C}_{-\frac{1}{2}}$
are the difference of the phase of the helicity amplitude $B$ and $C$,
respectively. For the charge conjugate mode,
$\bar{\Lambda}_{c}^{-}\to\bar{\Xi}^{0}K^{-}$, the formula of angular
distribution is same.

##  Appendix C iii. SYSTEMATIC UNCERTAINTY

The systematic uncertainties arise mainly from the reconstruction of final
states, $\Delta E$ requirement, $M_{\text{BC}}$ signal selection, the
background subtraction, the uncertainties from the quoted values of
$\alpha_{0}$, $\Delta_{0}$, $\alpha_{\Lambda\pi^{0}}$,
$\alpha_{\bar{\Lambda}\pi^{0}}$, $\Delta_{\Lambda\pi^{0}}$,
$\Delta_{\bar{\Lambda}\pi^{0}}$, $\alpha_{p\pi^{-}}$, and
$\alpha_{\bar{p}\pi^{+}}$, and the fit bias. Systematic uncertainties from
various sources are combined in quadrature to calculate the total systematic
uncertainties.

The reconstruction efficiency of charged kaon is studied with the control
sample of $J/\psi\to K_{S}^{0}K^{\pm}\pi^{\mp}$ events, that for $\pi^{0}$
with $\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ and $e^{+}e^{-}\to\omega\pi^{0}$, and
that for $\Lambda$ with $J/\psi\to\bar{p}K^{+}\Lambda$ and
$J/\psi\to\Lambda\bar{\Lambda}$ [4]. The signal MC samples are re-weighted
based on the data-MC differences in various momentum ranges resulting in new
MC integration and new fitting parameters. The uncertainties related to the
$\Delta E$ and $M_{\text{BC}}$ requirement are evaluated by smearing the
signal MC sample with a Gaussian resolution function. The changes of the fit
results based on new accepted signal MC events are taken as the systematic
uncertainties. All effects mentioned above are negligible except for $\Lambda$
reconstruction.

For the background subtraction, we consider both background size and the
background modelling. The background size including combinational background
and mis-reconstructed component is obtained from the fit to the
$M_{\text{BC}}$ spectrum. The relevant systematic uncertainties are examined
by repeating the fits with a alternative background size obtained from the
Gaussian sampling of the fitted parameters. The ensemble of fitted parameters
obtained will be fit to a Gaussian and the sum of the fitted Gaussian
resolution and the difference between the fitted Gaussian mean and the nominal
result are assigned as the systematic uncertainty for background size. The
uncertainty of the background modelling also considers both combinational
background and the mis-reconstructed component. The mis-reconstructed model is
examined by an alternative signal MC sample produced with the new input
parameters $\alpha_{\Xi^{0}K^{+}}$ and $\Delta_{\Xi^{0}K^{+}}$ which are
changed within $\pm 1\sigma$. The uncertainty due to the combinational
background model is estimated by varying the relative weights between
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs and other hadronic events based
on the uncertainties of their cross section ratio.

The systematic uncertainty due to the input parameters is evaluated by varying
these parameters within $\pm 1\sigma$ using a Gaussian sampling method. For
each parameter, the obtained results are fit to a Gaussian function and the
sum of the fitted Gaussian resolution and the difference between the fitted
Gaussian mean and the nominal result is taken as the systematic uncertainty.
The systematic uncertainties arising from fit bias are investigated via pull
distribution checks. The mean values of the pull distributions are used to
correct the nominal results. The differences between the uncorrected and
corrected results are assigned as systematic uncertainties.

All systematic uncertainties discussed above are summarized in Table 2.

Table 2: Systematic uncertainties in $\alpha_{\Xi^{0}K^{+}}$ and $\Delta_{\Xi^{0}K^{+}}$. Source | $\alpha_{\Xi^{0}K^{+}}$ | $\Delta_{\Xi^{0}K^{+}}[\text{rad}]$
---|---|---
Tracking and PID efficiencies | negligible | negligible
$\pi^{0}$ reconstruction | negligible | negligible
$\Lambda$ reconstruction | $0.01$ | $0.01$
$\Delta E$ and $M_{\text{BC}}$ signal regions | negligible | negligible
Background subtraction | 0.03 | $0.08$
Input parameters | $0.01$ | $0.14$
Fit bias | negligible | $0.05$
Total | $0.03$ | $0.17$

##  Appendix D iv. RELATION WITH WEAK DECAY AMPLITUDE

In the Standard Model, the amplitude for the two-body weak decay
$\Lambda_{c}^{+}\to\Xi^{0}\,K^{+}$ can be parameterized as
$\mathcal{M}=i\bar{u}_{\Xi^{0}}(A-B\gamma_{5})u_{\Lambda_{c}^{+}}$, based on
the $S$-wave and $P$-wave amplitudes, $s=A$ and $p=\kappa B$ [3]. Combination
the Eq. (i.1) and the relationship between the
$\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})$ and the amplitude, the decay
width and decay asymmetry can be obtained as follows:

$\begin{split}\Gamma=\frac{\mathcal{B}(\Lambda_{c}^{+}\to\Xi^{0}K^{+})}{\tau_{\Lambda_{c}^{+}}}&=\frac{|\vec{p}_{c}|}{8\pi}\Big{[}\frac{(m_{\Lambda_{c}^{+}}+m_{\Xi^{0}})^{2}-m_{K^{+}}^{2}}{m_{\Lambda_{c}^{+}}^{2}}|A|^{2}+\frac{(m_{\Lambda_{c}^{+}}-m_{\Xi^{0}})^{2}-m_{K^{+}}^{2}}{m_{\Lambda_{c}^{+}}^{2}}|B|^{2}\Big{]},\\\
\alpha_{\Xi^{0}K^{+}}&=\frac{2\kappa|A||B|\text{cos}(\delta_{p}-\delta_{s})}{|A|^{2}+\kappa^{2}|B|^{2}},\\\
\Delta_{\Xi^{0}K^{+}}&=\text{arctan}\frac{2\kappa|A||B|\text{sin}(\delta_{p}-\delta_{s})}{|A|^{2}-\kappa^{2}|B|^{2}},\\\
\end{split}$ (iv.1)

with
$\kappa=|\vec{p}_{c}|/(E_{\Xi^{0}}+m_{\Xi^{0}})=\sqrt{(E_{\Xi^{0}}-m_{\Xi^{0}})/(E_{\Xi^{0}}+m_{\Xi^{0}})}\approx
0.234582$ and $\vec{p}_{c}$ is the momentum of the $\Xi^{0}$ baryon in the
rest frame of $\Lambda_{c}^{+}$ particle [5].

Upon solving the system of equations presented in Eq. (iv.1), an exquisite
solution can be obtained. The complexity of the solution is such that its
exposition is not suitable here.

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