# Weak phases and CP-symmetry tests in sequential decays of entangled double-
strange baryons

BESIII Collaboration

###### Abstract

Using a sample of $1.31\times 10^{9}$ $J/\psi$ events collected with the
BESIII detector at the electron–positron collider BEPCII, we analyse the full
$J/\psi\to\Xi^{-}\overline{\Xi}^{+},\ \Xi^{-}\to\Lambda\pi^{-},\ \Lambda\to
p\pi^{-},\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+},\
\overline{\Lambda}\to\overline{p}\pi^{+}$ decay chain. A new method,
exploiting the fact that the $\Xi^{-}\overline{\Xi}^{+}$ pair is entangled and
sequentially decaying, and where the complete decay chains are reconstructed,
is applied for the first time. This enables precision measurements of the
decay parameters for the $\Xi^{-}\to\Lambda\pi^{-}$ decay ($\alpha_{\Xi}$,
$\phi_{\Xi}$) as well as the $\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+}$
decay ($\overline{\alpha}_{\Xi}$, $\overline{\phi}_{\Xi}$). From the decay
parameters, two independent CP tests were performed, quantified by the
observables $A_{\rm CP}^{\Xi}$ and $\Delta\phi_{\Xi}$. Our results, $A_{\rm
CP}^{\Xi}$ = $(6.0\pm 13.4\pm 5.6)\times 10^{-3}$ and
$\Delta\phi_{\Xi}=(-4.8\pm 13.7\pm 2.9)\times 10^{-3}~{}{\rm rad}$, are
consistent with CP symmetry. Furthermore, our method enables a separation of
strong and weak $\Xi\to\Lambda\pi$ decay amplitudes. This results in the first
direct measurement of the weak phase difference for any baryon decay. The
result is found to be $(\xi_{P}-\xi_{S})=(1.2\pm 3.4\pm 0.8)\times 10^{-2}$
rad and is one of the most precise tests of CP symmetry for strange baryons.
The strong phase difference is measured to be
$(\delta_{P}-\delta_{S})=(-4.0\pm 3.3\pm 1.7)\times 10^{-2}$ rad. In addition,
we provide an independent measurement of the recently debated $\Lambda$ decay
parameter, $\alpha_{\Lambda}=0.757\pm 0.011\pm 0.008$. The
$\Lambda\overline{\Lambda}$ asymmetry is measured to be $A_{\rm
CP}^{\Lambda}=(-3.7\pm 11.7\pm 9.0)\times 10^{-3}$.

M. Ablikim1, M. N. Achasov10,c, P. Adlarson67, S. Ahmed15, M. Albrecht4, R.
Aliberti28, A. Amoroso66A,66C, M. R. An32, Q. An63,49, X. H. Bai57, Y. Bai48,
O. Bakina29, R. Baldini Ferroli23A, I. Balossino24A, Y. Ban38,k, K.
Begzsuren26, N. Berger28, M. Bertani23A, D. Bettoni24A, F. Bianchi66A,66C, J.
Biernat67, J. Bloms60, A. Bortone66A,66C, I. Boyko29, R. A. Briere5, H. Cai68,
X. Cai1,49, A. Calcaterra23A, G. F. Cao1,54, N. Cao1,54, S. A. Cetin53A, J. F.
Chang1,49, W. L. Chang1,54, G. Chelkov29,b, D. Y. Chen6, G. Chen1, H. S.
Chen1,54, M. L. Chen1,49, S. J. Chen35, X. R. Chen25, Y. B. Chen1,49, Z. J
Chen20,l, W. S. Cheng66C, G. Cibinetto24A, F. Cossio66C, X. F. Cui36, H. L.
Dai1,49, X. C. Dai1,54, A. Dbeyssi15, R. E. de Boer4, D. Dedovich29, Z. Y.
Deng1, A. Denig28, I. Denysenko29, M. Destefanis66A,66C, F. De Mori66A,66C, Y.
Ding33, C. Dong36, J. Dong1,49, L. Y. Dong1,54, M. Y. Dong1,49,54, X. Dong68,
S. X. Du71, Y. L. Fan68, J. Fang1,49, S. S. Fang1,54, Y. Fang1, R.
Farinelli24A, L. Fava66B,66C, F. Feldbauer4, G. Felici23A, C. Q. Feng63,49, J.
H. Feng50, M. Fritsch4, C. D. Fu1, Y. Gao38,k, Y. Gao64, Y. Gao63,49, Y. G.
Gao6, I. Garzia24A,24B, P. T. Ge68, C. Geng50, E. M. Gersabeck58, A Gilman61,
K. Goetzen11, L. Gong33, W. X. Gong1,49, W. Gradl28, M. Greco66A,66C, L. M.
Gu35, M. H. Gu1,49, S. Gu2, Y. T. Gu13, C. Y Guan1,54, A. Q. Guo22, L. B.
Guo34, R. P. Guo40, Y. P. Guo9,h, A. Guskov29, T. T. Han41, W. Y. Han32, J.
Hansson67, X. Q. Hao16, F. A. Harris56, N Hüsken22,28, K. L. He1,54, F. H.
Heinsius4, C. H. Heinz28, T. Held4, Y. K. Heng1,49,54, C. Herold51, M.
Himmelreich11,f, T. Holtmann4, Y. R. Hou54, Z. L. Hou1, H. M. Hu1,54, J. F.
Hu47,m, T. Hu1,49,54, Y. Hu1, G. S. Huang63,49, L. Q. Huang64, X. T. Huang41,
Y. P. Huang1, Z. Huang38,k, T. Hussain65, W. Ikegami Andersson67, W. Imoehl22,
M. Irshad63,49, S. Jaeger4, S. Janchiv26,j, Q. Ji1, Q. P. Ji16, X. B. Ji1,54,
X. L. Ji1,49, Y. Y. Ji41, H. B. Jiang41, X. S. Jiang1,49,54, J. B. Jiao41, Z.
Jiao18, S. Jin35, Y. Jin57, T. Johansson67, N. Kalantar-Nayestanaki55, X. S.
Kang33, R. Kappert55, M. Kavatsyuk55, B. C. Ke43,1, I. K. Keshk4, A.
Khoukaz60, P. Kiese28, R. Kiuchi1, R. Kliemt11, L. Koch30, O. B. Kolcu53A,e,
B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc67, M. G. Kurth1,54, W. Kühn30,
J. J. Lane58, J. S. Lange30, P. Larin15, A. Lavania21, L. Lavezzi66A,66C, Z.
H. Lei63,49, H. Leithoff28, M. Lellmann28, T. Lenz28, C. Li39, C. H. Li32,
Cheng Li63,49, D. M. Li71, F. Li1,49, G. Li1, H. Li63,49, H. Li43, H. B.
Li1,54, H. J. Li16, H. J. Li9,h, J. L. Li41, J. Q. Li4, J. S. Li50, Ke Li1, L.
K. Li1, Lei Li3, P. R. Li31, S. Y. Li52, W. D. Li1,54, W. G. Li1, X. H.
Li63,49, X. L. Li41, Xiaoyu Li1,54, Z. Y. Li50, H. Liang1,54, H. Liang63,49,
H. Liang27, Y. F. Liang45, Y. T. Liang25, G. R. Liao12, L. Z. Liao1,54, J.
Libby21, C. X. Lin50, B. J. Liu1, C. X. Liu1, D. Liu63,49, F. H. Liu44, Fang
Liu1, Feng Liu6, H. B. Liu13, H. M. Liu1,54, Huanhuan Liu1, Huihui Liu17, J.
B. Liu63,49, J. L. Liu64, J. Y. Liu1,54, K. Liu1, K. Y. Liu33, Ke Liu6, L.
Liu63,49, M. H. Liu9,h, P. L. Liu1, Q. Liu54, Q. Liu68, S. B. Liu63,49, Shuai
Liu46, T. Liu1,54, W. M. Liu63,49, X. Liu31, Y. Liu31, Y. B. Liu36, Z. A.
Liu1,49,54, Z. Q. Liu41, X. C. Lou1,49,54, F. X. Lu16, F. X. Lu50, H. J. Lu18,
J. D. Lu1,54, J. G. Lu1,49, X. L. Lu1, Y. Lu1, Y. P. Lu1,49, C. L. Luo34, M.
X. Luo70, P. W. Luo50, T. Luo9,h, X. L. Luo1,49, S. Lusso66C, X. R. Lyu54, F.
C. Ma33, H. L. Ma1, L. L. Ma41, M. M. Ma1,54, Q. M. Ma1, R. Q. Ma1,54, R. T.
Ma54, X. X. Ma1,54, X. Y. Ma1,49, F. E. Maas15, M. Maggiora66A,66C, S.
Maldaner4, S. Malde61, Q. A. Malik65, A. Mangoni23B, Y. J. Mao38,k, Z. P.
Mao1, S. Marcello66A,66C, Z. X. Meng57, J. G. Messchendorp55, G. Mezzadri24A,
T. J. Min35, R. E. Mitchell22, X. H. Mo1,49,54, Y. J. Mo6, N. Yu. Muchnoi10,c,
H. Muramatsu59, S. Nakhoul11,f, Y. Nefedov29, F. Nerling11,f, I. B.
Nikolaev10,c, Z. Ning1,49, S. Nisar8,i, S. L. Olsen54, Q. Ouyang1,49,54, S.
Pacetti23B,23C, X. Pan9,h, Y. Pan58, A. Pathak1, P. Patteri23A, M. Pelizaeus4,
H. P. Peng63,49, K. Peters11,f, J. L. Ping34, R. G. Ping1,54, R. Poling59, V.
Prasad63,49, H. Qi63,49, H. R. Qi52, K. H. Qi25, M. Qi35, T. Y. Qi9, T. Y.
Qi2, S. Qian1,49, W. B. Qian54, Z. Qian50, C. F. Qiao54, L. Q. Qin12, X. P.
Qin9, X. S. Qin41, Z. H. Qin1,49, J. F. Qiu1, S. Q. Qu36, K. H. Rashid65, K.
Ravindran21, C. F. Redmer28, A. Rivetti66C, V. Rodin55, M. Rolo66C, G.
Rong1,54, Ch. Rosner15, M. Rump60, H. S. Sang63, A. Sarantsev29,d, Y.
Schelhaas28, C. Schnier4, K. Schoenning67, M. Scodeggio24A,24B, D. C. Shan46,
W. Shan19, X. Y. Shan63,49, J. F. Shangguan46, M. Shao63,49, C. P. Shen9, P.
X. Shen36, X. Y. Shen1,54, H. C. Shi63,49, R. S. Shi1,54, X. Shi1,49, X. D
Shi63,49, J. J. Song41, W. M. Song27,1, Y. X. Song38,k, S. Sosio66A,66C, S.
Spataro66A,66C, K. X. Su68, P. P. Su46, F. F. Sui41, G. X. Sun1, H. K. Sun1,
J. F. Sun16, L. Sun68, S. S. Sun1,54, T. Sun1,54, W. Y. Sun34, W. Y. Sun27, X
Sun20,l, Y. J. Sun63,49, Y. K. Sun63,49, Y. Z. Sun1, Z. T. Sun1, Y. H. Tan68,
Y. X. Tan63,49, C. J. Tang45, G. Y. Tang1, J. Tang50, J. X. Teng63,49, V.
Thoren67, Y. T. Tian25, I. Uman53B, B. Wang1, C. W. Wang35, D. Y. Wang38,k, H.
J. Wang31, H. P. Wang1,54, K. Wang1,49, L. L. Wang1, M. Wang41, M. Z.
Wang38,k, Meng Wang1,54, W. Wang50, W. H. Wang68, W. P. Wang63,49, X.
Wang38,k, X. F. Wang31, X. L. Wang9,h, Y. Wang50, Y. Wang63,49, Y. D. Wang37,
Y. F. Wang1,49,54, Y. Q. Wang1, Y. Y. Wang31, Z. Wang1,49, Z. Y. Wang1, Ziyi
Wang54, Zongyuan Wang1,54, D. H. Wei12, P. Weidenkaff28, F. Weidner60, S. P.
Wen1, D. J. White58, U. Wiedner4, G. Wilkinson61, M. Wolke67, L. Wollenberg4,
J. F. Wu1,54, L. H. Wu1, L. J. Wu1,54, X. Wu9,h, Z. Wu1,49, L. Xia63,49, H.
Xiao9,h, S. Y. Xiao1, Z. J. Xiao34, X. H. Xie38,k, Y. G. Xie1,49, Y. H. Xie6,
T. Y. Xing1,54, G. F. Xu1, Q. J. Xu14, W. Xu1,54, X. P. Xu46, Y. C. Xu54, F.
Yan9,h, L. Yan9,h, W. B. Yan63,49, W. C. Yan71, Xu Yan46, H. J. Yang42,g, H.
X. Yang1, L. Yang43, S. L. Yang54, Y. X. Yang12, Yifan Yang1,54, Zhi Yang25,
M. Ye1,49, M. H. Ye7, J. H. Yin1, Z. Y. You50, B. X. Yu1,49,54, C. X. Yu36, G.
Yu1,54, J. S. Yu20,l, T. Yu64, C. Z. Yuan1,54, L. Yuan2, X. Q. Yuan38,k, Y.
Yuan1, Z. Y. Yuan50, C. X. Yue32, A. Yuncu53A,a, A. A. Zafar65, Y. Zeng20,l,
B. X. Zhang1, Guangyi Zhang16, H. Zhang63, H. H. Zhang50, H. H. Zhang27, H. Y.
Zhang1,49, J. J. Zhang43, J. L. Zhang69, J. Q. Zhang34, J. W. Zhang1,49,54, J.
Y. Zhang1, J. Z. Zhang1,54, Jianyu Zhang1,54, Jiawei Zhang1,54, L. M. Zhang52,
L. Q. Zhang50, Lei Zhang35, S. Zhang50, S. F. Zhang35, Shulei Zhang20,l, X. D.
Zhang37, X. Y. Zhang41, Y. Zhang61, Y. H. Zhang1,49, Y. T. Zhang63,49, Yan
Zhang63,49, Yao Zhang1, Yi Zhang9,h, Z. H. Zhang6, Z. Y. Zhang68, G. Zhao1, J.
Zhao32, J. Y. Zhao1,54, J. Z. Zhao1,49, Lei Zhao63,49, Ling Zhao1, M. G.
Zhao36, Q. Zhao1, S. J. Zhao71, Y. B. Zhao1,49, Y. X. Zhao25, Z. G. Zhao63,49,
A. Zhemchugov29,b, B. Zheng64, J. P. Zheng1,49, Y. Zheng38,k, Y. H. Zheng54,
B. Zhong34, C. Zhong64, L. P. Zhou1,54, Q. Zhou1,54, X. Zhou68, X. K. Zhou54,
X. R. Zhou63,49, X. Y. Zhou32, A. N. Zhu1,54, J. Zhu36, K. Zhu1, K. J.
Zhu1,49,54, S. H. Zhu62, T. J. Zhu69, W. J. Zhu36, W. J. Zhu9,h, Y. C.
Zhu63,49, Z. A. Zhu1,54, B. S. Zou1, J. H. Zou1

1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9 Fudan University, Shanghai 200443, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
24 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
25 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27 Jilin University, Changchun 130012, People’s Republic of China
28 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31 Lanzhou University, Lanzhou 730000, People’s Republic of China
32 Liaoning Normal University, Dalian 116029, People’s Republic of China
33 Liaoning University, Shenyang 110036, People’s Republic of China
34 Nanjing Normal University, Nanjing 210023, People’s Republic of China
35 Nanjing University, Nanjing 210093, People’s Republic of China
36 Nankai University, Tianjin 300071, People’s Republic of China
37 North China Electric Power University, Beijing 102206, People’s Republic of
China
38 Peking University, Beijing 100871, People’s Republic of China
39 Qufu Normal University, Qufu 273165, People’s Republic of China
40 Shandong Normal University, Jinan 250014, People’s Republic of China
41 Shandong University, Jinan 250100, People’s Republic of China
42 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
43 Shanxi Normal University, Linfen 041004, People’s Republic of China
44 Shanxi University, Taiyuan 030006, People’s Republic of China
45 Sichuan University, Chengdu 610064, People’s Republic of China
46 Soochow University, Suzhou 215006, People’s Republic of China
47 South China Normal University, Guangzhou 510006, People’s Republic of China
48 Southeast University, Nanjing 211100, People’s Republic of China
49 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
50 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
51 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
52 Tsinghua University, Beijing 100084, People’s Republic of China
53 Turkish Accelerator Center Particle Factory Group, (A)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (B)Near East University, Nicosia,
North Cyprus, Mersin 10, Turkey
54 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
55 University of Groningen, NL-9747 AA Groningen, The Netherlands
56 University of Hawaii, Honolulu, Hawaii 96822, USA
57 University of Jinan, Jinan 250022, People’s Republic of China
58 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
59 University of Minnesota, Minneapolis, Minnesota 55455, USA
60 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
61 University of Oxford, Keble Rd, Oxford, UK OX13RH
62 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
63 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
64 University of South China, Hengyang 421001, People’s Republic of China
65 University of the Punjab, Lahore-54590, Pakistan
66 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
67 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
68 Wuhan University, Wuhan 430072, People’s Republic of China
69 Xinyang Normal University, Xinyang 464000, People’s Republic of China
70 Zhejiang University, Hangzhou 310027, People’s Republic of China
71 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Bogazici University, 34342 Istanbul, Turkey
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
d Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia
e Also at Istanbul Arel University, 34295 Istanbul, Turkey
f Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
g 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
h 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
i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA
j Currently at: Institute of Physics and Technology, Peace Ave.54B,
Ulaanbaatar 13330, Mongolia
k Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
l School of Physics and Electronics, Hunan University, Changsha 410082, China
m Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China

##  1 Introduction

The Standard Model (SM) of particle physics has been proven immensely
successful in describing the elementary particles and their interaction.
However, there are still questions that the SM cannot answer. One example is
why our Universe consists of so much more matter than antimatter. Unless fine-
tuned in the Big Bang, the matter abundance must have dynamical origin. This
dynamical generation, Baryogenesis, is however only possible if certain
criteria are fulfilled. One of these is the existence of processes that
violate charge conjugation and parity (CP) symmetry [1]. Small violations of
CP symmetry are predicted by the SM [2, 3] and are a well-established
phenomenon in weak decays of mesons. However, since these violations are far
too small to explain e.g the observed matter-antimatter asymmetry of the
Universe [1, 4, 5], CP tests are considered a promising area to search for
beyond-the-Standard-Model (BSM) physics. So far, no CP violating effects
beyond the SM have been observed, neither in the meson nor in the baryon
sector.

In general, CP symmetry is tested by comparing the decay patterns of a
particle to those of its antiparticle. Hadrons decay through an interplay of
strong and weak processes, quantified in terms of relative phases between the
amplitudes. Many CP-symmetry tests in hadron decays rely on a non-zero strong
phase to reveal the signal. This strategy is applied in the determination of
the ratio $\epsilon^{\prime}/\epsilon$, quantifying the difference between the
two-pion decay rates of the two weak eigenstates of neutral kaons. The
$\epsilon^{\prime}/\epsilon$ measurement constitutes the only observation of
direct CP violation for strange hadrons [6, 7] and provides the most stringent
test of BSM contributions in strange quark systems [8]. This strategy,
however, comes at a price: it is difficult to disentangle the weak/BSM
contributions from the strong processes in a model-independent way. Approaches
that do not rely on strong interactions require that the kaon decays into four
final state particles [9]. In contrast, baryons allow for this separation
through spin measurements. Known examples involving three-body decays are spin
correlations and polarisation in nuclear and neutron $\beta$ decays [10].
Sequential two-body decays of entangled multi-strange baryon-antibaryon pairs
provide another, hitherto unexplored diagnostic tool to separate the strong
and the weak phases. The approach, outlined in Ref. [11], requires
simultaneous but independent determination of spin-$\frac{1}{2}$ baryon and
antibaryon two-body decay parameters.

The spin direction of an unstable baryon manifests itself in the momentum
direction of its daughter particle, enabling straight-forward experimental
access to the spin properties. Spin-$\frac{1}{2}$ baryon decays are described
by a parity conserving (P-wave) and a parity violating (S-wave) amplitude,
quantified in terms of the decay parameters $\alpha_{Y}$, $\beta_{Y}$ and
$\gamma_{Y}$ [12]. The $Y$ refers to the decaying mother hyperon (e.g.
$\Lambda$ or $\Xi^{-}$). These parameters are constrained by the relation
$\alpha_{Y}^{2}+\beta_{Y}^{2}+\gamma_{Y}^{2}=1$. By defining the parameter
$\phi_{Y}$ according to

$\beta_{Y}=\sqrt{1-\alpha_{Y}^{2}}\sin\phi_{Y},~{}~{}~{}\gamma_{Y}=\sqrt{1-\alpha_{Y}^{2}}\cos\phi_{Y},$
(1)

the decay is completely described by two independent parameters $\alpha_{Y}$
and $\phi_{Y}$. In the standard experimental approach [13, 14, 15, 16, 17],
the initial baryon is produced in a well-defined spin polarised state which
allows access to the decay parameters through the angular distribution of the
final-state particles. For sequentially decaying baryons, e.g. the decay of
the double-strange $\Xi^{-}$ baryon into $\Lambda\pi^{-}$, two effects are
possible: 1) a polarised $\Xi^{-}$ transfers its polarisation $\bm{P}_{\Xi}$
to the daughter $\Lambda$; 2) a longitudinal component of the daughter
$\Lambda$ polarisation is induced by the $\Xi^{-}$ decay even if the $\Xi^{-}$
polarisation has no component in this direction. In a reference system with
the $\hat{\bf z}$ axis along the $\Lambda$ momentum in the $\Xi^{-}$ rest
frame and the $\hat{\bf y}$ axis along ${\bf P}_{\Xi}\times\hat{\bf z}$, the
$\Lambda$ polarisation vector is given by [12]

${\bf P}_{\Lambda}\cdot\hat{\bf z}=\frac{\alpha_{\Xi}+{\bf
P}_{\Xi}\cdot\hat{\bf z}}{1+\alpha_{\Xi}{\bf P}_{\Xi}\cdot\hat{\bf z}}\
;~{}~{}~{}\ {\bf P}_{\Lambda}\times\hat{\bf
z}={P}_{\Xi}\sqrt{1-\alpha_{\Xi}^{2}}\frac{\sin\phi_{\Xi}\hat{\bf
x}+\cos\phi_{\Xi}\hat{\bf y}}{1+\alpha_{\Xi}{\bf P}_{\Xi}\cdot\hat{\bf z}}\ ,$
(2)

as illustrated in Fig. 1. This means that the longitudinal ($\hat{\bf z}$)
component depends on $\alpha_{\Xi}$ while the transversal components are
rotated by the angle $\phi_{\Xi}$ with respect to the $\Xi^{-}$ polarisation.

Figure 1: Illustration of the relation between the polarisation vectors of
$\Xi$ and $\Lambda$ in terms of the decay parameters $\alpha_{\Xi}$,
$\beta_{\Xi}$ and $\gamma_{\Xi}$.

The decay parameter $\alpha_{\Xi}$ appears explicitly in the angular
distribution of the direct decay $\Xi^{-}\to\Lambda\pi^{-}$, whereas the
sequential decay distribution of the daughter $\Lambda$ depends on both
$\alpha_{\Lambda}$ and $\phi_{\Xi}$. CP symmetry implies that the baryon decay
parameters $\alpha$ and $\phi$ equal those of the antibaryon
$\overline{\alpha}$ and $\overline{\phi}$ but with opposite sign. Hence, CP
violation can be quantified in terms of the observables

$A_{\rm
CP}^{Y}=\frac{\alpha_{Y}+\overline{\alpha}_{Y}}{\alpha_{Y}-\overline{\alpha}_{Y}},~{}~{}~{}\Delta\phi_{\rm
CP}=\frac{\phi_{Y}+\overline{\phi}_{Y}}{2}.$ (3)

CP violation can only be observed if there is interference between CP-even and
CP-odd terms in the decay amplitude. Since the decay amplitude for
$\Xi^{-}\to\Lambda\pi^{-}$ consists of both a P-wave and an S-wave part, the
leading-order contribution to the CP asymmetry $A_{\rm CP}^{\Xi}$ can be
written as

$A_{\rm CP}^{\Xi}\approx-\tan(\delta_{P}-\delta_{S})\tan(\xi_{P}-\xi_{S}),$
(4)

where $\tan(\delta_{P}-\delta_{S})=\beta/\alpha$ denotes the strong phase
difference of the final-state interaction between the $\Lambda$ and $\pi^{-}$
from the $\Xi^{-}$ decay. CP-violating effects would manifest themselves in a
non-zero weak phase difference $\xi_{P}-\xi_{S}$ [18, 19], an observable that
is complementary to the kaon decay parameter $\epsilon^{{}^{\prime}}$ [20, 6,
7] since the latter only involves an $S$-wave. The strong phase difference can
be extracted from the $\phi_{\Xi}$ parameter, and is found to be small:
($-0.037\pm 0.014$) [17, 21]. Hence CP-violating signals in $A_{\rm CP}^{\Xi}$
are strongly suppressed and difficult to interpret in terms of the weak phase
difference.

An independent CP-symmetry test in $\Xi^{-}\to\Lambda\pi^{-}$ is provided by
determining the value of $\Delta\phi_{\rm CP}$. At leading order, this
observable is related directly to the weak-phase difference

$(\xi_{P}-\xi_{S})_{\rm{LO}}=\frac{\beta+\overline{\beta}}{\alpha-\overline{\alpha}}\approx\frac{\sqrt{1-\alpha^{2}}}{\alpha}\Delta\phi_{\rm
CP}$ (5)

and can be measured even if $\delta_{P}=\delta_{S}$.

This absence of a strong suppression factor therefore improves the sensitivity
to CP-violation effects by an order of magnitude with respect to that of the
$A_{\rm CP}^{\Xi}$ observable [23, 18]. To measure $\Delta\phi_{\rm CP}$ using
the aforementioned standard technique requires beams of polarised $\Xi^{-}$
and $\overline{\Xi}^{+}$. In such experiments the precision is limited by the
magnitude of the polarisation and the accuracy of the polarisation
determination, which in turn is sensitive to asymmetries in the production
mechanisms [24]. In fact, no experiment with a polarised $\overline{\Xi}^{+}$
has been performed, and the polarisation of the $\Xi^{-}$ beams were below 5%
[17]. Here, we present an alternative approach where the baryon–antibaryon
pair is produced in a spin-entangled CP eigenstate and all decay sequences are
analysed simultaneously.

Until now, no direct measurements of any of the asymmetries defined in Eq. (3)
have been performed for the $\Xi^{-}$ baryon. The HyperCP experiment, designed
for the purpose of CP tests in baryon decays, utilised samples of the order
$10^{7}-10^{8}$ $\Xi^{-}$ and $\overline{\Xi}^{+}$ events to determine the
products $\alpha_{\Xi}\alpha_{\Lambda}$ and
$\overline{\alpha}_{\Xi}\overline{\alpha}_{\Lambda}$ [25]. From these
measurements, the sum $A_{\rm CP}^{\Lambda}+A_{\rm CP}^{\Xi}$ was estimated to
be $(0.0\pm 5.5\pm 4.4)\times 10^{-4}$. In addition to the aforementioned
problem of the smallness of $\phi_{\Xi}$, which limits the sensitivity of
$A_{\rm CP}^{\Xi}$ to CP violation, an observable defined as the sum of
asymmetries comes with other drawbacks: should $A_{\rm CP}^{\Lambda}$ and
$A_{\rm CP}^{\Xi}$ have opposite signs, the sum could be consistent with zero
even in the presence of CP-violating effects. A precise interpretation
therefore requires an independent measurement of $A_{\rm CP}^{\Lambda}$ with
matching precision. The most precise result so far is a recent BESIII
measurement [14] where $A_{\rm CP}^{\Lambda}$ was found to be $(-6\pm 12\pm
7)\times 10^{-3}$. Furthermore, Ref. [14] revealed a 17% disagreement with old
measurements on the $\alpha_{\Lambda}$ parameter [21], a result that rapidly
gained some support from a re-analysis of CLAS data [22]. Though the CLAS
result is in better agreement with BESIII than with the PDG value from 2018
and earlier, there is a discrepancy between the CLAS and BESIII results that
needs to be understood. This is particularly important since many physics
quantities from various fields depend on the parameter $\alpha_{\Lambda}$.
Examples include baryon spectroscopy, heavy ion physics and hyperon-related
studies at the LHC.

##  2 Method

In this work, a novel method [26, 11] is applied for the first time to study
entangled, sequentially decaying baryon-antibaryon pairs in the process
$e^{+}e^{-}\to J/\psi\to\Xi^{-}\overline{\Xi}^{+}$. This approach enables a
direct measurement of all weak decay parameters of the
$\Xi^{-}\to\Lambda\pi^{-},\Lambda\to p\pi^{-}+c.c.$ decay chain. The
production and multi-step decays can be described by nine kinematic variables,
here expressed as the helicity angles
$\bm{\xi}=(\theta,\theta_{\Lambda},\varphi_{\Lambda},\theta_{\overline{\Lambda}},\varphi_{\overline{\Lambda}},\theta_{p},\varphi_{p},\theta_{\overline{p}},\varphi_{\overline{p}})$.
The first, $\theta$, is the $\Xi^{-}$ scattering angle with respect to the
$e^{+}$ beam in the centre-of-momentum (c.m.) system of the reaction. The
angles $\theta_{\Lambda}$ and $\varphi_{\Lambda}$
($\theta_{\overline{\Lambda}},\varphi_{\overline{\Lambda}}$) are defined by
the $\Lambda$ ($\overline{\Lambda}$) direction in a reference system denoted
${\cal R}_{\Xi}$ (${\cal R}_{\overline{\Xi}}$), where $\Xi^{-}$
($\overline{\Xi}^{+}$) is at rest and where the $\hat{z}$ axis points in the
direction of the $\Xi^{-}$ ($\overline{\Xi}^{+}$) in the c.m. system. The
$\hat{y}$ axis is normal to the production plane. The angles $\theta_{p}$ and
$\varphi_{p}$ ($\theta_{\overline{p}}$ and $\varphi_{\overline{p}}$) give the
direction of the proton (antiproton) in the $\Lambda$ ($\overline{\Lambda}$)
rest system, denoted ${\cal R}_{\Lambda}$ (${\cal R}_{\overline{\Lambda}}$),
with the $\hat{z}$ axis pointing in the direction of the $\Lambda$
($\overline{\Lambda}$) in the ${\cal R}_{\Xi}$ (${\cal R}_{\overline{\Xi}}$)
system and the $\hat{y}$ axis normal to the plane spanned by the direction of
the $\Xi^{-}$ ($\overline{\Xi}^{+}$) and the direction of the $\Lambda$
($\overline{\Lambda}$). The structure of the nine-dimensional angular
distribution is determined by eight global (i.e independent of the $\Xi^{-}$
scattering angle) parameters
$\bm{\omega}=(\alpha_{\psi},\Delta\Phi,\alpha_{\Xi},\phi_{\Xi},\overline{\alpha}_{\Xi},\overline{\phi}_{\Xi},\alpha_{\Lambda},\overline{\alpha}_{\Lambda})$
and can be written in a modular form as [26]:

${{\cal{W}}}(\bm{\xi};\bm{\omega})=\sum_{\mu,\nu=0}^{3}C_{\mu\nu}\sum_{\mu^{\prime}\nu^{\prime}=0}^{3}a_{\mu\mu^{\prime}}^{\Xi}a_{\nu\nu^{\prime}}^{\overline{\Xi}}a_{\mu^{\prime}0}^{\Lambda}a_{\nu^{\prime}0}^{\overline{\Lambda}}.$
(6)

Here $C_{\mu\nu}(\theta;\alpha_{\psi},\Delta\Phi)$ is a $4\\!\times\\!4$ spin
density matrix, defined in the aforementioned reference systems ${\cal
R}_{\Xi}$ and ${\cal R}_{\overline{\Xi}}$, describing the spin configuration
of the entangled hyperon–antihyperon pair. The parameters $\alpha_{\psi}$ and
$\Delta\Phi$ are related to two production amplitudes, where $\alpha_{\psi}$
parameterises the $\Xi^{-}$ angular distribution. The $\Delta\Phi$ is the
relative phase between the two production amplitudes (in the so-called
helicity representation, Ref. [27]) and governs the polarisation $P_{y}$ of
the produced $\Xi^{-}$ and $\overline{\Xi}^{+}$ as well as their spin
correlations $C_{ij}$. The matrix elements are related to
$P_{y}=P_{y}(\theta)$ and $C_{ij}=C_{ij}(\theta)$ in the following way:

$C_{\mu\nu}=(1+\alpha_{\psi}\cos^{2}\theta)\left(\begin{array}[]{cccc}1&0&P_{y}&0\\\
0&C_{xx}&0&C_{xz}\\\ -P_{y}&0&C_{yy}&0\\\ 0&-C_{xz}&0&C_{zz}\\\
\end{array}\right).$ (7)

The matrices $a^{Y}_{\mu\nu}$ in Eq. (6) represent the propagation of the spin
density matrices in the sequential decays. The elements of these 4$\times$4
matrices are parameterised in terms of the weak decay parameters $\alpha_{Y}$
and $\phi_{Y}$ as well as the helicity angles:
$a_{\mu\mu^{\prime}}^{\Xi}(\theta_{\Lambda},\varphi_{\Lambda};\
\alpha_{\Xi},\phi_{\Xi})$ in reference system ${\cal R}_{\Xi}$,
$a_{\nu\nu^{\prime}}^{\overline{\Xi}}(\theta_{\overline{\Lambda}},\varphi_{\overline{\Lambda}};\
\overline{\alpha}_{\Xi},\overline{\phi}_{\Xi})$ in system ${\cal
R}_{\overline{\Xi}}$, $a_{\mu^{\prime}0}^{\Lambda}(\theta_{p},\varphi_{p};\
\alpha_{\Lambda})$ in system ${\cal R}_{\Lambda}$, and
$a_{\nu^{\prime}0}^{\overline{\Lambda}}(\theta_{\overline{p}},\varphi_{\overline{p}};\
\overline{\alpha}_{\Lambda})$ in system ${\cal R}_{\overline{\Lambda}}$. The
full expressions of $C_{\mu\nu}$ and $a^{Y}_{\mu\nu}$ are given in Ref.[26].

##  3 Analysis

We have carried out our analysis on a data sample of $(1.3106\pm 0.0070)\times
10^{9}$ $J/\psi$ events collected in electron-positron annihilations with the
multi-purpose BESIII detector [28]. The $J/\psi$ resonance decays into the
$\Xi^{-}\overline{\Xi}^{+}$ final state with a branching fraction [21] of
$(9.7\pm 0.8)\times 10^{-4}$. Our method requires exclusively reconstructed
$\Xi^{-}\overline{\Xi}^{+}\to\Lambda\pi^{-}\overline{\Lambda}\pi^{+}\to
p\pi^{-}\pi^{-}\overline{p}\pi^{+}\pi^{+}$ events. The final state particles
are measured in the main drift chamber (MDC), where a superconducting solenoid
provides a magnetic field allowing for momentum determination with an accuracy
of 0.5% at 1.0 GeV/$c$. The $\Lambda(\overline{\Lambda})$ candidates are
identified by combining $p\pi^{-}(\overline{p}\pi^{+})$ pairs and the
$\Xi^{-}(\overline{\Xi}^{+})$ candidates by subsequently combining
$\Lambda\pi^{-}(\overline{\Lambda}\pi^{+})$ pairs. Since it was found that the
long-lived $\Xi^{-}$ and $\overline{\Xi}^{+}$ can only be reconstructed with
sufficient quality if they fulfil $|\cos\theta|<0.84$, only $\Xi^{-}$ and
$\overline{\Xi}^{+}$ within this range were considered. After applying all
selection criteria, 73 244 $\Xi^{-}\overline{\Xi}^{+}$ event candidates remain
in the sample. The number of background events in the signal is estimated to
be $187\pm 16$. More details are given in the Appendix.

For each event, the complete set of the kinematic variables ${\bm{\xi}}$ is
calculated from the intermediate and final-state particle momenta. The
physical parameters in $\bm{\omega}$ are then determined from ${\bm{\xi}}$ by
an unbinned maximum log-likelihood (MLL) fit where the multidimensional
reconstruction efficiency is taken into account. The details of the MLL fit
procedure and the systematic uncertainties are described in the Appendix.

##  4 Results

The results of the fit, i.e. the weak decay parameters
$\Xi^{-}\to\Lambda\pi^{-}$ and
$\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+}$, as well as the production
related parameters $\alpha_{\psi}$ and $\Delta\Phi$, are summarised in Table
1. To illustrate the fit quality, the diagonal spin correlations and the
polarisation defined in Eq. (7) are shown in Fig. 2. The upper left panel of
Fig. 2 shows that the $\Xi^{-}$ baryon is polarised with respect to the normal
of the production plane. The maximum polarisation is $\approx 30\%$, as shown
in the figure. The data points are determined by independent fits for each
$\cos\theta$ bin, without any assumptions on the $\cos\theta$ dependence of
$C_{\mu\nu}$. The red curves represent the angular dependence obtained with
the parameters $\alpha_{\psi}$ and $\Delta\Phi$ determined from the global MLL
fit. The independently determined data points agree well with the globally
fitted curves.

Figure 2: Polarisation and spin correlations and in the
$e^{+}e^{-}\to\Xi^{-}\overline{\Xi}^{+}$ reaction. (a) $P_{y}$, (b) $C_{xx}$,
(c) $C_{yy}$ and (d) $C_{zz}$ as functions of $\cos\theta$. The data points
are determined independently in each bin. The red curves represent the
expected angular dependence obtained with the values of $\alpha_{\psi}$ and
$\Delta\Phi$ from the global fit. The errors bars indicate the statistical
uncertainties.

The extracted values of $\alpha_{\Xi}$, $\overline{\alpha}_{\Xi}$,
$\phi_{\Xi}$, $\overline{\phi}_{\Xi}$, $\alpha_{\Lambda}$ and
$\overline{\alpha}_{\Lambda}$ and their correlations allow for three
independent CP symmetry tests. The asymmetry $A^{\Xi}_{\rm CP}$ is measured
for the first time and found to be $(6.0\pm 13.4\pm 5.6)\times 10^{-3}$, where
the first uncertainty is statistical and the second systematic. The
corresponding SM prediction [19] is $A_{\rm CP,SM}^{\Xi}=(-0.6\pm 1.6)\times
10^{-5}$.

The result for the decay parameter $\overline{\phi}_{\Xi}$ is the first
measurement of its kind for any weakly decaying antibaryon. By combining this
parameter with the corresponding $\phi_{\Xi}$ measurement, the CP asymmetry
$\Delta\phi_{\rm CP}^{\Xi}$ can be determined, and is found to be $(-4.8\pm
13.7\pm 2.9)\times 10^{-3}$. Since this result is consistent with zero, we can
improve our knowledge of the value of $\phi_{\Xi}$ by assuming CP symmetry and
then calculating the mean value of $\phi_{\Xi}$ and $\overline{\phi}_{\Xi}$.
This procedure yields $\left<\phi_{\Xi}\right>=0.016\pm 0.014\pm 0.007$ rad,
which differs from the HyperCP measurement, $\phi_{\Xi,\rm HyperCP}=-0.042\pm
0.011\pm 0.011$ rad, by 2.6 standard deviations [17]. It is noteworthy that
our method yields a precision in $\left<\phi_{\Xi}\right>$ which is similar to
that of the HyperCP result, despite the three orders of magnitude larger data
sample of the latter measurement. This demonstrates the intrinsically high
sensitivity that can be achieved with entangled baryon-antibaryon pairs.

The measurement of $\left<\phi_{\Xi}\right>$ allows for a direct determination
of the strong phase difference, which is found to be
$(\delta_{P}-\delta_{S})=(-4.0\pm 3.3\pm 1.7)\times 10^{-2}$ rad, consistent
with the SM predictions obtained in the framework of heavy-baryon chiral
perturbation theory [19] of $(1.9\pm 4.9)\times 10^{-2}$ rad but in
disagreement with the value $(10.2\pm 3.9)\times 10^{-2}$ rad extracted from
the HyperCP $\phi_{\Xi}$ measurement [17] using $\alpha_{\Xi}=-0.376$ from
this work. Since the $(\delta_{P}-\delta_{S})$ value obtained in our analysis
is consistent with zero, a calculation of the weak phase difference from Eq.
(4) is unfeasible. Instead, we apply Eq. (5), which yields
$(\xi_{P}-\xi_{S})=(1.2\pm 3.4\pm 0.8)\times 10^{-2}$ rad. This is one of the
most precise tests of the CP symmetry for strange baryons and the first direct
measurement of the weak phase difference for any baryon. The corresponding SM
prediction [19] is $(\xi_{P}-\xi_{S})_{\rm SM}=(1.8\pm 1.5)\times 10^{-4}$
rad.

Sequential $\Xi^{-}$ decays also provide an independent measurement of the
$\Lambda$ decay parameters $\alpha_{\Lambda}$ and
$\alpha_{\overline{\Lambda}}$. Being the lightest baryon with strangeness,
many other baryons ($\Sigma^{0}$, $\Xi^{0}$, $\Xi^{-}$, $\Omega$,
$\Lambda_{c}$ etc.) decay with an appreciable fraction into final states
containing $\Lambda$. The measurements of spin observables [29, 30, 31] and
decay parameters of heavier baryons [33, 32] therefore implicitly depend on
$\alpha_{\Lambda}$. Furthermore, since decaying $\Lambda$ and
$\overline{\Lambda}$ beams are used for producing polarised proton and
antiproton beams [34], all physics from such experiments rely on a correct
determination of $\alpha_{\Lambda}$. The value of $\alpha_{\Lambda}=0.757\pm
0.011\pm 0.008$ measured in this analysis is in excellent agreement with that
obtained from the $J/\psi\to\Lambda\overline{\Lambda}$ analysis of BESIII
[14], while it disagrees with the result from the re-analysis of CLAS data
[22]. The precision of our measurement is similar to that of the
$J/\psi\to\Lambda\overline{\Lambda}$ study [14], despite being based on a six
times smaller data sample. The larger sensitivity is primarily explained by
the fact that $\alpha_{\Lambda}$ in Eq. 6 appears in a product with the
polarisation, which is much larger in the case of $\Lambda$ baryons from
$\Xi^{-}$ decays compared to those directly produced in
$J/\psi\to\Lambda\overline{\Lambda}$. Furthermore, the multi-step process
enhances the angular correlations between the baryons and antibaryons to such
an extent that $\alpha_{\Xi}$ and $\alpha_{\Lambda}$ can be measured with the
same precision even if the $\Xi^{-}\overline{\Xi}^{+}$ pair is produced
unpolarised [11].

##  5 Conclusion

To summarise, this letter presents the first application of a very sensitive
test of CP symmetry. This test provides a place to search for BSM physics in
strange hadrons that is complementary to $\epsilon^{\prime}/\epsilon$
measurements in kaon decays. When applied to future measurements with larger
data sets at BESIII [35], the upcoming PANDA experiment at FAIR [36] and the
proposed Super Tau-Charm Factory projects in China and Russia [37, 38], our
method has potential to reach the required precision for CP violating signals,
provided such effects exist.

Table 1: Summary of results. The $J/\psi\to\Xi^{-}\overline{\Xi}^{+}$ angular distribution parameter $\alpha_{\psi}$, the hadronic form factor phase $\Delta\Phi$, the decay parameters for $\Xi^{-}\to\Lambda\pi^{-}$ ($\alpha_{\Xi},\phi_{\Xi}$),$\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+}$ ($\overline{\alpha}_{\Xi},\overline{\phi}_{\Xi}$) $\Lambda\to p\pi^{-}$ $(\alpha_{\Lambda})$ and $\overline{\Lambda}\to\overline{p}\pi^{+}$ $(\overline{\alpha}_{\Lambda})$; the $CP$ asymmetries $A_{\rm CP}^{\Xi}$, $\Delta\phi_{\rm CP}^{\Xi}$ and $A_{\rm CP}^{\Lambda}$ and the average $\left<\phi_{\Xi}\right>$. The first and second uncertainties, are statistical and systematic, respectively. Parameter | This work | Previous result
---|---|---
$\alpha_{\psi}$ | $\phantom{-}0.586\pm 0.012\pm 0.010$ |  $\phantom{-}0.58\pm 0.04\pm 0.08$ [39]
$\Delta\Phi$ |  $\phantom{-}1.213\pm 0.046\pm 0.016$ rad | –
$\alpha_{\Xi}$ | $-0.376\pm 0.007\pm 0.003$ |  $-0.401\pm 0.010$ [21]
$\phi_{\Xi}$ |  $\phantom{-}0.011\pm 0.019\pm 0.009$ rad |  $-0.037\pm 0.014$ rad [21]
$\overline{\alpha}_{\Xi}$ | $\phantom{-}0.371\pm 0.007\pm 0.002$ | –
$\overline{\phi}_{\Xi}$ |  $-0.021\pm 0.019\pm 0.007$ rad | –
$\alpha_{\Lambda}$ | $\phantom{-}0.757\pm 0.011\pm 0.008$ |  $\phantom{-}0.750\pm 0.009\pm 0.004$ [14]
$\overline{\alpha}_{\Lambda}$ | $-0.763\pm 0.011\pm 0.007$ |  $-0.758\pm 0.010\pm 0.007$ [14]
$\xi_{P}-\xi_{S}$ |  $\phantom{-}(1.2\pm 3.4\pm 0.8)\times 10^{-2}$ rad | –
$\delta_{P}-\delta_{S}$ |  $(-4.0\pm 3.3\pm 1.7)\times 10^{-2}$ rad |  $\phantom{-}(10.2\pm 3.9)\times 10^{-2}$ rad [17]
$A_{\rm CP}^{\Xi}$ | $\phantom{-}(6.0\pm 13.4\pm 5.6)\times 10^{-3}$ | –
$\Delta\phi_{\rm CP}^{\Xi}$ |  $(-4.8\pm 13.7\pm 2.9)\times 10^{-3}$ rad | –
$A_{\rm CP}^{\Lambda}$ | $(-3.7\pm 11.7\pm 9.0)\times 10^{-3}$ |  $(-6\pm 12\pm 7)\times 10^{-3}$ [14]
$\left<\phi_{\Xi}\right>$ |  $\phantom{-}0.016\pm 0.014\pm 0.007$ rad |

Acknowledgements

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key Research and Development Program of China under Contracts Nos.
2020YFA0406300, 2020YFA0406400; National Natural Science Foundation of China
(NSFC) under Contracts Nos. 11625523, 11635010, 11735014, 11822506, 11835012,
11905236, 11935015, 11935016, 11935018, 12075107, 11961141012; the Chinese
Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint
Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos.
U1732263, U1832207; CAS Key Research Program of Frontier Sciences under
Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS;
CAS President’s International Fellowship Initiative (PIFI) programme; INPAC
and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under
Contract No. 758462; European Union Horizon 2020 research and innovation
programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790;
German Research Foundation DFG under Contracts Nos. 443159800, Collaborative
Research Center CRC 1044, FOR 2359, GRK 214; Istituto Nazionale di Fisica
Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; National Science and Technology fund; Olle Engkvist
Foundation under Contract No. 200-0605; STFC (United Kingdom); The Knut and
Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal
Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research
Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374,
DE-SC-0012069; National Science Centre (Poland) under Contract No.
2019/35/O/ST2/02907.

Author Contributions

All authors have contributed to the publication, being variously involved in
the design and the construction of the detectors, in writing software,
calibrating sub-systems, operating the detectors and acquiring data and
finally analysing the processed data.

Additional Information

Correspondence and requests for materials should be addressed to
<EMAIL_ADDRESS>
This pre-print is based on the style template found at
github.com/kourgeorge/arxiv-style.

Data Availability Statement

The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

## Appendix

### Monte Carlo simulation

For the selection and optimization of the final event sample, estimation of
background sources as well as normalization for the fit method, Monte Carlo
simulations have been used. The simulation of the BESIII detector is
implemented in the simulation software `GEANT4`[40, 41]. `GEANT4` takes into
account the propagation of the particles in the magnetic field and particle
interactions with the detector material. The simulation output is digitized,
converting energy loss to pulse heights and points in space to channels. In
this way the Monte Carlo digitized data have the same format as the
experimental data. The production of the $J/\psi$ is simulated by the Monte
Carlo event generator `KKMC`[42]. Particle decays are simulated using the
package `BesEvtGen`[43, 44], where the properties of mass, branching ratios
and decay lengths come from the world averaged values[21]. We find that while
the mass of the $\Lambda$ in our data agrees with the established value, that
of the $\Xi$ is $95$ keV/$c^{2}$ above the central value of the world
average[21], $m_{\Xi,\rm PDG}$= $1321.71\pm 0.07$ MeV/$c^{2}$. Hence, we have
adjusted the value in the simulation accordingly.

The signal channels used for optimization and consistency checks are
implemented with the helicity formalism and with parameter values in close
proximity to the results presented in Table 1.

### Selection Criteria

The data were accumulated during two run periods, in 2009 and 2012, where the
later set is approximately five times larger compared to the earlier. For the
analysis all charged final state particles have to be reconstructed. The Main
Drift Chamber (MDC) of the BESIII experimental setup is used for
reconstructing the charged-particle tracks. At least three positively and
three negatively charged tracks are required, each track fulfilling the
condition that $|\cos\theta|<0.93$, where $\theta$ is the polar angle with
respect to the positron beam direction. The momentum distributions of protons
and pions from the signal process are well separated and do not overlap, as
shown in Figure (supplementary) 1. Therefore a simple momentum criterion
suffices for particle identification: $p_{pr}>0.32$ GeV/$c$ and $p_{\pi}<0.30$
GeV/$c$ for protons and pions, respectively. Only events with at least one
proton, one anti-proton, two negatively and two positively charged pions are
saved for further analysis. Each $\Xi$ decay chain is reconstructed
separately, and is here described for the sequence
$\Xi^{-}\to\Lambda\pi^{-}\to p\pi^{-}\pi^{-}$. To find the correct $\Xi^{-}$
and $\Lambda$ particles all proton and $\pi^{-}$ candidates are combined
together. The $\Lambda$ and $\Xi^{-}$ particles are reconstructed through
vertex fits by first combining the $p\pi^{-}_{i}$ pair to form a $\Lambda$ and
then the $\Lambda\pi^{-}_{j}$ ($i\neq j$) pair to form a $\Xi^{-}$. The fits
take into account the non-zero flight paths of the hyperons which can give
rise to different production and decay points. All vertex fits must converge
and the combination which minimises
$((m_{p\pi\pi}-m_{\Xi})^{2}+(m_{p\pi}-m_{\Lambda})^{2})^{1/2}$, where
$m_{\Xi}$ and $m_{\Lambda}$ are the nominal masses and $m_{p\pi\pi}$
($m_{p\pi}$) is the mass of the candidate $\Xi$ ($\Lambda$), is retained for
further analysis. The same procedure is performed for the $\overline{\Xi}$
decay chain. For each decay chain the probability that the pions from the
$\Xi\to\Lambda\pi$ and $\Lambda\to p\pi$ decays is wrongly assigned is found
to be 0.51% and 0.49% for $\pi^{+}$ and $\pi^{-}$, respectively, which is
negligible for the analysis. The $m_{\Lambda\pi^{-}}$ versus
$m_{\overline{\Lambda}\pi^{+}}$ scatter plot is shown in the left panel of
Figure (supplementary) 2. A four-constraint kinematic fit requiring energy and
momentum conservation ($4C$) is imposed on the $e^{+}e^{-}\to
J/\psi\to\Xi^{-}\overline{\Xi}^{+}$ system, and only events where
$\chi^{2}_{4C}<100$ are retained for further analysis. The kinematic fit is
effective for removing the background processes $e^{+}e^{-}\to
J/\psi\to\gamma\eta_{c}\to\gamma\Xi^{-}\overline{\Xi}^{+}$ and $e^{+}e^{-}\to
J/\psi\to\Xi(1530)^{-}\overline{\Xi}\to\pi^{0}\Xi^{-}\overline{\Xi}^{+}$ (and
its charge conjugate), which have the same charged final state topology as the
signal channel, but contain extra neutral particles.

The invariant masses of the $p\pi^{-}$ and $\overline{p}\pi^{+}$ pairs are
also required to fulfil $|m_{p\pi}-m_{\Lambda,\rm peak}|<11.5$ MeV/$c^{2}$,
where $m_{\Lambda,\rm peak}$ is the peak position of the $\Lambda$ mass
distribution. A similar mass window criterion, optimised to remove the broad-
resonance $\Sigma^{-}(1385)\overline{\Sigma}^{+}(1385)$ background
contribution, is imposed on the $\Xi$ particle, $|m_{\Lambda\pi}-m_{\Xi,\rm
peak}|<11.0$ MeV/$c^{2}$.

The decay length is defined as the distance between the point of origin and
the decay position of the decaying $\Lambda$ or $\Xi$ particle. If the hyperon
momentum points oppositely to the direction from the collision to the decay
point, then the decay length becomes negative in the vertex-fit algorithm.
These events are removed from the sample.

When comparing experimental data with Monte Carlo simulations there are
differences seen for large polar angles. This discrepancy is seen to induce
systematic biases on the parameter values. The differences can, however, be
isolated and removed by restricting the sample to events where
$|\cos\theta_{\Xi,\rm c.m.}|<0.84$. Here $\theta_{\Xi,\rm c.m.}$ is the polar
angle of the reconstructed $\Xi$ particle in the center-of-mass frame. After
applying all selection criteria given above, 73 244
$\Xi^{-}\overline{\Xi}^{+}$ event candidates remain in the final sample.

### The maximum log-likelihood fit procedure

The global fit is performed on the data through the joint angular
distribution. For $N$ events the likelihood function is given by

${\cal{L}}(\boldsymbol{\xi}_{1},\boldsymbol{\xi}_{2},...,\boldsymbol{\xi}_{N};\bm{\omega})=\prod_{i=1}^{N}{\cal{P}}(\boldsymbol{\xi}_{i};\bm{\omega})=\prod_{i=1}^{N}\frac{{\cal{W}}(\boldsymbol{\xi}_{i};\bm{\omega})\epsilon({\boldsymbol{\xi}_{i}})}{{\cal{N}}(\bm{\omega})},$
(8)

where $\epsilon({\boldsymbol{\xi}})$ is the efficiency,
${\cal{W}}(\boldsymbol{\xi};\bm{\omega})$ is the weight as specified in Eqn.
(6), and the normalization factor ${\cal N}(\bm{\omega})=\int{\cal
W}(\boldsymbol{\xi};\bm{\omega})\epsilon({\boldsymbol{\xi}}){\rm
d}\boldsymbol{\xi}$. The normalisation factor is approximated as ${\cal
N}(\bm{\omega})\approx\frac{1}{M}\sum_{j=1}^{M}{\cal
W}(\boldsymbol{\xi}_{j};\bm{\omega})$, using $M$ Monte Carlo events
$\boldsymbol{\xi}_{j}$ generated uniformly over phase space, propagated
through the detector and reconstructed in the same way as data. $M$ is chosen
to be significantly larger than the number of events in data $N$; our results
exploit a simulation sample where $M/N\sim 35$. By taking the natural
logarithm of the joint probability density, the efficiency function can be
separated and removed as it only affects the overall log-likelihood
normalization and is not dependent on the parameters in $\bm{\omega}$. To
determine the parameters, the MINUIT package from the CERN library is
used[46]. The minimised function is given by ${\cal S}=-\ln({\cal{L}})$. The
operational conditions were slightly different for the 2009 and 2012 data
sets, most notably in the nominal value of the magnetic field. For this reason
the likelihoods are constructed separately for the two different run periods.

The results of the simultaneous fit are shown in Table 1. Those results that
depend on combinations of decay parameters account for the correlations
between the parameters. The correlation coefficients between the decay
parameters are given in Table (supplementary) 1. Assuming that CP-symmetry is
conserved we find $\left<\alpha_{\Lambda}\right>=\phantom{-}0.760\pm 0.006\pm
0.003$ and $\left<\alpha_{\Xi}\right>=-0.373\pm 0.005\pm 0.002$, where the
latter result is in disagreement with the current standard value
$\alpha_{\Xi}=-0.401\pm 0.010$[21]. The parameter $\alpha_{\Xi}$ has
previously only been measured indirectly via the product
$\alpha_{\Xi}\alpha_{\Lambda}$ and the assumed value of $\alpha_{\Lambda}$.
The current standard value of $\alpha_{\Lambda}=0.732\pm 0.014$ is an average
based on the two incompatible results of BESIII and the re-analysed CLAS data
Ref.[14, 22], and in disagreement with the value found in this analysis. In
contrast, our measured value for the product
$\left<\alpha_{\Xi}\right>\left<\alpha_{\Lambda}\right>=-0.284\pm 0.004\pm
0.002$ is compatible with the world average
$\alpha_{\Xi}\alpha_{\Lambda}=-0.294\pm 0.005$[21].

Table (supplementary) 1: Correlation coefficients for the production and asymmetry decay parameters. | $\alpha_{\psi}$ | $\Delta\Phi$ | $\alpha_{\Xi}$ | $\phi$ | $\alpha_{\Lambda}$ | $\overline{\alpha}_{\Xi}$ | $\overline{\alpha}_{\Lambda}$ | $\overline{\phi}_{\Xi}$
---|---|---|---|---|---|---|---|---
$\alpha_{\psi}$ | 1.0 | 0.414 | -0.008 | -0.006 | -0.107 | 0.014 | 0.120 | 0.003
$\Delta\Phi$ |  | 1.0 | -0.016 | 0.016 | -0.133 | 0.008 | 0.138 | -0.029
$\alpha_{\Xi}$ |  |  | 1.0 | -0.000 | 0.280 | 0.024 | 0.071 | 0.010
$\phi_{\Xi}$ |  |  |  | 1.0 | -0.002 | -0.010 | -0.010 | 0.013
$\alpha_{\Lambda}$ |  |  |  |  | 1.0 | 0.070 | 0.401 | 0.014
$\overline{\alpha}_{\Xi}$ |  |  |  |  |  | 1.0 | 0.269 | 0.001
$\overline{\alpha}_{\Lambda}$ |  |  |  |  |  |  | 1.0 | 0.006
$\overline{\phi}_{\Xi}$ |  |  |  |  |  |  |  | 1.0

### Systematic uncertainties

The systematic uncertainties are assigned by performing studies related to the
kinematic fit, the $\Lambda$ and $\Xi$ mass window requirements, the $\Lambda$
and $\Xi$ decay length selection, and a combined test on the
$\Xi\overline{\Xi}$ fit reconstruction with the $p,\pi$ MDC track
reconstruction efficiency. Searches of systematic effects are tested by
varying the criteria above and below the main selection. For each test, $i$,
the parameter values are re-obtained, $\bm{\omega}_{\rm sys,i}$ and the
changes evaluated compared to the central values,
$\Delta_{i}=|\bm{\omega}-\bm{\omega}_{\rm sys,i}|$. Also calculated are the
uncorrelated uncertainties $\sigma_{\rm
uc,i}=\sqrt{|\sigma_{\omega}^{2}-\sigma_{\omega,\rm sys,i}^{2}|}$, where
$\sigma_{\omega}$ and $\sigma_{\omega,\rm sys,i}$ correspond to the fit
uncertainties of the main and systematic test results, respectively. If the
ratio $\Delta_{i}/\sigma_{\rm uc,i}$ shows a trending behaviour and larger
than two this is attributed to a systematic effect[47, 48]. For each
systematic effect the corresponding uncertainty is evaluated. The assigned
systematic uncertainties are given in Tables (supplementary) 2,
(supplementary) 3 and (supplementary) 4, where the individual systematic
uncertainties are summed in quadrature.

  1. 1.

Estimator. To test if the method produces systematically biased results a
large Monte Carlo data sample is produced with production and decay
distributions corresponding to those of the fit results to the data sample
($\sim$10 times the experimental data). The simulated data are divided into
subsamples with equal number of events as the experimental sample, and run
through the fit procedure. The obtained fit parameters are found to be
consistent within one standard deviation of the generated parameter values and
hence no bias is detected.

  2. 2.

Kinematic fit. The systematic differences from the kinematic fit are tested by
varying the kinematic fit $\chi^{2}$ value from 40 to 200, with an increment
of 20 in each step. Significant effects are seen for the parameters
$\Delta\Phi$, $\phi$ and $\overline{\phi}$ when $\chi^{2}_{4C}>100$. For
$\chi^{2}_{4C}<100$ systematic deviations occur for $\alpha_{\Xi}$ and
$\overline{\alpha}_{\Lambda}$. The difference in track resolution between data
and Monte Carlo is the likely cause for these change in the parameter values.
The systematic uncertainty is assigned to be the average difference of the
main result to a lower and upper limit, determined to be at $\chi^{2}_{4C}=60$
and $200$, respectively.

  3. 3.

$\Lambda$ and $\Xi$ mass window selection. Possible systematic effects due to
the $\Lambda$, $\overline{\Lambda}$, $\Xi^{-}$ and $\overline{\Xi}^{+}$ mass
windows are investigate by varying the selection criteria between 2-30
MeV/$c^{2}$ and 2-20 MeV/$c^{2}$ for the $\Lambda/\overline{\Lambda}$ and
$\Xi^{-}/\overline{\Xi}^{+}$ candidates, respectively. For the $\Lambda$
selection systematic deviations are seen for decreasing mass windows. The
uncertainty is assigned to be the difference of the nominal result to the
result when $95\%$ of the events are included, at $|m_{p\pi}-m_{\Lambda,\rm
peak}|<6.9$ MeV/$c^{2}$. For the $\Xi^{-}$ and $\overline{\Xi}^{+}$ mass
windows significant effects are seen for the parameters $\alpha_{J/\psi}$,
$\alpha_{\Lambda}$ and $\phi_{\Xi}$. The systematic uncertainties for these
parameters are assigned to be the difference of the main result and the
results obtained one standard deviation lower than to the main selection
window, estimated from the $m_{\Lambda\pi}$ line shape uncertainty.

  4. 4.

$\Lambda$ and $\Xi$ decay length. Possible systematic effects related to the
$\Lambda$ and $\Xi$ life times are studied by varying the decay length
selection criteria for the $\Lambda$ and $\Xi$ candidates. For $\Xi$ no strong
trending behaviors are seen, but for $\Lambda$ a dependence is seen for the
asymmetry parameters $\alpha_{\Lambda}$ and $\overline{\alpha}_{\Lambda}$
which is accounted for in the final systematic uncertainty.

  5. 5.

The combined efficiency of $\Xi^{-}\overline{\Xi}^{+}$ reconstruction and
$p,\pi^{-}$ tracking. For the study of systematic effects related to the
tracking and the $\Lambda$ and $\Xi$ reconstruction it is assumed that the
combined efficiency for proton, anti-proton and $\pi^{\pm}$ only depends on
the polar angle $\cos\theta$ and the transverse momentum, $p_{T}$. To study
the tracking efficiency the fitted probability density function is modified by
allowing for arbitrary efficiency corrections as a function of $\cos\theta$
and $p_{T}$ for each particle type in an iterative procedure. The correction
procedure is repeated until the maximum log likelihood is stable within
$\ln(2)$ between two successive iterations. The difference between the fit
results with and without the tracking correction is assigned as the systematic
uncertainty.

  6. 6.

$\cos\theta_{\Xi,\rm c.m.}$ scattering angle. From comparing data to MC
simulation a discrepancy is seen for charged tracks with polar angles
$|\cos\theta|>0.84$. The discrepancy is also seen to have a notable effect on
some of the the decay parameters. The effect can be isolated by only removing
the events where $|\cos\theta_{\Xi,\rm c.m.}|>0.84$. Although the observed
data-simulation differences are removed by requiring that
$|\cos\theta_{\Xi,\rm c.m.}|<0.84$, residual systematic effects are observed
for $\left<\alpha_{\Xi}\right>$ and
$\left<\alpha_{\Xi}\right>\left<\alpha_{\Lambda}\right>$, which are included
in the systematic uncertainty.

  7. 7.

Data set consistency. When comparing the statistically independent results of
the 2009 and 2012 data sets, all parameters are found to agree within two
standard deviations. As there is no evidence of systematic bias, no
uncertainty is assigned associated with possible data set differences.

Table (supplementary) 2: Contributing systematic uncertainties, and the sum in quadrature. First row: statistical uncertainty as reference. The uncertainties of $\Delta\Phi$ and $\phi$ are given in radians. All values multiplied by a factor $10^{2}$. $\times 10^{2}$ | $\alpha_{\psi}$ | $\Delta\Phi$ | $\alpha_{\Xi}$ | $\overline{\alpha}_{\Xi}$ | $\alpha_{\Lambda}$ | $\overline{\alpha}_{\Lambda}$ | $\phi_{\Xi}$ | $\overline{\phi}_{\Xi}$
---|---|---|---|---|---|---|---|---
Statistical | 1.2 | 4.6 | 0.70 | 0.70 | 1.05 | 1.06 | 1.91 | 1.93
Kin. fit | 0.36 | 1.5 | 0.18 | 0.17 | 0.21 | 0.43 | 0.77 | 0.44
mass win $\Lambda$ | 0.44 | 0.44 | 0.07 | 0.02 | 0.56 | 0.33 | 0.17 | 0.46
mass win $\Xi$ | 0.25 | - | - | - | 0.36 | - | 0.46 | -
dec. length $\Lambda$ | - | - | - | - | 0.30 | 0.40 | - | -
Track. eff. | 0.80 | 0.41 | 0.27 | 0.05 | 0.21 | 0.14 | 0.16 | 0.16
Sum syst. | 1.0 | 1.6 | 0.33 | 0.18 | 0.79 | 0.69 | 0.93 | 0.66
Table (supplementary) 3: Contributing systematic uncertainties to CP tests, and the sum in quadrature. First row: statistical uncertainty as reference. All values multiplied by a factor $10^{2}$. $\times 10^{2}$ | $A_{\Lambda,\rm CP}$ | $A_{\Xi,\rm CP}$ |  $\Delta\phi_{\rm CP}$ (rad) |  $\delta_{P}-\delta_{S}$ (rad) |  $\zeta_{P}-\zeta_{S}$ (rad)
---|---|---|---|---|---
Statistical | 1.17 | 1.34 | 1.37 | 3.3 | 3.4
Kin. fit | 0.32 | 0.47 | 0.16 | 1.3 | 0.4
mass win. $\Lambda$ | 0.59 | 0.07 | 0.14 | 0.8 | 0.4
mass win. $\Xi$ | 0.38 | - | 0.20 | 0.7 | 0.5
dec. length $\Lambda$ | 0.46 | - | - | - | -
Track. eff. | 0.05 | 0.29 | 0.003 | 0.4 | $2\cdot 10^{-3}$
Sum syst. | 0.90 | 0.56 | 0.29 | 1.7 | 0.75
Table (supplementary) 4: Contributing systematic uncertainties to average values of decay parameters, and the sum in quadrature. First row: statistical uncertainty as reference. All values multiplied by a factor $10^{2}$. $\times 10^{2}$ | $\left<\alpha_{\Xi}\right>$ | $\left<\alpha_{\Lambda}\right>$ |  $\left<\phi\right>$ (rad) | $\left<\alpha_{\Xi}\right>\cdot\left<\alpha_{\Lambda}\right>$
---|---|---|---|---
Statistical | 0.49 | 0.58 | 1.35 | 0.38
Kin. fit | 0.09 | 0.19 | 0.54 | 0.02
mass win. $\Lambda$ | 0.05 | 0.12 | 0.31 | $<10^{-2}$
mass win. $\Xi$ | - | 0.07 | 0.26 | 0.04
$\cos\theta_{\Xi,\rm c.m.}$ | 0.12 | - | - | 0.13
Track. eff. | 0.16 | 0.17 | 0.16 | 0.06
Sum syst. | 0.22 | 0.29 | 0.69 | 0.21
Figure (supplementary) 1: Final-state particle momenta of pions and protons
for experimental data (blue crosses) and simulated data (red, solid line).

Figure (supplementary) 2: (left) Distribution of the invariant masses
$m_{\Lambda\pi^{-}}$ versus $m_{\overline{\Lambda}\pi^{+}}$. The other
structure seen is from the reaction
$J/\psi\to\Sigma(1385)^{-}\overline{\Sigma}(1385)^{+}$. (right) The
$m_{\Lambda\pi}$ distribution for the experimental data sample before the
final event selection criteria on $m_{\Lambda\pi}$ has been applied. The final
requirement selects the events between the two lines.

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# Representing ELMo embeddings as two-dimensional text online

Andrey Kutuzov
University of Oslo
<EMAIL_ADDRESS>
Elizaveta Kuzmenko
University of Trento
<EMAIL_ADDRESS>

###### Abstract

We describe a new addition to the WebVectors toolkit which is used to serve
word embedding models over the Web. The new ELMoViz module adds support for
contextualized embedding architectures, in particular for ELMo models. The
provided visualizations follow the metaphor of ‘two-dimensional text’ by
showing lexical substitutes: words which are most semantically similar in
context to the words of the input sentence. The system allows the user to
change the ELMo layers from which token embeddings are inferred. It also
conveys corpus information about the query words and their lexical substitutes
(namely their frequency tiers and parts of speech). The module is well
integrated into the rest of the WebVectors toolkit, providing lexical
hyperlinks to word representations in static embedding models. Two web
services have already implemented the new functionality with pre-trained ELMo
models for Russian, Norwegian and English.

##  1 Introduction

In this demo paper we describe a new module recently added to the free and
open-source WebVectors toolkit (Kutuzov and Kuzmenko, 2017)111A screencast is
available at https://www.youtube.com/watch?v=dDugoV1r_wk.. WebVectors allows
to easily deploy services to demonstrate the abilities of static
distributional word representations (word embeddings) (Bengio et al., 2003;
Mikolov et al., 2013) via web browsers. It currently powers at least two
embedding model hubs:

  * •

NLPL WebVectors222http://vectors.nlpl.eu/explore/embeddings/, featuring models
for English, Norwegian and other languages, trained within the Nordic Language
Processing Laboratory initiative.

  * •

RusVectōrēs333https://rusvectores.org/, featuring models for the Russian
language.

The new module (we name it ELMoViz) adds the functionality to study, probe and
compare recently introduced contextualized embedding (or ‘token-based’) models
(Melamud et al., 2016). In particular, at this point we provide support for
the ELMo architecture (Peters et al., 2018a) based on deep recurrent neural
networks. In the future, we plan to add support for Transformer-based models
like BERT (Devlin et al., 2019) and GPT-3 (Brown et al., 2020). ELMo
architecture is significantly less computationally expensive than
Transformers, while being almost on par in terms of performance. Thus, it
yields rich possibilities in the context of non-commercial web services.

Figure 1: Metaphor of two-dimensional text; borrowed from Biemann and Riedl
(2013).

For analyzing ELMo representations of an arbitrary input text, we offer the
metaphor of ‘two-dimensional text’ first proposed in Biemann and Riedl (2013)
(see Figure 1). This allows a sort of ‘visualization’ for contextualized
embeddings through finding words which are most semantically similar to the
input words in their current contexts. From the linguistic point of view,
these are ‘paradigmatic replacements’ (Saussure, 1916) – words that can to
some extent substitute target words. The two dimensions here are the
syntagmatic one (horizontal) which describes the linear order of the sentence,
and the paradigmatic one (vertical) which describes semantic classes to which
the words in the sentence belong to. The generated substitutes in the vertical
axis can also be thought of as ‘semantic variations’ of the input sentence.

The rest of the paper is organized as follows. In Section 2 we describe the
background for this work, including the WebVectors framework, and explain the
need to develop additional functionality in order to handle contextualized
embeddings. Section 3 describes in detail this functionality, both from the
point of view of the end user and from the point of view of deployment
logistics. In Section 4, we conclude and outline future work.

##  2 Background

Since the widespread adoption of prediction-based word embeddings (Mikolov et
al., 2013) started, there has always been a need to efficiently serve and
demonstrate these representations over the Web. Researchers and practitioners
need this for quick experimentation and testing hypotheses by comparing
different distributional models. Those who teach natural language processing
and computational linguistics need ways to show the students how dense
distributional representations capture lexical semantics without installing
any software or downloading any models (often it is desirable that this is
shown for a particular language or domain).

In turn, language teachers value tools to demonstrate lexical variety and
degrees of similarity for words in a foreign language. To this extent, serving
word embeddings over the Web can help both the teachers with preparing
educational materials and the students with grasping the concepts in a foreign
language.

The WebVectors framework we presented in Kutuzov and Kuzmenko (2017) is aimed
at all these purposes. It allows to quickly deploy a stable and robust web
service featuring operations on vector semantic models, including querying,
visualization and comparison, all available to users of any computer literacy
level. It extended already existing embedding visualization services like
Embedding Projector444https://projector.tensorflow.org/ by providing users
with the ability to find nearest semantic neighbors of query words, perform
vector math operations over embeddings, etc. Since being first presented in
2016, WebVectors keeps adding new functionality, and now it offers filtering
nearest associates by part of speech tags or corpus frequency, and can
generate semantic ego graphs, among other features (see Figure 2).

Figure 2: Screenshot of a WebVectors instance at
http://vectors.nlpl.eu/explore/embeddings/

Until the introduction of ELMoViz, these features were limited to the so-
called ‘static word embeddings’, that is, architectures like word2vec (Mikolov
et al., 2013), fastText (Bojanowski et al., 2017) or GloVe (Pennington et al.,
2014). In these architectures, after the training is finished, each word type
in the vocabulary is rigidly associated with a single dense vector. However,
in the recent years NLP saw a surge of pre-trained ‘contextualized’ embedding
architectures, like ELMo (Peters et al., 2018a), BERT (Devlin et al., 2019),
GPT-3 (Brown et al., 2020) and many others. One of the changes these deep
learning models brought was that even at inference time, each word token
representation (embedding) depends on its immediate context. This means that
ambiguous words will receive different representations depending on the sense
in which they are used, which opens rich new possibilities for natural
language understanding.

Libraries used in WebVectors to deal with static word embeddings (Gensim,
(Řehůřek and Sojka, 2010)) were not fit to power operations on contextualized
models. That is why we decided to implement an entirely new WebVectors module,
which would take a query phrase as an input, and produce paradigmatic
replacements (lexical substitutions) for each content word in this phrase,
based on a given pre-trained contextualized ELMo language model.

One can find a number of existing frameworks for online experimentation with
contextualized models: among others, we should mention Language
Interpretability Tool (Tenney et al., 2020), exBert by Hoover et al. (2019)
and the hosted inference API at the HuggingFace Community Model Hub (Wolf et
al., 2020). However, these projects are aimed exclusively at the Transformer-
based architectures. The system we present in this demo paper, on the other
hand, is aimed more towards RNN-based architectures like ELMo. As it was
shown, for example, in the field of semantic change detection (Kutuzov and
Giulianelli, 2020), ELMo can often outperform BERT or be on par with it, while
requiring significantly less computational resources. We believe it is
especially important for teaching activities.

Additionally, our system is more lexically oriented and is integrated with the
existing WebVectors functionality, as we will show in the next section.

##  3 System description

After turning on the contextualized embedding related functionality in the
WebVectors configuration file,555In principle, it is also possible to use only
ELMoViz, without other WebVectors modules. the person deploying the service
has to provide three data sources for each ELMo model:

  1. 1.

a pre-trained ELMo model itself in the standard format (*.HDF5 file with the
weights and options.json file with the model architecture description);

  2. 2.

a tab-separated frequency dictionary file to use when determining the
frequency tier of word types (it is recommended to derive it from the same
corpus the ELMo model was trained on, but technically this is not required);

  3. 3.

a set of static (type-based) word embeddings produced by averaging
contextualized token embeddings inferred with the same ELMo model.

The last item of this list requires some explanation. Our aim is to provide
the end user with a set of lexical substitutes for each word token in context
from the input sentence (see Figure 3). With static embedding architectures,
this boils down to looking up the vector of the target word $x$ and then
finding $n$ other words in the model vocabulary with the vectors closest to
$x$. However, this is obviously impossible with contextualized language
models: there are no static vector lookup tables to begin with. One can easily
infer contextualized representations for each word in the input sentence: but
what to compare them with in order to illustrate their meaning?

To cope with this issue, we adopted the approach described in (Liu et al.,
2019). They employed the so called type-level context averaging in order to
align pre-trained contextualized models cross-linguistically. In our case, we
needed only the first stage of their workflow. The idea is to obtain static
type-level word representations located in the same vector space as the
contextualized embeddings. Given a large enough reference text corpus and a
pre-trained contextualized language model, one takes the average of all token
representations for each target word occurrence in the corpus. This averaged
type embedding is comparable to contextualized token embeddings routinely
produced by the model.

Figure 3: Examples of two-dimensional text inferred from an ELMo model
($n=5$).

In practice, we found that one does not even need to average token embeddings:
it is enough to sum them, and then unit-normalize the resulting summed vector.
As for the list of target words, we simply use top 10 000 (or any other amount
found suitable) most frequent words from the corresponding ELMo model
vocabulary or from a reference corpus (excluding functional parts of speech
and digits). Low frequent words are usually not needed in this case anyway,
since the quality of their embeddings is also lower. We provide a simple
script to extract type embeddings from an ELMo model and a given corpus in our
GitHub repository.666https://github.com/akutuzov/webvectors/tree/master/elmo/

As a result, when an end user enters an input phrase or sentence (typically
from 5 to 15 words), WebVectors produces contextualized token embeddings for
each token in the query, and finds top $n$ words in the type embedding model,
which are the closest (by cosine similarity) to each of the token embeddings.
These predictions are lexical substitutes or paradigmatic replacements; they
demonstrate what other words could fill these positions in the query,
depending on the context.

Another option to produce such substitutes would be to feed the input sentence
to the ELMo model and then for each word token choose the strongest
activations at the final softmax layer of the language model and map them to
words in the model vocabulary. However, in practice we found that this
approach is slightly slower than the one described above. Additionally, ELMo
models are often published online without the vocabulary they were trained on.
Since the input layer of ELMo is purely character-based, it does not hinder
inferring token embeddings, but it effectively blocks using these weights as
language models per se. Our approach allows one to use any given ELMo model
with any desired corpus to produce a set of reference type embeddings.

System maintainers can provide several models for the service to work with,
including models for different languages; one of the models should be
specified in the configuration files as the default one. When entering the
query sentence, users can choose the model which will process the input.

Apart from choosing between different models, WebVectors also allows users to
choose the exact ELMo layer from which token representations will be inferred;
it was shown in Peters et al. (2018b) that different neural network layers
convey information related to different linguistic tiers: syntax, semantics,
pragmatics, etc. At this point, one can choose between the top ELMo layer and
the average of all layers. Note that for all operations with pre-trained ELMo
models we use simple_elmo: a lightweight TensorFlow-based Python package also
developed by us.777https://pypi.org/project/simple-elmo/ If need be,
simple_elmo can also be used as a standalone library to handle ELMo models.

Both the words from the input sentence and the lexical substitutes are colored
according to their frequency tier in the reference corpus (green for ‘high’,
blue for ‘mid’ and red for ‘low’), in accordance with other WebVectors
components. Similarly, each word is hyperlinked to its ‘landing page’ bound to
one of the static embedding models served by a particular WebVectors
installation (like the one in Figure 2), allowing easy and playful exploration
of the semantic space. The font size of the lexical substitute corresponds to
cosine similarity between the token embedding and the substitute type
embedding: thus, users can instantly see what word tokens the model is unsure
about. The service performs fast under-the-hood part-of-speech tagging of the
query,888Using UDPipe (Straka and Straková, 2017). so for functional words we
always yield themselves as substitutes (see ‘her’, ‘that’ and ‘can’ in Figure
3). They are also uncolored and not hyperlinked, so that a user might focus on
content words, while at the same time still having an impression of ‘full
sentence variations’.

The users should be aware that the lexical substitutes potentially contain all
the biases inherited from the corpus the model was trained on. Thus, the
paradigmatic axis might include slander words and stereotypes, if they were
frequent enough in the data. We did not address this issue in the present
work, but we advise the users to take this into account when dealing with any
unsupervised language models.

Importantly, we keep a short history of substitute queries, so that it is
possible to see at a glance the changes brought by a different context, a
different word order or a different contextualized model (if the web service
offers several models). Figure 4 shows an example from our Russian live demo
at the RusVectōrēs web service. In the first sentence, the word закладку
‘zakladku’ is used in the newer sense of ‘a secret place to store illegal
drugs’, while in the second sentence it is used in the older sense of ‘the act
of founding a building’. The generated substitutes reflect the differences in
word meaning depending on the context. In the first example the substitutes
include such words as ‘meeting, sale, operation’, and in the second example
the substitutes are ‘opening, building, repair’.

Figure 4: History of lexical substitute queries with a Russian ELMo model.

##  4 Conclusion

The described system for generating two-dimensional text using pre-trained
ELMo models is now deployed at the two model hubs mentioned in Section 1. NLPL
WebVectors features ELMo models trained on English Wikipedia and on Norwegian
corpora999http://vectors.nlpl.eu/explore/embeddings/en/contextual, while
RusVectōrēs features a model trained on concatenated Russian Wikipedia and
Russian National Corpus.101010https://rusvectores.org/en/contextual/

The presented component for the WebVectors framework allows users to explore
pre-trained ELMo models and to visualize contextualized embeddings as a two-
dimensional text for faster analysis of early research prototypes. While
previously the framework provided interface only to static vector semantic
models, introducing support for contextualized architectures allows for more
intricate exploration of linguistic phenomena, such as lexical ambiguity and
contextual semantic change.

We hope that the new functionality will provide language teachers, NLP
researchers and practitioners with a powerful tool to study word meaning in
context and at the same time keep the audience up-to-date with recent advances
in the field of distributional semantics and deep learning based NLP. A
separate important contribution is our simple_elmo library which makes using
ELMo models in Python much easier, especially for researchers with linguistic
background.

In the future, we plan to add support for other contextualized embedding
architectures like BERT, to allow inter-architectural comparisons. Another
interesting room for future work is integrating with other exploratory
services for neural NLP models, like the ones mentioned in Section 2.

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