# Production and Decay of Polarized Hyperon-Antihyperon Pairs

Karin Schönning
Department of Physics and Astronomy
Uppsala University
Uppsala, Sweden
<EMAIL_ADDRESS>
Varvara Batozskaya
Institute of High Energy Physics
Beijing, China
and National Centre for Nuclear Research
Warsaw, Poland
Patrik Adlarson
Department of Physics and Astronomy
Uppsala University
Uppsala, Sweden
Xiaorong Zhou
University of Science and Technology in China
Hefei, China
Corresponding author

###### Abstract

Polarized hyperon-antihyperon pairs shed light on various unresolved puzzles
in contemporary physics: How the strong interaction confines quarks into
hadrons, how accurately the Standard Model describes microcosmos and even why
our universe consists of so much more matter than antimatter. Thanks to their
weak, parity violating decays, hyperons reveal their spin properties. This can
be exploited e.g. the decomposition of the electromagnetic structure of
hyperons, precision tests of flavour symmetry and searches for CP violation.
At the BESIII experiment at BEPC-II, Beijing, China, hyperon-antihyperon pairs
can be produced in abundance. Recently collected large data samples have
triggered the development of new methods that provide unprecedented precision
and a plethora of new results have emerged. When applied at future high-
intensity facilities like PANDA and STCF, precision physics will be taken to a
new level which can contribute to the solution to the aforementioned puzzles.

_Keywords_ Hyperon $\cdot$ Polarization $\cdot$ CP Violation $\cdot$ Form
Factors

##  1 Introduction

Hyperons offer a unique tool to study some of the most challenging problems in
contemporary physics. In this review, we will in particular demonstrate how
polarized and entangled hyperon-antihyperon pairs can be exploited as a
precision instrument to probe various aspects of the Standard Model (SM) of
particle physics. Though immensely successful in describing the elementary
particles and their fundamental interactions, the SM falls short in describing
the properties of complex systems such as nucleons, nuclei or even the
evolution and composition of our universe. Many puzzles in modern physics are
related to the limitations of the SM and they fall into two main categories:
those whose solution should be sought within the SM, and those where one needs
to go beyond.

Within the SM, it remains a challenge to quantitatively describe the main
building block of the visible universe: the nucleon. Its mass, size and
structure emerge from the complex dynamics of the strong interaction. The
theory of Quantum ChromoDynamics (QCD), an inherent part of the SM, describes
these interactions from first principles, but its non-Abelian nature makes it
difficult to make quantitative predictions at distances where quarks form
hadrons. Replacing one or several light quarks in the nucleon with heavier
strange or charm, gives deeper insights into these interactions.

Various experimental observations, such as neutrino oscillations, the need for
dark matter to explain cosmological observations and the matter-antimatter
asymmetry of the universe, raise the question of whether there is a more
fundamental theory beyond the SM that encompasses these phenomena. Intensive
searches for physics beyond the SM (BSM) are ongoing at the high energy
frontier, in a quest to find hitherto unknown, heavy particles manifesting a
new fundamental theory. A complementary approach is the precision frontier:
new aspects of particles whose existence is established since long. By
studying the properties of these particles, e.g. measuring quantities that
should either be zero, or that can be calculated with immense precision,
physics beyond the SM would reveal itself by significant deviations from these
predictions.

Figure 1: The $Y\to BM$ decay, with the spin direction of $Y$ along the
$y$-axis.

Hyperons are well suited for precision physics through their weak, parity
violating decays, giving straightforward experimental access to their spin
properties. This is in contrast to e.g. protons, for which dedicated
polarimeter detectors are required for this purpose. In hyperon decays, the
daughter particles are emitted according to the direction of the spin of the
mother hyperon. For example, consider a two-body decay $Y\to BM$, where $Y$ is
a spin 1/2 hyperon, $B$ a spin 1/2 baryon and $M$ a pseudoscalar meson as
illustrated in Fig. 1. The angular distribution of $B$ in the rest system of
$Y$ with respect to some reference axis $\vec{y}$ is given by [1, 2]

$W(\cos\theta_{B})=\frac{1}{4\pi}(1+\alpha P_{y}\cos\theta_{B}).$ (1)

The polarization $P_{y}=P_{y}(\cos\theta_{Y})$ carries information about the
production mechanism, while the decay asymmetry $\alpha$ only depends on the
decay process. The parameter $\alpha$ is proportional to the real part of the
product of the interfering parity violating and parity conserving decay
amplitudes [3]. The imaginary part is accessible in sequential hyperon decays,
i.e. hyperons that decay weakly into other hyperons [4, 5]. Eq. 1 demonstrates
an example of how physical parameters, such as $\alpha$ and $P_{y}$, can be
retrieved from measurable quantities such as $\cos\theta_{B}$ and
$\cos\theta_{Y}$. This feature is crucial in the work presented in this
review.

##  2 Fundamental questions

###  2.1 Structure at the femtometer scale

The nature of the strong interaction manifests in the properties of the
nucleon, such as its mass and size. These are connected since a mass-scale is
a prerequisite to confine colour charges into colour singlets of finite size
[6]. The structure, i.e. the distribution and motion of the quarks inside
hadrons, is crucial to understand the underlying mechanisms. Recently,
tremendous efforts in atomic and subatomic physics as well as hadron theory
have led to rapid progress in solving the so-called proton radius puzzle [7].
A coherent picture of the emergence of hadron properties requires
corresponding studies of similar but different quark systems, such as neutrons
[8, 9] and hyperons. The latter contains one or several heavy quarks such as
strange and charm, which should have an impact on their inner structure,
including properties like the charge radius. Measured radii of pions [10] and
kaons [11] indicate that this is indeed the case.

Electromagnetic form factors (EMFFs) are structure functions that have the
advantage that they are experimentally accessible for both protons, neutrons
and hyperons. Space-like electric $G_{E}$ and magnetic $G_{M}$ form factors,
probed in elastic electron-baryon ($e^{-}B\to e^{-}B$ ) scattering, are
related to the charge- and magnetisation densities, respectively [12].
However, since the hyperons are unstable, they are unfeasible as beams or
targets and as a consequence, their space-like EMFFs are hard to access
experimentally. Instead, the time-like EMFFs constitute the most viable
structure observables for hyperons [13]. These can be probed in either
$e^{+}e^{-}\to\gamma^{*}\to Y_{1}\,\bar{Y}_{2}$ reactions or in so-called
Dalitz decays $Y_{1}\to Y_{2}\gamma^{*},\gamma^{*}\to e^{+}e^{-}$. The
experimentally accessible time-like and the intuitive space-like EMFFs are
related via dispersion relations [14].

Space-like EMFFs are real functions of $q^{2}$, whereas the time-like ones are
complex. The electric and the magnetic form factor of a spin 1/2 hyperon have
a relative phase $\Delta\Phi$ [15], reflecting fluctuations of the
$\gamma^{*}$ into e.g. a $\pi\pi$ intermediate state. At a certain scale
$q^{2}_{asy}$, the time-like and the space-like EMFFs should converge to the
same real value, i.e. their phase will approach an integer multiple of $\pi$
[16, 17]. From the asymptotic behaviour of the EMFF phase, we can even extract
information about space-like quantities such as the charge radius [18]. But
how do we measure this phase?

In the $e^{+}e^{-}\to Y\bar{Y}$ reaction, a non-zero phase manifests in a
polarized final state, even if the colliding beams are unpolarized [15]. The
polarization has a well-defined dependence on the hyperon scattering angle,
the form factor ratio $R=|G_{E}/G_{M}|$ and the phase $\sin\Delta\Phi$ [19].
Therefore, the phase can be calculated from the polarization, which in turn
can be extracted from the angular distribution of the decay products,
according to Eq. 1.

In the vicinity of vector charmonium such as $J/\Psi$, $\Psi(3686)$ and
$\Psi(3770)$, the production of most hyperon-antihyperon pairs is strongly
influenced by processes involving these vector states. In some cases, in
particular for $J/\Psi\to Y\bar{Y}$, this process dominates. For others, such
as the isospin-breaking $J/\Psi\to\Lambda\bar{\Sigma}^{0}$, it is suppressed.
In the former cases, the form factors describe the vector decay and are
denoted psionic form factors, in contrast to electromagnetic form factors
[19]. Though psionic form factors are straightforward to measure with high
precision, their physical meaning is yet to be understood.

###  2.2 Searching Beyond the Standard Model

####  2.2.1 CP violation

Despite its great success, the SM does not provide an explanation for the
matter-antimatter asymmetry of the universe; Why does our universe consists of
matter, and not antimatter? Unless fine-tuned in the Big Bang, the matter
abundance must have been generated dynamically. The mechanism proposed by A.
Sakharov, Baryogenesis [20], has become the most long-standing candidate for
such a process. However, several criteria need to be fulfilled for this to be
possible: i) the existence of processes that violate baryon number, ii) the
existence of processes that violate the conservation of charge conjugation (C)
and the combined charge conjugation and parity (CP), and iii) the processes i)
and ii) must occur outside thermal equilibrium. Hyperons provide a sensitive
instrument to test criterion ii) as we will demonstrate in this report.

CP violation in weak interactions is accommodated in the SM through the
Cabibbo-Kobayasi-Maskawa (CKM) mechanism [21, 22]. However, the predicted CP
violations are not sufficient to explain the observed matter abundance of the
universe. Hence, the search for CP violation is connected to the search for
physics beyond the SM. Decays of entangled hyperon-antihyperon pairs provide a
unique hunting-ground for such signals. Hyperons decay through a complex
interplay of strong, CP conserving interactions and weak, possibly CP
violating ones and the challenge is to interpret the decay patterns of
hyperons and antihyperons in terms of CP violating amplitudes.

While predicting small CP violations in weak interactions, the SM should allow
for CP violating strong interactions. This would for instance manifest in a
non-zero electric dipole moment (EDM) of the neutron [23]. However, the upper
limit of the neutron EDM has been found to be extremely small [24], suggesting
an unnaturally small CP violation in the strong interaction. The fact that the
SM does not explain why this is so small, is referred to as the strong CP
problem [25]. Flavour conserving radiative hyperon decays are related to the
neutron EDM through flavour symmetry and can therefore constitute a potential
test of the strong CP problem [26]. In addition, they offer a probe for tests
of C conservation. Flavour conserving radiative decays of hyperons involve
both weak, electromagnetic and strong interactions and probe P and CP symmetry
[27].

####  2.2.2 Flavour Symmetry

In addition to the aforementioned CP tests, hyperon decays offer independent
ways to probe the SM at the precision frontier. In particular, the CKM matrix
elements, characterizing the quark flavour mixing, are crucial parameters of
the SM. Considering the three lightest quarks up ($u$), down ($d$) and strange
($s$), the matrix elements $|V_{ud}|$ and $|V_{us}|$ can be precisely
predicted and also tested independently in semi-leptonic decays ($H_{1}\to
H_{2}l\bar{\nu}_{l}$) of mesons as well as hyperons. In fact, the interplay
between the weak interaction and the hadronic structure is revealed with
greater richness in hyperon decays compared to meson dittos. This is thanks to
the presence of three valence quarks in the former case, compared to the
simpler quark-antiquark structure [21].

A peculiar feature of the SM is that the electroweak gauge bosons, i.e. the
$\gamma$, the $Z$ and the $W^{\pm}$, couple in the same way to all leptons,
irrespective of generation. This feature, referred to as lepton universality,
can be probed in semi-leptonic decays of hyperons [28].

##  3 The BESIII Experiment

The BEijing Spectrometer (BESIII) [29] at the Beijing Electron Positron
Collider (BEPC-II) is a multi-purpose experiment that offers unique
opportunities to explore strange and single-charm hyperons. The experiment
operates at CMS energies ranging from 2.0 GeV up to 4.95 GeV. Hyperon-
antihyperon pairs can be produced either in one-photon exchange processes
$e^{+}e^{-}\to\gamma^{*}\to Y\bar{Y}$ by off-resonance energy scans, or from
vector charmonium decays, e.g. $e^{+}e^{-}\to J/\Psi\to Y\bar{Y}$.

Since the operation commenced in 2009, BESIII has collected more than 40 fb-1
of data, comprising several world-leading data samples including $10^{10}$
$J/\psi$ events [30] and $2.7\cdot 10^{9}$ $\psi(3686)$ events. These samples
give unprecedented access to the hyperon pairs via $\psi\to Y\bar{Y}$ decays.
In addition, fine energy scans have been performed in a range covering the
production threshold regions of all strange ground-state hyperons. Recently,
more than 3 fb-1 of data have been collected above the
$\Lambda_{c}^{+}\bar{\Lambda_{c}}^{-}$ threshold.

The BESIII detector covers 93% of the 4$\pi$ solid angle. A small-cell,
helium-based main drift chamber (MDC) surrounding the $e^{+}e^{-}$ collision
point provides precise tracking of charged particles. The BESIII detector also
comprises a time-of-flight system (TOF) based on plastic scintillators, an
electromagnetic calorimeter (EMC) made of CsI(Tl) crystals, a muon counter
(MUC) made of resistive plate chambers, and a superconducting solenoid magnet
with a central field of 1.0 Tesla.

The work presented here is based on a low-energy scan from 2015, including a
large sample at 2.396 GeV, a high-energy scan from the
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ threshold near 4.6 GeV, collected
during 2014, a scan above 4.6 GeV during 2019 - 2021, a sample of $\Psi(3686)$
events collected during 2009 and 2012, $1.2\cdot 10^{9}$ $J/\Psi$ events
collected in 2009 and 2012, and $10^{10}$ $J/\Psi$ from 2018 and 2019.

##  4 Polarized and Entangled Hyperons

###  4.1 Hyperon-Antihyperon Pair Production

Figure 2: Illustration of the process $e^{+}e^{-}\to\gamma^{*}\to
Y\overline{Y}$, where $\gamma^{*}$ denotes a virtual photon (left) and the
$e^{+}e^{-}\to\Psi\to Y\overline{Y}$, where $\Psi$ a vector charmonium
resonance (right).

Consider the production of a pair of a spin 1/2 hyperon and antihyperon in
either the $e^{+}e^{-}\to\gamma^{*}\to Y\bar{Y}$ (Fig. 2, left panel) or
$e^{+}e^{-}\to\Psi\to Y\bar{Y}$ (Fig. 2, right panel), where $\Psi$ is any
vector charmonium resonance: $J/\Psi$, $\Psi(3686)$, $\Psi(3770)$ etc. In
$e^{+}e^{-}$ annihilations, processes with spin-parity $J^{P}=1^{-}$ dominate
completely which means that the spin-parity of the final state is considered
to be known. In the case of an intermediate virtual photon $\gamma^{*}$, the
process can be parameterised in terms of two complex electromagnetic form
factors $G_{E}(q^{2})$ and $G_{M}(q^{2})$ [15, 31, 32, 33]. The form factors
are functions of the momentum transfer $q^{2}$ carried by the virtual photon.
The Born cross section $\sigma_{B}$ is related to the effective form factor
$G_{\rm eff}(q^{2})$, a linear combination of $G_{E}(q^{2})$ and
$G_{M}(q^{2})$:

$G_{\rm
eff}(q^{2})=\sqrt{\frac{2\tau|G_{M}(q^{2})|^{2}+|G_{E}(q^{2})|^{2}}{1+2\tau}}=\sqrt{\frac{\sigma_{B}\cdot
3q^{2}}{1+\frac{1}{2\tau}(4\pi\alpha^{2}\beta)}}$ (2)

where $\tau=q^{2}/(4M_{Y}^{2})$, $\alpha$ is the electromagnetic fine
structure constant and $\beta$ is the velocity.

The modulus of their ratio $R=|G_{E}(q^{2})/G_{M}(q^{2})|$ governs the
scattering angle $\theta$ between the produced hyperon and the positron beam:

$\frac{d\sigma_{B}}{d\Omega}=\frac{2\pi\alpha^{2}\beta}{4q^{2}}(1+\cos^{2}\theta+\frac{R^{2}}{\tau}\sin^{2}\theta)=\frac{2\pi\alpha^{2}\beta}{4q^{2}}(1+\eta\cos^{2}\theta),$
(3)

where $\eta=(\tau-R^{2})/(\tau+R^{2})$. The relative phase $\Delta\Phi$ of
$G_{E}(q^{2})$ and $G_{M}(q^{2})$ governs the vector polarization and tensor
polarization (i.e. spin correlations) of the produced hyperon-antihyperon pair
[4]. If the beams are unpolarized, then a vector polarization of the final
state hyperons is only allowed in the direction normal to the plane spanned by
the incoming beam and the outgoing hyperon:

$P=\frac{\sqrt{(1-\eta^{2})}\sin\theta\cos\theta}{1+\eta\cos^{2}\theta}\sin\Delta\Phi.$
(4)

In addition, there are five non-zero spin correlations. In the case of an
intermediate vector charmonium, the electromagnetic form factors are replaced
by the so-called psionic form factors $G_{E}^{\Psi}$ and $G_{M}^{\Psi}$ [4].
The formalism is the same as for an intermediate $\gamma^{*}$, but by
convention, the $\eta$ in Eq. 3 is denoted $\alpha_{\Psi}$ and the phase
$\Delta\Phi_{\Psi}$.

If a spin 1/2-spin 3/2 hyperon-antihyperon pair is produced in an initial
$J^{P}=1^{-}$ state, an additional Coulomb quadrupole form factor
$G_{C}(q^{2})$ [34] enters the parameterization. An equivalent description is
to use three helicity parameters as shown in Ref. [35]. In a similar way, the
production of a hyperon-antihyperon pair which are both spin 3/2 can be
parameterized by $G_{E}(q^{2})$, $G_{M}(q^{2})$, $G_{C}(q^{2})$ and the
octupole form factor $G_{O}(q^{2})$, or equivalently by four helicity
amplitudes [35, 36].

###  4.2 Hadronic Hyperon Decays

Figure 3: The reaction $e^{+}e^{-}\to Y\overline{Y}$ which decays either
hadronically ($BM$, to the left) or semi-leptonically ($BW^{-}$, to the
right). Theplanes illustrate different reference frames, where those where
particles are emitted back-to-back represent the rest frame, or the helicity
frame, of the decaying particle.

The self-analysing hyperon decays, mentioned in the introduction, provide a
powerful diagnostic tool to study

  * •

Structure functions, such as electromagnetic form factors.

  * •

The spin of the mother hyperon.

  * •

Parity violating and parity conserving decay amplitudes.

In the following, we will look into two-body hadronic decays of spin 1/2 or
spin 3/2 hyperons decaying into a baryon and a pseudoscalar mesons.

####  4.2.1 Direct Decays

By direct decay, we consider the case when a hyperon $Y$ either decays into a
stable baryon $B$ and a pseudoscalar meson $M$, or the case when the hyperon
decays into another hyperon but where the subsequent decay of the daughter
hyperon is not measured or where the daughter is a stable baryon. This process
is schematically shown in Figure 3. The production plane of the $e^{+}e^{-}\to
Y\overline{Y}$ process is shown in the middle and the antihyperon decaying
into a two-body hadronic state is shown to the left; to the mid-left in the
centre-of-mass of the reaction and to the very left in the rest system of the
decaying antihyperon.

The decay of a spin $1/2$ baryon is described by a parity conserving
($P$-wave) and a parity violating ($S$-wave) amplitude. A spin 3/2 hyperon
decay can be described in a similar way, but with the $S$-wave replaced by a
$D$-wave. The real part of their products is the aforementioned $\alpha_{Y}$
decay parameter from Eq. 1, while the imaginary part is denoted $\beta_{Y}$
[3]. A third parameter, $\gamma_{Y}$, is defined by the difference of the
squares of the parity conserving and parity violating amplitudes. These three
so-called Lee-Yang parameters fulfil
$\alpha_{Y}^{2}+\beta_{Y}^{2}+\gamma_{Y}^{2}=1$. Furthermore, a parameter
$\phi_{Y}$ can be defined according to

$\beta_{Y}=\sqrt{1-\alpha_{Y}^{2}}\sin\phi_{Y},~{}~{}~{}\gamma_{Y}=\sqrt{1-\alpha_{Y}^{2}}\cos\phi_{Y}.$
(5)

Hence, the decay is completely described by two independent parameters
$\alpha_{Y}$ and either $\beta_{Y}$, $\gamma_{Y}$ or $\phi_{Y}$. The parameter
$\alpha_{Y}$ can be understood more intuitively as the longitudinal
polarization induced by the decay, as illustrated in Figure 4. The
longitudinal polarization of the daughter will hence be $\alpha_{Y}$ plus the
transferred polarization from the mother. The latter depends on the decay
angle of the daughter. Similarly, $\beta_{Y}$ and $\gamma_{Y}$ quantify the
transversal polarization transfer.

Figure 4: The daughter hyperon $Y_{2}$ receives longitudinal
[$\alpha_{Y_{1}}+P_{Y_{1}}\cos\theta_{Y_{2}}$]$\hat{\bf{z}}$ and transversal
polarization components $\beta_{Y_{1}}\hat{\bf{x}}$ and
$\gamma_{Y_{1}}\hat{\bf{y}}$ from the mother hyperon $Y_{1}$.

The formalism of the full process $e^{+}e^{-}\to\gamma^{*}/\Psi\to
Y\overline{Y},Y\to BM,\overline{Y}\to\overline{BM}$ has been outlined in Refs.
[19, 33] and demonstrates how the form factors and the decay asymmetry
parameters $\alpha_{Y}$ and $\alpha_{\bar{Y}}$ can be accessed from the joint
angular distribution of the decay products $B,M,\overline{B}$ and
$\overline{M}$. By so-called single-tag analysis, where either the hyperon or
the antihyperon decay is measured while the undetected part is reconstructed
from missing kinematics, the quantity $\sin\Delta\Phi$ can be measured [33,
37]. Although the phase $\Delta\Phi$ can only be determined with a
$\Delta\Phi/180^{\circ}-\Delta\Phi$ ambiguity, it still provides useful
information about the structure [18].

From the decay asymmetry parameters, we can construct a CP test: Exact CP
symmetry implies equal decay patterns for particles and antiparticles, i.e.
$\alpha_{Y}=-\alpha_{\bar{Y}}$. Hence, the asymmetry

$A_{CP}=\frac{\alpha_{Y}+\alpha_{\bar{Y}}}{\alpha_{Y}-\alpha_{\bar{Y}}}$ (6)

quantifies the degree of CP violation. The asymmetry $A_{CP}$ is
straightforward to access for all weakly decaying ground-state hyperons.
Strange hyperons have the advantage that they decay with an appreciable
fraction into two-body hadronic states, hence it is feasible to collect large
exclusive $Y\overline{Y}$ data samples where the decays of the hyperon and the
antihyperon can be studied simultaneously.

The disadvantage of the $A_{CP}$ is that its sensitivity to CP violation is
somewhat limited. This is because hadronic decays involve strong, CP
conserving amplitudes as well as weak, possibly CP violating, amplitudes. For
spin 1/2 hyperon decays, the former are quantified by strong phase shifts
$\delta_{S}$ and $\delta_{P}$ in the final state $BM$ interaction. Here, $S$
and $P$ denote $S$-waves and $P$-waves, respectively. The strangeness
($|\Delta S|=1$) and isospin ($\Delta I=1/2$) changing transition in the $Y\to
BM$ decay induce weak, CP violating phases $\xi_{S}$ and $\xi_{P}$. Hence, the
phase difference $\xi_{P}-\xi_{S}$ arises from the interference between
$S$-waves and $P$-waves and if different from zero, it indicates CP violation.
This is similar to kaon decays, where CP violation is induced by an
interference between the isospin transition amplitudes $\Delta I=1/2$ and
$\Delta I=3/2$. Hyperon decays also involve both transitions but the $\Delta
I=1/2$ is believed to dominate [38]. Assuming this is the case, the weak, CP
violating phase $(\xi_{P}-\xi_{S})$ can be extracted from the measurable
asymmetry $A_{CP}$ by [39, 38, 40]

$A_{CP}\approx-\tan(\delta_{P}-\delta_{S})\tan(\xi_{P}-\xi_{s}).$ (7)

However, we see that if there is no $BM$ final state interaction, i.e. if the
difference $(\delta_{P}-\delta_{S})$ is close to zero, the $A_{CP}$ would
vanish even if $(\xi_{P}-\xi_{S})$ is non-zero. A separation of strong and
weak contributions can hence increase the sensitivity to CP violation. As we
will discuss next, this is possible through sequential hyperon decays.

####  4.2.2 Sequential Decays

Multi-strange and charmed hyperons decay sequentially, i.e. into other
unstable hyperons. These decay chains give access to the additional decay
parameters $\beta$ and $\phi$ [3]. A formalism for sequentially decaying
hyperon-antihyperon pairs from $e^{+}e^{-}$ annihilations has been outlined
for single-tag measurements in Ref. [4] and for exclusive measurement in Ref.
[35]. The latter includes the decay chains of spin 3/2 as well as spin 1/2
hyperons and demonstrates how helicity amplitudes and decay parameters can be
extracted from the joint angular distribution of the hyperons and their decay
products.

From the decay parameters $\beta$ and $\phi$, an additional CP test can be
constructed:

$\Delta\phi_{CP}=\frac{\phi_{Y}+\phi_{\bar{Y}}}{2}.$ (8)

This test has the advantage that it, at first order, only depends on the weak
phase difference whereas strong phases do not appear:

$\Delta\phi_{CP}\approx\frac{\left<\alpha_{Y}\right>(\xi_{P}-\xi_{S})_{LO}}{\sqrt{1-\left<\alpha_{Y}\right>^{2}}},$
(9)

where

$\left<\alpha_{Y}\right>=\frac{\alpha_{Y}-\alpha_{\bar{Y}}}{2}.$ (10)

This separation of the weak phase difference increases the sensitivity to CP
violation by at least an order of magnitude [38].

####  4.2.3 Projected Decay Distributions

In hyperon beam experiments, where a high-energy hyperon is produced in $pp$
or $pA$ interactions, one can integrate out most angles in the multi-step
hyperon decay chain. In other words, one obtains a projection of the angular
distribution on the final state baryon angle. This allows for the extraction
of the product $\alpha_{Y1}\alpha_{Y2}$ from the final state baryon in the
rest frame of the intermediate hyperon $Y_{2}$:

$\frac{dn}{d\cos\theta_{B}}=\frac{1}{2}(1+\alpha_{Y1}\alpha_{Y2}\cos\theta_{B}).$
(11)

The CP odd quantity $A_{CP}^{\rm\it eff}$ can then be defined

$A_{CP}^{\it
eff}=\frac{\alpha_{Y1}\alpha_{Y2}+\alpha_{\bar{Y1}}\alpha_{\bar{Y1}}}{\alpha_{Y1}\alpha_{Y2}-\alpha_{\bar{Y1}}\alpha_{\bar{Y2}}}\approx
A_{CP}^{Y1}+A_{CP}^{Y2}.$ (12)

The advantage of $A_{CP}^{\it eff}$ is its simplicity, but it is also less
sensitive to CP violation: It is possible that non-zero CP violations of
hyperon $Y_{1}$ and $Y_{2}$ have opposite signs and therefore cancel out
completely or partly.

###  4.3 Radiative Hyperon Decays

Radiative decays of hyperons, i.e. $Y_{1}\to Y_{2}\gamma$, are either purely
electromagnetic (if $Y_{1}$ and $Y_{2}$ have the same flavour and hence do not
involve transitions between quark generations) or weak (if $Y_{1}$ and $Y_{2}$
have different flavour and hence involve transitions such as $s\to u$).
Radiative decays into nucleons, i.e. $Y\to N\gamma$ are always weak.
Electromagnetic decays, e.g. $\Sigma^{0}\to\Lambda\gamma$, is expected to be
parity-conserving, given that i) the $\Sigma^{0}\Lambda\gamma$ coupling is
related to the $nn\gamma$ coupling by SU(3) flavour symmetry and ii) the upper
limit of the neutron EDM is found to be extremely small [41]. Hence, the decay
asymmetry $\alpha_{EM}$, defined by the real part of the product of the
electric and magnetic transition dipole moments, should be very close to zero.
The $\alpha_{EM}$ is experimentally accessible through the angle between the
photon from the $\Sigma^{0}\to\Lambda\gamma$ decay and the proton from the
subsequent $\Lambda\to p\pi^{-}$ decay. In Ref. [26], it was shown that the
asymmetries

$A^{EM}_{CP}=\alpha^{EM}_{Y}+\alpha^{EM}_{\bar{Y}}$ (13)

indicates strong CP violation, while

$A^{EM}_{C}=\alpha^{EM}_{Y}-\alpha^{EM}_{\bar{Y}}$ (14)

indicates C violation. Predictions from the SM of $A^{EM}_{CP}$ are of the
order $10^{-14}$, i.e. far below any foreseeable experimental resolution.
However, this means that even tiny signals would indicate physics beyond the
SM.

Weak radiative decays, involving transitions between quark generations $s\to
d\gamma$, can be parameterized in a similar way as the weak hadronic decays,
i.e. in terms of a decay parameter $\alpha_{\gamma}$. However, since the
photon pseudoscalar meson spin are different, the polarization transfer has a
different sign. This means that for unpolarized hyperons, the longitudinal
polarization of the daughter baryon is $-\alpha_{\gamma}$ in contrast to
$+\alpha_{Y}$ for a hadronic decay, recalling Fig. 4. Weak radiative decays
contain additional complexity due to the involvement of strong, possibly
parity violating processes. According to Hara’s theorem [42], parity violating
amplitudes are predicted to vanish in the limit of SU(3) flavour symmetry,
under the assumption that the weak interaction is CP conserving and only
includes left-handed currents. This implies that the weak radiative decay
asymmetry $\alpha_{\gamma}$ would be zero. This is in contrast to SU(3)
breaking where we instead have

$\alpha_{\gamma}=\frac{m_{s}^{2}-m_{d}^{2}}{m_{s}^{2}+m_{d}^{2}},$ (15)

$m_{s}$ and $m_{d}$ being the mass of the strange and down quark,
respectively. Constituent quark masses ($m_{s}\sim 500$ MeV and $m_{d}\sim
330$ MeV) yields $\alpha_{\gamma}$ around +0.4 and +0.5. This is similar to
the $b\to s\gamma$ transition, where one replaces $m_{s}$ by $m_{b}$. Since
$m_{b}>>m_{d}$, the asymmetry parameter will then be close to +1. The $b\to
s\gamma$ transition has put constraints on physics beyond the SM that remove
the chirality suppression. In a similar way, it is possible to investigate
beyond SM effects in $s\to d\gamma$.

###  4.4 Semi-leptonic Hyperon Decays

Semi-leptonic decays of SU(3) octet hyperons can be parameterized in terms of
four weak form factors, which in the limit of zero momentum transfer reduces
to three: the vector $g_{V}(q^{2})$, the axial vector $g_{A}(q^{2})$ and the
induced tensor $f_{2}(q^{2})$ where $q^{2}$ is the momentum transfer squared
from the mother to the daughter hyperon [43, 44]. Since the momentum transfer
in semi-leptonic hyperon decays is small compared to the baryon masses
($q^{2}\to 0$), a systematic expansion can be performed in terms of the
symmetry breaking parameter $\delta$, defined by the mass difference of the
mother and daughter hyperon [45]: $\delta=(M_{Y1}-M_{Y2})/M_{Y1}$. This
simplifies significantly the dependence of the decay rate on the form factors.
The ratio between the vector and axial vector form factors can be written

$g_{AV}=\left|\frac{g_{A}(0)}{g_{V}(0)}\right|\cdot e^{i\phi_{AV}}.$ (16)

The phase $\phi_{AV}$ quantifies the presence of a triple correlation term in
the transition probability and a non-zero value would imply violation of time-
reversal invariance [1]. The phase is accessible for polarized mother and/or
daughter hyperons from the joint angular distribution of the decay [46]. The
form factors from different channels are connected through the approximate
SU(3) flavour symmetry.

The semi-leptonic hyperon decays from hyperons produced in $e^{+}e^{-}$
annihilations can be analysed analogous to hadronic hyperon decays (see Fig.
3) where the scalar particle is replaced by the off-shell gauge boson $W^{-}$
decaying into an electron-neutrino pair. The study of joint angular
distributions typically require large data samples ($N>1000$). Since semi-
leptonic hyperon decays are typically rare compared to hadronic ones, it is
challenging to collect the samples necessary for polarization studies.
However, also for the modest samples it is possible to gain valuable insights,
for example by constructing CP tests from the semi-leptonic decay rates of
hyperons and antihyperons:

$A_{CP}^{\rm\it SLW}=\frac{\Gamma(Y_{1}\to
Y_{2}l^{-}\bar{\nu}_{l})}{\Gamma(\bar{Y}_{1}\to\bar{Y}_{2}l^{+}\nu_{l})}$ (17)

or

$A_{CP}^{\rm\it SLW}=\frac{BR(Y_{1}\to
Y_{2}l^{-}\bar{\nu}_{l})-BR(\bar{Y}_{1}\to\bar{Y}_{2}l^{+}\nu_{l})}{BR(Y_{1}\to
Y_{2}l^{-}\bar{\nu}_{l})+BR(\bar{Y}_{1}\to\bar{Y}_{2}l^{+}\nu_{l})}.$ (18)

Furthermore, it is straight-forward to construct a test on lepton universality
from the ratio

$R_{\mu e}=\frac{\Gamma(Y_{1}\to Y_{2}\mu^{-}\bar{\nu}_{\mu})}{\Gamma(Y_{1}\to
Y_{2}e^{-}\bar{\nu}_{e})}$ (19)

of different lepton decay channels.

##  5 Experimental status

###  5.1 Hyperon Structure

Electromagnetic properties of hadrons have been studied in electron-hadron
scattering ever since the 1950s [47] which has led to important insights on
the structure and size of the proton [12, 13, 48, 49] and to some extent also
the neutron [9, 8]. Due to experimental challenges, corresponding progress for
hyperons is yet to come: so far, the only existing charge radius measurement
is for the $\Sigma^{-}$ hyperon [10]. However, the advent of modern, high-
intensity $e^{+}e^{-}$ colliders, has ignited a veritable revolution for
hyperon structure measurements.

####  5.1.1 Effective form factor

The BaBar experiment carried out a pioneering measurement of the $\Lambda$ and
$\Sigma^{0}$ form factors and the $\Sigma^{0}\Lambda$ transition form factor
by exploiting Initial State Radiation (ISR) on a 230 fb-1 sample collected at
the bottomium resonance $\Upsilon(4S)$ [50]. Though the resulting hyperon-
antihyperon samples were small, they were sufficient to extract the energy
dependence of the cross section and the corresponding effective form factor.
The CLEO-c experiment studied $\Lambda$, $\Sigma$, $\Xi$ and $\Omega$
production in the $e^{+}e^{-}\to Y\bar{Y}$ reaction [51, 52] at energies
corresponding to masses of the charmonium resonances $\Psi(3686)$,
$\Psi(3770)$ and $\Psi(4170)$. The results indicated the importance of diquark
correlations [53]. However, the interpretation in terms of electromagnetic
form factors rely on the assumption that the $Y\bar{Y}$ pair production is
dominated by one-photon exchange ($e^{+}e^{-}\to\gamma^{*}\to Y\bar{Y}$) with
a negligible contribution from charmonium decay ($e^{+}e^{-}\to\Psi\to
Y\bar{Y}$). This is in contrast to a recent measurement from the BESIII
collaboration [54], that shows that the branching fraction of the decay
$\Psi(3770)\to\Lambda\bar{\Lambda}$ is at least an order of magnitude larger
than the prediction used in the CLEO-c papers [51, 52]. Furthermore, the
BESIII results indicate that also other charmonium resonances have a
significant impact on the $Y\bar{Y}$ production. This, however, implies that
hyperon-antihyperon production in the vicinity of vector charmonium resonances
can be utilized for getting deeper insights into the properties of these
resonances [55].

Dedicated energy scan campaigns by BESIII have enabled increased precision in
effective form factor measurements for the full spin $1/2$ octet. A low-energy
scan near the $\Lambda\bar{\Lambda}$ threshold revealed an unexpected
enhancement, resembling a cusp or a peak [56]. Vector meson dominance, in
particular with strong influence by the $\phi(2170)$, has been suggested as an
explanation [57, 58]. Pioneering measurements have also been achieved for
$\Sigma^{\pm}$ [59], $\Sigma^{0}$ [60], $\Xi^{-}$ [61, 62] and $\Xi^{0}$ [63].
The production cross sections of these hyperons in the $e^{+}e^{-}\to
Y\bar{Y}$ process from all BESIII measurements, are summarized in the left
panel of Figure 5. These results allow for a systematic analysis of the
$G_{\rm eff}$ behaviour, similarly to what has recently been done for protons
[64, 65, 66] and neutrons [9]. It has been found [67, 68] that for nucleons,
the effective form factor can be parameterised according to

$G_{\rm\it eff}(q^{2})=G_{D}(q^{2})+G_{\rm osc}(q^{2}),$ (20)

where $G_{D}(q^{2})$ is a dipole function and $G_{\rm osc}(q^{2})$ describes
oscillations that are periodic in either $q^{2}$ or the relative momentum
between the nucleon and antinucleon, $p$. The oscillations of the proton and
neutron form factors have different amplitude and phase, but the same period
[9] and have been attributed to nucleon-antinucleon rescattering into an
intermediate hadron-antihadron state. In a recent paper [69], the $G_{\rm\it
eff}$ was analysed for neutral octet baryons $n,\Lambda,\Sigma^{0}$ and
$\Xi^{0}$ using all available data. The study indicated a universal behaviour
where $G_{\rm osc}(q^{2})$ can be parameterized with a common phase and
amplitude, as indicated with a red dashed line in the right panel of Figure5.
However, their conclusion is not supported by new low-$q^{2}$ neutron form
factor from the SND experiment [70]. Fitting the oscillation model to both
BESIII and SND neutron data results in a significantly lower oscillation
frequency than that of the proton. The question remains if the observed
oscillations in baryon form factors follow a pattern that can be explained
within e.g. SU(3) symmetry and hence have a common origin, or is rather
accidental.

The structure of the charm baryon $\Lambda_{c}^{+}$ has been studied by the
Belle experiment using the ISR method [71] and by BESIII in an energy scan
[72]. The Belle data revealed a peaking cross section near threshold, in
contrast to BESIII where a plateau was seen. However, it should be pointed out
that the differences between Belle and BESIII are within the experimental
uncertainties. Attempts have been made to understand the threshold behaviour
of the cross section in terms of vector charmonium resonances [73, 74] and
final state interactions [75].

The cross section of $\Omega^{-}$ production in
$e^{+}e^{-}\to\Omega^{-}\bar{\Omega}^{+}$ has been studied in a recent
analysis of energy scan data from BESIII [76]. Having spin 3/2, the
$\Omega^{-}$ effective form factor is interpreted as a linear combination of
the electric, magnetic, quadrupole and octupole form factors [77]. The energy
dependence is, within the uncertainties, consistent with the prediction from
perturbative QCD and from Ref. [77].

Figure 5: Left: The $e^{+}e^{-}\to Y\bar{Y}$ cross sections, measured by the
BESIII experiment for different ground-state hyperons, as a function of the
velocity $\beta$. Right: The effective form factor, after subtracting the
dipole term, for neutral baryons studied by BaBar, BESIII and SND experiments,
as a function of the square root of the $e^{+}e^{-}$ CMS energy. The solid
curve represents the fit to BESIII neutron data and the dashed a ft to BESIII
and SND data.

####  5.1.2 Separation of Electric and Magnetic Form Factors

The energy scan campaigns carried out by BESIII included particularly large
data samples at 2.396 GeV, 2.64 GeV, 2.9 GeV, 4.57 GeV and 4.59 GeV. Using
either double-tag ($e^{+}e^{-}\to Y\bar{Y}$) or single-tag ($e^{+}e^{-}\to
Y\bar{X}$ or $e^{+}e^{-}\to\bar{Y}X$, where $X$ refers to an unmeasured
hyperon) selection, sufficiently many events were found to reconstruct the
hyperon scattering angle distribution. This enabled a separation of the
electric and the magnetic form factors for $\Lambda$ [78], $\Sigma^{+}$ [59]
and $\Lambda_{c}^{+}$ [72], as summarized in Figure 6. For $\Lambda$ and
$\Lambda_{c}^{+}$, the $G_{E}/G_{M}$ ratio was found to be close to unity near
threshold, whereas for the $\Sigma^{+}$, it was found to be significantly
larger near threshold while approaching unity at larger energies.

The large sample at 2.396 GeV, in combination with the relatively large
$e^{+}e^{-}\to\Lambda\bar{\Lambda}$ cross section at this energy, resulted in
more than 500 exclusively reconstructed $\Lambda\to
p\pi^{-},\bar{\Lambda}\to\bar{p}\pi^{+}$ events. This gave access to the joint
angular distribution of the production and decays from which the relative
phase $\Delta\Phi$ between $G_{E}$ amd $G_{M}$ could be extracted. Hence, the
time-like form factors could be unambiguously determined for the first time
for any baryon – a complete snapshot from its time evolution [78]. This
measurement gave rise to theory activity world-wide [79, 80, 81, 82, 83, 84,
85, 69, 55]. In particular, a new method was outlined which allows for the
extraction of the charge radius squared $\left<r^{2}\right>$ from polarization
measurements [18]. Combining the $R$ measurement from BESIII and BaBar with
the $\Delta\Phi$ measurement from BESIII, a first calculation of
$\left<r^{2}\right>$ was carried out. However, since $\Delta\Phi$ is only
measured at one energy, the result turned out to be ambiguous and six
different solutions were presented. Phase measurements in a broader energy
range will be helpful in pinpointing the charge radius squared for the first
time for a neutral hyperon.

Figure 6:  The $|G_{E}/G_{M}|$ of $\Lambda$, $\Sigma^{+}$ and
$\Lambda_{c}^{+}$ hyperons as a function of the velocity $\beta$. The filled
dots represent data from BESIII and the open dots BaBar.

####  5.1.3 Hyperon Production at Vector Charmonia

The large data samples collected by BESIII at charmonium resonances such as
$J/\Psi$, $\Psi(3686)$ and $\Psi(3770)$ allow for a very precise determination
of the psionic form factor phase $\Delta\Phi_{\Psi}$ and the angular
distribution parameter $\alpha_{\Psi}$, as shown in Table 1. It is found that
$\Delta\Phi_{\Psi}$ varies substantially between resonances and hyperon types.
Summarising the results, we note that

  * •

The $\Sigma^{+}$ phase is negative at the $J/\Psi$ resonance [86] while
positive for $\Lambda$ [87] and $\Xi^{-}$ [88].

  * •

The $\Sigma^{+}$ phase changes sign when going from the $J/\Psi$ to the
$\Psi(3686)$ [86].

  * •

The phase of $\Xi^{-}$ is significantly larger at $J/\Psi$ [88] compared to at
$\Psi(3686)$ [89].

  * •

The angular distribution parameter $\alpha_{\Psi}$ of the
$\Psi(3686)\to\Sigma\bar{\Sigma}$ decay is significantly larger for
$\Sigma^{-}$ compared to $\Sigma^{+}$ and $\Sigma^{0}$ [90].

The interpretation of psionic form factors is still a matter of discussion,
but the possibility to perform precision measurements calls for a theory
framework in which the relation between psionic form factors and charmonium
decay is outlined. Hopefully, this could provide a helpful tool to understand
the properties of vector charmonia and their coupling to baryon-antibaryon
states.

Table 1: Properties of the $e^{+}e^{-}\to J/\psi\to B\overline{B}$ decays to the pairs of ground-state octet hyperons.  Final state | ${\cal B}(\times 10^{-4})$ | $\alpha_{\psi}$ |  $\Delta\Phi$ (rad) | Ref.
---|---|---|---|---
$J/\psi\to\Lambda\overline{\Lambda}$ | $19.43\pm 0.03\pm 0.33$ | $\phantom{-}0.4748\pm 0.0022\pm 0.0031$ | $\phantom{-}0.7521\pm 0.0042\pm 0.0066$ | [91, 92]
$\psi(3686)\to\Lambda\overline{\Lambda}$ | $3.97\pm 0.02\pm 0.12$ | $\phantom{-}0.82\pm 0.08\pm 0.02$ |  – | [91]
$\psi(3770)\to\Lambda\overline{\Lambda}$ |  $>0.024$ (90$\%$ CL) | $\phantom{-}0.85^{+0.12}_{-0.20}\pm 0.02$ | $\phantom{-}1.24^{+1.15}_{-0.80}\pm 0.09$ | [54, 93]
$J/\psi\to\Sigma^{+}\overline{\Sigma}\vphantom{X}^{-}$ | $10.61\pm 0.04\pm 0.36$ | $-0.508\pm 0.006\pm 0.004$ | $-0.270\pm 0.012\pm 0.009$ | [94, 86]
$\psi(3686)\to\Sigma^{+}\overline{\Sigma}\vphantom{X}^{-}$ | $2.52\pm 0.04\pm 0.09$ | $\phantom{-}0.682\pm 0.030\pm 0.011$ | $\phantom{-}0.379\pm 0.070\pm 0.014$ | [94, 86]
$\psi(3686)\to\Lambda\overline{\Sigma}\vphantom{X}^{0}$ | $0.0160\pm 0.0031\pm 0.0059$ | – | – | [95]
$J/\psi\to\Sigma^{-}\overline{\Sigma}\vphantom{X}^{+}$ | – | – | – | –
$\psi(3686)\to\Sigma^{-}\overline{\Sigma}\vphantom{X}^{+}$ | $2.82\pm 0.04\pm 0.08$ | $\phantom{-}0.96\pm 0.09\pm 0.03$ | – | [90]
$J/\psi\to\Sigma^{0}\overline{\Sigma}^{0}$ | $11.64\pm 0.04\pm 0.23$ | $-0.449\pm 0.020\pm 0.008$ | – | [91]
$\psi(3686)\to\Sigma^{0}\overline{\Sigma}^{0}$ | $2.44\pm 0.03\pm 0.11$ | $\phantom{-}0.71\pm 0.11\pm 0.04$ | – | [91]
$J/\psi\to\Xi^{0}\overline{\Xi}\vphantom{X}^{0}$ | $11.65\pm 0.04\pm 0.43$ | $\phantom{-}0.66\pm 0.03\pm 0.05$ | – | [96]
$\psi(3686)\to\Xi^{0}\overline{\Xi}\vphantom{X}^{0}$ | $2.73\pm 0.03\pm 0.13$ | $\phantom{-}0.65\pm 0.09\pm 0.14$ | – | [96]
$J/\psi\to\Xi^{-}\overline{\Xi}\vphantom{X}^{+}$ | $10.40\pm 0.06\pm 0.74$ | $\phantom{-}0.586\pm 0.012\pm 0.010$ | $\phantom{-}1.213\pm 0.046\pm 0.016$ | [97, 88]
$\psi(3686)\to\Xi^{-}\overline{\Xi}\vphantom{X}^{+}$ | $2.78\pm 0.05\pm 0.14$ | $\phantom{-}0.693\pm 0.048\pm 0.049$ | $\phantom{-}0.667\pm 0.111\pm 0.058$ | [97, 89]

###  5.2 Hadronic Hyperon Decays

####  5.2.1 Single-strange Hyperon Decays

Searches for CP violation in the hadronic $\Lambda\to p\pi^{-}$ and
$\bar{\Lambda}\to\bar{p}\pi^{+}$ have been carried out since the mid-1980s.
The first was performed at the CERN Intersecting Storage Rings in the
reactions $pp\to\Lambda X$ and $\bar{p}p\to\bar{\Lambda}X$ [98]. From the
measured $\alpha_{\Lambda}P_{\Lambda}$ and
$\alpha_{\bar{\Lambda}}P_{\bar{\Lambda}}$, where $P$ is the polarization, the
quantity $\alpha_{\Lambda}P_{\Lambda}/\alpha_{\bar{\Lambda}}P_{\bar{\Lambda}}$
was defined. Assuming $P_{\Lambda}=P_{\bar{\Lambda}}$, this quantity should be
1 when CP is conserved. However, it is straight-forward to construct a CP test
similar to Eq. 6

$A_{CP}^{\alpha
P}=\frac{\alpha_{\Lambda}P_{\Lambda}+\alpha_{\bar{\Lambda}}P_{\bar{\Lambda}}}{\alpha_{\Lambda}P_{\Lambda}-\alpha_{\bar{\Lambda}}P_{\bar{\Lambda}}}\approx\frac{\alpha_{\Lambda}+\alpha_{\bar{\Lambda}}}{\alpha_{\Lambda}-\alpha_{\bar{\Lambda}}}.$
(21)

The approximately $10^{4}$ events yielded the result $A_{CP}=0.02\pm 0.14$.
The DM2 collaboration measured the $A_{CP}$ using the
$J/\Psi\to\Lambda\bar{\Lambda}$ process, however overlooking the possibility
that the $\Lambda$ may be polarized. As a result, terms in the joint angular
distribution that are sensitive to $\alpha_{\Lambda}/\alpha_{\bar{\Lambda}}$
differences cancel when being integrated over [99]. The PS185 collaboration at
LEAR, CERN, provided the first essentially model-independent CP test by using
the $\bar{p}p\to\bar{\Lambda}\Lambda$ reaction and studying the quantity
defined in Eq. 21 as a function of the $\bar{\Lambda}$ scattering angle. This,
in combination with the large data sample, $96\cdot 10^{3}$ exclusively
reconstructed $\Lambda\bar{\Lambda}$ events, resulted in a significantly
improved precision [100] that remained unchallenged for more than 20 years.

The large $J/\Psi$ samples from the BESIII collaboration – $1.3\cdot 10^{9}$
$J/\Psi$ during 2009 and 2012 and $10^{10}J/\Psi$ during 2018 and 2019 –
started a new era in the search for CP violation in baryon decays. First, the
2009 and 2012 data resulted in $\approx 42\cdot 10^{4}$ exclusively
reconstructed $\Lambda\bar{\Lambda}\to p\pi^{-}\bar{p}\pi^{+}$ events [87] and
$\approx 88\cdot 10^{3}$ $\Sigma^{+}\bar{\Sigma}^{-}\to
p\pi^{0}\bar{p}\pi^{0}$ events [86]. This enabled the most precise $\Lambda$
CP test so far and the first CP test for $\Sigma^{+}$. In addition, the
asymmetry parameter of the $\bar{\Lambda}\to\bar{n}\pi^{0}$ was measured for
the first time [87]. A remarkable finding in the $\Lambda\to p\pi^{-}$ decay
study was that the asymmetry parameter was found to be 17% larger than the
previous world average, mainly based on proton polarimeter experiments in the
1970s. A re-analysis of CLAS data [101] revealed a value more consistent with
that obtained by BESIII than the old world average. The Particle Data Group
[1] has now abandoned the old measurements in favour of a world-average based
on Refs. [87] and [101]. A recent study of the
$J/\Psi\to\Xi^{-}\bar{\Xi}^{+}\to\Lambda\pi^{-}\bar{\Lambda}\pi^{+}\to
p\pi^{-}\pi^{-}\bar{p}\pi^{+}\pi^{+}$ provided an independent measurement of
$\alpha_{\Lambda}$, $\alpha_{\bar{\Lambda}}$ and $A_{CP}$, consistent with
previous results from BESIII. The recent $\alpha_{\Lambda}$ measurements, as
well as the previous PDG world average from older experiments, are summarised
in Figure 7.

The precision of the $\alpha_{\Lambda}$ measurement from $\approx 73\cdot
10^{3}$ $\Xi^{-}\bar{\Xi}^{+}$ [88] decays is about the same as the one
achieved from the almost one order of magnitude larger
$J/\Psi\to\Lambda\bar{\Lambda}$ sample [87]. This is because $\Lambda$
hyperons from $\Xi^{-}$ decays have a larger degree of polarization than
$\Lambda$ hyperons from $J/\Psi$ decays. Furthermore, $\Lambda$ hyperons from
a sequential decay can be polarized in several directions while those from a
decaying $J/\Psi$ can only be polarized along the production plane.

Even more recently, the full $10^{10}J/\Psi$ sample, including $3\cdot 10^{6}$
exclusively reconstructed $\Lambda\bar{\Lambda}$ decays, has been analyzed for
the purpose of achieving a high-precision CP test. The $\Lambda\to p\pi^{-}$
decay asymmetry was found to be consistent with the previous measurement and
the precision of $A_{CP}$ is the most precise so far: $0.0025\pm 0.0046\pm
0.0012$.

Figure 7:  Asymmetry parameter values $\alpha_{\Lambda}$ from the PDG from
2018 [102] and from new measurements by BESIII [87, 88, 92], CLAS [101] and
LHCb [103]. The PDG-2018 value is based on polarimeter measurements from the
1960s and 1970s. In the new BESIII measurements, the average value
$\left<\alpha_{\Lambda}\right>=(\alpha_{\Lambda}-\overline{\alpha}_{\Lambda})/2$
is displayed. The * and ** denote the 2009-2012 and 2009-2019 $J/\psi$
samples, respectively.

####  5.2.2 Double-strange Hyperon Decays

Sequential decays of multi-strange and charmed hyperons give, in addition to
the decay asymmetry $\alpha_{Y}$, also access to $\beta$ and $\gamma$ or
$\phi$ (see section 4.2.2). The studies of these parameters are not limited to
exclusive, double-tag $Y\bar{Y}$ measurements but can also be studied when it
is only possible to reconstruct either the hyperon decay chain or the
antihyperon ditto, i.e. using the single-tag approach.

The world data bank [1] of the $\Xi^{0}$ and $\Xi^{-}$ hyperons is dominated
by experiments of polarized or unpolarized secondary hyperon beams. Many of
these apply the formalism in Eq. 11. The most precise measurement of the
$\alpha_{\Xi^{0}}\alpha_{\Lambda}$ was achieved by the NA48 experiment at CERN
[104]. The $\phi_{\Xi 0}$ parameter was measured with large uncertainties at
various experiments [105, 106, 107].

The E756 collaboration at Fermilab achieved the best precision so far of the
$\alpha_{\Xi^{-}}\alpha_{\Lambda}$ product and also measured the
$A_{CP}^{\rm\it eff}$ defined in Eq. 12 for the first time [108]. Thanks to a
dedicated measurement, the HyperCP collaboration improved the precision of
$A_{CP}^{\rm\it eff}$ and their measurement is the most precise so far:
$(0.0\pm 5.1\pm 4.4)\times 10^{-4}$ [109].

By studying the joint angular distribution of polarized $\Xi^{-}$ hyperons,
the E756 [110] and HyperCP [111] collaborations also measured the $\phi_{\Xi}$
parameter, from which the strong phase difference ($\delta_{P}-\delta_{S}$)
was extracted. These two measurements implied a significant improvement of the
world data bank [112, 113, 105, 114, 115, 106, 107]. Indeed, the
($\delta_{P}-\delta_{S}$) was found to be very small, showing that even a CP
violating weak phase would be difficult to detect from $A_{CP}$ using Eq. 7.

A recent measurement from BESIII demonstrates the merits of polarized,
entangled and sequentially decaying hyperon-antihyperon pairs [92]. The full
process $e^{+}e^{-}\to\Xi^{-}\bar{\Xi}^{+},\Xi^{-}\to\Lambda\pi^{-},\Lambda\to
p\pi^{-}+c.c.$ was studied using $\approx 73\cdot 10^{3}$ exclusively
reconstructed, or double-tagged, $\Xi^{-}\bar{\Xi}^{+}$ pairs identified from
a sample of $1.3\cdot 10^{9}$ $J/\Psi$. The asymmetries $A_{CP}^{\Xi}$ and
$A_{CP}^{\Lambda}$ were determined separately. From $\Delta\phi_{\Xi}$, the
weak phase difference $\xi_{P}-\xi_{S}$ has been determined for the first time
in a direct measurement. The result, $(\xi_{P}-\xi_{S})=(1.2\pm 3.4\pm
0.8)\cdot 10^{-2}$ rad, is consistent with CP conservation. It is also
noteworthy that the strong phase difference $\delta_{P}-\delta_{S}$ was
measured with a similar precision as the one achieved by HyperCP [111] despite
the 2000 times larger data sample from HyperCP. The excellent sensitivity
achieved by the BESIII method can be attributed to the fact that the clean,
well-known production process and the subsequent decays can be treated in a
model-independent way: all angles are measured and no information is
integrated out from the differential cross sections. Hence, all quantum
correlations between the hyperon and the antihyperons, and their decay
products, are taken into account, as well as the polarization transfer in
multiple directions from the mother $\Xi^{-}$ to the daughter $\Lambda$
hyperon. Extending this analysis to the eight times larger data sample
collected in 2018 and 2019 will further improve the precision and enable
comparison with theory calculations of the strong phase from e.g. [116].

####  5.2.3 Triple-strange Hyperon Decays

The triple-strange $\Omega^{-}$ hyperon should, according to the quark model,
have spin 3/2. However, this was not empirically well-established until
recently; the first measurements from SPS at CERN [117, 118] were based on
very small data samples and therefore very simplified angular distributions.
In a later measurement by BaBar, $\Omega^{-}$ hyperons were studied in
$\Omega_{c}$ and $\Xi_{c}$ decays and the conclusions relied on assumptions of
the helicity of the mother baryons [119]. Recently, the BESIII collaboration
applied the model-independent formalism outlined in Ref. [35] on single-tagged
$\Omega^{-}\to\Lambda K^{-},\Lambda\to p\pi^{-}$ and
$\bar{\Omega}^{+}\to\bar{\Lambda}K^{+},\bar{\Lambda}\to\bar{p}\pi^{+}$ from
the $e^{+}e^{-}\to\Psi(3686)\to\Omega^{-}\Omega^{+}$ reaction. The sample
contained $\approx 450\cdot 10^{6}$ $\Psi(3686)$ events, including 2507
$\Omega^{-}$ and 2238 $\bar{\Omega}^{+}$ candidates. Two hypotheses were
tested: $\Omega^{-}$ having either spin 1/2 or 3/2. The spin 3/2 turned out to
be favoured over the spin 1/2 hypothesis by 14$\sigma$ [120], which is the
most precise, model-independent spin test for the $\Omega^{-}$ hyperon so far.
This validates not only the quark model but also the results based upon its
validity. In addition, the decay asymmetry parameter $\phi_{\Omega}$ was
measured for the first time.

In the past, the decay asymmetries of the decays $\Omega^{-}\to\Lambda K^{-}$,
$\Omega^{-}\to\Xi^{0}\pi^{-}$ and $\Omega^{-}\to\Xi^{-}\pi^{0}$ have been
studied with a secondary hyperon beam at CERN [121]. More recently, the
HyperCP experiment measured the decay asymmetry of $\Omega^{-}\to\Lambda
K^{-}$ and $\bar{\Omega}^{-}\to\bar{\Lambda}K^{+}$ and the CP asymmetry
$A_{CP}$. The HyperCP employed a secondary $\Omega$ beam produced in $pCu$
collisions and since all $\Omega$ hyperons were selected to be collinear with
the proton beam, they were unpolarized. This enabled the measurement of the
$\alpha_{\Omega}\alpha_{\Lambda}$ and
$\alpha_{\bar{\Omega}}\alpha_{\bar{\Lambda}}$ products. By combining these
quantities with $\alpha_{\Lambda}$ and $\alpha_{\bar{\Lambda}}$ from other
measurements, it was concluded that the $\Omega^{-}\to\Lambda K^{-}$ decay
conserve parity almost completely. Their measured values of $A_{CP}$ as
defined by Eq. 6 and the effective asymmetry $A_{CP}^{\rm\it eff}$ by Eq. 12
were both found to be consistent with CP conservation.

####  5.2.4 Charm Hyperon Decays

Charm hyperons can decay into many different final states and as a
consequence, the branching fraction of each channel is small. Therefore, charm
hyperon decays are almost exclusively studied using the single-tag approach.

Most measurements of the $\Lambda_{c}^{+}$ decay parameters, by e.g. ARGUS
[122], CLEO [123, 124] and FOCUS [125] collaborations only consider the
angular distribution of the final state hadron, i.e. not the joint decay of
the multi-step process. This means that the decay asymmetry
$\alpha_{\Lambda_{c}}$ can be retrieved, but not $\beta$ and $\gamma/\phi$. In
addition, the results rely on a precise knowledge of the decay parameter of
the hyperon in the second step, for example $\Lambda$ or $\Sigma^{+}$.
Nevertheless, the FOCUS collaboration performed the first CP test for a
charmed baryon, by measuring $A_{CP}$ for the
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$ decay to be $0.07\pm 0.19\pm 0.24$.

In a recent BESIII measurement, based on 0.5 fb-1 at a $e^{+}e^{-}$ CMS energy
of 4.6 GeV, we presented a proof-of-concept for measuring the decay parameter
$\alpha_{\Lambda_{c}}$, and for the first time also $\beta_{\Lambda_{c}}$ and
$\gamma_{\Lambda_{c}}$, of the $\Lambda_{c}^{+}$ decay into $\Lambda\pi^{+}$,
$\Sigma^{+}\pi^{0}$ and $\Sigma^{0}\pi^{+}$ [126], all containing hyperons
that decay subsequently. In addition, $\alpha_{\Lambda_{c}}$ of the
$\Lambda_{c}^{+}\to pK_{S}$ was measured for the first time, though with large
uncertainties. Due to limited sample size, it was assumed that
$\alpha_{\Lambda_{c}}=-\alpha_{\bar{\Lambda_{c}}}$ i.e. CP symmetry. The
BESIII collaboration has also carried out the first model-independent
experimental determination of the $\Lambda_{c}^{+}$ spin. The analysis
combined several sequential two-body decay chains and found that the spin 1/2
hypothesis was favoured over the spin 3/2 hypothesis by more than 6$\sigma$
[127].

The most precise measurements of $\alpha_{\Lambda_{c}}$ for several two-body
decays have been achieved by the Belle experiment in the
$e^{+}e^{-}\to\Lambda_{c}^{+}X+c.c.$ reaction [128]. The CP asymmetry $A_{CP}$
has been calculated and though the statistical precision is comparable to that
of strange hyperon decays, the total uncertainty is dominated by the
systematics. The Belle [129, 130] and CLEO [131] experiments have also
measured the $\alpha_{\Xi_{c}}$ of several $\Xi_{c}^{0}$ decays.

###  5.3 Radiative Hyperon Decays

From Hara’s theorem [27] and from the breaking of SU(3) symmetry in the quark
model, on expects a positive value of the weak radiative decay parameter
$\alpha_{\gamma}$ for decays involving $s\to d$ transitions. One example of
such a decay is $\Sigma^{+}\to p\gamma$. Therefore, it came as a surprise when
several experiments reported a large and negative value [132, 133, 134, 135].
Other experiments reported similar values for the $\Xi^{0}\to\Lambda\gamma$
and $\Xi^{0}\to\Sigma^{0}\gamma$ decay [136, 137, 104], in disagreement with
older measurements [138, 139]. The discrepancy between theory predictions and
experimental results sparked wide interests and various solutions to the
puzzle were proposed [140, 141, 142, 143]. It has been suggested that the
$\Lambda\to n\gamma$ can provide further insights [144, 145]: if positive, it
indicates a violation of Hara’s theorem.

Polarized and entangled hyperon-antihyperon pairs from vector charmonium
decays have the advantage that the polarization of the hyperon (antihyperon)
can be measured with high precision if one considers the "leg" of the reaction
where a hyperon (antihyperon) decays into a two-body hadronic final state. A
simultaneous measurement of the other radiative weak decay "leg" of the
antihyperon (hyperon) will then also be precise, thanks to constrained
production parameters such as the polarization and spin correlations. This has
been demonstrated in a recent study by the world-record data sample of
$10^{10}$ $J/\Psi$ decays from BESIII. These data contain an appreciable
fraction of $J/\Psi\to\Lambda\bar{\Lambda},\Lambda\to
n\gamma,\bar{\Lambda}\to\bar{p}\pi^{+}+c.c.$ events that can be exclusively
reconstructed. The large transverse polarization of $\Lambda$ [87] enabled the
first determination of the decay parameter $\alpha_{\gamma}$ of $\Lambda\to
n\gamma$. It was found to be $-0.16\pm 0.10_{\rm stat.}\pm 0.05_{\rm syst.}$
[146], indicating that Hara’s theorem still holds for hyperon radiative
decays. The results provide essential input to constrain the amplitudes of
parity violation contribution in theoretical calculations, and can be used to
validate the predictions and constrain the parameters of the various
theoretical models [147, 148, 149]. Recent calculations based on Chiral
Perturbation Theory are in good agreement with the BESIII data [149].

###  5.4 Semi-leptonic Hyperon Decays

The small branching ratios and the immense difficulty in detecting neutrinos
make the study of semi-leptonic hyperon decays a challenging task. On the one
hand, a semi-leptonic decay can be identified also without the neutrino thanks
to the special characteristic of a three-body decay compared to the more
abundant two-body hadronic decays. On the other hand, the angular analyses
from which the physics is revealed, require all other particles in the decay
to be accurately reconstructed. Nevertheless, experiments with polarized beams
made pioneering measurements of $\Lambda\to pe^{-}\bar{\nu}_{e}$ in the 1970s
[150, 151, 152, 153, 154, 155, 156, 157], though with relatively modest sample
sizes. Later measurements offered larger samples [158, 159, 160] and enabled
measurements of the axial vector form factor ratio $g_{A}/g_{V}$ for $\Lambda$
[1]. More recently, the KTeV [161, 162, 163] and NA48 [164, 165]
collaborations have studied the decay
$\Xi^{0}\to\Sigma^{+}e^{-}\bar{\nu}_{e}$, providing the form factor ratios
$g_{1}(0)/f_{1}(0)$ and $f_{2}(0)/f_{1}(0)$. In addition, the KTeV measured
$g_{2}(0)/f_{1}(0)$.

Semi-leptonic decays provide an independent way of measuring the CKM matrix
element $|V_{us}|$, in addition to kaon and $\tau$ decays. The current world-
average of $0.2243\pm 0.0008$ is essentially calculated from $K\to\pi
l\nu(\gamma)$ and $K\to\mu\nu(\gamma)$ decays [1]. Combining all available
data on semi-leptonic hyperon decays provide comparable precision as for kaon
decays and can thereby further improve the precision of the average. The fit
of previous measurements of semi-leptonic hyperon decays yields
$|V_{us}|=0.2250\pm 0.0027$ [43]. Further theoretical and experimental
progress should improve the precision in the semi-leptonic hyperon decays.

The abundant production of polarized hyperons and antihyperons in the BESIII
experiment, in combination with the versatile and near 4$\pi$ detector,
enables a new generation of semi-leptonic hyperon decay studies. Recent
measurements have been carried out of decays such as $\Lambda\to
p\mu^{-}\bar{\nu}_{\mu}$ [166] and $\Sigma^{+}\to\Lambda e^{+}\nu_{e}$ [167].
In the latter study, the CP asymmetry defined in Eq. 17 was measured for the
first time. Furthermore, searches have been performed for the expectedly rare
$\Xi^{-}\to\Xi^{0}e^{-}\bar{\nu}_{e}$ [168] and
$\Xi^{0}\to\Sigma^{-}e^{+}\nu_{e}$ [169] decays. The results from the BESIII
semi-leptonic studies are summarised in Table 2.

All aforementioned results are obtained by analyzing only the semi-leptonic
decay itself, i.e. not considering the production process. The ability of the
BESIII experiment to reconstruct the complete $Y\bar{Y}$ pair and its
subsequent hadronic and semi-leptonic decays, opens up for a new method where
the production is taken into account. The method outlined in Ref. [35] can be
generalized to describe the joint angular distributions [170]. It allows for
the extraction of spin correlations and polarization of the produced hyperon-
antihyperon system which improves the precision of the semi-leptonic decay
parameters (Eq. 16).

Table 2: BRs measurement of semi-leptonic hyperon decays in the $e^{+}e^{-}\to J/\psi\to Y\overline{Y}$ decays at BESIII. The data statistics are $10^{10}J/\psi$ events.  $Y\bar{Y}$ pair | Final state | ${\cal B}(\times 10^{-4})$ | $N^{\mathrm{SLW}}$ | $R_{\mu e}$ | $A_{CP}^{SLW}$ | Ref.
---|---|---|---|---|---|---
$J/\psi\to\Lambda\overline{\Lambda}$ | $\Lambda\to p\mu^{-}\bar{\nu}_{\mu}$ | $1.48\pm 0.21\pm 0.08$ | $64\pm 9$ | $0.178\pm 0.028$ |  | [166]
$J/\psi\to\Xi^{-}\overline{\Xi}^{+}$ | $\Xi^{-}\to\Xi^{0}e^{-}\bar{\nu}_{e}$ | $<2.59(90\%)$ | $-$ |  |  | [168]
$J/\psi\to\Xi^{0}\overline{\Xi}^{0}$ | $\Xi^{0}\to\Sigma^{-}e^{+}\nu_{e}$ | $<1.6(90\%)$ | $-$ |  |  | [169]
$J/\psi\to\Sigma^{+}\overline{\Sigma}^{-}$ | $\Sigma^{+}\to\Lambda e^{+}\nu_{e}$ | $29.3\pm 7.4\pm 1.3$ | $16\pm 4$ |  | $1.06\pm 0.28$ | [167]

A data collection campaign during 2019-2021, where 4.5 fb-1 of data were
collected between 4.6 GeV and 4.7 GeV, enabled a pioneering study of the
differential decay rate and weak form factors of the semi-leptonic charm decay
$\Lambda_{c}^{+}\to\Lambda e^{+}\nu_{e}$ [171]. The results were compared to
predictions from lattice QCD calculations [172]. By combining the measured
branching fraction, the life-time $\tau_{\Lambda_{c}}$ and the
$q^{2}$-integrated rate predicted by lattice QCD, the matrix element
$|V_{cs}|$ was determined to be $|V_{cs}|=0.936\pm 0.017_{\mathcal{B}}\pm
0.024_{\mathrm{LQCD}}\pm 0.007_{\tau_{\Lambda_{c}}}$. The semi-leptonic
$\Lambda_{c}^{+}$ decay measurement provides an important consistency test for
the SM and a probe for physics beyond the SM, complementary to $D$ meson
analyses [173, 174].

Searches for violations of lepton flavour universality in hyperon decays have
been studied in the numerous hyperon beam experiment in the 1970s [175, 176,
177, 178]. The most precise measurement was performed in 1972 [178] and was
reported as a relative $\Gamma(\Lambda\to
p\mu^{-}\bar{\nu}_{\mu})/\Gamma(\Lambda\to N\pi)=(1.4\pm 0.5)\cdot 10^{-4}$
based on 14 signal events which were selected from about $6\cdot 10^{5}$
bubble chamber pictures. The current world average of the observable $R_{\mu
e}$, defined in Eq. 19, is $0.189\pm 0.041$ for $\Lambda$ [1]. The large
experimental uncertainty is dominated by the $\mathcal{B}(\Lambda\to
p\mu^{-}\bar{\nu}_{\mu})$. Recently, the BESIII collaboration [166] reported a
value of $0.178\pm 0.028$ based on 31 signal events obtained from the
$10^{10}$ $J/\psi$ sample. All results so far are consistent with the SM
prediction, obtained at next-to-leading order, of $0.153\pm 0.008$ [28]. A
more accurate measurement of $R_{\mu e}$ will provide a more stringent test of
lepton flavour universality.

##  6 Discussion

The abundant production of polarized and entangled hyperon-antihyperon pairs
in the BESIII experiment has paved the way for a new generation of
measurements on structure and symmetry. In the following, we will summarise
the recent findings and formulate the next steps in the future roadmap of
hyperon physics.

What is the origin of the oscillatory features of the form factors (effective
form factor, $G_{E}/G_{M}$ ratio and magnetic form factor) observed for
protons, nucleons and possibly also hyperons? The first comparison between
protons [66] and neutrons [9] indicated oscillations with a common frequency,
a behaviour that possibly carries over to strange hyperons. However, a recent
measurement by the SND [70] instead reveals a smaller frequency for neutrons
which raises the question of whether the observed oscillatory patterns are
only similar by accident or if they indeed have a common origin. Mapping the
form factors over a broad energy range for the complete spin 1/2 octet would
be very illuminating and can help find an answer to this question. Data for
such measurements can either be obtained in dedicated energy scans with the
BESIII experiment or by utilizing the technique of initial state radiation
(ISR) [179] on data collected at charmonium or bottomium resonances with the
BESIII (charmonium) and Belle II (bottomium) experiments.

How well can we understand the structure of hyperons on the femtometer scale?
Can we extract space-like quantities using time-like measurements? A recent
publication [18] outlines a method to quantitatively relate the experimentally
accessible time-like region to space-like properties, such as the charge
radius, from the energy dependence of the form factor phase. So far, the only
existing conclusive, off-resonance form factor phase measurement is for the
$\Lambda$ hyperon [78]. This corresponds to a "snapshot" of the time evolution
of the hyperon-antihyperon pair. The energy dependence of the phase yields a
sequence of such snapshots, from which the space-like structure can be
revealed. Corresponding measurements for the full SU(3) octet or even to charm
hyperons, can reveal patterns from which we will gain a deeper understanding
of the complex dynamics that bind quarks into hadrons.

At $e^{+}e^{-}$ collision energies that coincide with vector charmonia such as
$J/\Psi$, $\Psi(3686)$ and $\Psi(3770)$, large data samples are available from
the BESIII experiment. This has enabled very precise measurements of the so-
called psionic form factors [19]. However, it is yet an open question how to
interpret these form factors and what they can reveal about the decaying
charmonium resonance. In particular, the close connection between the form
factors and polarization could be helpful in understanding the role of spin in
charmonium decay processes. However, in order to optimize the output from the
high-precision data, the measured parameters need a quantitative foundation in
the form of a phenomenological model. An increased exchange between the
experimental collaborations and the theory community can lead to new ideas and
help exploiting the richness of the hyperon-antihyperon decay distributions.

The abundant production of hyperon-antihyperon pairs whose decays constitute a
hunting-ground for CP violation, complementary to that of e.g. the $\epsilon$
and $\epsilon^{\prime}$ parameters in kaon decays. This is particularly true
in the search for BSM physics, since many such models which involve an
enhanced parity-even gluon dipole operator that does not contribute to
$\epsilon^{\prime}$. Parameters such as $A_{CP}$ and $(\xi_{P}-\xi_{S})$ are
sensitive to both parity-odd and parity-even operators and can hence be non-
zero even if $\epsilon^{\prime}$ is vanishingly small. For several
observables, the CP violating signal is expected to be up to two orders of
magnitude larger for certain classes of models beyond the SM compared to the
SM [180, 181, 182, 40, 183]. More precise CP tests will hence provide
constraints for BSM models and guidance for future theoretical efforts. In
addition, increased precision in the measurement of e.g. strong phases will be
helpful in theory calculations of weak phases. One example is the $\Sigma$
decay in Ref. [40] for which the theory uncertainty of the weak phase is
$\approx$200%. New, precise measurements of the decay amplitudes of various
hyperons will increase the precision of the predictions from theory. But how
precise measurements are possible?

Figure 8:  CP symmetry tests from Refs. [87, 94, 88, 92, 109, 129, 128]. The
$J/\psi\to\Lambda\overline{\Lambda}$ (1) and
$J/\psi\to\Xi^{-}\overline{\Xi}^{+}\to\Lambda\pi^{-}\overline{\Lambda}\pi^{+}$
(2) from BESIII have both been used to determine $A_{CP}$ for $\Lambda\to
p\pi^{-}$. The asterisks * and ** denote the 2009-2012 and 2009-2019 $J/\psi$
samples, respectively. In the sequential decay from HyperCP, the effective
asymmetry $A_{CP}^{\rm\it eff}$, defined in Eq. 12, was measured.

Recent measurements from the BESIII experiment, exploiting the polarization
and spin correlations, provide increased precision for a given sample size
[88] compared to techniques based on polarized beams of secondary hyperons
[111]. In particular, sequentially decaying multi-strange and charmed baryons
offer a way to separate strong and weak contributions to the decay amplitude,
giving clean access to CP-violating signals. In Figure 8, the CP asymmetry
$A_{CP}$ is shown for a number of hyperon decays. Though many of these are
very precise, it is not sufficient to test the aforementioned theoretical
predictions. In fact, the precision required is out of reach with currently
available data samples. The ongoing campaign of collecting 50 ab-1 of data
with Belle II, as well as the planned next-generation facilities PANDA and
STCF will extend the current data bank with at least two orders of magnitude.
Experiments with polarized electron-positron beams can further increase the
precision [184].

With current facilities such as BESIII and Belle II, methods based on
entangled hyperon-antihyperon pairs are in principle only possible to apply to
strange hyperons. Charmed baryons decay into a large number of final states,
all with relatively small branching fractions (at percent level or smaller).
The recent single-tag measurements by Belle II of $\Lambda_{c}^{+}$ and
$\Xi_{c}^{0}$ decays have in some cases a precision approaching that of
strange hyperons. However, a limiting factor is the systematic uncertainties.
This may be partly attributed to the method in which the polarization of the
initial and intermediate baryons in the chain $Y_{1}\to Y_{2}M_{1}\to
Y_{3}M_{1}M_{2}\to...$ are integrated out. The ongoing Belle II campaign,
where up to 50 ab-1 of data will be collected, opens up for methods in which
the richness of the joint sequential angular distribution can be exploited to
its fullest [4]. This can potentially improve the precision of charmed baryon
decay parameters.

The large transverse polarization of hyperons such as $\Lambda$, $\Sigma^{+}$
and $\Xi^{-}$ produced in vector charmonium decays, open up for precision
studies of decay parameters in weak, radiative $Y_{1}\to Y_{2}\gamma$ decays.
These decays provide a tool to understand the elementary process $s\to
d\gamma$ in a way that is complementary to radiative kaon decays, which in
turn serve as guidance in the search for physics beyond the SM [147, 148].

Due to the small branching fractions, the semi-leptonic data samples are
small. As a consequence, the focus of semi-leptonic studies being put on
observables based on decay widths rather than angular distributions and
polarization. Large samples from ongoing campaigns or future facilities can be
a game-changer in this regard, and lead to substantial advances.

##  7 Future Prospects

In the following, we will discuss the existing facilities BESIII and Belle II,
as well as the planned PANDA and STCF, where polarized and entangled hyperon-
antihyperon pairs can be studied.

###  7.1 Existing Facilities

####  7.1.1 BESIII at BEPCII

Figure 9: Left: The integrated luminosity collected with BESIII at BEPCII till
2024, as a function of the CMS energy $\sqrt{(}s)$. Right: The effective
integrated luminosity with ISR with a reduced $\sqrt{(}s)$ between 1.87 and
5.0 GeV with the Belle II experiment.

Data from the energy scan performed by BESIII in 2014-2015 have resulted in
several pioneering hyperon structure measurements so far [78, 60, 59], but the
story does not end here. Several analyses of these data are still ongoing and
the results will be published soon. These studies include $\Lambda$ and
$\Sigma^{+}$ form factor phase measurements at several energies.

A campaign to collect 20 fb-1 data at the $\Psi(3770)$, would enable precision
studies of all the SU(3) octet hyperons using the ISR technique. Furthermore,
a dedicated campaign has been carried out between CMS energies of 4.6 GeV and
4.95 GeV, with the purpose of a precision measurement of the $q^{2}$
dependence of the $\Lambda_{c}^{+}$ form factors.

At the time of this review, the most precise $\Lambda$ CP test so far has
emerged from the world-record data sample of $10^{10}$ events. However, data
from the same sample are being analysed for high-precision CP tests in
$\Sigma^{0}$, $\Sigma^{+}$, $\Xi^{-}$ and $\Xi^{0}$ decays. Since most of
these hyperons decay into an intermediate $\Lambda$, each of these studies
will provide an independent measurements of the $\alpha_{\Lambda}$ parameter.
In addition, this and future BESIII samples will enable searches for rare
hyperon decays with unprecedented precision. In addition to the aforementioned
radiative and semi-leptonic decays, it is possible to search for rare or
forbidden processes, for example those involving flavour changing neutral
currents.

The physics potential of BESIII will reach a new level with a planned upgrade
of the BEPCII collider to a maximum beam energy of 2.8 GeV. This corresponds
to a CMS energy of 5.6 GeV where the peak luminosity will be increased by a
factor of three compared to what is possible at CMS energies below 4.0 GeV. As
a result, BESIII will offer a great opportunity to study the production and
structure of various charmed baryons, as well as their decay properties. In
the left panel of Figure 9, the integrated luminosities previous and planned
data campaigns are summarized.

####  7.1.2 Belle II at SuperKEKB

The goal of the ongoing campaign with the Belle II experiment at SuperKEKB in
Japan [185] is to collect in total 50 ab-1 of data in the bottomium region,
with an intermediate milestone of 10 ab-1. This enables high-precision
structure studies of strange and charmed hyperons through the ISR technique:
although the radiator function in the ISR cross section results in a reduced
effective luminosity (see right panel of Figure 9). The huge integrated
luminosity will eventually outperform other off-resonance samples from other
$e^{+}e^{-}$ colliders.

The Belle II and its predecessor Belle have, with the data already on tape,
performed pioneering measurements of charmed baryon decay asymmetry
parameters, as well as CP tests. New data have been and are being collected by
Belle II and can provide better statistical precision. In addition, larger
samples enable studies of more complete decay chains, exploiting the richness
of the angular distributions. This could reduce the systematic uncertainties
which currently dominate the total uncertainty.

###  7.2 Future Facilities

The methods developed for hyperon-antihyperon pairs provide sensitive CP
tests, but are currently limited by statistical precision. Improving the
uncertainties by several orders of magnitude is not feasible with currently
operating facilities world-wide, but requires next-generation experiments
where polarized and entangled/quantum-correlated hyperon-antihyperon pairs are
produced in abundance. The PANDA experiment at FAIR, Germany and the proposed
STCF at USTC, China are two excellent candidates which will be presented in
the following.

####  7.2.1 PANDA

The future experiment antiProton ANihilation at DArmstadt (PANDA) will produce
hyperon-antihyperon pairs in abundance, thanks to the expected large
production cross section in $\bar{p}p$ annihilations and the large
luminosities enabled by the stored antiproton beam. The latter will be
delivered from the High Energy Storage Ring (HESR) within a momentum range
from 1.5 GeV/$c$ up to 15 GeV/$c$ and impinge on a hydrogen cluster or pellet
target. During the first phase of operation, Phase One [186], the HESR will
provide a luminosity of $\approx 10^{31}$ cm2s-1 whilst the design luminosity
of $\approx 2\cdot 10^{32}$ cm2s-1 will be achieved during Phase Three. Since
PANDA will cover nearly $4\pi$ of the solid angle, the prerequisites for
exclusive reconstruction of hyperon-antihyperon pairs and their subsequent
decays are well fulfilled. This has been demonstrated by simulation studies in
for example Refs. [187, 186], summarised in Table 3. We conclude that PANDA,
already during the first phase of operation, will be able to collect samples
several orders of magnitude larger than what is currently possible with
BESIII. What does this mean in terms of the precision of the CP asymmetries?

Table 3: Results from simulation studies of the various production reactions of ground state hyperons [187, 186]. Reaction |  $p_{\overline{p}}$ (GeV/$c$) | S/B | Events / day, Phase One | Events / day, full luminosity
---|---|---|---|---
$\overline{p}p\rightarrow\overline{\Lambda}\Lambda$ | 1.64 | 114 | $3.8\cdot 10^{6}$ | $7.6\cdot 10^{7}$
$\overline{p}p\rightarrow\overline{\Sigma}^{0}\Lambda$ | 6.0 | 21 | $4.3\cdot 10^{5}$ | $4.1\cdot 10^{6}$
$\overline{p}p\rightarrow\overline{\Xi}^{+}\Xi^{-}$ | 4.6 | 274 | $2.6\cdot 10^{4}$ | $5.2\cdot 10^{5}$

In $\bar{p}p$ annihilations, many partial waves contribute to the production.
This is in contrast to the $e^{+}e^{-}$ case, where the spin and parity of the
initial state is almost exclusively $J^{P}=1^{-}$. Hence, in $e^{+}e^{-}$
production, all production parameters such as polarization and spin
correlations are well-defined functions of the hyperon scattering angle
$\theta_{Y}$, governed by two global parameters $\alpha_{\Psi}$ and
$\Delta\Phi$ that depend on the collision energy only. Hyperon-antihyperon
pairs produced in $\bar{p}p\to\bar{Y}Y$ reactions are quantum correlated
rather than entangled since several possibilities exist for the initial spin-
parity state. Nevertheless, in the case of spin 1/2 hyperons, the reaction can
be parameterized in terms of five independent production parameters, each with
an unknown $\theta_{Y}$ dependence. Decay parameters and CP asymmetries,
therefore, need to be extracted from global fits. However, as demonstrated in
Ref. [100], the precision for a given sample size is about the same as in
$e^{+}e^{-}$ experiments [87]. The future CP tests by PANDA really have the
potential to significantly improve the precision of the world data and put SM
and BSM predictions to the test.

####  7.2.2 STCF

The Super Tau-Charm facility (STCF) proposed in China is a next-generation
electron-positron collider that will operate within a CMS energy range from
2.0 to 7.0 GeV [188]. The design luminosity is $0.5\cdot 10^{35}$ cm-2s-1 or
higher at the optimal CMS energy of 4.0 GeV. With this luminosity, it will be
possible to collect 1 ab-1 of data per year. At the $J/\Psi$ mass, this
corresponds to $3.4\cdot 10^{12}$ $J/\Psi$ events per year, given a production
cross-section of 3400 nb and a beam energy spread that equals that of BESIII.
This opens up for high-precision CP tests in decays of hyperon-antihyperon
pairs. Realistic projections of the precision can be obtained by scaling the
statistical uncertainty achieved with BESIII with at least two orders of
magnitude larger data sets expected with the STCF. The results are presented
in Table 4 and are discussed in detail in Ref. [184].

Table 4: Expected statistical uncertainty for the CP asymmetries $A_{CP}^{\Lambda}$, $A_{CP}^{\Xi^{-}}$ and $B_{CP}^{\Xi^{-}}$ at BESIII and the proposed STCF. BESIII has already published measurements on $A_{CP}^{\Lambda}$ [87, 92] and $A_{CP}^{\Xi^{-}}$ [88], the former with the full $10^{10}J/\Psi$ data sample. The projected uncertainties for future measurements at BESIII and STCF are obtained by scaling the statistical uncertainties with the number of events. The $\ast$ and $\ast\ast$ indicate the combination of BESIII measurements [87, 92] and projected precision, respectively. | $N_{J/\psi}$ | $\sigma(A_{CP}^{\Lambda})$ | $\sigma(A_{CP}^{\Xi^{-}})$ | $\sigma(B_{CP}^{\Xi^{-}})$
---|---|---|---|---
BESIII | $1.3\cdot 10^{9}$ | $1.0\cdot 10^{-2\ast}$ | $1.3\cdot 10^{-2}$ | $3.5\cdot 10^{-2}$
BESIII | $1.0\cdot 10^{10\ast\ast}$ | $3.6\cdot 10^{-3}$ | $4.8\cdot 10^{-3}$ | $1.3\cdot 10^{-2}$
STCF | $3.4\cdot 10^{12\ast\ast}$ | $2.0\cdot 10^{-4}$ | $2.6\cdot 10^{-4}$ | $6.8\cdot 10^{-4}$

Future upgrades may include even higher luminosities, a longitudinal
polarization of the electron beam and a reduced beam energy spread. It is
estimated that an electron beam polarization of about $80\%$ at $J/\psi$
energies can be obtained without reducing the beam current [189]. This
polarization is much larger than the observed natural transverse polarization
of hyperons and antihyperons (about 30% at its maximum) from $J/\psi$ decays
and will increase the sensitivity of most CP tests in hyperon decays [184].

The beam energy spread can be reduced by a collision scheme where electrons
(positrons) with higher momenta collide with positrons (electrons) with lower
momenta [190, 191, 192], similar to the Belle II experiment at SuperKEKB
[185]. Reducing the CMS energy spread compared to what is currently possible
with BEPC-II can further increase the data quality. In particular, if the
spread approaches the natural width of the $J/\psi$, $\rm{\Gamma}=90$ keV, the
number of the $J/\psi$ events produced with a given integrated luminosity will
increase. Furthermore, the off-resonance background can be reduced which would
improve the sensitivity to rare processes.

##  8 Summary

Polarized and entangled hyperon-antihyperon pairs offer a common platform to
study a wide range of topics including precision aspects of the strong
interaction, tests of fundamental symmetries and flavour physics. Recent
measurements from BESIII employ newly developed methods that benefit from the
possibility to precisely parameterize the complicated decay chains, including
particle-antiparticle correlations. This has resulted in pioneering
measurements of the hyperon structure at the femtometer scale, some of the
most precise CP tests for baryons so far, and studies of rare hyperon decays.
These methods can be extended and applied to larger data samples from BESIII
and Belle II, but also from future particle-antiparticle collider experiments
such as PANDA and STCF. The results can lead to a more complete understanding
of the Standard Model with all its features, including currently challenging
aspects such as the strong interaction in the confinement domain. Furthermore,
future hyperon-antihyperon studies have the potential to reveal physics beyond
the Standard Model.

## Acknowledgement

The authors are grateful to the Knut and Alice Wallenberg Foundation, Contract
No. 2016.0157 and 2021.0299 (Sweden), the Swedish Research Council, Contracts
No. 2019-04594 and 2021-04567 (Sweden), The Swedish Foundation for
International Cooperation in Research and Higher Education, Contract No.
CH2018-7756, the Olle Engkvist Foundation, Contract No. 200-0605 (Sweden),
Lundström-Åman Foundation (Sweden), the National Natural Science Foundation,
Contract Nos. 11935018, 12122509 (China), the CAS President’s International
Fellowship Initiative, Grant No. 2021PM0014 (China), Polish National Science
Centre, Grant 2019/35/O/ST2/02907 (Poland), the Double First-Class university
project foundation of USTC (China). Finally, we would like to thank Prof. Hai-
Bo Li for valuable input and encouragement during the writing process.

## References

  * [1] R. L. Workman et al., “Review of Particle Physics,” PTEP, vol. 2022, p. 083C01, 2022.
  * [2] I. I. Bigi and A. I. Sanda, CP violation, vol. 9.  Cambridge University Press, 9 2009.
  * [3] T. D. Lee and C.-N. Yang, “General Partial Wave Analysis of the Decay of a Hyperon of Spin 1/2,” Phys. Rev., vol. 108, pp. 1645–1647, 1957.
  * [4] G. Fäldt, “Production and sequential decay of charmed hyperons,” Phys. Rev. D, vol. 97, no. 5, p. 053002, 2018.
  * [5] E. Thomé, Multistrange and Charmed Antihyperon-Hyperon Physics for PANDA.  PhD thesis, Uppsala U., 2012.
  * [6] D. Binosi, “Emergent Hadron Mass in Strong Dynamics,” Few Body Syst., vol. 63, no. 2, p. 42, 2022.
  * [7] H. Gao and M. Vanderhaeghen, “The proton charge radius,” Rev. Mod. Phys., vol. 94, no. 1, p. 015002, 2022.
  * [8] H. Atac, M. Constantinou, Z. E. Meziani, M. Paolone, and N. Sparveris, “Measurement of the neutron charge radius and the role of its constituents,” Nature Commun., vol. 12, no. 1, p. 1759, 2021.
  * [9] M. Ablikim et al., “Oscillating features in the electromagnetic structure of the neutron,” Nature Phys., vol. 17, no. 11, pp. 1200–1204, 2021.
  * [10] I. M. Gough Eschrich et al., “Measurement of the Sigma- Charge Radius by Sigma- Electron Elastic Scattering,” Phys. Lett. B, vol. 522, pp. 233–239, 2001.
  * [11] S. Amendolia, G. Batignani, G. Beck, E. Bellamy, E. Bertolucci, G. Bologna, L. Bosisio, C. Bradaschia, M. Budinich, M. Dell’orso, B. D. Piazzoli, F. Fabbri, F. Fidecaro, L. Foa, E. Focardi, S. Frank, P. Gianetti, A. Giazzotto, M. Giorgi, M. Green, G. Heath, M. Landon, P. Laurelli, F. Liello, G. Mannocchi, P. March, P. Marrocchesi, A. Menzione, E. Meroni, P. Picchi, F. Ragusa, L. Ristori, L. Rolandi, A. Scribano, A. Stefanini, D. Storey, J. Strong, R. Tenchini, G. Tonelli, G. Triggiani, W. von Schlippe, and A. Zallo, “A measurement of the kaon charge radius,” Physics Letters B, vol. 178, no. 4, pp. 435–440, 1986.
  * [12] V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, “The Structure of the Nucleon: Elastic Electromagnetic Form Factors,” Eur. Phys. J. A, vol. 51, p. 79, 2015.
  * [13] S. Pacetti, R. Baldini Ferroli, and E. Tomasi-Gustafsson, “Proton electromagnetic form factors: Basic notions, present achievements and future perspectives,” Phys. Rept., vol. 550-551, pp. 1–103, 2015.
  * [14] M. A. Belushkin, H. W. Hammer, and U. G. Meissner, “Dispersion analysis of the nucleon form-factors including meson continua,” Phys. Rev. C, vol. 75, p. 035202, 2007.
  * [15] A. Z. Dubnickova, S. Dubnicka, and M. P. Rekalo, “Investigation of the nucleon electromagnetic structure by polarization effects in e+ e- —$>$ N anti-N processes,” Nuovo Cim. A, vol. 109, pp. 241–256, 1996.
  * [16] V. A. Matveev, R. M. Muradyan, and A. N. Tavkhelidze, “Automodelity in strong interactions,” Teor. Mat. Fiz., vol. 15, pp. 332–339, 1973.
  * [17] S. J. Brodsky and G. R. Farrar, “Scaling Laws at Large Transverse Momentum,” Phys. Rev. Lett., vol. 31, pp. 1153–1156, 1973.
  * [18] A. Mangoni, S. Pacetti, and E. Tomasi-Gustafsson, “The first exploration of the physical Riemann surfaces of the ratio $G_{E}^{\Lambda}/G_{M}^{\Lambda}$,” Phys. Rev. D, vol. 104, no. 11, p. 116016, 2021.
  * [19] G. Fäldt and A. Kupsc, “Hadronic structure functions in the $e^{+}e^{-}\rightarrow\bar{\Lambda}\Lambda$ reaction,” Phys. Lett. B, vol. 772, pp. 16–20, 2017.
  * [20] A. D. Sakharov, “Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe,” Pisma Zh. Eksp. Teor. Fiz., vol. 5, pp. 32–35, 1967.
  * [21] N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Phys. Rev. Lett., vol. 10, pp. 531–533, 1963.
  * [22] M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory of Weak Interaction,” Prog. Theor. Phys., vol. 49, pp. 652–657, 1973.
  * [23] J. Liu, C. Q. Geng, and J. N. Ng, “Strong cp violation and the neutron electric dipole moment,” Phys. Rev. Lett., vol. 63, pp. 589–592, Aug 1989\.
  * [24] C. Abel et al., “A search for neutron to mirror-neutron oscillations using the nedm apparatus at psi,” Physics Letters B, vol. 812, p. 135993, 2021.
  * [25] R. D. Peccei, “The Strong CP problem and axions,” Lect. Notes Phys., vol. 741, pp. 3–17, 2008.
  * [26] S. S. Nair, E. Perotti, and S. Leupold, “Constraining p and cp violation in the main decay of the neutral sigma hyperon,” Physics Letters B, vol. 788, pp. 535–541, 2019.
  * [27] Y. Hara, “Nonleptonic Decays of Baryons and the Eightfold Way,” Phys. Rev. Lett., vol. 12, pp. 378–379, 1964.
  * [28] H.-M. Chang, M. González-Alonso, and J. Martin Camalich, “Nonstandard Semileptonic Hyperon Decays,” Phys. Rev. Lett., vol. 114, no. 16, p. 161802, 2015.
  * [29] M. Ablikim et al., “Design and Construction of the BESIII Detector,” Nucl. Instrum. Meth. A, vol. 614, pp. 345–399, 2010.
  * [30] M. Ablikim et al., “Number of $J/\psi$ events at BESIII,” Chin. Phys. C, vol. 46, no. 7, p. 074001, 2022.
  * [31] G. I. Gakh and E. Tomasi-Gustafsson, “General analysis of polarization phenomena in e+ + e- —$>$ N + anti-N for axial parametrization of two-photon exchange,” Nucl. Phys. A, vol. 771, pp. 169–183, 2006.
  * [32] H. Czyz, A. Grzelinska, and J. H. Kuhn, “Spin asymmetries and correlations in lambda-pair production through the radiative return method,” Phys. Rev. D, vol. 75, p. 074026, 2007.
  * [33] G. Fäldt, “Polarization observables in the $e^{+}e^{-}\rightarrow\bar{\Lambda}\Lambda$ reaction,” Eur. Phys. J. A, vol. 52, no. 5, p. 141, 2016.
  * [34] H. Jones and M. Scadron, “Multipole $\gamma$n-$\delta$ form factors and resonant photo- and electroproduction,” Annals of Physics, vol. 81, no. 1, pp. 1–14, 1973.
  * [35] E. Perotti, G. Fäldt, A. Kupsc, S. Leupold, and J. J. Song, “Polarization observables in $e^{+}e^{-}$ annihilation to a baryon-antibaryon pair,” Phys. Rev. D, vol. 99, no. 5, p. 056008, 2019.
  * [36] V. Kopeliovich, “,” Soviet Journal of Nuclear Physics, vol. 12, p. 6, 1971\.
  * [37] V. Thorén, Hadron Physics in a Polarized World : Exploring Electromagnetic Interactions with Spin Observables.  PhD thesis, Uppsala U., 2022.
  * [38] J. F. Donoghue, X.-G. He, and S. Pakvasa, “Hyperon Decays and CP Nonconservation,” Phys. Rev. D, vol. 34, p. 833, 1986.
  * [39] J. F. Donoghue and S. Pakvasa, “Signals of CP Nonconservation in Hyperon Decay,” Phys. Rev. Lett., vol. 55, p. 162, 1985.
  * [40] J. Tandean and G. Valencia, “CP violation in hyperon nonleptonic decays within the standard model,” Phys. Rev. D, vol. 67, p. 056001, 2003.
  * [41] C. A. Baker, D. D. Doyle, P. Geltenbort, K. Green, M. G. D. van der Grinten, P. G. Harris, P. Iaydjiev, S. N. Ivanov, D. J. R. May, J. M. Pendlebury, J. D. Richardson, D. Shiers, and K. F. Smith, “Improved experimental limit on the electric dipole moment of the neutron,” Phys. Rev. Lett., vol. 97, p. 131801, Sep 2006.
  * [42] Y. Hara, “Nonleptonic decays of baryons and the eightfold way,” Phys. Rev. Lett., vol. 12, pp. 378–379, Mar 1964.
  * [43] N. Cabibbo, E. C. Swallow, and R. Winston, “Semileptonic hyperon decays,” Ann. Rev. Nucl. Part. Sci., vol. 53, pp. 39–75, 2003.
  * [44] S. Weinberg, “Charge symmetry of weak interactions,” Phys. Rev., vol. 112, pp. 1375–1379, 1958.
  * [45] A. Garcia and P. Kielanowski, “THE BETA DECAY OF HYPERONS,” 1985.
  * [46] P. H. Frampton and W.-K. Tung, “Hyperon beta decay,” Phys. Rev. D, vol. 3, pp. 1114–1121, 1971.
  * [47] R. Hofstaedter, “The electron-scattering method and its applications to the structure of tnuclei and nucleons,” in Nobel Lectures Physics 1942-1962, Amsterdam: Elsevier Publishing Company, 1964.
  * [48] S. Pacetti and E. Tomasi-Gustafsson, “Sensitivity of the elastic electron–proton cross section to the proton radius,” Eur. Phys. J. A, vol. 56, no. 3, p. 74, 2020.
  * [49] Y.-H. Lin, H.-W. Hammer, and U.-G. Meißner, “New Insights into the Nucleon’s Electromagnetic Structure,” Phys. Rev. Lett., vol. 128, no. 5, p. 052002, 2022.
  * [50] B. Aubert et al., “Study of $e^{+}e^{-}\to\Lambda\bar{\Lambda}$, $\Lambda\bar{\Sigma}^{0}$, $\Sigma^{0}\bar{\Sigma}^{0}$ using initial state radiation with BABAR,” Phys. Rev. D, vol. 76, p. 092006, 2007.
  * [51] S. Dobbs, A. Tomaradze, T. Xiao, K. K. Seth, and G. Bonvicini, “First measurements of timelike form factors of the hyperons, $\Lambda^{0}$, $\Sigma^{0}$, $\Sigma^{+}$, $\Xi^{0}$, $\Xi^{-}$, and $\Omega^{-}$, and evidence of diquark correlations,” Phys. Lett. B, vol. 739, pp. 90–94, 2014.
  * [52] S. Dobbs, K. K. Seth, A. Tomaradze, T. Xiao, and G. Bonvicini, “Hyperon Form Factors & Diquark Correlations,” Phys. Rev. D, vol. 96, no. 9, p. 092004, 2017.
  * [53] M. Anselmino, E. Predazzi, S. Ekelin, S. Fredriksson, and D. B. Lichtenberg, “Diquarks,” Rev. Mod. Phys., vol. 65, pp. 1199–1234, 1993.
  * [54] M. Ablikim et al., “Measurement of the cross section for $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ and evidence of the decay $\psi(3770)\rightarrow\Lambda\bar{\Lambda}$,” Phys. Rev. D, vol. 104, no. 9, p. L091104, 2021.
  * [55] R.-Q. Qian, Q. Huang, and X. Liu, “Predicted $\Lambda\bar{\Lambda}$ and $\Xi^{-}\bar{\Xi}^{+}$ decay modes of the charmoniumlike Y(4230),” Phys. Lett. B, vol. 833, p. 137292, 2022.
  * [56] M. Ablikim et al., “Observation of a cross-section enhancement near mass threshold in $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$,” Phys. Rev. D, vol. 97, no. 3, p. 032013, 2018.
  * [57] X. Cao, J.-P. Dai, and Y.-P. Xie, “Vector mesons and electromagnetic form factor of the $\Lambda$ hyperon,” Phys. Rev. D, vol. 98, no. 9, p. 094006, 2018.
  * [58] Z.-Y. Li, A.-X. Dai, and J.-J. Xie, “Electromagnetic Form Factors of $\Lambda$ Hyperon in the Vector Meson Dominance Model and a Possible Explanation of the Near-Threshold Enhancement of the Reaction,” Chin. Phys. Lett., vol. 39, no. 1, p. 011201, 2022.
  * [59] M. Ablikim et al., “Measurements of $\Sigma^{+}$ and $\Sigma^{-}$ time-like electromagnetic form factors for center-of-mass energies from 2.3864 to 3.0200 GeV,” Phys. Lett. B, vol. 814, p. 136110, 2021.
  * [60] M. Ablikim et al., “Measurement of the $e^{+}e^{-}\to\Sigma^{0}\bar{\Sigma}^{0}$ cross sections at center-of-mass energies from 2.3864 to 3.0200GeV,” Phys. Lett. B, vol. 831, p. 137187, 2022.
  * [61] M. Ablikim et al., “Measurement of the cross section for $e^{+}e^{-}\rightarrow\Xi^{-}\bar{\Xi}^{+}$ and observation of an excited $\Xi$ baryon,” Phys. Rev. Lett., vol. 124, no. 3, p. 032002, 2020.
  * [62] M. Ablikim et al., “Measurement of cross section for $e^{+}e^{-}\to\Xi^{-}\bar{\Xi}^{+}$ near threshold at BESIII,” Phys. Rev. D, vol. 103, no. 1, p. 012005, 2021.
  * [63] M. Ablikim et al., “Measurement of cross section for $e^{+}e^{-}\rightarrow\Xi^{0}\bar{\Xi}^{0}$ near threshold,” Phys. Lett. B, vol. 820, p. 136557, 2021.
  * [64] J. P. Lees et al., “Study of $e^{+}e^{-}\to p\bar{p}$ via initial-state radiation at BABAR,” Phys. Rev. D, vol. 87, no. 9, p. 092005, 2013.
  * [65] M. Ablikim et al., “Measurement of proton electromagnetic form factors in the time-like region using initial state radiation at BESIII,” Phys. Lett. B, vol. 817, p. 136328, 2021.
  * [66] M. Ablikim et al., “Measurement of proton electromagnetic form factors in $e^{+}e^{-}\to p\bar{p}$ in the energy region 2.00 - 3.08 GeV,” Phys. Rev. Lett., vol. 124, no. 4, p. 042001, 2020.
  * [67] A. Bianconi and E. Tomasi-Gustafsson, “Phenomenological analysis of near threshold periodic modulations of the proton timelike form factor,” Phys. Rev. C, vol. 93, no. 3, p. 035201, 2016.
  * [68] A. Bianconi and E. Tomasi-Gustafsson, “Periodic interference structures in the timelike proton form factor,” Phys. Rev. Lett., vol. 114, no. 23, p. 232301, 2015.
  * [69] A.-X. Dai, Z.-Y. Li, L. Chang, and J.-J. Xie, “Electromagnetic form factors of neutron and neutral hyperons in the oscillating point of view *,” Chin. Phys. C, vol. 46, no. 7, p. 073104, 2022.
  * [70] M. N. Achasov et al., “Experimental study of the $e^{+}e^{-}\rightarrow n{\bar{n}}$ process at the VEPP-2000 $e^{+}e^{-}$ collider with the SND detector,” Eur. Phys. J. C, vol. 82, no. 8, p. 761, 2022.
  * [71] G. Pakhlova et al., “Observation of a near-threshold enhancement in the e+e- —$>$ Lambda+(c) Lambda-(c) cross section using initial-state radiation,” Phys. Rev. Lett., vol. 101, p. 172001, 2008\.
  * [72] M. Ablikim et al., “Precision measurement of the $e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ cross section near threshold,” Phys. Rev. Lett., vol. 120, no. 13, p. 132001, 2018.
  * [73] L.-Y. Dai, J. Haidenbauer, and U. G. Meißner, “Re-examining the $X(4630)$ resonance in the reaction $e^{+}e^{-}\rightarrow\Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$,” Phys. Rev. D, vol. 96, no. 11, p. 116001, 2017.
  * [74] L.-Q. Song, D. Song, J.-T. Zhu, and J. He, “Possible $\Lambda$c$\Lambda$¯c molecular states and their productions in nucleon-antinucleon collision,” Phys. Lett. B, vol. 835, p. 137586, 2022.
  * [75] A. I. Milstein and S. G. Salnikov, “Final-state interaction in the process $e^{+}e^{-}\rightarrow\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$,” Phys. Rev. D, vol. 105, no. 7, p. 074002, 2022.
  * [76] M. Ablikim et al., “Study of $e^{+}e^{-}\rightarrow\Omega^{-}\bar{\Omega}^{+}$ at center-of-mass energies from 3.49 to 3.67 GeV,” 12 2022.
  * [77] G. Ramalho, “Electromagnetic form factors of the $\Omega^{-}$ baryon in the spacelike and timelike regions,” Phys. Rev. D, vol. 103, no. 7, p. 074018, 2021.
  * [78] M. Ablikim et al., “Complete Measurement of the $\Lambda$ Electromagnetic Form Factors,” Phys. Rev. Lett., vol. 123, no. 12, p. 122003, 2019.
  * [79] J. Haidenbauer, U.-G. Meißner, and L.-Y. Dai, “Hyperon electromagnetic form factors in the timelike region,” Phys. Rev. D, vol. 103, no. 1, p. 014028, 2021.
  * [80] Y. Yang, D.-Y. Chen, and Z. Lu, “Electromagnetic form factors of $\Lambda$ hyperon in the vector meson dominance model,” Phys. Rev. D, vol. 100, no. 7, p. 073007, 2019.
  * [81] T. Husek and S. Leupold, “Radiative corrections for the decay $\Sigma^{0}\to\Lambda e^{+}e^{-}$,” Eur. Phys. J. C, vol. 80, no. 3, p. 218, 2020.
  * [82] O. Junker, S. Leupold, E. Perotti, and T. Vitos, “Electromagnetic form factors of the transition from the spin-3/2 $\Sigma$ to the $\Lambda$ hyperon,” Phys. Rev. C, vol. 101, no. 1, p. 015206, 2020.
  * [83] G. Ramalho, M. T. Peña, and K. Tsushima, “Hyperon electromagnetic timelike elastic form factors at large $q^{2}$,” Phys. Rev. D, vol. 101, no. 1, p. 014014, 2020.
  * [84] A. Bianconi and E. Tomasi-Gustafsson, “Comparative analysis of time-like hyperon form factors,” Phys. Rev. C, vol. 105, no. 6, p. 065206, 2022\.
  * [85] Y.-H. Lin, H.-W. Hammer, and U.-G. Meißner, “Dispersion-theoretical analysis of the electromagnetic form factors of the $\Lambda$ hyperon,” Eur. Phys. J. C, vol. 82, no. 12, p. 1091, 2022.
  * [86] M. Ablikim et al., “$\Sigma^{+}$ and $\bar{\Sigma}^{-}$ polarization in the $J/\psi$ and $\psi(3686)$ decays,” Phys. Rev. Lett., vol. 125, no. 5, p. 052004, 2020.
  * [87] M. Ablikim et al., “Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron Annihilation,” Nature Phys., vol. 15, pp. 631–634, 2019.
  * [88] M. Ablikim et al., “Probing CP symmetry and weak phases with entangled double-strange baryons,” Nature, vol. 606, no. 7912, pp. 64–69, 2022\.
  * [89] M. Ablikim et al., “Observation of $\Xi$\- hyperon transverse polarization in $\psi$(3686)→$\Xi$-$\Xi$¯+,” Phys. Rev. D, vol. 106, no. 9, p. L091101, 2022.
  * [90] M. Ablikim et al., “Observation of the decay $\psi$(3686) →${\Sigma}^{-}{\overline{\Sigma}}^{+}$ and measurement of its angular distribution,” JHEP, vol. 12, p. 016, 2022.
  * [91] M. Ablikim et al., “Study of $J/\psi$ and $\psi(3686)$ decay to $\Lambda\bar{\Lambda}$ and $\Sigma^{0}\bar{\Sigma}^{0}$ final states,” Phys. Rev. D, vol. 95, no. 5, p. 052003, 2017.
  * [92] M. Ablikim et al., “Precise Measurements of Decay Parameters and $CP$ Asymmetry with Entangled $\Lambda-\bar{\Lambda}$ Pairs Pairs,” Phys. Rev. Lett., vol. 129, no. 13, p. 131801, 2022.
  * [93] M. Ablikim et al., “Measurement of $\Lambda$ baryon polarization in $e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ at $\sqrt{s}=3.773$ GeV,” Phys. Rev. D, vol. 105, no. 1, p. L011101, 2022.
  * [94] M. Ablikim et al., “Measurement of branching fractions of $J/\psi$ and $\psi(3686)$ decays to $\Sigma^{+}$ and $\overline{\Sigma}^{-}$,” JHEP, vol. 11, p. 226, 2021.
  * [95] M. Ablikim et al., “Measurements of the branching fractions of $\psi(3686)\rightarrow\bar{\Sigma}^{0}\Lambda+c.c.$ and $\chi_{cJ(J=0,1,2)}\rightarrow\Lambda\bar{\Lambda}$,” Phys. Rev. D, vol. 103, p. 112004, 2021.
  * [96] M. Ablikim et al., “Study of $J/\psi$ and $\psi(3686)\rightarrow\Sigma(1385)^{0}\bar{\Sigma}(1385)^{0}$ and $\Xi^{0}\bar{\Xi}^{0}$,” Phys. Lett. B, vol. 770, pp. 217–225, 2017.
  * [97] M. Ablikim et al., “Study of $\psi$ decays to the $\Xi^{-}\bar{\Xi}^{+}$ and $\Sigma(1385)^{\mp}\bar{\Sigma}(1385)^{\pm}$ final states,” Phys. Rev. D, vol. 93, no. 7, p. 072003, 2016.
  * [98] P. Chauvat et al., “Test of CP Invariance in $\Lambda^{0}$ Decay,” Phys. Lett. B, vol. 163, pp. 273–276, 1985.
  * [99] M. H. Tixier et al., “Looking at CP Invariance and Quantum Mechanics in $J/\psi\to\Lambda\bar{\Lambda}$ Decay,” Phys. Lett. B, vol. 212, pp. 523–527, 1988.
  * [100] P. D. Barnes et al., “Observables in high statistics measurements of the reaction anti-p p –$>$ anti-Lambda Lambda,” Phys. Rev. C, vol. 54, pp. 1877–1886, 1996.
  * [101] D. G. Ireland, M. Döring, D. I. Glazier, J. Haidenbauer, M. Mai, R. Murray-Smith, and D. Rönchen, “Kaon Photoproduction and the $\Lambda$ Decay Parameter $\alpha_{-}$,” Phys. Rev. Lett., vol. 123, no. 18, p. 182301, 2019.
  * [102] M. Tanabashi et al., “Review of Particle Physics,” Phys. Rev. D, vol. 98, no. 3, p. 030001, 2018.
  * [103] R. Aaij et al., “Measurement of the $\Lambda^{0}_{b}\rightarrow J/\psi\Lambda$ angular distribution and the $\Lambda^{0}_{b}$ polarisation in $pp$ collisions,” JHEP, vol. 06, p. 110, 2020.
  * [104] J. R. Batley et al., “New Precise Measurements of the $\Xi^{0}->\Lambda\gamma$ and $\Xi^{0}->\Sigma^{0}\gamma$ Decay Asymmetries,” Phys. Lett. B, vol. 693, pp. 241–248, 2010.
  * [105] C. Baltay, A. Bridgewater, W. A. Cooper, L. K. Gershwin, M. Habibi, M. Kalelkar, N. Yeh, and A. Gaigalas, “Properties of ${\Xi}^{-}$ and ${\Xi}^{0}$ hyperons,” Phys. Rev. D, vol. 9, pp. 49–62, Jan 1974.
  * [106] P. M. Dauber, J. P. Berge, J. R. Hubbard, D. W. Merrill, and R. A. Muller, “Production and decay of cascade hyperons,” Phys. Rev., vol. 179, pp. 1262–1285, Mar 1969.
  * [107] J. P. Berge, P. Eberhard, J. R. Hubbard, D. W. Merrill, J. Button-Shafer, F. T. Solmitz, and M. L. Stevenson, “Some properties of ${\Xi}^{-}$ and ${\Xi}^{0}$ hyperons produced in ${K}^{-}p$ interactions between 1.05 and 1.7 bev/c,” Phys. Rev., vol. 147, pp. 945–961, Jul 1966.
  * [108] K. B. Luk et al., “Search for direct CP violation in nonleptonic decays of charged Xi and Lambda hyperons,” Phys. Rev. Lett., vol. 85, pp. 4860–4863, 2000.
  * [109] T. Holmstrom et al., “Search for CP violation in charged-Xi and Lambda hyperon decays,” Phys. Rev. Lett., vol. 93, p. 262001, 2004.
  * [110] A. Chakravorty, H. T. Diehl, J. Duryea, G. Guglielmo, K. Heller, P. M. Ho, C. James, K. Johns, D. M. Kaplan, M. J. Longo, K. B. Luk, R. Rameika, H. A. Rubin, S. Teige, G. B. Thomson, and Y. Zou, “Measurement of decay parameters for ${\Xi}^{-}\rightarrow\Lambda{\pi}^{-}$ decay,” Phys. Rev. Lett., vol. 91, p. 031601, Jul 2003.
  * [111] M. Huang et al., “New measurement of Xi- —$>$ Lambda pi- decay parameters,” Phys. Rev. Lett., vol. 93, p. 011802, 2004.
  * [112] D. Aston, R. K. Carnegie, W. Dunwoodie, S. Durkin, P. G. Estabrooks, R. J. Hemingway, A. Honma, D. Hutchinson, W. B. Johnson, P. F. Kunz, T. Lasinski, D. W. G. S. Leith, L. Levinson, R. McKee, B. T. Meadows, A. C. McPherson, W. T. Meyer, G. K. Oakham, B. N. Ratcliff, S. Shapiro, T. Shimomura, S. Suzuki, J. Va’Vra, and S. Williams, “Inclusive production of multistrange hyperons from 11-gev/c ${K}^{\mathrm{-}}$p interactions,” Phys. Rev. D, vol. 32, pp. 2270–2287, Nov 1985.
  * [113] J. R. Bensinger et al., “Upper Limit for the Xi0 $\to\Sigma^{0}\gamma$ Radiative Decay,” Phys. Lett. B, vol. 215, pp. 195–198, 1988.
  * [114] R. L. Cool, G. Giacomelli, E. W. Jenkins, T. F. Kycia, B. A. Leontic, K. K. Li, and J. Teiger, “Measurement of the Magnetic Moment and of the Decay Parameters of the xi- Hyperon,” Phys. Rev. D, vol. 10, p. 792, 1974.
  * [115] R. L. Cool, G. Giacomelli, E. W. Jenkins, T. F. Kycia, B. A. Leontić, K. K. Li, and J. Teiger, “Measurement of the ${\Xi}^{-}$ magnetic moment,” Phys. Rev. Lett., vol. 29, pp. 1630–1634, Dec 1972.
  * [116] B.-L. Huang, J.-S. Zhang, Y.-D. Li, and N. Kaiser, “Meson-baryon scattering to one-loop order in heavy baryon chiral perturbation theory,” Phys. Rev. D, vol. 96, p. 016021, Jul 2017.
  * [117] M. Deutschmann et al., “Spin and Lifetime of the omega- Hyperon,” Phys. Lett. B, vol. 73, pp. 96–98, 1978.
  * [118] M. Baubillier et al., “A Study of the Lifetime and Spin of $\Omega^{-}$ Produced in $K^{-}p$ Interactions at 8.25-GeV/$c$,” Phys. Lett. B, vol. 78, pp. 342–346, 1978.
  * [119] B. Aubert et al., “Measurement of the spin of the Omega- hyperon at BABAR,” Phys. Rev. Lett., vol. 97, p. 112001, 2006.
  * [120] M. Ablikim et al., “Model-Independent Determination of the Spin of the $\Omega^{-}$ and Its Polarization Alignment in $\psi(3686)\to\Omega^{-}\bar{\Omega}^{+}$,” Phys. Rev. Lett., vol. 126, no. 9, p. 092002, 2021.
  * [121] M. Bourquin et al., “MEASUREMENT OF OMEGA- DECAY PROPERTIES IN THE CERN SPS HYPERON BEAM,” Nucl. Phys. B, vol. 241, pp. 1–47, 1984.
  * [122] H. Albrecht et al., “A Measurement of asymmetry in the decay lambda(c)+ —$>$ lambda pi+,” Phys. Lett. B, vol. 274, pp. 239–245, 1992\.
  * [123] P. Avery et al., “Study of the decays lambda(c)+ —$>$ xi0 K+, lambda(c)+ —$>$ sigma+ K+ K- and lambda(c)+ —$>$ xi- K+ pi+,” Phys. Rev. Lett., vol. 71, pp. 2391–2395, 1993.
  * [124] M. Bishai et al., “Measurement of the decay asymmetry parameters in Lambda(c)+ —$>$ Lambda pi+ and Lambda(c)+ —$>$ Sigma+ pi0,” Phys. Lett. B, vol. 350, pp. 256–262, 1995.
  * [125] J. M. Link et al., “Study of the decay asymmetry parameter and CP violation parameter in the Lambda(c)+ —$>$ Lambda pi+ decay,” Phys. Lett. B, vol. 634, pp. 165–172, 2006.
  * [126] M. Ablikim et al., “Measurements of Weak Decay Asymmetries of $\Lambda_{c}^{+}\to pK_{S}^{0}$, $\Lambda\pi^{+}$, $\Sigma^{+}\pi^{0}$, and $\Sigma^{0}\pi^{+}$,” Phys. Rev. D, vol. 100, no. 7, p. 072004, 2019.
  * [127] M. Ablikim et al., “Determination of the $\Lambda_{c}^{+}$ spin via the reaction $e^{+}e^{-}\to\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$,” Phys. Rev. D, vol. 103, no. 9, p. L091101, 2021.
  * [128] “Measurement of branching fractions and decay asymmetry parameters for $\Lambda_{c}^{+}\to\Lambda h^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{0}h^{+}$ ($h=K,\pi$), and search for $CP$ violation in baryon decays,” 8 2022.
  * [129] Y. B. Li et al., “Measurements of the branching fractions of the semileptonic decays $\Xi_{c}^{0}\to\Xi^{-}\ell^{+}\nu_{\ell}$ and the asymmetry parameter of $\Xi_{c}^{0}\to\Xi^{-}\pi^{+}$,” Phys. Rev. Lett., vol. 127, no. 12, p. 121803, 2021.
  * [130] S. Jia et al., “Measurements of branching fractions and asymmetry parameters of $\Xi^{0}_{c}\to\Lambda\bar{K}^{*0}$, $\Xi^{0}_{c}\to\Sigma^{0}\bar{K}^{*0}$, and $\Xi^{0}_{c}\to\Sigma^{+}K^{*-}$ decays at Belle,” JHEP, vol. 06, p. 160, 2021.
  * [131] S. Chan et al., “A Measurement of the decay asymmetry parameters in Xi0(c) —$>$ Xi- pi+,” Phys. Rev. D, vol. 63, p. 111102, 2001\.
  * [132] L. K. Gershwin, M. Alston-Garnjost, R. O. Bangerter, A. Barbaro-Galtieri, T. S. Mast, F. T. Solmitz, and R. D. Tripp, “Asymmetry parameter and branching ratio of ${\Sigma}^{+}\rightarrow p\gamma$,” Phys. Rev., vol. 188, pp. 2077–2080, Dec 1969.
  * [133] M. Foucher, I. F. Albuquerque, N. F. Bondar, R. Carrigan, D. Chen, L. Chengze, P. S. Cooper, A. S. Denisov, A. V. Dobrovolsky, T. Dubbs, A. M. F. Endler, C. O. Escobar, T. Fukun, V. L. Golovtsov, P. A. Goritchev, H. Gottschalk, P. Gouffon, V. T. Grachev, S. Huanzhang, Y. Jie, A. V. Khanzadeev, M. A. Kubantsev, N. P. Kuropatkin, J. Lach, M. Luksys, V. N. Lebedenko, D. Lisheng, J. R. P. Mahon, E. McCliment, A. Morelos, C. Newsom, L. Pengfei, M. C. Pommot Maia, V. M. Samsonov, Z. Shuchen, V. J. Smith, N. K. Terentyev, S. Timm, I. I. Tkatch, L. N. Uvarov, A. A. Vorobyov, Z. Wenheng, Z. Yuanyuan, and L. Yunshan, “Measurement of the asymmetry parameter in the hyperon radiative decay ${\mathrm{\Sigma}}^{+}$$\rightarrow$p$\gamma$,” Phys. Rev. Lett., vol. 68, pp. 3004–3007, May 1992.
  * [134] M. Kobayashi, J. Haba, T. Homma, H. Kawai, K. Miyake, T. S. Nakamura, N. Sasao, and Y. Sugimoto, “New measurement of the asymmetry parameter for the ${\Sigma}^{+}$$\rightarrow$p$\gamma$ decay,” Phys. Rev. Lett., vol. 59, pp. 868–871, Aug 1987.
  * [135] A. Manz, S. Reucroft, R. Settles, G. Wolf, J. Marraffino, C. Roos, J. Waters, and M. Webster, “A new measurement of $\sigma^{+}\to p\gamma$ decay properties,” Physics Letters B, vol. 96, no. 1, pp. 217–222, 1980.
  * [136] A. Lai et al., “Measurement of the $\xi^{0}\to\lambda\gamma$ decay asymmetry and branching fraction,” Physics Letters B, vol. 584, no. 3, pp. 251–259, 2004.
  * [137] A. Alavi-Harati et al., “A Measurement of the branching ratio and asymmetry of the decay Xi0 —$>$ Sigma0 gamma,” Phys. Rev. Lett., vol. 86, pp. 3239–3243, 2001.
  * [138] C. James et al., “Branching ratio and asymmetry for Xi0 —$>$ Lambda gamma,” Phys. Rev. Lett., vol. 64, pp. 843–846, 1990.  [Erratum: Phys.Rev.Lett. 89, 169901 (2002)].
  * [139] S. Teige et al., “Measurement of the Xi0 $\to\Sigma^{0}\gamma$ Branching Ratio and Asymmetry Parameter,” Phys. Rev. Lett., vol. 63, pp. 2717–2720, 1989.  [Erratum: Phys.Rev.Lett. 89, 169902 (2002)].
  * [140] M. Gavela, A. Le Yaouanc, L. Oliver, O. Pène, J. Raynal, and T. Pham, “Parity violating radiative weak decays and the quark model,” Physics Letters B, vol. 101, no. 6, p. 417 – 422, 1981.  Cited by: 80.
  * [141] P. Zenczykowski, “Chiral approach to weak radiative hyperon decays and the $\Xi^{0}\to\Lambda\gamma$ asymmetry,” Phys. Rev. D, vol. 62, p. 014030, 2000.
  * [142] B. Borasoy and B. R. Holstein, “Resonances in radiative hyperon decays,” Phys. Rev. D, vol. 59, p. 054019, 1999.
  * [143] P. Zenczykowski, “Joint description of weak radiative and nonleptonic hyperon decays in broken SU(3),” Phys. Rev. D, vol. 73, p. 076005, 2006.
  * [144] K. D. Larson et al., “Weak radiative decay Lambda —$>$ n gamma and the radiative capture reaction K- p —$>$ Sigma (1385) gamma.,” Phys. Rev. D, vol. 47, pp. 799–810, 1993.
  * [145] P. Zenczykowski, “Revisiting Weak Radiative Decays of Hyperons,” Acta Phys. Polon. B, vol. 51, p. 2111, 2020.
  * [146] M. Ablikim et al., “Measurement of the Absolute Branching Fraction and Decay Asymmetry of $\Lambda$→n$\gamma$,” Phys. Rev. Lett., vol. 129, no. 21, p. 212002, 2022.
  * [147] X.-G. He and G. Valencia, “Constraints on s —$>$ d gamma from radiative hyperon and kaon decays,” Phys. Rev. D, vol. 61, p. 075003, 2000\.
  * [148] J. Tandean, “New physics and short distance s —$>$ d gamma transition in Omega- —$>$ Xi-gamma decay,” Phys. Rev. D, vol. 61, p. 114022, 2000.
  * [149] R.-X. Shi, S.-Y. Li, J.-X. Lu, and L.-S. Geng, “Weak radiative hyperon decays in covariant baryon chiral perturbation theory,” Sci. Bull., vol. 67, pp. 2298–2304, 2022.
  * [150] J. Lindquist et al., “Measurement of the up-down asymmetries in the beta decay of polarized lambda hyperons,” Phys. Rev. Lett., vol. 27, pp. 612–616, 1971.
  * [151] J. Lindquist et al., “Measurement of the Angular Correlation Parameters in the beta Decay of Polarized Lambda Hyperons,” J. Phys. G, vol. 2, p. L211, 1976.
  * [152] T. H. Burnett, W. R. Innes, G. Masek, T. Maung, E. S. Miller, H. Ruderman, J. Smith, and W. Vernon, “A Measurement of the Form-Factor g(1)/f(1) in Lambda beta Decay,” Nuovo Cim. A, vol. 34, pp. 14–20, 1976.
  * [153] K. H. Althoff, R. M. Brown, D. Freytag, K. S. Heard, J. Heintze, R. Mundhenke, H. Rieseberg, V. Soergel, H. Stelzer, and A. Wagner, “Beta decay of polarized lambda hyperons. iii. correlations in the lambda centre-of-mass system and the proton recoil spectrum,” Phys. Lett. B, vol. 43, pp. 237–239, 1973.
  * [154] K. H. Althoff, R. M. Brown, D. Freytag, K. S. Heard, J. Heintze, R. Mundhenke, H. Rieseberg, V. Soergel, H. Stelzer, and A. Wagner, “Beta-decay of polarized lambda hyperons. i. measurement of the decay rate,” Phys. Lett. B, vol. 37, pp. 531–534, 1971.
  * [155] M. Baggett, N. Baggett, F. Eisele, H. Filthuth, H. Frehse, V. Hepp, R. Howard, E. Leitner, and G. Zech, “Measurement of the ratio axial vector to vector coupling constant (g(1)/f(1) in lambda —$>$ p e- anti-nu decay,” Z. Phys., vol. 249, pp. 279–285, 1972.
  * [156] J. Canter, J. Cole, J. Lee-Franzini, R. J. Loveless, and P. Franzini, “Beta decay of the lambda hyperon,” Phys. Rev. Lett., vol. 26, pp. 868–870, 1971.
  * [157] J. E. Maloney and B. Sechi-Zorn, “Measurements of the branching ratio and form of interaction for the beta decay of the lambda hyperon,” Phys. Rev. Lett., vol. 23, pp. 425–427, 1969.
  * [158] J. Wise, D. A. Jensen, M. N. Kreisler, F. Lomanno, R. Poster, M. S. Z. Rabin, K. Raychaudhuri, M. Way, and J. Humphrey, “Precise Measurement of the Ratio of the Axial Vector Coupling to Vector Coupling in $\Lambda^{0}\to pe^{-}\bar{\nu}$,” Phys. Lett. B, vol. 98, p. 123, 1981.  [Erratum: Phys.Lett.B 100, 519 (1981)].
  * [159] M. Bourquin et al., “Measurements of Hyperon Semileptonic Decays at the CERN Super Proton Synchrotron. 2. The $\Lambda\to pe$ Anti-neutrino, Xi- $\to\Lambda e$ Anti-neutrino, and Xi- $\to\Sigma^{0}e$ Anti-neutrino Decay Modes,” Z. Phys. C, vol. 21, p. 1, 1983.
  * [160] J. Dworkin et al., “High statistics measurement of g(a) / g(v) in Lambda —$>$ p + e- + anti-neutrino,” Phys. Rev. D, vol. 41, pp. 780–800, 1990.
  * [161] A. A. Affolder et al., “Observation of the decay Xi0 —$>$ Sigma+ e- anti-neutrino,” Phys. Rev. Lett., vol. 82, pp. 3751–3754, 1999\.
  * [162] E. Abouzaid et al., “Observation of the decay Xi0 —$>$ Sigma+ mu- anti-nu(mu),” Phys. Rev. Lett., vol. 95, p. 081801, 2005.
  * [163] A. Alavi-Harati et al., “First Measurement of Form-Factors of the Decay $\Xi^{0}\to\Sigma^{+}e^{-}\bar{\nu}_{e}$,” Phys. Rev. Lett., vol. 87, p. 132001, 2001.
  * [164] J. R. Batley et al., “Measurement of the branching ratios of the decays Xi0 —$>$ Sigma+ e- anti-nu(e) and anti-Xi0 —$>$ anti-Sigma+ e+ nu(e),” Phys. Lett. B, vol. 645, pp. 36–46, 2007.
  * [165] J. R. Batley et al., “Measurement of the branching ratio of the decay $\Xi^{0}\rightarrow\Sigma^{+}\mu^{-}\bar{\nu}_{\mu}$,” Phys. Lett. B, vol. 720, pp. 105–110, 2013.
  * [166] M. Ablikim et al., “First Measurement of the Absolute Branching Fraction of $\Lambda\to p\mu^{-}\bar{\nu}_{\mu}$,” Phys. Rev. Lett., vol. 127, no. 12, p. 121802, 2021.
  * [167] M. Ablikim et al., “First Direct Measurement of the Absolute Branching Fraction of $\Sigma^{+}\to\Lambda e^{+}\nu_{e}$,” 12 2022.
  * [168] M. Ablikim et al., “Search for the hyperon semileptonic decay $\Xi^{-}\rightarrow\Xi^{0}e^{-}\bar{\nu_{e}}$,” Phys. Rev. D, vol. 104, no. 7, p. 072007, 2021.
  * [169] M. Ablikim et al., “Search for hyperon $\Delta S=\Delta Q$ violating decay $\Xi^{0}\to\Sigma^{-}e^{+}\nu_{e}$,” 8 2022.
  * [170] V. Batozskaya, A. Kupsc, N. Salone, and J. Wiechnik, “Semileptonic decays of spin-entangled baryon-antibaryon pairs,” 2 2023.
  * [171] M. Ablikim et al., “Study of the Semileptonic Decay $\Lambda$c+→$\Lambda$e+$\nu$e,” Phys. Rev. Lett., vol. 129, no. 23, p. 231803, 2022.
  * [172] S. Meinel, “$\Lambda_{c}\to\Lambda l^{+}\nu_{l}$ form factors and decay rates from lattice QCD with physical quark masses,” Phys. Rev. Lett., vol. 118, no. 8, p. 082001, 2017.
  * [173] J. D. Richman and P. R. Burchat, “Leptonic and semileptonic decays of charm and bottom hadrons,” Rev. Mod. Phys., vol. 67, pp. 893–976, 1995.
  * [174] Y. Aoki et al., “FLAG Review 2021,” Eur. Phys. J. C, vol. 82, no. 10, p. 869, 2022.
  * [175] B. Ronne, C. Baglin, J. Six, W. L. Knight, F. R. Stannard, and A. Haatuft, “Branching Ratio for the Muonic Decay Mode of the Lambda Hyperon,” Phys. Lett., vol. 11, pp. 357–360, 1964.
  * [176] V. G. Lind, T. O. Binford, M. L. Good, and D. Stern, “Experimental Investigation of V-A in Leptonic Lambda Decay,” Phys. Rev., vol. 135, p. B1483, 1964.
  * [177] J. Canter, J. Cole, J. Lee-Franzini, R. J. Loveless, and P. Franzini, “Branching ratio for lambda —$>$ p mu nu,” Phys. Rev. Lett., vol. 27, pp. 59–61, 1971.
  * [178] M. Baggett, N. Baggett, F. Eisele, H. Filthuth, H. Frehse, V. Hepp, R. Howard, and E. Leitner, “Measurement of the lambda —$>$ p mu- anti-nu branching ratio,” Z. Phys., vol. 252, pp. 362–370, 1972.
  * [179] G. Bonneau and F. Martin, “Hard photon emission in e+ e- reactions,” Nucl. Phys. B, vol. 27, pp. 381–397, 1971.
  * [180] X. G. He, H. Steger, and G. Valencia, “Status of CP violation in hyperon decays,” Phys. Lett. B, vol. 272, pp. 411–418, 1991.
  * [181] X.-G. He and G. Valencia, “CP violation in Lambda —$>$ p pi- beyond the Standard Model,” Phys. Rev. D, vol. 52, pp. 5257–5268, 1995\.
  * [182] X.-G. He, H. Murayama, S. Pakvasa, and G. Valencia, “CP violation in hyperon decays from supersymmetry,” Phys. Rev. D, vol. 61, p. 071701, 2000.
  * [183] X.-G. He, J. Tandean, and G. Valencia, “Pursuit of cp violation in hyperon decays at e+e- colliders,” Science Bulletin, vol. 67, no. 18, pp. 1840–1843, 2022.
  * [184] N. Salone, P. Adlarson, V. Batozskaya, A. Kupsc, S. Leupold, and J. Tandean, “Study of CP violation in hyperon decays at super-charm-tau factories with a polarized electron beam,” Phys. Rev. D, vol. 105, no. 11, p. 116022, 2022.
  * [185] Y. Ohnishi et al., “Accelerator design at SuperKEKB,” PTEP, vol. 2013, p. 03A011, 2013.
  * [186] G. Barucca et al., “PANDA Phase One,” Eur. Phys. J. A, vol. 57, no. 6, p. 184, 2021.
  * [187] G. Barucca et al., “The potential of $\Lambda$ and $\Xi^{-}$ studies with PANDA at FAIR,” Eur. Phys. J. A, vol. 57, no. 4, p. 154, 2021.
  * [188] H. P. Peng et al., “Physics Potential of a Super tau-Charm Facility,” Snowmass2021-Letter of Interest, 2021.
  * [189] I.A.Koop and A.V.Otboev, “Longitudinal Polarization in Novosibirsk c-tau factory,” in Joint Workshop on Future charm-tau Factory, 2019.
  * [190] A. Renieri, “Possibility of Achieving Very High-Energy Resolution in electron-Positron Storage Rings,” 1975.
  * [191] A. A. Avdienko, G. A. Kornyukhin, I. Y. Protopopov, A. N. Skrinsky, A. B. Temnykh, G. M. Tumaikin, and A. A. Zholents, “THE PROJECT OF MODERNIZATION OF THE VEPP-4 STORAGE RING FOR MONOCHROMATIC EXPERIMENTS IN THE ENERGY RANGE OF PSI AND UPSILON MESONS,” Conf. Proc. C, vol. 830811, pp. 186–189, 1983\.
  * [192] V. I. Telnov, “Monochromatization of $e^{+}e^{-}$ colliders with a large crossing angle,” 8 2020.