Testing of $P$ and $CP$ Symmetries with $e^{+}e^{-}\to
J/\psi\to\Lambda\bar{\Lambda}$

X.G. He1,2,3 and J.P. Ma4,5,6,

1 Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai 200240, China
2Key Laboratory for Particle Astrophysics and Cosmology (MOE) & Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University,
Shanghai 200240, China
3 Department of Physics and National Center for Theoretical Sciences, National
Taiwan University, Taipei 10617
4 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
P.O. Box 2735, Chinese Academy of Sciences, Beijing 100190, China
5 School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
6 School of Physics and Center for High-Energy Physics, Peking University,
Beijing 100871, China

###### Abstract

We propose to test $P$\- and $CP$-symmetries with the process $e^{+}e^{-}\to
J/\psi\to\Lambda\bar{\Lambda}$. The general form of the angular distribution
of the process is derived and several observables for the proposed test are
introduced. With these observables one can distinguish the effects of
$P$-violations in the production of $J/\psi$ from its decay into a pair of
$\Lambda\bar{\Lambda}$. Numerical estimations for proposed asymmetries of
$P$-violation due to $Z$ exchange in SM are given. Our results show that
BESIII has reached the sensitivity to probe these effects. At the proposed
super-tau-charm factory with much enhanced luminosity it will be able to test
the SM predictions with high precision. $CP$ violating effect in
$J/\psi\to\Lambda\bar{\Lambda}$ can also be tested. In particular, BESIII data
already reached the sensitivity to prove $CP$ violating effect due to current
upper limit of electric dipole moment of $\Lambda$. At the proposed super-tau-
charm factory the upper limit can be improved by an order of magnitude.

1\. Introduction

The symmetries of Parity ($P$), Charge conjugation ($C$) and Time-reversal
($T$), are fundamental space-time discrete symmetries. Their conservation and
violation play an important role in understanding the laws of Nature. The
$P$\- and $CP$-violation have been already found in several places and can be
described in the framework of the standard model (SM)[1]. It is important to
further test these symmetries in all possible ways to probe new physics beyond
SM. Several facilities are actively carrying out such precision tests with
large data samples. In this letter we focus on possible tests of $P$\- and
$CP$-symmetry using the resonant production of $J/\psi$ at a $e^{+}e^{-}$
collider which subsequently decays into a pair of $\Lambda\bar{\Lambda}$.

Tests of the discrete symmetries require large data samples in experiment.
There are large data samples of $e^{+}e^{-}\to J/\psi\to\Lambda\bar{\Lambda}$
collected by BESIII at BEPCII. At Super-Tau-Charm Factory (STCF) proposed in
[5, 6] the data samples will be more larger than those collected by BESIII.
Data from these experiments can provide much needed information. This process
has been studied in BESIII experiment, but for testing discrete symmetries
only for $CP$-violation in the decays of the produced $\Lambda$ and
$\bar{\Lambda}$ [2, 3, 4]. There are no tests with the production of $J/\psi$
and its decay into $\Lambda\bar{\Lambda}$. In general the violation of
discrete symmetries in the chain process can also happen in these two places.

It is noted that without observation of polarizations of the final- and
initial state one can not test discrete symmetry with $e^{+}e^{-}\to
J/\psi\to\Lambda\bar{\Lambda}$[7]. Thanks to the weak decays of $\Lambda$ and
$\bar{\Lambda}$, the polarization of the produced $\Lambda$ and
$\bar{\Lambda}$ can be observed[8]. This makes the tests of symmetries
possible in $e^{+}e^{-}\to J/\psi\to\Lambda\bar{\Lambda}$, where the
polarization of the initial state is not observed. $CP$-symmetry test in the
decay of $J/\psi\to\Lambda\bar{\Lambda}$ has been studied in [10]. However,
the results only apply for the unpolarized $J/\psi$. In the process considered
here, $J/\psi$ is polarized. We propose several observables for the symmetry
test without observing the polarization of the initial state. Using these
observables effects due to $P$-violation in $e^{+}e^{-}\to J/\psi$ and in
$J/\psi\to\Lambda\bar{\Lambda}$ can be separately measured. $CP$-violation
appearing in $J/\psi\to\Lambda\bar{\Lambda}$ can also be detected with our
observables. The proposed observables can already be measured with the data
sample at BES. Using polarized beams to study $P$-violation in the process for
measuring Weinberg angle has been studied in [9].

In the following we will first give the most general form of the angular
distribution only based on covariance and discuss the constraints of discrete
symmetries in Sect.2. In Sect.3 we give each component in the angular
distribution with a general parametrization of the amplitude. In Sect.4
Observables and corresponding asymmetries for testing discrete symmetries are
suggested and their results are given. Numerical results and discussions are
provided in Sect.5. Finally we will give our summary in Sect.6.

2\. The General Angular Distributions and Its Constraints from Symmetries

We consider the production of $J/\psi$ through the annihilation of an
$e^{+}e^{-}$ pair and its sequential decay into a $\Lambda\bar{\Lambda}$ pair:

$e^{-}(p_{1})+e^{+}(p_{2})\to
J/\psi\to\Lambda(k_{1},s_{1})+\bar{\Lambda}(k_{2},s_{2}),$ (1)

where the spins of the initial state are averaged. Momenta and spins of
$\Lambda$ and $\bar{\Lambda}$ are indicated in the brackets. In the center of
mass frame (CMF), the momenta are

$p_{1}^{\mu}=(E_{c},{\bm{p}}),\quad p_{2}^{\mu}=(E_{c},-{\bm{p}}),\quad
k_{1}^{\mu}=(k^{0},{\bm{k}}),\quad k_{2}^{\mu}=(k^{0},-{\bm{k}}).$ (2)

In this frame the $J/\psi$ is at rest and $E_{c}$ is the half of the
$J/\psi$-mass $M_{J/\psi}$. We define the spin vector $s_{1}$ of $\Lambda$ as
the spin vector in its rest frame which is obtained through a Lorentz boost
without any spacial rotation from the frame of CMF. Hence, the spin vector has
only nonzero spacial components, i.e., $s_{1}^{\mu}=(0,{\bm{s}}_{1})$. The
spin vector $s_{2}$ of $\bar{\Lambda}$ is defined similarly. For convenience
we introduce three unit vectors and the variable $\omega$:

$\hat{\bm{p}}=\frac{{\bm{p}}}{|{\bm{p}}|},\quad\hat{\bm{k}}=\frac{{\bm{k}}}{|{\bm{k}}|},\quad\hat{\bm{n}}=\frac{{\bm{p}}\times{\bm{k}}}{|{\bm{p}}\times{\bm{k}}|},\quad\omega=\hat{\bm{p}}\cdot\hat{\bm{k}},$
(3)

where the first two vectors are for the momentum direction of $e^{-}$ and
$\Lambda$, respectively. $\hat{\bm{n}}$ is the direction normal to the
production plane.

For the $J/\psi$ decay we can write the decay amplitude ${\mathcal{T}}$ and
the density matrix $R$ for the decay as:

${\mathcal{T}}=\epsilon^{\mu}{\mathcal{A}}_{\mu},\quad
R(\hat{\bm{p}},\hat{\bm{k}},{\bm{s}}_{1},{\bm{s}}_{2})={\mathcal{T}}{\mathcal{T}}^{\dagger}=\rho^{ij}{\mathcal{M}}^{ij},$
(4)

with

${\mathcal{M}}^{ij}={\mathcal{A}}^{i}{\mathcal{A}}^{*j},\quad\rho^{ij}=\epsilon^{i}\epsilon^{*j}.$
(5)

where $\epsilon^{\mu}$ is the polarization vector of $J/\psi$ in its rest
frame with $\epsilon^{0}=0$. $\rho^{ij}$ is the density matrix for the
production of $J/\psi$, ${\mathcal{M}}$ is the density matrix of the decay. It
is clear that ${\mathcal{M}}$ only depend on ${\bm{k}}$ and ${\bm{s}}_{1,2}$.
$\rho^{ij}$ contains the information about the production. In our case that
$J/\psi$ is produced though $e^{+}e^{-}$ annihilation, it only depends on
${\bm{p}}$. With rotational covariance, $\rho$ can be decomposed as

$\rho^{ij}(\hat{\bm{p}})=\frac{1}{3}\delta^{ij}-id_{J}\epsilon^{ijk}\hat{p}^{k}-\frac{c_{J}}{2}\biggr{(}\hat{p}^{i}\hat{p}^{j}-\frac{1}{3}\delta^{ij}\biggr{)},$
(6)

where we have normalized $\rho$ so that ${\rm Tr}\rho=1$. $d_{J}$ and $c_{J}$
are real constants. Parity conservation implies that there is no asymmetric
part in $\rho$, i.e., $d_{J}=0$. $CP$-symmetry gives no constraint. If the
annihilation is into one virtual photon in the first step and the photon is
then converting to $J/\psi$, we have $c_{J}=1$. The density matrix $R$
contains all information about the considered process.

The decomposition of ${\mathcal{M}}^{ij}$ with the three vector $\hat{\bm{k}}$
and ${\bm{s}}_{1,2}$ is complicated. However, the spin-dependence of
${\mathcal{M}}$ or $R$ can be easily found as:

$R\left(\hat{\bm{p}},\hat{\bm{k}},{\bm{s}}_{1},{\bm{s}}_{2}\right)=a(\omega)+{\bm{s}}_{1}\cdot{\bm{B}}_{1}(\hat{\bm{p}},\hat{\bm{k}})+{\bm{s}}_{2}\cdot{\bm{B}}_{2}(\hat{\bm{p}},\hat{\bm{k}})+s_{1}^{i}s_{2}^{j}C^{ij}(\hat{\bm{p}},\hat{\bm{k}}).$
(7)

With rotational covariance ${\bm{B}}_{1,2}$ and $C^{ij}$ can be decomposed as:

$\displaystyle{\bm{B}}_{1}(\hat{\bm{p}},\hat{\bm{k}})$ $\displaystyle=$
$\displaystyle\hat{\bm{p}}b_{1p}(\omega)+\hat{\bm{k}}b_{1k}(\omega)+\hat{\bm{n}}b_{1n}(\omega),\quad{\bm{B}}_{2}(\hat{\bm{p}},\hat{\bm{k}})=\hat{\bm{p}}b_{2p}(\omega)+\hat{\bm{k}}b_{2k}(\omega)+\hat{\bm{n}}b_{2n}(\omega),$
$\displaystyle C^{ij}(\hat{\bm{p}},\hat{\bm{k}})$ $\displaystyle=$
$\displaystyle\delta^{ij}c_{0}(\omega)+\epsilon^{ijk}\biggr{(}\hat{p}^{k}c_{1}(\omega)+\hat{k}^{k}c_{2}(\omega)+\hat{n}^{k}c_{3}(\omega)\biggr{)}$
(8)
$\displaystyle+\left(\hat{p}^{i}\hat{p}^{j}-\frac{1}{3}\delta^{ij}\right)c_{4}(\omega)+\left(\hat{k}^{i}\hat{k}^{j}-\frac{1}{3}\delta^{ij}\right)c_{5}(\omega)+\left(\hat{p}^{i}\hat{k}^{j}+\hat{k}^{i}\hat{p}^{j}-\frac{2}{3}\omega\delta^{ij}\right)c_{6}(\omega)$
$\displaystyle+\left(\hat{p}^{i}\hat{n}^{j}+\hat{n}^{i}\hat{p}^{j}\right)c_{7}(\omega)+\left(\hat{k}^{i}\hat{n}^{j}+\hat{n}^{i}\hat{k}^{j}\right)c_{8}(\omega).$

Without constraints from $P$ and $CP$ symmetries, there are 16 scalar
functions charactering the process.

The spin of $\Lambda$ or $\bar{\Lambda}$ can be measured with its weak decay
$\Lambda\to p+\pi$ or $\bar{\Lambda}\to\bar{p}+\pi$, respectively. It is noted
that the spin of $\Lambda$ or $\bar{\Lambda}$ measured through their decays
are not exactly the same as those in the production. This is due to Larmor
precession in the magnetic field of the detector. The effect of Larmor
precession is small[11]. We will neglect the effect. The differential
distributions of the decays in the rest frame can be written as:

$\frac{d\Gamma_{\Lambda}}{d\Omega_{p}}({\bm{s}}_{1},\hat{\bm{l}}_{p})\propto
1+\alpha{\bm{s}}_{1}\cdot\hat{\bm{l}}_{p},\quad\frac{d\Gamma_{\bar{\Lambda}}}{d\Omega_{\bar{p}}}({\bm{s}}_{2},\hat{\bm{l}}_{\bar{p}})\propto
1-\bar{\alpha}{\bm{s}}_{2}\cdot\hat{\bm{l}}_{\bar{p}},$ (9)

where $\hat{\bm{l}}_{p}$ and $\hat{\bm{l}}_{\bar{p}}$ is the direction of the
momentum of the proton or anti-proton in the rest frame of $\Lambda$ or
$\bar{\Lambda}$, respectively. $\Omega_{p,\bar{p}}$ is the corresponding solid
angle. $\alpha$ and $\bar{\alpha}$ are constants. $CP$-conservation gives that
$\alpha=\bar{\alpha}$. Taking the distributions and $R$ as density matrices,
the spin vector ${\bm{s}}_{1,2}$ should be understood as Pauli matrix
${\bm{\sigma}}_{1,2}$, respectively.

With the above discussion, the general angular distribution for the process
where the produced $\Lambda$ and $\bar{\Lambda}$ decays through weak
interaction sequentially, is given by:

$\displaystyle\frac{d\sigma}{d\Omega_{k}d\Omega_{p}d\Omega_{\bar{p}}}$
$\displaystyle\propto$ $\displaystyle{\mathcal{W}}(\Omega),$
$\displaystyle{\mathcal{W}}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{1}{4}{\rm
Tr}\biggr{(}R(\hat{\bm{p}},\hat{\bm{k}},{\bm{\sigma}}_{1},{\bm{\sigma}}_{2})(1+\alpha{\bm{\sigma}}_{1}\cdot\hat{\bm{l}}_{p})(1-\bar{\alpha}{\bm{\sigma}}_{2}\cdot\hat{\bm{l}}_{\bar{p}})\biggr{)}$
(10) $\displaystyle=$ $\displaystyle
a(\omega)+\alpha\hat{\bm{l}}_{p}\cdot{\bm{B}}_{1}(\hat{\bm{p}},\hat{\bm{k}})-\bar{\alpha}{\hat{\bm{l}}}_{\bar{p}}\cdot{\bm{B}}_{2}(\hat{\bm{p}},\hat{\bm{k}})-\alpha\bar{\alpha}\hat{l}_{p}^{i}\hat{l}_{\bar{p}}^{j}C^{ij}(\hat{\bm{p}},\hat{\bm{k}}),$

where $\Omega_{k}$ is the solid angle of $\hat{\bm{k}}$. We use $\Omega$ to
denote the set of solid angles $\Omega_{k,p,\bar{p}}$ as variables of
${\mathcal{W}}$. With this general form and the decomposition in Eq.(8) one
can derive the general angular distribution in any coordinate system which is
convenient for analyzing experimental data. The angular distribution with the
constraints of discrete symmetries have been given in [9, 12] in a specified
coordinate system.

The general angular distribution is derived only by using rotational
covariance. Discrete symmetries give constraints on it. In the following we
discuss on the constraints for $R$. If $C$-symmetry holds, we have:

$R\left(\hat{\bm{p}},\hat{\bm{k}},{\bm{s}}_{1},{\bm{s}}_{2}\right)=R\left(-\hat{\bm{p}},-\hat{\bm{k}},{\bm{s}}_{2},{\bm{s}}_{1}\right)$
(11)

This implies for the functions in Eq.(8):

$b_{1m}(\omega)=-b_{2m}(\omega),\quad m=k,p.\quad
b_{1n}(\omega)=b_{2n}(\omega),\quad
c_{3}(\omega)=c_{7}(\omega)=c_{8}(\omega)=0.$ (12)

If parity is conserved, we have

$R\left(\hat{\bm{p}},\hat{\bm{k}},{\bm{s}}_{1},{\bm{s}}_{2}\right)=R\left(-\hat{\bm{p}},-\hat{\bm{k}},{\bm{s}}_{1},{\bm{s}}_{2}\right).$
(13)

This indicates that the following functions must be zero:

$b_{1p}(\omega)=b_{2p}(\omega)=b_{1k}(\omega)=b_{2k}(\omega)=c_{1}(\omega)=c_{2}(\omega)=c_{7}(\omega)=c_{8}(\omega)=0.$
(14)

Therefore, if $C$\- and $P$ are conserved, $R$ is characterized only by 6
functions of $\omega$ instead of 16 in Eq.(8). If CP-symmetry helds, we have
the following relations:

$\displaystyle b_{1m}(\omega)=b_{2m}(\omega),\quad m=p,k,n.\quad
c_{i}(\omega)=0,\quad i=1,2,3.$ (15)

The symmetry of time-reversal does not give constrains as strong as those
discussed in the above. It only indicates that the difference

$R\left(\hat{\bm{p}},\hat{\bm{k}},{\bm{s}}_{1},{\bm{s}}_{2}\right)-R\left(-\hat{\bm{p}},-\hat{\bm{k}},-{\bm{s}}_{1},-{\bm{s}}_{2}\right)$
(16)

is proportional to the interference between absorptive- and dispersive part of
the amplitude. The difference gives the so-called $T$-odd effects. E.g., it is
easy to find that nonzero $b_{1n,2n}$ is from the interference and results in
that the $\Lambda$ is polarized along the $\hat{\bm{n}}$-direction as observed
in experiment.

With the constrains given in the above, tests of discrete symmetries of $C$,
$P$ and $CP$ are to check the relations in Eqs.(12), (14) and (15),
respectively.

3\. The General Form of the Decay Amplitude and Results for $R$

The general form of the amplitude of $J/\psi\to\Lambda\bar{\Lambda}$ can be
written as:

$\displaystyle{\mathcal{A}}^{\mu}=\bar{u}(k_{1})\biggr{[}\gamma^{\mu}F_{V}+\frac{i}{2m_{\Lambda}}\sigma^{\mu\nu}q_{\nu}H_{\sigma}+\gamma^{\mu}\gamma_{5}F_{A}+\sigma^{\mu\nu}q_{\nu}\gamma_{5}H_{T}\biggr{]}v(k_{2}),$
(17)

where $F_{V},F_{A},H_{\sigma}$ and $H_{T}$ are in general complex numbers, or
form factors at fixed $q^{2}$ with $q^{2}=M^{2}_{J/\psi}$. $m_{\Lambda}$ is
the mass of $\Lambda$. $P$ conservation implies that $F_{A}=0$. If $CP$ is
conserved, we have $H_{T}=0$. Since it is expected that $F_{A}$ and $H_{T}$
will be much smaller than other two constants in the amplitude, we will only
take the interference effect from the last two terms in Eq.(17) into account.
In calculating of $R$ we need the density matrix $\rho$ in Eq.(6). We assume
that the dominant contribution for the production of $J/\psi$ is from one-
photon exchange. With this assumption we take $c_{J}$ as $1$.

In Eq.(17) the form factor $F_{V}$ or $H_{\sigma}$ is the analogy to the
Dirac- and Pauli form factor of a bayron, respectively. In [12] different form
factors are introduced. We find that with the form factors in [12] the results
are simpler than those expressed with $F_{V}$ and $H_{\sigma}$. Hence, we
introduce two form factors $G_{1,2}$ which are related to $F_{V}$ and
$H_{\sigma}$ as:

$G_{1}=F_{V}+H_{\sigma},\quad
G_{2}=G_{1}-\frac{(k_{1}-k_{2})^{2}}{2m_{\Lambda}^{2}}H_{\sigma},$ (18)

where $G_{1}(G_{2})$ is proportional to $G_{M}^{\psi}(G_{E}^{\psi})$ defined
in [12]. It is noted that $G_{1}(G_{2})$ is not an analogy to Sachs form
factors.

The $P$\- and $C$\- conserving parts of $R$ is given by the following nonzero
functions:

$\displaystyle a(\omega)$ $\displaystyle=$ $\displaystyle
E_{c}^{2}\biggr{[}|G_{1}|^{2}+y_{m}^{2}|G_{2}|^{2}+\omega^{2}(|G_{1}|^{2}-y_{m}^{2}|G_{2}|^{2})\biggr{]},\quad$
$\displaystyle b_{1n}(\omega)$ $\displaystyle=$ $\displaystyle
b_{2n}(\omega)=2|\hat{\bm{p}}\times\hat{\bm{k}}|E_{c}^{2}\omega y_{m}{\rm
Im}(G_{1}G_{2}^{*}),\quad c_{0}(\omega)=\frac{1}{3}a(\omega),\quad
c_{4}(\omega)=2E_{c}^{2}|G_{1}|^{2},$ $\displaystyle c_{5}(\omega)$
$\displaystyle=$ $\displaystyle
2E_{c}^{2}\biggr{[}|G_{1}|^{2}-y_{m}^{2}|G_{2}|^{2}+\omega^{2}|G_{1}-y_{m}G_{2}|^{2}\biggr{]},\quad
c_{6}(\omega)=-2E_{c}^{2}\omega\biggr{[}|G_{1}|^{2}-y_{m}{\rm
Re}(G_{1}^{*}G_{2})\biggr{]},$ (19)

with $y_{m}=m_{\Lambda}/E_{c}$ and $E_{c}=M_{J/\psi}/2$.

The $P$\- and $CP$\- violating part of $R$ is given by the following nonzero
functions:

$\displaystyle b_{1p}(\omega)$ $\displaystyle=$ $\displaystyle
2E_{c}^{2}\biggr{(}-2\omega\beta E_{c}{\rm Im}(H_{T}G_{1}^{*})+\omega\beta
y_{m}{\rm Re}(F_{A}G_{2}^{*})+2y_{m}d_{J}{\rm Re}(G_{1}G_{2}^{*})\biggr{)},$
$\displaystyle b_{1k}(\omega)$ $\displaystyle=$ $\displaystyle
2E_{c}^{2}\biggr{(}-2\beta E_{c}{\rm
Im}\biggr{[}H_{T}(\omega^{2}G_{1}+(1-\omega^{2})y_{m}G_{2})^{*}\biggr{]}+\beta{\rm
Re}\biggr{[}F_{A}((1+\omega^{2})G_{1}-\omega^{2}y_{m}G_{2})^{*}\biggr{]}$
$\displaystyle+2\omega d_{J}\biggr{(}|G_{1}|^{2}+y_{m}{\rm
Re}(G_{1}G_{2}^{*})\biggr{)}\biggr{)},$ $\displaystyle c_{1}(\omega)$
$\displaystyle=$ $\displaystyle-4\omega\beta E_{c}^{3}{\rm
Re}(H_{T}G_{1}^{*}),\quad c_{2}(\omega)=4\beta E_{c}^{3}{\rm
Re}\biggr{[}H_{T}(\omega^{2}G_{1}+(1-\omega^{2})y_{m}G_{2})^{*}\biggr{]},$
$\displaystyle c_{7}(\omega)$ $\displaystyle=$ $\displaystyle-2\beta
E_{c}^{2}|\hat{\bm{p}}\times\hat{\bm{k}}|{\rm
Im}\biggr{(}F_{A}G_{1}^{*}\biggr{)},$ $\displaystyle c_{8}(\omega)$
$\displaystyle=$ $\displaystyle
2E_{c}^{2}|\hat{\bm{p}}\times\hat{\bm{k}}|\biggr{(}\beta\omega E_{c}^{2}{\rm
Im}\biggr{[}F_{A}(G_{1}-y_{m}G_{2})^{*}\biggr{]}+2y_{m}d_{J}{\rm
Im}(G_{1}^{*}G_{2})\biggr{)},$ (20)

with $\beta=\sqrt{1-y_{m}^{2}}$. $b_{2p}(\omega)$ and $b_{2k}(\omega)$ are
obtained from $b_{1p}(\omega)$ and $b_{1k}(\omega)$ by replacing $H_{T}$ with
$-H_{T}$, respectively. It is interesting to note that $c_{3}(\omega)$ is
zero. Therefore, in the general form of the angular distribution one can
discard the contribution from $c_{3}(\omega)$. The results in this and the
last section can be extended to the case that the production is in continuum
regions.

4\. Integrated Observables and Asymmetries

The various functions of $\omega$ charactering the general angular
distribution in Eqs.(8) and (10) can be projected out by studying the angular
distributions of $\hat{\bm{l}}_{p}$ and $\hat{\bm{l}}_{\bar{p}}$ in the space
spanned by $\hat{\bm{p}}$, $\hat{\bm{k}}$ and $\hat{\bm{n}}$. Because of the
structure of $\rho$ in Eq.(6), these functions of $\omega$ can only be one of
the three forms which are a constant, a constant multiplied with $\omega$ or
$\omega^{2}$. Since the effects of symmetry-violation are expected to be
small, it will be convenient to use integrated observables or asymmetries to
directly measure the effects.

The integrated observables can be built with the four unit vectors
$\hat{\bm{l}}_{p}$, $\hat{\bm{l}}_{\bar{p}}$, $\hat{\bm{p}}$ and
$\hat{\bm{k}}$. For any observable ${\mathcal{O}}$, its average which can be
measured, is given by:

$\langle{\mathcal{O}}\rangle=\frac{1}{{\mathcal{N}}}\int\frac{d\Omega_{k}d\Omega_{p}d\Omega_{\bar{p}}}{(4\pi)^{3}}{\mathcal{O}}{\mathcal{W}}(\Omega),$
(21)

with the normalization factor ${\mathcal{N}}$:

$\quad{\mathcal{N}}=\int\frac{d\Omega_{k}d\Omega_{p}d\Omega_{\bar{p}}}{(4\pi)^{3}}{\mathcal{W}}(\Omega)=\frac{1}{3}E_{c}^{2}\biggr{[}4(|G_{1}|^{2}+y_{m}^{2}|G_{2}|^{2})+(|G_{1}|^{2}-y_{m}^{2}|G_{2}|^{2})\biggr{]}.$
(22)

In the following, we will only consider these observables whose nonzero values
indicate $P$\- or $CP$\- violation.

We first consider a set of observables in which only $\hat{\bm{l}}_{p}$ is
involved. These observables and their results are:

$\displaystyle\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{p}}\rangle$
$\displaystyle=$
$\displaystyle\frac{4\alpha}{9{\mathcal{N}}}E_{c}^{2}d_{J}\biggr{(}4y_{m}{\rm
Re}(G_{1}G_{2}^{*})+|G_{1}|^{2}\biggr{)},$
$\displaystyle\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{k}}\rangle$
$\displaystyle=$
$\displaystyle\frac{4\alpha\beta}{9{\mathcal{N}}}E_{c}^{2}\biggr{[}2{\rm
Re}\biggr{(}F_{A}G_{1}^{*}\biggr{)}-E_{c}{\rm
Im}\biggr{(}H_{T}G_{1}^{*}+y_{m}H_{T}G_{2}^{*}\biggr{)}\biggr{]},$ (23)

These observables receive contributions only from $b_{1p}(\omega)$ and
$b_{1k}(\omega)$. If any of the above observables is nonzero, it implies $P$
violation. It is noted that the first observable is only sensitive to the
parity-violation in the $J/\psi$ production. It is proportional to $d_{J}$.
The second observable is only sensitive to the parity violation in the decay
of $J/\psi\to\Lambda\bar{\Lambda}$. Therefore, using these two observables one
can distinguish where parity is violated, if the effect of parity violation is
observed.

Another set of observables we consider involve both of $\hat{\bm{l}}_{p}$ and
$\hat{\bm{l}}_{\bar{p}}$. This type of observables directly measures the
quantum entanglement of spins in the process, or the matrix $C^{ij}$ in
Eq.(8). We introduce the following observables as:

$\displaystyle\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{k}}\hat{\bm{l}}_{\bar{p}}\cdot\hat{\bm{n}}+\hat{\bm{l}}_{p}\cdot\hat{\bm{n}}\hat{\bm{l}}_{\bar{p}}\cdot\hat{k}\rangle$
$\displaystyle=$
$\displaystyle-\frac{2\alpha\bar{\alpha}\pi}{9{\mathcal{N}}}y_{m}E_{c}^{2}d_{J}{\rm
Im}(G_{1}^{*}G_{2}),$
$\displaystyle\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{p}}\hat{\bm{l}}_{\bar{p}}\cdot\hat{\bm{n}}+\hat{\bm{l}}_{p}\cdot\hat{\bm{n}}\hat{\bm{l}}_{\bar{p}}\cdot\hat{\bm{p}}\rangle$
$\displaystyle=$
$\displaystyle\frac{\alpha\bar{\alpha}\pi}{36{\mathcal{N}}}E_{c}^{2}\beta{\rm
Im}\biggr{[}F_{A}\biggr{(}3G_{1}^{*}+y_{m}G_{2}^{*}\biggr{)}\biggr{]}.$ (24)

These two observables only receive contributions from $c_{7,8}(\omega)$. The
first observable measures only the $P$ violation in the production of
$J/\psi$, while the second measures the $P$ violation in the decay.

The observables discussed in the above can be used for testing $P$\- symmetry.
To test CP symmetry one can use the second observable in Eq.(23) and that of
$\bar{\Lambda}$ to check the following sum:

$\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{k}}\rangle+\langle\hat{\bm{l}}_{\bar{p}}\cdot\hat{\bm{k}}\rangle=-\frac{8\alpha\beta}{9{\mathcal{N}}}E_{c}^{3}{\rm
Im}\biggr{(}H_{T}G_{1}^{*}+y_{m}H_{T}G_{2}^{*}\biggr{)}+{\mathcal{O}}(\bar{\alpha}-\alpha).$
(25)

If $CP$ is conserved in in the decay of $\Lambda$ and $\bar{\Lambda}$, i.e.,
$\alpha=\bar{\alpha}$, $CP$-violation in the sum is only from $J/\psi$ decays.
The following observable will only be sensitive to $CP$-violation in the decay
of $J/\psi\to\Lambda\bar{\Lambda}$:

$\langle(\hat{\bm{l}}_{p}\times\hat{\bm{l}}_{\bar{p}})\cdot\hat{\bm{k}}\rangle=-\frac{8\alpha\bar{\alpha}}{27{\mathcal{N}}}\beta
y_{m}E_{c}^{3}{\rm Re}(H_{T}G_{2}^{*}).$ (26)

This observable only receives contribution from $c_{1,2}(\omega)$. It is noted
that
$\langle(\hat{\bm{l}}_{p}\times\hat{\bm{l}}_{\bar{p}})\cdot\hat{\bm{p}}\rangle=0$
after integration over $\Omega_{k}$.

One can use any observables in Eqs.(23, 24, 25,26) for probing $P$\- or $CP$\-
violating effects. For these observable ${\mathcal{O}}$ one can always define
corresponding asymmetries in events as:

${\mathcal{A}}({\mathcal{O}})=\frac{{\mathcal{N}}_{\rm
event}({\mathcal{O}}>0)-{\mathcal{N}}_{\rm
event}({\mathcal{O}}<0)}{{\mathcal{N}}_{\rm
event}({\mathcal{O}}>0)+{\mathcal{N}}_{\rm
event}({\mathcal{O}}<0)}=\frac{1}{{\mathcal{N}}}\int\frac{d\Omega_{k}d\Omega_{p}d\Omega_{\bar{p}}}{(4\pi)^{3}}\biggr{(}\theta({\mathcal{O}})-\theta(-{\mathcal{O}})\biggr{)}{\mathcal{W}}(\Omega).$
(27)

We introduce the following asymmetries and give their results:

$\displaystyle\mathcal{A}_{d_{J}}={\mathcal{A}}(\hat{\bm{l}}_{p}\cdot\hat{\bm{p}})=\frac{3}{2}\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{p}}\rangle,\quad\mathcal{A}_{F_{A}}={\mathcal{A}}(\hat{\bm{l}}_{p}\cdot\hat{\bm{k}}-\hat{\bm{l}}_{\bar{p}}\cdot\hat{\bm{k}})=\frac{3}{2}\langle\hat{\bm{l}}_{p}\cdot\hat{\bm{k}}-\hat{\bm{l}}_{\bar{p}}\cdot\hat{\bm{k}}\rangle,$
$\displaystyle\mathcal{A}_{H_{T}}={\mathcal{A}}((\hat{\bm{l}}_{p}\times\hat{\bm{l}}_{\bar{p}})\cdot\hat{k})=\frac{3}{2}\langle(\hat{\bm{l}}_{p}\times\hat{\bm{l}}_{\bar{p}})\cdot\hat{\bm{k}}\rangle.$
(28)

The statistical error of any asymmetry is given by $1/\sqrt{N_{event}}$ with
$N_{event}$ as the number of events.

5\. Numerical Results and Discussion

The four form factors $G_{1}$, $G_{2}$, $F_{A}$ and $H_{T}$ in the decay
introduced in Eq.(17) are in general complex. One of them can be taken as real
and positive because the overall phase has no effect. We will give our
numerical results under the assumption that $G_{1}$ is real and positive, and
$F_{A}$ and $H_{T}$ are real. With this assumption we have:

$\displaystyle\mathcal{A}_{d_{J}}\approx
0.60d_{J},\quad\mathcal{A}_{F_{A}}\approx
606F_{A},\quad\mathcal{A}_{H_{T}}\approx-46(m_{\Lambda}H_{T}).$ (29)

In the above we have used the central values of $G_{1}\approx 1.14\times
10^{-3}$, $|G_{2}/G_{1}|^{2}\approx 0.686$, the phase difference
$\Delta\Phi=\Phi_{G_{2}}-\Phi_{G_{1}}=0.752$ and the averaged $\Lambda$
polarization $\alpha=0.742$ extracted from BESIII data [2].

Non-zero values of $d_{J}$ or $F_{A}$ imply $P$ violation, and a non-zero
$H_{T}$ implies $CP$ violation. therefore experimental measurements of
${\mathcal{A}}_{d_{J}}$, ${\mathcal{A}}_{F_{A}}$ and ${\mathcal{A}}_{H_{T}}$
will provide new information about $P$\- and $CP$\- symmetries at the resonant
production of $e^{+}e^{-}\to J/\psi$ followed by the subsequent decay of
$J/\psi\to\Lambda\bar{\Lambda}$. Since nonperturbative effects are dominant in
the process, it is in general difficult to estimate the contributions to these
parameters from SM or new physics beyond SM, where interactions violate $P$
and $CP$ are given at fundamental quark level. E.g., if these symmetries in
the formation of $J/\psi$ from a $c\bar{c}$-pair are violated, the produced
$J/\psi$ is no longer a $P$\- or $CP$\- eigenstate, but a state mixed with
states of different quantum numbers. The effects due to the mixed state are
contained in $F_{A}$ and $H_{T}$. It is noted that $d_{J}$ is not affected by
symmetry-violations in the formation. The leading contribution to $d_{J}$ in
SM comes from $Z$-boson exchange, where the $e^{+}e^{-}$ pair annihilates into
a virtual $Z$ and the $Z$ decays into a $c\bar{c}$ pair. We obtain:

$d_{J}=\frac{3-8\sin^{2}\theta_{W}}{32\cos^{2}\theta_{W}\sin^{2}\theta_{W}}\frac{M^{2}_{J/\psi}}{M_{Z}^{2}}\approx
2.53\times 10^{-4},$ (30)

where the on-shell scheme value[1] of $0.223$ for $\sin^{2}\theta_{W}$ is
used.

Similarly, $F_{A}$ in SM also receives a nonzero contributions from $Z$-boson
exchange in SM, where $J/\psi$ decays into $\Lambda\bar{\Lambda}$ through the
$Z$-boson exchange between $c\bar{c}$ and light quark pairs $q\bar{q}$. Its
contribution to the decay amplitude can be written:

$\displaystyle{\mathcal{T}}_{Z}=-{g^{2}g_{V}^{c}\over
4m^{2}_{Z}\cos^{2}\theta_{W}}\langle
0|\bar{c}\gamma^{\mu}c|J/\psi\rangle\langle\bar{\Lambda}\Lambda|g_{A}^{q}\bar{q}\gamma_{\mu}\gamma_{5}q|0\rangle\;.$
(31)

The matrix element of the charm quark current is given by $\langle
0|\bar{c}\gamma_{\mu}c|J/\psi>=g_{V}\epsilon_{\mu}$ with
$g_{V}=1.25GeV^{2}$[10]. The sum over light flavors $q=(u,d,s)$ is implied
with $g^{c}_{V}=1/2-4\sin^{2}\theta_{W}/3$ and
$g^{d}_{A}=g^{s}_{A}=-g^{u}_{V}=-1/2$. To give an estimate of
${\mathcal{T}}_{Z}$ or $F_{A}$ we relate the matrix element
$\langle\bar{\Lambda}\Lambda|\bar{q}\gamma_{\mu}\gamma_{5}q|0\rangle$ to
$\langle\bar{\Lambda}|\bar{q}\gamma_{\mu}\gamma_{5}q|\Lambda\rangle$ by using
crossing symmetry. Neglecting the momentum difference between $\Lambda$ and
$\bar{\Lambda}$, the sum
$\langle\bar{\Lambda}|g_{A}^{q}\bar{q}\gamma_{\mu}\gamma_{5}q|\Lambda\rangle$
is then given by $-D/3\bar{\Lambda}\gamma_{\mu}\gamma_{5}\Lambda$ with
$D=0.80$ in the light quark flavor $SU(3)$ limit[13]. Under the approximation
we have the estimation:

$\displaystyle F_{A}\approx-{1\over 6}Dg_{V}{g^{2}\over
4\cos^{2}\theta_{W}}{(1-8\sin^{2}\theta_{W}/3)\over
m^{2}_{Z}}\approx-1.07\times 10^{-6}\;.$ (32)

Unlike $d_{J}$, there are other possible contributions to $F_{A}$ which we
neglected in the above estimation.

The $CP$-violating $H_{T}$ can only be generated in SM at three loops or
higher and is extremely small. When going beyond SM, a sizable $H_{T}$ can be
generated. For example, a non-zero $\Lambda$ electric dipole moment
$d_{\Lambda}$ induces a non-zero $H_{T}$ which has been calculated in [10]:

$\displaystyle H_{T}={2e\over 3m^{2}_{J/\psi}}g_{V}d_{\Lambda}\;.$ (33)

It should be kept in mind that form factors have a $q^{2}$-dependence in
general. In our case $q^{2}$ is given by $q=M_{J/\psi}$. We neglect this
dependence. Then the form factor $H_{T}$ is in our approximation related to
$d_{\Lambda}$ which is in fact defined at $q^{2}=0$. The best experiment upper
bound of $d_{\Lambda}$ is[1] $1.5\times 10^{-16}$ ecm at the 95% C.L.. Taking
$d_{\Lambda}$ as the source for $H_{T}$, we obtain the upper bound for $H_{T}$
to be

$\displaystyle m_{\Lambda}H_{T}=2.23\times 10^{-4}\;.$ (34)

Using the above estimated $d_{J}$, $F_{A}$ in the SM and the upper bound of
$H_{T}$, we obtain the following theoretical predictions:

$\displaystyle\mathcal{A}_{d_{J}}\approx 1.52\times
10^{-4}\;,\;\;\mathcal{A}_{F_{A}}\approx-6.51\times
10^{-4}\;,\;\;\mathcal{A}_{H_{T}}\leq-1.02\times 10^{-2}\;.$ (35)

It should noted that the last two numerical results should be taken as order
of magnitude estimates, because they are obtained under the assumption of
relative phases of form factors and approximations mentioned before in this
section.

For an asymmetry ${\mathcal{A}}$ its statistical error or sensitivity is given
by $\delta{\mathcal{A}}=1/\sqrt{{\mathcal{N}}_{\rm event}}$, where
${\mathcal{N}}_{\rm event}$ is the number of events used to measure
${\mathcal{A}}$. In experiment there are other sources of errors beside
statistical error. It is clear that an asymmetry can not be measured if its
value is smaller than its statistical error. With a given event number, one
can estimate how well these asymmetries in the above or the parameters in the
asymmetries in Eq.(29) can be measured. Currently, BESIII has collected
$4.6\times 10^{6}$ events of the studied process[2] which implies a statistic
sensitivity of about $4.6\times 10^{-4}$, the data can already be used to
probe $P$-violation due to $F_{A}$ and to improve the bound on $d_{\Lambda}$
by a factor of a few. At the proposed STCF[5, 6], where the proposed
luminosity will be enhanced to $0.5\times 10^{35}cm^{-2}s^{-1}$, $3.4\times
10^{12}$ $J/\psi$ with one year’s running can be produced. It is expected that
there will be about $6.4\times 10^{9}$ $e^{+}e^{-}\to
J/\psi\to\Lambda\bar{\Lambda}$ data sample for analysis. The statistical error
of an asymmetry will be reduced to $1.2\times 10^{-5}$. The $P$-violating
effects due to non-zero $d_{J}$ and $F_{A}$ can be detected to a high
precision, and the bound on $\Lambda$’s edm will be improved by an order of
magnitude or better.

When going beyond the SM, the size of $d_{J}$ and $F_{A}$ may be larger. For
example, if there is a leptoquark $S_{1}:(\bar{3},3)(-5/6)$ couple to e and c,
$\lambda_{ec}\bar{L}^{c}_{L}Q_{L}S_{1}$, a new ontribution to $d_{J}$ will be
generated. The strongest constrain on the coupling $\lambda_{ec}$ come from
electron $a_{e}=(g_{e}-2)/2$ by exchange $c$ and $S_{1}$ in the loop. Using
$\Delta a_{e}=a^{exp}_{e}-a^{SM}_{e}=(48\pm 30)\times 10^{-14}$ from data[14],
we find that within the allowed error range, $\Delta d_{J}$ can be an order of
magnitude larger than the SM predictions. Exchange of a di-quark scalar
$S_{D}:(3,1)(-4/3)$ may enhance $F_{A}$. $S_{D}$ can have Yukawa coupling to
quarks, $\lambda_{uc}\bar{u}^{c}_{R}c_{R}S_{D}$. Exchange $S_{D}$ at tree
level will produce an effective $F_{A}\sim 1.4\times
10^{-6}|\lambda_{uc}|^{2}\left({100\mbox{GeV}\over m_{S_{D}}}\right)^{2}$. If
the coupling $\lambda_{uc}(100\mbox{GeV}/m_{S_{D}})$ is not too much smaller
than 1, there may be some enhancement compared with SM value. This may be
possible if $\lambda_{uc}$ is the only non-zero coupling and close to its
unitarity bound of $4\pi$ or so.

We have only considered the asymmetries for testing discrete symmetries in the
above. For an asymmetry the statistical error is simply determined by number
of events. One can also consider the corresponding observables introduced in
the last section. The statistical error of an ${\mathcal{O}}$ is given by:

$\delta\langle{\mathcal{O}}\rangle=\sqrt{\frac{\langle{\mathcal{O}}^{2}\rangle-\langle{\mathcal{O}}\rangle^{2}}{{\mathcal{N}}_{\rm
event}}}$ (36)

where ${\mathcal{N}}_{\rm event}$ is the number of events. We have also
considered these integrated observables. Their statistical sensitivities are
roughly at the same orders as the asymmetries discussed in the above. However,
from the integrated observables it is possible to enhance statistical
sensitivities through constructing the so-called observables so that the value
of $\delta{\langle{\mathcal{O}}\rangle}/\langle{\mathcal{O}}\rangle$ is
minimized[15]. To perform such a study it is better to include errors of other
sources to have a realistic estimate of sensitivities.

6\. Summary

We have studied how to test discrete $P$\- and $CP$\- space-time symmetries in
the process $e^{+}e^{-}\to J/\psi\to\Lambda\bar{\Lambda}$, where the
polarization of the initial state is unobserved and that of $\Lambda$ and
$\bar{\Lambda}$ is measured through their weak decays. The general angular
distribution is derived for the chain process. The constraints on the
distribution from $C$-, $P$\- and $CP$-symmetry are given. From a general
parametrization of the decay amplitude of $J/\psi$, we derive each component
in the distribution. We find that effects due to $P$-violation in the
$J/\psi$-production through $e^{+}e^{-}$ annihilation and $J/\psi$ decays into
$\Lambda\bar{\Lambda}$ can be measured separately with integrated observables
proposed in this work. The effects in $P$\- violating observables are
estimated by considering $Z$-boson exchange in SM. $CP$-violating observables
are predicted in term of electric dipole moment of $\Lambda$. Our results show
that BESIII has already reached the sensitivity to probe the estimated
$P$-violating effects. At the proposed STCF’s they can be probed with high
precision. $CP$\- violating effect in $J/\psi\to\Lambda\bar{\Lambda}$ can also
be tested. In particular, BESIII data already reached the level to improve
$CP$\- the current upper limit of $\Lambda$’s electric dipole moment and the
limit can be improved by an order of magnitude at the proposed STCF. We urge
our experimental colleagues at BESIII to carry out measurements of observables
proposed here.

Acknowledgments

We thank Dr. H.B. Li for very useful discussions. The work of J.P. Ma was
supported by National Natural Science Foundation of P.R. China(No.12075299,
11821505, 11847612 and 11935017) and by the Strategic Priority Research
Program of Chinese Academy of Sciences, Grant No. XDB34000000. The work of
X.G. He was supported by the Fundamental Research Funds for the Central
Universities, by National Natural Science Foundation of P.R.
China(No.12090064, 11735010, and 11985149) and by MOST 109–2112-M-002–017-MY3.

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