# Fundamental Tests of P and CP Symmetries
Using Octet Baryons at the $J/\psi$ Threshold

Yong Du<EMAIL_ADDRESS>Xiao-Gang He<EMAIL_ADDRESS>Tsung-Dao Lee
Institute, Shanghai Jiao Tong University, Shanghai 200240, China  Key
Laboratory for Particle Astrophysics and Cosmology (MOE) & Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University,
Shanghai 200240, China  Jian-Ping Ma<EMAIL_ADDRESS>CAS Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, P.O. Box 2735, Chinese
Academy of Sciences, Beijing 100190, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  School of
Physics and Center for High-Energy Physics, Peking University, Beijing 100871,
China  Xin-Yu Du<EMAIL_ADDRESS>Tsung-Dao Lee Institute, Shanghai Jiao
Tong University, Shanghai 200240, China  Key Laboratory for Particle
Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China

###### Abstract

We investigate tests of the parity P and the combined parity and charge-
conjugate CP symmetries from differential angular distributions of $J/\psi$
decaying into the lowest-lying baryon pairs at BESIII and the next-generation
super tau-charm facilities (STCFs). Large corrections from $Z$ and $W$
exchange induced parity violating effects are found for $J/\psi$ decays with
large logarithms resummed up to $\mathcal{O}(\alpha_{s})$. The parity-
violating asymmetries on the production and the decay sides of $J/\psi$ are
both found to be of $\mathcal{O}(10^{-4})$, thus barely observable with the 10
billion $J/\psi$ events currently collected at BESIII. Nevertheless, these
asymmetries utilizing the current BESIII data already permit a determination
of the weak mixing angle with an absolute uncertainty $\delta s_{w}^{2}\approx
0.08$, corresponding to the first determination of $s_{w}^{2}$ at the $J/\psi$
threshold. While limited by statistics currently, STCFs are estimated to
improve this precision by a factor of $\sim\,20$ to $\delta s_{w}^{2}\approx
0.004$ within one year based on luminosity rescaling. We also obtain the 95%
confidence level upper bounds on the electric dipole moments of the octet
baryons, which are of $\mathcal{O}(10^{-18})\,e{\rm\,cm}$ for BESIII and
$\mathcal{O}(10^{-19})\,e{\rm\,cm}$ for STCFs. These bounds are improved by
two to three orders of magnitude in comparison with the only existing one on
$\Lambda$ from Fermilab. The method discussed in this work also paves a way
for a first and direct measurement of the $\Xi$ and $\Sigma$ electric dipole
moments.

##  I Introduction

Symmetries, along with the tests in terms of their violation and/or
conservation at different energies and processes, have played an essential
role in understanding the laws of nature. Starting with the pioneering work by
Lee and Yang Lee and Yang (1956) in 1956, the conservation of parity P, in
beta, hyperon and meson decays was questioned, whose violation was confirmed
one year later by the now widely known Wu experiment using polarized Co60 Wu
et al. (1957) and also soon confirmed in pion and muon decays Garwin et al.
(1957); Friedman and Telegdi (1957). The breakthrough in this regard, on the
one hand, opened a completely new chapter in physics, and on the other hand,
has been driving physicists in questioning the conservation of the other
fundamental symmetries in physics such as the time reversal and the charge-
conjugate symmetries Lee et al. (1957).

The dedicated efforts in this regard had led to the first observation of the
violation of the combined symmetry of charge-conjugate and parity CP at
Brookhaven Christenson et al. (1964) in 1964, through the 2$\pi$ decay mode of
neutral Kaons. Subsequent studies at SLAC from the Babar Aubert et al. (2001)
and at KEK from the BELLE Abe et al. (2001) collaborations also revealed the
same phenomenology with the $B^{0}$ meson. Recently the LHCb collaboration
also observed this violation in charm decays with the $D^{0}$ meson Aaij et
al. (2019).

Theoretically, the lasting and keen interest in CP violation can be traced
back to the observed baryon-antibaryon asymmetry of the Universe Hu and
Dodelson (2002); Cyburt (2004); Aghanim et al. (2020), rooted in the violation
of this combined symmetry as one of the necessary conditions from Sakharov
Sakharov (1967). It has been well-known that the established and well tested
Standard Model (SM) of particle physics is insufficient in producing the
aforementioned asymmetry of our Universe Cline (2006), thus new sources of CP
violation beyond the SM have to be identified along with the widely studied
meson systems presented in the last paragraph.

The weak decay of hyperons provides such an alternative. Theoretically, using
the branching ratios of different decay modes of $\Sigma^{\pm}$, the
observation of CP violation was firstly investigated by Okubo in 1958 Okubo
(1958). Soon after, a short discussion on parity and CP violation for anti-
baryon interactions was presented in Pais (1959). A systematical study on CP
violation using the rate differences between particles and antiparticles of
the octet baryons was performed in Brown et al. (1983), where upper bounds
within the quark model were estimated. Using a different approach, ref. Chau
and Cheng (1983) explored the possibility of CP violation measurements from
the angular distributions of non-leptonic decay modes of the octet baryons,
with a systematic study on all the nonleptonic decay modes of the hyperons
performed in Donoghue and Pakvasa (1985); Donoghue et al. (1986). In the
context of new physics where non-standard semileptonic interactions exist, the
SM Effective Field Theory (SMEFT) or its low-energy variants for example,
hyperon decays have been studied in Chang et al. (2015) in combination with
data from the Large Hadron Collider (LHC) at CERN. See also Li (2017);
Goudzovski et al. (2023) and references therein in this context.

Experimentally, searches of CP violation with hyperons had been carried out at
Fermilab in 1981 Pondrom et al. (1981), where the only existing electric
dipole moment result for the $\Lambda$ baryon was reported. Studies on CP
violation were later also carried out by the R608 collaboration at CERN, where
the $\Lambda$ baryon as a target of CP violation was produced from $p\bar{p}$
collision at $\sqrt{s}=30.8\rm\,GeV$ Chauvat et al. (1985) and no CP violation
was observed. Subsequent studies at DCI Orsay from the DM2 collaboration
further investigated the hyperon system by studying
$J/\psi\to\Lambda\bar{\Lambda}$ Tixier et al. (1988) and $J/\psi\to
p\bar{p},\,\Lambda\bar{\Lambda},\,\Sigma^{0}\bar{\Sigma}^{0}$, where the CP
parameters Lee and Yang (1957) of these baryons were reported in Pallin et al.
(1987). CP violation in the $\Xi^{-}$ baryon was carried out by the HyperCP
collaboration at Fermilab, where the hyperons were produced with an 800 GeV
$p$ beam on the copper target. The produced hyperons were then used to measure
the CP-violating asymmetries in $\Xi^{-}\to\Lambda\pi^{-}\to p\pi^{-}\pi^{-}$
and $\bar{\Xi}^{-}\to\bar{\Lambda}\pi^{+}\to\bar{p}\pi^{+}\pi^{+}$ by the E871
experiment Holmstrom et al. (2004). However, no CP violation was observed due
to limited statistics, as well as very large systematical uncertainties. At
higher energies, the LHCb Aaij et al. (2017) recently reported the evidence of
CP violation at the $3.3\sigma$ level from beauty baryon decays of
$\Lambda_{b}^{0}\to p\pi^{-}\pi^{+}\pi^{-}$. On the other hand, with the very
rich events accumulated at BESIIII at the $J/\psi$ mass threshold, the recent
analysis on $J/\psi\to\Lambda\bar{\Lambda}$ from the BESIII collaboration
shows no signal of CP violation and also presents very precise decay
parameters for the $\Lambda$ that are dramatically different from the world
average at that time Ablikim et al. (2019a). These results from BESIII further
modify the other experiments using the proton to reconstruct $\Lambda$, refs.
Bunce et al. (1976); Pondrom (1985); Chakravorty et al. (2003); Aad et al.
(2015); Adamczyk et al. (2017) for example, and are also favored by an
independent analysis from the CLAS collaboration Ireland et al. (2019).

Using the $\Lambda$ baryon and the other lowest-lying baryons for testing the
CP symmetry has been further investigated at BESIII since then Ablikim et al.
(2019b, 2020a, 2022a, 2022b, 2022c, 2022d, 2023a, 2023b), and theoretically in
He et al. (1993); He and Ma (2023) for $J/\psi\to\Lambda\bar{\Lambda}$ and in
Hong et al. (2023) for $J/\psi\to\Xi^{0}\bar{\Xi}^{0}$. In view of this
progress and the tens of billions $J/\psi$ particles already collected at
BESIII, we think it timely to systematically investigate tests of the parity
and the CP symmetries with $J/\psi$ decay. This study is further motivated by
the STCFs Bondar et al. (2013); Achasov et al. (2024) that is expected to
deliver $3.4\times 10^{12}$ $J/\psi$ particles at its mass threshold each
year. The rich physics of $J/\psi$ has been reviewed in Kopke and Wermes
(1989), and for this work, we only focus on $J/\psi$ produced at rest at
BESIII and STCFs that subsequently decays into the lowest-lying baryons for
testing the parity and the CP symmetries. Furthermore, we only consider the
unpolarized beam option of these facilities and we refer the readers to Bondar
et al. (2020); Zeng et al. (2023); Fu et al. (2023); Cao et al. (2024) for
recent studies with polarized beams. From this study, we find that:

  * •

In calculating the parity violating hadronic form factors of $J/\psi$ decay,
we find some generic modifications by a factor of a few when renormalization
group evolved from the weak scale down to the decay scale in the SM. In
particular, the axial-vector form factor of $\Sigma^{0}$ is found to be
reduced by a factor of $\sim\,40$ as summarized in table 2 for all the octet
baryons;

  * •

The parity-violating asymmetries introduced in this work are predicted to be
of $\mathcal{O}(10^{-4})$, as seen in table 4, which are slightly below the
statistical uncertainties with the 10 billion $J/\psi$ particles collected up
to now at BESIII. A combined analysis of these parity-violating asymmetries
from both the production and the decay of $J/\psi$, on the other hand, already
leads to a determination of the weak mixing angle with an absolute uncertainty
of $\delta s_{w}^{2}\approx 0.08$, corresponding to the first measurement at
the mass pole of $J/\psi$, as summarized in table 7 and shown in figure 1.
Based on luminosity rescaling, we also estimate that this precision reach in
$s_{w}^{2}$ is expected to be improved by about a factor of $\sim$20 at STCFs
within one year;

  * •

The 95% confidence level sensitivity for the upper bounds on the electric
dipole moments of the octet baryons are found to be of
$\mathcal{O}(10^{-18})\,e{\rm\,cm}$ at BESIII, about two orders of magnitude
stronger than the only measured one on $\Lambda$ from Fermilab more than 40
years ago. These upper bounds from BESIII are expected to be further improved
to $\mathcal{O}(10^{-19})\,e{\rm\,cm}$ at STCFs with just one-year data
collection, as summarized in table 4.

In the next sections, we provide details of our calculations. The rest of the
paper is organized as follows: We briefly review the theoretical formalism for
testing P and the CP symmetries with threshold $J/\psi$ production and decay
in section II, based on which we define the asymmetric observables that can be
directly connected with experiments. These observables depend on hadronic form
factors, whose derivations are detailed in section III, where their
renormalization group evolution is discussed. The results from this study are
presented in section IV, and we draw our conclusions in section V.

##  II Theoretical setup

As mentioned in the Introduction, the abundant production of $J/\psi$ at
BESIII and STCFs makes $J/\psi$ an ideal testbed for fundamental symmetries
such as P and the CP symmetries. In this section, we discuss how to test these
fundamental symmetries in $J/\psi$ decaying into the baryon-antibaryon pairs
$B\bar{B}$ with $B=\Lambda,\Sigma^{\pm,0},\Xi^{\pm,0}$, where $B$ or $\bar{B}$
subsequently undergoes two-body weak decays.

We consider the production of a $B\bar{B}$ system through the following
process:

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

In the center-of-mass frame, we define the momenta of the initial leptons and
the baryons as

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

with $E_{c}=m_{J/\psi}/2$. $s_{1,2}$ are spin vectors of the final state
particles in their respective rest frames as

$\displaystyle s_{1}^{\mu}=(0,{\bm{s}_{1}}),\quad
s_{2}^{\mu}=(0,{\bm{s}_{2}}).$ (3)

In this work, we only consider the case where the initial electron and
positron are unpolarized. However, the analysis can be directly generalized to
the polarized case such as STCF Achasov et al. (2024), Belle-II Altmannshofer
et al. (2019), ILC ILC (2013); Aryshev et al. (2022), CLIC Accomando et al.
(2004); Robson and Roloff (2018), and CEPC Abdallah et al. (2023). In this
unpolarized scenario, the probability of this considered process can only
depend on ${\bm{p}}$, ${\bm{k}}$ and ${\bm{s}_{1,2}}$:

$R({\bm{p}},{\bm{k}},{\bm{s}_{1}},{\bm{s}_{2}})=\frac{1}{4}\sum|\bra{f}\mathcal{T}\ket{i}|^{2},$
(4)

where $i$ and $f$ are particle collections in the initial and final states,
respectively, and the sum is over the spins of the initial states. If P or CP
is conserved, $R$ has the following properties:

$\displaystyle
R({\bm{p}},{\bm{k}},{\bm{s}_{1}},{\bm{s}_{2}})=R(-{\bm{p}},-{\bm{k}},{\bm{s}_{1}},{\bm{s}_{2}}),\quad
R({\bm{p}},{\bm{k}},{\bm{s}_{1}},{\bm{s}_{2}})=R({\bm{p}},{\bm{k}},{\bm{s}_{2}},{\bm{s}_{1}}),$
(5)

where the first and the second equalities are from P and CP conservation,
respectively. From these relations, it is required to measure the spins of
final state particles for testing P and CP symmetries Bernreuther et al.
(1989). The probability $R$ can be decomposed as

$R({\bm{p}},{\bm{k}},{\bm{s}_{1}},{\bm{s}_{2}})=a(\hat{\bm{p}}\cdot\hat{\bm{k}})+{\bm{s}}_{1}\cdot\bm{B_{1}}(\bm{\hat{p}},\bm{\hat{k}})+{\bm{s}}_{2}\cdot\bm{B_{2}}(\bm{\hat{p}},\bm{\hat{k}})-s_{1}^{i}s_{2}^{j}C^{ij}(\bm{\hat{p}},\bm{\hat{k}}),$
(6)

where

$\hat{\bm{p}}=\frac{{\bm{p}}}{|{\bm{p}}|},\quad\hat{\bm{k}}=\frac{{\bm{k}}}{|{\bm{k}}|}.$
(7)

The vector functions ${\bm{B}}_{1,2}(\bm{\hat{p}},\bm{\hat{k}})$ and the
tensor function $C^{ij}(\bm{\hat{p}},\bm{\hat{k}})$ can be further decomposed
with rotation covariance, with constraints from eq.(5) under P and CP
conservation. Details about this decomposition and constraints can be found in
He and Ma (2023).

The general form of the decay amplitude $J/\psi\to B\bar{B}$ can be written
with Lorentz invariance as:

$\displaystyle\mathcal{A}^{\mu}_{J/\psi\to
B\bar{B}}=\bar{u}(k_{1})\left[\gamma^{\mu}F_{V}^{B}(q^{2})+\frac{i}{2m_{\Lambda}}\sigma^{\mu\nu}q_{\nu}H_{\sigma}^{B}(q^{2})+\gamma^{\mu}\gamma_{5}F_{A}^{B}(q^{2})+i\sigma^{\mu\nu}q_{\nu}\gamma_{5}H_{T}^{B}(q^{2})\right]v(k_{2}),$
(8)

with $q=k_{1}+k_{2}$ the momentum transfer, and form factors $F_{A}^{B}$ and
$H_{T}^{B}$ the P and the CP violating strength, respectively.
$H_{\sigma}^{B}$ is the magnetic moment of baryon $B$, whose value is known
for all the lowest-lying baryons Workman et al. (2022). Practically, we find
the following combinations convenient for expressing the differential angular
distributions:

$\displaystyle G_{1}^{B}=F_{V}^{B}+H_{\sigma}^{B},\quad
G_{2}^{B}=G_{1}^{B}-\frac{(k_{1}-k_{2})^{2}}{4m_{B}^{2}}H_{\sigma}^{B},$ (9)

which are related to the electric and magnetic form factors $G_{E,M}$ and
those defined in Fäldt and Kupsc (2017) as
$G_{1}=G_{M}={G_{M}^{\psi}}/{e_{g}}$ and $G_{2}=G_{E}={G_{E}^{\psi}}/{e_{g}}$.
To simplify the notations in the subsequent discussion, we leave out the
superscript $B$ in the form factors from here on. These form factors $G_{1,2}$
can be determined from the decay parameters of $J/\psi$ as will be explained
in section III.3.

The information of the spin of $B$ or $\bar{B}$ is revealed in their
corresponding two-body weak decays, i.e., $B\to b+\pi$ and its charge
conjugated decay $\bar{B}\to\bar{b}+\bar{\pi}$, where $b$ stands for a baryon.
For instance, $b$ is the proton for $B=\Lambda$ and $\Lambda$ for $B=\Xi$. The
differential decay rates take the following forms Lee and Yang (1957):

$\frac{d\Gamma_{B}}{d\Omega_{b}}({\bm{s}}_{1},\hat{\bm{l}}_{b})\propto
1+\alpha_{B}{\bm{s}}_{1}\cdot\hat{\bm{l}}_{b},\quad\frac{d\Gamma_{\bar{b}}}{d\Omega_{\bar{b}}}({\bm{s}}_{2},\hat{\bm{l}}_{\bar{b}})\propto
1-\bar{\alpha}_{B}{\bm{s}}_{2}\cdot\hat{\bm{l}}_{\bar{b}},$ (10)

where $\hat{\bm{l}}_{b}$ and $\hat{\bm{l}}_{\bar{b}}$ are the momentum
directions of the secondary baryon $b$ and anti-baryon $\bar{b}$ in the rest
frames of $B$ and $\bar{B}$, respectively. $\Omega_{b,\bar{b}}$ are the
corresponding solid angles. $\alpha_{B}$ and $\bar{\alpha}_{B}$ are constants,
which satisfies $\alpha_{B}=\bar{\alpha}_{B}$ in the CP conserving limit.

From the differential decay width and $R$ in Eq.(4), the angular distribution
of $e^{-}e^{+}\to J/\psi\to B\bar{B}$ followed by $B\to b\pi$ and
$\bar{B}\to\bar{b}\pi$ is given by:

$\displaystyle\frac{d\sigma}{d\Omega_{k}d\Omega_{b}d\Omega_{\bar{b}}}\propto\mathcal{W}({\hat{\bm{p}}},{\hat{\bm{k}}},\hat{l}_{b},\hat{l}_{\bar{b}}),$
(11)

where the general from of the angular distribution is given by

$\displaystyle\mathcal{W}=a(\hat{\bm{p}}\cdot\hat{\bm{k}})+\alpha_{B}\hat{\bm{l}}_{b}\cdot\bm{B_{1}}(\bm{\hat{p}},\bm{\hat{k}})-\bar{\alpha}_{B}\hat{\bm{l}}_{\bar{b}}\cdot\bm{B_{2}}(\bm{\hat{p}},\bm{\hat{k}})-\alpha_{B}\bar{\alpha}_{B}\hat{l}_{b}^{i}\hat{l}_{\bar{b}}^{j}C^{ij}(\bm{\hat{p}},\bm{\hat{k}}),$
(12)

and $\Omega_{k}$ is the solid angle of ${\bm{k}}$. The proportional factors in
Eq.(10-11) are irrelevant in this work as they cancel in observables, which we
will discuss in the following.

In this work we will use integrated observables to study P and CP violation.
The four unit vectors $\hat{\bm{p}}$, $\hat{\bm{k}}$ and
$\hat{\bm{l}}_{b,\bar{b}}$ can be used to construct any observable
${\mathcal{O}}$ for our purpose, and its expectation value is given by:

$\displaystyle\langle{\mathcal{O}}\rangle=\frac{1}{{\mathcal{N}}}\int\frac{d\Omega_{k}d\Omega_{b}d\Omega_{\bar{b}}}{(4\pi)^{3}}{\mathcal{O}}\cdot\mathcal{W}({\hat{\bm{p}}},{\hat{\bm{k}}},\hat{l}_{b},\hat{l}_{\bar{b}}),$
(13)

where ${\mathcal{N}}$ is a normalization factor determined by $\langle
1\rangle=1$. The following expectation values of observables can be used for
extracting the spin of $B$ or $\bar{B}$:

$\displaystyle\langle\hat{\bm{l}}_{b}\cdot\hat{p}\rangle$
$\displaystyle\,=\frac{4\alpha_{B}}{9\mathcal{N}}E_{c}^{2}d_{J}\left(4y_{m}{\rm
Re}\left(G_{1}G_{2}^{\ast}\right)+\left|G_{1}\right|^{2}\right),$ (14)
$\displaystyle\langle\hat{\bm{l}}_{\bar{b}}\cdot\hat{\bm{p}}\rangle$
$\displaystyle\,=-\frac{4\bar{\alpha}_{B}}{9\mathcal{N}}E_{c}^{2}d_{J}\left(4y_{m}{\rm
Re}\left(G_{1}G_{2}^{\ast}\right)+\left|G_{1}\right|^{2}\right),$ (15)
$\displaystyle\langle\hat{\bm{l}}_{b}\cdot\hat{\bm{k}}\rangle$
$\displaystyle\,=\frac{8\alpha_{B}\beta}{9\mathcal{N}}E_{c}^{2}\left[{\rm
Re}\left(F_{A}G_{1}^{\ast}\right)-E_{c}{\rm
Im}\left(H_{T}G_{1}^{\ast}+y_{m}H_{T}G_{2}^{\ast}\right)\right],$ (16)
$\displaystyle\langle\hat{\bm{l}}_{\bar{b}}\cdot\hat{\bm{k}}\rangle$
$\displaystyle\,=-\frac{8\bar{\alpha}_{B}\beta}{9\mathcal{N}}E_{c}^{2}\left[{\rm
Re}\left(F_{A}G_{1}^{\ast}\right)+E_{c}{\rm
Im}\left(H_{T}G_{1}^{\ast}+y_{m}H_{T}G_{2}^{\ast}\right)\right],$ (17)
$\displaystyle\langle(\hat{\bm{l}}_{b}\times\hat{l}_{\bar{b}})\cdot\hat{\bm{k}}\rangle$
$\displaystyle\,=-\frac{16\alpha_{B}\bar{\alpha}_{B}}{27\mathcal{N}}\beta
y_{m}E_{c}^{3}{\rm Re}\left(H_{T}G_{2}^{\ast}\right),$ (18)

with $\beta=\sqrt{1-y_{m}^{2}}$ and $y_{m}=m_{B}/E_{c}$. $d_{J}$ is a measure
of P violation on the production side of $J/\psi$, as explained below in eq.
(44), whose form will be derived in the subsequent section. The normalization
factor is given by:

$\displaystyle\mathcal{N}=\frac{2}{3}E_{c}^{2}\left(2\left|G_{1}\right|^{2}+y_{m}^{2}\left|G_{2}\right|^{2}\right).$
(19)

We comment that there are some errors in the results of the above expectation
values in He and Ma (2023). They are corrected here. Note also that all
expectation values in Eq.(14-18) would vanish if $\alpha_{B}$ and
$\bar{\alpha}_{B}$ are zero, i.e., the produced baryon $B$ and $\bar{B}$ are
unpolarized.

Asymmetries can be obtained by the above observables for probing P or CP
violating effects. For any ${\mathcal{O}}$ one can always define corresponding
asymmetries in events as:

${\mathcal{A}}({\mathcal{O}})=\frac{{\mathcal{N}}_{\rm
event}({\mathcal{O}}>0)-{\mathcal{N}}_{\rm
event}({\mathcal{O}}<0)}{{\mathcal{N}}_{\rm
event}({\mathcal{O}}>0)+{\mathcal{N}}_{\rm
event}({\mathcal{O}}<0)}=\frac{1}{{\mathcal{N}}}\int\frac{d\Omega_{k}d\Omega_{p}d\Omega_{\bar{p}}}{(4\pi)^{3}}\biggr{(}\theta({\mathcal{O}})-\theta(-{\mathcal{O}})\biggr{)}{\mathcal{W}}(\Omega).$
(20)

The statistical error of any asymmetry is given by $1/\sqrt{N_{event}}$ with
$N_{event}$ the number of events for the $B\bar{B}$ final state. In this work
we will focus on the following asymmetries

$\begin{split}\mathcal{A}(\hat{\bm{l}}_{b}\cdot\hat{\bm{p}}-\hat{\bm{l}}_{\bar{b}}\cdot\hat{\bm{p}})&\,\equiv
A_{\rm
PV}^{(1)}\simeq\frac{4\alpha_{B}}{3\mathcal{N}}E_{c}^{2}d_{J}\left(4y_{m}{\rm
Re}\left(G_{1}G_{2}^{\ast}\right)+\left|G_{1}\right|^{2}\right),\\\
\mathcal{A}(\hat{\bm{l}}_{b}\cdot\hat{\bm{k}}-\hat{\bm{l}}_{\bar{b}}\cdot\hat{\bm{k}})&\,\equiv
A_{\rm PV}^{(2)}\simeq\frac{8\alpha_{B}\beta}{3\mathcal{N}}E_{c}^{2}\,{\rm
Re}\left(F_{A}G_{1}^{\ast}\right),\end{split}$ (21)

for testing the P symmetry on the production and the decay sides of $J/\psi$,
respectively, and the following asymmetries

$\begin{split}\mathcal{A}(\hat{\bm{l}}_{b}\cdot\hat{\bm{k}}+\hat{\bm{l}}_{\bar{b}}\cdot\hat{\bm{k}})&\,\equiv
A_{\rm CPV}^{(1)}\simeq-\frac{4\alpha_{B}\beta}{3\mathcal{N}}E_{c}^{3}\,{\rm
Im}\left(H_{T}G_{1}^{\ast}+y_{m}H_{T}G_{2}^{\ast}\right),\\\
\mathcal{A}((\hat{\bm{l}}_{b}\times\hat{\bm{l}}_{\bar{b}})\cdot\hat{\bm{k}})&\,\equiv
A_{\rm CPV}^{(2)}\simeq-\frac{8\alpha_{B}\bar{\alpha}}{9\mathcal{N}}\beta
y_{m}E_{c}^{3}{\rm Re}\left(H_{T}G_{2}^{\ast}\right),\end{split}$ (22)

for testing the CP symmetry. We comment that, given the opposite signs in eqs.
(14-15), we introduce the difference, i.e., $A_{\rm PV}^{(1)}$, as a practical
measurement of P violation to enhance its magnitude by a factor of 2.
Experimentally, this will require the reconstruction of both
$\bm{\hat{l}_{b}}$ and $\bm{\hat{l}_{\bar{b}}}$. This reconstruction, however,
can be performed individually instead of simultaneously such that this
enhancement can be achieved without introducing any extra technical
difficulties. This is similarly true for $A_{\rm PV}^{(2)}$ and $A_{\rm
CPV}^{(1)}$, with the exception for $A_{\rm CPV}^{(2)}$ that requires a
simultaneous determination of $\bm{\hat{l}_{b}}$ and $\bm{\hat{l}_{\bar{b}}}$.

Quantitative predictions for these asymmetries rely on prior knowledge of
these form factors and the decay parameters. The latter has been measured, at
BESIII for instance, for some of the lowest-lying baryons as we will summarize
in section IV, the former needs to be derived theoretically to connect
theories with experiments, to which we devote the next section.

##  III Form factors

The asymmetries in eqs. (21-22) depend on the form factors $F_{A}$, $G_{1,2}$
and $H_{T}$ defined in last section, which play a key role in reinterpreting
data in theories. The form factors describing $J/\psi\to B\bar{B}$ at low
energies generically receive non-perturbative contributions, and can be
calculated with nonperturbative methods. In this section, we give a derivation
for these form factors from a more phenomenological approach.

Table 1: Contributions to $d_{B}$ from quark EDMs, i.e., $d_{q}$, in the quark model (second column) and quark CDMs, i.e., $f_{q}$, in the non-relativistic (NR) quantum chromodynamics (QCD) limit in the quark model (fifth column). Given the current constraints on $d_{n,p}$ as detailed in the main text, the reduced results are given, respectively, in the third and the sixth columns. $d_{B}$ | QM | Reduced Results | $d_{B}$ | NR QCD & QM | Reduced Results
---|---|---|---|---|---
$d_{p}^{\rm qEDM}$ | $\frac{1}{3}(4d_{u}-d_{d})$ | — | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(4Q_{d}f_{d}-Q_{u}f_{u}\right)$ | —
$d_{n}^{\rm qEDM}$ | $\frac{1}{3}(4d_{d}-d_{u})$ | — | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(4Q_{u}f_{u}-Q_{d}f_{d}\right)$ | —
$d_{\Sigma^{+}}^{\rm qEDM}$ | $\frac{1}{3}(4d_{u}-d_{s})$ | $-\frac{1}{3}d_{s}$ | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(4Q_{u}f_{u}-Q_{s}f_{s}\right)$ | $-\frac{1}{9}ef_{s}$
$d_{\Sigma^{0}}^{\rm qEDM}$ | $\frac{1}{3}(2d_{u}+2d_{d}-d_{s})$ | $-\frac{1}{3}d_{s}$ | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(2Q_{u}f_{u}+2Q_{d}f_{d}-Q_{s}f_{s}\right)$ | $-\frac{1}{9}ef_{s}$
$d_{\Sigma^{-}}^{\rm qEDM}$ | $\frac{1}{3}(4d_{d}-d_{s})$ | $-\frac{1}{3}d_{s}$ | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(4Q_{d}f_{d}-Q_{s}f_{s}\right)$ | $-\frac{1}{9}ef_{s}$
$d_{\Xi^{0}}^{\rm qEDM}$ | $\frac{1}{3}(4d_{s}-d_{u})$ | $\frac{4}{3}d_{s}$ | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(4Q_{s}f_{s}-Q_{u}f_{u}\right)$ | $\frac{4}{9}ef_{s}$
$d_{\Xi^{-}}^{\rm qEDM}$ | $\frac{1}{3}(4d_{s}-d_{d})$ | $\frac{4}{3}d_{s}$ | $d_{p}^{\rm\,qCDM}$ | $-\frac{1}{3}\left(4Q_{s}f_{s}-Q_{d}f_{d}\right)$ | $\frac{4}{9}ef_{s}$
$d_{\Lambda^{0}}^{\rm qEDM}$ | $d_{s}$ | $d_{s}$ | $d_{p}^{\rm\,qCDM}$ | $-Q_{s}f_{s}$ | $\frac{1}{3}ef_{s}$

###  III.1 $H_{T}$ from electric dipole interactions

The form factor $H_{T}$ depends on the momentum transfer $q^{2}$ and is
related to the electric dipole moment (EDM) of $B$ in the vanishing $q^{2}$
limit. For this work, we ignore the $q^{2}$ dependence of $H_{T}$ and
approximate it as the real EDM of baryon $B$ in the final state of the
decaying $J/\psi$, i.e.,

$\displaystyle\mathcal{L}_{B_{\rm
EDM}}=-i\frac{d_{B}}{2}\bar{B}\sigma_{\mu\nu}\gamma_{5}BF^{\mu\nu},$ (23)

with $d_{B}$ the baryon EDM, $F^{\mu\nu}$ the electromagnetic field tensor,
and $B$ the baryon field. In this approximation, the decay amplitude of
$J/\psi\to B\bar{B}$ is given by

$\displaystyle i\mathcal{M}_{\rm decay}$
$\displaystyle\,=\frac{ieQ_{c}d_{B}}{m_{J/\psi}^{2}}\cdot\left(\bar{B}\sigma^{\alpha\beta}q_{\beta}\gamma_{5}B\right)\cdot
g_{V}\cdot\epsilon_{\alpha}^{J/\psi},$ (24)

where $Q_{c}=2/3$ is the electric charge of the $c$ quark, and we use
$\bra{0}(\bar{c}\gamma^{\alpha}c)\ket{J/\psi}=g_{V}\cdot\epsilon_{\alpha}^{J/\psi}$
for the charm quark vector current with the form factor $g_{V}=\left(1.286\pm
0.012\right){\rm\,GeV^{2}}$ as determined from the branching ratio of
$J/\psi\to\ell^{\pm}\ell^{\mp}$ He et al. (1993):

$\displaystyle\Gamma_{J/\psi\to\ell^{\pm}\ell^{\mp}}=\frac{16g_{V}^{2}\pi\alpha_{\rm
EM}^{2}}{27m_{J/\psi}^{3}}\left(1+\frac{2m_{\ell}^{2}}{m_{J/\psi}^{2}}\right)\sqrt{1-\frac{4m_{\ell}^{2}}{m_{J/\psi}^{2}}},$
(25)

with $\alpha_{\rm EM}=e^{2}/4\pi$ the fine structure constant and $e$ the
electron charge magnitude. From eq. (24), its matching with eq. (8) leads to

$\displaystyle H_{T}=\frac{e\cdot Q_{c}\cdot g_{V}\cdot
d_{B}}{m_{J/\psi}^{2}}.$ (26)

To connect with the quark-level interactions, we further estimate the baryon
EDM $d_{B}$ by decomposing it into contributions from its quark components in
the quark model (QM). In this respect, we consider two different sources: (1)
EDMs of quarks ($d_{q}$), and (2) chromo-dipole moments (CDMs) of quarks
($f_{q}$). The results are summarized in the second and fifth columns of table
1, whose derivation is given in appendix A.

Recall that the current upper bounds on $d_{n}$, $d_{p}$, and $d_{\Lambda}$
are, respectively, $|d_{n}|<1.8\times 10^{-26}\,e\,{\rm cm}$ Abel et al.
(2020), $|d_{p}|<2.1\times 10^{-25}\,e\,{\rm cm}$ Workman et al. (2022), and
$|d_{\Lambda}|<1.5\times 10^{-16}\,e\,{\rm cm}$ Pondrom et al. (1981), the
different combinations of $d_{n,p}$ in terms of $d_{u,d}$ or $f_{u,d}$ would
suggest the dominance of the strange quark in contributing to the baryon EDMs
in the quark model. Upon this observation, one approximately finds the
following “reduced” relations

$\displaystyle d_{\Sigma^{0,\pm}}^{\rm
qEDM,\,qCDM}\simeq-\frac{1}{3}d_{\Lambda^{0}}^{\rm qEDM,\,qCDM},\quad
d_{\Xi^{0,-}}^{\rm qEDM,\,qCDM}\simeq\frac{4}{3}d_{\Lambda^{0}}^{\rm
qEDM,\,qCDM},$ (27)

as shown in the third and sixth columns of table 1. Based on these reduced
results, the 95% confidence level (CL) upper bounds on $d_{B}$ are thus
estimated to be of $\mathcal{O}(10^{-17}\sim 10^{-16})\,e{\rm\,cm}$ based on
that of $d_{\Lambda}$ in Pondrom et al. (1981). As we will see shortly in
section IV.2, these upper bounds are already expected to be improved at BESIII
or STCFs utilizing $A_{\rm CPV}^{(1,\,2)}$ introduced in this work.

###  III.2 $F_{A}$ from P violating interactions

The form factor $F_{A}$ originates from the parity-violating decay of
$J/\psi$, which is rooted in an intermediate $s$-channel $Z$ and an
intermediate $t$-channel $W$ boson exchanges in the SM. The quark level
Lagrangian in the low momentum transfer limit, as is the case for $J/\psi$
production and decay considered in this work, is given by

$\displaystyle\mathcal{L}_{Z+W}=$
$\displaystyle\,-4\sqrt{2}G_{F}\cdot\sum_{q=u,d,s}\left[g_{V-A}^{q}g_{V-A}^{c}\left(\bar{q}_{R}\gamma_{\mu}q_{R}\right)\left(\bar{c}_{R}\gamma_{\mu}c_{R}\right)+g_{V-A}^{q}g_{V+A}^{c}\left(\bar{q}_{R}\gamma_{\mu}q_{R}\right)\left(\bar{c}_{L}\gamma_{\mu}c_{L}\right)\right.$
$\displaystyle\,\left.\qquad\qquad\quad+g_{V+A}^{q}g_{V-A}^{c}\left(\bar{q}_{L}\gamma_{\mu}q_{L}\right)\left(\bar{c}_{R}\gamma_{\mu}c_{R}\right)+g_{V+A}^{q}g_{V+A}^{c}\left(\bar{q}_{L}\gamma_{\mu}q_{L}\right)\left(\bar{c}_{L}\gamma_{\mu}c_{L}\right)\right]$
$\displaystyle\,-2\sqrt{2}G_{F}\left[\left|V_{cd}\right|^{2}\left(\bar{d}_{L}\gamma_{\alpha}c_{L}\right)\left(\bar{c}_{L}\gamma^{\alpha}d_{L}\right)+\left|V_{cs}\right|^{2}\left(\bar{s}_{L}\gamma_{\alpha}c_{L}\right)\left(\bar{c}_{L}\gamma^{\alpha}s_{L}\right)\right],$
(28)

where the first term is from an intermediate $Z$ exchange and the second that
from an intermediate $W$. $G_{F}$ is the Fermi constant as determined from the
muon lifetime, $V$ is the CKM matrix, and the couplings are all defined at the
matching scale $m_{Z}$ as given by

$\displaystyle g_{V\pm A}^{q}\equiv g_{V}^{q}\pm g_{A}^{q},\quad
g_{V}^{q}=\frac{1}{2}T_{q}^{3}-Q_{q}s_{w}^{2},\quad
g_{A}^{q}=\frac{1}{2}T_{q}^{3}.$ (29)

Here, $c_{w}\equiv\cos\theta_{w}$ and $s_{w}\equiv\sin\theta_{w}$ with
$\theta_{w}$ the weak mixing angle, $T_{q}^{3}$ and $Q_{q}$ are, respectively,
the iso-spin and the electric charge of quark $q$. Since $J/\psi\to B\bar{B}$
occurs at a much lower energy scale than $m_{Z}$, the running effects on these
operators will have to be included due to significant QCD modifications. These
have been well documented in literature, see for example, Refs. Gaillard and
Lee (1974, 1974); Ecker and Grimus (1985); He et al. (1988); Buchalla et al.
(1996) for kaon decay and ref. Jenkins et al. (2018) for the weak effective
field theory. For this work, we extract the anomalous dimension from Jenkins
et al. (2018) by focusing on the dominant pure QCD corrections only. Since the
singlet operators in eq (28) will mix with the octet ones through the
anomalous dimension even under this approximation, we define the following
linear combinations to decouple them such that the anomalous dimension in this
basis is diagonal:

$\begin{split}&\,\mathcal{O}_{\pm}^{LL\,(RR)}=\left(\bar{d}\gamma_{\mu}P_{L\,(R)}c\right)\left(\bar{c}\gamma_{\mu}P_{L\,(R)}d\right)\pm\left(\bar{d}\gamma_{\mu}P_{L\,(R)}d\right)\left(\bar{c}\gamma_{\mu}P_{L\,(R)}c\right),\\\
&\,\mathcal{O}_{+}^{LR\,(RL)}=\left(\bar{d}\gamma_{\mu}P_{L\,(R)}c\right)\left(\bar{c}\gamma_{\mu}P_{R\,(L)}d\right)+\frac{2}{3}\left(\bar{d}P_{R\,(L)}d\right)\left(\bar{c}P_{L\,(R)}c\right),\\\
&\,\mathcal{O}_{-}^{LR\,(RL)}=\frac{2}{3}\left(\bar{d}P_{R\,(L)}d\right)\left(\bar{c}P_{L\,(R)}c\right),\end{split}$
(30)

for $W$, and

$\displaystyle\,\mathcal{O}_{uu+}^{LR\,(RL)}=\frac{1}{N_{c}}\left(\bar{u}_{R\,(L)}\gamma_{\mu}u_{R\,(L)}\right)\left(\bar{c}_{L\,(R)}\gamma_{\mu}c_{L\,(R)}\right)+2\left(\bar{u}_{R\,(L)}\gamma_{\mu}T^{A}u_{R\,(L)}\right)\left(\bar{c}_{L\,(R)}\gamma_{\mu}T^{A}c_{L\,(R)}\right),$
$\displaystyle\,\mathcal{O}_{uu-}^{LR\,(RL)}=-\frac{4C_{F}}{N_{c}}\left(\bar{u}_{R\,(L)}\gamma_{\mu}u_{R\,(L)}\right)\left(\bar{c}_{L\,(R)}\gamma_{\mu}c_{L\,(R)}\right)+\frac{4}{N_{c}}\left(\bar{u}_{R\,(L)}\gamma_{\mu}T^{A}u_{R\,(L)}\right)\left(\bar{c}_{L\,(R)}\gamma_{\mu}T^{A}c_{L\,(R)}\right),$
$\displaystyle\,\mathcal{O}_{ud+}^{LR}=\frac{1}{N_{c}}\left(\bar{d}_{R}\gamma_{\mu}d_{R}\right)\left(\bar{c}_{L}\gamma_{\mu}c_{L}\right)+2\left(\bar{d}_{R}\gamma_{\mu}T^{A}d_{R}\right)\left(\bar{c}_{L}\gamma_{\mu}T^{A}c_{L}\right),$
(31)
$\displaystyle\,\mathcal{O}_{ud-}^{LR}=-\frac{4C_{F}}{N_{c}}\left(\bar{d}_{R}\gamma_{\mu}d_{R}\right)\left(\bar{c}_{L}\gamma_{\mu}c_{L}\right)+\frac{4}{N_{c}}\left(\bar{d}_{R}\gamma_{\mu}T^{A}d_{R}\right)\left(\bar{c}_{L}\gamma_{\mu}T^{A}c_{L}\right),$
$\displaystyle\,\mathcal{O}_{du+}^{LR}=\frac{1}{N_{c}}\left(\bar{d}_{L}\gamma_{\mu}d_{L}\right)\left(\bar{c}_{R}\gamma_{\mu}c_{R}\right)+2\left(\bar{d}_{L}\gamma_{\mu}T^{A}d_{L}\right)\left(\bar{c}_{R}\gamma_{\mu}T^{A}c_{R}\right),$
$\displaystyle\,\mathcal{O}_{du-}^{LR}=-\frac{4C_{F}}{N_{c}}\left(\bar{d}_{L}\gamma_{\mu}d_{L}\right)\left(\bar{c}_{R}\gamma_{\mu}c_{R}\right)+\frac{4}{N_{c}}\left(\bar{d}_{L}\gamma_{\mu}T^{A}d_{L}\right)\left(\bar{c}_{R}\gamma_{\mu}T^{A}c_{R}\right),$

for $Z$, with $T^{A}$ the Gell-Mann matrices normalized as ${\rm
Tr}(T^{A}T^{B})=\delta^{AB}/2$, $C_{F}=(N_{c}^{2}-1)/(2N_{c})$, $N_{c}=3$ the
number of colors, and $f_{L\,(R)}\equiv P_{L\,(R)}f$ with $P_{L,\,(R)}$ the
chiral projectors.

Using these combinations, we rewrite eq. (28) as

$\begin{split}-\mathcal{L}_{Z+W}=&\,4\sqrt{2}G_{F}\cdot\sum_{q=u,d,s}\left[\frac{1}{2}g_{V-A}^{q}g_{V-A}^{c}\left(C_{+}^{R}\mathcal{O}_{+}^{RR}-C_{-}^{R}\mathcal{O}_{-}^{RR}\right)+\frac{1}{2}g_{V+A}^{q}g_{V+A}^{c}\left(C_{+}^{L}\mathcal{O}_{+}^{LL}-C_{-}^{L}\mathcal{O}_{-}^{LL}\right)\right]\\\
&\,+4\sqrt{2}G_{F}\cdot\left[\frac{g_{V-A}^{u}g_{V+A}^{c}}{N_{c}}C_{uu+}^{LR}\mathcal{O}_{uu+}^{LR}-\frac{g_{V-A}^{u}g_{V+A}^{c}}{2}C_{uu-}^{LR}\mathcal{O}_{uu-}^{LR}\right.\\\
&\,\left.\qquad\qquad\qquad+\frac{g_{V+A}^{u}g_{V-A}^{c}}{N_{c}}C_{uu+}^{RL}\mathcal{O}_{uu+}^{RL}-\frac{g_{V+A}^{u}g_{V-A}^{c}}{2}C_{uu-}^{RL}\mathcal{O}_{uu-}^{RL}\right]\\\
&\,+4\sqrt{2}G_{F}\cdot\sum_{q=d,s}\left[\frac{g_{V-A}^{q}g_{V+A}^{c}}{N_{c}}C_{ud+}^{LR}\mathcal{O}_{ud+}^{LR}-\frac{g_{V-A}^{q}g_{V+A}^{c}}{2}C_{ud-}^{LR}\mathcal{O}_{ud-}^{LR}\right.\\\
&\,\left.\qquad\qquad\qquad+\frac{g_{V+A}^{q}g_{V-A}^{c}}{N_{c}}C_{du+}^{LR}\mathcal{O}_{du+}^{LR}-\frac{g_{V+A}^{q}g_{V-A}^{c}}{2}C_{du-}^{LR}\mathcal{O}_{du-}^{LR}\right]\\\
&\,+\sqrt{2}G_{F}\sum_{q=d,s}\left|V_{cq}\right|^{2}\left(C_{+}^{L}\mathcal{O}_{+}^{LL}+C_{-}^{L}\mathcal{O}_{-}^{LL}\right),\end{split}$
(32)

with all the $C$’s being unity at scale $m_{Z}$. Running to an arbitrary scale
$\mu$, we find

$\begin{split}&\,C_{+}^{L,R}(\mu^{2})\equiv
C_{1}\left(\mu^{2}\right)=\left(\frac{\alpha_{s}\left(\mu^{2}\right)}{\alpha_{s}\left(m_{Z}^{2}\right)}\right)^{-\frac{3(N_{c}-1)}{bN_{c}}},\quad
C_{-}^{L,R}(\mu^{2})\equiv
C_{2}\left(\mu^{2}\right)=\left(\frac{\alpha_{s}\left(\mu^{2}\right)}{\alpha_{s}\left(m_{Z}^{2}\right)}\right)^{\frac{3(1+N_{c})}{bN_{c}}},\\\
&\,C_{uu+,ud+,du+}^{LR}\left(\mu^{2}\right)\equiv
C_{3}\left(\mu^{2}\right)=\left(\frac{\alpha_{s}\left(\mu^{2}\right)}{\alpha_{s}\left(m_{Z}^{2}\right)}\right)^{\frac{6C_{F}}{b}},\quad
C_{uu-,ud-,du-}^{LR}\left(\mu^{2}\right)\equiv
C_{4}\left(\mu^{2}\right)=\left(\frac{\alpha_{s}\left(\mu^{2}\right)}{\alpha_{s}\left(m_{Z}^{2}\right)}\right)^{-\frac{3}{bN_{c}}},\end{split}$
(33)

with the strong coupling $\alpha_{s}$ and its $\beta$ function $b$ given by

$\displaystyle\alpha_{s}(\mu^{2})=\frac{4\pi}{b\log\left(\frac{\mu^{2}}{\Lambda_{f}^{2}}\right)},\quad
b=\frac{11}{3}N_{c}-\frac{2}{3}N_{f},$ (34)

where $\Lambda_{f}$ is the QCD parameter for $N_{f}$ dynamical quark flavors
under consideration in the $\overline{\rm MS}$ scheme, and $\mu$ is the ’t
Hooft scale below $m_{Z}$. For $J/\psi$ decay as considered in this work,
since $m_{J/\psi}\approx m_{b}$ with $m_{b}$ the bottom quark mass, we
consider the running of these parameters down to the scale $\mu=m_{J/\psi}$
with $N_{f}=5$, $\Lambda_{5}=225\rm\,MeV$ Buchalla et al. (1996), and we use
the average $\alpha_{s}(m_{Z}^{2})=0.1180$ as reported in Workman et al.
(2022). Numerically, we find

$\displaystyle C_{1}(m_{J/\psi}^{2})=0.78,\quad
C_{2}(m_{J/\psi}^{2})=1.66,\quad C_{3}(m_{J/\psi}^{2})=2.76,\quad
C_{4}(m_{J/\psi}^{2})=0.88\,,$ (35)

which are significantly modified compared with their values at $\mu=m_{Z}$. On
the other hand, since $m_{b}$ is slightly above $m_{J/\psi}$, integrating also
the $b$ quark out leads to a slightly different setup with

$\displaystyle C_{1}(m_{J/\psi}^{2})=0.78,\quad
C_{2}(m_{J/\psi}^{2})=1.65,\quad C_{3}(m_{J/\psi}^{2})=2.72,\quad
C_{4}(m_{J/\psi}^{2})=0.88,$ (36)

with $N_{f}=4$ and $\Lambda_{4}=325\rm\,MeV$ Buchalla et al. (1996). The
axial-vector form factors with this setup, which will be discussed shortly
below in eq. (40), are found mildly modified, and our main conclusion from
this study is thus barely affected. In the following, to simplify the
notations, we rename

$\displaystyle C_{1}(m_{J/\psi}^{2})=c_{1},\quad
C_{2}(m_{J/\psi}^{2})=c_{2},\quad C_{3}(m_{J/\psi}^{2})=c_{3},\quad
C_{4}(m_{J/\psi}^{2})=c_{4},$ (37)

and use their values in eq. (35) for the following discussion. Then by
including the running effects and focusing on the parity violating part of the
decay amplitude for the extraction of $F_{A}$, we obtain

$\begin{split}i\mathcal{M}_{F_{A}}=&\,4\sqrt{2}G_{F}\cdot
ig_{V}\cdot\epsilon_{\alpha}^{J/\psi}\\\
&\,\times{\bra{B\bar{B}}\left[\frac{1}{2}\sum_{q=u,d,s}\left(g_{V}^{c}g_{A}^{q}+g_{A}^{c}g_{V}^{q}\right)\mathcal{R}_{Z}+\frac{1}{2}\sum_{q=u,d,s}\left(g_{V}^{c}g_{A}^{q}-g_{A}^{c}g_{V}^{q}\right)\mathcal{\tilde{R}}_{Z}+\sum_{q=d,s}\frac{\left|V_{cq}\right|^{2}}{8}\mathcal{R}_{W}\right]\bar{q}\gamma^{\alpha}\gamma_{5}q\ket{0}},\end{split}$
(38)

with

$\displaystyle\mathcal{R}_{Z}=\frac{1}{2}\left[(c_{1}+c_{2})+\frac{c_{1}-c_{2}}{N_{c}}\right]\approx
1.07,\quad\mathcal{\tilde{R}}_{Z}=c_{4}+\frac{c_{3}-c_{4}}{N_{c}^{2}}\approx
1.09,\quad\mathcal{R}_{W}=\frac{1}{2}\left[(c_{1}-c_{2})+\frac{c_{1}+c_{2}}{N_{c}}\right]\approx-0.03,$
(39)

corresponding to about 7%, 9% enhancement and a factor of 10 reduction,
respectively, compared with their values at $m_{Z}$. It is straightforward to
check that one recovers the quark level Lagrangian in eq. (28) by taking the
unity limits of $c_{i}$’s at $\mu=m_{Z}$ in eq. (38), which also provides a
cross check of our results. We comment that contributions from
$\mathcal{O}_{uu\pm}^{LR\,(RL)}$ completely cancel out due to lepton flavor
universality, we nevertheless include them in eq. (38) for compactness and a
straightforward generalization to new physics analysis where lepton flavor
universality is not necessarily respected. It is also noted that compared with
the scenario without QCD running where only $g_{V}^{c}g_{A}^{q}$’s (with
$q=u,d,s$) contribute to $F_{A}$ in the $Z$ process, the axial-vector current
of the $c$ quark also gives a non-vanishing contribution when the
renormalization group evolution is included.

The evaluation of
$\bra{B\bar{B}}\left(\bar{q}\gamma^{\alpha}\gamma_{5}q\right)\ket{0}$ is
realized by computing the crossed matrix element
$\bra{B}\left(\bar{q}\gamma^{\alpha}\gamma_{5}q\right)\ket{B}$ using chiral
perturbation theory ($\chi$PT). The details can be found in appendix B, and we
finally obtain

$\displaystyle F_{A}^{n}$
$\displaystyle\,=\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot\left\\{D\cdot\left[\left(1-2s_{w}^{2}\right)\mathcal{R}_{Z}-\frac{2}{3}s_{w}^{2}\mathcal{\tilde{R}}_{Z}\right]-\left[D+\left(\left|V_{cd}\right|^{2}-\left|V_{cs}\right|^{2}\right)F\right]\mathcal{R}_{W}\right\\},$
$\displaystyle\quad F_{A}^{p}$
$\displaystyle\,=-\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot\left\\{\left(F-\frac{D+7F}{3}s_{w}^{2}\right)\mathcal{R}_{Z}+\frac{D-F}{3}s_{w}^{2}\mathcal{\tilde{R}}_{Z}+(D-F)\left|V_{cs}\right|^{2}\mathcal{R}_{W}\right\\},$
$\displaystyle F_{A}^{\Sigma^{+}}$
$\displaystyle\,=-\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot\left\\{\left(F-\frac{D+7F}{3}s_{w}^{2}\right)\mathcal{R}_{Z}+\frac{D-F}{3}s_{w}^{2}\mathcal{\tilde{R}}_{Z}+(D-F)\left|V_{cd}\right|^{2}\mathcal{R}_{W}\right\\},$
$\displaystyle F_{A}^{\Sigma^{0}}$
$\displaystyle\,=\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot
D\cdot\left\\{\frac{1}{3}s_{w}^{2}\left(\mathcal{R}_{Z}-\mathcal{\tilde{R}}_{Z}\right)-\left|V_{cd}\right|^{2}\mathcal{R}_{W}\right\\},$
(40) $\displaystyle F_{A}^{\Sigma^{-}}$
$\displaystyle\,=\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot\left\\{\left[\left(\frac{D-7F}{3}s_{w}^{2}+F\right)\mathcal{R}_{Z}-\frac{D+F}{3}s_{w}^{2}\mathcal{\tilde{R}}_{Z}\right]-(D+F)\left|V_{cd}\right|^{2}\mathcal{R}_{W}\right\\},$
$\displaystyle F_{A}^{\Xi^{0}}$
$\displaystyle\,=\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot\left\\{D\cdot\left[\left(1-2s_{w}^{2}\right)\mathcal{R}_{Z}-\frac{2}{3}s_{w}^{2}\mathcal{\tilde{R}}_{Z}\right]-\left[D-\left(\left|V_{cd}\right|^{2}-\left|V_{cs}\right|^{2}\right)F\right]\mathcal{R}_{W}\right\\},$
$\displaystyle F_{A}^{\Xi^{-}}$
$\displaystyle\,=\left(\frac{G_{F}g_{V}}{2\sqrt{2}}\right)\cdot\left\\{\left[\left(\frac{D-7F}{3}s_{w}^{2}+F\right)\mathcal{R}_{Z}-\frac{D+F}{3}s_{w}^{2}\mathcal{\tilde{R}}_{Z}\right]-(D+F)\left|V_{cs}\right|^{2}\mathcal{R}_{W}\right\\},$
$\displaystyle F_{A}^{\Lambda}$
$\displaystyle\,=\left(\frac{G_{F}g_{V}}{3\sqrt{2}}\right)\cdot
D\cdot\left\\{\left[\left(1-\frac{11}{6}s_{w}^{2}\right)\mathcal{R}_{Z}-\frac{5}{6}s_{w}^{2}\mathcal{\tilde{R}}_{Z}\right]-\frac{1}{2}\left(\left|V_{cd}\right|^{2}+4\left|V_{cs}\right|^{2}\right)\mathcal{R}_{W}\right\\},$

where the unitary condition
$\left|V_{cd}\right|^{2}+\left|V_{cs}\right|^{2}\approx 1$ is applied in
obtaining the results above. Note that the contribution to
$F_{A}^{\Sigma^{0}}$ from the $s$-channel $Z$ exchange from the
renormalization group evolution would vanish at $\mu=m_{Z}$ since at which
$\mathcal{R}_{Z}=\mathcal{\tilde{R}}_{Z}=1$. As a consequence,
$F_{A}^{\Sigma^{0}}$ would be sensitive to the underlying $Z$ exchange
process, especially given the $V_{cd}$ suppression for the $W$ channel.

Table 2: Numerical results of axial-vector form factors for different baryons, where the first numeric line with label “No” indicates results where the running effects discussed in the main text are excluded from the calculations. Corrections to the $t$-channel $W$ process are shown in the “$t$” row, and to both the $t$-channel $W$ and the $s$-channel $Z$ processes in the “$t+s$” row. Running? $F_{A}^{B}$ | $F_{A}^{n}\,(\times 10^{-6})$ | $F_{A}^{p}\,(\times 10^{-7})$ | $F_{A}^{\Sigma^{+}}\,(\times 10^{-7})$ | $F_{A}^{\Sigma^{0}}\,(\times 10^{-9})$ | $F_{A}^{\Sigma^{-}}\,(\times 10^{-7})$ | $F_{A}^{\Xi^{0}}\,(\times 10^{-6})$ | $F_{A}^{\Xi^{-}}\,(\times 10^{-7})$ | $F_{A}^{\Lambda}\,(\times 10^{-7})$
---|---|---|---|---|---|---|---|---
No | 0.79 | $-11.0$ | $-8.22$ | $-49.9$ | 7.22 | $-0.55$ | $-8.93$ | $-5.84$
$t$ | 1.18 | $-7.78$ | $-8.06$ | $4.99$ | 8.16 | 1.31 | 9.77 | 8.95
$t+s$ | 1.25 | $-8.39$ | $-8.66$ | $1.25$ | 8.69 | 1.39 | 10.3 | 9.44

Numerically, form factors for different baryons are listed in table 2, where
the “No” row corresponds to results without considering any renormalization
group evolution. The “$t$” row represents results with the $t$-channel $W$
exchange process improved by the renormalization group equations, and the
“$s+t$” row that where the additional $s$-channel $Z$ exchange process is also
improved. As is clear from the table, the parity-violating form factors
$F_{A}$ from the decay of $J/\psi$ are generically modified by a factor of a
few when the running effects are included, dominantly rooted in large QCD
corrections for the non-local $t$-channel process. In contrast, the additional
$s$-channel corrections turn out to be relatively small, as can be seen from
the “$s+t$” row of table 2. This observation is true for almost all the octet
baryons with the only exception being $\Sigma^{0}$, and the reason is clear
from a closer look at eqs. (39-40): When the running effects are not included,
$c_{i}=1$ ($i=1,\cdots,4$) such that
$F_{A}^{\Sigma^{0}}\propto\mathcal{R}_{W}=1/N_{c}$. In contrast, if only the
“$t$” process is improved by the renormalization group equations, even though
it still hods that $F_{A}^{\Sigma^{0}}\propto\mathcal{R}_{W}$, the accidental
cancellation between the first term $(c_{1}-c_{2})$ and the second term
$(c_{1}+c_{2})/N_{c}$ in $\mathcal{R}_{W}$ of eq. (39) reduces its value by a
factor of 10 in addition to the sign flip. Furthermore, $F_{A}^{\Sigma^{0}}$
is further bated by a factor of $\sim 4$ when the renormalization group
improved $s$-channel $Z$ exchange process is included, such that its magnitude
becomes $\sim 10^{3}$ smaller than its charged counterparts, i.e.,
$\Sigma^{\pm}$. This makes it very challenging to test the P symmetry through
$J/\psi\to\Sigma^{0}\bar{\Sigma}^{0}$ unless augmented by new P violating
sources.

###  III.3 $G_{1,2}$ from the decay of $J/\psi$

We discuss the phenomenological determination of the remaining decay form
factors $G_{1,2}$ in this section. Due to the smallness of P and P violation,
contributions from the $F_{A}$ and the $H_{T}$ terms in eq. (8) to the decay
width of $J/\psi$ can be safely neglected such that

$\displaystyle\Gamma_{J/\psi\to
B\bar{B}}=\frac{|G_{1}|^{2}m_{J/\psi}}{12\pi}\sqrt{1-\frac{4m_{B}^{2}}{m_{J/\psi}^{2}}}\left(1+\frac{2m_{B}^{2}}{m_{J/\psi}^{2}}\left|\frac{G_{2}}{G_{1}}\right|^{2}\right).$
(41)

Recall that

$\displaystyle\frac{G_{2}}{G_{1}}=\frac{G_{E}^{\psi}}{G_{M}^{\psi}}=\frac{G_{E}}{G_{M}}=\left|\frac{G_{E}}{G_{M}}\right|e^{i\Delta\Phi}\equiv
e^{i\Delta\Phi}\mathtt{R},$ (42)

with $\Delta\Phi$ the relative phase between $G_{2}$ and $G_{1}$, and the
ratio $\mathtt{R}$ experimentally determined from the angular distribution
parameter $\alpha_{J/\psi}^{B}$ defined as the ratio of electromagnetic form
factors Fäldt and Kupsc (2017)

$\displaystyle{\alpha_{J/\psi}^{B}}=\frac{s\left|G_{M}^{\psi}\right|^{2}-4m_{B}^{2}\left|G_{E}^{\psi}\right|^{2}}{s\left|G_{M}^{\psi}\right|^{2}+4m_{B}^{2}\left|G_{E}^{\psi}\right|^{2}}=\frac{s-4m_{B}^{2}\mathtt{R}^{2}}{s+4m_{B}^{2}\mathtt{R}^{2}},$
(43)

where $s=4E_{c}^{2}$, one can readily determine $G_{1}$ from
$\Gamma_{J/\psi\to B\bar{B}}$. Since only the relative phase $\Delta\Phi$ is
of interest for this study, without loss of generality, $G_{1}$ can be taken
as a real parameter as will be adopted in the following.

###  III.4 $d_{J}$ from the production of $J/\psi$

Finally, we derive the parity violating parameter $d_{J}$ from the production
of on-shell $J/\psi$ particles in this section. We start by parameterizing the
production amplitude of $J/\psi$ as

$\displaystyle
i\mathcal{M}=\frac{ie^{2}Q_{c}Q_{e}}{s}\left[\bar{v}_{e^{+}}\left(\gamma_{\mu}+d_{J}\gamma_{\mu}\gamma_{5}\right)u_{e^{-}}\right]\left[\bar{u}_{c}\gamma^{\mu}v_{\bar{c}}\right],$
(44)

with $d_{J}$ the parity-violating part of the amplitude. Clearly, the only
tree-level contribution to $d_{J}$ in the SM comes from the $s$-channel $Z$
exchange diagram that violates parity, and a direct matching between the
amplitudes leads to

$\displaystyle d_{J}=\frac{\sqrt{2}sG_{F}}{32\pi\alpha_{\rm
EM}}\cdot(3-8s_{w}^{2}).$ (45)

It shall be understood that $d_{J}$ above is obtained at scale $m_{Z}$ and
thus shall also be evolved to scale $m_{J/\psi}$ in analogy to our discussion
on $F_{A}$ using the renormalization group equations. Recall that since the
quark bilinear is an SU(3)c singlet, pure QCD corrections at the leading order
will be absent in this case. Thus to the leading order, these semi-leptonic
operators only receive mixed quantum electrodynamics (QED) and QCD corrections
from the dipole operators that arise at the loop order in the SM and are also
model-independently very stringently constrained experimentally. We therefore
ignore the running effects on $d_{J}$ and interpret it as the P violating
parameter at the desired scale.

With these results, we close the discussion on form factors and further
rewrite P and CP violating asymmetries defined above into the following
compact forms utilizing these results:

$\begin{split}A_{\rm
PV}^{(1)}&\,=2\alpha_{B}d_{J}\times\frac{1+4y_{m}\mathtt{R}\cos\Delta\Phi}{2+y_{m}^{2}\mathtt{R}^{2}},\qquad\qquad
A_{\rm
PV}^{(2)}=2F_{A}\alpha_{B}\beta^{\frac{3}{2}}\cdot\sqrt{\frac{E_{c}}{3\pi\,\Gamma_{J/\psi\to
B\bar{B}}}}\times\frac{1}{\sqrt{2+y_{m}^{2}\mathtt{R}^{2}}},\\\ A_{\rm
CPV}^{(1)}&\,=\frac{(\beta
E_{c})^{\frac{3}{2}}y_{m}\alpha_{B}\sin\Delta\Phi}{\sqrt{3\pi\,\Gamma_{J/\psi\to
B\bar{B}}}}H_{T}\times\mathcal{F}(\mathtt{R}),\quad A_{\rm
CPV}^{(2)}=-\frac{2(\beta
E_{c})^{\frac{3}{2}}y_{m}\alpha_{B}\bar{\alpha}_{B}\cos\Delta\Phi}{3\sqrt{3\pi\,\Gamma_{J/\psi\to
B\bar{B}}}}H_{T}\times\mathcal{F}(\mathtt{R}),\end{split}$ (46)

with

$\displaystyle\mathcal{F}(\mathtt{R})\equiv\frac{\mathtt{R}}{\sqrt{2+y_{m}^{2}\mathtt{R}^{2}}},$
(47)

where the assumption of real $G_{1}$ is adopted as commented earlier. We
comment that the parameters in these expressions all depend on the type of $B$
in the final state of $J/\psi$ decay except for $E_{c}$ and $d_{J}$,
$\mathtt{R}$ is in short of $\mathtt{R}_{B}$ for instance. Similar to the
hadronic form factors of $J/\psi$ decay, we leave out these
superscripts/subscripts to keep the results neat. It is also interesting to
note that, for both P and CP violating asymmetries from the decay of $J/\psi$,
i.e., $A_{\rm PV}^{(2)}$ and $A_{\rm CPV}^{(1,2)}$, are inversely proportional
to the decay width $\Gamma_{J/\psi\to B\bar{B}}$. This in turn suggests the
possibility of testing the P and the CP symmetries from rare decay processes
of $J/\psi$ that inherit similar decay patterns as discussed above, especially
given that the loss in statistics for these rare processes could possibly be
compensated by the very high luminosity of STCFs. It is noted this enhancement
might be eliminated/further enhanced by the other decay parameters such as
$\alpha$ and $\mathtt{R}$, and a further investment on this possibility is
left for a future work.

Table 3: Decay parameters for $e^{+}e^{-}\to J/\psi\to B\bar{B}$, with different final states shown in the first row. Blue numbers in this table are from Workman et al. (2022). Brown numbers are from table 2 of ref. Dobbs et al. (2014) at $\sqrt{s}=3.770$ GeV and momentum transfer $|Q^{2}|=14.2\rm\,GeV^{2}$, assuming $|G_{E}|=|G_{M}|$. Purple numbers are from the DM2 collaboration in Pallin et al. (1987). The remaining numbers in black are from the corresponding references in the first row. Whenever given, the first number for each parameter corresponds to its central value, followed by its uncertainties added in quadrature from the original references. “(derived)” means results derived from the discussion in section III.3 will be used for the calculation of asymmetries. Parameters |  $\Sigma^{+}\bar{\Sigma}^{-}$ Ablikim et al. (2024) | $\Sigma^{-}\bar{\Sigma}^{+}$ |  $\Sigma^{0}\bar{\Sigma}^{0}$ Ablikim et al. (2017) |  $\Lambda\bar{\Lambda}$ Ablikim et al. (2022c) |  $p\bar{p}$ Ablikim et al. (2020b) |  $\Xi^{0}\bar{\Xi}^{0}$ Ablikim et al. (2023c, d) |  $\Xi^{-}\bar{\Xi}^{+}$ Ablikim et al. (2022a)
---|---|---|---|---|---|---|---
$\sqrt{s}\,(\mathrm{GeV})$ | 2.9000 | — | $m_{J/\psi}$ | $m_{J/\psi}$ | $3.0800$ | $m_{J/\psi}$ | $m_{J/\psi}$
$\alpha_{J/\psi}^{B}$ | $0.35\pm 0.23$ | — | $-0.449\pm 0.022$ | $0.4748\pm 0.0038$ | — | $0.514\pm 0.016$ | $0.586\pm 0.016$
$\alpha_{B}$ | $-0.982\pm 0.14$ | $-0.068\pm 0.008$ | $0.22\pm 0.31$ | $0.7519\pm 0.0043$ | $0.62\pm 0.11$ | $-0.3750\pm 0.0038$ | $-0.376\pm 0.008$
$\bar{\alpha}_{B}$ | $-0.99\pm 0.04$ | — | — | $0.7559\pm 0.0078$ | — | $-0.3790\pm 0.0040$ | $-0.371\pm 0.007$
$\Delta\Phi\left({\rm\,rad.}\right)$ | $1.3614\pm 0.4149$ | — | — | $0.7521\pm 0.0066$ | — | $1.168\pm 0.026$ | $1.213\pm 0.049$
$\left|{G_{E}}/{G_{M}}\right|\equiv\mathtt{R}$ | $0.85\pm 0.22$ | — | $1.04\pm 0.37$ | $0.96\pm 0.14$ | $0.80\pm 0.15$ | $1$ | $1$
$\left|G_{M}\right|(\times 10^{-2})$ | (derived) | — | $0.71\pm 0.09$ | (derived) | $3.47\pm 0.18$ | $0.81\pm 0.21$ | $1.14\pm 0.10$

##  IV Results

With the form factors derived in last section, we present our results for
testing P and CP symmetries in this section based on the most precise
measurements as reported by the BESIII collaboration, as well as the prospects
at STCFs for $J/\psi$ decaying into the octet baryon pair final states.

###  IV.1 Experimental inputs

The experimental inputs we use to estimate the asymmetries are summarized in
table 3, where the numbers in black are from BESIII as referenced in the first
row, those in blue are instead taken from Workman et al. (2022), in brown from
Dobbs et al. (2014) and in purple from Pallin et al. (1987) due to their
limited statistics at BESIII. In particular, for $\Sigma^{+}$ and
$p$,111Results for $\Sigma^{+}$ can also be found in Ablikim et al. (2021a),
which is consistent with those in table 3. We choose the latest results in
Ablikim et al. (2024), which will not modify our conclusion of this work. This
strategy in data selection is similarly adopted for $p$ in Patterion (2017);
Larin (2021); Ablikim et al. (2021b). we select numbers in the original
references for $\sqrt{s}$ close to $m_{J/\psi}$. In table 3, whenever given,
the first number corresponds to its central value, followed by its
uncertainties obtained by adding the reported systematical and statistical
ones in quadrature. In particular, the current precision of these measurements
are dominantly statistical and will thus be further reduced at STCFs.222We
also comment that for the proton $p$, the value $\mathtt{R}=0.47\pm 0.45$ as
reported by the BESIII collaboration Ablikim et al. (2020b) is consistent with
that from the DM2 collaboration in Pallin et al. (1987) but with a larger
uncertainty. For this reason, the result in Pallin et al. (1987) is chosen for
presentation in table 3. In addition, the CP parameters of
$J/\psi\to\Lambda\bar{\Sigma}/\bar{\Lambda}\Sigma^{0}$ are recently reported
in Ablikim et al. (2023b).

###  IV.2 Results for testing P and CP symmetries

Using measurements summarized in table 3, the form factors discussed in last
section and the asymmetries in eq. (46), $A_{\rm PV}^{(1,2)}$ and the baryon
EDMs can be directly obtained for $J/\psi\to B\bar{B}$ with $B=\Lambda$,
$\Sigma^{+}$, and $\Xi^{0,-}$ where the necessary information is known at this
stage. While the current precision is limited by statistics, the STCFs are
expected to improve the precision significantly with their luminosities and
different center-of-mass energy runs. In particular, a complete measurement of
the decay parameters of $\Sigma^{0,-}$ may become possible, the full
information of which is now missing due to limited statistics.

Table 4: P violating asymmetries and baryon EDMs from current measurements summarized in table 3. For the latter, $d_{B}^{(1)}$ ($d_{B}^{(2)}$) corresponds to the upper bounds at 95% CL resulted from $A_{\rm CPV}^{(1)}$ ($A_{\rm CPV}^{(2)}$), assuming statistics dominates the uncertainties at BESIII/STCFs. The last two columns show the statistical uncertainties $\delta$ with 10 billion events from a 12-year running time for BESIII and $t=1$ for one-year data collection at STCF Achasov et al. (2024), assuming the systematical errors are well under control and a detector efficiency of $\epsilon$ indicated in the first column Schönning et al. (2023). P/CP violation | $A_{\rm PV}^{(1)}\,(\times 10^{-4})$ | $A_{\rm PV}^{(2)}\,(\times 10^{-4})$ | ${\sqrt{\epsilon\cdot t}}\cdot d_{B}^{(1)}\,(\times 10^{-18}\,e{\rm\,cm})$ | ${\sqrt{\epsilon\cdot t}}\cdot d_{B}^{(2)}\,(\times 10^{-18}\,e{\rm\,cm})$ | ${\sqrt{\epsilon\cdot t}}\cdot{\delta}\,(\times 10^{-4})$
---|---|---|---|---|---
BESIII | STCF | BESIII | STCF | BESIII | STCF
$\Lambda\,(\epsilon=0.4)$ | 4.42 | 5.45 | 4.64 | 0.25 | 8.64 | 0.47 | 2.30 | 0.13
$\Sigma^{+}\,(\epsilon=0.2)$ | $-3.02$ | 7.80 | 2.58 | 0.14 | 18.4 | 1.00 | 3.06 | 0.17
$\Xi^{0}\,(\epsilon=0.2)$ | $-1.55$ | $-3.23$ | 8.85 | 0.47 | 82.6 | 4.41 | 2.92 | 0.16
$\Xi^{-}\,(\epsilon=0.2)$ | $-1.45$ | $-2.55$ | 8.93 | 0.48 | 95.9 | 5.20 | 3.21 | 0.17

The results for $A_{\rm PV}^{(1,2)}$ and the baryon EDMs are summarized in
table 4, where the 95% CL upper bounds on $d_{B}^{(1)}$ ($d_{B}^{(2)}$) are
obtained from $A_{\rm CPV}^{(1)}$ ($A_{\rm CPV}^{(2)}$) assuming the
statistical uncertainty dominates at BESIII/STCFs with vanishing central
values and a Gaussian distribution. Several comments are in order:

  * •

The P violating asymmetry $A_{\rm PV}^{(1)}$ extracted from the production
side of $J/\psi$ is close to but about a factor of two smaller than that
extracted from the decay of $J/\psi$, i.e., $A_{\rm PV}^{(2)}$, for $\Lambda$,
$\Sigma^{+}$, and $\Xi^{0,-}$. As a result, $A_{\rm PV}^{(2)}$ is practically
more convenient in testing the P symmetry. However, given the detector
efficiency, one realizes that both the two P violating asymmetries are not yet
measurable at BESIII due to the limited statistics. This situation will change
at the next-generation STCFs as indicated in the last column, where all these
four channels can be utilized to test the weak theory from P violation.

  * •

For testing the CP symmetry, we find the baryon EDMs as resulted from $A_{\rm
CPV}^{(1)}$ by projecting the momenta of the secondary baryons along the
direction of the primary baryons from $J/\psi$ decay, i.e., $d_{B}^{(1)}$,
lead to a factor of a few or even one order of magnitude stronger upper bounds
compared with $d_{B}^{(2)}$ from $A_{\rm CPV}^{(2)}$ as measured by projecting
the primary baryon direction onto the decay plane of the secondary baryons.
Recall that, as discussed in section II, the measurement of $A_{\rm
CPV}^{(1)}$ is technically relatively simpler as it does not require a
simultaneous reconstruction of the secondary baryon momenta in contrast with
that of $A_{\rm CPV}^{(2)}$, $A_{\rm CPV}^{(1)}$ would thus be favored for
practical convenience in testing the CP symmetry and measuring the baryon
EDMs. On the other hand, the weaker bound on $d_{B}^{(2)}$, as a consequence
of a relative smaller $A_{\rm CPV}^{(2)}$, renders $A_{\rm CPV}^{(2)}$ more
sensitive to new CP violating sources beyond the SM that are indispensable for
our understanding on the matter-antimatter asymmetry. We also notice that the
estimated 95% CL upper bounds on $d_{\Lambda}$ at BESIII is already about two
orders of magnitude stronger than that in Pondrom et al. (1981) from
Fermilab.333An earlier measurement was presented in Baroni et al. (1971), but
the constraint is weaker. This upper bound can be further improved by about
one more order of magnitude with just one-year data collection at STCFs, which
is also consistent with a recent analysis in Fu et al. (2023). For the other
baryons, the upper bounds on their EDMs are of the same order as that of
$d_{\Lambda}$, but slightly above those predicted directly within the quark
model as summarized in the “Reduced Results” columns in table 1. We emphasis
that at present there are no direct measurements of $\Xi$ and $\Sigma$ EDMs as
can be seen in table 4. The method discussed here will render their direct
measurements possible.

  * •

From eq. (46), the P violating asymmetry $A_{\rm PV}^{(2)}$ is a function of
the hadronic decay parameter $\alpha$, $\mathtt{R}$, and the axial-vector form
factor $F_{A}$ where itself depends on the low-energy constants ($D$, $F$),
the CKM matrix elements $V_{cd,\,cs}$, the number of colors $N_{c}$, and the
weak mixing angle $\theta_{w}$. As a consequence, by measuring $A_{\rm
PV}^{(2)}$ for these octet baryons, together with their correlations, it would
be possible to perform a global analysis based on these asymmetries for a
global determination of these parameters. The implication from this global
analysis could in turn be used for both the unitary test of the second row of
the CKM matrix, and the test of the $\chi$PT/SU(3)${}_{L,\,R}$ symmetry. While
it remains unclear at this stage how significant the implication would be, we
encourage our experimental colleagues in performing such an analysis,
especially for the STCFs due to the foreseen unprecedented precision reach.

  * •

The CP violating asymmetries can also be applied in constraining new CP
violating sources. Given the tininess of the baryon EDMs as predicted in the
SM, a model-independent approach can be adopted within the SMEFT. In this
framework, contact CP violating operators can be interpreted as the sources of
baryon EDMs, see for example ref. Engel et al. (2013) for a thorough review.
As discussed earlier in section III.1 and summarized in table 1, our
decomposition of the baryon EDMs in terms of the quark EDMs and CDMs can be
understood as the dipole operators in this SMEFT framework. Subject to
corrections, for example, from field redefinition and Yukawa modifications
which we ignore in the following, these dipole operators are:

$\begin{split}Q_{fW}=&\,\left\\{\begin{array}[]{cl}(\bar{\mathtt{F}}\sigma^{\mu\nu}f_{R})\tau^{I}{\varphi}W_{\mu\nu}^{I},&T_{3}^{{\mathtt{F}}}=-\frac{1}{2}\\\
(\bar{\mathtt{F}}\sigma^{\mu\nu}f_{R})\tau^{I}\widetilde{\varphi}W_{\mu\nu}^{I},&T_{3}^{{\mathtt{F}}}=\frac{1}{2}\end{array}\right.,\quad
Q_{fB}=\left\\{\begin{array}[]{cl}(\bar{\mathtt{F}}\sigma^{\mu\nu}f_{R}){\varphi}B_{\mu\nu},&T_{3}^{{\mathtt{F}}}=-\frac{1}{2}\\\
(\bar{\mathtt{F}}\sigma^{\mu\nu}f_{R})\widetilde{\varphi}B_{\mu\nu},&T_{3}^{{\mathtt{F}}}=\frac{1}{2}\end{array}\right.\\\
Q_{uG}=&\,(\bar{Q}\sigma^{\mu\nu}T^{A}u_{R})\widetilde{\varphi}G_{\mu\nu}^{A},\qquad\qquad\qquad\quad\,Q_{dG}=(\bar{Q}\sigma^{\mu\nu}T^{A}d_{R}){\varphi}G_{\mu\nu}^{A},\end{split}$
(48)

with ${\mathtt{F}}$ the SM fermion fields and $f_{R}$ the corresponding SU(2)L
singlet fermions, $I$ the SU(2)L index, $u_{R}$ the up-type and $d_{R}$ the
down-type SU(2)L singlet quarks. $\varphi$ is the Higgs doublet with
$\widetilde{\varphi}=i\sigma_{2}\varphi^{\ast}$ and $\sigma_{2}$ the second
Pauli matrix. We then parameterize the dipole Lagrangian as

$\displaystyle\mathcal{L}_{\rm
dipole}=c_{fB}g_{1}Q_{fB}+c_{fW}g_{2}Q_{fW}+c_{uG}g_{3}Q_{uG}+c_{dG}g_{3}Q_{dG}+{\rm
h.c.},$ (49)

where $g_{1}$, $g_{2}$ and $g_{3}$ are, respectively, the U(1)Y, SU(2)L and
SU(3)c gauge couplings. $c_{fW,fB,uG,dG}$ are Wilson coefficients with mass
dimension $-2$, and ${\rm h.c.}$ stands for hermitian conjugate. Then after
the spontaneous symmetry breaking and by matching onto the quark EDM and CDM
operators as parameterized in eqs. (53) and (55), one readily finds the quark
EDMs $d_{q}$ and CDMs $f_{q}$ as given by

$\displaystyle d_{q}=-\sqrt{2}v\,e\,{\rm Im}\left(c_{qB}-c_{qW}\right),\quad
f_{q}=-\sqrt{2}v\,{\rm Im}\left(c_{qG}\right),$ (50)

with $v\simeq 246\rm\,GeV$ the Higgs vacuum expectation value. The strongest
upper bounds from $d_{\Lambda}^{(1)}$ as summarized in table 4 would then
imply, as estimated in the quark model summarized in table 1,

$\begin{split}&\,\left|{\rm Im}\left(c_{sB}-c_{sW}\right)\right|\lesssim
1.07\,[{\rm\,TeV}^{-2}],\quad\left|{\rm Im}\left(c_{sG}\right)\right|\lesssim
3.21\,[{\rm\,TeV}^{-2}],\qquad\text{(For BESIII)}\\\ &\,\left|{\rm
Im}\left(c_{sB}-c_{sW}\right)\right|\lesssim
0.06\,[{\rm\,TeV}^{-2}],\quad\left|{\rm Im}\left(c_{sG}\right)\right|\lesssim
0.17\,[{\rm\,TeV}^{-2}],\qquad\text{(For STCFs)}\\\ \end{split}$ (51)

corresponding to the sensitivity to new physics around the TeV scale for
$\mathcal{O}(1)$ Wilson coefficients.

###  IV.3 Weak mixing angle determination

The P violating asymmetries $A_{\rm PV}^{(1,\,2)}$ from the production and
decay of $J/\psi$ are sensitive to the weak mixing angle $\theta_{w}$ as seen
in eq. (40) and (45), whose precision measurements have played a key role in
both the development and the subsequent precision tests of the SM. Therefore,
we think it interesting to investigate the precision to which this parameter
can be determined with these P violating asymmetries from $J/\psi$ production
and decay, which will be our focus of this subsection.

Table 5: Precision determination of the weak mixing angle at BESIII using $A_{\rm PV}^{(1)}$ from the production of $J/\psi$, along with the components of its uncertainty. Baryons | $\mathtt{a}_{0}\,(\times 10^{-2})$ | $\frac{\delta m_{B}}{m_{B}}\,(\times 10^{-6})$ | $\mathtt{a}_{1}$ | $\frac{\delta m_{J/\psi}}{m_{J/\psi}}\,(\times 10^{-6})$ | $\mathtt{a}_{2}\,(\times 10^{-2})$ | $\frac{\delta\mathtt{R}}{\mathtt{R}}$ | $\mathtt{a}_{3}$ | $\frac{\delta{\alpha}}{\alpha}\,(\times 10^{-2})$
---|---|---|---|---|---|---|---|---
$\Lambda$ | 16.7 | 5.38 | 1.02 | 1.94 | 16.7 | 0.15 | 0.59 | 0.34
$\Sigma^{+}$ | $2.84\times 10^{-2}$ | 58.9 | 1.18 | 1.94 | $2.84\times 10^{-2}$ | 0.26 | 0.59 | 14.3
$\Xi^{0}$ | 2.43 | 152 | 1.16 | 1.94 | 2.43 | — | 0.59 | 1.46
$\Xi^{-}$ | 0.62 | 53.0 | 1.18 | 1.94 | 0.62 | — | 0.59 | 2.85
(continued) | $\mathtt{a}_{4}$ | $\frac{\delta\Delta\Phi}{\Delta\Phi}\,(\times 10^{-2})$ | $\mathtt{a}_{5}$ | $\frac{\sqrt{\epsilon}\delta A_{\rm PV}^{(1)}}{A_{\rm PV}^{(1)}}$ | $(s_{w}^{2}\pm\delta s_{w}^{2})_{\rm BESIII}$
$\Lambda$ | 0.28 | 0.88 | 0.59 | 0.52 | $0.2355\pm 0.1149$
$\Sigma^{+}$ | 1.33 | 30.5 | 0.59 | 1.01 | $0.2355\pm 0.3308$
$\Xi^{0}$ | 0.93 | 2.23 | 0.59 | 1.88 | $0.2355\pm 0.5866$
$\Xi^{-}$ | 1.05 | 4.04 | 0.59 | 2.22 | $0.2355\pm 0.6924$

Due to the simpler form, we start with $A_{\rm PV}^{(1)}$ by taking
$(\alpha_{\rm EM},\,m_{Z},\,G_{F})$ as our inputs and ignoring their
uncertainties due to the precision. The uncertainty in $s_{w}^{2}$ in this
case can then be decomposed as follows

$\displaystyle\frac{\delta s_{w}^{2}}{s_{w}^{2}}=\mathtt{a}_{0}\frac{\delta
m_{B}}{m_{B}}\oplus\mathtt{a}_{1}\,\frac{\delta
m_{J/\psi}}{m_{J/\psi}}\oplus\mathtt{a}_{2}\,\frac{\delta\mathtt{R}}{\mathtt{R}}\oplus\mathtt{a}_{3}\,\frac{\delta\alpha}{\alpha}\oplus\mathtt{a}_{4}\,\frac{\delta\Delta\Phi}{\Delta\Phi}\oplus\mathtt{a}_{5}\frac{\delta
A_{\rm PV}^{(1)}}{A_{\rm PV}^{(1)}},$ (52)

with $\delta x$ the absolute uncertainty in $x$ and the coefficients of each
contributing uncertainty summarized in table 5. The absolute uncertainties in
$s_{w}^{2}$ for each baryon final state can then be calculated by adding all
the uncertainties in quadrature with the experimental inputs summarized in
table 3. For the asymmetries, as commented above, we take their respective
statistical uncertainties as dominant. The results are then shown in the last
column, along with the decomposition of $\delta s_{w}^{2}$ into its
components. Recall that since we use $s_{w}^{2}$ at $m_{J/\psi}$ as our
benchmark, the central values of $s_{w}^{2}$ from different decay channels all
align with the SM prediction $s_{w}^{2}(m_{J/\psi})=0.2355$. Clearly, it is
seen from table 3 that the uncertainties in $s_{w}^{2}$ from both $\delta
m_{B}$ and $\delta m_{J/\psi}$ are generically negligible. In contrast,
uncertainties from the other sources, viz, the baryon decay parameters and the
P violating asymmetries, are comparable and dominate the final uncertainty in
$s_{w}^{2}$, which are all expected to be reduced at STCFs. The current
precision of these decay parameters and the asymmetries hinders the
determination $s_{w}^{2}$, whose absolute uncertainty is found to be quite
large, with the best result $\delta s_{w}^{2}=0.1149$ from the $\Lambda$
channel.

Figure 1: Weak mixing angle determination from different experiments. The blue
curve represents the SM prediction of the running of the $s_{w}^{2}$ in Erler
and Ramsey-Musolf (2005). Black data points are from existing experiments as
indicated by the legends, and the result from BESIII as obtained in this work
is shown in purple with its central value from Erler and Ramsey-Musolf (2005)
as a benchmark, and similarly for the prospect of STCFs as shown in orange.
$s_{w}^{2}$ from Tevatron and the LHC are both obtained at the $Z$ pole but
shifted horizontally for clarity as in literature.

A similar analysis can be performed using $A_{\rm PV}^{(2)}$, as well as a
combined analysis of $A_{\rm PV}^{(1)}$ and $A_{\rm PV}^{(2)}$ fully utilizing
the data from both the production and the decay sides of $J/\psi$. Both these
two cases are investigated and we refer the readers to appendix C for details.
In each case, new parametric uncertainties enter the total error in
$s_{w}^{2}$, those for the low-energy constants $D$ and $F$ for instance. With
$A_{\rm PV}^{(2)}$ alone and as can be seen in table 6, we find the
uncertainties in $s_{w}^{2}$ comparable but slightly smaller than the
corresponding ones from $A_{\rm PV}^{(1)}$, partially due to the enhancement
from relatively larger $A_{\rm PV}^{(2)}$’s. On the other hand, a combined
analysis with both $A_{\rm PV}^{(1)}$ and $A_{\rm PV}^{(2)}$, as shown in
table 7, leads to $\delta s_{w}^{2}=0.0788$ as the most precise determination
of $s_{w}^{2}$ from the $\Lambda$ baryon decay channel compared with all the
other processes considered in this work, corresponding to a relative
determination at the $\sim 30\%$ level. Though not yet competitive with the
per mille level precision reach of the next-generation dedicated low-energy
experiments such as the MOLLER experiment at the Jefferson Lab Benesch et al.
(2014) and the P2 experiment at Mainz Berger et al. (2016), this measurement
at BESIII provides an independent test of the SM at the $J/\psi$ threshold,
where our analysis provides a guidance for devoting efforts in reducing the
dominant uncertainties.444It is also noted that a comparable precision with
MOLLER/P2 could be achieved at the STCFs using the left-right asymmetry with
80% beam polarization Bondar et al. (2020); Fu et al. (2023). Furthermore, at
STCFs, the precision reach in $s_{w}^{2}$ is expected to be further improved
by a factor of about 20 within just one year based on rescaling. For
comparison, this determination of $s_{w}^{2}$ from
$J/\psi\to\Lambda\bar{\Lambda}$ at BESIII and its prospect at STCFs are also
pictorially shown in figure 1 respectively in purple and orange, with the blue
curve the SM prediction from Erler and Ramsey-Musolf (2005) and the
experimental data points in black taken from the summary report in Workman et
al. (2022).

##  V Conclusions

In this work, we focus on threshold production of $J/\psi$ at unpolarized
electron-positron colliders and its subsequent decay into the lowest-lying
baryons for testing the P and the CP symmetries. Since the production and
decay of $J/\psi$ occur at a very low energy scale compared with $m_{Z}$, the
large logarithms resulted from the large energy gap between the weak scale and
the $J/\psi$ decay scale are resummed by solving the renormalization group
equations up to $\mathcal{O}(\alpha_{s})$. The effects from the running are
found significant, and are also quantitatively summarized in table 2 for all
the lowest-lying baryons. As a consequence, the axial-vector form factor are
either reduced or enhanced by a factor of a few, in addition with sign changes
in some cases such as $F_{A}^{\Xi^{0}}$ and $F_{A}^{\Lambda}$. In paticular,
$F_{A}^{\Sigma^{0}}$ is remarkably reduced by a factor of about $40$ such that
its magnitude is two to three orders of magnitude smaller than the other octet
baryons, rendering it challenging as a candidate for testing the P symmetry
unless enhanced by new mechanism(s) beyond the SM.

With all the form factors and the decay parameters of $\Lambda$, $\Sigma^{+}$,
and $\Xi^{0,-}$ as reported by the BESIII collaboration using the 10 billion
$J/\psi$ events collected at BESIII up to now, we find both P violating
asymmetries $A_{\rm PV}^{(1,\,2)}$ of $\mathcal{O}(10^{-4})$. These
asymmetries are currently slightly smaller than their statistical
uncertainties such that they are not yet observable. This will change with the
increased luminosity of STCFs as seen in the last column of table 4, where the
statistical uncertainties are expected to be reduced by at least a factor of
10. On the other hand, as presented in the third and fourth columns of table
4, the 95% CL upper bounds on the EDMs of the octet baryons are found to be of
$\mathcal{O}(10^{-18})\,e\rm\,cm$ at BESIII. Interestingly, these are already
two orders of magnitude stronger than that on $\Lambda$ from Fermilab.
Assuming similarly that statistics dominates the uncertainties at STCFs, these
upper bounds on the baryon EDMs can be further lifted by about a factor of 10
with just one-year data collection, corresponding to three orders of magnitude
improvement compared with that from Fermilab. Especially, a direct probe of
the $\Xi$ and $\Sigma$ EDMs can also be achieved with the method discussed in
this work.

Finally, we also estimate the uncertainty in the weak mixing angle $s_{w}$
from the P violating asymmetries $A_{\rm PV}^{(1,\,2)}$. In each case,
$J/\psi\to\Lambda\bar{\Lambda}$ leads to the most precise determination of
$s_{w}$ compared with the other baryon final states, thanks to the abundant
$J/\psi$ production and the large detector efficiency for $\Lambda$ at BESIII.
While the absolute uncertainty in $s_{w}^{2}$ is $\delta s_{w}^{2}\approx 0.1$
using either $A_{\rm PV}^{(1)}$ from the production or $A_{\rm PV}^{(2)}$ from
the decay of $J/\psi$, a combined analysis using both data sets reduces this
uncertainty down to $\delta s_{w}^{2}\approx 0.08$ as summarized in table 7
and pictorially shown in figure 1. Though this current uncertainty is still
large, it provides an independent measurement of $s_{w}^{2}$ at the mass pole
of $J/\psi$ and our analysis also provides a direction for reducing the
dominant uncertainties in the determination of $s_{w}$ in the future. We also
expect the precision of $s_{w}$ to be improved at the next-generation STCFs to
$\delta s_{w}^{2}\approx 0.004$ as a result of higher luminosities. In this
regard, the higher luminosities of next-generation STCFs also render it
possible to both improve the current precision in determining the decay
parameters of $\Lambda$, $\Sigma^{+}$, and $\Xi^{0,-}$ and the measurements of
the other baryons, i.e., $\Sigma^{0,+}$ that are currently rather limited by
statistics. As a consequence, a global analysis of $J/\psi$ decaying into all
the hyperon final states might become possible, along with a unitary test of
the second row of the CKM matrix.

## Acknowledgements

YD is indebted to the committee and participants of the Workshop on Hyperon
Physics hyp  in Huizhou for communications on BESIII and STCF, especially to
Prof. Shuangshi Fang for the discussion on detector efficiencies. YD also
thanks Prof. Michael Ramsey-Musolf for helpful discussions, and Dr. Chia-Wei
Liu for the discussion on the $t$-channel $W$ exchange process and its
renormalization group evolution. This work was partially supported by the
Fundamental Research Funds for the Central Universities. XGH was also
partially supported by NSFC grant under numbers 12375088 and 12090064. 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. YD was partially supported by NSFC Special Funds for Theoretical
Physics under grant number 12347116, by the China Postdoctoral Science
Foundation under grant number 2023M732255, and by the Postdoctoral Fellowship
Program of CPSF under number GZC20231613. YD also acknowledges support from
the Shanghai Super Postdoc Incentive Plan, and the T.D. Lee Postdoctoral
Fellowship at the Tsung-Dao Lee Institute, Shanghai Jiao Tong University.

##  Appendix A Baryon EDMs in the Quark Model

As discussed in the main text, we consider the baryon EDMs $d_{B}$ as sourced
from their quark components. In the effective language, the quarks have both
electric and chromo dipole moments, which we discuss case by case below.

####  A.0.1  Contributions from quark EDMs

The quark electric dipole moment (EDM) Lagrangian can be parameterized as

$\displaystyle\mathcal{L}_{q}=-\frac{i}{2}d_{q}\bar{q}\sigma^{\mu\nu}\gamma_{5}qF_{\mu\nu},$
(53)

with $d_{q}$ the quark EDM. In this work, we use the quark model to find their
contributions to $d_{B}$, which predicts

$\begin{split}d_{p}^{\rm qEDM}&\,=\frac{1}{3}(4d_{u}-d_{d}),\quad d_{n}^{\rm
qEDM}=\frac{1}{3}(4d_{d}-d_{u}),\\\ d_{\Sigma^{+}}^{\rm
qEDM}&\,=\frac{1}{3}(4d_{u}-d_{s}),\quad d_{\Sigma^{0}}^{\rm
qEDM}=\frac{1}{3}(2d_{u}+2d_{d}-d_{s}),\quad d_{\Sigma^{-}}^{\rm
qEDM}=\frac{1}{3}(4d_{d}-d_{s}),\\\ d_{\Xi^{0}}^{\rm
qEDM}&\,=\frac{1}{3}(4d_{s}-d_{u}),\quad d_{\Xi^{-}}^{\rm
qEDM}=\frac{1}{3}(4d_{s}-d_{d}),\quad d_{\Lambda^{0}}^{\rm
qEDM}=d_{s}.\end{split}$ (54)

####  A.0.2  Contributions from quark CDMs

Similarly, we write the quark chromo dipole moment (CDM) Lagrangian as

$\displaystyle-\mathcal{L}_{g}=\mathcal{H}_{g}=g_{s}f_{q}\frac{1}{2}\bar{q}\sigma_{\mu\nu}\gamma_{5}T^{A}qG^{\mu\nu}_{A},$
(55)

with $g_{s}$ the SU(3)c gauge coupling, $f_{q}$ the quark CDM, $T^{A}$ the
Gell-Mann matrices, and $G$ the gluon field. To find its contribution to
$d_{B}$, we utilize the non-relativistic quark model and treat
$\mathcal{H}_{g}$ perturbatively Khriplovich and Zhitnitsky (1981); He et al.
(1989). In this approximation, one neglects the “induced” gluon fields as well
as its time variation in the non-relativistic limit in analogy with classical
electrodynamics, and the perturbation part $\mathcal{H}_{g}$ reduces to

$\displaystyle\mathcal{H}_{g}=-g_{s}\sum_{k}f_{q}\bm{\sigma}_{k}(-\nabla_{\bm{x}_{k}}G_{0}^{A}(\bm{x}_{k}))T_{k}^{A},$
(56)

with $G_{0}^{a}$ the time component of the gluon field potential,
$\bm{\sigma}$ the Pauli matrices, and the summation over the internal degrees
of freedom. In this non-relativistic picture, recall that the unperturbed QCD
Hamiltonian is given by

$\displaystyle\mathcal{H}_{0}=\sum_{k}\left(\frac{p_{k}^{2}}{2m}+g_{s}T_{k}^{A}G_{0}^{A}(\bm{x}_{k})\right),$
(57)

one can readily find

$\displaystyle\mathcal{H}_{g}=i\sum_{k}f_{q}\bm{\sigma}_{k}\cdot[\bm{p}_{k},\mathcal{H}_{0}],$
(58)

such that the modified baryon wave function becomes

$\displaystyle\ket{\tilde{B}}$
$\displaystyle\,=\left(1+i\sum\limits_{k}f_{q}\bm{\sigma}_{k}\cdot\bm{p}_{k}\right)\ket{B},$
(59)

with $\ket{\tilde{B}}$ and $\ket{B}$ the perturbed and unperturbed
wavefunction for $B$. It is then straightforward to calculate $d_{B}$ from
$f_{q}$ as given below:

$\begin{split}d_{n}^{\rm
qCDM}&\,=\frac{1}{3}\left(-4Q_{d}f_{d}+Q_{u}f_{u}\right),\quad d_{p}^{\rm
qCDM}=\frac{1}{3}\left(-4Q_{u}f_{u}+Q_{d}f_{d}\right),\\\ d_{\Sigma^{+}}^{\rm
qCDM}&\,=\frac{1}{3}\left(-4Q_{u}f_{u}+Q_{s}f_{s}\right),\quad
d_{\Sigma^{0}}^{\rm
qCDM}=\frac{1}{3}\left(-2Q_{u}f_{u}-2Q_{d}f_{d}+Q_{s}f_{s}\right),\quad
d_{\Sigma^{-}}^{\rm qCDM}=\frac{1}{3}\left(-4Q_{d}f_{d}+Q_{s}f_{s}\right),\\\
d_{\Xi^{0}}^{\rm
qCDM}&\,=\frac{1}{3}\left(-4Q_{s}f_{s}+Q_{u}f_{u}\right),\quad
d_{\Xi^{-}}^{\rm qCDM}=\frac{1}{3}\left(-4Q_{s}f_{s}+Q_{d}f_{d}\right),\quad
d_{\Lambda^{0}}^{\rm qCDM}=-Q_{s}f_{s},\end{split}$ (60)

with $Q_{q}$ the electric charge of quark $q$ in unit of $e$.

##  Appendix B $F_{A}$ derivation in $\chi$PT

Under ${\rm SU(3)}_{L}\otimes{\rm SU(3)}_{R}$, the lowest-lying baryons form
an octet and can be parameterized as555We follow the notations established in
Scherer and Schindler (2012) in this work.

$\displaystyle B$
$\displaystyle\,=\sum_{a=1}^{8}\frac{B_{a}\lambda_{a}}{\sqrt{2}}=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda\end{array}\right),$ (64)

and the lowest-order Lagrangian respecting this symmetry is given by

$\displaystyle\mathcal{L}_{MB}^{(1)}={\rm
Tr}\left[\bar{B}(i\not{D}-M_{0})B\right]-\frac{D}{2}{\rm
Tr}\left[\bar{B}\gamma^{\mu}\gamma_{5}\left\\{u_{\mu},B\right\\}\right]-\frac{F}{2}{\rm
Tr}\left[\bar{B}\gamma^{\mu}\gamma_{5}\left[u_{\mu},B\right]\right],$ (65)

with $F$ and $D$ constants determined from semi-leptonic $B$ decays. From
tree-level fitting, it was found that $F=0.50$ and $D=0.80$ Borasoy (1999),
and a recent next-to-leading order analysis up to $\mathcal{O}(p^{3})$ in
$\chi$PT obtained $F=0.441(47)(2)$ and $D=0.623(61)(17)$ Ledwig et al. (2014),
the latter of which will be used in this work. The unitary square root $u$,
the chiral connection $\Gamma_{\mu}$, the chiral vielbein $u_{\mu}$, and the
baryon covariant derivative $D_{\mu}B$, are defined, respectively, as

$\begin{split}&\,u=U^{\frac{1}{2}}=\exp\left(i\frac{\phi}{2F_{0}}\right),\quad\Gamma_{\mu}=\frac{1}{2}\left[u^{\dagger}\left(\partial_{\mu}-ir_{\mu}\right)u+u\left(\partial_{\mu}-il_{\mu}\right)u^{\dagger}\right],\\\
&\,u_{\mu}\equiv
i\left[u^{\dagger}\left(\partial_{\mu}-ir_{\mu}\right)u-u\left(\partial_{\mu}-il_{\mu}\right)u^{\dagger}\right],\quad
D_{\mu}B=\partial_{\mu}B+[\Gamma_{\mu},B],\end{split}$ (66)

with $F_{0}\approx F_{\pi}=92.4\rm\,MeV$ and $F_{\pi}$ the pion decay
constant, and $U$ the transformation matrix for the meson fields $\phi$
parameterized as

$\displaystyle\phi$
$\displaystyle\,=\sum_{a=1}^{8}\phi_{a}\lambda_{a}=\left(\begin{array}[]{ccc}\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-\frac{2}{\sqrt{3}}\eta\end{array}\right).$
(70)

$r_{\mu}$ and $l_{\mu}$ account for the external sources. Plugging the quark
currents as represented by terms inside the square bracket of eq. (38) into
eq. (65) and then matching with eq. (8), one obtains results in eq. (40).

Table 6: Precision determination of the weak mixing angle at BESIII using $A_{\rm PV}^{(2)}$ from the decay of $J/\psi\to B\bar{B}$ with $B=\Lambda,\Sigma^{+},\Xi^{0,-}$, along with the components of its uncertainty. Baryons | $\mathtt{b}_{0}$ | $\frac{\delta m_{B}}{m_{B}}\,(\times 10^{-6})$ | $\mathtt{b}_{1}$ | $\frac{\delta m_{J/\psi}}{m_{J/\psi}}\,(\times 10^{-6})$ | $\mathtt{b}_{2}$ | $\frac{\delta\mathtt{R}}{\mathtt{R}}$ | $\mathtt{b}_{3}$ | $\frac{\delta{\alpha}}{\alpha}\,(\times 10^{-2})$ | $\mathtt{b}_{4}$ | $\frac{\delta V_{cs}}{V_{cs}}$ | $\mathtt{b}_{5}$ | $\frac{\delta{\Gamma_{J/\Psi\to B\bar{B}}}}{{\Gamma_{J/\Psi\to B\bar{B}}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\Lambda$ | 1.24 | 5.38 | 1.58 | 1.94 | 0.13 | 0.15 | 0.68 | 0.34 | 0.14 | 0.011 | 0.34 | 0.048
$\Sigma^{+}$ | 1.38 | 58.9 | 1.68 | 1.94 | 0.10 | 0.26 | 0.59 | 14.3 | 0.04 | 0.011 | 0.30 | 0.037
$\Xi^{0}$ | 2.77 | 152 | 3.10 | 1.94 | 0.18 | — | 0.67 | 1.46 | 0.14 | 0.011 | 0.33 | 0.034
$\Xi^{-}$ | 3.01 | 53.0 | 3.36 | 1.94 | 0.19 | — | 0.70 | 2.85 | 0.23 | 0.011 | 0.35 | 0.082
(continued) | $\mathtt{b}_{6}$ | $\frac{\sqrt{\epsilon}\delta A_{\rm PV}^{(2)}}{A_{\rm PV}^{(2)}}$ | $\mathtt{b}_{7}\,(\times 10^{-2})$ | $\frac{\delta D}{D}$ | $\mathtt{b}_{8}$ | $\frac{\delta F}{F}$ | $\mathtt{b}_{9}$ | $\frac{\delta g_{V}}{g_{V}}\,(\times 10^{-3})$ | $(s_{w}^{2}\pm\delta s_{w}^{2})_{\rm BESIII}$
$\Lambda$ | 0.52 | 0.42 | 68.1 | 0.10 | 0 | 0.11 | 0.68 | 9.59 | $0.2355\pm 0.1086$
$\Sigma^{+}$ | 0.57 | 0.39 | 0.07 | 0.10 | 0.59 | 0.11 | 0.59 | 9.59 | $0.2355\pm 0.1250$
$\Xi^{0}$ | 0.53 | 0.91 | 63.6 | 0.10 | 0.03 | 0.11 | 0.67 | 9.59 | $0.2355\pm 0.3192$
$\Xi^{-}$ | 0.54 | 1.26 | 6.52 | 0.10 | 0.64 | 0.11 | 0.70 | 9.59 | $0.2355\pm 0.4647$
Table 7: Precision determination of the weak mixing angle at BESIII from a combined analysis $A_{\rm PV}^{(1,\,2)}$, along with the components of its uncertainty. Baryons | $\mathtt{c}_{0}$ | $\frac{\delta m_{B}}{m_{B}}\,(\times 10^{-6})$ | $\mathtt{c}_{1}$ | $\frac{\delta m_{J/\psi}}{m_{J/\psi}}\,(\times 10^{-6})$ | $\mathtt{c}_{2}$ | $\frac{\delta{\Gamma_{J/\Psi\to B\bar{B}}}}{{\Gamma_{J/\Psi\to B\bar{B}}}}$ | $\mathtt{c}_{3}$ | $\frac{\delta\mathtt{R}}{\mathtt{R}}$ | $\mathtt{c}_{4}$ | $\frac{\delta{\alpha}}{\alpha}$ | $\mathtt{c}_{5}$ | $\frac{\delta V_{cs}}{V_{cs}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\Lambda$ | 0.56 | 5.38 | 1.31 | 1.94 | 0.18 | 0.048 | 0.01 | 0.15 | 0.64 | 0.003 | 0.07 | 0.011
$\Sigma^{+}$ | 2.26 | 58.9 | 1.99 | 1.94 | 0.48 | 0.037 | 0.17 | 0.26 | 0.59 | 0.14 | 0.06 | 0.011
$\Xi^{0}$ | 1.79 | 152 | 2.42 | 1.94 | 0.22 | 0.034 | 0.11 | — | 0.64 | 0.01 | 0.09 | 0.011
$\Xi^{-}$ | 1.80 | 53.0 | 2.48 | 1.94 | 0.21 | 0.082 | 0.11 | — | 0.66 | 0.03 | 0.14 | 0.011
(continued) | $\mathtt{c}_{6}$ | $\frac{\delta D}{D}$ | $\mathtt{c}_{7}$ | $\frac{\delta F}{F}$ | $\mathtt{c}_{8}$ | $\frac{\delta\Delta\Phi}{\Delta\Phi}$ | $\mathtt{c}_{9}$ | $\frac{\delta g_{V}}{g_{V}}$ | $\mathtt{c}_{10}$ | $\frac{\sqrt{\epsilon}\delta A_{\rm PV}^{(1)}}{A_{\rm PV}^{(1)}}$ | $\mathtt{c}_{11}$ | $\frac{\sqrt{\epsilon}\delta A_{\rm PV}^{(2)}}{A_{\rm PV}^{(2)}}$
$\Lambda$ | 0.35 | 0.10 | 0 | 0.11 | 0.13 | 0.01 | 0.35 | 0.01 | 0.29 | 0.56 | 0.35 | 0.45
$\Sigma^{+}$ | 0.001 | 0.10 | 0.10 | 0.11 | 0.84 | 0.31 | 0.97 | 0.01 | 0.37 | 1.09 | 0.97 | 0.42
$\Xi^{0}$ | 0.41 | 0.10 | 0.02 | 0.11 | 0.33 | 0.02 | 0.43 | 0.01 | 0.21 | 2.02 | 0.43 | 0.98
$\Xi^{-}$ | 0.04 | 0.10 | 0.38 | 0.11 | 0.42 | 0.04 | 0.42 | 0.01 | 0.24 | 2.39 | 0.42 | 1.36
$(s_{w}^{2}\pm\delta s_{w}^{2})_{\rm BESIII}$ | $\left(0.2355\pm 0.0788\right)_{\Lambda}\quad\left(0.2355\pm 0.1296\right)_{\Sigma^{+}}\quad\left(0.2355\pm 0.2923\right)_{\Xi^{0}}\quad\left(0.2355\pm 0.3932\right)_{\Xi^{-}}$

##  Appendix C Weak mixing angle from $A_{\rm PV}^{(2)}$ and the combination
of $A_{\rm PV}^{(1,\,2)}$

Similar to the discussion on weak mixing angle determination with $A_{\rm
PV}^{(1)}$, we present our discussion on this point utilizing $A_{\rm
PV}^{(2)}$ and a combined analysis of $A_{\rm PV}^{(1,\,2)}$ in the following.
For that with $A_{\rm PV}^{(2)}$, we decompose the uncertainty in $s_{w}$ as
follows:

$\displaystyle\frac{\delta s_{w}^{2}}{s_{w}^{2}}=\mathtt{b}_{0}\frac{\delta
m_{B}}{m_{B}}\oplus\mathtt{b}_{1}\,\frac{\delta
m_{J/\psi}}{m_{J/\psi}}\oplus\mathtt{b}_{2}\,\frac{\delta\mathtt{R}}{\mathtt{R}}\oplus\mathtt{b}_{3}\,\frac{\delta\alpha}{\alpha}\oplus\mathtt{b_{4}}\,\frac{\delta
V_{cs}}{V_{cs}}\oplus\mathtt{b}_{5}\,\frac{\delta{\Gamma_{J/\Psi\to
B\bar{B}}}}{{\Gamma_{J/\Psi\to B\bar{B}}}}\oplus\mathtt{b}_{6}\frac{\delta
A_{\rm PV}^{(2)}}{A_{\rm PV}^{(2)}}\oplus\mathtt{b}_{7}\frac{\delta
D}{D}\oplus\mathtt{b}_{8}\frac{\delta F}{F}\oplus\mathtt{b}_{9}\frac{\delta
g_{V}}{g_{V}},$ (71)

and summarize the decomposition and the final uncertainty in $s_{w}$ in table
6. Similar to the analysis for $A_{\rm PV}^{(1)}$, it is clear that the total
uncertainty in $s_{w}^{2}$ is about 0.1 and quite large, dominated by the
precision in the decay parameters, the asymmetry $A_{\rm PV}^{(2)}$, and the
low-energy constants. While largely rooted in limited statistics at BESIII for
the first two sources, the uncertainty in the last one, i.e., the low-energy
constant, can be improved either through a lattice calculation, or a global
fit to data with more statistics in the future.

The large uncertainty in $s_{w}$ from either $A_{\rm PV}^{(1)}$ or $A_{\rm
PV}^{(2)}$ can be reduced by combining the two asymmetries from the production
and the decay of $J/\psi$. In this case, the uncertainty in $s_{w}^{2}$ can be
decomposed as:

$\begin{split}\frac{\delta s_{w}^{2}}{s_{w}^{2}}=&\,\mathtt{c}_{0}\frac{\delta
m_{B}}{m_{B}}\oplus\mathtt{c}_{1}\,\frac{\delta
m_{J/\psi}}{m_{J/\psi}}\oplus\mathtt{c}_{2}\frac{\delta{\Gamma_{J/\Psi\to
B\bar{B}}}}{{\Gamma_{J/\Psi\to
B\bar{B}}}}\oplus\mathtt{c}_{3}\,\frac{\delta\mathtt{R}}{\mathtt{R}}\oplus\mathtt{b_{4}}\,\frac{\delta\alpha}{\alpha}\oplus\mathtt{c}_{5}\frac{\delta
V_{cs}}{V_{cs}}\\\ &\,\oplus\mathtt{c}_{6}\frac{\delta
D}{D}\oplus\mathtt{c}_{7}\frac{\delta
F}{F}\oplus\mathtt{c}_{8}\frac{\delta\Delta\Phi}{\Delta\Phi}\oplus\mathtt{c}_{9}\frac{\delta
g_{V}}{g_{V}}\oplus\mathtt{c}_{10}\frac{\delta A_{\rm PV}^{(1)}}{A_{\rm
PV}^{(1)}}\oplus\mathtt{c}_{11}\frac{\delta A_{\rm PV}^{(2)}}{A_{\rm
PV}^{(2)}},\end{split}$ (72)

with the coefficients and the final uncertainty from different decay channels
summarized in table 7. Note that to enhance the asymmetry, we sum over these
two asymmetries for $\Lambda$ and $\Xi^{0,-}$ while subtract them for
$\Sigma^{+}$ for this combined analysis. Once again, we find
$J/\psi\to\Lambda\bar{\Lambda}$ leads to the most precise determination of
$s_{w}^{2}$ with an absolute uncertainty $\delta s_{w}^{2}=0.0788$ compared
with the other baryon channels.

## References

  * Lee and Yang (1956) T. D. Lee and C.-N. Yang, Phys. Rev. 104, 254 (1956).
  * Wu et al. (1957) C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 105, 1413 (1957).
  * Garwin et al. (1957) R. L. Garwin, L. M. Lederman, and M. Weinrich, Phys. Rev. 105, 1415 (1957).
  * Friedman and Telegdi (1957) J. I. Friedman and V. L. Telegdi, Phys. Rev. 106, 1290 (1957).
  * Lee et al. (1957) T. D. Lee, R. Oehme, and C.-N. Yang, Phys. Rev. 106, 340 (1957).
  * Christenson et al. (1964) J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. Lett. 13, 138 (1964).
  * Aubert et al. (2001) B. Aubert et al. (BaBar), Phys. Rev. Lett. 87, 091801 (2001), eprint hep-ex/0107013.
  * Abe et al. (2001) K. Abe et al. (Belle), Phys. Rev. Lett. 87, 091802 (2001), eprint hep-ex/0107061.
  * Aaij et al. (2019) R. Aaij et al. (LHCb), Phys. Rev. Lett. 122, 211803 (2019), eprint 1903.08726.
  * Hu and Dodelson (2002) W. Hu and S. Dodelson, Ann. Rev. Astron. Astrophys. 40, 171 (2002), eprint astro-ph/0110414.
  * Cyburt (2004) R. H. Cyburt, Phys. Rev. D 70, 023505 (2004), eprint astro-ph/0401091.
  * Aghanim et al. (2020) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A1 (2020), eprint 1807.06205.
  * Sakharov (1967) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  * Cline (2006) J. M. Cline, in _Les Houches Summer School - Session 86: Particle Physics and Cosmology: The Fabric of Spacetime_ (2006), eprint hep-ph/0609145.
  * Okubo (1958) S. Okubo, Phys. Rev. 109, 984 (1958).
  * Pais (1959) A. Pais, Phys. Rev. Lett. 3, 242 (1959).
  * Brown et al. (1983) T. Brown, S. F. Tuan, and S. Pakvasa, Phys. Rev. Lett. 51, 1823 (1983).
  * Chau and Cheng (1983) L.-L. Chau and H.-Y. Cheng, Phys. Lett. B 131, 202 (1983).
  * Donoghue and Pakvasa (1985) J. F. Donoghue and S. Pakvasa, Phys. Rev. Lett. 55, 162 (1985).
  * Donoghue et al. (1986) J. F. Donoghue, X.-G. He, and S. Pakvasa, Phys. Rev. D 34, 833 (1986).
  * Chang et al. (2015) H.-M. Chang, M. González-Alonso, and J. Martin Camalich, Phys. Rev. Lett. 114, 161802 (2015), eprint 1412.8484.
  * Li (2017) H.-B. Li, Front. Phys. (Beijing) 12, 121301 (2017), [Erratum: Front.Phys.(Beijing) 14, 64001 (2019)], eprint 1612.01775.
  * Goudzovski et al. (2023) E. Goudzovski et al., Rept. Prog. Phys. 86, 016201 (2023), eprint 2201.07805.
  * Pondrom et al. (1981) L. Pondrom, R. Handler, M. Sheaff, P. T. Cox, J. Dworkin, O. E. Overseth, T. Devlin, L. Schachinger, and K. J. Heller, Phys. Rev. D 23, 814 (1981).
  * Chauvat et al. (1985) P. Chauvat et al. (R608), Phys. Lett. B 163, 273 (1985).
  * Tixier et al. (1988) M. H. Tixier et al. (DM2), Phys. Lett. B 212, 523 (1988).
  * Lee and Yang (1957) T. D. Lee and C.-N. Yang, Phys. Rev. 108, 1645 (1957).
  * Pallin et al. (1987) D. Pallin et al. (DM2), Nucl. Phys. B 292, 653 (1987).
  * Holmstrom et al. (2004) T. Holmstrom et al. (HyperCP), Phys. Rev. Lett. 93, 262001 (2004), eprint hep-ex/0412038.
  * Aaij et al. (2017) R. Aaij et al. (LHCb), Nature Phys. 13, 391 (2017), eprint 1609.05216.
  * Ablikim et al. (2019a) M. Ablikim et al. (BESIII), Nature Phys. 15, 631 (2019a), eprint 1808.08917.
  * Bunce et al. (1976) G. Bunce et al., Phys. Rev. Lett. 36, 1113 (1976).
  * Pondrom (1985) L. G. Pondrom, Phys. Rept. 122, 57 (1985).
  * Chakravorty et al. (2003) A. Chakravorty et al. (FNAL E756), Phys. Rev. Lett. 91, 031601 (2003), eprint hep-ex/0306047.
  * Aad et al. (2015) G. Aad et al. (ATLAS), Phys. Rev. D 91, 032004 (2015), eprint 1412.1692.
  * Adamczyk et al. (2017) L. Adamczyk et al. (STAR), Nature 548, 62 (2017), eprint 1701.06657.
  * Ireland et al. (2019) D. G. Ireland, M. Döring, D. I. Glazier, J. Haidenbauer, M. Mai, R. Murray-Smith, and D. Rönchen, Phys. Rev. Lett. 123, 182301 (2019), eprint 1904.07616.
  * Ablikim et al. (2019b) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 123, 122003 (2019b), eprint 1903.09421.
  * Ablikim et al. (2020a) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 124, 032002 (2020a), eprint 1910.04921.
  * Ablikim et al. (2022a) M. Ablikim et al. (BESIII), Nature 606, 64 (2022a), eprint 2105.11155.
  * Ablikim et al. (2022b) M. Ablikim et al. (BESIII), Phys. Rev. D 105, L011101 (2022b), eprint 2111.11742.
  * Ablikim et al. (2022c) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 129, 131801 (2022c), eprint 2204.11058.
  * Ablikim et al. (2022d) M. Ablikim et al. (BESIII), Phys. Rev. D 106, L091101 (2022d), eprint 2206.10900.
  * Ablikim et al. (2023a) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 131, 191802 (2023a), eprint 2304.14655.
  * Ablikim et al. (2023b) M. Ablikim et al. (BESIII) (2023b), eprint 2309.04139.
  * He et al. (1993) X.-G. He, J. P. Ma, and B. McKellar, Phys. Rev. D 47, R1744 (1993), eprint hep-ph/9211276.
  * He and Ma (2023) X. G. He and J. P. Ma, Phys. Lett. B 839, 137834 (2023), eprint 2212.08243.
  * Hong et al. (2023) P.-C. Hong, R.-G. Ping, T. Luo, X.-R. Zhou, and H. Li, Chin. Phys. C 47, 093103 (2023), eprint 2306.08517.
  * Bondar et al. (2013) A. E. Bondar et al. (Charm-Tau Factory), Phys. Atom. Nucl. 76, 1072 (2013).
  * Achasov et al. (2024) M. Achasov et al., Front. Phys. (Beijing) 19, 14701 (2024), eprint 2303.15790.
  * Kopke and Wermes (1989) L. Kopke and N. Wermes, Phys. Rept. 174, 67 (1989).
  * Bondar et al. (2020) A. Bondar, A. Grabovsky, A. Reznichenko, A. Rudenko, and V. Vorobyev, JHEP 03, 076 (2020), eprint 1912.09760.
  * Zeng et al. (2023) S. Zeng, Y. Xu, X. R. Zhou, J. J. Qin, and B. Zheng, Chin. Phys. C 47, 113001 (2023), eprint 2306.15602.
  * Fu et al. (2023) J. Fu, H.-B. Li, J.-P. Wang, F.-S. Yu, and J. Zhang, Phys. Rev. D 108, L091301 (2023), eprint 2307.04364.
  * Cao et al. (2024) X. Cao, Y.-T. Liang, and R.-G. Ping (2024), eprint 2404.00298.
  * Altmannshofer et al. (2019) W. Altmannshofer et al. (Belle-II), PTEP 2019, 123C01 (2019), [Erratum: PTEP 2020, 029201 (2020)], eprint 1808.10567.
  * ILC (2013) (2013), eprint 1306.6352.
  * Aryshev et al. (2022) A. Aryshev et al. (ILC International Development Team) (2022), eprint 2203.07622.
  * Accomando et al. (2004) E. Accomando et al. (CLIC Physics Working Group), in _11th International Conference on Hadron Spectroscopy_ (2004), CERN Yellow Reports: Monographs, eprint hep-ph/0412251.
  * Robson and Roloff (2018) A. Robson and P. Roloff (2018), eprint 1812.01644.
  * Abdallah et al. (2023) W. Abdallah et al. (CEPC Study Group) (2023), eprint 2312.14363.
  * Bernreuther et al. (1989) W. Bernreuther, U. Low, J. P. Ma, and O. Nachtmann, Z. Phys. C 43, 117 (1989).
  * Workman et al. (2022) R. L. Workman et al. (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Fäldt and Kupsc (2017) G. Fäldt and A. Kupsc, Phys. Lett. B 772, 16 (2017), eprint 1702.07288.
  * Abel et al. (2020) C. Abel et al., Phys. Rev. Lett. 124, 081803 (2020), eprint 2001.11966.
  * Gaillard and Lee (1974) M. K. Gaillard and B. W. Lee, Phys. Rev. Lett. 33, 108 (1974).
  * Ecker and Grimus (1985) G. Ecker and W. Grimus, Nucl. Phys. B 258, 328 (1985).
  * He et al. (1988) X.-G. He, B. H. J. McKellar, and S. Pakvasa, Phys. Rev. Lett. 61, 1267 (1988).
  * Buchalla et al. (1996) G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996), eprint hep-ph/9512380.
  * Jenkins et al. (2018) E. E. Jenkins, A. V. Manohar, and P. Stoffer, JHEP 01, 084 (2018), [Erratum: JHEP 12, 042 (2023)], eprint 1711.05270.
  * Dobbs et al. (2014) S. Dobbs, A. Tomaradze, T. Xiao, K. K. Seth, and G. Bonvicini, Phys. Lett. B 739, 90 (2014), eprint 1410.8356.
  * Ablikim et al. (2024) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 132, 081904 (2024), eprint 2307.15894.
  * Ablikim et al. (2017) M. Ablikim et al. (BESIII), Phys. Rev. D 95, 052003 (2017), eprint 1701.07191.
  * Ablikim et al. (2020b) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 124, 042001 (2020b), eprint 1905.09001.
  * Ablikim et al. (2023c) M. Ablikim et al. (BESIII), Phys. Rev. D 108, L011101 (2023c), eprint 2302.09767.
  * Ablikim et al. (2023d) M. Ablikim et al. (BESIII), Phys. Rev. D 108, L031106 (2023d), eprint 2305.09218.
  * Ablikim et al. (2021a) M. Ablikim et al. (BESIII), Phys. Lett. B 814, 136110 (2021a), eprint 2009.01404.
  * Patterion (2017) P. Patterion (BESIII), Nuovo Cim. C 40, 28 (2017).
  * Larin (2021) P. Larin (BESIII), Nucl. Part. Phys. Proc. 312-317, 63 (2021).
  * Ablikim et al. (2021b) M. Ablikim et al. (BESIII), Phys. Lett. B 817, 136328 (2021b), eprint 2102.10337.
  * Schönning et al. (2023) K. Schönning, V. Batozskaya, P. Adlarson, and X. Zhou, Chin. Phys. C 47, 052002 (2023), eprint 2302.13071.
  * Baroni et al. (1971) G. Baroni, S. Petrera, and G. Romano, Lett. Nuovo Cim. 2S2, 1256 (1971).
  * Engel et al. (2013) J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Prog. Part. Nucl. Phys. 71, 21 (2013), eprint 1303.2371.
  * Erler and Ramsey-Musolf (2005) J. Erler and M. J. Ramsey-Musolf, Phys. Rev. D 72, 073003 (2005), eprint hep-ph/0409169.
  * Benesch et al. (2014) J. Benesch et al. (MOLLER) (2014), eprint 1411.4088.
  * Berger et al. (2016) N. Berger et al., J. Univ. Sci. Tech. China 46, 481 (2016), eprint 1511.03934.
  * (87) https://indico.ihep.ac.cn/event/21135/.
  * Khriplovich and Zhitnitsky (1981) I. B. Khriplovich and A. R. Zhitnitsky, Sov. J. Nucl. Phys. 34, 95 (1981).
  * He et al. (1989) X.-G. He, B. H. J. McKellar, and S. Pakvasa, Int. J. Mod. Phys. A 4, 5011 (1989), [Erratum: Int.J.Mod.Phys.A 6, 1063–1066 (1991)].
  * Scherer and Schindler (2012) S. Scherer and M. R. Schindler, _A Primer for Chiral Perturbation Theory_ , vol. 830 (2012), ISBN 978-3-642-19253-1.
  * Borasoy (1999) B. Borasoy, Phys. Rev. D 59, 054021 (1999), eprint hep-ph/9811411.
  * Ledwig et al. (2014) T. Ledwig, J. Martin Camalich, L. S. Geng, and M. J. Vicente Vacas, Phys. Rev. D 90, 054502 (2014), eprint 1405.5456.