CP asymmetries in Strange Baryon Decays^*

First we describe the 'landscape' and the tools one can use. The super narrow vector resonance ${\rm{J}}/{\rm{\psi }}$ produces a connection of the spins of a pair of hyperons Y of Λ, Σ, and Ξ and final state X of p, Λ:

$$\begin{eqnarray}&&{\rm{J}}/{\rm{\psi }}\to \bar{Y}Y\to [\bar{X}\bar{{\rm{\pi }}}][X{\rm{\pi }}]\end{eqnarray} \tag{ 13 }$$

One can measure T-odd moments:

$$\begin{eqnarray}&&{\alpha }_{Y}^{(X)}=\langle {\mathop{\sigma }\limits^{\rightarrow}}_{Y}\cdot ({\mathop{\sigma }\limits^{\rightarrow}}_{X}\times {\mathop{{\rm{\pi }}}\limits^{\rightarrow}}_{X})\rangle,{\alpha }_{\bar{Y}}^{(\bar{X})}=\langle {\mathop{\sigma }\limits^{\rightarrow}}_{\bar{Y}}\cdot ({\mathop{\sigma }\limits^{\rightarrow}}_{\bar{X}}\times {\mathop{{\rm{\pi }}}\limits^{\rightarrow}}_{\bar{X}})\rangle,\end{eqnarray} \tag{ 14 }$$

where $\mathop{\sigma }\limits^{\rightarrow}$ and $\mathop{{\rm{\pi }}}\limits^{\rightarrow}$ describe the spins and the momenta of the baryons in the rest frames of Y/$\bar{Y}$. It is not surprising to find sizable or even large non-zero values of T-odd moments; below we will give an example from data with large values. It shows the impact of final state interactions (FSI). Non-zero values of T-odd moments do not mean by themselves we have found T violation or CP violation using CPT invariance. However, one can probe direct CP asymmetries

$$\begin{eqnarray}&&\langle {A}_{{CP}}^{(X)}\rangle =\displaystyle \frac{{\alpha }_{Y}^{(X)}+{\alpha }_{\bar{Y}}^{(\bar{X})}}{{\alpha }_{Y}^{(X)}-{\alpha }_{\bar{Y}}^{(\bar{X})}}\end{eqnarray} \tag{ 15 }$$

without polarized Y and $\bar{Y}$. The crucial point is to use the connection of the spin-1 of the initial state ${\rm{J}}/{\rm{\psi }}$ with the final state with two spin-1/2. The goal is to find $\langle {A}_{{CP}}^{(X)}\rangle \ne 0$ without directly measuring polarization of $Y/\bar{Y}$ and $X/\bar{X}$ baryons and their correlations due to the very narrow resonance ${\rm{J}}/{\rm{\psi }}$ in their production.

We measure correlations between the pair of final state baryons and pions. In the rest frame of ${\rm{J}}/{\rm{\psi }}$ one can define ${C}_{T}=({\mathop{p}\limits^{\rightarrow}}_{X}\times {\mathop{p}\limits^{\rightarrow}}_{{\rm{\pi }}})\cdot {\mathop{p}\limits^{\rightarrow}}_{\bar{X}}$ and conjugate transitions ${\bar{C}}_{T}=({\mathop{p}\limits^{\rightarrow}}_{\bar{X}}\times {\mathop{p}\limits^{\rightarrow}}_{\bar{{\rm{\pi }}}})\cdot {\mathop{p}\limits^{\rightarrow}}_{X}$, and compare the event numbers (N) with positive and negative values

$$\begin{eqnarray}&&\langle {A}_{T}\rangle =\displaystyle \frac{N({C}_{T}\gt 0)-N({C}_{T}\lt 0)}{N({C}_{T}\gt 0)+N({C}_{T}\lt 0)}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\langle {\bar{A}}_{T}\rangle =\displaystyle \frac{N({\bar{C}}_{T}\gt 0)-N({\bar{C}}_{T}\lt 0)}{N({\bar{C}}_{T}\gt 0)+N({\bar{C}}_{T}\lt 0)}.\end{eqnarray} \tag{ 17 }$$

Thus

$$\begin{eqnarray}&&{{\mathcal{A}}}_{T}=\displaystyle \frac{1}{2}\left[\langle {A}_{T}\rangle +\langle {\bar{A}}_{T}\rangle \right]=\langle {A}_{T}\rangle \ne 0\end{eqnarray} \tag{ 18 }$$

are observable CP asymmetries based on CPT invariance. We have taken a different convention for ${\bar{A}}_{T}$ compared to Ref. [8].

One can compare to its charge-conjugate channel to get rid of a fake CP asymmetry induced by the FSI effect. However, the charge-conjugate of the process ${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{\Lambda }}}\to [{\rm{p}}{{\rm{\pi }}}^{-}][\bar{{\rm{p}}}{{\rm{\pi }}}^{+}]$ considered here is itself; or it is an untagged sample that cannot be distinguished by experiment. Such a situation is, to some extent, different from a measurement of ${{\rm{D}}}^{0}({\bar{{\rm{D}}}}^{0})\to {\rm{K}}\bar{{\rm{K}}}{\rm{\pi }}{\rm{\pi }}/4{\rm{\pi }}$.

The landscapes are quite different for ${\rm{\Lambda }}\to {\rm{p}}{{\rm{\pi }}}^{-}$ and ${\rm{\Sigma }}\to {\rm{p}}{\rm{\pi }}$ and even more so for ${\rm{\Xi }}\to {\rm{\Lambda }}{\rm{\pi }}$, with many paths for transitions: ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{\Lambda }}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{+}][{\rm{p}}{{\rm{\pi }}}^{-}];$ ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}];$ ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Xi }}}}^{0}{{\rm{\Xi }}}^{0}\to [\bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{0}][{\rm{\Lambda }}{{\rm{\pi }}}^{0}];$ ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Xi }}}}^{+}{{\rm{\Xi }}}^{-}\to [\bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{+}][{\rm{\Lambda }}{{\rm{\pi }}}^{-}]$²⁾.

• 1)  
  One can calibrate those transitions with ${\rm{J}}/{\rm{\psi }}\to {\rm{\Delta }}(1232)\bar{{\rm{\Delta }}}(1232)$, where CPV cannot appear.
• 2)  
  That is not the end of the impact of strong resonances of ${\rm{\Delta }}(1232)$ with width ∼117 MeV and N(1440) with width 250–450 MeV. They will affect the lessons we learn from future data about possible CP violations in the transitions of strange baryons.
• 3)  
  The "duality" between the worlds of quarks and hadrons is very subtle, in particular close to the thresholds of resonances.

Here we estimate the sensitivities for measuring such observables using the collected ${\rm{J}}/{\rm{\psi }}$ sample. By the end of 2018, ${10}^{10}\,{\rm{J}}/{\rm{\psi }}$ events will be accumulated by the BESIII experiment [15]. The detection efficiency for pions, protons, and kaons with momentum larger than 100 MeV can reach 98%. As for particle identification (PID), pions, kaons and protons can be distinguished with $3\sigma$ (three standard deviations) uncertainty below a momentum of 1.0 GeV. Considering the branching fractions of ${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{\Lambda }}}\to [{\rm{p}}{{\rm{\pi }}}^{-}][\bar{{\rm{p}}}{{\rm{\pi }}}^{+}]$ and ${\rm{J}}/{\rm{\psi }}\to {{\rm{\Xi }}}^{-}{\bar{{\rm{\Xi }}}}^{+}\to [{\rm{\Lambda }}{{\rm{\pi }}}^{-}][\bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{+}]$ [4] etc., we can estimate the expected numbers of events and the further sensitivities, as shown in Table 1. Probing such a large data sample with refined tools will give us rich information about the underlying dynamics.

Table 1.  The numbers of reconstructed events after considering decay branching fractions, tracking, and particle identification. The sensitivity is estimated without considering possible background dilutions, which should be small at the BESIII experiment. Estimations are based on the ${10}^{10}\,{\rm{J}}/{\rm{\psi }}$ data which will be collected by the BESIII collaboration by the end of 2018 (and the branching fractions from PDG2016). Systematic uncertainties are expected to be of the same order as the statistical uncertainties shown in the table.

channel	# of events	sensitivity on ${{\mathcal{A}}}_{T}$
${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{\Lambda }}}\to [{\rm{p}}{{\rm{\pi }}}^{-}][\bar{{\rm{p}}}{{\rm{\pi }}}^{+}]$	$2.6\times {10}^{6}$	0.06%
${\rm{J}}/{\rm{\psi }}\to {{\rm{\Sigma }}}^{+}{\bar{{\rm{\Sigma }}}}^{-}\to [{\rm{p}}{{\rm{\pi }}}^{0}][\bar{{\rm{p}}}{{\rm{\pi }}}^{0}]$	$2.5\times {10}^{6}$	0.06%
${\rm{J}}/{\rm{\psi }}\to {{\rm{\Xi }}}^{-}{\bar{{\rm{\Xi }}}}^{+}\to [{\rm{\Lambda }}{{\rm{\pi }}}^{-}][\bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{+}]$	$1.1\times {10}^{6}$	0.1%
${\rm{J}}/{\rm{\psi }}\to {{\rm{\Xi }}}^{0}{\bar{{\rm{\Xi }}}}^{0}\to [{\rm{\Lambda }}{{\rm{\pi }}}^{0}][\bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{0}]$	$1.6\times {10}^{6}$	0.08%

Measuring ${{\mathcal{A}}}_{T}$ is the first step to probe CP asymmetries. There are various options for intermediate steps like:

$$\begin{eqnarray}&&{{\mathcal{A}}}_{T}(d)=\displaystyle \frac{N({C}_{T}\gt | d| )-N({C}_{T}\lt -| d| )}{N({C}_{T}\gt | d| )+N({C}_{T}\lt -| d| )}.\end{eqnarray} \tag{ 19 }$$

This is a very promising way to go beyond ${{\mathcal{A}}}_{T}$ in Eq. (18). By the end of 2018 we can expect that BESIII can probe CPV in the decays of strange baryons to the level of ${10}^{-4}$ sensitivity, as shown in Table 1 (neglecting systematic errors).

The above method can also be applied to ${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{\Lambda }}}{\rm{\pi }}{\rm{\pi }}$ to probe CPV in Ξ decays. Interestingly, for the case of Ξ baryons, CPV can also be probed by polarized Ξ thanks to the decay chain ${{\rm{\Xi }}}^{0}\to {\rm{\Lambda }}{\rm{\pi }}\to ({\rm{p}}{\rm{\pi }}){\rm{\pi }}$, where the CP-violating observable can be related to the helicity amplitudes. A similar proposal was given in Ref. [16] for ${{\rm{\Lambda }}}_{c}$ decay. Such a CP-violating signal can be extracted by performing an angular analysis, which is again accessible in the BESIII experiment due to the large ${\rm{J}}/{\rm{\psi }}$ data sample. However, for Λ decays the interference of the helicity amplitude is absent in the angular observable [14], which handicaps accessing CP the same way as in polarized Ξ decay, i.e., measuring the interference of helicity amplitudes.

The DM2 Collaboration measured CP invariance (and quantum mechanics) in ${{\rm{e}}}^{+}{{\rm{e}}}^{-}\to {\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{\Lambda }}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{+}][{\rm{p}}{{\rm{\pi }}}^{-}]$ in 1988 without polarized Λ and $\bar{{\rm{\Lambda }}}$. Their result was ${A}_{{CP}}=0.01\pm 0.10$ [17]. This was the first step in an important direction.

Now we describe the situation 30 years later, looking at what BESIII could achieve by 2018. We use the Jacob-Wick helicity formalism [18], as shown in Fig. 3 of Ref. [14]; it was also applied in Refs. [19, 20]:

• 1)  
  The ${\rm{J}}/{\rm{\psi }}$ rest frame is along the Λ out-going direction, and the solid angle ${\varOmega }_{0}(\theta,\phi )$ is between the incoming ${e}^{+}$ and the out-going Λ.
• 2)  
  For ${\rm{\Lambda }}\to {\rm{p}}{{\rm{\pi }}}^{-}$ the solid angle of the 'daughter' particle ${\varOmega }_{1}({\theta }_{1},{\phi }_{1})$ is in reference to the Λ rest frame (although in the out-going direction); likewise for $\bar{{\rm{\Lambda }}}\to \bar{{\rm{p}}}{{\rm{\pi }}}^{+}$.

We describe the angular distribution for this process following Ref. [21]:

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\varGamma }{{\rm{d}}\varOmega } & \propto & (1-\alpha ){\sin }^{2}\theta \cdot \left[1+{\alpha }_{{\rm{\Lambda }}}{\alpha }_{\bar{{\rm{\Lambda }}}}\left(\cos {\theta }_{1}\cos {\bar{\theta }}_{1}\right.\right.\\ & & \left.\left.+\sin {\theta }_{1}\sin {\bar{\theta }}_{1}\cos ({\phi }_{1}+{\bar{\phi }}_{1})\right)\right]\\ & & -(1+\alpha )(1+{\cos }^{2}\theta )\left({\alpha }_{{\rm{\Lambda }}}{\alpha }_{\bar{{\rm{\Lambda }}}}\cos {\theta }_{1}\cos {\bar{\theta }}_{1}-1\right),\end{eqnarray} \tag{ 20 }$$

where ${\rm{d}}\varOmega \equiv {\rm{d}}{\varOmega }_{0}{\rm{d}}{\varOmega }_{1}{\rm{d}}{\bar{\varOmega }}_{1}$ and:

• 1)  
  α is the angular distribution parameter for Λ;
• 2)  
  ${\alpha }_{{\rm{\Lambda }}}[{\alpha }_{\bar{{\rm{\Lambda }}}}]$ is the ${\rm{\Lambda }}[\bar{{\rm{\Lambda }}}]$ decay parameter;
• 3)  
  these data depend only on the product of ${\alpha }_{{\rm{\Lambda }}}{\alpha }_{\bar{{\rm{\Lambda }}}}$, see Eq. (20).

By fitting Eq. (20) to the data, one can determine ${\alpha }_{J/\psi }$ and ${\alpha }_{{\rm{\Lambda }}}{\alpha }_{\bar{{\rm{\Lambda }}}}$. One can make a substitution:

$$\begin{eqnarray}&&{\alpha }_{{\rm{\Lambda }}}{\alpha }_{\bar{{\rm{\Lambda }}}}\equiv \displaystyle \frac{A-1}{A+1}{\alpha }_{{\rm{\Lambda }}}^{2},\end{eqnarray} \tag{ 21 }$$

where A describes a CP asymmetry observable.

As mentioned before, published 2010 data from the BES collaboration show [14] ${\alpha }_{\bar{{\rm{\Lambda }}}}^{(\bar{p})}=-0.755\pm 0.083\pm 0.063$, based on previously measured ${\alpha }_{{\rm{\Lambda }}}^{(p)}=0.642\pm 0.013$. Their non-zero numbers show the impact of re-scattering; it is large, which is not surprising. We also note that Eq. (20) is derived from considering only spin projection ${J}_{z}=\pm 1$ for ${\rm{J}}/{\rm{\psi }}$, which is a consequence of the QED process ${{\rm{e}}}^{+}{{\rm{e}}}^{-}\to \gamma \to {\rm{c}}\bar{{\rm{c}}}({\rm{J}}/{\rm{\psi }})$ [22]. Thus, only the product term ${\alpha }_{{\rm{\Lambda }}}{\alpha }_{\bar{{\rm{\Lambda }}}}$ can be measured¹⁾.

The present limit on direct CP asymmetry is around a few percent. ND cannot even produce effects close to 1% here. To reach ${\mathcal{O}}(0.1 \% )$ would be a considerable achievement for $\langle {A}_{{CP}}^{(p)}\rangle$ based on ${\alpha }_{{\rm{\Lambda }}}^{(p)}\sim {\alpha }_{\bar{{\rm{\Lambda }}}}^{(\bar{p})}\sim 0.64$. Measuring semi-local asymmetry would give us new lessons about non-perturbative QCD.

It has been said very recently in Ref. [23] that Ref. [14] missed some contributions for on-shell ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{\Lambda }}\to \bar{{\rm{p}}}{{\rm{\pi }}}^{+}{\rm{p}}{{\rm{\pi }}}^{-};$ so far we are not convinced. Of course, experimental data from BESIII are needed in order to test this.

Even now BESIII has much more data, and by the end of 2018 we will have about $2.6\times {10}^{6}$ events for the ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{\Lambda }}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{+}][{\rm{p}}{{\rm{\pi }}}^{-}]$ decay chain, after considering the efficiencies for tracking, particle identification and Λ pair reconstruction [15]. The sensitivity might reach $6\times {10}^{-4}$ by the end of 2018, as listed in Table 1. Now the landscape is different, with some hope to find CP asymmetry here, and also for ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Sigma }}}{\rm{\Sigma }}$, which will be discussed below.

Some general comments are as follows. (a) Except $n-\bar{n}$ (or maybe, even ${\rm{\Lambda }}-\bar{{\rm{\Lambda }}}$) oscillations [25], one probes only direct CPV with baryons. (b) It is well-known that the impact (local) penguin operators are crucial for ${\rm{\Delta }}S=1$ transitions for strange mesons, in particular for the non-zero value for ${\epsilon }^{\prime}$. What about decays of strange baryons? The transitions of pairs of strange baryons are not far from the thresholds; thus one has to think about the "duality" between the world of hadrons and that of quarks and gluons. We discuss this further below.

To probe CP asymmetries with accuracy, BESIII can calibrate with transitions where CP asymmetries cannot happen. We have two examples with broad resonances, where the situations are complex: ${\rm{J}}/{\rm{\psi }}\to \bar{N}(1440)N(1440)$ with ${\varGamma }_{N(1440)}\sim (250-450)$ MeV carrying isospin I = 1/2, and ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1223)$ with ${\varGamma }_{{\rm{\Delta }}(1232)}\sim$ 117 MeV carrying isospin I = 3/2. The BESIII experiment has measured the background very well and continues to do so. The total background is very small, which provides good opportunity for a clean probe of the CPV signal.

From knowledge of the previous BES measurement [14] (the main background channels are also listed in this reference) and the ongoing measurement¹⁾, we can conservatively estimate that the number of combinatorial background events is roughly ${10}^{-3}$ of the signal events, a very small fraction, such that CPV can be cleanly probed. This is one of the strengths of the BESIII collaboration. We can show this more transparently with an example. By the end of 2018 we can expect $2.6\times {10}^{6}$ signal events, and assuming the CPV is on the level of ${10}^{-4}$, one has ${\rm{\Delta }}N={N}_{+}-{N}_{-}=260$. The background can induce ${\rm{\Delta }}{N}_{\mathrm{bkg}}=\sqrt{2.6\times {10}^{3}}\approx 51$. This illustrates that a nonzero A_T will really indicate the observation of CP asymmetry. However, if the CPV ${{\mathcal{A}}}_{T}$ is below the level of ${10}^{-5}$, the impact of the background is important.

Above, we discussed ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232)$ as background for ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{\Lambda }}$. We also said that it is very important to analyze pairs of baryons. However, the situation here is even more complex, as we have discussed just above:

• 1)  
  ${{\rm{\Sigma }}}^{+}$ carries isospin $(I,{I}_{3})=(1,+1)$, while Λ has isospin zero.
• 2)  
  Looking at straightforward diagrams we get ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}]$ vs. ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232)\to \bar{{\rm{p}}}{\rm{\pi }}{\rm{p}}{\rm{\pi }}$ as a background.
• 3)  
  However, we have to go beyond that, as you can see by comparing $M({{\rm{\Sigma }}}^{+})\simeq 1189\,\mathrm{MeV}$ vs. $M({\rm{\Delta }}(1232))\simeq 1232\,\mathrm{MeV}$ with width ${\rm{\Delta }}(1232)\simeq 117\,\mathrm{MeV}$. Off-shell intermediate amplitudes sizably affect total amplitudes and also measurable CP asymmetries in $[\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}]$ final states.In other words, there should be a sizable overlap between the waves of $\bar{{\rm{\Sigma }}}$/Σ and $\bar{{\rm{\Delta }}}(1232)$/${\rm{\Delta }}(1232)$ due to $\varGamma ({\rm{\Delta }}(1232))$, which cannot be ignored.
• 4)  
  Therefore, we use two different diagrams for transitions: "$\Rightarrow$" describes the amplitudes due to QCD(×QED) that conserve P and C symmetries; "$\to$" includes ${SU}{(2)}_{L}$ with weak dynamics, including sources of P, C and CP asymmetries. The direct path is

  $$\begin{eqnarray}&&{\rm{J}}/{\rm{\psi }}\Rightarrow {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}],\end{eqnarray} \tag{ 22 }$$

  while a somewhat indirect path can happen due to off-shell intermediate amplitudes:

  $$\begin{eqnarray}&&{\rm{J}}/{\rm{\psi }}\Rightarrow {\unicode{x201C}}\bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232){\unicode{x201D}}\Rightarrow {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}].\end{eqnarray} \tag{ 23 }$$

  As said before, the impact of re-scattering is complex, as one can see from comparing $\mathrm{BR}({{\rm{\Sigma }}}^{+}\to {\rm{p}}{{\rm{\pi }}}^{0})\simeq 0.52\,\&\,\mathrm{BR}({{\rm{\Sigma }}}^{+}\to n{{\rm{\pi }}}^{+})\simeq 0.48$ vs. $\mathrm{BR}({\rm{\Lambda }}\to {\rm{p}}{{\rm{\pi }}}^{-})\simeq 0.64\,\&\,\mathrm{BR}({\rm{\Lambda }}\to n{{\rm{\pi }}}^{0})\simeq 0.36$.
• 5)  
  There is another possible amplitude that is even more complex, in particular with

  $$\begin{eqnarray}&&{\rm{J}}/{\rm{\psi }}\Rightarrow {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to {\unicode{x201C}}\bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232){\unicode{x201D}}\Rightarrow [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}].\end{eqnarray} \tag{ 24 }$$

We hypothesize off-line exchanges of kaons or new dynamics.

The data produced by the end of 2018 will be analyzed by the BESIII collaboration with the best available tools. It is possible that off-line resonances like $\bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232)$ might impact the CP asymmetries more in ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}]$ than in ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{\Lambda }}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{+}][{\rm{p}}{{\rm{\pi }}}^{-}]$, since off-shell $\bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232)$ are closer to the on-shell ${\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+};$ the second vertex technique will help greatly to distinguish them from the background from $\bar{{\rm{\Delta }}}(1232){\rm{\Delta }}(1232)$.

Using the ${10}^{10}{\rm{J}}/{\rm{\psi }}$ events, we expect $2.5\times {10}^{6}$ signal events, with sensitivity at 0.06%, as shown in Table 1.

It is also interesting to probe CP violation by using the decays ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Xi }}}}^{+}{{\rm{\Xi }}}^{-}$ and ${\bar{{\rm{\Xi }}}}^{0}{{\rm{\Xi }}}^{0}$. One can reconstruct both Ξs in the ${\rm{\Xi }}\to {\rm{\Lambda }}{\rm{\pi }}$ mode. For the neutral channel ${\rm{\Xi }}\to {\rm{\Lambda }}{{\rm{\pi }}}^{0}$, the ${{\rm{\pi }}}^{0}$ in ${\bar{{\rm{\Xi }}}}^{0}$ decay can be easily separated from that in ${{\rm{\Xi }}}^{0}$ decay without ambiguity, since both Ξs are back to back and strongly boosted in the rest frame of the ${\rm{J}}/{\rm{\psi }}$. That is, the ${{\rm{\Xi }}}^{0}\to {\rm{\Lambda }}{{\rm{\pi }}}^{0}$ is reconstructed in the opposite decay hemisphere against the decay hemisphere for the $\bar{{\rm{\Xi }}}\to \bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{0}$. This situation will be similar to that in Section 2.4 for ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Sigma }}}}^{-}{{\rm{\Sigma }}}^{+}\to [\bar{{\rm{p}}}{{\rm{\pi }}}^{0}][{\rm{p}}{{\rm{\pi }}}^{0}]$. By the end of 2018 we will have data for ${\rm{J}}/{\rm{\psi }}\to {\bar{{\rm{\Xi }}}}^{+}{{\rm{\Xi }}}^{-}\to \bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{+}{\rm{\Lambda }}{{\rm{\pi }}}^{-}$ and ${\bar{{\rm{\Xi }}}}^{0}{{\rm{\Xi }}}^{0}\to \bar{{\rm{\Lambda }}}{{\rm{\pi }}}^{0}{\rm{\Lambda }}{{\rm{\pi }}}^{0}$ with $1.1\times {10}^{6}$ and $1.6\times {10}^{6}$ events, respectively [15]. Thus, the reaches for the triple-product asymmetry are about $10\times {10}^{-4}$ and $8\times {10}^{-4}$ for the charged and neutral Ξ, respectively.

Once non-zero values for CP asymmetries are found, one can attempt another challenge: do they come from the transition of ${\rm{\Xi }}\to {\rm{\Lambda }}{\rm{\pi }}$ or ${\rm{\Lambda }}\to {\rm{p}}{{\rm{\pi }}}^{-}$, or the interferences between them?

One can also compare the transitions ${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{X}}}$ vs. ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{X}}$, in particular ${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{p}}}{{\rm{K}}}^{+}$ vs. ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\mathrm{pK}}^{-}$. The 2016 data tell us BR$({\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{p}}}{{\rm{K}}}^{+}/\bar{{\rm{\Lambda }}}{\mathrm{pK}}^{-})=(0.89\pm 0.16)\cdot {10}^{-3}$. By the end of 2018 the expected number of events is about $9\times {10}^{6}$, which has sensitivity for CP asymmetry of $2.4\times {10}^{-4}$ by comparing the partial widths between ${\rm{J}}/{\rm{\psi }}\to {\rm{\Lambda }}\bar{{\rm{X}}}$ and ${\rm{J}}/{\rm{\psi }}\to \bar{{\rm{\Lambda }}}{\rm{X}}$ decays. The systematic uncertainties are expected to be larger than in the previous cases discussed above, but not by an order of ten.

We are studying non-perturbative QCD in a novel situation; it is not trivial to know how much to apply it and where. Just below we give comments about available theoretical tools. The hope is to find signs of CP asymmetries in the decays of strange baryons; therefore the goal is not to go for accuracy. We 'paint' the landscape to find CP asymmetries. Once our community has found a non-zero value somewhere, one can discuss the correlations with other situations.