⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 062301 Angular Distribution of Production Planes in $J/\psi\rightarrow{\it \Lambda}\bar{\it \Lambda}$ Decay * Qiaorong Shen (沈巧蓉), Xinyan Tong (童心言)**, Yunfei Long (龙云飞), Yajun Mao (冒亚军) Affiliations School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871 Received 21 February 2019, online 18 May 2019 *Supported by the National Natural Science Foundation of China under Grant No 11661141008.
**Corresponding author. Email: txyz_cc@pku.edu.cn Citation Text: Chen Q R, Tong X Y, Long Y F and Mao Y J 2019 Chin. Phys. Lett. 36 062301    Abstract A new observable, the angle between ${\it \Lambda}$ and $\bar{\it \Lambda}$ decay planes, is proposed to test the theoretical predictions on the spin correlation. With 10 billion $J/\psi$ events collected with the BESIII detector at the BEPCII $e^+e^-$ collider, the distribution of the angle could be measured to verify whether or not a correlation exists. DOI:10.1088/0256-307X/36/6/062301 PACS:23.20.En, 13.25.Gv, 14.20.Jn © 2019 Chinese Physics Society Article Text Research into $J/\psi \rightarrow {\it \Lambda}\bar{\it \Lambda}$ decay is an important probe for abundant physics, such as understanding baryon properties,[1] testing of the CP violation,[2] measurement of the ${\it \Lambda}$ electric dipole moment[3] and searching for the ${\it \Lambda}\bar{\it \Lambda}$ oscillation.[4] The decay branching ratio and ${\it \Lambda}$ polar angular distribution have been measured with high precision by the Mark II,[5] DM2,[6] BESIII,[7] and BABAR[8] collaborations. Furthermore, it is well known that weak decay ${\it \Lambda}$($\bar{\it \Lambda}$) is its own spin analyzer and this special feature could be used to test the paradox of EPR[9] with $J/\psi \rightarrow {\it \Lambda}\bar{\it \Lambda} \rightarrow p \bar{p}\pi^-\pi^+$ as suggested by Törnqvist[10] and measured by the DM2 collaboration.[11] Alternatively, we propose a new observable, the angle between ${\it \Lambda}$ and $\bar{\it \Lambda}$ decay planes, rather than the angle between the momenta of $\pi^-$ and $\pi^+$ in the rest frames of their mother particles (${\it \Lambda}$ and $\bar{\it \Lambda}$), donated as $\alpha$ (shown in Fig. 1), to test the quantum mechanics predictions on the spin correlation in the decay.
cpl-36-6-062301-fig1.png
Fig. 1. Schematic drawing of $J/\psi\to{\it \Lambda}\bar{\it \Lambda}\to p\pi^-\bar{p}\pi^+$ decay.
Figure 1 shows a schematic drawing of $J/\psi\to{\it \Lambda}\bar{\it \Lambda}\to p\pi^-\bar{p}\pi^+$ decay. The coordinate system is similar to that defined in Ref. [12] and can be described as follows: assuming $\hat{i},\hat{j},\hat{k}$ is the unit vector of the $x,y,z$ coordinate system respectively, we choose the direction of the momentum of ${\it \Lambda}$ as the direction of $\hat{k}$ in the ${\it \Lambda}$'s helicity rest frame, that is to say, $\hat{k}=\hat{p}$. As for $\hat{\jmath}$ and $\hat{\imath}$, we have $\hat{\jmath}=\hat{k_0}\times \hat{k},\hat{\imath}=\hat{\jmath}\times \hat{k}$, where $\hat{k_0}$ is a common direction in the $J/\psi$ rest frame, and here the $e^+e^-$ beam direction is selected as $\hat{k_0}$ as an example. In the $J/\psi$ rest frame, ${\it \Lambda}\bar{\it \Lambda}$ emit in the opposite direction, thus $\hat{p}_{\it \Lambda}=-\hat{p}_{\bar{\it \Lambda}}$. Then we have $\hat{\imath}({\it \Lambda})=\hat{\imath}(\bar{\it \Lambda})$, $\hat{\jmath}({\it \Lambda})=-\hat{\jmath}(\bar{\it \Lambda})$ and $\hat{k}({\it \Lambda})=-\hat{k}(\bar{\it \Lambda} )$. As can be seen in Fig. 1, $\hat{k}({\it \Lambda})$ and $\hat{k}(\bar{\it \Lambda})$ are collinear, thus the value of $\alpha$ depends on both the azimuth angles $\phi_{\pi^\pm}\in(0,2\pi)$ of the momentum of $\pi^\pm$ in the helicity rest frame of their own parent particles (${\it \Lambda}$ and $\bar{\it \Lambda}$). Requiring $\alpha\in[0,\pi]$, the relation between $\alpha$ and $\phi_{\pi^\pm}$ can be written as $$\begin{alignat}{1} \alpha=\begin{cases} \phi_{\pi^-}+\phi_{\pi^+}, & \phi_{\pi^-}+\phi_{\pi^+}\in[0,\pi],\\ 2\pi-\phi_{\pi^-}-\phi_{\pi^+}, & \phi_{\pi^-}+\phi_{\pi^+}\in(\pi,2\pi], \\ \phi_{\pi^-}+\phi_{\pi^+}-2\pi, & \phi_{\pi^-}+\phi_{\pi^+}\in(2\pi,3\pi],\\ 4\pi-\phi_{\pi^-}-\phi_{\pi^+},& \phi_{\pi^-}+\phi_{\pi^+}\in(3\pi,4\pi]. \\ \end{cases}~~ \tag {1} \end{alignat} $$ Assuming that $W'(\phi_{\pi^-},\phi_{\pi^+})$ represents the probability density function of $\phi_{\pi^\pm}$, the distribution of $\alpha$ can be expressed as $$ W(\alpha)=\int_{0}^{2\pi}W^\prime(\phi_{\pi^-},\phi_{\pi^+})d\phi_{\pi^-}.~~ \tag {2} $$ With the relation between $\alpha$ and $\phi_{\pi^\pm}$ in Eq. (1), Eq. (2) becomes $$\begin{alignat}{1} W(\alpha)=\,&\int_{0}^{\alpha}W'(\phi_{\pi^-},\alpha-\phi_{\pi^-})d\phi_{\pi^-}\\ &+\int_{0}^{2\pi-\alpha}W'(\phi_{\pi^-},2\pi-\phi_{\pi^-}-\alpha)d\phi_{\pi^-}\\ &+\int_\alpha^{2\pi}W'(\phi_{\pi^-},\alpha+2\pi-\phi_{\pi^-})d\phi_{\pi^-}\\ &+\int_{2\pi-\alpha}^{2\pi} W'(\phi_{\pi^-},4\pi-\alpha-\phi_{\pi^-})d\phi_{\pi^-}.~~ \tag {3} \end{alignat} $$ Generally, we have the probability density function of $\theta_{\pi^\pm}$ and $\phi_{\pi^\pm}$, donated as $W^{\prime\prime}(\theta_{\pi^-},\phi_{\pi^-};\theta_{\pi^+},\phi_{\pi^+})$, where $\theta_{\pi^\pm}$ are the polar angles of the momentum of $\pi^\pm$ in the helicity rest frame of their own parent particles. Once having $\theta_{\pi^\pm}$ of integration, $W^\prime(\phi_{\pi^-},\phi_{\pi^+})$, the distribution of $\phi_{\pi^\pm}$ can be deduced. In the decay chain $J/\psi\to{\it \Lambda}\bar{\it \Lambda}\to p\pi^-\bar{p}\pi^+$, the angle distribution is[12] $$\begin{align} &W^{\prime\prime}(\theta_{\pi^-},\phi_{\pi^-};\theta_{\pi^+},\phi_{\pi^+})\\ =\,&\frac{1}{4\pi}\sum_{L_1\geqslant0}^{2J_1}\sum_{L_2\geqslant0}^{2J_2}C_{L_1} (0,0;0,0)C_{L_2}(0,0;0,0)\\ &\times\sum_{M_1,M_2}t_{M_1M_2}^{L_1L_2}(C_1,C_2)Y_{L_1M_1} (\theta_{\pi^-},\phi_{\pi^-})\\ &\cdot Y_{L_2M_2}(\theta_{\pi^+},\phi_{\pi^+}),~~ \tag {4} \end{align} $$ where $C_{0{\it \Lambda}}(0,0;0,0)=C_{0\bar{\it \Lambda}}(0,0;0,0)=\frac{2}{\pi}$, $C_{1{\it \Lambda}}(0,0;0,0)=\frac{2}{\pi}a_{\it \Lambda}$, $C_{1\bar{\it \Lambda}}(0,0;0,0)=\frac{2}{\pi}a_{\bar{\it \Lambda}}$, $a_{\it \Lambda}$ ($a_{\bar{\it \Lambda}}$) is the decay parameter of ${\it \Lambda}$ ($\bar{\it \Lambda}$),[13] $t_{M_1M_2}^{L_1L_2}$ are the joint multipole parameters of the ${\it \Lambda}\bar{\it \Lambda}$ system, which can be derived directly from the density matrix of the initial state, and $Y_{LM}$ are the spherical harmonic functions. The density matrices of the initial state of ${\it \Lambda}$ and $\bar{\it \Lambda}$ rely on the initial polarization of $J/\psi$. The spin parity of $J/\psi$ is $J^P=1^-$, and parity conservation and angular momentum conservation require that only the S-wave and D-wave exist in the decay $J/\psi\to{\it \Lambda}\bar{\it \Lambda}$. To simplify the following discussion, we assume that the initial polarization of $J/\psi$ is isotropic, although it is different from the $e^+e^-\to J/\psi$ process. Therefore, the partition of the $J/\psi$ particles with the spin projection $-1$, $0$, $+1$ on the $z$-axis in the helicity rest frame are all one-third and the density matrix of the ${\it \Lambda}\bar{\it \Lambda}$ system is $$ \rho=\frac{1}{3}(\rho_{-1}+\rho_0+\rho_1).~~ \tag {5} $$ Considering both the S-wave and D-wave, the density matrix of the system of ${\it \Lambda}$ and $\bar{\it \Lambda}$ becomes $$ \rho=\frac{1}{6}\left(\begin{matrix} 1 & 0 &0 & 1 \\ 0 & 2 &0 & 0 \\ 0 & 0 &2 & 0 \\ 1 & 0 &0 & 1 \end{matrix}\right).~~ \tag {6} $$ Here we use the initial spin projection $+1$ on the $z$-axis of $J/\psi$ in the D-wave as an example. The wave function before and after decay of the D-wave can be written as $$\begin{alignat}{1} \!\!\!\!\!&\chi_{J/\psi}(S=1,S_z=1)\\ \!\!\!\!\!=\,&\sqrt{\frac{3}{5}}\chi_{{\boldsymbol L}}(L=2,L_z=2)\chi_{{\it \Lambda}\bar{\it \Lambda}}(S=1,S_z=-1)\\ \!\!\!\!\!&-\sqrt{\frac{3}{10}}\chi_{{\boldsymbol L}}(L=2,L_z=1)\chi_{{\it \Lambda}\bar{\it \Lambda}}(S=1,S_z=0)\\ \!\!\!\!\!&+\sqrt{\frac{1}{10}}\chi_{{\boldsymbol L}}(L=2,L_z=0)\chi_{{\it \Lambda}\bar{\it \Lambda}}(S=1,S_z=1),~~ \tag {7} \end{alignat} $$ where $\chi_{{\boldsymbol L}}(L,L_z)$ is the orbital angular momentum between ${\it \Lambda}\bar{\it \Lambda}$, and $\chi_{{\it \Lambda}\bar{\it \Lambda}}(S,S_z)$ is the total spin angular momentum of ${\it \Lambda}\bar{\it \Lambda}$. The joint density matrix of ${\it \Lambda}$ and $\bar{\it \Lambda}$ is $$\begin{align} \rho^{{\it \Lambda}\bar{\it \Lambda}}=\,&\frac{3}{5}|1,-1\rangle\langle1,-1|+\frac{1}{10}|1,1\rangle\langle1,1| +\frac{3}{10}|1,0\rangle\langle1,0|\\ =\,&\frac{3}{5}\left(\begin{matrix} 0 & 0 \\0 &1\end{matrix}\right) \otimes\left(\begin{matrix} 1 & 0 \\0 & 0\end{matrix}\right) \\ &+\frac{1}{10}\left(\begin{matrix} 1 & 0 \\0 & 0\end{matrix}\right) \otimes\left(\begin{matrix} 0 & 0 \\0 & 1\end{matrix}\right)\\ &+\frac{3}{20}\Big[\left(\begin{matrix} 1 & 0 \\0 & 0\end{matrix}\right)\otimes\left(\begin{matrix} 1 & 0 \\0 & 0\end{matrix}\right)\\ &+\left(\begin{matrix} 0 & 0 \\0 & 1\end{matrix}\right)\otimes\left(\begin{matrix} 0 & 0 \\0 & 1\end{matrix}\right)\\ &+\left(\begin{matrix} 0 & 0 \\1 & 0\end{matrix}\right) \otimes\left(\begin{matrix} 0 & 0 \\1 & 0\end{matrix}\right)\\ &+\left(\begin{matrix} 0 & 1 \\0 & 0\end{matrix}\right)\otimes\left(\begin{matrix} 0 & 1 \\0 & 0\end{matrix}\right)\Big]\\ =\,&\frac{1}{20}\left(\begin{matrix} 3 & 0 &0 & 3 \\ 0 & 2 &0 & 0 \\0 & 0 &12 & 0 \\3 & 0 &0 & 3 \end{matrix}\right),~~ \tag {8} \end{align} $$ where the terms before (after) $\otimes$ are the density matrices of ${\it \Lambda}$ ($\bar{\it \Lambda}$) in ${\it \Lambda}$ ($\bar{\it \Lambda}$)'s helicity rest frame, respectively. The expression of joint multipole parameters of the initial states can be written as $$ t^{LL^{\prime}}_{MM^{\prime}}=tr(\rho^{{\it \Lambda}\bar{\it \Lambda}}\hat{T}({\it \Lambda})_{M}^L\otimes \hat{T}(\bar{\it \Lambda})_{M^{\prime}}^{L^{\prime}}),~~ \tag {9} $$ where $\hat{T}_{M}^L$ is the spherical tensor operator, $0\leq L\leq2s$, $-L\leq M\leq L$, $s$ is the spin of the particle, and $L$ is the rank of the tensor operator. Then we put the density matrix of the S-wave and D-wave in. The joint multipole parameters are listed in Table 1.
Table 1. The joint multipole parameters $t^{LL^{\prime}}_{MM^{\prime}}$.
$L,M=0$,0 $L,M=1$,0 $L,M=1$,1 $L,M=1$,$-$1
$L^{\prime},M^{\prime}=0$,0 1 0 0 0
$L^{\prime},M^{\prime}=1$,0 0 $-\frac{1}{9}$ 0 0
$L^{\prime},M^{\prime}=1$,1 0 0 $\frac{1}{9}$ 0
$L^{\prime},M^{\prime}=1$,$-$1 0 0 0 $\frac{1}{9}$
Having $\theta_{\pi^-}$ and $\theta_{\pi^+}$ of integration, the distribution of $W(\phi_{\pi^-},\phi_{\pi^+})$ can be derived from Eq. (4) and Table 1, $$\begin{alignat}{1} \!\!\!\!\!W'(\phi_{\pi^-},\phi_{\pi^+})=\frac{1}{4\pi^2}+\frac{a_{\it \Lambda} a_{\bar{\it \Lambda}}}{3\pi^4}\cos(\phi_{\pi^-}+\phi_{\pi^+}).~~ \tag {10} \end{alignat} $$ Combining Eqs. (3) and (10) and after normalization, the result can be obtained as $$ W(\alpha)=\frac{1}{\pi}+\frac{4a_{\it \Lambda} a_{\bar{\it \Lambda}}}{3\pi^3}\cos\alpha.~~ \tag {11} $$ The latest study suggests that $J/\psi$ produced from $e^+e^-$ is partially polarized.[14] Compared with two transverse polarized components, the longitudinal one can be neglected. Taking into account the above situation, with a similar procedure we can obtain the distribution of $\alpha$ as follows: $$ W(\alpha)=\frac{1}{\pi}+\frac{4(5+b)}{5(3+b)}\frac{a_{\it \Lambda} a_{\bar{\it \Lambda}}}{\pi^3}\cos\alpha,~~ \tag {12} $$ where $b$ is the decay parameter of $J/\psi\to B\bar{B}$,[15] equaling 0.51 in $J/\psi \rightarrow {\it \Lambda}\bar{\it \Lambda}$ theoretically, which is negligible. According to the estimation, polarization and no polarization conditions cannot be distinguished at the 1$\sigma$ confidence interval with the data available now. However, without the consideration of the quantum mechanical correlation of ${\it \Lambda}$ and $\bar{\it \Lambda}$, the two decays of hyperons should be taken as independent. Then there is no correlation between the angular distribution of $\pi^-$ and $\pi^+$ on both sides. $W^\prime(\phi_{\pi^-},\phi_{\pi^+})$ in Eq. (2) can be written as the product of the respective probability density function of $\phi_{\pi^-}$ and $\phi_{\pi^+}$. Therefore, $\alpha$ is expected to be uniformly distributed, $$ W(\alpha)=\frac{1}{\pi}.~~ \tag {13} $$ From the BESIII detector at the BEPCII collider, about 10 billion $J/\psi$ events are collected and the branching ratio of $J/\psi\to{\it \Lambda}\bar{\it \Lambda}$ is about $1.6\times 10^{-3}$.[16] Thus 16 million $J/\psi\to{\it \Lambda}\bar{\it \Lambda}$ events could be used to test the theoretical prediction with a great deal of statistics. Combining the efficiency of the detector, which can be estimated from the MC simulation (here we take 40$\%$ as an approximation), we can calculate the error band with 40 intervals, as shown in Fig. 2. If we consider the new results on the ${\it \Lambda}$($\bar{\it \Lambda} $) decay asymmetry parameter from the BESIII collaboration,[7] the spin correlation is strengthened slightly. The obvious discrepancy between the two distributions of $\alpha$ allows the experiments to verify the two conditions.
cpl-36-6-062301-fig2.png
Fig. 2. The uncertainty band of two distributions with 10 billion $J/\psi$ divided into 40 intervals ($a_{\it \Lambda}=0.642$, $a_{\bar{\it \Lambda}}=-0.642$).[16]
This decay channel can also be used to check the decay parameters of ${\it \Lambda}$ and $\bar{\it \Lambda}$ in CP violation, since the value of $a_{\it \Lambda} a_{\bar{\it \Lambda}}$ can be obtained from the final distribution, Eq. (11). The fit results from the experiment could be compared with the theoretical expectation. This method can be universally used in all of the two-body decay chains $A\to C_1+C_2,C_1\to E_1+F_1,C_2\to E_2+F_2$ as long as the angle between the two decay planes can be reconstructed, which is feasible to be carried out in experiments, although the final distribution relies on the spin and parity of the mother particle and daughter particles. Compared with the angle between the momenta of $\pi^-$ and $\pi^+$ in their mother particles' (${\it \Lambda}$ and $\bar{\it \Lambda}$) rest frames defined by Törnqvist, the angle that we proposed between the ${\it \Lambda}$ and $\bar{\it \Lambda}$ decay planes is more rigorous, as the angle can be strictly deduced theoretically and is feasible to be measured by experiments. After all the above discussion, we may safely draw the conclusion that we have developed a new method to derive the distribution of an observable to test quantum mechanical correlation. The parameter we use, which is the angle between two decay planes, is accessible from the experiment and varies clearly under two conditions. The authors thank Dr. Xun Chen for the valuable discussion. Our thanks also go to Bingtian Ye, who greatly inspired the authors with the method from quantum information. References Charm Physics CP violation in J ψ → Λ Λ ¯ CP asymmetries in Strange Baryon DecaysStudy of Λ − Λ ¯ oscillation in quantum coherent Λ Λ ¯ by using J / ψ → Λ Λ ¯ decayDecays of the ψ ( 3097 ) to baryon-antibaryon final statesBaryon pair production in decaysStudy of J / ψ and ψ ( 3686 ) decay to Λ Λ ¯ and Σ 0 Σ ¯ 0 final statesStudy of e + e − → Λ Λ ¯ , Λ Σ ¯ 0 , Σ 0 Σ ¯ 0 using initial state radiation with BABARCan Quantum-Mechanical Description of Physical Reality Be Considered Complete?Suggestion for Einstein-Podolsky-Rosen experiments using reactions like $$e^ + e^ - \to \Lambda \bar \Lambda \to \pi ^ - p\pi ^ + \bar p$$Looking at CP invariance and quantum mechanics in decayGeneral Partial Wave Analysis of the Decay of a Hyperon of Spin ½Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron AnnihilationHelicity selection rules and tests of gluon spin in exclusive quantum-chromodynamic processesReview of Particle Physics
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