Manipulating Bell nonlocality and entanglement in polarized electron-positron annihilation

TL;DR

This work develops a spin-density-matrix framework for hyperon–antihyperon pairs produced in $e^{+}e^{-}$ annihilation and analyzes how lepton-beam polarization manipulates Bell nonlocality and entanglement. By expressing the system in Bloch–Fano form and, for certain kinematics, as an $X$-state, the authors derive closed-form expressions for the CHSH parameter $\mathcal{B}$, concurrence $\mathcal{C}$, and negativity $\mathcal{N}$ as functions of longitudinal $P_L$ and transverse $P_T$ polarization, scattering angle $\theta$, and azimuth $\phi$. They show that longitudinal polarization broadens the angular region with Bell nonlocality and enhances entanglement but does not yield maximal entanglement, whereas transverse polarization can achieve maximal entanglement across all angles for suitable $\phi$ (and $P_T$ up to 1), with distinct separability regimes arising at special $\phi^*$ and $\theta_{Sep}$. These results elucidate the hierarchy of quantum correlations in high-energy processes and propose polarized beams as a practical tool to study and control quantum correlations in $Y\bar{Y}$ systems at current or future facilities like BESIII and a potential Super $\tau$-Charm facility.

Abstract

The hyperon-antihyperon pairs produced in electron-positron annihilation as a massive two-qubit quantum system can be used to study the quantum correlations at high energies. This paper is theoretically dedicated to how polarization of lepton beams manipulate the Bell nonlocality and entanglement of hyperon pairs system. The response of CHSH parameter, concurrence, and negativity to the polarization degree is numerically calculated by exploiting the joint spin density matrix of $Y\bar{Y}$ system. Different influences of longitudinal and transverse polarization of beams on entanglement are found and compared. The results provide alternative perspectives for the decay of charmoniun to hyperon pairs.

Figures (19)

• Figure 1: The process of $e^{+}e^{-} \rightarrow \gamma^* /\psi \rightarrow Y\bar{Y}$ in the c.m. frame. The coordinate system of $Y$ and $\bar{Y}$ are chosen to be of the same chirality with $\{\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}\}$ being three axes in the helicity rest frame of $Y$ and $\{\hat{\mathbf{x}}',\hat{\mathbf{y}}',\hat{\mathbf{z}}'\}$ of $\bar{Y}$. For hyperon $Y$, $\hat{\mathbf{y}}={(\hat{\mathbf{p}}_{e}\times\hat{\mathbf{p}}_{Y})}/{\left|\hat{\mathbf{p}}_{e}\times\hat{\mathbf{p}}_{Y}\right|},\ \hat{\mathbf{z}}=\hat{\mathbf{p}}_{Y},\ \hat{\mathbf{x}}=\hat{\mathbf{y}}\times\hat{\mathbf{z}}$, where $\hat{\mathbf{p}}_{e}$ and $\hat{\mathbf{p}}_{Y}$ are unit vectors along momentum directions of the electron and hyperon respectively. The axes of antihyperon $\bar{Y}$ hold the transformation: $\{\hat{\mathbf{x}}',\hat{\mathbf{y}}',\hat{\mathbf{z}}'\}=\{\hat{\mathbf{x}},-\hat{\mathbf{y}},-\hat{\mathbf{z}}\}$.
• Figure 2: The CHSH parameter $\mathcal{B}[\rho^{P_L}_{Y \bar{Y}}]$ as a function of $\cos\theta$ ($\theta$ is the scattering angle) and longitudinal beam polarization degree $P_L$ in $J/\psi\to Y{\bar{Y}}$ for $Y = \Lambda$, $\Sigma^{+}$, $\Xi^{-}$ and $\Xi^{0}$. The dashed curve is for $\mathcal{B}[\rho^{P_L}_{Y \bar{Y}}] = 2$.
• Figure 3: The $\max_{\theta}\mathcal{B}[\rho^{P_L}_{Y \bar{Y}}]$ (upper panel) and $\cos\theta^*$ (lower panel, see main text for defination of $\theta^*$) as a function of $P_L$ in $J/\psi\to Y{\bar{Y}}$ for $Y = \Lambda$, $\Sigma^{+}$, $\Xi^{-}$ and $\Xi^{0}$.
• Figure 4: The concurrence $\mathcal{C}[\rho^{P_L}_{Y \bar{Y}}]$ as a function of $\cos\theta$ and $P_L$ in $J/\psi\to Y{\bar{Y}}$ for $Y=\Lambda$, $\Sigma^{+}$, $\Xi^{-}$ and $\Xi^{0}$.
• Figure 5: The $\cos \theta_{\max}$ (see main text for defination of $\theta_{\max}$) as a function of $P_L$ in $J/\psi\to Y{\bar{Y}}$ for $Y=\Xi^{-}$ and $\Xi^{0}$.