Quantum entanglement and Bell nonlocality in top-quark pair production at a photon linear collider

Abstract

A photon linear collider, the two-photon collision mode of an $e^+e^-$ linear collider, uses high-energy laser photons backscattered off the incoming electrons and positrons. The colliding-photon polarization is fully controllable through the polarizations of the initial electron and positron beams and laser photons. We investigate the impact of colliding-photon polarization on the observability of quantum entanglement in top-quark pair production at a photon linear collider. Constructing the spin density matrix of the $t\bar{t}$ two-qubit system from the helicity amplitudes, we demonstrate that a photon linear collider is an ideal machine to probe quantum entanglement and Bell nonlocality across the broad phase space of the process.

Figures (8)

• Figure 1: The $t\bar{t}$ production plane with the definition of the scattering angle $\Theta$ for the process $\gamma(k_1, \lambda_1)\, +\, \gamma(k_2,\lambda_2) \to t(p,\sigma)\, + \, \bar{t}(\bar{p}, \bar{\sigma})$. The four-momentum $k_1$ ($k_2$) is assigned to the photon backscattered off the incoming $e^-$ ($e^+$) at the PLC. The transverse polarization vectors $P_T$ and $\bar{P}_T$ have azimuthal angles $\alpha$ and $\bar{\alpha}$ appearing in Eq. \ref{['eq:PDM_ttbar']}, respectively, measured from the production plane. The $\{\hat{n}\,,\hat{r}\,,\hat{k}\}$ basis is defined by $\hat{k} = {\vec{p}}/|{\vec{p}}\,|$, $\hat{n} = {\vec{k}}_1\times{\vec{p}}/|{\vec{k}}_1\times{\vec{p}}\,|$, and $\hat{r} = \hat{k}\times\hat{n}$. Note that, in the $\{\hat{n}\,,\hat{r}\,,\hat{k}\}$ basis, $P_L = P_k$, $P_T c_\alpha = -P_r$, and $P_T s_\alpha = P_n$ for the top quark, whereas for the anti-top quark, $\bar{P}_L = - \bar{P}_k$, $\bar{P}_T c_{\bar{\alpha}} = -\bar{P}_r$, and $\bar{P}_T s_{\bar{\alpha}} = \bar{P}_n$.
• Figure 2: (Left) The helicity-dependent luminosities $L^{\lambda_1\lambda_2}=\frac{1}{{\cal L}_{ee}} \frac{{\rm d}{\cal L}^{\lambda_1\lambda_2}_{\gamma\gamma}}{{\rm d}\sqrt\tau}$ taking $P_eP_c=\tilde{P}_e\tilde{P}_c=-1$ and $x=4.8$. From left to right, $(\lambda_1,\lambda_2) =(+,+)$, $(-,-)$, $(-,+)$, and $(+,-)$ while, from top to bottom, $(P_e=-P_c,\tilde{P}_e=-\tilde{P}_c)=(+,+)$, $(-,-)$, $(-,+)$, and $(+,-)$. In each frame, we also show the average luminosity $\frac{1}{4}\sum_{\lambda_1,\lambda_2=\pm}L^{\lambda_1\lambda_2}$ and the unpolarized one $\frac{1}{{\cal L}_{ee}} \frac{{\rm d}{\cal L}^{\rm unp}_{\gamma\gamma}}{{\rm d}\sqrt\tau}$ in dashed red and blue lines, respectively, for comparison. (Right) The weight functions $w^{\lambda_1\lambda_2}$ for $(\lambda_1,\lambda_2)=(P_e,\tilde{P}_e)$ (black solid), $(\lambda_1,\lambda_2)=(-P_e,-\tilde{P}_e)$ (green dashed), and $(\lambda_1,\lambda_2)=(\mp P_e,\pm\tilde{P}_e)$ (orange dotted). The vertical lines at $\sqrt{\tau}=0.345$ and $0.69$ mark the $2M_t$ threshold for $\sqrt{s}=1~\mathrm{TeV}$ and $500~\mathrm{GeV}$, respectively.
• Figure 3: Quantum entanglement in the unpolarized case: (Left) Quantum entanglement for the unpolarized case $P_e=P_c=\tilde{P}_e=\tilde{P}_c=0$, shown in the $(\cos\Theta,\sqrt{\hat{s}})$ plane. Shaded regions satisfy $\mathcal{N}[\rho]>0$, $\mathcal{C}[\rho]>0$, and $D<-1/3$, indicating entanglement of the $t\bar{t}$ spin state. The densely shaded red regions additionally satisfy $m_{12}>1$, indicating violation of the Bell inequality. (Right) Magnified view of the region $2M_t < \sqrt{\hat{s}} < 500$ GeV.
• Figure 4: Quantum entanglement with $w^{\pm\pm}=1$: (Left and Middle) Contour regions in the $(\cos\Theta,\sqrt{\hat{s}})$ plane for negativity ${\cal N}[\rho]$ (upper-left), concurrence ${\cal C}[\rho]$ (upper-right), the Bell nonlocality parameter $m_{12}$, and the entanglement marker $D$ (lower-right). (Right) Bell nonlocality parameter $m_{12}$, concurrence $\mathcal{C}[\rho]$, negativity $\mathcal{N}[\rho]$, and entanglement marker $D$ as functions of $\sqrt{\hat{s}}$, from top to bottom. Shaded regions satisfy $m_{12}>1$, $\mathcal{C}[\rho]>0$, $\mathcal{N}[\rho]>0$, and $D<-1/3$, indicating quantum entanglement and violation of the Bell inequality. The lower frame magnifies the region $2M_t<\sqrt{\hat{s}}<500$ GeV.
• Figure 5: Quantum entanglement with $w^{\pm\mp}=1$: Same as in Fig. \ref{['fig:QE_perfectPP']} but for $w^{\pm\mp}=1$ and, in the right panel, for $\cos\Theta=0$ and $0.5$.