Survey of top quark polarization at a polarized linear $e^{+}{e}^{-}$ collider

We discuss in detail top quark polarization in above-threshold $(t\overline{t})$ production at a polarized linear $e^{+}{e}^{-}$ collider. We pay particular attention to the minimization and maximization of the polarization of the top quark by tuning the longitudinal polarization of the $e^{+}$ and $e^{-}$ beams. The polarization of the top quark is calculated in full next-to-leading order QCD. We also discuss the beam polarization dependence of the longitudinal spin-spin correlations of the top and antitop quark spins.

Figure 1

Average (a) longitudinal polarization $⟨P^{(l)}⟩$ and (b) transverse polarization $⟨P^{(\mathrm{tr})}⟩$ as a function of the c.m. energy $\sqrt{s}$, for the values $P_{\mathrm{eff}}=-1$ (dashed), $P_{\mathrm{eff}}=-0.5$ (dash-dotted), $P_{\mathrm{eff}}=0$ (solid), $P_{\mathrm{eff}}=+0.5$ (dash-dotted), and $P_{\mathrm{eff}}=+1$ (dashed). Averaging is over $\cos \theta$.

Figure 2

A generic configuration for top pair production and top polarization at a polarized $e^{+}{e}^{-}$ collider with a $(e_L^{-},e_R^{+})$ polarization. The positive $z$ axis points into the direction of the electron momentum. The angle $\alpha$ is the polar angle of the top quark polarization relative to the top quark momentum measured anticlockwise from the direction of the top quark.

Figure 3

Contour plots in the $(h_{-},h_{+})$ plane (a) for fixed values of the effective polarization $P_{\mathrm{eff}}=(h_{-}-h_{+})/(1-h_{-}{h}_{+})$ and (b) for fixed values of the gain factor $K_G=(1-h_{-}{h}_{+})$. The two points marked off in the plots correspond to $(h_{-},h_{+})=(-0.8,+0.625)$ and $(+0.8,-0.625)$, respectively.

Figure 4

The total cross section $\sigma$ at the one-loop level as a function of the beam energy $\sqrt{s}$ and (a) the electron polarization $h_{-}$ $(h_{+}=0)$; (b) for three values of the positron polarization $h_{+}=0$, 0.4, 0.6, and with the fixed electron beam polarization of $h_{-}=-0.9$ (solid lines). In Fig. 4b we also show the respective LO rates (dashed lines).

Figure 5

Polar angle dependence of the differential cross section for (a) $\sqrt{s}=500\mathrm{GeV}$ and $P_{\mathrm{eff}}=-1,0,+1$ and (b) $P_{\mathrm{eff}}=0$ for beam energies $\sqrt{s}=360\mathrm{GeV}$ (dotted line), 500 GeV (dashed line), 1000 GeV (dash-dotted line), and 3000 GeV (solid line).

Figure 6

Left-right polarization asymmetry $A_{LR}$ for $\sqrt{s}=360$, 500, 1000, and 3000 GeV (notation as in Fig. 5).

Figure 7

Correlation of the angles $\alpha$ and $\theta$ in dependence on the effective beam polarization $P_{\mathrm{eff}}$ (a) for threshold energies $\sqrt{s}=2m_t$ and (b) for $s\to \infty$.

Figure 8

The top quark polarization angle $\alpha$ for a scattering angle of $\theta =90°$ as a function of the beam energy for (a) negative values of $P_{\mathrm{eff}}=-1,-0.5,-0.25$ and (b) positive values of $P_{\mathrm{eff}}=+1,+0.5,+0.25$.

Figure 9

Correlation of angles $\alpha$ and $\theta$ for (a) $P_{\mathrm{eff}}=-1$ ($\alpha ={\alpha }_{LR}$), (b) $P_{\mathrm{eff}}=+1$ ($\alpha ={\alpha }_{RL}$), and (c) $P_{\mathrm{eff}}=+0.5$ for different values of the c.m. energy $\sqrt{s}=360$, 500, 1000, and 3000 GeV (notation as in Fig. 5).

Figure 10

Difference $\Delta {\alpha }_{LR}={\alpha }_{LR}(\mathrm{NLO})-{\alpha }_{LR}(\mathrm{LO})$ of NLO and LO polarization angles for $\sqrt{s}=360$, 500, 1000, and 3000 GeV (notation as in Fig. 5).

Figure 11

NLO longitudinal top polarization as a function of $\cos \theta$ drawn (a) for effective beam polarizations $P_{\mathrm{eff}}=-1,-0.5,0,+0.5,+1$ (notation as in Fig. 1) at $\sqrt{s}=500\mathrm{GeV}$; (b) at beam energies $\sqrt{s}=360$, 500, 1000, and 3000 GeV (notation as in Fig. 5) and $P_{\mathrm{eff}}=-1$.

Figure 12

Average (a) longitudinal polarization $⟨P^{(l)}⟩$ and (b) transverse polarization $⟨P^{(\mathrm{tr})}⟩$ in the forward and backward hemispheres for $P_{\mathrm{eff}}=-1$.

Figure 13

NLO transverse top polarization as a function of the scattering angle $\theta$ drawn (a) for effective beam polarizations $P_{\mathrm{eff}}=-1,-0.5,0,+0.5,+1$ (notation as in Fig. 1) at $\sqrt{s}=500\mathrm{GeV}$; (b) at beam energies $\sqrt{s}=360$, 500, 1000, and 3000 GeV (notation as in Fig. 5) and $P_{\mathrm{eff}}=-1$.

Figure 14

$O({\alpha }_s)$ normal polarization $P^{(n)}$ of the top quark as a function of $\cos \theta$ (a) for effective beam polarizations $P_{\mathrm{eff}}=-1,-0.5,0,+0.5,+1$ (notation as in Fig. 1) at $\sqrt{s}=500\mathrm{GeV}$; (b) for $P_{\mathrm{eff}}=-1$ at beam energies $\sqrt{s}=360$, 500, 1000, and $\sqrt{s}=3000\mathrm{GeV}$ (notation as in Fig. 5).

Figure 15

Total NLO top quark polarization as a function of $\cos \theta$ for beam energies $\sqrt{s}=360$, 500, 1000, and 3000 GeV (notation as in Fig. 5) and (a) $P_{\mathrm{eff}}=-1$, (b) $P_{\mathrm{eff}}=+1$, and (c) $P_{\mathrm{eff}}=0$.

Figure 16

Average NLO top quark polarization $⟨|\overrightarrow{P}|⟩$ for a scattering angle of $\theta =90°$ as a function of the beam energy $\sqrt{s}$ for $P_{\mathrm{eff}}=-1,-0.5,0,+0.5,+1$ (notation as in Fig. 1).

Figure 17

Total NLO top quark polarization averaged over the forward and backward hemispheres for $P_{\mathrm{eff}}=-1,0,+1$ as a function of $\sqrt{s}$.

Figure 18

(a) $P_{\mathrm{eff}}$ values required for minimal top quark polarization $|\overrightarrow{P}|$ and (b) minimal values for $|\overrightarrow{P}|$, plotted against $\cos \theta$, for $\sqrt{s}=360$, 500, 1000, and 3000 GeV notation as in Fig. 5).

Figure 19

NLO Parametric plot of the orientation and the length of the polarization vector in dependence on the c.m. energy $\sqrt{s}$ for values $\theta =60°$, 90°, 120°, and 150° for (a) $P_{\mathrm{eff}}=-1$ (solid lines) and $P_{\mathrm{eff}}=-0.95$ (dashed lines), and (b) $P_{\mathrm{eff}}=+1$ (solid lines) and $P_{\mathrm{eff}}=+0.95$ (dashed lines). The dots on the trajectories from the border to the central line stand for $\sqrt{s}=500$, 1000, and 3000 GeV.

Figure 20

NLO beam energy dependence of the polar average of the longitudinal spin-spin correlation function $⟨P^{({\ell }_1{\ell }_2)}⟩$.