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4.7.1 CP violation studies in $e^+e^- \rightarrow t\overline {t}$

CP violation in $e^+e^- \rightarrow$ $t \overline {t}$ can mainly arise through the couplings of the top quark to a virtual photon and a virtual Z, which are responsible for $t \overline {t}$ production, and the tbW coupling responsible for the dominant decay of the top quark in to a b quark and a W. The CP-violating couplings of a $t \overline {t}$ current to $\gamma $ and Z can be written as $ie\Gamma_\mu^j$, where

\begin{displaymath}\Gamma_\mu^j\;=
\;\frac{c_d^j}{2\,m_t}\,i\gamma_5\,
(p_t\,-\,p_{\overline{t}})_{\mu},\;\;j\;=\;\gamma,Z,
\end{displaymath} (4.40)

where $ec_d^{\gamma}/m_t$ and ecdZ/mt are the electric and ``weak' dipole couplings. The tbW vertex is parametrized as in eqs. (1.34) and (1.35).

While these CP-violating couplings may be studied using CP-violating correlations among momenta and spins which include the t and $\overline{t}$momenta and spins [42], it may be much more useful to study asymmetries and correlations constructed out of the initial e+/e- momenta and the momenta of the decay products, which are more directly observable. In addition, the observables using top spin depend on the basis chosen [43,42], and would require reconstruction of the basis which has the maximum sensitivity. In studying decay distributions, this problem is avoided.

Correlations of optimal CP-violating observables have been studied by Zhou [44]. Using purely hadronic or hadronic-leptonic variables, limits on the dipole moment of the order of 10-18 e cm are shown to be possible with $\sqrt {s}=500$ GeV and integrated luminosity of 50 fb-1.

Examples of CP-violating asymmetries using single-lepton angular distributions and the lepton energy correlations have been discussed in Sec. 1.6.3. In addition, we have studied, in [45], additional CP-violating asymmetries which are functions of lepton energy. Using suitable ranges for the lepton energy, it is possible to enhance the relative contributions of CP violation in production and CP violation in decay.

One-loop QCD corrections can contribute as much as 30% to $t \overline {t}$ production cross section at $\sqrt {s}=500$ GeV [46]. It is therefore important to include these in estimates of sensitivities of CP-violating observables. The effect of QCD corrections in the soft-gluon approximation in decay lepton distributions in $e^+e^- \rightarrow$ $t \overline {t}$ were discussed in [47]. These were incorporated in CP-violating leptonic angular asymmetries and corresponding limits possible at JLC with longitudinal beam polarization were presented in [48]. These are in the laboratory frame, do not need accurate detailed top energy-momentum reconstruction, and are insensitive to CP violation (or other CP-conserving anomalous effects) in the tbW vertex.

Four different asymmetries have been studied in [48]. In addition to two asymmetries where the azimuthal angles are integrated over, and which are exactly the ones defined in [49], there are two others which depend on azimuthal distributions of the lepton. A cut-off $\theta_0$ in the forward and backward directions is assumed in the polar angle of the lepton. The up-down asymmetry is defined by

\begin{displaymath}A_{ud}(\theta_0)=\frac{1}{2\,\sigma (\theta_0)}\int_{\theta_0...
...d\,\sigma^-_{\rm down} } {d\,\theta_l}
\right] {d\,\theta_l} ,
\end{displaymath} (4.41)

Here up/down refers to $(p_{l^{\pm}})_y\;\raisebox{-1.0ex}{$\stackrel{\textstyle>}{<}$ }\;0,\;
\:(p_{l^{\pm}})_y$ being the ycomponent of $\vec{p}_{l^{\pm}}$ with respect to a coordinate system chosen in the $e^+\,e^-$ center-of-mass (cm) frame so that the z-axis is along $\vec{p}_e$, and the y-axis is along $\vec{p}_e\,\times\,\vec{p}_t$. The $t\bar{t}$ production plane is thus the xz plane. Thus, ``up" refers to the range $0<\phi_l<\pi$, and ``down" refers to the range $\pi<\phi_l<2\pi$.

The left-right asymmetry is defined by

\begin{displaymath}A_{lr}(\theta_0)=\frac{1}{2\,\sigma (\theta_0)}\int_{\theta_0...
...\,\sigma^-_{\rm right} } {d\,\theta_l}
\right] {d\,\theta_l} ,
\end{displaymath} (4.42)

Here left/right refers to $(p_{l^{\pm}})_x\;\raisebox{-1.0ex}{$\stackrel{\textstyle>}{<}$ }\;0,\;
\:(p_{l^{\pm}})_x$ being the xcomponent of $\vec{p}_{l^{\pm}}$ with respect to the coordinate system system defined above. Thus, ``left" refers to the range $-\pi /2<\phi_l<\pi /2$, and ``right" refers to the range $\pi /2<\phi_l<3\pi /2$.

The simultaneous independent 90% CL limits on the couplings $c_d^{\gamma,Z}$ which can be obtained at a linear collider with $\sqrt {s}=500$ GeV with integrated luminosity 200 fb-1, and for $\sqrt{s}=1000$ GeV with integrated luminosity 1000 fb-1and using only e- longitudinal beam polarization $\pm 0.9$ are given in Table 4.2.


 
Table 4.2: Simultaneous limits on dipole couplings combining data from polarizations Pe=0.9 and Pe=-0.9, using separately Aud and Alr. Values of $\sqrt {s}$ and integrated luminosities as in the previous tables.
  Aud Alr
$\sqrt {s}$ (GeV) $\theta_0$ Re $c_d^{\gamma}$ RecdZ $\theta_0$ Im $c_d^{\gamma}$ ImcdZ
500 25$^\circ $ 0.031 0.045 $35^{\circ}$ 0.031 0.056
1000 30$^\circ $ 0.0085 0.013 $60^{\circ}$ 0.028 0.052

As can be seen from the table, the limits on the dipole couplings are of the order of a few times 10-17 e cm for $\sqrt {s}=500$ GeV.


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ACFA Linear Collider Working Group
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