Decay-lepton angular distribution in polarized $e^+e^- \to \tt$ and $CP$-violating dipole couplings of the top quark

In the presence of an electric dipole coupling of $t \bar t$ to a photon, and an analogous ``weak" dipole coupling to the $Z$, $CP$ violation in the process $e^+e^- \rightarrow t \bar t$ leads to the polarization of the top and anti-top. This polarization can be analyzed by studying the energy and angular distributions of a decay charged lepton (anti-lepton) when the top (anti-top) decays leptonically. We have obtained analytic expressions for these distributions when either $t$ or $\overline t$ decays leptonically. We study two types of simple $CP$-violating angular asymmetries which do not need tof simple $CP$-violating angular asymmetries which do not need the full reconstruction of the $t$ or $\bar t$. These together can help to determine the electric and weak dipole form factors independently. We have also shown how the use of longitudinal beam polarization can help to do this with only one asymmetry measurement, and to imprhelp to do this with only one asymmetry measurement, and to improve the sensitivity.