Effect of longitudinal electron polarization in the measurement of the $\tau$ weak dipole moment in $e^+e^-$ collisions

We show that certain CP-odd momentum correlations in the production and subsequent decay of tau pairs in $e^+e^-$ collisions get enhanced when the electron beam is longitudinally polarized. Analytic expressions for these correlations are obtained when $\tau^+\tau^-$ have a ``weak" dipole form factor (WDFF) coupling to $Z$ in the case of two-body decay modes of the $\tau$. Expressions of the variance of these correlations due to the standard model interactions are also presented. For $e^+e^-$ collisions at the $Z$ peak, a sensitivity of about $10^{-17} e$ cm for the real part of the $\tau$ WDFF and $3\times 10^{-17} e$ cm for the imaginary part of the $\tau$ WDFF can be reached using $\pi$ and $\rho$ channels (which are the most sensitive channels) in $\tau$ decay, with $10^6\, Z$'s likely to be available at the SLC at Stanford with $e^-$ polarization of 62\%-75\%.