  Elastic scattering of transversely polarized electron antineutrino beam off
unpolarized electrons can be used to detect the CP-violating effects by
measuring the azimuthal asymmetry of recoil electrons caused by the
interference terms between the standard vector $c_{V}^{L}$, axial $c_{A}^{L}$
couplings of left-chirality antineutrinos and exotic scalar $c_{S}^{R}$
coupling of right-chirality ones in the differential cross section. It would be
a positive evidence for the existence of the exotic antineutrino states.
Moreover, we also show that the differential cross section for the
$\overline{\nu} e^- $ scattering can be obtained from the one for the $\nu e$
scattering, if one makes the substitution $c_{T}^{R} \to -c_{T}^{R}$,
$c_{A}^{L} \to -c_{A}^{L}$, ${\bf q}\to -{\bf q}$, ${\boldmath
$\hat{\eta}_{\nu}$}\to -{\boldmath $\hat{\eta}_{\nu}$}$.
