  In this paper a scenario admitting the participation of the exotic scalar
coupling of the right-handed neutrinos in addition to the standard vector and
axial couplings of the left-handed neutrinos in the weak interactions is
considered. The research is based on the low-energy $(\nu_{\mu} e^{-})$ and
$(\nu_{e} e^{-})$ scattering processes. The main goal is to show how the
presence of the right-handed neutrinos in the above processes changes the
laboratory differential cross section in relation to the Standard Model
prediction. Both processes are studied at the level of the four-fermion point
interaction. Neutrinos are assumed to be polarized Dirac fermions and to be
massive. In the laboratory differential cross section, the new interference
term between the standard vector coupling of the left-handed neutrinos and
exotic scalar coupling of the right-handed neutrinos appears which does not
vanish in the limit of massless neutrino. This additional contribution,
including information about the transverse components of neutrino polarization,
generates the azimuthal asymmetry in the angular distribution of the recoil
electrons. This regularity would be a signature of the participation of the
right-handed neutrinos in the neutrino-electron scattering. The future
low-energy high-precision neutrino-electron scattering experiments using the
strong and polarized artificial neutrino source would allow to search for the
exotic effects coming from the R-handed neutrinos.
