  The Standard Model differential cross section for $\bar{\nu}_e-e^-$ elastic
scattering vanishes exactly, at lowest order, for forward electrons and
incident $\bar{\nu}_e$ energy close to the rest energy of the electron. This
dynamical zero is not induced by a fundamental symmetry of the Lagrangian but
by a destructive interference between the left- and right-handed chiral
couplings of the electron in the charged and neutral current amplitudes. We
show that lowest order analyses based on this favorable kinematic configuration
are only mildly affected by the inclusion of the $O(\alpha)$ radiative
corrections in the $\bar{\nu}_e-e^-$ differential cross section, thus providing
an excellent opportunity for the search of ``new physics''. In the light of
these results, we discuss possible methods to improve the upper limits on the
neutrino magnetic moment by selecting recoil electrons contained in a forward
narrow cone. We conclude that, in spite of the obvious loss in statistics, one
may have a better signal for small angular cones.
