  Zero modes of massive standard model fermions have been found on electroweak
Z-strings. A zero mode solution for a massless left-handed neutrino is also
known, but was thought to be non-normalizable. Here we show that although this
mode is not discretely normalizable, it is delta-function normalizable and the
correct interpretation of this solution is within the framework of the
continuum spectrum. We also analyze an extension of the standard model
including right-handed neutrinos in which neutrinos have Dirac masses, arising
from a Yukawa coupling to the usual SU(2) Higgs doublet, and right-handed
Majorana masses. The Majorana mass terms are taken to be spatially homogeneous
and are presumed to arise from the vacuum expectation value of some field
acquired in a phase transition well above the electroweak phase transition. The
resulting zero energy equations have a discrete zero mode.
