  At the interface between two massless Dirac models with opposite helicity a
paradoxical situation arises: A transversally impinging electron can seemingly
neither be transmitted nor reflected, due to the locking between spin and
momentum. Here we investigate this paradox in one spatial dimension where,
differently from higher dimensional realizations, electrons cannot leak along
the interface. We show that models involving only massless Dirac modes lead to
either no solutions or to trivial solutions to the paradox, depending on how
the helicity change across the interface is modeled. However, non trivial
scattering solutions to the paradox are shown to exist when additional massive
Dirac modes are taken into account. Although these modes carry no current for
energies within their gap, their interface coupling with the massless modes can
induce a finite and tunable transmission. Finally, we show that such
massless+massive Dirac model can be realized in suitably gated spin-orbit
coupled nanowires exposed to an external Zeeman field, where the transmission
coefficient can be controlled electrically.
