  In a brane (domain wall) scenario with an infinite extra dimension and
localized gravity, bulk fermions and scalars often have bound states with zero
4-dimensional mass. In this way massless matter residing on the brane may be
obtained. We consider what happens when one tries to introduce small, but
non-vanishing mass to these matter fields. We find that the discrete zero modes
turn into quasi-localized states with finite 4-dimensional mass and finite
width. The latter is due to tunneling of massive matter into extra dimension.
We argue that this phenomenon is generic to fields that can have bulk modes. We
also point out that in theories meant to describe massive scalars, the
4-dimensional scalar potential has, in fact, power-law behavior at large
distances.
