  The scale of fermion mass generation can, as shown by Appelquist and
Chanowitz, be bounded from above by relating it to the scale of unitarity
violation in the helicity nonconserving amplitude for fermion-anti-fermion
pairs to scatter into pairs of longitudinally polarized electroweak gauge
bosons. In this paper, we examine the process t tbar -> W_L W_L in a family of
phenomenologically-viable deconstructed Higgsless models and we show that scale
of unitarity violation depends on the mass of the additional vector-like
fermion states that occur in these theories (the states that are the
deconstructed analogs of Kaluza-Klein partners of the ordinary fermions in a
five-dimensional theory). For sufficiently light vector fermions, and for a
deconstructed theory with sufficiently many lattice sites (that is,
sufficiently close to the continuum limit), the Appelquist-Chanowitz bound can
be substantially weakened. More precisely, we find that, as one varies the mass
of the vector-like fermion for fixed top-quark and gauge-boson masses, the
bound on the scale of top-quark mass generation interpolates smoothly between
the Appelquist-Chanowitz bound and one that can, potentially, be much higher.
In these theories, therefore, the bound on the scale of fermion mass generation
is independent of the bound on the scale of gauge-boson mass generation. While
our analysis focuses on deconstructed Higgsless models, any theory in which
top-quark mass generation proceeds via the mixing of chiral and vector fermions
will give similar results.
