  Some years ago Appelquist and Chanowitz considered the scattering of
fermion--anti-fermion into a pair of longitudinal gauge bosons. Their
calculation established that unitarity implies that the physics giving mass to
a quark of mass $m_f$ must be below a mass scale of $16 \pi v^2 / m_f$ ($v=246$
GeV). This bound is a bit difficult to interpret, because the unitarity of
gauge boson scattering requires in any case that there be new physics, such as
a Higgs boson, with a mass lighter than this. This paper re-examines the
Appelquist-Chanowitz bound in order to clarify its meaning. This work uses toy
models with a singlet Higgs boson to unitarize gauge boson scattering, and
considers other possibilities for the new physics affecting the fermion mass.
This new physics has the effect of changing the Higgs
boson--fermion--anti-fermion coupling. New physics cannot significantly alter
this coupling unless it is substantially lighter than the Appelquist-Chanowitz
bound.
