Unitarity and Fermion Mass Generatation

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.