Chiral Hierarchies and Flavor-Changing Neutral Currents in Hypercolor

Dynamical chiral-symmetry breaking in asymptotically free theories with a slowly running coupling is analyzed. When the confinement scale $\ensuremath{\Lambda}$ is much less than the cutoff $M$ beyond which the theory cannot be used in isolation, the dynamical mass $\ensuremath{\Sigma}(p)$ starts from a value $\ensuremath{\approx}\ensuremath{\Lambda}$ for momenta $p\ensuremath{\lesssim}\ensuremath{\Lambda}$ and falls slowly for a significant range $p>\ensuremath{\Lambda}$. It then takes on the asymptotic form $\frac{{(\mathrm{ln}p)}^{a}}{{p}^{2}}$ where $a>1$. In hypercolor theories this behavior generates sufficiently large fermion masses for a higher value of $M$ than naively expected. This in turn could allow for an adequate suppression of flavor-changing neutral currents.