Horizontal-quantum-flavor-dynamics approach to the fermion mass spectrum

Abstract

We show that the important features associated with the fermion mass spectrum can be explained within the framework of an ${[\mathrm{S}\mathrm{U}(2)\phantom{\rule{0ex}{0ex}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]}_{\mathrm{WS}}$ \ifmmode\times\else\texttimes\fi{} U'(1) gauge theory subject to a generalized generation structure, without assuming any hierarchy among the various Yukawa coupling constants. At the two-generation level, we show that (i) the only allowed nondiagonal mass matrix is of the canonical type which characterizes certain discrete-symmetry models, and (ii) $(\frac{{m}_{u}}{{m}_{c}})\ensuremath{\simeq}{(\frac{{m}_{d}}{{m}_{s}})}^{2}$. However, we argue that the approximate equality (within a factor of 1.5-2) in masses of the two lightest quarks cannot be explained without invoking at least one additional generation of particles. Moreover, the naturalness of ${m}_{u}\ensuremath{\simeq}{m}_{d}$ completely determines the structure of the two charged-quark mass matrices in the six-flavor scheme. It follows then that ${m}_{c}\ensuremath{\gg}{m}_{s}\ensuremath{\gtrsim}{({m}_{c}{m}_{u,d})}^{\frac{1}{2}}$, and the mass scale of the third-generation quarks is fixed by the masses of the two lighter generations to be of order $\frac{{{m}_{c}}^{2}}{{m}_{s}}$. A new relation for the Cabibbo angle, namely ${\ensuremath{\theta}}_{C}\ensuremath{\simeq}{(\frac{{m}_{d}}{{m}_{s}^{\mathrm{eff}}})}^{\frac{1}{2}}$ with ${m}_{s}^{\mathrm{eff}}=\frac{{{m}_{c}}^{3}}{{{m}_{b}}^{2}}$, emerges. We also calculate the main $\mathrm{CP}$-violating parameter associated with the ${K}^{0}\ensuremath{-}{\overline{K}}^{0}$ system to be ${\ensuremath{\epsilon}}_{K}\ensuremath{\simeq}{(\frac{{m}_{s}}{{m}_{c}})}^{3}\ensuremath{\sim}{10}^{\ensuremath{-}3}$ as observed. Finally we discuss the effects of the horizontal gauge symmetry on the leptonic sector and establish the possibility of having a full quarklepton correspondence without conflicting with any known experimental considerations.