Minimal anomaly-free electroweak model for several generations

A horizontal U′(1) gauge symmetry is invoked to distinguish several fermion generations which have the one and the same Weinberg-Salam (WS) structure. It is assumed that both weak hypercharges associated with the WS U(1) and the new U′(1) factorize, i. e., they are of the form ${\lambda }_{i}y$ and ${{\lambda }^{\prime }}_i{y}^{\prime }$, where the scales ${\lambda }_i$,${{\lambda }^{\prime }}_i$ are the so-called seriality numbers of the $i\mathrm{th}$ generation. The motivation for such an assumption comes from the standard WS model, where the anomaly constraints determine the $Y$ quantum numbers for a given generation up to a common scale. The generation structure is extended to the Higgs system as well, i.e., associated with the $i\mathrm{th}$ generation, there is an $i\mathrm{th}$ Higgs doublet whose U′(1) hypercharge is ${{\lambda }^{\prime }}_{i}h$ where $h$ is a number. We study as a prototype the simplest but the fundamental two-generation scheme in detail and show how when the above assumptions are combined with (a) anomaly-free conditions, (b) conservation of lepton numbers, and (c) the emergence of a proper Cabibbo structure, we are led to a unique choice for $Y^{\prime }$ which can be expressed as $Y^{\prime }=\mp [\frac{Y}{3}+\frac{2(B-L)\mathrm{}}{3}]$ with the (∓) signs being the seriality numbers for the two generations, respectively. A hierarchy in the vacuum expectation values (VEV's) of the Higgs doublets produces the desired hierarchy in the fermion masses, and correlates the smallness of $m_e$ with the smallness of $m_u$, $m_d$, and ${\theta }_C$. The gauge boson associated with U′(1) produces of course the undesirable flavor-changing neutral currents and the violation of $e-\mu$ universality. However, an additional Higgs singlet with zero $Y$ and with a large VEV suffices to make the mass of $Z^{\prime }$ heavy enough to suppress the unwanted features to any desirable degree. From the $K_L-K_S$ mass difference we estimate $M\left(Z^{\prime }\right)>\left(\frac{1}{\alpha }\right)M\left(Z\right)\mathrm{}$. We investigate briefly a three-generation scheme within the framework of the present model and argue that it fits more naturally in an ${\left[\mathrm{SU}\mathrm{}\right(2)\mathrm{}\times \mathrm{U}\mathrm{}(1\left)\right]}_{\mathrm{WS}}\times {\mathrm{U}}^{\prime }{\mathrm{}\left(1\right)\mathrm{}}^N$ model for $2^N$ generations. Thus, from our point of view, the existence of a third generation suggests the existence of a fourth one as well.