Understanding of spin ½, family structure, and mass pattern of leptons and quarks in a composite model

Abstract

A relativistic three-preon interaction with a spin dependence ${\ensuremath{\Sigma}}^{a}\ensuremath{\equiv}{\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\nu}}^{(1)}{\ensuremath{\sigma}}_{\ensuremath{\nu}\ensuremath{\lambda}}^{(2)}{\ensuremath{\sigma}}_{\ensuremath{\lambda}\ensuremath{\mu}}^{(3)}$ is proposed for an SU(2) preon ($t, v$) model. The ${\ensuremath{\Sigma}}^{a}$ operator (i) gives zero force for $S=\frac{3}{2}$ states, thus rendering them dynamically irrelevant, and (ii) has the complex combinations ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}\ifmmode\pm\else\textpm\fi{}i{\ensuremath{\chi}}^{\ensuremath{'}}$ of the two spin-\textonehalf{} functions (${\ensuremath{\chi}}^{\ensuremath{'}}, {\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$) as eigenstates with eigenvalues of opposite signs so that only one combination is relevant for confinement. This feature in turn leads to just three ${\mathrm{S}}_{3}$-symmetry classes (out of six available ones) of spatial wave function, again in complex form which are sought to be identified with the generation structures. Their spatial functional forms are such as to obey severe selection rules preventing electromagnetic transitions such as $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$. Apart from these qualitative features which are shown to remain valid within a fairly general (Bethe-Salpeter-type) dynamical framework, a quantitative model of confinement with a very steep potential ($\ensuremath{\sim}{R}^{12}$) is proposed for a unified description of lepton and quark spectra. This leads to a universal ($l, q$) mass formula for all three generations with only two free parameters, in rather good agreement with the observed pattern. In particular the lepton mass ratios which do not involve any free parameter are predicted as $\frac{{m}_{\ensuremath{\mu}}}{{m}_{e}}=196.7$ and $\frac{{m}_{\ensuremath{\tau}}}{{m}_{\ensuremath{\mu}}}=14.3$.