The symmetry of $4\times 4$ mass matrices predicted by the spin-charge-family theory --- $SU(2)\times SU(2)\times U(1)$ --- remains in all loop corrections

Abstract

The spin-charge-family theory predicts the existence of the fourth family to the observed three. The $4\times 4$ mass matrices --- determined by the nonzero vacuum expectation values of the two triplet scalars, the gauge fields of the two groups of $\widetilde{SU}(2)$ determining family quantum numbers, and by the contributions of the dynamical fields of the two scalar triplets and the three scalar singlets with the family members quantum numbers (${\tau }^{\alpha }=(Q,Q^{\prime },Y^{\prime })$) --- manifest the symmetry $\widetilde{SU}(2)\times \widetilde{SU}(2)\times U(1)$. All scalars carry the weak and the hyper charge of the standard model higgs field ($\pm \frac{1}{2},\mp \frac{1}{2}$, respectively). It is demonstrated, using the massless spinor basis, that the symmetry of the $4\times 4$ mass matrices remains $SU(2)\times SU(2)\times U(1)$ in all loop corrections, and it is discussed under which conditions this symmetry is kept under all corrections, that is with the corrections induced by the repetition of the nonzero vacuum expectation values included.

Comments: 55 pages, 4 figures, appeared in the Proceedings to the $21^{st}$ Workshop "What Comes Beyond the Standard Models", Bled, 23 June - 1 of July, 2018, DMFA Ljubljana, December 2018