Neutrino Mass Matrix in a gauge group $SU(2{)}_L\times U(1{)}_e\times U(1{)}_{\mu }\times U(1{)}_{\tau }$

Abstract

The electroweak unification group $G\equiv SU(2{)}_L\times U(1{)}_e\times U(1{)}_{\mu }\times U(1{)}_{\tau }$ in which each fermion multiplet has its own $U(1)$ factor was proposed in 1986 to get the neutrino mass matrix. In this paper, the gauge group G is restricted to lepton section only, leaving quark multiplets as in the standard model. In addition to lepton multiplets $L_e$, $L_{\mu }$ and $L_{\tau }$, there are three $SU(2)$singlet right handed neutrinos $N_R^{(i)}$'s. WIth the breaking of G to $SU(2{)}_L\times U(1)$, the right handed neutrinos acquire heavy Majorana masses. Three heavy right handed neutrinos $N_R^{(i)}$'s are available to generate a $3\times 3$ non-diagonal neutrino mass matrix in terms of three Yukawa couplings $h_1^{(2)}$, $h_2^{(3)}$, $h_3^{(1)}$ of the Higgs scalar doublet to $L_e$, $L_{\mu }$, $L_{\tau }$ with $N_R^{(1)}$, $N_R^{(2)}$ and $N_R^{(3)}$ respectively. Three Yukawa couplings can be arranged and expressed in terms of masses $m_e$, $m_{\mu }$, $m_{\tau }$ in three different ways to obtain the results of interest for Case 1: (${\nu }_e\to {\nu }_{\tau }$); Case 2: (${\nu }_e\to {\nu }_{\mu }$); Case 3: (${\nu }_{\mu }\to {\nu }_{\tau }$). The results obtained for the three cases are compared with the experimental data from neutrino oscillations. Cases 1 and 2 are relevant for solar neutrino oscillations whereas Case 3 is relevant for atmospheric neutrino oscillations.