Neutrino Mass Matrix in a gauge group $SU(2)_L \times U(1)_e \times U(1)_μ\times U(1)_τ$

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 $h^{(2)}_{1}$, $h^{(3)}_{2}$, $h^{(1)}_{3}$ 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: ($\nu_e \rightarrow \nu_\tau$); Case 2: ($\nu_e \rightarrow \nu_\mu$); Case 3: ($\nu_\mu \rightarrow \nu_\tau$). 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.