Neutrino mass in a gauged $L_\mu - L_\tau$ model

We study the origin of neutrino mass through lepton-number violation and spontaneous $U(1)_{L_\mu-L_\tau}$ symmetry breaking. To accomplish the purpose, we include one Higgs triplet, two singlet scalars, and two vector-like doublet leptons in the $U(1)_{L_\mu-L_\tau}$ gauge extension of the standard model. To completely determine the free parameters, we employ the Frampton-Glashow-Marfatia (FGM) two-zero texture neutrino mass matrix as a theoretical input. It is found that when some particular Yukawa couplings vanish, an FGM pattern can be achieved in the model. Besides the explanation of neutrino data, we find that the absolute value of neutrino mass $m_j$ can be obtained in the model, and their sum can satisfy the upper bound of the cosmological measurement with $\sum_j |m_j| < 0.12$ eV. The effective Majorana neutrino mass for neutrinoless double-beta decay is below the current upper limit and is obtained as $\langle m_{\beta \beta} \rangle =(0.34,\, 2.3)\times 10^{-2}$ eV. In addition, the doubly charged Higgs $H^{\pm\pm}$ decaying to $\mu^\pm \tau^\pm$ final states can be induced from a dimension-6 operator and is not suppressed, and its branching ratio is compatible with the $H^{\pm \pm}\to W^\pm W^\pm$ decay when the vacuum expectation value of Higgs triplet is $O(0.01)$ GeV.