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In this paper, we propose a $G_{L-R}\times U(1)_{L_\mu-L_\tau}$gauge-symmetric model where $G_{L-R}$ is the left-right gauge symmetry and$L_i$ is the $i-$flavor lepton number. We use the spontaneous breaking (SSB) of$U(1)_{L_\mu-L_\tau}$ to explain two discrepancies in the standard model: muonanomalous magnetic moment and light neutrinos and its oscillations. The massiveneutral gauge boson, $Z_{\mu\tau}$, arising from the SSB can provide additionalcontributions to the muon anomalous magnetic moment. In order to explainneutrino masses, we employ the low-energy inverse seesaw mechanism by addingthree $G_{L-R}$ singlet fermions, $S_{e,\mu,\tau}$. The light neutrino massmatrix from the inverse seesaw formula has a specific two-zero texture patternreferred in the literature as the Type-C two-zero texture due to the$U(1)_{L_\mu-L_\tau}$ symmetry. This allows us to predict the values of theCP-violating Dirac phase, Majorana phases, and the absolute value of lightneutrino masses in terms of the precisely measured mixing angles and masssquared differences. The model accommodates a quasi-degenerate spectrum ofneutrino masses with inverted ordering. The calculated best-fit value of$\delta_{CP}$ surprisingly matches with the current experimentally measuredbest-fit value of $\delta_{CP}$. At $1\sigma$, the measured value of$\delta_{CP}$ favors a $\theta_{23}>\pi/4$. At $1\sigma$, most of the parameterspace is within the cosmological bound on the sum of neutrino mass and thebound on the effective Majorana mass from neutrinoless beta decay.

Gauged $L_\mu$-$L_\tau$ Model with an Inverse Seesaw Mechanism for Neutrino Masses