\begin{document}$ U(1)_{L_\alpha- L_\beta} $\end{document} symmetry. A neutrino mass matrix is formulated for these models via a type-I seesaw mechanism by introducing right-handed neutrinos. The neutrino mass matrix has more degrees of freedom than minimal scenarios that have only one new scalar field, but its structure is still restricted by symmetry. Subsequently, we find that the sum of neutrino mass can be lower than that of minimal scenarios and easily satisfies observed constraints. In addition, we can fit neutrino data for \begin{document}$ U(1)_{L_e - L_{\mu(\tau)}} $\end{document} cases that are disfavored in minimal models. Furthermore, some correlations among sum of neutrino mass and CP violating phases are still observed, although we have more free parameters."> Neutrino observables in gauged <inline-formula><tex-math id="Z-20250318162927">\begin{document}${\boldsymbol U({\bf 1})_{\boldsymbol L_{\boldsymbol\alpha}- {\boldsymbol L}_{\boldsymbol\beta}} }$\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/f73595a6-f963-457c-badd-3e9ca097caec/CPC-2024-0362_Z-20250318162927.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="//www.macurncorp.com/hepnp/article/app/id/f73595a6-f963-457c-badd-3e9ca097caec/CPC-2024-0362_Z-20250318162927.png"/></alternatives></inline-formula> models with two Higgs doublet and one singlet scalars -
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