Minimal Gauged U(1)$_{L_\alpha - L_\beta}$ Models Driven into a Corner

It is well known that the differences between the lepton numbers can be gauged with the Standard Model matter content. Such extended gauge theories, dubbed as the gauged U(1)$_{L_\alpha - L_\beta}$ models, have been widely discussed so far as potential candidates for physics beyond the Standard Model. In this work, we study the minimal versions of these gauge theories, where three right-handed neutrinos as well as a single U(1)$_{L_\alpha - L_\beta}$ symmetry breaking Higgs field---an SU(2)$_L$ singlet or doublet---are introduced. In these minimal models, the neutrino mass terms are constrained by the gauge symmetry, which result in the two-zero texture or two-zero minor structure of neutrino mass matrices. Such restrictive forms of neutrino mass matrices lead to non-trivial predictions for the neutrino oscillation parameters as well as the size of the mass eigenvalues. We find that due to this restriction the minimal gauged U(1)$_{L_\alpha - L_\beta}$ models are either incompatible with the observed values of the neutrino parameters or in strong tension with the Planck 2018 limit on the sum of the neutrino masses. Only the U(1)$_{L_\mu-L_\tau}$ model with an SU(2)$_L$ singlet U(1)$_{L_\mu-L_\tau}$-breaking field barely evades the limit, which can be tested in the future neutrino experiments.