Minimal gauged ${\mathrm{U}(1)}_{L_{\alpha }-L_{\beta }}$ 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 $\mathrm{U}(1{)}_{L_{\alpha }-L_{\beta }}$ 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 $\mathrm{U}(1{)}_{L_{\alpha }-L_{\beta }}$ symmetry breaking Higgs field—an ${\mathrm{SU}(2)}_L$ 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 nontrivial 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 $\mathrm{U}(1{)}_{L_{\alpha }-L_{\beta }}$ 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 ${\mathrm{U}(1)}_{L_{\mu }-L_{\tau }}$ model with an ${\mathrm{SU}(2)}_L$ singlet ${\mathrm{U}(1)}_{L_{\mu }-L_{\tau }}$-breaking field barely evades the limit, which can be tested in the future neutrino experiments.

Figure 1

The sum of the neutrino masses as a function of ${\theta }_{23}$ predicted in the singlet model with $M_{\ell }=\mathrm{diag}(m_e,m_{\mu },m_{\tau })$ or $\mathrm{diag}(m_e,m_{\tau },m_{\mu })$ for NO. The vertical gray dashed line represents the best fit value of ${\theta }_{23}$, while the vertical gray dotted lines (the plot range) indicate the $1\sigma$ ($3\sigma$) range. The dark (light) red band represents the uncertainty coming from the $1\sigma$ ($3\sigma$) range of ${\theta }_{13}$. We also show in the horizontal gray dashed line the limit imposed by the Planck experiment: $\sum _i{m}_i<0.12\mathrm{eV}$ ($\mathrm{Planck}\mathrm{TT}+\mathrm{lowP}+\mathrm{lensing}+\mathrm{ext}$) [77].

Figure 2

The sum of the neutrino masses as a function of ${\theta }_{23}$ predicted in the doublet models. The dark (light) red bands represent the $1\sigma$ ($3\sigma$) uncertainty coming from the $1\sigma$ ($3\sigma$) range of $\mathrm{\Delta }{m}_{32}^2$. The vertical and horizontal lines are the same as in Fig. 1.

Figure 3

The predictions for (a) the Dirac $CP$ phase $\delta$ and (b) the effective Majorana neutrino mass $⟨m_{\beta \beta }⟩$ in the singlet case. The red lines show the predictions as functions of ${\theta }_{23}$, and the dark (light) red bands show the uncertainty coming from the $1\sigma$ ($3\sigma$) errors in the other parameters. The vertical gray dashed lines represent the best fit value of ${\theta }_{23}$, while the vertical gray dotted lines (the plot range) indicate the $1\sigma$ ($3\sigma$) range. In (a), we also show the $1\sigma$ ($3\sigma$) favored region of $\delta$ in the dark (light) horizontal green bands. In (b), the light blue band represents the limit from KamLAND-Zen, $⟨m_{\beta \beta }⟩<0.061–0.165\mathrm{eV}$ [99], where the band indicates uncertainty from the nuclear matrix element.