Compromise between neutrino masses and collider signatures in the type-II seesaw model

A natural extension of the standard $\mathrm{SU}(2{)}_{\mathrm{L}}\times \mathrm{U}(1{)}_{\mathrm{Y}}$ gauge model to accommodate massive neutrinos is to introduce one Higgs triplet and three right-handed Majorana neutrinos, leading to a $6\times 6$ neutrino mass matrix which contains three $3\times 3$ submatrices, $M_{\mathrm{L}}$, $M_{\mathrm{D}}$ and $M_{\mathrm{R}}$. We show that three light Majorana neutrinos (i.e., the mass eigenstates of ${\nu }_e$, ${\nu }_{\mu }$, and ${\nu }_{\tau }$) are exactly massless in this model, if and only if $M_{\mathrm{L}}=M_{\mathrm{D}}{M}_{\mathrm{R}}^{-1}{M}_{\mathrm{D}}^T$ exactly holds. This no-go theorem implies that small but nonvanishing neutrino masses may result from a significant but incomplete cancellation between $M_{\mathrm{L}}$ and $M_{\mathrm{D}}{M}_{\mathrm{R}}^{-1}{M}_{\mathrm{D}}^T$ terms in the Type-II seesaw formula, provided three right-handed Majorana neutrinos are of $\mathcal{O}(1)\mathrm{TeV}$ and experimentally detectable at the LHC. We propose three simple Type-II seesaw scenarios with the $A_4\times \mathrm{U}(1{)}_{\mathrm{X}}$ flavor symmetry and its explicit breaking to interpret the observed neutrino mass spectrum and neutrino mixing pattern. Such a TeV-scale neutrino model can be tested in two complementary ways: (1) searching for possible collider signatures of lepton number violation induced by the right-handed Majorana neutrinos and doubly-charged Higgs particles; and (2) searching for possible consequences of unitarity violation of the $3\times 3$ neutrino mixing matrix in the future long-baseline neutrino oscillation experiments.