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We propose a simple model for Dirac neutrinos where the smallness of neutrino mass follows from a parameter κ whose absence enhances the symmetry of the theory. Symmetry breaking is performed in a two-doublet Higgs sector supplemented by a gauge singlet scalar, realizing an accidental global U(1) symmetry. Its spontaneous breaking at the few TeV scale leads to a physical Nambu–Goldstone boson – the Diracon, denoted D – which is restricted by astrophysics and induces invisible Higgs decays such as h→DD. The scheme provides a rich, yet very simple scenario for symmetry breaking studies at colliders such as the LHC.