We study a model in which lepton number violation is solely triggered by a dimension 4 hard breaking term in the scalar potential. A minimal model, which contains a SU(2) triplet with hypercharge Y=2 and a pair of singlet doubly charged scalar fields in addition to the standard model (SM) Higgs doublet, is constructed. The model is technically natural in the sense that lepton number is preserved in the limit that the hard term vanishes. SM phenomenology restricts the vacuum expectation value of the triplet scalar field vT<5.78 GeV. Neutrino masses controlled by vT are generated at the two loop level and are naturally in the sub-eV range. In general they exhibit normal hierarchy structure. Here the neutrino mass term does not dominate neutrinoless double beta decays of nuclei. Instead the short-distance physics with doubly charged Higgs exchange gives the leading contribution. We expect weak scale singly and doubly charged Higgs bosons to make their appearances at the LHC and the ILC.