Radiative Neutrino Mass in Type III Seesaw Model

The simplest type III seesaw model as originally proposed introduces one lepton triplet. It thus contains four active neutrinos, two massive and two massless at tree level. We determine the radiative masses that the latter receive first at two loops. The masses are generally so tiny that they are definitely excluded by the oscillation data, if the heavy leptons are not very heavy, say, within the reach of LHC. To accommodate the data on masses, the seesaw scale must be as large as the scale of grand unification. This indicates that the most economical type III model would entail no new physics at low energies beyond the tiny neutrino masses.