Cosmological Constraint on the Zee model

It is well known that the Zee model induces small neutrino masses by radiative corrections, where the bi-maximal flavor mixing is possible. We analyze the cosmological condition in order for the baryon asymmetry generated in the early universe not to be washed out in the Zee model. Since the lepton number is violated explicitly in the Zee model, the baryon asymmetry might be washed out through the sphaleron processes together with the lepton-number violating interactions. In this letter, we will show that the baryon asymmetry is {\it not} washed out, although it has been said that the Zee model cannot preserve the baryon asymmetry generated in the early universe. This can be seen by considering an approximately conserved number, $L’ \equiv L_e-L_\mu-L_\tau$.