Affleck-Dine leptogenesis with an ultralight neutrino

We perform a detailed analysis on Affleck-Dine leptogenesis taking into account the thermal effects on the dynamics of the flat direction field phi. We find that an extremely small mass for the lightest neutrino nu (1), m(nu1)less than or similar to 10(-8) eV is required to produce enough lepton-number asymmetry to explain the baryon asymmetry in the present Universe. We impose here the reheating temperature after inflation T-R to be T-R less than or similar to 10(8) GeV to solve the cosmological gravitino problem. The required value of the neutrino mass seems to be very unlike the case since the recent SuperKamiokande experiments suggest the masses of the heavier two neutrinos nu (2) and nu (3) to be in a range of 10(-1)-10(-3) eV. We also propose a model to avoid this difficulty based on the Peccei-Quinn symmetry, where the required neutrino mass can be as large as m(nu1)similar or equal to 10(-4) eV.