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 $\nu_1$, ${m_\nu}_1\lsim 10^{-8}$ 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 $T_R\lsim 10^8$ GeV to solve the cosmological gravitino problem. The required value of neutrino mass seems to be very unlikely the case since the recent Superkamiokande experiments suggest the masses of 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 ${m_\nu}_1\simeq 10^{-4}$ eV.