Q-ball formation: Obstacle to Affleck-Dine baryogenesis in the gauge-mediated SUSY breaking ?

We consider the Affleck-Dine baryogenesis comprehensively in the minimal supersymmetric standard model with the gauge-mediated supersymmetry breaking. Considering the high temperature effects, the Affleck-Dine field is naturally deformed into the form of the Q ball. In the natural scenario when the initial amplitude of the field and the A-terms are both determined by the nonrenormalizable superpotential, we obtain only a few parameter sets for explaining the baryon number of the universe for the situations that the Q-ball formation occurs just after the baryon number production. Moreover, most of the parameter sets suited for the natural scenario have already been excluded by currents experiments. We also find the new situations in which the Q-ball formation takes place rather late compared with the baryon number creation. This situation is more preferable, since it allows wider parameter region for naturally consistent scenarios, although it is still difficult to realize in the actual cosmological scenario