We numerically study the formation of the gauge-mediation type $Q$ balls in the logarithmic square potential on three-dimensional lattices. We obtain the broad charge distribution of the $Q$ ball of this type for the first time, to the best of our knowledge. The charge of the $Q$ ball at the peak of the distribution is smaller than what we estimated as the average of the largest tens of the $Q$ balls in the logarithmic potential for the same initial amplitude of the field at the onset of its oscillation. We also discuss some impacts of the broad distribution on cosmology and astrophysics. In the $B$ ball ($Q$ being the baryon number) case, the broad distribution would lead to the coexistence of both stable and unstable $B$ balls. We find that stable $B$ balls can account for the dark matter of the Universe without affecting successful big bang nucleosynthesis by the decay of the unstable $B$ balls, but the baryon number of the Universe cannot be explained by them. On the other hand, the large $L$ balls ($Q$ being the lepton number) would be the dark matter as well while avoiding the constraints on the x- and/or $\gamma$-rays from the decay of the smaller $L$ balls.

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• Classical solutions in field theory
• Cosmology
• Dark matter
• Supersymmetric models