  Affleck-Dine (AD) baryogenesis along a d=6 flat direction in gauge-mediated
supersymmetry-breaking (GMSB) models can produce unstable Q-balls which
naturally have field strength similar to the messenger scale. In this case a
new kind of Q-ball is formed, intermediate between gravity-mediated and
gauge-mediated type. We study in detail these new Q-ball solutions, showing how
their properties interpolate between standard gravity-mediated and
gauge-mediated Q-balls as the AD field becomes larger than the messenger scale.
It is shown that E/Q for the Q-balls can be greater than the nucleon mass but
less than the MSSM-LSP mass, leading to Q-ball decay directly to Standard Model
fermions with no MSSM-LSP production. More significantly, if E/Q is greater
than the MSSM-LSP mass, decaying Q-balls can provide a natural source of
non-thermal MSSM-LSPs, which can subsequently decay to gravitino dark matter
without violating nucleosynthesis constraints. The model therefore provides a
minimal scenario for baryogenesis and gravitino dark matter in the
gauge-mediated MSSM, requiring no new fields.
