The $\mu$-problem and axion in gauge mediation

We revisit the idea of generating the Higgs {mu} parameter through a spontaneously broken Peccei-Quinn (PQ) symmetry in a gauge-mediated supersymmetry breaking scenario. For the messenger scale of gauge mediation higher than the PQ scale, the setup naturally generates {mu}{approx}m{sub soft} and the Higgs soft parameter B < or approx. O(m{sub soft}) with the CP phase of B aligned to the phase of gaugino masses, while giving the PQ scale v{sub PQ{approx}{radical}}(m{sub soft{Lambda}}), where m{sub soft} denotes the gauge-mediated gaugino or sfermion masses and {Lambda} is the cutoff scale which can be identified as the Planck scale or the grand unified theory scale. The PQ sector of the model results in distinctive cosmology including a late thermal inflation. We discuss the issue of dark matter and baryogenesis in the resulting thermal inflation scenario, and find that a right amount of gravitino dark matter can be produced together with a successful Affleck-Dine leptogenesis, when the gravitino mass m{sub 3/2}=O(100) keV.