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 gauge-mediated supersymmetry breaking scenario. For the messenger scale of gauge mediation higher than the PQ scale, the setup naturally generates mu ~ m_soft and the Higgs soft parameter B \leq O(m_soft) with the CP phase of B aligned to the phase of gaugino masses, while giving the PQ scale v_PQ ~ \surd (m_soft {\Lambda}), where m_soft denote the gauge-mediated gaugino or sfermion masses and {\Lambda} is the cutoff scale which can be identified as the Planck scale or the GUT 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_{3/2} = O(100) keV.