The μ problem and the axion in gauge mediation

We revisit the idea of generating the Higgs μ 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 μ∼m _soft and the Higgs soft parameter BO(m_soft) with the CP phase of B aligned to the phase of gaugino masses, while giving the PQ scale v_PQ∼√m_softΛ, where m_soft denotes the gauge-mediated gaugino or sfermion masses and Λ 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_3/2=O(100)keV.