Dynamically Solving the $mu/B_mu$ Problem in Gauge-mediated Supersymmetry Breaking

We provide a simple solution to the $mu/B_mu$ problem in the gauge-mediated Next-to-Minimal Supersymmetric Standard Model. In this model the messenger sector contains one pair of $3+bar 3$ and one pair of $2+bar 2$ messengers. These two messenger pairs couple to different gauge singlets in the hidden sector in which supersymmetry (SUSY) is broken. Such a gauge-mediation structure can naturally arise in many backgrounds. Because of the two effective SUSY breaking scales $frac{< F_i>}{< M_i>}$ in the messenger sector, the renormalization group evolutions of the soft SUSY breaking parameters can be properly modified, leading to a negative enough singlet soft mass square $m_N^2(Lambda_{EW})$ and hence reasonable $mu/B_mu$ values. In most of the perturbative (up to the GUT scale) parameter region, as a result, the electroweak scale is stabilized and phenomenologically interesting mass spectra of particles and superparticles are obtained. In addition, this model favors large values of $tanbeta$: $5 sim 50$ and a heavy scalar spectrum. With the relatively large $tanbeta$, the light $U(1)_R$ pseudoscalar (mainly appearing in the low-scale gauge-mediated SUSY breaking models) becomes extremely singlet-like, and is no longer a problem in this model. These features apply to all cases of low-, intermediate- and high-scale gauge-mediated SUSY breaking.