Next-to-minimal supersymmetric standard model with the gauge mediation of supersymmetry breaking

We study the next-to-minimal supersymmetric standard model (NMSSM) as the simplest candidate solution to the μ-problem in the context of the gauge mediation of supersymmetry breaking (GMSB). We first review various proposals to solve the μ-problem in models with the GMSB. We find none of them entirely satisfactory and point out that many of the scenarios still lack quantitative studies, and motivate the NMSSM as the simplest possible solution. We then study the situation in the minimal supersymmetric standard model (MSSM) with the GMSB and find that an order 10% cancellation is necessary between the μ-parameter and the soft SUSY-breaking parameters to correctly reproduce $M_Z.$ Unfortunately, the NMSSM does not give a phenomenologically viable solution to the μ-problem. We present quantitative arguments which apply both for the low-energy and high-energy GMSB and prove that the NMSSM does not work for either case. Possible modifications to the NMSSM are then discussed. The NMSSM with additional vector-like quarks works phenomenologically, but requires an order of a few percent cancellation among parameters. We point out that this cancellation has the same origin as the cancellation required in the MSSM.