Solving the μ Problem in Gauge Mediated Supersymmetry Breaking Models with Flavor Symmetry

We suggest a solution to the μ problem of gauge mediated supersymmtery breaking models based on flavor symmetries. In this scenario the μ term arises through the vacuum expectation value of a singlet scalar field which is suppressed by a flavor symmetry factor relative to the scale of dynamical SUSY breaking. The same flavor symmetry also ensures that the soft SUSY breaking parameter Bμ is not much larger than μ, a necessary condition for the stability of electroweak symmetry breaking. Explicit examples where Bμ ∼ μ and Bμ ≪ μ are presented. The latter case provides a natural solution to the supersymmetric CP problem. We show that the same flavor symmetry that suppresses the μ and the Bμ parameters can also play a role in explaining the fermion mass and mixing hierarchy puzzle. email: [email protected] email: [email protected] §1 There is a huge hierarchy between the electroweak scale (∼ 10 GeV) and the Planck scale (∼ 10 GeV). Supersymmetric theories are excellent candidates which can stabilize this hierarchy against quantum corrections. Furthermore, these theories can also explain the origin of such a huge hierarchy in scenarios where the electroweak symmetry is broken radiatively [1]. However, supersymmetry (SUSY) alone is not sufficient to explain electroweak symmetry breaking. For it to be successful, the supersymmetry breaking mass parameters should lie around the electroweak breaking scale. This is needed for a natural resolution of the hierarchy problem. In the minimal supersymmetric standard model (MSSM) with radiative electroweak symmetry breaking, the stability of the Higgs potential requires the SUSY breaking masses as well as the supersymmtric Higgs mass parameter μ to be around the electroweak scale. Here the superpotential term W ∋ μHuHd, where Hu and Hd are the two Higgs doublets of MSSM, defines the μ parameter. The soft SUSY breaking Bμ parameter, defined through V ∋ BμHuHd, should also lie around the electroweak scale. These requirements are evident from the two minimization conditions of the MSSM Higgs potential, which read at tree–level as: M Z = −μ + m2Hd −mHu tan β tan β − 1 , (1) sin 2β = 2Bμ 2μ2 +mHu +m 2 Hd . (2) Here we have used standard notation with tanβ = vu/vd, vu and vd being the vacuum expectation values (VEVs) of the Higgs fields Hu and Hd. Eq. (1) clearly shows that μ 2 cannot be much larger than M Z without fine–tuning, Eq. (2) shows that Bμ is bounded by μ + (m2Hu +m 2 Hd )/2. Since the μ parameter is part of the supersymmetric Lagrangian, a question arises as to why it is not much above the electroweak scale, say near the Planck scale. Such a large value of μ is of course inconsistent with symmetry breaking requirements. This is the so–called μ problem. There is a good solution [2] for this problem in scenarios where supersymmetry breaking is communicated to the standard model sector through gravity [3]. In supergravity models, the supersymmetry breaking masses are given by mSUSY ∼ FX/MP , where FX , the order parameter for SUSY breaking, is the F component