F-theory, GUTs, and the Weak Scale
  In this paper we study a deformation of gauge mediated supersymmetry breaking in a class of local F-theory GUT models where the scale of supersymmetry breaking determines the value of the mu term. Geometrically correlating these two scales constrains the soft SUSY breaking parameters of the MSSM. In this scenario, the hidden SUSY breaking sector involves an anomalous U(1) Peccei-Quinn symmetry which forbids bare mu and B mu terms. This sector typically breaks supersymmetry at the desired range of energy scales through a simple stringy hybrid of a Fayet and Polonyi model. A variant of the Giudice-Masiero mechanism generates the value mu ~ 10^2 - 10^3 GeV when the hidden sector scale of supersymmetry breaking is F^(1/2) ~ 10^(8.5) GeV. Further, the B mu problem is solved due to the mild hierarchy between the GUT scale and Planck scale. These models relate SUSY breaking with the QCD axion, and solve the strong CP problem through an axion with decay constant f_a ~ M_(GUT) * mu / L, where L ~ 10^5 GeV is the characteristic scale of gaugino mass unification in gauge mediated models, and the ratio \mu / L ~ M_(GUT)/M_(pl) ~ 10^(-3). We find f_a ~ 10^12 GeV, which is near the high end of the phenomenologically viable window. Here, the axino is the goldstino mode which is eaten by the gravitino. The gravitino is the LSP with a mass of about 10^1 - 10^2 MeV, and a bino-like neutralino is (typically) the NLSP with mass of about 10^2 - 10^3 GeV. Compatibility with electroweak symmetry breaking also determines the value of tan(beta) ~ 30 +/- 7. 
Jonathan J. Heckman and Cumrun Vafa