Abstract
We study a simple framework for gauge-mediated supersymmetry breaking in local grand unified theory (GUT) models based on $F$-theory 7-branes and demonstrate that a mechanism for solving both the $\ensuremath{\mu}$ and $\ensuremath{\mu}/{B}_{\ensuremath{\mu}}$ problems emerges in a natural way. In particular, a straightforward coupling of the messengers to the Higgs sector leads to a geometry which not only provides us with an approximate $U(1{)}_{\mathrm{PQ}}$ symmetry that forbids the generation of $\ensuremath{\mu}$ at the GUT scale, it also forces the SUSY-breaking spurion field to carry a nontrivial Peccei-Quinn (PQ) charge. This connects the breaking of supersymmetry (SUSY) to the generation of $\ensuremath{\mu}$ so that the same scale enters both. Moreover, the messenger sector naturally realizes the $D3$-instanton triggered SUSY-breaking model of [J. Heckman, J. Marsano, N. Saulina, S. Schafer-Nameki, and C. Vafa, arXiv:0808.1286.] so this scale is exponentially suppressed relative to ${M}_{\mathrm{GUT}}$. The effective action at low scales is in fact precisely of the form of the ``sweet spot supersymmetry'' scenario studied by Ibe and Kitano in [M. Ibe and R. Kitano, J. High Energy Phys. 08 (2007) 016.].
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