Surveying extended GMSB models with mh=125 GeV

In order to achieve maximal stop mixing and m(h) = 125 GeV, we consider extensions of minimal GMSB that include marginal MSSM-messenger superpotential interactions. Using a new approach to analytic continuation in superspace, we derive general formulas for the soft masses in the presence of such interactions, correctly taking into account the role of MSSM-messenger mixing in a general framework for the first time. We classify and catalog all possible such interactions consistent with perturbative SU(5) unification, and we survey the impact of turning on one interaction at a time, from the point of view of fine tuning, spectrum and phenomenology. We find that the best models are fine-tuned to the sub-percent level and are accessible at the 14 TeV LHC. We highlight potential search strategies that can probe the characteristic spectra of these models.