A 125 GeV Fat Higgs at large $\tan \beta$

We study the viability of regions of large $\tan \beta$ within the framework of Fat Higgs/$\lambda$-SUSY Models. We compute the one-loop effective potential to find the corrections to the Higgs boson mass due to the heavy non-standard Higgs bosons. As the tree level contribution to the Higgs boson mass is suppressed at large $\tan \beta$, these one-loop corrections are crucial to raising the Higgs boson mass to the measured LHC value. By raising the Higgsino and singlino mass parameters, typical electroweak precision constraints can also be avoided. We illustrate these new regions of Fat Higgs/$\lambda$-SUSY parameter space by finding regions of large $\tan \beta$ that are consistent with all experimental constraints including direct dark matter detection experiments, relic density limits and the invisible decay width of the $Z$ boson. We find that there exist regions around $\lambda = 1.25, \tan \beta = 50$ and a uniform psuedo-scalar $4~{\rm TeV} \lsim M_A \lsim 8$~TeV which are consistent will all present phenomenological constraints. In this region the dark matter relic abundance and direct detection limits are satisfied by a lightest neutralino that is mostly bino or singlino. As an interesting aside we also find a region of low $\tan \beta$ and small singlino mass parameter where a well-tempered neutralino avoids all cosmological and direct detection constraints.