Solving the mu problem with a heavy Higgs boson

We discuss the generation of the mu-term in a class of supersymmetric models characterized by a low energy effective superpotential containing a term lambda S H_1 H_2 with a large coupling lambda~2. These models generically predict a lightest Higgs boson well above the LEP limit of 114 GeV and have been shown to be compatible with the unification of gauge couplings. Here we discuss a specific example where the superpotential has no dimensionful parameters and we point out the relation between the generated mu-term and the mass of the lightest Higgs boson. We discuss the fine-tuning of the model and we find that the generation of a phenomenologically viable mu-term fits very well with a heavy lightest Higgs boson and a low degree of fine-tuning. We discuss experimental constraints from collider direct searches, precision data, thermal relic dark matter abundance, and WIMP searches finding that the most natural region of the parameter space is still allowed by current experiments. We analyse bounds on the masses of the superpartners coming from Naturalness arguments and discuss the main signatures of the model for the LHC and future WIMP searches.Comment: Extended discussion of the LHC phenomenology, as published on JHEP plus an addendum on the existence of further extremal points of the potential. 47 pages, 16 figure