Two Higgs Models for Large Tan Beta and Heavy Second Higgs

We study two Higgs models for large $\tanβ$ and relatively large second Higgs mass. In this limit the second heavy Higgs should have small vev and therefore couples only weakly to two gauge bosons. Furthermore, the couplings to down type quarks can be significantly modified (so long as the second Higgs is not overly heavy). Both these facts have significant implications for search strategies at the LHC and ILC. We show how an effective theory and explicit fundamental two Higgs model approach are related and consider the additional constraints in the presence of supersymmetry or $Z_2$ flavor symmetries. We argue that the best tests of the two Higgs doublet potential are likely to be measurements of the light Higgs branching fractions. We show how higher dimension operators that have recently been suggested to raise the light Higgs mass are probably best measured and distinguished in this way.