Naturalness, b to s gamma, and SUSY Heavy Higgses

We explore naturalness constraints on the masses of the heavy Higgs bosons H^0, H^+/-, and A^0 in supersymmetric theories. We show that, in any extension of MSSM which accommodates the 125 GeV Higgs at the tree level, one can derive an upper bound on the SUSY Higgs masses from naturalness considerations. As is well-known for the MSSM, these bounds become weak at large tan beta. However, we show that measurements of b to s gamma together with naturalness arguments lead to an upper bound on tan beta, strengthening the naturalness case for heavy Higgs states near the TeV scale. The precise bound depends somewhat on the SUSY mediation scale: allowing a factor of 10 tuning in the stop sector, the measured rate of b to s gamma implies tan beta < 30 for running down from 10 TeV but tan beta < 4 for mediation at or above 100 TeV, placing m_A near the TeV scale for natural EWSB. Because the signatures of heavy Higgs bosons at colliders are less susceptible to being "hidden" than standard superpartner signatures, there is a strong motivation to make heavy Higgs searches a key part of the LHC's search for naturalness. In an appendix we comment on how the Goldstone boson equivalence theorem links the rates for H to hh and H to ZZ signatures.