Implications of Direct Dark Matter Constraints for Minimal Supersymmetric Standard Model Higgs Boson Searches at the Tevatron

Searches for the Minimal Supersymmetric Standard Model (MSSM) Higgs bosons are among the most promising channels for exploring new physics at the Tevatron. In particular, interesting regions of large $\tan \beta$ and small $m_A$ are probed by searches for heavy neutral Higgs bosons, A and H, when they decay to $\tau^+ \tau^-$ and $b\bar{b}$. At the same time, direct searches for dark matter, such as CDMS, attempt to observe neutralino dark matter particles scattering elastically off nuclei. This can occur through t-channel Higgs exchange, which has a large cross section in the case of large $\tan \beta$ and small $m_A$. As a result, there is a natural interplay between the heavy, neutral Higgs searches at the Tevatron and the region of parameter space explored by CDMS. We show that if the lightest neutralino makes up the dark matter of our universe, current limits from CDMS strongly constrain the prospects of heavy, neutral MSSM Higgs discovery at the Tevatron (at 3 sigma with 4 fb^-1 per experiment) unless $|\mu| \gsim$ 400 GeV. The limits of CDMS projected for 2007 will increase this constraint to $|\mu| \gsim$ 800 GeV. On the other hand, if CDMS does observe neutralino dark matter in the near future, it will make the discovery of heavy, neutral MSSM Higgs bosons far more likely at the Tevatron.