Direct production of a light CP-odd Higgs boson at the Tevatron and LHC

We show that the existing CDF $L=630 pb^{-1}$ Tevatron data on $pp\to \mu^+\mu^- + X$ places substantial limits on a light CP-odd Higgs boson $a$ with $m_a<2m_B$ produced via $gg\to a$, even for $m_a>2m_\tau$ for which $BR(a\to \mu^+\mu^-)$ is relatively small. Extrapolation of this existing CDF analysis to $L=10 fb^{-1}$ suggests that Tevatron limits on the $a b\anti b$ coupling strength in the region $m_a>8\gev$ could be comparable to or better than limits from Upsilon decays in the $m_a<7\gev$ region. We also give rough estimates of future prospects at the LHC, demonstrating that early running will substantially improve limits on a light $a$ (or perhaps discover a signal). In particular, outside the Upsilon peak region, integrated luminosity of only $5-20 fb^{-1}$ (depending on $m_a$ and $\sqrt s$) could reveal a peak in $M_{\mu^+\mu^-}$ and will certainly place important new limits on a light $a$. The importance of such limits in the context of NMSSM Higgs discovery and $(g-2)_\mu$ are outlined.