NLO Supersymmetric QCD Corrections to $t \bar t h^0$ Associated Production at Hadron Colliders

We calculate NLO QCD corrections to the lightest neutral Higgs boson production associated with top quark pair at hadron colliders in the minimal supersymmetric standard model(MSSM). Our calculation shows that the total QCD correction significantly reduces its dependence on the renormalization/factorization scale. The relative correction from the SUSY QCD part approaches to be a constant, if either $M_S$ or $m_{\tilde{g}}$ is heavy enough. The corrections are generally moderate(in the range of few percent to 20%) and under control in most of the SUSY parameter space. The relative correction is obviously related to $m_{\tilde{g}}$, $A_t$ and $\mu$, but not very sensitive to $\tan\beta$, $M_S$ at both the Tevatron and the LHC with our specified parameters.