The ATLAS and CMS experiments observed a particle at the LHC with a mass $approx 126$ GeV, which is compatible with the Higgs boson of the Standard Model. A crucial question is, if for such a Higgs mass value, one could extrapolate the model up to high scales while keeping the minimum of the scalar potential that breaks the electroweak symmetry stable. Vacuum stability requires indeed the Higgs boson mass to be $M_H gsim 129 pm 1$ GeV, but the precise value depends critically on the input top quark pole mass which is usually taken to be the one measured at the Tevatron, $m_t^{rm exp}=173.2 pm 0.9$ GeV. However, for an unambiguous and theoretically well-defined determination of the top quark mass one should rather use the total cross section for top quark pair production at hadron colliders. Confronting the latest predictions of the inclusive $p bar p to tbar t +X$ cross section up to next-to-next-to-leading order in QCD to the experimental measurement at the Tevatron, we determine the running mass in the $bar{rm MS}$-scheme to be $m_t^{bar{rm MS}}(m_t) = 163.3 pm 2.7$ GeV which gives a top quark pole mass of $m_t^{rm pole}= 173.3 pm 2.8$ GeV. This leads to the vacuum stability constraint $M_H geq 129.8 pm 5.6$ GeV to which a $approx 126$ GeV Higgs boson complies as the uncertainty is large. A very precise assessment of the stability of the electroweak vacuum can only be made at a future high-energy electron-positron collider, where the top quark pole mass could be determined with a few hundred MeV accuracy.
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