  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 $p \bar p \to t\bar t +X$ 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.
