Stability of the Electroweak Vacuum: Gauge Independence and Advanced Precision

We perform a manifestly gauge-independent analysis of the vacuum stability in the Standard Model (SM) including two-loop matching, three-loop renormalization group evolution, and pure QCD corrections through four loops. All these ingredients are exact, except that light-fermion masses are neglected. We in turn apply the criterion of nullifying the $\overline{\mathrm{MS}}$ Higgs self-coupling and its beta function and a recently proposed consistent method for determining the true minimum of the effective Higgs potential that also avoids gauge dependence. Exploiting our knowledge of the Higgs-boson mass, we derive an upper bound on the pole mass of the top quark by requiring that the SM be stable all the way up to the Planck mass scale and conservatively estimate the theoretical uncertainty. This bound is compatible with Monte Carlo mass quoted by the Particle Data Group at the $1.3σ$ level.