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 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 Higgs self-coupling and its beta function in the modified minimal-subtraction scheme 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 standard model be stable all the way up to the Planck mass scale and conservatively estimate the theoretical uncertainty. This bound is compatible with the Monte Carlo mass quoted by the Particle Data Group at the $1.3\sigma$ level.

Figure 1

RG evolution of $\lambda (\mu )$ from ${\mu }^{\text{thr}}$ to ${\mu }^{\text{cri}}$ and beyond in the $(\lambda ,{\beta }_{\lambda })$ plane for default input values and matching scale (red solid line), effects of $1\sigma$ (brown solid lines) and $3\sigma$ (blue solid lines) variation in $M_t^{\mathrm{MC}}$, theoretical uncertainty due to the variation of $\xi$ from $1/2$ to 2 (upper and lower black dashed lines with asterisks in the insets), and results for $M_t^{\text{cri}}$ (green dashed line) and $M_H^{\text{cri}}$ (purple dashed line). The $1\sigma$ (brown ellipses) and $3\sigma$ (blue ellipses) contours due to the errors in $M_t^{\mathrm{MC}}$ and $M_H$ are indicated for selected values of $\mu$. The insets in the upper right and lower left corners refer to $\mu =M_t^{\mathrm{MC}}$ and $\mu =1.55\times {10}^{10}\mathrm{GeV}$, respectively.

Figure 2

Phase diagram of vacuum stability (light-green shaded area), metastability, and instability (pink shaded area) in the $(M_H,M_t)$ plane, contours of $\lambda ({\mu }^0)=0$ for selected values of ${\mu }^0$ (purple dotted lines), contours of ${\beta }_{\lambda }({\mu }^0)=0$ for selected values of ${\mu }^0$ (solid parabolalike lines) with uncertainties due to $1\sigma$ error in ${\alpha }_s^{(5)}(M_Z)$ (dashed and dot-dashed lines), critical line of Eq. (2) (solid green line) with uncertainty due to $1\sigma$ error in ${\alpha }_s^{(5)}(M_Z)$ (orange shaded band), and critical points with $M_t^{\text{cri}}$ (lower red bullet) and $M_H^{\text{cri}}$ (right red bullet). The present world average of $(M_t^{\mathrm{MC}},M_H)$ (upper left red bullet) and its $1\sigma$ (purple ellipse), $2\sigma$ (brown ellipse), and $3\sigma$ (blue ellipse) contours are marked for reference.