Consistent Use of the Standard Model Effective Potential

The stability of the standard model is determined by the true minimum of the effective Higgs potential. We show that the potential at its minimum when computed by the traditional method is strongly dependent on the gauge parameter. It moreover depends on the scale where the potential is calculated. We provide a consistent method for determining absolute stability independent of both gauge and calculation scale, order by order in perturbation theory. This leads to a revised stability bounds $m_h^{\text{pole}}>(129.4\pm 2.3)\mathrm{GeV}$ and $m_t^{\text{pole}}<(171.2\pm 0.3)\mathrm{GeV}$. We also show how to evaluate the effect of new physics on the stability bound without resorting to unphysical field values.

Figure 1

Gauge dependence of the absolute stability bound with $m_t^{\text{pole}}=173.34\mathrm{GeV}$.

Figure 2

Gauge dependence of the instability scale ${\mathrm{\Lambda }}_I$, defined by $V({\mathrm{\Lambda }}_I)=0$, at 1 loop in the traditional approach. There is no known way to make this scale gauge invariant.

Figure 3

Boundaries of absolute stability (lower band, NLO) and metastability (upper line, LO). The thickness of the lower boundary indicates perturbative and ${\alpha }_s$ uncertainty. The theoretical uncertainty of the metastability boundary is unknown. The elliptical contours are 68%, 95% and 99% confidence bands on the Higgs and top masses: $m_h^{\text{pole}}=(125.14\pm 0.23)\mathrm{GeV}$ and $m_t^{\text{pole}}=(173.34\pm 1.12)\mathrm{GeV}$. Dotted lines are scales in GeV at which $V_{\text{min}}$ can be lifted positive by new physics.