Stationary configurations of the Standard Model Higgs potential: Electroweak stability and rising inflection point

We study the gauge-independent observables associated with two interesting stationary configurations of the Standard Model Higgs potential (extrapolated to high energy according to the present state of the art, namely the next-to-next-to-leading order): i) the value of the top mass ensuring the stability of the SM electroweak minimum and ii) the value of the Higgs potential at a rising inflection point. We examine in detail and reappraise the experimental and theoretical uncertainties which plague their determination, finding that i) the stability of the SM is compatible with the present data at the $1.5\sigma$ level and ii) despite the large theoretical error plaguing the value of the Higgs potential at a rising inflection point, the application of such a configuration to models of primordial inflation displays a $3\sigma$ tension with the recent bounds on the tensor-to-scalar ratio of cosmological perturbations.

Figure 1

Sketch of the shape of SM effective potential at high energy in the Landau gauge, for increasing values of the top mass from top to bottom. Two peculiar stationary configurations can be identified: two degenerate vacua and a rising point of inflection, associated, respectively, to $m_t^c$ and $m_t^i$.

Figure 2

Lines for which the Higgs potential develops a second degenerate minimum at high energy. The solid line corresponds to the central value of ${\alpha }_s^{(5)}$; the dashed lines are obtained by varying ${\alpha }_s^{(5)}$ in its experimental range, up to $3\sigma$. The (red) arrow represents the theoretical error in the position of the lines. The (green) shaded regions are the covariance ellipses obtained combining $m_t^{\mathrm{MC}}=173.34\pm 0.76\mathrm{GeV}$ and $m_H^{\exp }=125.09\pm 0.24\mathrm{GeV}$; the probability of finding $m_t^{\mathrm{MC}}$ and $m_H$ inside the inner (central, outer) ellipse is equal to 68.2% (95.5%, 99.7%).

Figure 3

Dependence of $m_t^c$ on $\alpha$, for the increasing level of the effective potential expansion. We choose the central values for $m_H$ and ${\alpha }_s^{(5)}$.

Figure 4

Lines for which the Higgs potential develops a second degenerate minimum at high energy. The solid line corresponds to the central value of ${\alpha }_s^{(5)}$; the dashed lines are obtained by varying ${\alpha }_s^{(5)}$ in its experimental range, up to $3\sigma$. The (red) arrow represents the theoretical error in the position of the lines. The (green) shaded regions are the covariance ellipses obtained combining $m_t^{\mathrm{CMS}}=172.38\pm 0.66\mathrm{GeV}$ [49] and $m_H^{\exp }=125.09\pm 0.24\mathrm{GeV}$.

Figure 5

Dependence of ${\overline{V}}_i^{1/4}$ on $m_H$ for fixed values of ${\alpha }_s^{(5)}$. The (red) arrow and solid lines show the theoretical error due to the matching of $\lambda$ (positive contribution to $\lambda$ for the upper line, negative for the lower one). The right vertical axis displays the associated value of the tensor-to-scalar ratio $r$, according to Eq. (24).

Figure 6

Dependence of ${\overline{V}}_i^{1/4}$ on $\alpha$. For definiteness, ${\alpha }_s^{(5)}$ and $m_H$ are assigned to their central values.

Figure 7

Contour levels of $r$ in the plane $(m_H,{\alpha }_s^{(5)})$. The theoretical uncertainty corresponding to $r=0.12$ is shown by means of the (red) dashed lines. The shaded regions are the covariance ellipses indicating that the probability of finding the experimental values of $m_H$ and ${\alpha }_s^{(5)}$ inside the ellipses are, respectively, 68.2%, 95.4%, and 99.7%.