Higgs boson and top quark masses as tests of electroweak vacuum stability

The measurements of the Higgs boson and top-quark masses can be used to extrapolate the Standard Model Higgs potential at energies up to the Planck scale. Adopting a next-to-next-to-leading-order renormalization procedure, we (i) find that electroweak vacuum stability is at present allowed and discuss the associated theoretical and experimental errors and the prospects for its future tests, (ii) determine the boundary conditions allowing for the existence of a shallow false minimum slightly below the Planck scale, which is a stable configuration that might have been relevant for primordial inflation, and (iii) derive a conservative upper bound on type-I seesaw right-handed neutrino masses, following from the requirement of electroweak vacuum stability.

Figure 1

Value of $\lambda (m_H)$ obtained by performing the matching at different scales $\mu$—indicated by the labels—as a function of $m_H$. The solid (dashed) lines are obtained by including corrections up to the two-loop (one-loop) level. We fixed $m_t=172\mathrm{GeV}$ [for different values see Eq. (5)].

Figure 2

Values of $h_t(m_t)$ and ${\overline{m}}_t(m_t)$ as a function of $m_t$. The curves are obtained by matching at different scales, which are indicated by the labels. We fixed $m_H=126\mathrm{GeV}$ for definiteness but the results do not significantly dependent on $m_H$, provided it is chosen in its experimental range.

Figure 3

The SM Higgs potential (left) and the quartic Higgs coupling (right) as functions of the renormalization scale $\mu$, for $m_H=126\mathrm{GeV}$ and different values of ${\overline{m}}_t(m_t)$, increasing from top to bottom by the amount indicated by the labels. The dashed curve in the right plot shows the associated value of ${\beta }_{\lambda }(\mu )$. The other input parameters are fixed at the central values discussed in the previous section.

Figure 4

The scale ${\mu }_{\beta }$ as a function of ${\overline{m}}_t(m_t)$ and for different values of $m_H$, as indicated by the labels.

Figure 5

The solid (black) line marks the points in the plane $[m_H,{\overline{m}}_t(m_t)]$ where a second vacuum, degenerate with the electroweak one, is obtained just below the Planck scale. The (red) diagonal arrow shows the effect of varying ${\alpha }_3(m_Z)=0.1196\pm 0.0017$ [19]; the (blue) horizontal arrow shows the effect of varying ${\mu }_{\lambda }$ (the matching scale of $\lambda$) from $m_Z$ up to $2m_H$. The shaded (yellow) vertical region is the $2\sigma$ ATLAS [1] and CMS [2] combined range, $m_H=125.65\pm 0.85\mathrm{GeV}$; the shaded (green) horizontal region is the range ${\overline{m}}_t(m_t)=163.3\pm 2.7\mathrm{GeV}$, equivalent to $m_t=173.3\pm 2.8\mathrm{GeV}$ [15].

Figure 6

The transition line between stability and metastability in the plane $[m_H,m_t]$ and for fixed values of ${\alpha }_3(m_Z)=0.1196\pm 0.0017$ [19]. The thickness of the lines represent the theoretical error due to the variation of ${\mu }_{\lambda }$ (the matching scale of $\lambda$) from $m_Z$ up to $2m_H$. The shaded region is obtained by intersecting the $2\sigma$ ATLAS [1] and CMS [2] combined range ($m_H=125.65\pm 0.85\mathrm{GeV}$) with the running top mass range given by Ref. 15, ${\overline{m}}_t(m_t)=163.3\pm 2.7\mathrm{GeV}$, equivalent to $m_t=173.3\pm 2.8\mathrm{GeV}$.

Figure 7

Left: Theoretical uncertainty in the determination of the transition line between stability and metastability according to our Eq. (9) (thinner) and Eq. 10 of Ref. 14 (thicker). For definiteness we choose ${\alpha }_3(m_Z)=0.1196$. Right: Transition line between stability and metastability according to Eq. 10 of Ref. 14; the thickness of the band accounts for both the 1 GeV theoretical error and the experimental error due to the variation of ${\alpha }_3(m_Z)$ in the range $0.1184\pm 0.0007$, as was done in Ref. 14. The (brown) shaded disks represent the $1\sigma$ and $2\sigma$ combined ranges for $m_t$ and $m_H$ used in Ref. 14 (see their Fig. 5). The (green) rectangle allows for the comparison with the ranges of $m_t$ and $m_H$ used here (see Fig. 6).

Figure 8

Values of ${\mu }_i$ (left) and ${\mu }_{\beta }$ (right) as a function of $m_H$. For the solid lines, the input parameters are fixed at their central values and the matching of $\lambda$ is done at $\mu =m_H$. The shaded regions show the uncertainty induced by the experimental error of ${\alpha }_3(m_Z)=0.1196\pm 0.0017$: the short and long dashed curves refer to the lower and upper value at $1\sigma$, respectively.

Figure 9

Left: The value of $\lambda ({\mu }_{\beta })$ as a function of $m_H$. Right: The Higgs potential at ${\mu }_i$ and the associated prediction for $r$ as a function of $m_H$. The short and long dashed curves refer to the $1\sigma$ lower and upper values of ${\alpha }_3(m_Z)$, respectively. The shaded (yellow) vertical region marks the preferred range of $m_H$ at $2\sigma$ [1, 2]. The upper region in the right plot is excluded because $r\lesssim 0.2$ [37].

Figure 10

Upper bound on $M_{\nu }$ as a function of the running top mass, following from the requirement that the electroweak vacuum is not destabilized because of the inclusion of the seesaw, for $m_H=126\mathrm{GeV}$. The shaded region is obtained by varying ${\alpha }_3(m_Z)$ in its $1\sigma$ range.

Figure 11

The dashed curve represents the Higgs potential as a function of the renormalization scale, for $m_H=126\mathrm{GeV}$, ${\alpha }_3(m_Z)=0.1196$, and $m_t=171.56\mathrm{GeV}$ (the value of the top mass leading to an inflection-point configuration in the SM case with the former two parameters fixed). The solid lower curves display the effect of adding the seesaw, with three increasing values of $M_{\nu }$ from top to bottom.