The Probable Fate of the Standard Model

TL;DR

The paper interrogates whether the Standard Model can be extrapolated to the Planck scale without new physics by tracking the Higgs sector’s renormalisation-group evolution and the effective potential. It combines two-loop RGEs with vacuum-stability, metastability, and perturbativity bounds and integrates this with a global electroweak fit via Gfitter, incorporating LEP/Tevatron Higgs searches. The main result is that the blow-up scenario is excluded at about 99% CL, while the SM remains compatible with surviving up to the Planck scale or existing a metastable vacuum depending on the Higgs mass; precise MH measurements could pin down the fate by constraining the cutoff Λ. The work provides quantitative MH–Λ mappings that connect collider data to high-scale implications for the SM’s validity and the necessity of new physics before $M_P \, (\sim 2\times10^{18}$ GeV).

Abstract

Extrapolating the Standard Model to high scales using the renormalisation group, three possibilities arise, depending on the mass of the Higgs boson: if the Higgs mass is large enough the Higgs self-coupling may blow up, entailing some new non-perturbative dynamics; if the Higgs mass is small the effective potential of the Standard Model may reveal an instability; or the Standard Model may survive all the way to the Planck scale for an intermediate range of Higgs masses. This latter case does not necessarily require stability at all times, but includes the possibility of a metastable vacuum which has not yet decayed. We evaluate the relative likelihoods of these possibilities, on the basis of a global fit to the Standard Model made using the Gfitter package. This uses the information about the Higgs mass available directly from Higgs searches at LEP and now the Tevatron, and indirectly from precision electroweak data. We find that the `blow-up' scenario is disfavoured at the 99% confidence level (96% without the Tevatron exclusion), whereas the `survival' and possible `metastable' scenarios remain plausible. A future measurement of the mass of the Higgs boson could reveal the fate of the Standard Model.

Figures (7)

• Figure 1: Dependence on $M_H$ of the $\Delta \chi^2$ function obtained from the global fit of the SM parameters to precision electroweak data gfitter, excluding (left) or including (right) the results from direct searches at LEP and the Tevatron.
• Figure 2: The scale $\Lambda$ at which the two-loop RGEs drive the quartic SM Higgs coupling non-perturbative, and the scale $\Lambda$ at which the RGEs create an instability in the electroweak vacuum ($\lambda < 0$). The width of the bands indicates the errors induced by the uncertainties in $m_t$ and $\alpha_{ S}\xspace$ (added quadratically). The perturbativity upper bound (sometimes referred to as 'triviality' bound) is given for $\lambda = \pi$ (lower bold line [blue]) and $\lambda =2\pi$ (upper bold line [blue]). Their difference indicates the size of the theoretical uncertainty in this bound. The absolute vacuum stability bound is displayed by the light shaded [green] band, while the less restrictive finite-temperature and zero-temperature metastability bounds are medium [blue] and dark shaded [red], respectively. The theoretical uncertainties in these bounds have been ignored in the plot, but are shown in Fig. \ref{['fig:bounds_zoom']} (right panel). The grey hatched areas indicate the LEP Higgs-LEP and Tevatron Higgs-Tev exclusion domains.
• Figure 3: Lower bounds on the Higgs mass due to absolute vacuum stability (light shaded [green]), finite-temperature (medium shaded [blue]) and zero-temperature metastability (dark shaded [red]), as functions of the cut-off scale $\Lambda$. The bands indicate the errors induced by the uncertainties in $m_t$ and $\alpha_{ S}\xspace$ (added quadratically). The left plot is thus identical to Fig. \ref{['fig:bounds']}, but with a zoomed ordinate. The right plot includes theoretical uncertainties, which treated as an offset, i.e., they are not quadratically added to the other errors ( cf. Sec \ref{['sec:combinedLikelihoodAnalysis']}). At $\Lambda=M_P$, the bounds correspond to Eqs. (\ref{['eq:stability']}), (\ref{['eq:thermmetastab']}) and (\ref{['eq:metastab']}), respectively.
• Figure 4: The levels of 1 $-$ CL versus $M_H$ for the different scenarios defined by the ultraviolet behaviour of the Higgs potential. The regions are (from left to right): the 'collapse region' (light [red] shaded/hatched) corresponding to $M_H$ violating the metastability bound (\ref{['eq:metastab']}) and thus vulnerable to quantum tunneling of the electroweak vacuum in a time shorter than the age of the Universe; the 'zero-temperature metastability' region ([blue] dotted) corresponding to values of $M_H$ between the bounds (\ref{['eq:metastab']}) and (\ref{['eq:stability']}), where quantum tunneling is acceptably slow; the 'finite-temperature metastability' region (dark [green] hatched), defined by the lower bound (\ref{['eq:thermmetastab']}), where the local SM minimum is stable against thermal fluctuations up to temperatures equal to $M_P$; the 'stability' region (darker [green] shaded) delimited by the bounds (\ref{['eq:stability']}) and (\ref{['eq:blow-up']}); and finally the 'non-perturbativity' region (light [grey] shaded/hatched), bound by Eq. (\ref{['eq:blow-up']}), where the Higgs self-coupling becomes non-perturbative at some scale smaller than $M_P$. The slopes of the 'pyramids' representing the boundaries of the different regions reflect the uncertainties in $m_t$ and $\alpha_{ S}\xspace(M_Z^2)\xspace$ which lead, together with the theoretical errors affecting the bounds, to apparent overlaps between the regions. Also shown is the 1 $-$ CL function for the combination of current constraints on $M_H$ equivalent to the right plot of Fig. \ref{['fig:basis']} (bold solid [blue] line).
• Figure 5: Contours of 40%, 68%, 95% and 99% CL obtained from scans of fits with fixed values of the variables $M_H$ and $\log_{10}(\Lambda/\hbox{\rm:GeV}\xspace)$. The fits include the electroweak precision data and the bounds from the perturbativity and stability requirements shown in Fig. \ref{['fig:bounds']}. The lower plot also incorporates the direct Higgs boson searches at LEP and the Tevatron (corresponding to the complete fit scenario in Ref. gfitter). Their respective 95% CL exclusion domains are depicted by the hatched bands.