Investigating the near-criticality of the Higgs boson

TL;DR

The paper performs a high-precision NNLO analysis of the Standard Model Higgs sector, extracting MSbar parameters from electroweak observables and evolving them with 3-loop RGEs up to the Planck scale. It demonstrates that the measured Higgs mass places the SM near the boundary between vacuum stability and metastability, with λ crossing near 10^10–10^12 GeV and a slow βλ around M_Pl. The authors map detailed phase diagrams in terms of both weak-scale and Planck-scale couplings and discuss multiple high-energy interpretations, including high-scale matching scenarios and multiverse-based criticality attractors. The results suggest near-criticality of the Higgs potential as a potentially fundamental clue about physics beyond the weak scale, even in the absence of new low-energy signals. The work emphasizes the interplay between precise SM parameter determinations, high-scale dynamics, and philosophical implications for naturalness and the structure of fundamental theories.

Abstract

We extract from data the parameters of the Higgs potential, the top Yukawa coupling and the electroweak gauge couplings with full 2-loop NNLO precision, and we extrapolate the SM parameters up to large energies with full 3-loop NNLO RGE precision. Then we study the phase diagram of the Standard Model in terms of high-energy parameters, finding that the measured Higgs mass roughly corresponds to the minimum values of the Higgs quartic and top Yukawa and the maximum value of the gauge couplings allowed by vacuum metastability. We discuss various theoretical interpretations of the near-criticality of the Higgs mass.

Figures (7)

• Figure 1: Renormalisation of the SM gauge couplings $g_1=\sqrt{5/3}g_Y, g_2, ~g_3$, of the top, bottom and $\tau$ couplings ($y_t$, $y_b$, $y_\tau$), of the Higgs quartic coupling $\lambda$ and of the Higgs mass parameter $m$. All parameters are defined in the $\overline{\hbox{\sc ms}}$ scheme. We include two-loop thresholds at the weak scale and three-loop RG equations. The thickness indicates the $\pm1\sigma$ uncertainties in $M_t,M_h,\alpha_3$.
• Figure 2: Upper: RG evolution of $\lambda$ ( left) and of $\beta_\lambda$ ( right) varying $M_t$, $\alpha_3(M_Z)$, $M_h$ by $\pm 3\sigma$. Lower: Same as above, with more "physical" normalisations. The Higgs quartic coupling is compared with the top Yukawa and weak gauge coupling through the ratios ${\rm sign}(\lambda)\sqrt{4|\lambda|}/y_t$ and ${\rm sign}(\lambda)\sqrt{8|\lambda|}/g_2$, which correspond to the ratios of running masses $m_h/m_t$ and $m_h/m_W$, respectively ( left). The Higgs quartic $\beta$-function is shown in units of its top contribution, $\beta_\lambda$(top contribution) $=-3y_t^4/8\pi^2$ ( right). The grey shadings cover values of the RG scale above the Planck mass $M_{\rm Pl} \approx 1.2\times 10^{19}\,{\rm GeV}$, and above the reduced Planck mass $\bar{M_{\rm Pl}} = M_{\rm Pl} /\sqrt{8\pi}$.
• Figure 3: Left: SM phase diagram in terms of Higgs and top pole masses. The plane is divided into regions of absolute stability, meta-stability, instability of the SM vacuum, and non-perturbativity of the Higgs quartic coupling. The top Yukawa coupling becomes non-perturbative for $M_t>230\, \,{\rm GeV}$. The dotted contour-lines show the instability scale $\Lambda_I$ in $\,{\rm GeV}$ assuming $\alpha_3(M_Z)=0.1184$. Right: Zoom in the region of the preferred experimental range of $M_h$ and $M_t$ (the grey areas denote the allowed region at 1, 2, and 3$\sigma$). The three boundary lines correspond to 1-$\sigma$ variations of $\alpha_3(M_Z)=0.1184\pm 0.0007$, and the grading of the colours indicates the size of the theoretical error.
• Figure 4: Left: SM phase diagram in terms of quartic Higgs coupling $\lambda$ and top Yukawa coupling $y_t$ renormalised at the Planck scale. The region where the instability scale $\Lambda_I$ is larger than $10^{18}\, \,{\rm GeV}$ is indicated as 'Planck-scale dominated'. Right: Zoom around the experimentally measured values of the couplings, which correspond to the thin ellipse roughly at the centre of the panel. The dotted lines show contours of $\Lambda_I$ in $\,{\rm GeV}$.
• Figure 5: SM phase diagram in terms of the Higgs quartic coupling $\lambda (M_{\rm Pl} )$ and of the gauge coupling $g_2(M_{\rm Pl} )$. Left: A common rescaling factor is applied to the electro-weak gauge couplings $g_1$ and $g_2$, while $g_3$ is kept constant. Right: A common rescaling factor is applied to all SM gauge couplings $g_1,g_2,g_3$, such that a $10\%$ increase in the strong gauge coupling at the Planck scale makes $\Lambda_{\rm QCD}$ larger than the weak scale. The measured values of the couplings correspond to the small ellipse marked as 'SM'.