The quantum criticality of the Standard Model and the hierarchy problem

Abstract

The naturalness principle has long guided efforts to understand physics beyond the Standard Model, with the hierarchy problem as the central issue. We revisit the role of quantum corrections in the fine-tuning of the low-energy effective description and its phase structure. We implement, for the first time in this context, the full Standard Model within the Wilsonian functional renormalization group. Crucially, this method captures conveniently both logarithmic and quadratic scalings, which must both be considered in the tuning, and allows us to provide a new generic and quantitative study of fine-tuning and its interpretation in terms of critical phenomena. We emphasize on the connection between the hierarchy problem and the near-criticality of the Standard Model and extract scheme-independent information on the infrared Higgs phases and the associated quantum phase transition as well as discuss a related enhanced fine-tuning usually not considered in tuning estimates. Finally, we illustrate the framework's versatility by exploring new physics coupled to the Higgs sector that can soften high-scale sensitivity, recovering also the large-anomalous-dimension solution to the hierarchy problem.

Figures (10)

• Figure 1: In the top panel, we show the trajectories of the SM gauge ($g_3,\,g_2,\,g_1$), top ($y_{\rm t}$) and bottom ($y_{\rm b}$) Yukawa couplings. These span from the Planck scale ($k_\textrm{Planck}\approx 10^{19}$ GeV) down to below the dynamical chiral symmetry breaking scale ($k_{\chi\textrm{SB}}\approx 0.1$ GeV), over the EW ($k_\textrm{SSB,1}\approx 10^{3}$ GeV) scale. In the bottom panel, we display the trajectories of the dimensionless Higgs curvature mass ($\bar{\mu}^2$ in blue) and the quartic coupling ($\bar{\lambda}$ in orange). These two parameters define, in the present truncation, the shape of the average potential and hence the effective phases of the Higgs field. Three different regions are marked by plain ($\bar{\mu}^2=0$) and dashed ($\bar{\lambda}=0$) vertical lines, depending on the change of the small or large field curvature.
• Figure 2: Phase diagram of the SM class of theories projected on the plane of scalar sector parameters ($\bar{\mu}^2$--$\bar{\lambda}$). The left-most cartoon depicts the flow in a sub-plain of theory space. On the plots, the black arrowed lines indicate the flows from UV to IR for the parameter space. The yellow dot and red lines correspond to the partial fixed-point solutions and the separatrix as defined in \ref{['eq:FPHiggs']}, \ref{['eq:separatrix1']} and \ref{['eq:separatrix2']}, respectively. From left to right, we display slices of theory space at $k=10^3$, $10^{5}$ and $10^{17}$ GeV. The pink star indicates the position of the SM trajectory at the respective scales and the white dashed line the full trajectory in the multi-dimensional volume projected on the plane of couplings.
• Figure 3: Total ($\partial_{t} \bar{\mu}^2$, plain black line) and partial (plain blue and green lines) contributions of the flow of the Higgs curvature mass for the SM trajectory. The power-law contribution or critical surface is depicted in blue ($\left.\partial_{\bar{\rho}} \,\,\overline{\text{Flow}}\left[ V_{\textrm{eff}}\right]\right|_{\bar{\rho}_0=0}$), and the logarithmic ($(-2+\eta_H)\bar{\mu}^2$) in green. The symmetric (shaded green) and broken (shaded blue) regimes are marked, where logarithmic and power scalings respectively dominate. The dashed lines show a neighbouring solution to the SM one which leads to no EWSB, as discussed in \ref{['fig:SM phase diagram mHiggs no EWSB']}.
• Figure 4: In the left-most panels, the Higgs curvature mass trajectories for various boundary conditions (yellow to blue lines) around the physical one (pink lines and stars) are shown. In the middle (rightmost) panel, we show the Higgs vacuum expectation value $v$ at $k\to0$ (value of the $k_{\textrm{SSB},1}$ scale), as a function of the relative variation of the boundary condition with respect to the physical trajectory ($\Delta \bar{\mu}^2_\textrm{rel} (\Lambda_{\textrm{UV}})$). From top to bottom, each row corresponds to a different choice of $\Lambda_{\textrm{UV}}=\{10^4,10^8,10^{10}\}$ GeV, respectively. See text for details.
• Figure 5: On the left plot we show the phase diagram for the SM class of theories projected onto the $\bar{\mu}^2$--$\bar{\lambda}$ plane at a particular scale. We distinguish the SM trajectory (solid purple line) from a neighbouring one with no Higgs mechanism (dashed blue line). The red lines indicate the partial fixed points in \ref{['eq:partialFPmu']}, and the vertical dashed white line marks the zero crossing of the curvature mass, signalling the critical scale. The two shaded regions on either side of the separatrix highlight the two possible solutions. On the right plot we show the cutoff dependence of the Higgs Euclidean mass, as defined in \ref{['eq:EuclideanmH']}, for the two trajectories depicted in the left plot (same colour coding) and for the massless condition (red dotted) given by \ref{['eq:partialFPmu']}.