Fermi scale from quantum gravity scaling solution

TL;DR

This work examines whether the observed gauge hierarchy can be explained by a quantum-gravity scaling solution in a scale-invariant standard model augmented by a cosmon. Using a functional renormalization group framework, it studies the two-field potential $U(H,\chi;k)$ through its dimensionless form $u(\tilde{\rho},\tilde{h})$, deriving a scaling equation and exploring local and global scaling solutions. A central focus is the cosmon-Higgs coupling $\lambda_m$ and its flow, which—under a UV fixed point and gravity-induced anomalous dimensions—becomes an irrelevant parameter, enabling self-organized criticality to fix the ratio $\varphi_0/M_p$. The analysis suggests that, for a wide class of UV scenarios with a largest intrinsic mass scale well below $\varphi_0$, the Fermi scale is predicted in units of the Planck mass, providing a potential dynamical explanation for the gauge hierarchy; numerical results in a truncated model corroborate the qualitative picture and illustrate how the IR value can align with observations given plausible UV data.

Abstract

Fundamental scale invariance implies the scale invariant standard model. Both the Fermi scale and the Planck mass are given by fields, and their ratio is dictated by a dimensionless cosmon-Higgs coupling. For an ultraviolet fixed point of quantum gravity this coupling is an irrelevant parameter of the renormalization flow and becomes predictable. An analytic scaling solution for quantum gravity admits no free parameter for the mass term of the Higgs boson. If the largest intrinsic mass scale generated by the renormalisation flow away from the fixed point is sufficiently below the Fermi scale, the couplings of the scale invariant standard model are determined by the scaling solution. For a given short distance model remaining valid to infinitely small distances the ratio Fermi scale over Planck mass can be predicted. With reasonable assumptions for an ultraviolet fixed point a numerical solution finds a tiny value for the ratio between the Fermi and Planck scales, very close to a second order quantum electroweak phase transition. This could explain the observed gauge hierarchy.

Figures (5)

• Figure 1: Critical trajectory. We plot $g$ as a function of $x$, for two initial values $\sigma_m (x_F)=-0.1$ (blue) and $\sigma_m (x_F)= 0.0009$ (orange). The green curve shows the critical curve.
• Figure 2: Infrared flow of cosmon-Higgs coupling. We plot $\sigma_m = \tilde{\rho} \lambda_m$ as a function of $x=\ln \tilde{\rho}$. The two curves correspond to initial values $\sigma_m(x_F) = -0.1$ (blue) and $\sigma_m(x_F) = 0.4$ (orange).
• Figure 3: Flow of cosmon-Higgs coupling $\lambda_m$ in the Planck region as a function of $x = \ln \tilde{\rho}$. Two of the three curves correspond to the initial values $\sigma_m(x_F)=-0.1$ and $\sigma_m(x_F)=0.4$ as for Fig. \ref{['fig:1']}. The third curve is on approximation to $\lambda_m^{(\text{cr})}$. The three curves almost coincide.
• Figure 4: Field-dependent couplings of the standard model. As a function of $x= \ln \tilde{\rho}$ we plot the squared top-quark Yukawa coupling $y^2_t$ (blue), the squared gauge couplings $g^2_3$ (orange), $g^2_2$ (green) and $g^2_1$ (red), as well as the quartic scalar coupling $\lambda_h$ (violet).
• Figure 5: Ultraviolet flow of cosmon-Higgs coupling. We plot $\lambda_m$ as a function of $x = \ln \tilde{\rho}$, for two initial values $\sigma_m(x_F)=-0.1$ (green) and $\sigma_m(x_F)=0.4$ (orange). The curve for $\sigma_m(x_F)=-0.1$ coincides with $\lambda_m^{(\text{cr})}(\tilde{\rho})$.