Dilaton-assisted generation of the Fermi scale from the Planck scale

In scale-invariant theories of gravity the Planck mass $M_P$, which appears due to spontaneous symmetry breaking, can be the only scale at the classical level. It was argued that the second scale can be generated by a quantum nonperturbative gravitational effect. The new scale, associated with the Higgs vacuum expectation value, can be orders of magnitude below $M_P$, leading to the hierarchy between the Fermi and the Planck scales. We study a theory in which the nonperturbative effect is sensitive both to the physics at energy scales as high as $M_P$ and to the low-energy, Standard Model physics. This makes it possible to constrain the mechanism from experiment. We find that the crucial ingredients of the mechanism are nonminimal coupling of the scalar fields to gravity, the approximate Weyl invariance at high energies, and the metastability of the low-energy vacuum.

Figure 1

The classical solutions in the model (6) satisfying the boundary conditions (20) at infinity. We take $\lambda (\theta )=0.01\cos 4\theta$, ${\xi }_{\chi }=0.2$, ${\xi }_h=0.02$. Left panel: The solid line represents the bounce, and the dashed line is a circle of radius $M_P^{*}$ [see Eqs. (24) and (25)]. Right panel: The solid line represents the singular instanton solution. Its trajectory, if drawn from $r=\infty$, follows the path of the bounce until the turning point, after which $\theta$ starts decreasing according to the asymptotics (33), while $\rho$ starts growing according to Eq. (30).

Figure 2

Left panel: The profile of the angular field component of the bounce (the solid red line with nonzero asymptotics at $r\to 0$) and of the instanton (the solid blue line with zero asymptotics at $r\to 0$). One sees that at $r{M}_P\gg 1$ the two Euclidean solutions almost coincide. The dashed line denotes the upper limit of $\theta$, above which the kinetic term of the scalar fields of the model (6) is not positive definite. Right panel: The solid line is the Euclidean Lagrangian (13) computed on the bounce or on the instanton. The dashed line is the correction to the Euclidean Lagrangian, $\delta {\mathcal{L}}_E$, due to nonzero $\delta$, $\delta B_{\text{bounce}}=\int dr{M}_P\delta {\mathcal{L}}_E$ [see Eq. (27)]. Here we take $\delta =1$. The parameters of the model are $\lambda (\theta )=0.01\cos 4\theta$, ${\xi }_{\chi }=0.2$, and ${\xi }_h=0.02$.

Figure 3

Left panel: The low-energy contribution to the instanton action (28) as a function of ${\xi }_h$ and for different values of ${\xi }_{\chi }$. From the top to the bottom line, $6{\xi }_{\chi }-1={10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$. We see that $B_{\mathrm{LE}}$ is independent of ${\xi }_h$, at least when ${\xi }_h\lesssim {10}^{-2}$. Here $\lambda (\theta )=0.01\cos 4\theta$. Right panel: The high-energy contribution to the instanton action (28) as a function of $\delta$ and for different values of ${\xi }_{\chi }$. From the low to the large inclination, $6{\xi }_{\chi }-1={10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$. We see that $|B_{\mathrm{HE}}|$ shows powerlike dependence on the deviation of ${\xi }_{\chi }$ from the conformal limit, in accordance with Eq. (43), while the sign of $B_{\mathrm{HE}}$ is different for the different values of $\delta$, and small $\delta$ are needed to obtain $B_{\mathrm{HE}}<0$.

Figure 4

The instanton action $B$ plotted against $m_t$ and ${\xi }_{\chi }$, with ${\xi }_h=0.02$ and $\delta ={10}^{-18}$ (left panel), and $\delta ={10}^{-20}$ (right panel). The grey dashed isoparametric curves are plotted with steps $\mathrm{\Delta }B=\pm 2000$, starting from $B=\log M_P/v\approx 37$. Here we take the function $\lambda (\theta )$ as the Higgs self-coupling constant undergoing RG running within the SM and under the SI renormalization scheme; see the text for details. The vertical dashed line marks the central value of the top mass $m_t=172.25\mathrm{GeV}$ [68], and the running of $\lambda$ is computed at the central value of the Higgs mass $m_H=125.09\mathrm{GeV}$ [69].

Figure 5

Schematic form of diagrams providing the leading correction to $\delta$ in perturbation theory above the classical vacuum (10). Here ${\tilde{h}}_{\mu \nu }$ is the metric fluctuation above the flat background.