Lifetime of the electroweak vacuum and sensitivity to Planck scale physics

If the Standard Model (SM) is valid up to extremely high energy scales, then the Higgs potential becomes unstable at approximately $10^{11}\mathrm{GeV}$. However, calculations of the lifetime of the SM vacuum have shown that it vastly exceeds the age of the Universe. It was pointed out by two of us (V. B., E. M.) that these calculations are extremely sensitive to effects from Planck scale higher-dimensional operators and, without knowledge of these operators, firm conclusions about the lifetime of the SM vacuum cannot be drawn. The previous paper used analytical approximations to the potential and, except for Higgs contributions, ignored loop corrections to the bounce action. In this work, we do not rely on any analytical approximations and consider all contributions to the bounce action, confirming the earlier result. It is surprising that the Planck scale operators can have such a large effect when the instability is at $10^{11}\mathrm{GeV}$. There are two reasons for the size of this effect. In typical tunneling calculations, the value of the field at the center of the critical bubble is much larger than the point of the instability; in the SM case, this turns out to be numerically within an order of magnitude of the Planck scale. In addition, tunneling is an inherently nonperturbative phenomenon and may not be as strongly suppressed by inverse powers of the Planck scale. We include effective ${\mathrm{\Phi }}^6$ and ${\mathrm{\Phi }}^8$ Planck-scale operators and show that they can have an enormous effect on the tunneling rate.

Figure 1

The stability phase diagram according to the standard analysis, i.e. in the absence of new interactions at the Planck scale. The $M_H\text{−}{M}_t$ plane is divided into three sectors: absolute stability, metastability, and instability regions. The dot indicates $M_H\sim 125.7\mathrm{GeV}$ and $M_t\sim 173.34\mathrm{GeV}$. The ellipses take into account $1\sigma$, $2\sigma$, and $3\sigma$, according to the current experimental errors.

Figure 2

The potential in the Standard Model, for $M_H=125.7\mathrm{GeV}$ and $M_t=173.34\mathrm{GeV}$, is sketched (figure not to scale). The potential goes negative at a scale of $10^{11}\mathrm{GeV}$ and reaches a new minimum at roughly $10^{30}\mathrm{GeV}$. The tunneling through the barrier goes from the base of the arrow [$\varphi (r=\infty )$] to the tip [$\varphi (0)$], which turns out to be close to or above the Planck scale.

Figure 3

Profile of the bounce solution that enters in the computation of the electroweak vacuum lifetime $\tau$ for $M_H=125.7\mathrm{GeV}$ and $M_t=173.34\mathrm{GeV}$, the present central experimental values of $M_H$ and $M_t$. The value of the field at the center of the bounce ($r=0$) is ${\varphi }_b(0)=0.34M_P$, very close to the Planck scale.

Figure 4

Profile of the bounce solution that enters in the computation of the electroweak vacuum lifetime $\tau$ for values of $M_H$ and $M_t$ slightly different from those of Fig. 3. Actually, $\pm 2\sigma$ (current experimental errors) for $M_t$ in the left panel (with $M_H$ kept fixed to the central value $M_H=125.7\mathrm{GeV}$), and $\pm 2\sigma$ (current experimental errors) for $M_H$ in the right panel (with $M_t$ kept fixed to the central value $M_t=173.34\mathrm{GeV}$). As in Fig. 3, the values of the field at the center of the bounce, ${\varphi }_b(0)$, turn out to be very close to the Planck scale, sometimes even above this scale.

Figure 5

Profile of the bounce solution found with the forward-backward method described in Appendix pp2 for the potential of Eq. (22), with $\lambda =-0.01345$, ${\lambda }_6=-2$, and ${\lambda }_8=2.1$.

Figure 6

The solid line shows the potential $V_{\text{new}}(\varphi )$ of Eq. (22) with $\lambda =-0.01345$, ${\lambda }_6=-2$, and ${\lambda }_8=2.1$. The dotted line is the plot of the approximation to $V_{\text{new}}(\varphi )$ given in Eq. (33), with $\eta \simeq 0.7912M_P$ (determined self-consistently in the text), ${\lambda }_{\text{eff}}=\lambda +\frac{2}{3}{\lambda }_6\frac{{\eta }^2}{M_P^2}+\frac{1}{2}{\lambda }_8\frac{{\eta }^4}{M_P^4}=-0.4366$, and $\gamma =-{\lambda }_{\text{eff}}{\eta }^3{(\lambda {\eta }^3+{\lambda }_6\frac{{\eta }^5}{M_P^2}+{\lambda }_8\frac{{\eta }^7}{M_P^4})}^{-1}=-0.987$. As explained in the text, the latter provides a good approximation to $V_{\text{new}}(\varphi )$ for values of $\varphi$ around $\eta$. The dashed line is the potential in the absence of new physics interactions (${\lambda }_6=0$ and ${\lambda }_8=0$).

Figure 7

The analytical bounce solution, Eq. (34), when $V_{\text{new}}(\varphi )$ is approximated as in Eq. (33), for the values of the parameters considered in the text (see also Fig. 6). In particular, from Eq. (35), we have $\overline{r}=0.61M_P^{-1}$, and for the bounce size, $\overline{R}=5.33M_P^{-1}$.

Figure 8

Plot of $x^2\phi (x)$, for three different solutions of Eq. (25), with $\lambda =-0.01345$, ${\lambda }_6=-2$, and ${\lambda }_8=2.1$. The $x$ range goes from $x=x_{\min }=6\times 10^{-2}$ to $x=50$, although the numerical integration is performed up to $x_{\max }=10^2$. This figure well illustrates the forward shooting. Equation (25) is integrated starting with the initial values (b5) for $\phi (x_{\min })$ and ${\phi }^{\prime }(x_{\min })$ at $x=x_{\min }=6\times 10^{-2}$. The parameter $B$ is tuned until $x^2\phi (x)$ saturates to a plateau for values of $x$ greater than $x_{\min }$ and at least up to $x_{\max }$. We see that for $B=0.967$ (dotted line) and $B=0.9665$ (dashed line), $x^2\phi (x)$ diverges downwards and upwards, respectively. For $B=0.966777$ (solid line), the plateau is reached and our first approximation to the bounce is obtained.

Figure 9

The backward shooting with a plot of ${\phi }^{\prime }(x)/x$ for three different solutions of Eq. (25) ($\lambda =-0.01345$, ${\lambda }_6=-2$, ${\lambda }_8=2.1$). The $x$ range goes from $x_{\min }=10^{-2}$ to $x=0.15$, although the numerical integration is performed from $x_{\max }=10^2$ down to $x_{\min }=10^{-2}$. Equation (25) is integrated with initial values (b6) for $\phi (x_{\max })$ and ${\phi }^{\prime }(x_{\max })$. The parameter $A$ is tuned until ${\phi }^{\prime }(x)/x$ saturates to a plateau for small values of $x$. For $A=13.37292731$ (dotted line) and $A=13.37292732$ (dashed line), ${\phi }^{\prime }(x)/x$ diverges downwards and upwards, respectively. Finally, for $A=13.372927315215$ (solid line), the plateau is reached. We have then, to a very high degree of numerical accuracy, the bounce solution to our equation.

Figure 10

Diagrams leading to higher dimensional operators in the low energy theory. $\varphi$ is the Standard Model Higgs and $\mathrm{\Psi }$ is the 24-plet.