Standard model vacuum decay in a de Sitter background

We present a calculation of thick-wall Coleman-de Luccia (CdL) bounces in the standard model effective potential in a de Sitter background. The calculation is performed including the effect of the bounce backreaction on the metric, which we compare with the case of a fixed de Sitter background, and with similar full-backreaction calculation in a model polynomial potential. The results show that the standard model potential exhibits nontrivial behavior: rather than a single CdL solution, there are multiple (nonoscillating) bounce solutions which may contribute to the decay rate. All the extra solutions found have higher actions than the largest amplitude solution, and thus would not contribute significantly to the decay rate, but their existence demonstrates that CdL solutions in the standard model potential are not unique, and the existence of additional, lower action, solutions cannot be ruled out. This suggests that a better understanding of the appearance and disappearance of CdL solutions in de Sitter space is needed to fully understand the vacuum instability issue in the standard model.

Figure 1

Polynomial potential of Eq. (34) for ${\lambda }_4=-1,{\lambda }_6=+1,m^2=0.1M_{\mathrm{P}}^2,M=M_{\mathrm{P}}$.

Figure 2

Overshoot/undershoot structure for the polynomial potential Eq. (34), above and below the critical threshold at $V_{0\mathrm{crit}}=0.01482M_{\mathrm{P}}^4$. Below the threshold there are undershoot solutions for $\varphi (0)$ sufficiently close to the top of the barrier: above it these solution disappear and all the solutions found are overshoots. This implies that no CdL-type solution exists above $H_{\mathrm{crit}}$, and the Hawking-Moss solution is the only contributor.

Figure 3

Plots of the CdL solutions for various values of $V_0$ in the polynomial potential, showing how they gradually approach the Hawking-Moss solution as $V_0\to V_{0\mathrm{crit}}$. For reference, $V_{0\mathrm{crit}}=0.014822M_{\mathrm{P}}^4$.

Figure 4

Left: Action of the CdL and Hawking-Moss solutions for the polynomial potential Eq. (34). The CdL action continuously approaches the flat false-vacuum $V_0=0$ result below $V_{0\mathrm{crit}}$, but appears to merge with the Hawking-Moss at the critical threshold. Right: Difference between Hawking-Moss and CdL decay exponents, showing that this falls to zero precisely at the critical threshold.

Figure 5

Scan through possible $\varphi (0)$ values in the standard model effective potential, giving the resulting $\varphi ({\chi }_{\max })$ at which $\dot{\varphi }({\chi }_{\max })=0$ if the solution is an undershoot. Straight lines are drawn between the start and ends of each bounce solution at an overshoot/undershoot boundary. When crossing the critical threshold of $H_{\mathrm{crit}}=1.1931\times {10}^8\mathrm{GeV}$, ($V_{0\mathrm{crit}}=7.203\times {10}^{-21}{M}_{\mathrm{P}}^4$) here is a dramatic change in the nature of the noninstanton solutions which start close to the barrier (region A, see Fig. 6), and for those near the top of the CdL bounce (region B, see Fig. 7).

Figure 6

Zoomed scan plot for region A on Fig. 5. An additional narrow overshoot region on the left is exaggerated to improve visibility.

Figure 7

Zoomed scan plot for region B on Fig. 5. Notice the narrow region of overshoots which is not large enough to be visible on Fig. 5.

Figure 8

Plot of the overshoot/undershoot behavior about the top of the barrier of the standard model effective potential, for $V_0=7.210\times {10}^{-21}{M}_{\mathrm{P}}^4$, $H_0=1.1937\times {10}^8\mathrm{GeV}>H_{\mathrm{crit}}$, confirming that solutions sufficiently close to the barrier are overshoots.

Figure 9

Left: CdL bounce solutions for $V_0=7.210\times {10}^{-21}{M}_P^4$ ($H_0=1.1937\times {10}^8\mathrm{GeV}$). Three solutions are similar to CdL solution 1 in their interior, but on the false-vacuum side they approach the barrier, while the CdL solution 1 reaches much further down. A further solution (CdL solution 2) straddles the barrier with small amplitude. Right: Zoomed view of bounce solutions around the barrier. Initial and final values of these solutions are given in Table 1. CdL solutions are numbered by order of the proximity of their false vacuum end value to the false vacuum, thus CdL 1 always corresponds to the largest amplitude solution. All solutions ultimately cross the barrier once.

Figure 10

Plot of the CdL and Hawking-Moss decay exponent, $B$, and its three components $B_1$, $B_2$, $B_3$ [see Eqs. (27)–(29) for definitions], for different values of $V_0$, compared to the flat false-vacuum ($V_0=0$) case.

Figure 11

Plot of $H_0{\chi }_{\max }(H_0)-\pi$ against $H_0$ for a range of CdL bounces in the standard model, computed numerically (crosses) and compared to the analytic prediction ${\chi }_{\max }\approx \frac{\pi }{H_0}-0.2559M_P^{-1}+\cdots$ of Eq. (40).