False vacuum decay catalyzed by black holes

False vacuum states are metastable in quantum field theories, and true vacuum bubbles can be nucleated due to the quantum tunneling effect. It was recently suggested that an evaporating black hole (BH) can be a catalyst of bubble nucleations and dramatically shortens the lifetime of the false vacuum. In particular, in the context of the Standard Model valid up to a certain energy scale, even a single evaporating BH may spoil the successful cosmology by inducing the decay of our electroweak vacuum. In this paper, we reinterpret catalyzed vacuum decay by BHs, using an effective action for a thin-wall bubble around a BH to clarify the meaning of bounce solutions. We calculate bounce solutions in the limit of a flat spacetime and in the limit of negligible backreaction to the metric, where it is much easier to understand the physical meaning, and compare these results with the full calculations done in the literature. As a result, we give a physical interpretation of the enhancement factor: it is nothing but the probability of producing states with a finite energy. This makes it clear that all the other states such as plasma should also be generated through the same mechanism, and calls for finite density corrections to the tunneling rate, which tend to stabilize the false vacuum. We also clarify that the dominant process is always consistent with the periodicity indicated by the BH Hawking temperature after summing over all possible remnant BH masses or bubble energies, although the periodicity of each bounce solution as a function of a remnant BH can be completely different from the inverse temperature of the system, as mentioned in the previous literature.

Figure 1

Example of bubble potential as a function of its radius $r_{*}$.

Figure 2

Plot of $B(E)$ (blue line) and $-\mathrm{d}B(E)/\mathrm{d}E$ (pink line) as a function of $E$ for a thin-wall bubble in quantum field theory. Here, we have normalized quantities by those at the zero-temperature: $B_0=27{\pi }^2{\sigma }^4/2{\epsilon }^3$ and $r_{*,0}=3\sigma /\epsilon$. The results with these combinations are independent of $\sigma$ and $\epsilon$.

Figure 3

Left: $B_{\text{bubble}}$ (blue line) as a function of $\mathrm{\Delta }M(\equiv M_{+}-M_{-})$, and Right: $\mathrm{d}{B}_{\text{bubble}}/\mathrm{d}{M}_{-}$ (pink line) as a function of $\mathrm{\Delta }M$, for a thin-wall bubble in the Schwarzschild–de Sitter spacetime. In the left panel, we also plot $B$ (black thick line) and $B_{\text{boundary}}$ (yellow dashed line); in the right panel, $2\mathrm{\Delta }{\tau }_{-}$ (black dashed line) is shown. We take $R_{\mathrm{BH},+}=0.1r_{*,0}$, ${\mathrm{\Lambda }}_{+}=0$, and ${\mathrm{\Lambda }}_{-}=-0.03/r_{*,0}^2$. In this case, one can show that $l=10r_{*,0}$ and $\sigma l/M_{\mathrm{Pl}}^2=r_{*,0}/l=0.1\ll 1$. We take $r_{*,0}=1/M_{\mathrm{Pl}}$ to plot the results, but the $r_{*,0}$ dependence can be trivially factorized such that $B$, $B_{\mathrm{CdL}}\propto r_{*,0}^2$, $\mathrm{\Delta }M,M_{+}\propto r_{*,0}$, and $\mathrm{d}B/\mathrm{d}{M}_{-}\propto r_{*,0}$. Note that $\mathrm{d}{B}_{\text{bubble}}/\mathrm{d}{M}_{-}=-\mathrm{d}{B}_{\text{bubble}}/\mathrm{d}\mathrm{\Delta }M$.

Figure 4

Left: $B$ as a function of $\mathrm{\Delta }M(\equiv M_{+}-M_{-})$, and Right: $\mathrm{d}{B}_{\text{bubble}}/\mathrm{d}{M}_{-}$ as a function of $\mathrm{\Delta }M$, for a thin-wall bubble in the Schwarzschild–de Sitter spacetime. We take the parameters to be the same as in Fig. 3 except that we take $R_{\mathrm{BH},+}/r_{*,0}=0.01$ (red line), 0.1 (pink line), 0.2 (violet line), and 0.3 (blue line).

Figure 5

Relation between $2\mathrm{\Delta }\tilde{\lambda }$ and $k_2$ for static solutions. The red dashed line represents the asymptotic value of $2\mathrm{\Delta }\tilde{\lambda }$ for $k_2\to \pm \infty$. The green dashed line represents an upper bound of $k_2$ for the case of ${\mathrm{\Lambda }}_{+}=0$.

Figure 6

Plot of $B(E_{*})$ (left panel) and $-\mathrm{d}B(E_{*})/\mathrm{d}{E}_{*}$ (right panel) as a function of $E_{*}$ for a thin-wall bubble in the Schwarzschild spacetime without the backreaction of the bubble to the metric. In the left panel, we also plot $B+E_{*}/T_{\mathrm{BH}}$ (black thick line) and $E_{*}/T_{\mathrm{BH}}$ (yellow dashed line). We take $R_{\mathrm{BH},+}=0.1r_{*,0}$ and $l=10r_{*,0}$. We also take $r_{*,0}=1/M_{\mathrm{Pl}}$ though the $r_{*,0}$ dependence can be trivially factorized as $\mathrm{d}B/\mathrm{d}{E}_{*}\propto r_{*,0}$. One can see that the result almost coincides with Fig. 3.

Figure 7

Plot same as Fig. 6 but with $R_{\mathrm{BH},+}/r_{*,0}=0.01$ (red line), 0.1 (pink line), 0.2 (violet line), and 0.3 (blue line). One can see that the result almost coincides with Fig. 4.

Figure 8

Schematic figures that describe the excitation of a sphaleron bubble in the thermal plasma with an infinite volume (left panel) and in the thermal plasma with a finite volume whose size is smaller than the sphaleron solution (right panel). The process that the bubble nucleation at the BH radius by the Hawking radiation corresponds to is described in the right panel.