2. False vacuum decay

Our effective description is general enough to include bubble dynamics in the absence of gravity as well. The limit $G _{N} \rightarrow 0$ represent not only a necessary consistency check of our model, but also a worth investigating special case by itself.

The correct limiting procedure requires to express the two cosmological constants in terms of the corresponding vacuum energy densities $\epsilon _{in}$, $\epsilon _{out}$: $${\Lambda _{in}= 8 \pi G _{N} \epsilon _{in}}$$, $${\Lambda _{out}= 8 \pi G _{N} \epsilon _{out}}$$. In a single Minkowski domain $\sigma _{in}= \sigma _{out}= 1$. Then, by expanding $L ^{\mathrm{eff}}$ up to the first order in $G _{N} / R ^{2}$ we get

$$L^{\mathrm{eff}} = - 4 \pi \rho R ^{2} - \frac{4 \pi}{3}... ... \epsilon \equiv \epsilon _{in} - \epsilon _{out} \quad ,$$ (8)

and represents the minisuperspace approximation of the gauge action for the membrane

 

$$I = - \rho \int d ^{3} \xi \sqrt{-\gamma} - \frac{e}{3!} \int d ^{4} x \, J ^{\mu \nu \rho} A _{\mu \nu \rho}$$

$$- \frac{1}{2 \cdot 4!} \int d ^{4} x F _{\mu \nu \rho \sigma} F ^{\... ...ial _{\mu} \left( F ^{\mu \nu \rho \sigma} A _{\nu \rho \sigma} \right) \quad .$$ (9)

where, $\gamma$ is the determinant of the induced metric on the $2$-brane world tube $x ^{\mu} = x ^{\mu} ( \xi ^{a})$, $a=0,1,2$, and $J ^{\mu \nu \rho} (x)$ is the $2$-brane current. The last term in (9) is a total divergence ensuring that variations of the gauge potential $A _{\mu \nu \rho}$ will not produce unusual boundary terms due to the presence of the $2$-brane world tube. The corresponding hamiltonian is

 

$${\mathcal{H}} = 4 \pi \rho R ^{2} + \frac{4 \pi}{3} \Delta \epsilon R ^{3} \, \sqrt{1 - (3 P _{R}/ 4 \pi \rho R ^{3}) ^{2}} \quad ,$$ (10)

$$P _{R} = - \frac{4 \pi}{3} \Delta \epsilon R ^{3} \frac{\dot{R}}{\sqrt{1 + \dot{R}^{2}}} \quad .$$ (11)

The classical trajectories describing true vacuum bubbles are solutions of the hamiltonian constraint $${{\mathcal{H}} = 0}$$ stating that the total mass energy of a vacuum bubble is vanishing. From equation (10) we see that classical solutions, corresponding to positive surface tension bubbles, are allowed only if $\Delta \epsilon < 0$, i.e. the internal energy density (of the true vacuum) must be smaller than the external energy density (of the false vacuum), the net amount of energy released in the transition being converted into (positive) kinetic energy of the bubble wall. The semi-classical picture of the true vacuum domain nucleation corresponds to a classically forbidden motion. The classically unphysical tunneling trajectory is a solutions of the euclidean, equation of motion obtained via the Wick rotation $${\tau _{E} \equiv i \tau}$$, $${P _{E}\equiv i P}$$

$${\mathcal{H}} _{E} = 0 \quad \Rightarrow \quad P _{E}= 4 ... ...uad , \qquad R _{0} = \frac{3 \rho}{\Delta \epsilon} \quad .$$ (12)

Then, we can use the classical solution (12) for $P _{E}$ in the WKB integral for the calculation of the nucleation probability through tunnel effect:

$$B = 2 \int _{0} ^{R _{0}} dR P _{E}(R) = 8 \pi \rho \i... ...right) ^{1/2} = \frac{\pi ^{2}}{2} \rho R ^{3} _{0} \quad .$$ (13)

The usual result (13) is in agreement with the Coleman-DeLuccia $B$ decay coefficient [18].