Musings

Tunnelling branes

It’s well-known that, in certain respects, low-energy brane dynamics can differ markedly from naïve field theory expectations. An important example is D-brane inflation, where the DBI action allows much steeper potentials to be compatible with inflation.

There’s a very interesting recent paper which argues that the DBI action, similarly, modifies the Coleman-de Luccia tunnelling rate, governing the decay of the false vacuum by bubble nucleation. The main effect is that the DBI action modifies the domain wall tension

where the potential has a true minimum, $V(\phi_-)=V_-$ and a local minimum, $V(\phi_+)=V_+$. $$V_0(\phi) = V(\phi) + V_+/f(\phi)$$ $f(\phi)$ is the warp factor, which appears in the Euclidean DBI action as $$S_E = 2\pi^2 \int \rho^3 d\rho \left(\frac{1}{f(\phi)} \sqrt{1+f(\phi) \dot{\phi}^2}- \frac{1}{f(\phi)} +V(\phi)\right)$$ Dot represents derivative with respect to $\rho$ and we’ve taken an $O(4)$ symmetric ansatz. In the absence of warping, $f(\phi)={\alpha'}^2$. In the limit $f\to 0$, we recover the usual thin-wall formula. But as $V_0 f\to 2$, the tension can be much smaller than expected.

In the absence of gravity, the decay rate per unit volume $$\Gamma/V \sim e^{- \pi^2 T \overline{\rho}^3/2} = e^{-27\pi^2 T^4/2\epsilon^3}$$ where $\epsilon = V_+-V_-$ and the radius of the bubble is $\overline{\rho}=3T/\epsilon$. So (1) can have a huge effect on the lifetime of the false vacuum.

Gravitational corrections modify $\overline{\rho}$ $$\overline{\rho} = \frac{3 T M_p}{\sqrt{\epsilon^2 M_p^2 +3 T^2 (V_++V_-)/2}}$$ and contribute an “extra” gravitational contribution in the exponent, just as is the usual case. But the qualitative effect remains: the lifetime of these metastable open string vacua can be *much* shorter than the naïve CDL result.