Vacuum transitions with the Gauss-Bonnet term in $D$ dimensions

In our previous paper [1,2], we proposed a probabilistic argument to explain the reason why the cosmological constant is very small in $4D$. We can ask a question: if the behavior of tunneling exponent $B$ can be generalized to $D$-dimension. Moreover, in higher dimensional theory motivated by string theory the Gauss-Bonnet term plays an important role. Therefore, in this paper, we generalize our result in [1,2] to arbitrary $D$ dimensions including the Gauss-Bonnet term. As a result, we have two main results. We find that the Euclidean action of the bounce, $B$, describing the decay of a de Sitter vacuum, is proportional to $k_{+}^{-(D-2)}$, which has a pole as $k_{+}^2\to 0$ where $k_{+}^2$ is the curvature of the parent vacuum. This result is similar to the result in $4D$. The other result is that we find a new decay channel, describing up-tunneling from anti-de Sitter into de Sitter. The meaning of this new decay channel in the string landscape should be explored in the future.