Inspired by the question How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?, I decided to come back to the first paper by Coleman, Fate of the false vacuum: Semiclassical theory, in order to better understand the limits of validity of the approximations made.
Going through the calculations, I stopped reflecting on the thin wall approximation. I found two “not-so-clear”(to my level of comprehension) points:
I read a lot of articles in the literature, but none of them describe in a clear manner why the instanton solution $\phi_1$ can be approximated by the expression (4.10) of the original paper. It is an important statement because equation (4.18) is often cited as the condition for the validity of the thin-wall approximation. Someone can help me in explaining this approximation?
Furthermore, I see that often is it said that inside the bubble there is the true vacuum, and outside there is the false one. However this is a huge simplification, since the wall is thin but finite. How can the bubble thickness be expressed in terms of the problem variables?