  We present a fully analytical calculation of the false vacuum decay rate for
a self-interacting scalar field in the thin-wall approximation. We obtain the
bounce solution, together with the Euclidean action, counter-terms and
renormalization group running, and we extract the functional determinant via
the Gel'fand-Yaglom theorem. Our procedure is valid for a generic spacetime
dimension $D$, and we provide an explicit finite renormalized decay rate in
$D=3,4$.
