Bifurcation of a periodic instanton in a decay-rate transition

We investigate a bifurcation of a periodic instanton in the Euclidean action-temperature diagram in quantum mechanical models. It is analytically shown that multiple zero modes of the fluctuation operator should arise at bifurcation points. This fact is used to derive a condition for the appearance of bifurcation points in the action-temperature diagram. This condition enables one to compute the number of bifurcation points for a given quantum mechanical system and hence, to understand the whole behaviour of the decay rate. It is explicitly shown that the previous criterion derived by nonlinear perturbation or negative-mode considerations is a special limit of our case.