Negative modes and the decay-rate transition

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Abstract
We investigate the relationship between the number of negative modes around the periodic instanton solution and the type of decay-rate transition. It is shown that for the case of the first-order decay-rate transition the lowest positive mode in the low-energy periodic instanton becomes an additional negative mode in the high-energy regime, while in the second-order case there is only one negative mode in the full range of energy. This kind of analysis of the negative modes makes it possible to derive the sufficient criterion for the first-order transition.
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