Singular instantons with finite action

Recently, it was shown that in the absence of gravity there exist non-$O(4)$-symmetric instanton solutions with finite action beyond Coleman's instantons. In this paper, focusing on the false-vacuum decay in a single scalar field in flat Euclidean space, we provide a general discussion on $O(4)$-symmetric instantons that are singular at the true-vacuum bubble. We find that, for the action to remain finite without introducing a UV cutoff, the potential must be unbounded from below, thereby evading Coleman's theorem. We then consider two explicit examples of such instantons and perturbatively analyze the dynamics of small deformations around them. We find that one of them does not allow regular deformations, which indicates that the $O(4)$ symmetric solution still gives the minimum action, while the other one is found to allow regular deformations that cost no additional action at second order in perturbation. The latter example opens up the possibility of the existence of non-linear non-$O(4)$-symmetric solutions with lower action if we allow singular instantons with finite action.