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On the Imaginary Part of the Effective Action in de Sitter Spacetime with Different Regularization Schemes

The imaginary part of the effective action encodes vacuum instability and particle production in the background field. Two standard approaches are commonly used to derive it: the Bogoliubov method and the Green's function method, which are usually expected to agree. However, in de Sitter spacetime they yield different results. We revisit this problem by introducing explicit time and momentum cutoffs in the Green's function representation of the effective action. The apparent discrepancy is found to be due to the different limiting procedures in regularization, which reproduces the Bogoliubov result and the Green's function result respectively. Therefore, the two approaches are understood to be different regularization limits of the same expression, which clarifies the origin of their disagreement.