Behaviors of two supersymmetry breaking scales in $N$ = 2 supergravity

We study the supersymmetry breaking patterns in four-dimensional $N$ = 2 gauged supergravity. The model contains multiple (Abelian) vector multiplets and a single hypermultiplet which parametrizes SO(4, 1)/SO(4) coset. We derive the expressions of two gravitino masses under general gaugings and prepotential based on the embedding tensor formalism, and discuss their behaviors in some concrete models. Then we confirm that in a single vector multiplet case, the partial breaking always occurs when the third derivative of the prepotential exists at the vacuum, which is consistent with the result of ref. [1], but we can have several breaking patterns otherwise. The discussion is also generalized to the case of multiple vector multiplets, and we found that the full ($N$ = 0) breaking occurs even if the third derivative of the prepotential is nontrivial.