Four-fermion interactions and sgoldstino masses in models with a superlight gravitino

We discuss the role of the effective interactions among four matter fermions in supersymmetric models with a very light gravitino. We show that, from a field-theoretical viewpoint, no model-independent bound on the gravitino mass can be derived from such interactions. Making use of a naturalness criterion, however, we are able to derive some interesting but not very stringent bounds, complementary to those obtained from the direct production of supersymmetric particles. We also show that, generically, masses for the spin-0 partners of the goldstino (sgoldstinos) of the order of the gravitino mass and much smaller than squark and slepton masses do not obey a naturalness criterion.