  In supersymmetric extensions of the Standard Model with a very light
gravitino, the effective theory at the weak scale should contain not only the
goldstino, but also its supersymmetric partners, the sgoldstinos. In the
simplest case, the goldstino is a gauge-singlet and its superpartners are two
neutral spin-0 particles, S and P. We study possible signals of massive
sgoldstinos at hadron colliders, focusing on those that are most relevant for
the Tevatron. We show that inclusive production of sgoldstinos, followed by
their decay into two photons, can lead to observable signals or to stringent
combined bounds on the gravitino and sgoldstino masses. Sgoldstino decays into
two gluon jets may provide a useful complementary signature.
