Effective Lagrangians and light gravitino phenomenology

We construct the low-energy effective Lagrangian for a light gravitino coupled to the fields of the minimal supersymmetric standard model with soft supersymmetry breaking. Our effective Lagrangian is written in terms of the spin-$\frac{1}{2}$ Goldstino (the longitudinal component of the gravitino) transforming under a nonlinear realization of supersymmetry. The Goldstino is derivatively coupled. We use this Lagrangian to place bounds on the supersymmetry-breaking scale $\sqrt{F}$ from astrophysics, cosmology, and particle physics. The leading interactions of the Goldstino at low energies arise from Fayet-Iliopoulos terms induced by supersymmetry breaking, which give rise to a coupling of a single photon or $Z$ to Goldstino pairs. We argue that it is unnatural for the induced Fayet-Iliopoulos term to have a coefficient much less than $(50\mathrm{}\mathrm{GeV}{)}^2.$ On this basis, the most stringent limits come from the cooling of supernova SN 1987A, which gives $\sqrt{F}\gtrsim 240\mathrm{GeV}$ or, equivalently, $m_{3/2}\gtrsim 1.4\times {10}^{-5}\mathrm{eV}.$