New Phases in O'Raifeartaigh-like Models and $R$-symmetry Breaking

We study $\mathcal{N}=1$ models in which supersymmetry and $R$-symmetry are broken spontaneously. We find that even the simplest well-known O’Raifeartaigh-like models exhibit previously unnoticed phases and dynamics. D-terms play an important role in the dynamics, creating a runaway behavior at tree-level which is stabilized radiatively at a large scale. This leads to a dynamically-generated hierarchy of scales. Additionally, this mechanism generically leads to a spontaneously broken $R$-symmetry without fine tuning the couplings of the theory. These are arguably the simplest known models that break the $R$-symmetry spontaneously.