Generalized Gaugino Condensation in Super Yang-Mills Theories: Discrete R-Symmetries and Vacua

One can define generalized models of gaugino condensation as theories which dynamically break a discrete R-symmetry, but do not break supersymmetry. We consider general examples consisting of gauge and matter fields, and the minimal number of gauge singlet fields to avoid flat directions in the potential. We explore which R-symmetries can arise, and their spontaneous breaking. In general, we find that the discrete symmetry is $\mathbb{Z}_{2b_0R}$ and the number of supersymmetric vacua is $b_0$, where $b_0$ is the coefficient of the one-loop beta function. Results are presented for various groups, including $SU(N_c), SO(N_c), Sp(2N_c)$, and $G_2$, for various numbers of flavors, $N_f$, by several methods. This analysis can also apply to the other exceptional groups, and thus all simple Lie groups. We also comment on model building applications where a discrete R-symmetry, broken by the singlet vevs, can account for $\mu$-type terms and allow a realistic Higgs spectrum naturally.