Holomorphic selection rules, the origin of the mu term, and thermal inflation

When an abelian gauge theory with integer charges is spontaneously broken by the expectation value of a charge Q field, there remains a Z_Q discrete symmetry. In a supersymmetric theory, holomorphy adds additional constraints on the operators that can appear in the effective superpotential. As a result, operators with the same mass dimension but opposite sign charges can have very different coupling strengths. In the present work we characterize the operator hierarchies in the effective theory due to holomorphy, and show that there exist simple relationships between the size of an operator and its mass dimension and charge. Using such holomorphy-induced operator hierarchies, we construct a simple model with a naturally small supersymmetric mu term. This model also provides a concrete realization of late-time thermal inflation, which has the ability to solve the gravitino and moduli problems of weak-scale supersymmetry.