Renormalization of supersymmetric chiral theories in rational spacetime dimensions

We renormalize models with scalar chiral superfields with an odd superpotential to several orders in perturbation theory. These extensions of the cubic Wess-Zumino model are renormalizable in spacetime dimensions which are rational. When endowed with an $O(N)$ symmetry it is shown that they share the same property as their non-supersymmetric counterparts in that at a particular fixed point there is an emergent $OSp(1|n-1)$ symmetry, where $n$ is the power of the superpotential. This is shown at a loop order beyond that for which it was established in the parallel non-supersymmetric theory.