A note on supersymmetric WZW term in four dimensions

We reconsider the supersymmetric Wess-Zumino-Witten (SWZW) term in four dimensions. It has been known that the manifestly supersymmetric form of the SWZW term includes derivative terms on auxiliary fields, the highest components of chiral superfields, and then we cannot eliminate them by their equations of motion. We discuss a possibility for the elimination of such derivative terms by adding total derivative terms. Although most of the derivative terms can be eliminated in this way, we find that all the derivative terms can be canceled, if and only if an anomalous term in SWZW term vanishes. As a by-product, we find the first example of a higher derivative term free from such problem.