The non-anticommutative supersymmetric Wess-Zumino model

We discuss the non-anticommutative (N=1/2) supersymmetric Wess-Zumino model in four dimensions. Firstly we introduce differential operators which implement the non-anticommutative supersymmetry algebra acting on the component fields and action. Then we perform the renormalisation of the model up to two-loop order, including the complete set of terms necessary for renormalisability. We determine the form such an action must take in order to be preserved under renormalisation and show that only two such forms are possible, in one of which the action reduces to the ordinary N=1 action upon eliminating the auxiliary fields.