action we used the background field method and the supergraph technique. Like in the commutative Wess-Zumino model, there is no tadpole contribution. There is no mass counterterm which is again the same as in the undeformed Wess-Zumino model. However, the divergent part of the two-point function cannot be canceled and we have to conclude that our model is not renormalizable. Calculating divergent parts of the three-point and higher functions does not make sense and it is technically very demanding.

Having in mind results of [@on-shell], we also investigated on-shell renormalizability of our model. In general, on-shell renormalizability leads to a one-loop renormalizable $S$-matrix. On the other hand, one-loop on-shell renormalizable Green functions may spoil renormalizability at higher loops. After using the equations of motion which follow form the action (\[SGaction\]) to obtain the on-shell divergent terms we see that the divergences in the two-point function remain and therefore the model is also not on-shell renormalizable.

In our previous work [@D-def-us] we had a similar problem, a deformed model which was not renormalizable. To obtain a renormalizable model we had to relax the condition of minimality of deformation and to include non-minimal terms. Also, in [@Penati] new terms of the form 