a space where $\bar{\theta} \neq (\theta)^*$. Since we want to work in Minkowski space-time and keep the usual complex conjugation, we use the $\star$-product (\[star\]) and decompose the $\star$-products of superfields into their irreducible components using the projectors defined in [@wessbook]. In that way (\[phistarphi\]) becomes $$\Phi\star \Phi = P_1(\Phi\star \Phi) + P_2 (\Phi\star \Phi) 
+ P_T(\Phi\star \Phi),$$ with antichiral, chiral and transversal projectors given by $$\begin{aligned}
P_1 &=& \frac{1}{16} \frac{D^2 \bar{D}^2}{\Box}, \label{P1}\\
P_2 &=& \frac{1}{16} \frac{\bar{D}^2 D^2 }{\Box}, \label{P2}\\
P_T &=& -\frac{1}{8} \frac{D \bar{D}^2 D}{\Box}. \label{PT}\end{aligned}$$

Finally, the deformed Wess-Zumino action is constructed requiring that the action is invariant under the deformed SUSY transformations (\[defsusytr\]) and that in the commutative limit it reduces to the undeformed Wess-Zumino action. In addition, we require that deformation is minimal: We deform only those terms that are present in the commutative Wess-Zumino model. We do not, for the time being, add the terms whose commutative limit is zero.

Taking these requirements into account we propose the following action $$\begin{aligned}
S &=& \int {\mbox{d}}^4 x\hspace{1mm} \Big{\{} \Phi^+\star\Phi\Big|_{\theta\theta\bar\theta\bar\theta}
+ \Big[ \frac{m}{2}P_2\big( \Phi\star\Phi\big)
\Big|_{\theta\theta} \nonumber\\
&& + \frac{\lambda}{3}P_2\Big( \Phi\star
P_2\big( \Phi\star\Phi\big)\Big) \Big |_{\theta\theta} + {\mbox{
c.c }} \Big] \Big{\}}, \label{S}\end{aligned}$$ where $m$ and $\lambda$ are real 