noncommutative and/or a nonanticommutative space and to formulate a physical model on it, see [@NCbooks] and [@NCreview]. We shall follow the approach of [@NCbookMi] and [@miSUSY].

Let us first fix the notation and the conventions which we use. The superspace is generated by supercoordinates $x^{m}$, $\theta^{\alpha}$ and $\bar{\theta}_{\dot{\alpha}}$ which fulfill $$\begin{aligned}
\lbrack x^m, x^n \rbrack = \lbrack x^m,
\theta^\alpha \rbrack = \lbrack x^m, \bar{\theta}_{\dot\alpha} \rbrack = 0 ,\quad
\{ \theta^\alpha , \theta^\beta\} = \{ \bar{\theta}_{\dot\alpha}
, \bar{\theta}_{\dot\beta}\} = \{ \theta^\alpha ,
\bar{\theta}_{\dot\alpha}\} = 0 , \label{undefsupsp}\end{aligned}$$ with $m=0,\dots ,3$ and $\alpha, \beta =1,2$. To $x^m$ we refer as bosonic and to $\theta^\alpha$ and $\bar{\theta}_{\dot\alpha}$ we refer as fermionic coordinates. We work in Minkowski space-time with the metric $(-,+,+,+)$ and $x^m x_m = -(x^0)^2 + (x^1)^2 + (x^2)^2 +
(x^3)^2$.

A general superfield ${\rm F}(x,\theta, \bar{\theta})$ can be expanded in powers of $\theta$ and $\bar{\theta}$, $$\begin{aligned}
{\rm F}(x, \theta, \bar{\theta}) &=&\hspace*{-2mm} f(x) + \theta\phi(x)
+ \bar{\theta}\bar{\chi}(x)
+ \theta\theta m(x) + \bar{\theta}\bar{\theta} n(x) + \theta\sigma^m\bar{\theta}v_m(x)\nonumber\\
&& + \theta\theta\bar{\theta}\bar{\lambda}(x) + \bar{\theta}\bar{\theta}\theta\varphi(x)
+ \theta\theta\bar{\theta}\bar{\theta} d(x) .\label{F}\end{aligned}$$ Under the infinitesimal ${\cal N} = 1$ SUSY transformations it transforms as $$\delta_\xi {\rm F} = \big(\xi Q + \bar{\xi}\bar{Q} \big) {\rm F}, \label{susytr}$$ where $\xi^{\alpha}$ and $\bar{\xi}_{\dot{\alpha}}$ 