are constant anticommuting parameters and the SUSY generators $Q^{\alpha}$ and $\bar{Q}_{\dot\alpha}$ are given by, $$\begin{aligned}
Q_\alpha = \p_\alpha
- i\sigma^m_{\ \alpha\dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\p_m, \quad \bar{Q}_{\dot{\alpha}} = -\bar{\p}_{\dot{\alpha}} + i\theta^\alpha \sigma^m_{\
\alpha\dot{\alpha}}\p_m
.\label{q,barq}\end{aligned}$$ Transformations (\[susytr\]) close in the algebra $$[\delta_\xi, \delta_\eta] = -2i(\eta\sigma^m \bar{\xi} 
- \xi\sigma^m \bar{\eta})\p_m. \label{xietaalg}$$ The product of two superfields is a superfield again; its transformation law is given by $$\begin{aligned}
\delta_\xi ({\rm F}\cdot {\rm G}) &=& \big(\xi Q + \bar{\xi}\bar{Q} \big) 
({\rm F}\cdot {\rm G}), \nonumber\\
&=& (\delta_\xi {\rm F})\cdot {\rm G} + {\rm F}\cdot(\delta_\xi {\rm G}).
\label{undefLrule}\end{aligned}$$ The last line is the undeformed Leibniz rule for the infinitesimal SUSY transformation $\delta_\xi$.

Nonanticommutativity is introduced following the twist approach [@NCbookMi]. For the twist $\cal{F}$ we choose $${\cal F} = e^{\frac{1}{2}C^{\alpha\beta}\p_\alpha \otimes\p_\beta
+ \frac{1}{2}\bar{C}_{\dot{\alpha}\dot{\beta}}\bar{\p}^{\dot{\alpha}}\otimes\bar{\p}^{\dot{\beta}} } ,\label{twist}$$ with the complex constant matrix $C^{\alpha\beta} = C^{\beta\alpha}$. Note that $C^{\alpha\beta}$ and $\bar{C}^{\dot{\alpha}\dot{\beta}}$ are related by the usual complex conjugation. It can be shown that the twist (\[twist\]) satisfies all necessary requirements [@chpr].

The inverse of the twist (\[twist\]) $${\cal F}^{-1} = e^{-\frac{1}{2}C^{\alpha\beta}\p_\alpha
\otimes\p_\beta -
\frac{1}{2}\bar{C}_{\dot{\alpha}\dot{\beta}}\bar{\p}^{\dot{\alpha}}\otimes\bar{\p}^{\dot{\beta}}
} ,\label{invtwist}$$ defines a new product in the algebra of superfields called the $\star$-product. For two arbitrary superfields ${\rm F}$ and ${\rm G}$ the $\star$-product is defined as 