constants. To rewrite (\[S\]) in terms of component fields and as compact as possible, we introduce the following notation $$\begin{aligned}
C_{\alpha\beta}&=& K_{ab}(\sigma^{ab}\varepsilon)_{\alpha\beta},\label{n1}\\
\bar{C}_{\dot{\alpha}\dot{\beta}}&=&
K^*_{ab}(\varepsilon\bar{\sigma}^{ab})_{\dot{\alpha}\dot{\beta}},\label{n2}\end{aligned}$$ where $K_{ab}=-K_{ba}$ is an antisymmetric self-dual complex constant matrix. Then we have $$\begin{aligned}
C^2 = 2K_{ab}K^{ab}, && \bar{C}^2 = 2K^*_{ab}K^{*ab}, \quad
K^{ab}K^*_{ab}= 0.\label{n4} \\
K^*_{cd}K_{ab}\big( \sigma^n\bar{\sigma}^{cd}\bar{\sigma}^m
\sigma^{ab}\big)_\alpha^{\ \beta} &=&
-4\delta^\beta_\alpha K^{ma}K^{*n}_{\ \ a} + 8 K^{ma}K^{*nb}(\sigma_{ba})_\alpha^{\ \beta} ,\label{n5}\\
C^{\alpha\beta}\bar{C}^{\dot{\alpha}\dot{\beta}} \sigma^m_{\
\alpha\dot{\alpha}}\sigma^l_{\ \beta\dot{\beta}} &=& 8K^{am}K^{*\
l}_a. \label{n6}\end{aligned}$$ Using these formulas and expanding (\[S\]) in component fields we obtain $$\begin{aligned}
S &=& \int {\mbox{d}}^4 x\hspace{1mm} \Big{\{}
\Phi^+\star\Phi\Big|_{\theta\theta\bar\theta\bar\theta}\nonumber\\
&& + \Big[ \frac{m}{2}P_2\big( \Phi\star\Phi\big)
\Big|_{\theta\theta} + \frac{\lambda}{3}P_2\Big( \Phi\star P_2\big(
\Phi\star\Phi\big)\Big) \Big |_{\theta\theta} + {\mbox{ c.c }} \Big] 
\Big{\}}\nonumber\\
&=&
\int {\mbox{d}}^4 x\hspace{1mm} \Big{\{}
A^*\Box A + i(\p_m\bar\psi)\bar{\sigma}^m\psi + F^*F\nonumber\\
&& + \Big[ \frac{m}{2}\big( 2AF - \psi\psi \big) 
+\lambda\big( FA^2 - A\psi\psi  \big)\nonumber\\
&& -\frac{\lambda}{3}\Big(K^{m}_{\ a}K^{*na}\psi(\p_n\psi) - 2K^m_{\
a}K^{*n}_{\ \ b}(\p_n\psi)\sigma^{ba}\psi\Big) (\p_m A)\nonumber\\
&& -\frac{\lambda}{12}K^{mn}K_{mn}F^3
+\frac{\la}{6}\KK F(\p_m A)(\p_n A) \label{Sincompfields}\\
&& +\frac{\la}{3}\KK F\frac{1}{\Box}\p_m\Big( (\p_n A)\Box
A\Big) +\frac{\la}{192}\KKt\KKtc  F(\Box A)^2  + c.c \Big]
\Big{\}}. \nonumber\end{aligned}$$ Partial integration was used to rewrite some of the terms in (\[Sincompfields\]) in a more compact way. Note that this it the complete action; there are no higher order terms in the deformation parameter $K^{ab}$. However, for 