simplicity in the following sections we shall keep only terms up to second order in the deformation parameter.

One-loop effective action
=========================

In this section we look at the quantum properties of our model. We calculate the one-loop divergent part of the one-point and the two-point functions up to second order in the deformation parameter. We use the background field method, dimensional regularization and the supergraph technique. The supergraph technique significantly simplifies calculations. However, we cannot directly apply this technique since our action (\[Sincompfields\]) is not written as an integral over the whole superspace and in terms of the chiral field $\Phi$ and its derivatives. This is a consequence of the particular deformation (\[F\]) and differs from [@D-def-us].

In order to be able to use the supergraph technique we notice the following: From (\[chiral\]), see also [@wessbook], it follows that the fields $A$, $\psi$ and $F$ can be written as $$A=\Phi|_{\theta, \bar{\theta} =0}, \quad
\psi_{\alpha}=\frac{1}{\sqrt{2}}D_{\alpha}\Phi|_{\theta, \bar{\theta} 
=0}, \quad
F=-\frac{1}{4}D^{2}\Phi|_{\theta, \bar{\theta} =0}. \label{cfieldsPhi}$$ Inserting this in (\[Sincompfields\]) we obtain $$\begin{aligned}
S &=& \int {\mbox{d}}^8 z\hspace{1mm} \Big{\{} \Phi^+\Phi +
\Big[ -\frac{m}{8}\Phi\frac{D^2}{\Box}\Phi -\lambda\Phi^2\frac{D^2}{12\Box}
\Phi\nonumber\\
&& +\lambda\theta\theta\bar{\theta}\bar{\theta} \Big( 
\frac{1}{768} K^{mn}K_{mn}(D^{2}\Phi)^{3} \nonumber \\
& &-\frac{1}{6}\left(K^{m}_{\ a}K^{*na}
(D^{\alpha}\Phi)(\partial_{n}D_{\alpha}\Phi)- 2K^{m}_{\ a}
 K^{*n}_{\ b}
(\partial_{n}D^{\alpha}\Phi)(\sigma^{ba})_{\alpha}^{\ \beta}D_{\beta}\Phi\right)(\partial_{m}\Phi)\nonumber\\
& & 