.\label{F2'}\end{aligned}$$

The one-loop effective action is then $$\G=S_0+S_{int}+\frac{i}{2}\tr\log(1+{\mathcal M}^{-1}{\mathcal V}).
\label{ED-opsta}$$ The last term in (\[ED-opsta\]) is the one-loop correction to the effective action and ${\cal M}^{-1}$ is the inverse of (\[M\]) given by $${\mathcal M}^{-1}=\left(\begin{array}{cc} A & B\\ \bar{B} &
\bar{A} \end{array}\right)=\left(\begin{array}{cc}
\frac{mD^2}{4\square (\square-m^2)} &
\frac{D^2\bar{D}^2}{16\square (\square-m^2)}+\frac{\bar{D}^2 D^2-2\bar{D}D^2\bar{D}}{16\xi\square^2}\\
\frac{\bar{D}^2 D^2}{16\square
(\square-m^2)}+\frac{D^2\bar{D}^2-2D\bar{D}^2 D}{16\xi\square^2} &
\frac{m\bar{D}^2}{4\square (\square-m^2)}
\end{array}\right) .\label{Minv}$$ Expansion of the logarithm in (\[ED-opsta\]) leads to the one-loop corrections $$\Gamma_1=\frac{i}{2}\tr\sum_{n=1}^{\infty}
\frac{(-1)^{n+1}}{n}({\mathcal
M}^{-1}\mathcal{V})^{n}=\sum^{\infty}_{n=1} \Gamma_{1}^{(n)}.
\label{1LoopExp}$$

One-point and two-point functions
=================================

The first term in the expansion (\[1LoopExp\]) gives the divergent part of the one-point functions, the tadpole contribution. We obtain $$\Gamma^{(1)}_{1}=\frac{i}{2} \tr ({\mathcal
M}^{-1}{\mathcal V})=\frac{i}{2} \tr
\big( AF+\bar{A}\bar{F}\big) = 0. \label{1point}$$ Therefore just like in the commutative Wess-Zumino model there is no tadpole contribution.

Next we calculate the divergent part of the two point functions. It is given by $$\begin{aligned}
\Gamma^{(2)}_{1} &=& -\frac{i}{4} \tr ({\mathcal
M}^{-1}{\mathcal V})^{2} \nonumber\\
&=& -\frac{i}{4} \tr \big( AFAF + 2
\bar{B}FB\bar{F} + \bar{A}\bar{F}\bar{A}\bar{F} \big) .\label{2point1}\end{aligned}$$ First we calculate the $AFAF$ contributions. They are given by (remember that $F^{(i)}$ is the i-th element of the expansions (\[F1\]) and (\[F2’\])) $$\begin{aligned}
{\rm Tr} (AF^{(0)}AF^{(0)})&=&0 ,\nonumber\\
{\rm Tr} (AF^{(1)}AF^{(0)})\bigg |_{d.p.}&=& -\frac{i m^{2}
\lambda^{2}K^{m}_{\ a}K^{*na}}{6\pi^{2}\varepsilon} \int 
{\mbox{d}}^4 x\hspace{1mm} 
\partial_{m}A\partial_{n}A ,\nonumber\\
