-
\frac{1}{24} K^{m}_{\ a}K^{*na}(D^{2}\Phi)(\partial_{m}\Phi)
 (\partial_{n}\Phi) \nonumber \\ 
& & -\frac{1}{12}K^{m}_{\ a} K^{*na}(D^{2}\Phi)\frac{1}{\Box}
 \partial_{m}\big( (\partial_{n}\Phi)(\square\Phi) \big) \Big)
 +c.c. \Big] \Big{\}}, \label{SGaction}\end{aligned}$$ with $f(x)\frac{1}{\Box}g(x) = 
f(x)\int {\mbox{d}}^4 y\hspace{1mm} G(x-y)g(y)$. Notice that two spurion fields $$U_{(1)\ \ ab}^{mn} = K^{m}_{\ a} K^{*n}_{\ b}\theta\theta\bar{\theta}\bar{\theta}, \quad
U_{(2)} = K^{mn}K_{mn}\theta\theta\bar{\theta}\bar{\theta}$$ appear in (\[SGaction\]). This is a consequence of rewriting the action (\[Sincompfields\]) as an integral over the whole superspace.

Now we can start the machinery of the background field method. First we split the chiral and antichiral superfields into their classical and quantum parts $$\begin{aligned}
\Phi\to \Phi+ \Phi_q ,\quad \Phi^+ \to \Phi^{+}+\Phi^{+}_q\end{aligned}$$ and integrate over the quantum superfields in the path integral. Since $\Phi_q$ and $\Phi_q^{+}$ are chiral and antichiral fields, they are constrained by $$\bar D_{\dot\a}\Phi_q= D_{\a}\Phi_q^+=0 .$$ To simplify the supergraph technique we introduce the unconstrained superfields $\S$ and $\S^+$, $$\Phi_{q} = -\frac{1}{4}\bar{D}^2\Sigma ,\quad
\Phi_{q}^{+} = -\frac{1}{4}D^2\Sigma^{+}\ .\label{abel-gaug-tr}$$ Note that we do not express the background superfields $\Phi$ and $\Phi^+$ in terms of $\S$ and $\S^+$, only the quantum parts $\Phi_q$ and $\Phi_q^+$. After the integration of quantum superfields, the result is expressed in terms of the (anti)chiral superfields. This is a big advantage of the background field method and of the supergraph technique. 