The unconstrained superfields are determined up to a gauge transformation $$\S \to \S+\bar
D_{\dot\a}\bar\Lambda^{\dot\a}, \quad 
\S^+ \to \S^++D^{\a}\Lambda_{\a} , \label{gf1}$$ with the gauge parameter $\Lambda$. This additional symmetry has to be fixed, so we add a gauge-fixing term to the action. For the gauge functions we choose $$\chi_\a = D_\a\S ,\quad \bar \chi_{\dot\a} = \bar
D_{\dot\a}\S^+ \ . \label{gf2}$$ The product $\d(\chi)\d(\bar\chi)$ in the path integral is averaged by the weight $e^{-i\xi \int d^8z\bar{f}Mf}$: $$\int {\mbox{d}}f {\mbox{d}}\bar f \hspace{1mm}
\d(\chi_\a-f_\a)\d(\bar\chi^{\dot\a}-\bar f^{\dot\a})e^{-i\xi\int
d^8 z\bar f^{\dot\a}M_{\dot\a \a}f^\a}$$ where $$\bar f^{\dot \a}M_{\dot\a\a}f^\a=\frac{1}{4}\bar f^{\dot
\a}(D_\a\bar D_{\dot\a}+\frac{3}{4}\bar D_{\dot\a}D_\a)f^\a$$ and the gauge-fixing parameter is denoted by $\xi$. The gauge-fixing term becomes $$S_{gf}=-\xi\int
{\mbox{d}}^8 z\hspace{1mm} (\bar D_{\dot\a}\bar\S)(\frac{3}{16}\bar
D^{\dot\a}D^{\a}+\frac 14 D^\a\bar D^{\dot\a})(D_\a\S) .$$ One can easily show that the ghost fields are decoupled.

After the gauge-fixing, the part of the classical action quadratic in quantum superfields is given by $$S^{(2)} = S^{(2)}_0 + S^{(2)}_{int}, \label{S2}$$ with $$S^{(2)}_0=\frac{1}{2}\int {\mbox{d}}^8 z\hspace{1mm} 
\left(\begin{array}{cc}
\Sigma & \Sigma^{+} \end{array} \right){\mathcal
M} \left(\begin{array}{l}\Sigma\\
\Sigma^{+} \end{array} \right) \label{Snula}$$ and $$S^{(2)}_{int}=\frac{1}{2}\int {\mbox{d}}^8 z{\mbox{d}}^8 z'\hspace{1mm} 
\left(\begin{array}{cc}
\Sigma & \Sigma^{+} \end{array} \right)(z){\mathcal
V}(z,z') \left(\begin{array}{l}\Sigma\\
\Sigma^{+} \end{array} \right) (z'). \label{Sint}$$ Kinetic and interaction terms are collected in the matrices $\cal{M}$ and $\cal{V}$ respectively. The matrix $\cal{M}$ is given by 