<PERSON>'s theorem on a Four point QED Green function in 2nd order perturbation theory
Problem 13.1 in Mandl & Shaw's QFT. I need to calculate the second order contributions to the four point Green function
$$ \langle A^{\mu}(x_1)A^{\nu}(x_2)\psi(x_3)\bar{\psi}(x_4) \rangle, $$
where $ A^{\mu} $ and $ \psi $ are electromagnetic and fermion field operators. With the method I am using, I will need to evaluate the free field Green function
$$ \frac{i^2}{2}\langle \mathcal{L}_I(y)\mathcal{L}_I(z) A^{\mu}(x_1)A^{\nu}(x_2)\psi(x_3)\bar{\psi}(x_4) \rangle_0, $$
where $\mathcal{L}I(x) = e\bar{\psi}(x)\gamma{\alpha}A^{\alpha}(x)\psi(x)$ is the interaction part of the QED Lagrangian. Substituting the explicit expressions for the interaction Lagrangians and rearranging some terms gives
$$ \frac{(ie)^2}{2}\big\langle A^{\alpha}(y) A^{\beta}(z) A^{\mu}(x_1)A^{\nu}(x_2) \big[\bar{\psi}\gamma_{\alpha}\psi\big]y \big[\bar{\psi}\gamma{\beta}\psi\big]_z \psi(x_3)\bar{\psi}(x_4) \big\rangle_0 $$
Applying <PERSON>'s theorem makes it possible to express this Green function in terms of non-vanishing contractions of the field operators, and thus in terms of the photon and fermion Feynman propagators $D_F^{\alpha\beta}$ and $S_F$. I think applying <PERSON>'s theorem is difficult and here I am not sure if am proceeding correctly. 324
My approach to this problem was to find the non-vanishing contractions of the photon and fermion field operators on their own, and then combine the results.
For the photon field operators I find three combinations which yield three terms, each with two $D_F$ propagator functions
$$ i^2 \big(D_F^{\alpha\beta}(y-z)D_F^{\mu\nu}(x_1-x_2) + D_F^{\alpha\mu}(y-x_1)D_F^{\beta\nu}(z-x_2) + D_F^{\alpha\nu}(y-x_2)D_F^{\beta\mu}(z-x_1)\big).$$
For the fermion field operators I also find three combinations which yield the three terms, each with three $S_F$ propagator functions
$$ i^3 \big( S_F(y-x_4)S_F(z-y)S_F(x_3-z) - S_F(y-z)S_F(x_3-y)S_F(z-x_4) - S_F(y-z)S_F(z-y)S_F(x_3-x_4) \big). $$
Since the photon and fermion field operator contractions give three terms each, there will be a total number of 9 terms after multiplying everything together. I am supposed to draw the corresponding configuration space <PERSON> diagrams, does this mean I have to draw 9 separate diagrams, or can any of these terms be combined in some way? I am also very unsure about the somewhat ad hoc approach to finding these terms. Is there in general a more structured way to use <PERSON>'s theorem where I would feel less worried about missing contributions? The examples in my book are always for a small number of field operators which usually only yield a couple non-vanishing contractions.
Edit: I should add I am looking for the terms that give contributions to the connected Green function. How can I see which terms correspond to disconnected diagrams? 324