Calculating a four-point Green function using <PERSON>'s theorem (problem 12.1 in Mandl & Shaw)
In problem 12.1 in Quantum Field Theory, Mandl & Shaw the aim is to calculate the four point green function
$$ G^{\mu\nu}(x,y,z,w) = \frac{\langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S\big)| 0 \rangle}{\langle 0 | S | 0 \rangle} $$
in second order perturbation theory, where the interaction Lagrangian in S is the one for QED, $\mathcal{L}I = eN\big(\bar{\psi}\gamma{\alpha}A^{\alpha}\psi \big)$, N is the normal ordering operator. I have seen similar examples and done this for some simpler numerators, but the number of factors in the time-ordered product causes the number of terms to increase significantly. This makes me question if I am applying <PERSON>'s theorem correctly and if there are any shortcuts I am missing and should be aware of.
I start by taking $ S = S^{(0)} + S^{(1)} + S^{(2)} $ in second order, which gives the denominator as
$$ \langle 0 | S | 0 \rangle = 1 + \frac{ie^2}{2} \int d^4x_1d^4x_2 \big( \bar{\psi}^{\bullet} \gamma_{\alpha}A^{\bullet \bullet\alpha} \psi^{\bullet \bullet \bullet} \big){x_1} \big( \bar{\psi}^{\bullet \bullet \bullet} \gamma{\beta}A^{\bullet \bullet\beta} \psi^{\bullet} \big)_{x_2}, $$
where the term with $S^{(1)}$ gives no contribution due to it having an odd number of electromagnetic field operators.
The numerator is similarly expanded as
$$ \langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S\big)| 0 \rangle = \ \langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S^{(0)}\big)| 0 \rangle + \langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S^{(1)}\big)| 0 \rangle + \langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)S^{(2)}\big)| 0 \rangle = \ \langle 0 |F^{(0)} 0| \rangle + \langle 0 |F^{(1)} 0| \rangle + \langle 0 |F^{(2)}0| \rangle.$$
Then I evaluate the terms in different order of perturbation theory one at a time.
$$ \langle 0 |F^{(0)} 0| \rangle = \langle 0 | T\big(A^{\mu}A^{\nu}\psi(z)\bar{\psi}(w)\big)| 0 \rangle = iD_F^{\mu\nu}(x-y)iS_F(z-w),$$
these I believe are the only non-vanishing contractions. $D_F$ and $S_F$ are the <PERSON> photon and fermion propagators. 324
Then once again due to $S^{(1)}$ having an odd number of electromagnetic field operators
$$ \langle 0 |F^{(1)} 0| \rangle = 0. $$
Hopefully so far the calculations make sense. It is when evaluating $ \langle 0 |F^{(2)} 0| \rangle $ I start encountering problems. Applying <PERSON>'s theorem to this expression seems to generate a very large number of terms and in general I find it difficult to keep track of all the possible contractions in problems like these. Could someone who is more comfortable with this theorem for problems like these list all the contractions or at least tell me how many there are supposed to be? I know some of these will probably be identical under a change of integration variables, which makes evaluating them at the end a lot simpler. 324