Understand Mandl & Shaw's Derivation of Moller Scattering

[jdstokes]

I'm trying to understand Mandl and Shaw's derivation of the part of the S-operator which affects Moller scattering.

Starting with the second term in the Dyson series expansion for the S-matrix, they extract using Wick's theorem the term with a contracted electromagnetic field and four uncontracted fermion fields.

Then using the fact that two electrons must be annihilated and then created they conclude that the relevant term is

$S(2e^-\to 2e^-) = -\frac{e^2}{2!} \iint d^4x_1 d^4x_2 : (\bar{\psi}^-\gamma^\alpha\psi^+)_{x_1}(\bar{\psi}^-\gamma^\beta \psi^+)_{x_2}:\mathrm{i} D_{\mathrm{F}\alpha\beta}(x_1 -x_2)$

where D_F is the Feynman propagator representing transmission of a virtual photon between the spacetime points x_1 and x_2.

Now here comes the tricky bit. They claim that the above expression gives four contributions to the Moller transition since each initial (final) electron (positron) can be annihilated (created) by either of the $\psi^+$ ($\psi^-$) operators.

I'm having trouble seeing why we should get four terms out of this. If we sandwich $S(2e^-\to 2e^-)$ between initial and final Fock states, how can this result in a sum of four terms mathematically?

 

[jdstokes]

I'm slowly working my way towards a resolution of this problem.

I am pretty sure it is to do with the fact that the initial and final Fock states consist of linear superpositions because of the fact that the fermions are identical e.g.

$|initial\rangle = \frac{1}{\sqrt{2}}(|p_1\rangle p_2\rangle - |p_2 \rangle p_1\rangle)$.

So far I have not been able to account for the minus sign prefactor in the first term on the RHS of equation (7.18).

 

[jdstokes]

After a lot of anguish, I managed to get Mandl and Shaw's (7.18) to within an overall phase factor of negative one. Could this be an error in the text?