Electron-muon scattering

Question

I'm reading Mandl & Shaw Quantum field theory and am currently working my way trough chapter 7.4. Here they are talking about electron - muon scattering. They start of with the following expression

$$S^{(2)}_{\mu e} = -e^2 \int d^4x_1d^4x_2 T \{ N( \bar{\psi}_\mu \gamma^\mu A_\mu \psi_\mu)_{x_1}N( \bar{\psi}_e \gamma^\mu A_\mu \psi_e)_{x_2}\} .\tag{7.56}$$

From this they then go on and say that by using Wick's theorem we expand the time ordering and that the only contraction will be between the 2 photon fields. I do not really see why this is the case. I guess I can kinda see that having a propagator between 2 different flavours of particles is weird but I'm more looking for an actual mathematical/theory backed up explanation.

Answer 1

1. Mandl & Shaw are in section 7.4 discussing leptons minimally coupled to QED.

   The action has a global $U(1)$ flavor symmetry for each lepton species, so a non-vanishing $n$-point correlation function must preserve the lepton number for each species, cf. my Phys.SE answer here.

   In particular, a 2-point function between 2 different leptons must vanish, cf. OP's question.

2. Concerning Wick's theorem, see e.g. this related Phys.SE post.