# Wick theorem for particle-antiparticle annihilation

I am trying to write down the LSZ formula for e- e+ -> 2 gamma:

$$S_{fi} \propto \langle 0| T \{\bar{\Psi}(x_1) \vec{A}(x_2) \Psi (x_4) \vec{A}(x_3) \left\lbrack \frac{(-i)^2}{2} \int d^4x A_\mu (x) \bar{\Psi}(x) \gamma^\mu \Psi (x) \int d^4y A_\nu (y) \bar{\Psi}(y) \gamma^\nu \Psi (y) \right\rbrack \} |0\rangle$$

So I have electron at ##x_1## and positron ##x_4##.

But now if i try to contract I can contract ##\bar{\Psi}(x_1)## with one at x or at y, and same with ##\vec{A}(x_2)## and this would produce a factor of 4 which would cancel with 1/2 from exponential expantion giving a factor 2, but this is wrong, there should not be a factor of 2.

Any ideas? thanks.

Sfi for two photon annihilation is probably
(-e)2∫d4x1t2<t1d4x2<2γ|ψ-(x1μψ(x1)Aμψ-(x2vψ(x2)Av(x2)|e-e+>,then you will have to introduce the decompositon of fermionic field into two parts and also use the creation operator on vacuum states to obtain the |e-e+> and |2γ> states.

No, i used LSZ, but i got it, the factor of 2 i am getting is correct. Thanks.

In any case,the final state represents two photons and initial one is |e-e+>,while your is only having sandwiched between vacuum states.so try to interpret it.

The leptons and photons are generated by the first four operators. I just used LSZ reduction formula.

you might like to give a reference for spinor electrodynamics lsz reduction. the term in square bracket contains the full interaction term of qed which is somewhat uneasy to me.

Greiner "Field quantization", chapter on LSZ for fermions.