Wick theorem for particle-antiparticle annihilation

[LayMuon]

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 expansion giving a factor 2, but this is wrong, there should not be a factor of 2.

Any ideas? thanks.

 

[andrien]

S_fi for two photon annihilation is probably
(-e)²∫d⁴x₁∫_t2<t1d⁴x₂<2γ|ψ^-(x₁)γ_μψ(x₁)A_μψ^-(x₂)γ_vψ(x₂)A_v(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.

 

[LayMuon]

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

 

[andrien]

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.

 

[LayMuon]

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

 

[andrien]

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.

 

[LayMuon]

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