Sunday, 14 February 2016

quantum field theory - Optical theorem for the given diagram


Assume we have the abelian gauge theory with single fermion. Suppose the following diagram: enter image description here


Here the initial "photon" $\gamma$ is in the same state as the final one, so this is the diagram of self-energy. I want to check whether the optical theorem holds for it.


The optical theorem for the total amplitude reads $$ \tag 1 2\text{Im}M_{\gamma \to \gamma}= (2\pi)^{4}\sum_{n}|M_{\gamma \to n}|^{2}\delta (p_{n} - p_{\gamma}) \simeq \Gamma_{\gamma} $$


The question: have I also take into account the diagram enter image description here which is also 6th order of perturbation theory?


An edit


It seems that the formal answer is indeed yes. Really, the rhs of $(1)$ contains the half of the squared diagram, and the diagram above is needed in the lhs.





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