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The coupling constant in the QED lagrangian is clearly the electric charge $e$. However, one often hears the statement that the expansion parameter in QED is the fine structure constant $\alpha = e^2/4\pi$, not $e$.

Unfortunately, I've never seen a formal proof that the sum of all contributions to S-matrix which are proportional to the given odd power of the electric charge $e$ must vanish.

My question is whether this is really true?

E.g. why the combined third order contributions to the proces of the scattering of an electron and a photon into an electron and two photons vanish?

Maybe what is meant by the statement that the expansion parameter in QED is $\alpha$ is simply that the perturbation series has the form $\sum_{L=0}^\infty e^{E-2+2L} a_L = e^{E-2}\sum_{L=0}^\infty \alpha^L a_L$ ($L$ - number of loops, $E$ - number of external lines), which is not difficult to prove.

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Your latter option is what is meant - the perturbation series is an expansion in loop orders, and the power of $\alpha$ is what counts the loop order.

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Every perturbative correction of a diagram comes with an internal photon which contributes two other vertices to the diagram. Hence the corrections should be of order $\alpha$, $\alpha^2$ $\dots$

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