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When we evaluate the amplitude of the three-photon interaction in QED, we find out that the amplitude of the following diagrams cancel out.enter image description here

In the limit of $m_e \rightarrow0$ it boils down into trace identities to show that the amplitude is indeed zero. I can do that, but my question is why is this a significant result for us to assume that QED is renormalizable. I have read that it is esential for this interaction to vanish. Why ?

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    $\begingroup$ Ma&ybe because the divergence of each diagram is too "severe" for getting removed with renormalization of "something". $\endgroup$ Dec 29, 2019 at 19:04
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    $\begingroup$ By the way, this amplitude vanishes trivially by Furry's Theorem $\endgroup$
    – nox
    Dec 29, 2019 at 20:59
  • $\begingroup$ Do you have any further reading material(or explanation) on the subject?? $\endgroup$ Dec 29, 2019 at 22:55
  • $\begingroup$ Mostly I'd like to know why this process is important for the renormalizability of QED $\endgroup$ Dec 29, 2019 at 22:56

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I'm extremely late. The reason is that otherwise one would have to insert a counterterm in the Lagrangian that is cubic in the electromagnetic field. This term would break gauge invariance.

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