In Peskin and Shcroeder, when calculating the one-loop vertex correction, the line above Eq. (6.38) reads

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$$ \rightarrow \int \frac{d^4 k}{(2\pi)^4} \frac{-ig_{\nu\rho}}{(k-p)^2 + iϵ} \bar{u}(p') (-ie\gamma^\nu) \frac{i(\displaystyle{\not} k' + m)}{k'^2 - m^2 + i\epsilon} \gamma^\mu \frac{i(\displaystyle{\not} k + m)}{k^2 - m^2 + i\epsilon}(-e\gamma^\rho)u(p)$$

I am trying to figure out where does the $\gamma^\mu$ between the $k'$ and $k$ propagators come from. What I see is that this should be the propagator of the real photon, so the term should be $-ie\gamma^\mu$. Where did the $-ie$ go?

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    $\begingroup$ It comes from the photon-electron-electron vertex non-perturbative Feynman rule. He writes in the line above (6.38), $\Gamma^\mu = \gamma^\mu + \delta \Gamma^\mu$. PS - For all future questions, please explicitly write down the equation and context for your question. No one would (and should) spend time looking up texts to understand the question. $\endgroup$ – Prahar Mar 25 '18 at 18:52
  • $\begingroup$ Thanks. I did write it now. I also reformulated my question because I wasn’t very clear. $\endgroup$ – Y2H Mar 26 '18 at 8:35
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    $\begingroup$ Look carefully at what PS is calculating. He never says that the diagram above is equal to the integral. Instead, he says that $\delta \Gamma^\mu = $ the integral. Of course, $\delta \Gamma^\mu$ has the $-ie$ factored out in its definition, since we define the non-perturbative Feynman rule as $ee\gamma$-vertex = $- i e \Gamma^\mu$. $\endgroup$ – Prahar Mar 26 '18 at 16:28
  • $\begingroup$ I just thought of that last night. Very glad to see that someone agrees with me. Thank you. Why don’t you post this as an answer. $\endgroup$ – Y2H Mar 27 '18 at 7:26
  • $\begingroup$ @Prahar Why do you say non-perturbative? Isn't $\Gamma^\mu$ supposed to be the all-orders perturbative contribution to the electron vertex function? $\endgroup$ – gj255 Mar 28 '18 at 9:32

I found the answer. The original expression for all vertex corrections is $-ie\Gamma^\mu$. The expression on the left should hence be proportional to $-ie\delta\Gamma^\mu$ and the $-ie$ term on the left is removed with the same term from the photon propagator on the right.


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