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I might be making great confusion in trying to interpret proper vertex function. I'm studying QED vertex correction. I'm just going to write down the pieces of the puzzle.

So I know that the effective action is (for real scalar field)

$\Gamma[\phi_{cl}]= \frac{1}{2} \int \frac{dk}{(2 \pi)^4}[\phi _{cl} (k^2+m^2- \Pi (k^2))\phi_{cl}]+\sum^{\infty}_{n=3} \frac{1}{n!}\int \frac{dk_1}{(2 \pi)^4}...\int \frac{dk_n}{(2 \pi)^4} V_n(k_1 ... k_n) \phi_{cl}(k_1)... \phi_{cl}(k_2)$

But also $ \Gamma[ \phi_{cl}]= \sum^{\infty}_{n=2} \frac{ \bar{h} }{n!} \int dx_1... dx_n \phi_{cl}(x_1)... \phi_{cl}(x_n) \Gamma^{(2)} (x_1,...,x_n) ... \Gamma^{(n)}(x_1 ,..., x_n)$

Therefore we can identify

$\Gamma^{(2)} \rightarrow k^2+m^2- \Pi (k^2)$

Ok, now let's pass to the QED vertex

Here I obviously have

$\Gamma^{(2)}(k)= \gamma^{\mu} k_{\mu}-m - \Sigma(k^2)$

I know that in the amplitude of the process I have at leading order $\gamma^{\mu}$, so with the correction I have (p-electron incoming, p'-electron outgoing)

$-ie \bar{u}(p') \Gamma^{\mu} u(p) A^{cl}_{\mu}$

with $\Gamma^{\mu} = \gamma^{\mu}+ \delta \Gamma ^ {mu}$

so that $j^{\mu}= -ie \bar{\psi}(x) \Gamma^{\mu} \psi(x)$

Now, hoping not to have written something wrong, my question is:

Am I right to understand that, since I can expand the self energy function on the proper vertex function as $\Sigma (k^2)= \sum_{l>1} \Gamma^{(2)}_l(k) =\sum_{l>1} \lambda^l \Sigma_l(k^2) $, I can make the correspondence:

$\Gamma^{(2)}_l \leftrightarrow \delta \Gamma ^{\mu}$

I'm a bit confused as the QED vertex correction takes the tree point proper vertex function, not the two point one.

And what about $\gamma ^{\mu}$? With what does it identify? Did I write any nonsense? I'm sorry but I'm a bit puzzled

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