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- What is exactly the renormalization condition for the QED vertex?

Hello, I'm studying the renormalization of QED. I have the Lagrangian

$$\mathscr{L}_{QED}=\mathscr{L}_{physical}+\mathscr{L}_{counterterms}$$

where

$$\mathscr{L}_{physical}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi - e \bar{\psi}\gamma^\mu\psi A_\mu$$

$$\mathscr{L}_{counterterms}=-\frac{1}{4}\delta_3 F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\delta_2\gamma^\mu\partial_\mu - \delta_m)\psi - e \delta_1\bar{\psi}\gamma^\mu\psi A_\mu$$

with ##\delta_k## the counterterms, fixed by the renormalization conditions.

I don't have problems with the mass and field-strengths renormalization conditions, my problem is with the condition for the electric charge renormalization.

If ##-ie\Gamma^\mu(p', p)## is the amplitude for the 1PI vertex diagrams with ##p'## and ##p## the electron momenta, the renormalization condition is usually stated by imposing that, when the photon is on-shell (##q^2=0##) then this amplitude must reduce to ##\Gamma^\mu = \gamma^\mu##.

My question is: In general, the momenta of the electrons in the vertex don't need to be on-shell, right? Then, do I need to impose on-shell electrons to determine the value of ##\delta_1##? Because, if I understand this properly, ##\delta_1## should be independent of ##p## and ##p'## and I don't see how this is possible if I don't fix them.

My question arises because all the calculations that I've seen assume on-shell electrons, and I don't understand if:

1- By definition ##\Gamma^\mu## must have on-shell electrons or

2- The renormalization condition imposes on-shell electrons in addition to the on-shell photon.

Thank you very much!

$$\mathscr{L}_{QED}=\mathscr{L}_{physical}+\mathscr{L}_{counterterms}$$

where

$$\mathscr{L}_{physical}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi - e \bar{\psi}\gamma^\mu\psi A_\mu$$

$$\mathscr{L}_{counterterms}=-\frac{1}{4}\delta_3 F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i\delta_2\gamma^\mu\partial_\mu - \delta_m)\psi - e \delta_1\bar{\psi}\gamma^\mu\psi A_\mu$$

with ##\delta_k## the counterterms, fixed by the renormalization conditions.

I don't have problems with the mass and field-strengths renormalization conditions, my problem is with the condition for the electric charge renormalization.

If ##-ie\Gamma^\mu(p', p)## is the amplitude for the 1PI vertex diagrams with ##p'## and ##p## the electron momenta, the renormalization condition is usually stated by imposing that, when the photon is on-shell (##q^2=0##) then this amplitude must reduce to ##\Gamma^\mu = \gamma^\mu##.

My question is: In general, the momenta of the electrons in the vertex don't need to be on-shell, right? Then, do I need to impose on-shell electrons to determine the value of ##\delta_1##? Because, if I understand this properly, ##\delta_1## should be independent of ##p## and ##p'## and I don't see how this is possible if I don't fix them.

My question arises because all the calculations that I've seen assume on-shell electrons, and I don't understand if:

1- By definition ##\Gamma^\mu## must have on-shell electrons or

2- The renormalization condition imposes on-shell electrons in addition to the on-shell photon.

Thank you very much!