How to understand Schwartz chapter19 equation (19.85)? I am reading Schwartz's QFT books, chapter 19.
In section 19.5, he claims,in equation 19.85/19.86, that there is a simpler way to prove that $Z_1=Z_2$ in all orders of perturbation theory.
He first claims that there is an effective Lagrangian, quote,"which produces at tree-level the identical 2- and 3-point functions that renormalized QED
produces at loop level"
$$L_{eff}=-\frac{1}{4}F_{\mu\nu}^2+i\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-m_{R}\overline{\psi}\psi-e_{R}A_{\mu}\overline{\psi}\Gamma^{\mu}(i\partial)\psi\tag{19.83}$$
which can be matched on to the original QED lagrangian at large distance
$$L=-\frac{1}{4}Z_3F_{\mu\nu}^2+iZ_2\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-Z_2Z_mm_{R}\overline{\psi}\psi-e_{R}Z_1A_{\mu}\overline{\psi}\gamma^{\mu}\psi.\tag{19.84}$$
Then he claim that the condition $\Gamma^{\mu}(0)=\gamma^{\mu}$ implies that
$$\lim_{p\to0}p_{\mu}\Gamma^{\mu}(p)=Z_1\gamma_{\mu}p^{\mu}\tag{19.85}$$
and the onshell renormalized electron propagator is
$$iG(\gamma^{\mu}q_{\mu})=\frac{i}{\gamma^{\mu}q_{\mu}-m_{R}}=\frac{1}{Z_2}\frac{i}{\gamma^{\mu}q_{\mu}-m_{0}}.\tag{19.86}$$
What I don't understand is the following:

*

*in 19.85, he seems to be matching the vertex(3-point) function of the effective theory with the orginal theory. On the LHS, he has effective theory at the tree level. Since he claims that effective theory at tree level is equivalent to original theory to loop level, then I expect that on the RHS, there should be original theory to loop level. However, what he has is $Z_1$, i.e. he only has the counter term, but not the loop correction from the original theory.


*Similar question for 19.86, I don't understand how the second = arises from the first place.
update:
3.I suspect that (19.85) and (19.86) are related with how an effective theory is matched on to the original theory, but I am not sure
 A: The answer to the first question is simple and direct, it is such because the on-shell renormalization scheme is "defined" to be so!
In other words, it should take into account all the quantum corrections at its tree level in the on-shell limit. It is defined such that there are no loop corrections at this certain energy level and the result is exact. As you make the mediator particle more and more off-shell(as one is interested in higher energy level amplitudes in terms of the current energy level amplitude) one is supposed to take into account new terms.
An example would be: "Why not include loop corrections to the pole mass ?"! Pole mass is defined not to receive any loop corrections(On-shell condition) while renormalized mass does receive loop corrections, as there can be always a finite difference between the pole mass $m_P$ and the renormalized mass $m_R$.
Nonetheless, one can write down the one-loop vertex amplitude and observe the anti-symmetricity of the magnetic term will vanish remove the loop correction when contracted with the momentum.
The only surviving term can be reabsorbed in the $Z_1$ term to be canceled out by fermion self-interaction loop correction.
Imagine it's trying to define $Z_1$ formally.
And keep in mind that $p^2=0$ doesn't necessarily imply that $p^{\mu}$ is zero.
