Renormalization (Electron self energy)

[Aleolomorfo]

Hello everybody!
I have a big question about the renormalization: I do not understand why the "renormalization condition" is to impose the tree level result. Now I will explain it better.
Let's take, for example, the electron self energy. The tree-level contribution is the simple fermionic propagator $\frac{i}{\displaystyle{\not}p-m}$.
If I calculate the complete propagator (using the sum of all 1-particle-irreducible), the result is
$$\frac{i}{\displaystyle{\not}p-m-\Sigma(\displaystyle{\not}p,m)}$$
which is like the tree level result but the pole is shifted.
Renormalizing the electron self energy implies to redefine the electron mass. I define the "renormalized mass" as the position of the pole in the propagator. Consequently, to define the "physical measurable mass" I need to find the pole:
$$\displaystyle{\not}p-m-\Sigma(\displaystyle{\not}p,m)|_{\displaystyle{\not}p=m_R}=0$$
After finding the renormalized mass I Taylor-expand the full propagator arounf $m_R$ and to find exactly the tree level result (mutatis mutandis with the renormalized mass) I will need also to redefine the electronic field with $Z_2$.
If I am not wrong this is the idea behind the renormalization of the electron self-energy. However, I do not understand why I want the tree level result and I redefine things to obtaint it. For example, why the "physical mass" , the one that I measure in an experiment, is the shifted pole of the propagator? I think I understand how to do the things but I do not understand why.
Thanks in advance!

 

[DarMM]

What textbook are you using? It should contain a proof that the two point function has a momentum-space pole at the single particle mass. It's nothing to do with matching the tree-level result.

 

[Aleolomorfo]

I am using Peskin and Schwartz (QFT and the SM)

 

[DarMM]

In Peskin and Schroeder section 7.1 has the proof that the interacting two point function has the pole I mentioned above.

 

[Reggid]

I do not understand why the "renormalization condition" is to impose the tree level result. Now I will explain it better.

This is not THE renormalization condition, it is A renormalization condition, that was chosen in this example. The mass scheme that you get using this renormalization condition is called the "pole mass" of the particle.

You can chose different renormalization conditions, leading to different mass schemes.