The logic behind the renormalization group of a quantum field theory

https://en.wikipedia.org/wiki/Renormalization_group

We are currently studying renormalization groups, in order to understand why gravity is non-renormalizable.

https://arxiv.org/abs/0709.3555

Assaf Shomer has written a 10 page explanation of the non-renormalizability of gravity.

Let us calculate a Feynman path integral, using some large number Λ as a cutoff for momenta.

In QED, the integral over a (vacuum polarization) loop diverges badly, but by setting a cutoff we can calculate results which are empirically correct! Why is that? What is going on?

Shomer requires that the partition function (= the generator for all correlation functions) stays the same regardless of the cutoff. The correlation functions tell us the physical behavior of the system. Why would a relatively arbitrary cutoff Λ give the correct behavior and not another slightly different cutoff Λ'?

Maybe the right model is to adjust the values of coupling constants for various cutoff sizes, in a way that the integral which yields the partition function has the same value regardless of the cutoff size?

Shomer derives in his paper the equation (13), which determines the RG flow, that is, the dependency we must set on the coupling constants on the cutoff Λ, in order to have the partition function integral the same regardless of the cutoff.

How does this compare to the intuitive idea of scaling self-similar systems, as outlined in the Wikipedia article?

In what way does a higher Λ mean analyzing the system in more precise detail? Like we would analyze the block spin example of Leo P. Kadanoff in the Wikipedia article?

The analogy between a higher cutoff and more detail is not clear. We have to think more about this.

Bare charge and a dressed electron

The Wikipedia article contains the familiar claim that virtual electron-positron pairs around an electron can "screen" some of the  charge of the electron.

As if a "dressed" electron would appear to have a smaller charge.

This claim is misleading. Consider an electron in a classical polarized media. It is the polarization of the media close to the the observer which does screen some of the electron charge. That is, the molecules close to the observer are polarized, and cancel some of the electric field of the electron.

Suppose then that the observer moves closer to the electron. Does the electron charge appear larger then? That depends on the amount of polarization close to the electron. It is somewhat misleading to say, as in Wikipedia, that the observer "bypasses a screen of virtual particles" as he moves closer. The bypassing is not relevant but the magnitude of polarization closer to the charge.

Let us analyze a scattering experiment of an electron and a positron using Feynman diagrams. In the diagrams, there is a vacuum polarization loop which makes the interaction weaker. The integral for the loop diverges.

But in the diagram there is no cloud of virtual pairs which would screen the charge. Why would we invent an artificial "explanation" using imagined virtual pairs?