We use Feynman diagrams to calculate some scattering or interaction, as we show in Fig 1. We try to maximize our knowledge of interaction by doing some integrals and eventually getting the matrix elements, which then is used for calculating the cross-sections ($\sigma$) and many more things.
Fig 1. Interaction of leptons through the neutral weak force.
There is a level, too basic but realistic in the classical sense, called "Tree Level," we compute anything first at the tree level to ignore the divergences, which will come if we go beyond (or under) the tree level. Where is this tree-level defined? It is defined differently. Fig 1 is a first-order Feynman Diagram. However, when we try to compute some higher-order diagrams at short distances, which inductively means high energy, we will get a diagram somewhat like the following.
Fig 2. Feynman Diagram of $e^- e^+ \rightarrow e^-e^+$ with the loops of pair production $\gamma \gamma \rightarrow e^- e^+$
The circles [1] are screening and background process; those loops are still the first order Feynman diagram, but now built inside some other diagram. If we try to calculate the matrix elements of higher-order diagrams, just like Fig 2, we will get some divergences. This was then a big problem; whenever doing higher-order scattering, they got these divergences, which diverged the mainstream calculations to a more typical solution to these divergences.
As we go through the history of the physics of renormalization group (RG), there is a whole group theory of it, we see they built renormalization on the idea of field theory, which is where divergences appeared in those days, and still field theory [2] dominates the renormalization group studies. Murray Gell-Mann and Francis Low gave the solution, however, not the first in 1954. The solution then gave a whole new subject to the field theory. However, we can say Gell-Mann and Low solution is old-school after a detailed thesis on the Renormalization Group. They were not on the thought of giving much importance to the study of renormalization. Instead, they were working to eliminate the infinities in QED and Strong Confinement [3].
The solution which Gell-Mann and Low had given was a differential equation. Before we jump to that differential equation, there is more to discuss. The solution which Gell-Mann and Low had given was a differential equation. Before we jump to that differential equation, there is more to discuss. There is a shorter, yet too wrong, way to understand RG. Let us say a theory is divergent after some scale M, and then we impose a cutoff $\Lambda$ which describes the theory under scale M. It is called Effective Field Theory, a hot topic of study whenever doing some speculative field theory.
Then, it was introduced that there are two notions of one measurement, one bare and second renormalized. Speaking of which, let us say an electron which we denote here as $e_{\lambda}$ oscillates between two values. The first is the bare charge, and the second is the physical charge. When $\lambda \rightarrow 0$ the case is of former, and for $\lambda \rightarrow \infty$ latter takes the command. We use bare quantities, which are renormalized contents, in measurements. So, renormalized charges are those charges which are defined at some energy level which in turn is defined using $\Lambda$. The remaining quantities are called counter-terms. With these counter-terms, the theory is appropriately not divergent now.
We have now touched the RG [4]. There is a lot more motivation for RG, including practical application in high-energy physics and condensed matter.
Footnotes;
[1] These are called self-interactions. We try to minimize these self-interactions. However, studying these loops gives us more intuition of particular theory when needed.
[2] Both Quantum Field Theory and Statistical Field Theory.
[3] This quest was why quarks could not be found in open space in free form, while leptons are found in their free form.
[4] This post was just a visualization of a brief history and crux of renormalization. There are many counter-intuitive feelings in a person who is studying RG. There is much more to discuss and much more to doubt.
Feel free to comment about questions and typos.
