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I am learning about Renormalization and I am reading various scripts/ books and online texts. In the wikipedia article with the same name:

https://en.wikipedia.org/wiki/Renormalization#Divergences_in_quantum_electrodynamics

The following is said:

Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:

  • a) photon creates a virtual electron–positron pair, which then annihilates. This is a vacuum polarization diagram.

  • b) An electron quickly emits and reabsorbs a virtual photon, called a self-energy.

  • c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 2, and it is called a vertex renormalization. The Feynman diagram for this is also called a “penguin diagram” due to its shape resembling a penguin.

The three divergences correspond to the three parameters in the theory under consideration:

  1. The field normalization Z.

  2. The mass of the electron.

  3. The charge of the electron.

I don't understand how we come up with the correspondence as described here. Could someone explain this to me? Is it the result of calculations or it can be somehow be seen/noted that this is the case without the need of calculations?

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  • $\begingroup$ The QED vertex diagram and the penguin diagram in strong-interaction physics are not the same—not even close, in fact. $\endgroup$ Commented Sep 16 at 0:38
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    $\begingroup$ I mean realistically this is an entire chapter or more in a QFT textbook. What is your starting point in terms of background? $\endgroup$ Commented Sep 16 at 0:59
  • $\begingroup$ I mean, we were told what renormalization and regularization are. The different schemes, and in the lecture we considered the scalar field. In the exercises we considered phi^4 at 1 loop order and did the MS bar renormalization. But that's about it really $\endgroup$ Commented Sep 16 at 20:47

2 Answers 2

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Briefly speaking, renormalization is structured as follows:

I) On one hand, any diagram can be built from 1-particle irreducible (1PI) diagrams. There are 2 kinds$^1$ of 1PI correlation functions:

  1. 1PI self-energy/vacuum polarization 2-pt function.

  2. 1PI $(n\geq 3)$-pt vertex function.

II) On the other hand, there are 3 kinds of $Z$-factors:

  1. A $Z$-factor $Z_{\phi}$ associated with wave function renormalization $\phi_0=Z_{\phi}^{1/2}\phi$ of each field $\phi$.

  2. A $Z$-factor $Z_m$ associated with each mass $m$.

  3. A $Z$-factor $Z_g$ associated with each interaction coupling constant $g$.

III) The two first types $Z_{\phi}$ & $Z_m$ are (the third type $Z_g$ is) associated with renormalization conditions of the 1PI self-energy/vacuum polarization (vertex function), respectively.

IV) Concerning QED, see e.g. my Phys.SE answer here.

$^1$ We assume for simplicity that there are no non-zero 1PI tadpole 1-pt functions.

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You regularize the integrals, sum then with the tree level diagrams of the same process, but taking the parameters of the theory not as fixed constants, but as functions of the regulators. You set what should some process with specific external moments gave you (you can think "I adjust our parameters with specific experiments"), in a way that is independent of the regularization. That makes posible to have the charge and mass (or the normalizations) depend on the regularization (that can be thinker to be linked to some scale, in some sense). Then the amplitudes will become functions in which you can take the limit giving results that depend on those fixed things. If you call charge, mass, etc to the parameters of a pseudo-tree level diagram that would be related to the fixed process but forgetting the loop diagrams, you could talk about the renormalized parameters (talking about the charge, mass, etc that appear in the Lagrangian as the bare parameters)

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