There are many resources on renormalization in Quantum Field Theory (QFT). I have been reading a few of them and I will review them here. This is roughly in the order I recommend reading them (as a complete beginner), with the first few being the best.

Find my recommended reading order below.

Note this blog doesn’t cover renormalization group and solid state physics. I will cover that in a separate post. The focus here is QFT.

Zee QFT

Anthony Zee’s textbook Quantum Field Theory in a Nutshell Part III Renormalization has great exposition. I especially like the dialogue between the experimentalist and the theorist.

It is worth mentioning Zee’s Lectures although it is abit blurry.

Chap III.1: Conceptual

In $\phi^4$ theory, 2 particle scattering up to 1 loop with hard cutoff $\Lambda$:

\[\mathcal{M}(s,t,u,\Lambda,\lambda)=-i \lambda+i C \lambda^2\left[\log \left(\frac{\Lambda^2}{s}\right)+\log \left(\frac{\Lambda^2}{t}\right)+\log \left(\frac{\Lambda^2}{u}\right)\right]+O\left(\lambda^3\right)\]

(equation 2)

Zee then explains that we can measure $\mathcal{M}$ at momenta $s_0, t_0, u_0$, and we call that $\lambda_P := \mathcal{M}(s_0,t_0,u_0, \Lambda, \lambda)$. To make sure that $\lambda_P$ is independent of $\Lambda$, Zee says that $\lambda(\Lambda)$ must satisfy

\[\Lambda \frac{d \lambda}{d \Lambda}=6 C \lambda^2+O\left(\lambda^3\right)\]

(equation 16)

What I especially like about Zee’s exposition is that he writes out the full derivation for how to invert $\lambda_P(\lambda)$ to get $\lambda(\lambda_P)$ (equation 7). This is something that is often glossed over in other lectures. Although it is just 2 lines, it is not a common procedure at all, and I think it is important to see it done explicitly the first time.

\[\begin{aligned} \mathcal{M}&=-i \lambda+i C \lambda^2 L+O\left(\lambda^3\right) \\ -i \lambda_P&=-i \lambda+i C \lambda^2 L_0+O\left(\lambda^3\right) \\ -i \lambda&=-i \lambda_P-i C \lambda^2 L_0+O\left(\lambda^3\right)\\ &=-i \lambda_P-i C \lambda_P^2 L_0+O\left(\lambda_P^3\right) \end{aligned}\]

(equations 5-7)

We then substitute $\lambda(\lambda_P)$ back into $\mathcal{M}$ to get

\[\begin{aligned} \mathcal{M}&=-i \lambda+i C \lambda^2 L+O\left(\lambda^3\right)\\ &=-i \lambda_P-i C \lambda_P^2 L_0+i C \lambda_P^2 L+O\left(\lambda_P^3\right) \\ &= -i \lambda_P+i C \lambda_P^2\left[\log \left(\frac{s_0}{s}\right)+\log \left(\frac{t_0}{t}\right)+\log \left(\frac{u_0}{u}\right)\right]+O\left(\lambda_P^3\right) \end{aligned}\]

(equation 9)

As Nabil Iqbal puts it, “you want to relate measureable quantities to the measureable quantities”, like pressure to temperature.

Zee then covers Pauli-Villars regularization, and dimensional regularization in the Appendix of Chapter III.1.

Chap III.2: Renormalizable vs Non-renormalizable (using Dimensional Analysis)

Zee covers it quite well. I especially like his example of a vector boson theory being the UV completion of Fermi theory (equation 1, page 157).

Chap III.3: Counterterms and Physical Perturbation Theory

I’ll be honest I don’t really like his explanation of mass renormalization that much.

Chap III.7: Polarizing the Vacuum and Renormalizing the Charge

No comments

Nabil Iqbal Lectures

Nabil Iqbal Lectures from QFT2 at Durham.

Lecture 3d: Intro to Renormalization

He mentions the problem of infinity if we take the limit $\Lambda \rightarrow \infty$ in the $\phi^4$ theory.

2:59 I especially like his analogy of ideal gas law and how we relate 2 observable things to each other.

7:10 I also like that he explicitly says that $\lambda_P \equiv i\mathcal{M}(s_0,t_0,u_0)$ is a definition. In Zee’s book, he says $\lambda_P$ is measured, but doesn’t explicitly say that it is definitionally equal to $\mathcal{M}(s_0,t_0,u_0)$, and it must instead be inferred by looking at the equations 3-4.

10:20 He inverts $\lambda_P(\lambda)$ to get $\lambda(\lambda_P)$, but leaves it to the viewer to verify. I prefer Zee for the explicit working in this regard.

Lecture 4a: Renormalized Perturbation Theory

To be honest, I don’t like this lecture that much. He explains the renormalization conditions (pole of 2-point function at $p^2 = m^2$ must have residue 1).

He derives the same result

\[i \mathcal{M}\left(s, t, u\right)=-i \lambda+\frac{i \lambda^2}{32 \pi^2} \left[ \log \left(\frac{s_0}{s}\right)+\log \left(\frac{t_0}{t}\right)+\log \left(\frac{u_0}{u}\right) \right]\]

(equation at 18:12)

using counterterms and renormalization conditions.

Lecture 4b: Renormalization of the two-point function

No comments

Lecture 4c: Which Theories Are Renormalizable

No comments

Lecture 4d: non-renormalizable theories

I like the examples he introduced.

Tobias Osborne Lectures

Tobias Osborne Lectures from his Advanced QFT Series. Tobias approaches renormalization in a very philosophical manner. He covers $\phi^4$ theory.

Lecture 8: Philosophy of Renormalization

He spends a good 40 minutes talking about philosophy of physics. I especially like 56:00 where he says we can declare victory even if the (bare) parameters of our theory are not necessarily observables, and they can totally depend on a cut-off $\Lambda$, and he labels them $z_i(\Lambda)$ to emphasise it. My takeaway is that the unrenormalized/bare parameters are merely a means to an end, and the end is to predict observables.

Lecture 9: Renormalization of $\phi^4$ theory

No comments

Lancaster QFT

Quantum Field Theory for the Gifted Amateur Part VIII Renormalization. He introduces renormalization using counterterms first, then explaining the philosophy of “what is physically observable”, mirroring the historical development of renormalization.

I think the historical way is not a good way to learn it though. I much rather learn it from the modern perspective of effective field theory first, and then learn the historical development. This is because physicists were confused about renormalization for a long time, before a drastic change in perspective made it clearer. It would thus make more sense to use the benefit of hindsight when teaching renormalization.

Chap 31: Quasiparticles

I don’t really like the explanations here.

Chap 32: Renormalization

He works with $\phi^4$ theory. While I like that he covers a lot of detailed mathematical steps, I don’t really like the explanations here. This is a useful resource if you are already familiar with the conceptual understanding and wish to revise it.

Chap 33: Propagators and Feynman Diagrams

I somewhat like his derivation of mass renormalization, as well as renormalization conditions for counterterms.

Chap 34 & 35: Renormalization Group

I like his use of examples, especially examples in solid state physics.

== Detailed Calculations ==

From here onwards, the resources are more detailed and less conceptual. They are good if you already know the conceptual idea and want to apply it to real calculations.

Lewis Ryder QFT

Lewis Ryder’s textbook Quantum Field Theory writes mostly about renormalization of gauge theories in Chapter 9. This is good if you already have a decent conceptual understanding of renormalization and want to see how it works in phenomenology.

Ricardo D. Matheus Lectures

QFT2 from IFT UNESP (Sao Paulo) is a great series if you want to dive into QED and QCD calculations in gory detail. I especially like his Kallen-Lehmann Lecture and his LSZ Lecture.

wky54321 Lecture (Summary)

Renormalization of QED is a standalone lecture summarising QED calculations. I think the first half is good for beginners, but the back half is hard to follow if you didn’t already know the material.

Francois David Lectures

Francois David Lectures from QFT2 at Saclay.

Ashoke Sen Lectures

Ashoke Sen Lectures from QFT2

== Recommended Reading Order ==

If you are completely new to renormalization, I recommend the following resources in this order:

Conceptual Understanding / Vertex Renormalization

Zee QFT Chapter III.1 (Conceptual)
Zee QFT Chapter III.2 (Dimensional Analysis)
Nabil Iqbal Lecture 3d (Intro to Renormalization)
Tobias Osborne Lecture 8 (Philosophy of Renormalization)
Tobias Osborne Lecture 9 (Dimensional Analysis)

Counterterms and Renormalization Conditions

Lancaster QFT Chap 32 (Renormalization)

Mass Renormalization

idk, I don’t like any of the explanations so far