# Renormalisation group transformation- Peskin

• muppet
In summary, in chapter 12 of Peskin's treatment of the Wilsonian approach to renormalisation, the question of why integrating out high-momentum modes generates all possible interactions is addressed. The answer involves a coupling of high- and low-frequency modes and doing the path integral over the high-frequency modes, resulting in terms such as (\phi\phi\phi\hat{\phi})^2 generating a phi^6 interaction. The role of the momentum of external particles is also discussed, with Peskin suggesting a more exact treatment by Taylor expanding in the external momenta of diagrams. This is a similar concept to Wilsonian renormalization in statistical physics, as explained in lecture notes and a reference provided.
muppet
Hi all,

I'm trying to understand Peskin's treatment of the Wilsonian approach to renormalisation, in chapter 12. The essential (i.e textbook-independent) question I have is: why does integrating out the high-momentum modes generate all possible interactions?

I understand part of the answer- one has a coupling of high-and low frequency modes, and doing the path integral over the high frequency modes (denoted by a circumflex) means that terms like
$$(\phi\phi\phi\hat{\phi})^2$$
will generate a phi^6 interaction.

What I think it is that I don't understand is the role played by the momentum of the external particles. Peskin argues that "a more exact treatment would taylor expand in [the external momenta of the diagrams]", but it isn't clear to me what's being expanded (a diagram? the n-point correlation function?) or why we have to expand in this Wilsonian treatment when we wouldn't ordinarily. To be honest, some complementary references would be good.

I'm not sure this is close enough, but something similar happens in Wilsonian renormalization applied to statistical physics.

http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/
The big picture is given in L7, III.E, p44.
The details as to how the additional terms are generated are given in L11, IV.F p64, 65.

I imagine a translation to HEP is something like that on p22 of http://arxiv.org/abs/hep-lat/9807028

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## 1. What is the Renormalisation Group (RG) transformation in quantum field theory?

The RG transformation is a mathematical tool used to study the behavior of a quantum field theory at different energy scales. It allows us to understand how the coupling constants and other parameters of the theory change as we move from high to low energies.

## 2. Why is the RG transformation important in particle physics?

The RG transformation is important because it helps us understand the behavior of quantum field theories at different energy scales, which is crucial in understanding the fundamental interactions between particles at the smallest scales. It also allows us to make predictions and calculations in theories that would otherwise be too complex to solve.

## 3. How is the RG transformation applied in Peskin's work?

In his work, Peskin applies the RG transformation to study the behavior of quantum electrodynamics (QED) at different energy scales. He uses it to calculate the running of the QED coupling constant, which describes the strength of the electromagnetic interaction between particles.

## 4. What are the benefits of using the RG transformation in particle physics?

One of the main benefits of using the RG transformation is that it allows us to make predictions and calculations in theories that would otherwise be too complex to solve. It also helps us to understand the behavior of quantum field theories at different energy scales, which is crucial in understanding the fundamental interactions between particles.

## 5. Are there any limitations to using the RG transformation?

Yes, there are limitations to using the RG transformation. It is most effective in theories with a small number of fields and interactions. In theories with a large number of fields, the calculations become more complicated and less accurate. Additionally, the RG transformation is not applicable to theories with strong interactions, such as quantum chromodynamics (QCD).

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