$\phi^4$ in $4 - \epsilon$ dimension renormalization beta function

In summary, the conversation discusses the topic of renormalization group and beta functions. The speaker is seeking feedback and clarification on their understanding of the beta function and its application in dimensional regularization and momentum cut-off schemes. They also mention a chapter on perturbative RG in their notes and express difficulty in understanding the physical picture and implications of the Wilson Fisher fixed point and running couplings. Overall, the conversation highlights the complexities and challenges of understanding and applying renormalization group techniques in quantum field theory.
  • #1
paralleltransport
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TL;DR Summary
I'd like to check my personal understanding of renormalization group and beta function.
Hi all,

I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback.

PART I:

I wrote this short summary of what I understand of the beta function:
1635740724986.png


Is this reasoning correct? Are there fine points that I am missing that needs refinement?PARTII:

The scheme presented above uses dimensional regularization to extract the divergent counterterms. It is a bit physically hard to visualize. When I use a momentum cut-off scheme, I feel like taking the derivative of the counter-term with respect to the logΛ, the momentum cut-off should give me the beta function (up to a sign). I'm not sure how to motivate this. If there's a text that does extract RG beta function in this perspective it would help.

Thank you.
 
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  • #3
Hi vanhess, thank you for the notes. The details in the notes are worked in pretty gory details, however I'm having difficulties understanding the physical picture being emphasized here.

We first start from equation 5.275, a perfectly reasonable condition: cross section amplitudes measured should be independent of our renormalization scale. From there we get the standard callan symanzik equation (5.279). I suppose there should be one for each observable (2 point, 4 point, etc...) vertex.

However, it's not obvious to me from the calculation how one can visualize the wilson fisher fixed point for phi^4 theory, and the physical meaning, if at all of the running couplings. What does γϕγϕ (wave function renormalization flow) mean if I were to simulate the QFT on a lattice at different scales? etc...
 
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1. What is $\phi^4$ theory in $4 - \epsilon$ dimensions?

$\phi^4$ theory is a quantum field theory that describes the interactions between particles in a four-dimensional space. In $4 - \epsilon$ dimensions, the theory is expanded to include a small deviation ($\epsilon$) from the usual four dimensions. This allows for a more accurate description of the behavior of particles at high energies.

2. What is renormalization in $\phi^4$ theory?

Renormalization is a mathematical technique used in quantum field theory to remove divergences that arise in calculations of physical quantities. In $\phi^4$ theory, renormalization is necessary to make sense of the theory at high energies, where infinities arise due to the interactions between particles.

3. What is the beta function in $\phi^4$ theory?

The beta function in $\phi^4$ theory is a mathematical expression that describes how the coupling constant (which determines the strength of interactions between particles) changes as the energy scale changes. It is an important quantity in renormalization, as it allows us to predict the behavior of the theory at different energy scales.

4. How is the beta function calculated in $\phi^4$ theory in $4 - \epsilon$ dimensions?

The beta function in $\phi^4$ theory in $4 - \epsilon$ dimensions is calculated using a technique called dimensional regularization. This involves performing calculations in a space with a small deviation from the usual four dimensions, and then taking the limit as this deviation goes to zero. This allows for the removal of infinities and the calculation of a finite beta function.

5. What is the significance of the beta function in $\phi^4$ theory?

The beta function in $\phi^4$ theory is significant because it allows us to understand how the theory behaves at different energy scales. By studying the behavior of the beta function, we can make predictions about the behavior of the theory at high energies, and test these predictions experimentally. The beta function also plays a crucial role in the renormalization of $\phi^4$ theory, allowing us to remove infinities and make meaningful calculations in the theory.

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