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

[paralleltransport]

TL;DR

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:

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.

 

[vanhees71]

I don't know, whether it's what you look for, but there's a chapter on the perturbative RG of $\phi^4$ theory in my notes on QFT:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

 

[paralleltransport]

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...