# About calculation of anomalous dimension in Peskin and Schroeder's book.

This question is in reference to question 13.2 in the QFT book by Peskin and Schroeder.

To put it in general - I would like to know how does one define "anomalous dimensions" if one is given the wave-function renormalization in the "epsilon" regularization scheme? (..without having to redo the whole calculation again!..)

The only way I know of defining the anomalous dimension is when one does the regularization in the MS-bar scheme. Is there a simple/obvious way to interchange between the two schemes?

• And in general is there a reference which does the anomalous dimensions calculation for O(N) vector model/linear sigma model and the non-linear sigma model?
• I'm a little confused about your reference to an "epsilon" regularization scheme. I usually interpret MS-bar as being dimensional regularization (but with some phase factors subtracted), which is one where you take dimensionality to be $4 - \epsilon$ and then you subtract the simple poles in $\epsilon$. How is this different from what you call the "epsilon regularization"? Jean Zinn-Justin's textbook, Quantum Field Theory and Critical Phenomena, should contain the information you've asked for in your final bullet. May 5, 2020 at 19:34

$$\gamma = \frac{1}{2} \frac{M}{Z} \frac{\partial Z}{\partial M} = \frac{1}{2} \frac{\partial \log Z}{\partial \log M}$$.
Sorry, I can't help you with the $$O(N)$$ vector model...
• Definitely thats the definition of $\gamma$ that I have in mind. But if lets say one is given the 2-loop calculation of the 2-point function in say $\phi^4$ theory in the $\epsilon$ regularization then can one use that to get $\gamma$ from the above definition? Thats my question. Sep 12, 2012 at 20:46
• @user6818 - To my knowledge, the safest way to get $\gamma$ is to calculate the bare couplings and extremize them with respect to the RG scale $\mu$. Then you will get a set of coupled first order equations $\gamma$, $\beta$ and $\gamma_m$( the anomalous dimension for the mass term) which you can solve. Ramond goes through this in his text - ' Field Theory: A moden Primer.' Peskin has a few tricks to get just $\beta$ or $\gamma$ for some particular cases in particular limits, but I don't think that they work in general. Sep 13, 2012 at 13:23