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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?
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  • $\begingroup$ 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. $\endgroup$ May 5, 2020 at 19:34

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The anomalous dimension for the field strength is defined as (eqn 12.63 Peskin):

$\gamma = \frac{1}{2} \frac{M}{Z} \frac{\partial Z}{\partial M} = \frac{1}{2} \frac{\partial \log Z}{\partial \log M} $.

This definition always holds. What you actually calculate for the right-hand side of the above equation once you have a Z within a particular scheme will be in general scheme-dependent.

Sorry, I can't help you with the $O(N)$ vector model...

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  • $\begingroup$ 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. $\endgroup$
    – user6818
    Sep 12, 2012 at 20:46
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    $\begingroup$ @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. $\endgroup$
    – DJBunk
    Sep 13, 2012 at 13:23

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