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

Question

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?

Answer 1

The anamolous 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} $.

That definintion 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...