I am trying to understand the general strategy and technical details of calculating $\beta$-function at higher orders. $\beta$-function is the anomalous dimension of the coupling constant and there is a complete set of anomalous dimensions corresponding to different fields, propagators (gauge field), and vertices.

The anomalous dimension corresponding to a renormalization constant could be defined as

$$\gamma=-\mu^2\frac{d \log Z}{d\mu^2}$$

And in the minimal subtraction scheme, one could expand the renormalization constants as

$$Z=1+\sum_{i=1}^\infty \frac{z_i (a_s,\xi)}{\epsilon^i}$$

where $\xi$ is the gauge parameter, i.e. we do not need to fix the gauge before computing the anomalous dimension, and the number of space-time dimensions $D=4-2\epsilon$. Now, let's consider the anomalous dimension of the coupling constant, and assume that scale dependence of the corresponding renormalization constant $Z_{a_s}$ happens through $a_s$ and $\xi$. How does the following relation hold?

$$-\beta(a_s)=\left(-\epsilon + \beta(a_s) \right) a_s \frac{\partial \log Z_{a_s}}{\partial a_s} $$

What about the anomalous dimension of $\xi$. What would the relation be?

note: Please refer to http://arxiv.org/abs/hep-ph/0405193v3 for the conventions; This article http://link.springer.com/article/10.1007%2FBF01079292 also contains valuable points such as (2.4) and (2.6) which I believe are related to the problem at hand.

This post imported from StackExchange Physics at 2016-07-18 16:03 (UTC), posted by SE-user moha