Here is the anomalous dimension:
$$
\gamma_{\Gamma}(t, g) = \left[\frac{\partial }{\partial t}\ln \left(Z_{\Gamma}(t , g) \right)\right]_{t = 1},
$$
where $Z_{\Gamma}$ is renormalization factor which arises in n-point functions $\Gamma $, $t$ denotes change of renormalization parameter $t = \frac{\mu{'}}{\mu}$. $Z_{\Gamma}$ arises explicitly after making shift of renormalization parameter $\mu$ (for fixed type of renormalization):

$$
\Gamma (xt , g) = Z_{\Gamma}^{-1}(t , g) \Gamma (x, \bar{g}(t , g)), \quad x = \frac{k}{\mu}, \quad t = \frac{\mu}{\mu{'}}.
$$

Let's change type of regularization (coupling constant will change to $g \to \tilde {g}(g)$. Then n-point function will change as
$$
\Gamma \left(\frac{k}{\mu} , g \right) = q(g) \tilde {\Gamma}\left( \frac{k}{\mu} , \tilde {g}(g) \right).
$$
How to get from these equations that $\gamma_{\Gamma}$ will change to
$$
\tilde{\gamma}_{\Gamma}(\tilde {g}(g)) = \gamma_{\Gamma}(g) - \beta (g)\frac{d\ln (q(g))}{dg}
$$
(the definition for $\beta$-function see here)?

This post imported from StackExchange Physics at 2014-09-18 08:02 (UCT), posted by SE-user PhysiXxx