Computation of anomalous dimension in MS scheme

[Luca_Mantani]

Hi,
I am computing the anomalous dimension of a mass operator in the MSbar scheme, but i have a doubt. The following is the solution of an exercise given by a professor but i don't understand a passage. I have computed the counterterm $\delta$ and i have the formula
$$\gamma=-\mu \frac{d\delta}{d\mu}$$

Substituting my calculation and dropping all the constants (which i need but are not relevant for the question) i have

$$\gamma\propto \mu \frac{de^2(\mu)}{d\mu}\frac{1}{\epsilon}$$

Then I use the formula

$$\mu \frac{de(\mu)}{d\mu}=-\epsilon e + \beta(e)$$

The solution says that if I substitute and drop the $\beta(e)$ i get the finite result we need, since the $\epsilon$ in the numerator cancels the one in the denominator. But why we drop the $\beta$ term, which is finite? If we keep it, we are not able to cancel the $1/\epsilon$ and we get a divergent term, even if we know that the anomalous dimension is finite. What am i missing in this calculation?

Thanks in advance for the help!

 

[vanhees71]

You find some calculations of anomalous dimensions in dim. reg. in my notes on QFT

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf