# What justifies the dependence of the coupling renormalization constant in the dimensional regularization regulator?

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I wanna clarify some issues about renormalization in the $\bar{MS}$ scheme that I glossed over when I first learnt about this stuff. I am following http://arxiv.org/abs/1411.7853 section 3.1. The gluon part of the QCD Lagrangian is considered and the renormalized coupling and gluon field are written

$$g=\bar{\mu}^{\epsilon}Z_gg_R\qquad{}A_{\mu}=\sqrt{Z_A}A_{\mu}^R$$

where $\bar{\mu}=\frac{\mu}{\sqrt{2\pi}}e^{\gamma_E/2}$. It is immediately stated that the renormalization constant takes the form

$Z_g=1+\frac{\alpha_s(\mu)}{4\pi}\frac{Z_{11}}{\epsilon}+\bigg(\frac{\alpha_s(\mu)}{4\pi}\bigg)^2\bigg(\frac{Z_{22}}{\epsilon^2}+\frac{Z_{21}}{\epsilon}\bigg)\\ +\bigg(\frac{\alpha_s(\mu)}{4\pi}\bigg)^3\bigg(\frac{Z_{33}}{\epsilon^3}+\frac{Z_{32}}{\epsilon^2}+\frac{Z_{31}}{\epsilon}\bigg)+\ldots{}$

I don't see why it should be obvious that $Z_g$ should take this form. What justifies this?

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