# The scale dependence of scale anomaly

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The scale anomaly states that if we have renormalizable theory without dimensionful function, which is scale invariant, then corresponding quantum theory may lost this symmetry because of regularization. Corresponding current, called dilatation current, isn't conserved:
$$\tag 1 \partial_{\nu}\langle |\hat{\theta}^{\nu}|\rangle = F[\beta (\mu), \mu],$$
where $F[...]$ is called the scale anomaly function, $\beta (\mu)$ is the coupling beta function, and $\mu$ is the renormalization scale.
For example, in QCD
$$\partial_{\nu}\langle |\hat{\theta}^{\nu}|\rangle \sim \frac{\beta (g_{s})}{g_{s}}G_{\mu \nu}^{a}G^{\mu \nu}_{a},$$

where $G_{\mu \nu}^{a}$ is gluons field strength.

The question: does dependence of $F$ on the fictious scale $\mu$ imply that the scale anomaly equation $(1)$ is scale dependent? Or, when we talk about the scale dependence of anomaly equation, we mean the principal existence of anomaly function $F$ in the rhs of Eq.$(1)$?

edited Mar 25, 2016

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