Hello and thank for the time you will take reading my question.
The $Z(3)$ symmetry can be defined as a simple global phase transformation: $$Z(3)=\left\{1, e^{\frac{i2 \pi}{3}}, e^{\frac{-i2 \pi}{3}}\right\}.$$ Lets call z one element of Z(3). If we make the following transformations:
- $A^\mu \rightarrow z A^\mu z^{-1} $
- $ \Psi \rightarrow z \Psi $ and $ \bar{\Psi} \rightarrow \bar{\Psi} z^{-1} $
The pure gluon term is invariant. The interaction term is of the form $ \bar{\Psi} T^a A^a _\mu \gamma ^\mu \Psi$. z should commute with all gamma matrices and SU(3) elements.
I am under the impression that the Lagrangian is invariant. I can't find where my mistake is. I am new to QCD so maybe there is something obvious I am missing? Because papers talking about QCD at finite temperature like this affirm that :
In full QCD, the theory is not invariant under the replacement of $ \Psi \rightarrow z \Psi $, where z $\in Z(3)$ and, therefore, the gauge group associated with full QCD is $SU(3)$.
