# If the $SU(3)$ Noether charge is not gauge-invariant how can we say Hadrons are colorless?

The Noether charge associated with global $SU(3)$ invariance is $$J^{\mu}_a=-f_{abc}F_c^{\nu\mu}A_{b\mu}-i\frac{\delta \mathcal{L}_{Matter}}{\delta D_{\nu}\psi}t_a\psi$$ and is conserved along the equations of motion $\partial_{\mu}J^{\mu}_a=0$ therefore giving rise to the conserved charges: $$Q^a=\int d^3x\,\, J^0_a$$ However the current is not gauge invariant (nor gauge covariant) so how can we say that Hadrons are colorless? In particular, can one say that the charge operator annihilates Hadrons? $$\hat{Q}^a|Hadron\rangle=0$$

$Q^a=Q^a_0 \neq 0$ for fixed $Q^a_0$ is not a gauge-invariant statement $Q^a=0$ is gauge invariant.