If I understand correctly, in gauge theory a field is defined to be "charged" under the gauge field if it transforms nontrivially under gauge transformations. (For example, in the abelian case a field $\Psi$ is defined to have charge $q$ if, under a gauge transformation in which $A^\mu \to A^\mu - \partial^\mu \Gamma$, it goes to $\exp(-i q \Gamma) \Psi$.) This then implies that any gauge-invariant quantity cannot be charged under the gauge field. For example, the electric currents $J^\mu := e \bar{\Psi} \gamma^\mu \Psi$ in spinor QED and $J^\mu := -i(\varphi^\dagger D^\mu \varphi - \varphi (D^\mu \varphi)^\dagger)$ in scalar QED are both uncharged under this definition.
Is this the standard definition? It seems funny to me to say that electric current is uncharged, even if it's made up of fundamental fields that are charged.
