I have seen several questions regarding the difference between global and gauge charges, but I don't really get the physical implications.
The sQED lagrangian is:
$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}^2 + |D_{\mu}\phi|^2-m^2 \phi^+\phi$
Which has a gauge symmetry:
$A^{\mu}\rightarrow A^{\mu}+ \partial^{\mu}\alpha(x)$
$\phi \rightarrow e^{-i \alpha(x)} \phi $
But if we take $\alpha(x)$ the symmetry becomes global and then:
$A^{\mu}\rightarrow A^{\mu}$
$\phi \rightarrow e^{-i \alpha} \phi $
Since we have a global symmetry we will have a conserved Noether current and a conserved charge. The same happens for QED, when we have a global $U(1)$ symmetry and therefore a Noether current. My question is:
When we have a gauge boson and a global symmetry (which comes from taking $\alpha(x)$ as a constant), is our conserved charge the electric charge? Moreover, if we hadn't the gauge boson in our theory, would the conserved charge be the number of particles?
