How to reconcile scaling dimension and gauge transformation in 2+1D U(1) Maxwell theory

Question

In 2+1D (free) $U(1)$ Maxwell theory, power-counting in the action implies that the gauge potential $A_{\mu}$ should have scaling dimension 1/2. This is borne out by the propagator $\langle F_{\mu \nu}(x)F_{\lambda \sigma}(0) \rangle \sim 1/x^3$ or, more schematically, by the (gauge-dependent) propagator $\langle A_{\mu}(x) A_{\nu}(0) \rangle \sim 1/x$. However, the gauge transformation rule $A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \alpha$ means that $A_{\mu}$ should have dimension 1, as $\alpha$ must be dimensionless so that $e^{i\alpha}$ is a well-defined element of the gauge group $U(1)$.

Why do these power counting arguments give different results? Is that fact meaningful?

Note that in Chern-Simons theory or in 3+1D $U(1)$ Maxwell theory, there are no such issues; power-counting in the action also leads to $A_{\mu}$ having dimension 1, consistent with the gauge transformation law.

Thanks!

Comments

If your convention is such that  $ A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \alpha$  is the gauge transformation for gauge fields, then on the matter field the gauge transformation should be a multiplication by $e^{ig \alpha}$ where $g$ is the gauge coupling. Remember the gauge coupling is dimensionful in 2+1 dimension, which means $\alpha$ is also dimensionful.