Landau Pole and gauging of Dirac fermions

Let us focus on 4d spacetime dimensions. It is well known that $N$ free massless Dirac fermions is conformal invariant. This theory has a $U(1)$ symmetry under which all the fermions have charge 1, i.e. $psi_ito e^{ialpha}psi_i, i=1, …, N$. Now, I’d like to consider another theory by gauging this $U(1)$. I just promote the $U(1)$ background field to dynamical field, and integrate it over in the path integral. The partition function is
$$int Da expleft(-sum_{i=1}^{N}int barpsi_i gamma^mu (partial_{mu}- ia_{mu})psi_i right). tag{*}$$
Note that there is no kinetic term for the gauge field from this naive gauging, i.e. the coupling constant $e$ is infinity for the Maxwell term $frac{1}{e^2}int f_{munu}f^{munu}$. Usually it is said that there is a Landau pole in $(*)$. Then after RG, the Maxwell term will be dynamically generated, and eventually $e$ will flow to $0$ logarithmically in the IR.

The questions I have are:

• (I) Is the theory $(*)$ at infinity coupling a well-defined QFT?
• (II) Is it a conformal field theory? Because $(*)$ is directly obtained by gauging a $U(1)$ global symmetry from a CFT, and moreover there is no obvious dimensionful coupling constant, so I wonder whether it is a CFT as well.