Perturbative computation in a QED$_3$-inspired conformal abelian gauge model on the lattice

We perform perturbative computations in a lattice gauge theory with a conformal measure that is quadratic in a non-compact abelian gauge field and is nonlocal, as inspired by the induced gauge action in massless QED$_3$. In a previous work, we showed that coupling fermion sources to the gauge model led to nontrivial conformal data in the correlation functions of fermion bilinears that are functions of charge $q$ of the fermion. In this paper, we compute such gauge invariant fermionic observables to order $q^2$ in lattice perturbation theory with the same conformal measure. We reproduce the expectations for scalar anomalous dimension from previous estimates in dimensional regularization. We address the issue of the lattice regulator dependence of the amplitudes of correlation functions.