Conformally Invariant Regularization and Skeleton Expansions in Gauge Theory

We consider a conformally invariant regularization of an Abelian gauge theory in an Euclidean space of even dimension $D\geq4$ and regularized skeleton expansions for vertices and higher Green's functions. We set the respective regularized fields $A^\varepsilon_\mu$ and $j^\varepsilon_\mu$ with the scaling dimensions $l^\varepsilon_A=1-\varepsilon$, and $l^\varepsilon_j=D-1+\varepsilon$ into correspondence to the gauge field $A_\mu$ and Euclidean current $j_\mu$. We postulate special rules for the limiting transition $\varepsilon\to0$. These rules are different for the transversal and longitudinal components of the field $A^\varepsilon_\mu$ and the current $j^\varepsilon_\mu$. We show that in the limit $\varepsilon\to0$, there appear conformally invariant fields $A_\mu$ and $j_\mu$ each of which is transformed by a direct sum of two irreducible representations of the conformal group. Removing the regularization, we obtain a well-defined skeleton theory constructed from conformal two- and three-point correlation functions. We consider skeleton equations on the transversal component of the vertex operator and of the spinor propagator in conformal quantum electrodynamics. For simplicity, we restrict the consideration to an Abelian gauge field $A_\mu$, but generalization to a non-Abelian theory is straightforward.