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Electromagnetic current correlations in reduced quantum electrodynamics

We consider a theory of massless reduced quantum electrodynamics (RQED$_{d_γ,d_e}$), e.g., a quantum field theory where the U(1) gauge field lives in $d_γ$-spacetime dimensions while the fermionic field lives in a reduced spacetime of $d_e$ dimensions ($d_e \leqslant d_γ$). In the case where $d_γ=4$ such RQEDs are renormalizable while they are super-renormalizable for $d_γ<4$. The 2-loop electromagnetic current correlation function is computed exactly for a general RQED$_{d_γ,d_e}$. Focusing on RQED$_{4,3}$, the corresponding $β$-function is shown to vanish which implies the scale invariance of the theory. Interaction correction to the 1-loop vacuum polarization, $Π_1$, of RQED$_{4,3}$ is found to be: $Π= Π_1 (1 + 0.056 \al)$ where $\al$ is the fine structure constant. The scaling dimension of the fermion field is computed at 1-loop and is shown to be anomalous for RQED$_{4,3}$.