Electromagnetic current correlations in reduced quantum electrodynamics

Abstract

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