Electromagnetic current correlations in reduced quantum electrodynamics

We consider a theory of massless reduced quantum electrodynamics (${\mathrm{RQED}}_{d_{\gamma },d_e}$), e.g., a quantum field theory where the $U(1)$ gauge field lives in $d_{\gamma }$-space-time dimensions while the fermionic field lives in a reduced space-time of $d_e$ dimensions ($d_e⩽d_{\gamma }$). In the case where $d_{\gamma }=4$ such RQEDs are renormalizable while they are super-renormalizable for $d_{\gamma }<4$. The 2-loop electromagnetic current correlation function is computed exactly for a general ${\mathrm{RQED}}_{d_{\gamma },d_e}$. Focusing on ${\mathrm{RQED}}_{4,3}$, the corresponding $\beta$-function is shown to vanish which implies the scale invariance of the theory. Interaction correction to the 1-loop vacuum polarization, ${\Pi }_1$, of ${\mathrm{RQED}}_{4,3}$ is found to be $\Pi ={\Pi }_1(1+0.056\alpha )$ where $\alpha$ is the fine structure constant. The scaling dimension of the fermion field is computed at 1-loop and is shown to be anomalous for ${\mathrm{RQED}}_{4,3}$.

Figure 1

Feynman rules for massless RQED.

Figure 2

One-loop diagrams: (a) gauge field self-energy, (b) fermion self-energy, and (c) fermion-gauge field vertex.

Figure 3

2-loop diagrams contributing to the vacuum polarization ($k_{12}=k_1-k_2$).