A simple UV completion of QED in 5 dimensions

We construct a Lifshitz-like version of five-dimensional (5D) QED which is UV completed and reduces at low energies to ordinary 5D QED. The UV quantum behavior of this theory is very smooth. In particular, the gauge coupling constant is finite at all energy scales and at all orders in perturbation theory. We study the IR properties of this theory, when compactified on a circle, and compare the one-loop energy dependence of the coupling in the Lifshitz theory with that coming from the standard 5D QED effective field theory. The range of validity of the 5D effective field theory is found to agree with the more conservative version of naive dimensional analysis.

Figure 1

Feynman rules of the Lifshitz 5D QED theory in the Coulomb gauge. The momenta are all incoming.

Figure 2

One-loop graphs contributing to the correction of the fermion propagator.

Figure 3

Left panel: $g_5^{-2}$ as a function of the energy scale $E$ in the uncompactified Lifshitz theory. Right panel: Comparison between the Lifshitz behavior (blue line) with the 5D linear regime (4.7) (straight red line). The energy is in units of $\Lambda$. We have taken $E_0=0$, $g_5^{-2}(0)=1$, and $a_{\psi }=1$.

Figure 4

Comparison between the exact Lifshitz behavior as given by Eq. (4.14) (blue line) with the effective coupling (4.4) for periodic fermions with $M=0$ (straight red line). The energy is in units of $1/R$. We have taken $E_0=1/(10R)$, $g_4^{-2}(1/(10R))=1$, $a_{\psi }=1$, and $\Lambda =25/R$.

Figure 5

Comparison between $g_4^{-2}$ as a function of the energy scale $E$ in the Lifshitz theory (blue lines), as given by Eq. (a2), and the effective 5D coupling (4.7) (straight red lines). The energy is in units of $1/R$. We have taken $E_0=1/R$ and $g_4^{-2}(1/R)=1$. (a) $a_{\psi }=1$ and $\Lambda =25/R$. (b) $a_{\psi }=10$ and $\Lambda =100/R$. Despite the higher cutoff, the range of validity of the 5D effective theory is lower in (b) than in (a).