A simple UV completion of QED in 5 dimensions

Abstract

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 behaviour 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.

Weighted power counting and perturbative unitarity

“…12) which is the condition for weighted renormalizability from before. So, for an N-point interaction, the unitarity condition is equivalent to the renormalizability condition.…”

“…Unfortunately, Lifshitz type UV-completions of these theories are not necessarily without problems. The proposed UV completion of 5d QED exhibits a fine tuning problem [12], and in the case of Hořava-Lifshitz gravity there has been concern over the consistency of various versions of the theory [13][14][15][16][17][18]. Some of these versions become strongly coupled at a certain scale and there is a breakdown of the perturbative expansion (for recent reviews related to this problem see [19,20]).…”

“…The anisotropic scaling leads to a modification of power counting arguments for renormalizability, and also changes the relativistic phase space factor thereby altering the condition for perturbative unitarity. Many theories can be constructed in which Lifshitz-point scaling restores renormalizability; for example, consider gauge theories in higher dimensions [8][9][10][11][12]. Unfortunately, Lifshitz type UV-completions of these theories are not necessarily without problems.…”

Cutoff estimate in Lifshitz five-dimensional field theories

“…The total mass matrix for the gauge fields is given by 10) with M 2 as in (5.4). By explicit computation, we find that the mass matrix (5.9) has eigenvalues…”

“…Second, Lifshitz field theories are known to possibly have an improved UV behaviour with respect to ordinary theories, exploiting the improved UV behaviour of the particle propagators. In fact, simple UV-completions of 5D theories based on Lifshitz field theories have been shown to be possible [10]. The price to be paid is however high.…”