Lifshitz theories with extra dimensions and3+1-d Lorentz invariance

Abstract

We construct Lifshitz field theories in $4+1$ dimensions which retain $3+1$-d Lorentz invariance and therefore ensure a unique limiting speed in the $3+1$-d world. Such a construction is potentially useful in developing field-theoretic ultraviolet completions of extra-dimensional field theories. The extra dimension $y$ is treated asymmetrically from the usual three spatial dimensions by introducing derivatives of order $2n$ with respect to $y$ in the action. We show that $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ theory becomes progressively less nonrenormalizable by power counting as $n$ is increased. This suggests that the nonlocal theory obtained in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit may be complete in the ultraviolet.