Asymptotically free lattice gauge theory in five dimensions

A lattice formulation of Lifshitz-type gauge theories is presented. While the Lorentz-invariant Yang-Mills theory is not renormalizable in five dimensions, non-Abelian Lifshitz-type gauge theories are renormalizable and asymptotically free. We construct a lattice gauge action and numerically examine the continuum limit and the bulk phase structure.

Figure 1

A $2\times 1$ twisted Wilson loop $T_{\mu \nu }(x)$.

Figure 2

The flow diagram of $e$ and $g$ for the $\mathrm{SU}(N)$ gauge group. The origin is a UV fixed point.

Figure 3

Action density with isotropic coupling constants $\beta (\equiv {\beta }_e={\beta }_g)$. The data of the Hořava-Lifshitz theory and the isotropic Yang-Mills theory on a $6^5$ lattice are plotted. The dashed lines are analytic results in the strong coupling limit ($\beta \to 0$) and the weak coupling limit ($\beta \to \infty$).

Figure 4

Action density of the Hořava-Lifshitz theory as a function of ${\beta }_e$ and ${\beta }_g$. The data on a $6^5$ lattice are plotted. Statistical error bars are omitted.

Figure 5

Effective mass $V_{\text{eff}}(t,x)$ at $x/a=2$. The data from simulations on a ${10}^5$ lattice with $\beta (\equiv {\beta }_e={\beta }_g)=9$ are plotted.

Figure 6

Color singlet potential $V(x)$. The data from simulations on a ${10}^5$ lattice with $\beta (\equiv {\beta }_e={\beta }_g)=9$ are plotted.

Figure 7

A Wilson loop on a lattice which reproduces $\mathrm{Tr}\{(D_k^{\mathrm{ad}}{F}_{ij})(D_k^{\mathrm{ad}}{F}_{ij})\}$ in the continuum limit.