One loop calculation of the renormalized anisotropy for improved anisotropic gluon actions on a lattice

Abstract

Using the infrared dispersion relation of the on-shell gluon, we calculate the renormalization of the anisotropy, \ensuremath{\chi}, to one loop in perturbation theory for lattice Yang-Mills theories, including the Wilson action and actions with Symanzik and/or tadpole improvement. Using twisted boundary conditions as a gauge invariant infrared regulator, we show for an SU(3) gauge group in $D=3+1$ dimensions that the one loop anisotropy is accurate to $O(3%)$ for a range of ${g}^{2}$ and $\ensuremath{\chi}$ covering current simulations. In doing so we also present Feynman rules for $\mathrm{SU}(N)$ gauge groups with generic anisotropy structure (including ``3+1'' and ``2+2'' cases) for both twisted and untwisted boundary conditions.