Improved Gauge Actions on Anisotropic Lattices I

TL;DR

This paper calculates key parameters of improved gauge actions on anisotropic lattices using perturbative methods and confirms findings with numerical simulations, revealing how different actions affect anisotropy ratios and coupling derivatives.

Contribution

It provides a comprehensive perturbative analysis of improved gauge actions on anisotropic lattices, including the behavior of the anisotropy ratio and coupling derivatives, and confirms results with numerical simulations.

Findings

The anisotropy ratio η varies with the action type, being greater than 1 for Wilson and Symanzik, and less than 1 for Iwasaki and DBW2.

The derivatives of the coupling constants with respect to anisotropy change sign depending on the action.

Sum of the derivatives is independent of the action type due to Karsch's sum rule.

Abstract

On anisotropic lattices with the anisotropy $\xi=a_\sigma/a_\tau$ the following basic parameters are calculated by perturbative method: (1) the renormalization of the gauge coupling in spatial and temporal directions, $g_\sigma$ and $g_\tau$, (2) the $\Lambda$ parameter, (3) the ratio of the renormalized and bare anisotropy $\eta=\xi/\xi_B$ and (4) the derivatives of the coupling constants with respect to $\xi$, $\partial g_\sigma^{-2}/\partial \xi$ and $\partial g_\tau^{-2}/\partial \xi$. We employ the improved gauge actions which consist of plaquette and six-link rectangular loops, $c_0 P(1 \times 1)_{\mu \nu} + c_1 P(1 \times 2)_{\mu \nu}$. This class of actions covers Symanzik, Iwasaki and DBW2 actions. The ratio $\eta$ shows an impressive behavior as a function of $c_{1}$, i.e.,$\eta>1$ for the standard Wilson and Symanzik actions, while $\eta<1$ for Iwasaki and DBW2 actions. This is confirmed non-perturbatively by numerical simulations in weak coupling regions. The derivatives $\partial g^{-2}_{\tau}/\partial \xi$ and $\partial g^{-2}_{\sigma}/\partial \xi$ also changes sign as $-c_{1}$ increases. For Iwasaki and DBW2 actions they become opposite sign to those for standard and Symanzik actions. However, their sum is independent of the type of actions due to Karsch's sum rule.