Improved Gauge Actions on Anisotropic Lattices I

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.