The Anisotropic Wilson Gauge Action

Abstract

Anisotropic lattices, with a temporal lattice spacing smaller than the spatial one, allow precision Monte Carlo calculations of problems that are difficult to study otherwise: heavy quarks, glueballs, hybrids, and high temperature thermodynamics, for example. We here perform the first step required for such studies with the (quenched) Wilson gauge action, namely, the determination of the renormalized anisotropy $ξ$ as a function of the bare anisotropy $ξ_0$ and the coupling. By, essentially, comparing the finite-volume heavy quark potential where the quarks are separated along a spatial direction with that where they are separated along the time direction, we determine the relation between $ξ$ and $ξ_0$ to a fraction of 1% for weak and to 1% for strong coupling. We present a simple parameterization of this relation for $1\leq ξ\leq 6$ and $5.5 \leq β\leq \infty$, which incorporates the known one-loop result and reproduces our non-perturbative determinations within errors. Besides solving the problem of how to choose the bare anisotropies if one wants to take the continuum limit at fixed renormalized anisotropy, this parameterization also yields accurate estimates of the derivative $\partialξ_0/\partialξ$ needed in thermodynamic studies.